南京大学：凝聚态物理学导论讲义英文版PartII（金国均）：凝聚态物理学导论讲义-金国均2004

分类：物理格式：pdf 日期：2009年02月25日

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Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
February 12,2004
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Part II
Wave Behavior in
Various Structures
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Solid State Physics,Revisit and Extension
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Like as the waves make towards the pebbled shore,
So do our minutes hasten to their end,
Each changing place with that which goes before,
In sequent toil all forward to contend.
— William Shakespeare
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Waves always behave in a similar way,whether
they are longitudinal or transverse,elastic or
electric,Scientists of last century always kept
this idea in mind ···,This general philosophy of
wave propagation,forgotten for a time,has been
strongly revived in the last decade ···
— L,Brillouin (1946)
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5,Wave Propagation in Periodic and
Quasiperiodic Structures
6,Dynamics of Bloch Electrons
7,Surface and Impurity Effects
8,Transport Properties
9,Wave Localization in Disordered Systems
10,Mesoscopic Quantum Transport
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Contents
II Wave Behavior inVarious Structures 3
5 Wave Propagation in Periodic and Quasiperiodic Structures 3
5.1 Unity of the Concept for Wave Propagation,,,,,,5
5.1.1 Wave Equations and Periodic Potentials,,,,5
5.1.2 Bloch Waves,,,,,,,,,,,,,,,,,,,,8
5.1.3 Revival of the Study on Classical Waves,,,,11
5.2 Electrons in Crystals,,,,,,,,,,,,,,,,,,,14
5.2.1 Free Electron Gas Model,,,,,,,,,,,,,14
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5.2.2 Nearly-Free Electron Model,,,,,,,,,,,17
5.2.3 Tight-Binding Electron Model,,,,,,,,,,23
5.2.4 Kronig-Penney Model for Superlattices,,,,,27
5.2.5 Density of States and Dimensionality,,,,,,35
5.3 Lattice Waves and Elastic Waves,,,,,,,,,,,,38
5.3.1 Dispersion Relation of Lattice Waves,,,,,,38
5.3.2 Frequency Spectrum of Lattice Waves,,,,,,38
5.3.3 Elastic Waves in Periodic Composites,,,,,,38
5.4 Electromagnetic Waves in PeriodicStructures,,,,,,39
5.4.1 Photonic Bandgaps in Layered PeriodicMedia,39
5.4.2 Dynamical Theory of X-Ray Diffraction,,,,39
5.4.3 Bandgaps in Three-DimensionalPhotonic Crystals 39
5.4.4 Quasi Phase Matching in NonlinearOptical Crystals 39
5.5 Quasiperiodic Structures,,,,,,,,,,,,,,,,,40
5.5.1 One-Dimensional Quasiperiodic Structure,,,40
5.5.2 Two-Dimensional Quasiperiodic Structures,,,40
5.5.3 Three-Dimensional Quasicrystals,,,,,,,,40
5.6 Waves in Quasiperiodic Structures,,,,,,,,,,,,41
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5.6.1 Electronic Spectra in a One-DimensionalQuasilattice 41
5.6.2 Wave Transmission through ArtificialFibonacci Structures 41
5.6.3 Pseudogaps in Real Quasicrystals,,,,,,,,41
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Chapter 5
Wave Propagation in Periodic and
Quasiperiodic Structures
Newton,speed of sound,one-dimensional lattice model
Beginning of the 20th century,Cambell,periodic LC circuit theory,fil-
ter electric waves,proposed cut-off frequency,pass band and forbidden
band
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Conventional solid state physics:
electrons and lattice vibration
Recent years,classical waves:
electromagnetic wave and elastic wave
in periodic composites
In 1980s,quasicrystals
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5.1,Unity of the Concept for Wave Prop-
agation
Formal analogy and similarity between three types of waves
5.1.1,Wave Equations and Periodic Potentials
An electron,λ = 2pi/k,p = planckover2pi1k
Schr¨odinger equation
iplanckover2pi1tψ =
bracketleftbigg
planckover2pi1
2
2m?
2 +V(r)
bracketrightbigg
ψ (5.1.1)
V(r+l) = V(r) (5.1.2)
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Thermal fluctuation at finite temperature
u—atomic displacement,Φ—potential energy
Newton equation
Ms?
2
t2ulsα =?
summationdisplay
lprimesprimeβ
Φlsα,lprimesprimeβulprimesprimeβ (5.1.3)
Force constant—Φlsα,lprimesprimeβ ≡?2Φ/?ulsα?ulprimesprimeβ|0,periodicity
Φlsα,lprimesprimeβ = Φ0sα,ˉlsprimeβ,(5.1.4)
with ˉl = lprime?l
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Electromagnetic waves,Maxwell equations
×E =?1c?B?t,?·B = 0
×H = 1c?D?t + 4pic j,?·D = 4piρ
Four coefficients related to materials,epsilon1,μ,χ,χm
D = epsilon1E = E +4piP = (1+ 4piχ)E
B = μH = H + 4piM = (1 + 4piχm)H
In general,epsilon1,μ,χandχm,tensors; Strong fields,nonlinear polarization
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Electromagnetic waves,Maxwell equations
×E =?1c?B?t,?·B = 0
×H = 1c?D?t + 4pic j,?·D = 4piρ
Four coefficients related to materials,epsilon1,μ,χ,χm
D = epsilon1E = E +4piP = (1+ 4piχ)E
B = μH = H + 4piM = (1 + 4piχm)H
In general,epsilon1,μ,χandχm,tensors; Strong fields,nonlinear polarization
Electric displacement vector D,wave equation
1
c2
2D
t2 +?×
parenleftbigg 1
μ(r)?×
D
epsilon1(r)
parenrightbigg
= 0 (5.1.5)
μ(r) and epsilon1(r),second-order tensors → scalars
μ(r+l) = μ(r),epsilon1(r+l) = epsilon1(r) (5.1.6)
Three other field quantities,E,B,H,equivalent
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5.1.2,Bloch Waves
Some common traits for wave propagation in periodic structures
Tuning condition:
Lattice separation a and Characteristic wavelengths λ
Electrons:
λ = (planckover2pi12/2mE)1/2,ranges from lattice spacing to bulk size
Lattice vibrations:
acoustic branches and optical branches
Wavelengths ranges from lattice spacing to bulk size
Electromagnetic radiation:
γ-ray (< 0.4?A),X-ray (0.4 ～ 50?A),ultraviolet ray (50 ～ 4000?A),vis-
ible light (4000 ～ 7000?A),infrared ray (0.76 ～ 600μm),to microwave
and radio wave (> 0.1 mm)
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Periodic structures,Bloch waves,an electron stationary equation
bracketleftbigg
planckover2pi1
2
2m?
2 +V(r)
bracketrightbigg
ψ(r) = Eψ(r) (5.1.7)
ψk(r) = uk(r)fk(r) (5.1.8)
uk(r +l) = uk(r) (5.1.9)
To determine fk(r),consider |ψk(r)|2,note
|ψk(r)|2 = |ψk(r+l)|2
get
|fk(r+l)|2 = |fk(r)|2
Bloch function
ψk(r) = uk(r)eik·r (5.1.10)
Bloch theorem
ψk(r+l) = ψk(r)eik·l (5.1.11)
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Discussed above can be generalized to other cases of wave equations
with periodic potentials.
Periodicity → Fourier transformation→ Reciprocal lattice
Dispersion relation,Bandgaps,Separated bands
Brillouin zones (BZ)–Wigner-Seitz cells of the reciprocal lattice
Born-von Karman cyclic boundary conditions
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5.1.3,Revival of the Study on Classical Waves
Classical waves—an old scientific problem:
Elastic waves and Electromagnetic waves
Waves in water—Solitons,Nonlinear physics
X-ray diffraction in crystals demonstrated:
wave property of the rays and periodic structures of crystals
Bragg equation,kinematical theory
2dsinθ = nλ (5.1.12)
wavelength λ and lattice parameter d
Can be used to X rays,electrons,and neutrons diffraction in crystals
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Bandgaps for X rays in crystals?
Frequency-wavevector relation of electromagnetic waves
stationary equation for electric displacement vector D
2D×?×[χ(r)D] = ω
2
c2 D (5.1.13)
χ(r) = 1? 1epsilon1(r) (5.1.14)
Photonic bandgaps:
a window of frequencies,electromagnetic wave cannot propagate
Crucial step to find the propagation gaps
Theoretically,John,Yablonovitch,in 1987;
Experimentally,Yablonovitchand Gmitter,in 1989,microwaves,three-
dimensional dielectric structures,called photonic crystals,pushed in
the late 1990s to light waves,application in photonics
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Renewed interests:
Classical waves in both Periodic and Aperiodic structures
Electronic wave,Band structure (Bloch,1928) → localization (Ander-
son,1958; Edwards,1958);
Classical wave,localization (John,1984; Anderson,1985) → band
structure (Yabolonovitch,1987; John,1987)
Electron and photon both have characteristics of waves
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Renewed interests:
Classical waves in both Periodic and Aperiodic structures
Electronic wave,Band structure (Bloch,1928) → localization (Ander-
son,1958; Edwards,1958);
Classical wave,localization (John,1984; Anderson,1985) → band
structure (Yabolonovitch,1987; John,1987)
Electron and photon both have characteristics of waves
There are some basic differences:
Dispersion relation,electrons is parabolic,photons is linear;
Angular momentum,electrons is 1/2,scalar-wave,photons have spin
1,vector-wave;
Band theory,electrons,approximated,Coulomb interactions,photons,
exact,no interactions
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5.2,Electrons in Crystals
A large number of electrons in a solid
Independent electron model
Periodic ionic potential exists
5.2.1,Free Electron Gas Model
For alkali metals,such as Li,Na,K,noble metals Cu,Ag,Au
V(r) = 0,
planckover2pi1
2
2m?
2ψ(r) = Eψ(r)
ψk(r) =1/2eik·r (5.2.1)
E(k) = planckover2pi12k2/2m (5.2.2)
Fermi surface
EF = E(kF),(5.2.3)
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Fermi wavevector kF
N =
summationdisplay
k
= 2?(2pi)3
integraldisplay
dk,(5.2.4)
integraltext dk = 4pik3
F/3
kF =
parenleftbigg
3pi2N?
parenrightbigg1/3
(5.2.5)
EF = planckover2pi1
2
2m
parenleftbigg
3pi2N?
parenrightbigg2/3
,(5.2.6)
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Fermi wavevector kF
N =
summationdisplay
k
= 2?(2pi)3
integraldisplay
dk,(5.2.4)
integraltext dk = 4pik3
F/3
kF =
parenleftbigg
3pi2N?
parenrightbigg1/3
(5.2.5)
EF = planckover2pi1
2
2m
parenleftbigg
3pi2N?
parenrightbigg2/3
,(5.2.6)
En(k),ψnk(r),?(pi/a) <k< (pi/a)
Fig,5.2.1
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0 pi/a 2pi/a 3pi/a-pi/a-2pi/a-3pi/a k k
EE
pi/a-pi/a
Figure 5.2.1 Dispersion curves of one-dimensional free electron gas for (a)
extended-zone scheme,and (b) reduced-zone scheme.
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5.2.2,Nearly-Free Electron Model
V(r) as a perturbation
ψ(0)nk,E(0)n (k)
En(k) = E(0)n (k) +
angbracketleftBig
ψ(0)nk |V|ψ(0)nk
angbracketrightBig
+
summationdisplay
nprimekprime
prime
vextendsinglevextendsingle
vextendsingle
angbracketleftBig
ψ(0)nprimekprime |V|ψ(0)nk
angbracketrightBigvextendsinglevextendsingle
vextendsingle
2
E(0)n (k)?E(0)nprime (kprime)
(5.2.7)
ψnk = ψ(0)nk +
summationdisplay
nprimekprime
prime
angbracketleftBig
ψ(0)nprimekprime |V|ψ(0)nk
angbracketrightBig
E(0)n (k)?E(0)nprime (kprime)
ψ(0)nprimekprime,(5.2.8)
one-dimensional case,n = 1,nprime = 2
E1(k) similarequalE(0)1 (k) + |V?2pi/a|
2
E(0)1 (k)?E(0)2 (k)
,(5.2.9)
V?2pi/a = 1L
integraldisplay
V(x)ei2pix/adx
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In the case k similarequal pi/a,E(0)1 = planckover2pi12k2/2m,E(0)2 = planckover2pi12(k? 2pi/a)2/2m?→
E(0)1 similarequalE(0)2,
degenerate perturbation theory
E±(k) = 12
braceleftBigg
E(0)1 (k) +E(0)2 (k)±
bracketleftbiggparenleftBig
E(0)2 (k)?E(0)1 (k)
parenrightBig2
+ 4|V?2pi/a|2
bracketrightbigg1/2bracerightBigg
(5.2.10)
Eg = E+(k)?E?(k) = 2|V?2pi/a| (5.2.11)
Fig,5.2.2
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0 pi/a 2pi/a 3pi/a-pi/a-2pi/a-3pi/a k 0 pi/a-pi/a k
E E
(a) (b)
Figure 5.2.2 Bands and gaps in one-dimensional nearly-free electron model for (a)
extended zone scheme,and (b) reduced zone scheme.
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Bragg diffraction for?pi/a<k<pi/a
ψ1k = ψ(0)1k + V?2pi/a
E(0)1 (k)?E(0)2 (k)
ψ(0)2k,(5.2.12)
right travelling wave ψ(0)1k = L?1/2 exp(ikx)
left travelling wave ψ(0)2k = L?1/2 exp[i(k?2pi/a)x]
wavefunctions at BZ boundaries
ψ±(x) = 1√2L
bracketleftBig
ψ(0)1,pi/a(x)±ψ(0)2,pi/a(x)
bracketrightBig
= 1√2L
parenleftBig
eipix/a ±e?ipix/a
parenrightBig
(5.2.13)
standing waves ψ+ = (2/L)1/2cos(pix/a) and ψ? = (2/L)1/2sin(pix/a)
Fig,5.2.3.
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V (x)
| |2ψ?| |2ψ+
x
Figure 5.2.3 Bragg Reflection of electron in a periodic structure.
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viewpoint of scattering:
Periodic potential → Strong scattering at k = pi/a
Gaps open at BZ boundaries → electronic band structure
In high-dimensions,Bragg condition,energy gaps at BZ also create
Quantitative details depend on the specific periodical potential →
various energy band structures
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5.2.3,Tight-Binding Electron Model
3d bands in transition metals
V(r) very strong
Wannier function
wn(r?l) = N?1/2
summationdisplay
k
e?ik·lψnk(r) (5.2.14)
ψnk(r) = N?1/2
summationdisplay
l
eik·lwn(r?l) (5.2.15)
Wannier functions,orthogonalized for different n and l,
wn(r?l) is a localized function
Take a Bloch function in a band
ψk(r) = N?1/2u(r)eik·r,(5.2.16)
assume same u(r)
For cubic lattice,a,Wannier function at the origin
w(r) = sin(pix/a)sin(piy/a)sin(piz/a)(pix/a)(piy/a)(piz/a) u(r) (5.2.17)
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decreasing oscillated
When a not small,to use atomic wavefunction
w(r?l) similarequalφa(r?l)
ψk(r) = N?1/2
summationdisplay
l
eik·lφa(r?l) (5.2.18)
E(k) =
integraltext ψ?
k
braceleftBig
planckover2pi122m?2 +V(r)
bracerightBig
ψkdr
integraltext ψ?
kψkdr
similarequal
summationdisplay
h
eik·hEh (5.2.19)
Eh = 1?
c
integraldisplay
φ?a(r+h)
braceleftbigg
planckover2pi1
2
2m?
2 +V(r)
bracerightbigg
φa(r)dr (5.2.20)
bracketleftbigg
planckover2pi1
2
2m?
2 +va(r)
bracketrightbigg
φa(r) = Eaφa(r) (5.2.21)
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decreasing oscillated
When a not small,to use atomic wavefunction
w(r?l) similarequalφa(r?l)
ψk(r) = N?1/2
summationdisplay
l
eik·lφa(r?l) (5.2.18)
E(k) =
integraltext ψ?
k
braceleftBig
planckover2pi122m?2 +V(r)
bracerightBig
ψkdr
integraltext ψ?
kψkdr
similarequal
summationdisplay
h
eik·hEh (5.2.19)
Eh = 1?
c
integraldisplay
φ?a(r+h)
braceleftbigg
planckover2pi1
2
2m?
2 +V(r)
bracerightbigg
φa(r)dr (5.2.20)
bracketleftbigg
planckover2pi1
2
2m?
2 +va(r)
bracketrightbigg
φa(r) = Eaφa(r) (5.2.21)
For a simple cubic lattice with h = (a,0,0),(0,a,0) and (0,0,a)
E(k) similarequalEa + 2E100(cosakx +cosaky +cosakz),(5.2.22)
BZ is a cube,bandwidth 12|E100|,lowest energy Ea + 6E100.
Fig,5.2.4.
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atom solid
E
1/a
Figure 5.2.4 Atomic levels spreading into bands as lattice separation decreases.
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5.2.4,Kronig-Penney Model for Superlattices
semiconductor superlattice,GaAs-Ga1?xAlxAs
An electron in the direction of superlattice
one-dimensional periodic rectangular well potential
Fig,5.2.5,lattice spacing d = d1 +d2,barrier height V0
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V(z)
V0
0-d2
z
d1 d1+d2
Figure 5.2.5 One-dimensional superlattice potential.
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d2
dz2ψ+
2m
planckover2pi12 Eψ = 0,for GaAs 0 <z <d1 (5.2.23)
d2
dz2ψ+
2m
planckover2pi12 [E?V0]ψ = 0,for AlGaAs?d2 <z < 0 (5.2.24)
For E <V0
α2 = 2mEplanckover2pi12,β2 = 2m(V0?E)planckover2pi12
d2
dz2ψ+α
2ψ = 0,for 0 <z <d1 (5.2.25)
d2
dz2ψ?β
2ψ = 0 for?d2 <z < 0 (5.2.26)
ψ(z) = uk(z)eikz (5.2.27)
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d2
dz2u+2ik
d
dzu+ (α
2?k2)u = 0,for 0 <z <d1,(5.2.28)
d2
dz2u+ 2ik
d
dzu?(β
2 +k2)u = 0,for?d2 <z < 0 (5.2.29)
u1 = Aei(α?k)z +Be?i(α+k)z,(5.2.30)
u2 = Ce(β?ik)z +De?(β+ik)z (5.2.31)
u1(z)|0 = u2(z)|0,du1(z)dz
vextendsinglevextendsingle
vextendsinglevextendsingle
0
= du2(z)dz
vextendsinglevextendsingle
vextendsinglevextendsingle
0
u1(z)|d1 = u2(z)|?d2,du1(z)dz
vextendsinglevextendsingle
vextendsinglevextendsingle
d1
= du2(z)dz
vextendsinglevextendsingle
vextendsinglevextendsingle
d2
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β2?α2
2αβ sinhβd2 sinαd1 + coshβd2 cosαd1 = cosk(d1 +d2) (5.2.32)
Problem,to deduce this formula,numerical solution!!
1 ≤
parenleftbiggV
0
2E?1
parenrightbiggparenleftbiggV
0
E?1
parenrightbigg?1/2
sin
parenleftbiggd
1(2mE)1/2
planckover2pi1
parenrightbigg
×sinh
parenleftbiggd
2[2m(V0?E)]1/2
planckover2pi1
parenrightbigg
+cos
parenleftbiggd
1(2mE)1/2
planckover2pi1
parenrightbigg
cosh
parenleftbiggd
2[2m(V0?E)]1/2
planckover2pi1
parenrightbigg
≤ 1(5.2.33)
Fig,5.2.6
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E4
0 1 2 3 4 5 6 7 8 9 10
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
Ener
gy
(eV)
Well or Barrier Width (nm)
Splitting
Levels for
Single Well
Allowed Bands
for Superlattice
E3
E2
E1
Figure 5.2.6 The subbands of a GaAs-AlxGa1?xAs semiconductor superlattice.
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E
-pi/a -pi/d pi/d pi/a k
Figure 5.2.7 Folding of the Brillouin zone in a superlattice.
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5.2.5,Density of States and Dimensionality
d-dimensional box with length L,periodic boundary condition
k =
parenleftbigg2pi
L
parenrightbiggd
(5.2.34)
k → k+dk
dN = 2
parenleftbiggL
2pi
parenrightbiggdintegraldisplay k+dk
k
dk (5.2.35)
A parabolic dispersion relation E(k) = planckover2pi12k2/2m
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g(E) = 1Ld dNdE =

1
2pi2
parenleftBig2m
planckover2pi12
parenrightBig3/2
E1/2,for d = 3;
m
piplanckover2pi12,for d = 2;parenleftBig
m
2pi2planckover2pi12
parenrightBig1/2
E?1/2,for d = 1.
(5.2.36)
g(Exy) = m/piplanckover2pi12
g(E) = nm/piplanckover2pi12 (5.2.37)
Density of states of superlattice
g(E) =
parenleftBig m
piplanckover2pi12
parenrightBig 1
2pikz(E)×2 (5.2.38)
Fig,5.2.8.
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f
3D:
Density of States
2D
a b c
d e
Superlattice
Energy
1D:
m /2mE piplanckover2pi3
2m/piplanckover2pi2
m/piplanckover2pi2
m/piplanckover2pi 2E
Figure 5.2.8 The densities of states of one-,two- and three-dimensional electron
systems with that of a superlattice in the energy range including first three subbands,
such as E1 between (a) and (b); E2 between (c) and (d); E3 between (e) and (f).
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5.3,Lattice Waves and Elastic Waves
5.3.1,Dispersion Relation of Lattice Waves
5.3.2,Frequency Spectrum of Lattice Waves
5.3.3,Elastic Waves in Periodic Composites
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5.4,Electromagnetic Waves in Periodic
Structures
5.4.1,Photonic Bandgaps in Layered Periodic
Media
5.4.2,Dynamical Theory of X-Ray Diffraction
5.4.3,Bandgaps in Three-Dimensional
Photonic Crystals
5.4.4,Quasi Phase Matching in Nonlinear
Optical Crystals
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5.5,Quasiperiodic Structures
5.5.1,One-Dimensional Quasiperiodic Structure
5.5.2,Two-Dimensional Quasiperiodic Structures
5.5.3,Three-Dimensional Quasicrystals
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5.6,Waves in Quasiperiodic Structures
5.6.1,Electronic Spectra in a One-Dimensional
Quasilattice
5.6.2,Wave Transmission through Artificial
Fibonacci Structures
5.6.3,Pseudogaps in Real Quasicrystals
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Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
February 16,2004
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Part II
Wave Behavior in
Various Structures
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Contents
II Wave Behavior inVarious Structures 3
5 Wave Propagation in Periodic and Quasiperiodic Structures 3
5.1 Unity of the Concept for Wave Propagation,,,,,,4
5.1.1 Wave Equations and Periodic Potentials,,,,4
5.1.2 Bloch Waves,,,,,,,,,,,,,,,,,,,,4
5.1.3 Revival of the Study on Classical Waves,,,,4
5.2 Electrons in Crystals,,,,,,,,,,,,,,,,,,,5
5.2.1 Free Electron Gas Model,,,,,,,,,,,,,5
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5.2.2 Nearly-Free Electron Model,,,,,,,,,,,5
5.2.3 Tight-Binding Electron Model,,,,,,,,,,5
5.2.4 Kronig-Penney Model for Superlattices,,,,,5
5.2.5 Density of States and Dimensionality,,,,,,5
5.3 Lattice Waves and Elastic Waves,,,,,,,,,,,,6
5.3.1 Dispersion Relation of Lattice Waves,,,,,,6
5.3.2 Frequency Spectrum of Lattice Waves,,,,,,12
5.3.3 Elastic Waves in Periodic Composites,,,,,,17
5.4 Electromagnetic Waves in PeriodicStructures,,,,,,22
5.4.1 Photonic Bandgaps in Layered PeriodicMedia,22
5.4.2 Dynamical Theory of X-Ray Diffraction,,,,29
5.4.3 Bandgaps in Three-DimensionalPhotonic Crystals 35
5.4.4 Quasi Phase Matching in NonlinearOptical Crystals 42
5.5 Quasiperiodic Structures,,,,,,,,,,,,,,,,,47
5.5.1 One-Dimensional Quasiperiodic Structure,,,47
5.5.2 Two-Dimensional Quasiperiodic Structures,,,47
5.5.3 Three-Dimensional Quasicrystals,,,,,,,,47
5.6 Waves in Quasiperiodic Structures,,,,,,,,,,,,48
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5.6.1 Electronic Spectra in a One-DimensionalQuasilattice 48
5.6.2 Wave Transmission through ArtificialFibonacci Structures 48
5.6.3 Pseudogaps in Real Quasicrystals,,,,,,,,48
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Chapter 5
Wave Propagation in Periodic and
Quasiperiodic Structures
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5.1,Unity of the Concept for Wave Prop-
agation
5.1.1,Wave Equations and Periodic Potentials
5.1.2,Bloch Waves
5.1.3,Revival of the Study on Classical Waves
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5.2,Electrons in Crystals
5.2.1,Free Electron Gas Model
5.2.2,Nearly-Free Electron Model
5.2.3,Tight-Binding Electron Model
5.2.4,Kronig-Penney Model for Superlattices
5.2.5,Density of States and Dimensionality
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5.3,Lattice Waves and Elastic Waves
Thermal motion → Vibrations of atoms → Lattice waves → Thermo-
dynamic properties
Vibrations of long wavelengths,elastic wave propagation in artificial
periodic composites
5.3.1,Dispersion Relation of Lattice Waves
From (5.1.3),Stationary equation
Msω2ulsα =
summationdisplay
lprimesprimeβ
Φlsα,lprimesprimeβulprimesprimeβ (5.3.1)
Periodicity of Φlsα,lprimesprimeβ → Bloch theorem
A special solution for wavevector k,Bloch function
ulsα = M?1/2s Usα(k)eik·l (5.3.2)
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Eigenvalue equations for different s and α,n×3
ω2(k)Usα(k) =
summationdisplay
sprimeβ
Dsα,sprimeβ(k)Usprimeβ(k) (5.3.3)
Coefficients,i.e.,matrix elements
Dsα,sprimeβ(k) = (MsMsprime)?1/2
summationdisplay
l
Φ0sα,lprimesprimeβeik·l (5.3.4)
Secular equations
ω2(k)U(k) = D(k)U(k) (5.3.5)
D(k),3n×3n dynamic matrix; U(k),1×3n state vector
Secular determinant
|D(k)?ω2(k)I| = 0 (5.3.6)
→ Eigenfrequencies ω2j(k),j = 1,···,3n
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Symmetric properties
Dsα,sprimeβ(k) = D?sprimeβ,sα(k)
D(k) = D(k+G),D(k) = D(?k),D(k) = D(gk)
ω2j(k) → Usα,j(k)
3n wavelike solutions for each k
3nN independent solutions in total
General solution
ulsα = (NMs)?1/2
summationdisplay
kj
Usα,j(k)exp{i[k·l?ωj(k)t]} (5.3.7)
3nN eigenfrequencies form the continuous curves as a function of k
ω = ωj(k) (j = 1,···,3n)
3n branches of the dispersion curves
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A linear diatomic chain,M1 and M2
Force constant f
Potential energy
Φ = 12
summationdisplay
m
f(um?um+1)2 (5.3.8)
Equations of motion
M1 d
2
dt2u2m+1 =?f(2u2m+1?u2m?u2m+2)
M2 d
2
dt2u2m+2 =?f(2u2m+2?u2m+1?u2m+3) (5.3.9)
A set of trial solutions
u2m+1 = U1ei[k(2m+1)?ωt],
u2m+2 = U2ei[k(2m+2)?ωt] (5.3.10)
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Secular equations
ω2M1U1 =?2fU1 +2f cos(ka)U2
ω2M2U2 =?2fU2 +2f cos(ka)U1 (5.3.11)
Secular determinant
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
2f?M1ω2?2f cos(ka)
2f cos(ka) 2f?M2ω2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= 0 (5.3.12)
Two eigenfrequencies
ω2± = f
parenleftbigg 1
M1 +
1
M2
parenrightbigg
±f
braceleftBiggparenleftbigg
1
M1 +
1
M2
parenrightbigg2
4sin
2(ka)
M1M2
bracerightBigg1/2
(5.3.13)
Fig,5.3.1,acoustic modes and optical modes
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k0 pi/2a-pi/2a
ω
Figure 5.3.1 Pass-band and stop-band for a linear diatomic lattice.
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5.3.2,Frequency Spectrum of Lattice Waves
Frequency distribution function
g(ω) = lim
ω→0
1
ω
ω≤ωj(k)≤ω+?ωsummationdisplay
j
summationdisplay
k
=
summationdisplay
j
summationdisplay
k
δ(ω?ωj(k)) (5.3.14)
Total number of frequencies
integraldisplay ∞
0
g(ω)dω = 3nN (5.3.15)
Sum rule summationdisplay
k
→?(2pi)3
integraldisplay
BZ
dk (5.3.16)
Integral form
g(ω) = lim
ω→0
1
ω
(2pi)3
summationdisplay
j
integraldisplay
ω≤ωj(k)≤ω+?ω
dk (5.3.17)
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Volume integral → a surface integral
dS,elementary area,on a constant frequency surface S,
corresponding to same ω
kωj(k) →?ω/?k
k =?ω/|?ωj(k)|
integraldisplay
ω≤ωj(k)≤ω+?ω
dk =
integraldisplay
S
dS?k =?ω
integraldisplay
S
dS
|?ωj(k)| (5.3.18)
g(ω) =
summationdisplay
j
gj(ω) =?(2pi)3
summationdisplay
j
integraldisplay
S
dS
|?ωj(k)| (5.3.19)
|?ωj(k)| = 0,van Hove singularity
Frequency distribution g(ω),related to thermodynamic properties
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Two approximate approaches:
(1) Einstein model (1907),discrete lattice
Each atom with equivalent vibration
Einstein frequency
ω = ωE,or 0
Density of states
g(ω) = 3nNδ(ω?ωE) (5.3.20)
→ High temperature thermodynamic properties
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(2) Debye model (1912),isotropic continuum
Three acoustic branches,long waves,elastic waves
Dispersion relation
ωl = vlk,ωt = vtk (5.3.21)
Density of states
g(ω)dω =?(2pi)34pik2dk =?(2pi)34pi
parenleftBigω
v
parenrightBig2 dω
v
g(ω) =?2pi2 ω
2
v3 =
2pi2
parenleftbigg1
v3l +
2
v3t
parenrightbigg
ω2 (5.3.22)
→ Low temperature specific heat T3
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To determine Debye frequency ωD
integraldisplay ωD
0
g(ω)dω = 3nN (5.3.23)
ωD =
parenleftbigg6pi2nN
parenrightbigg1/3
va,(5.3.24)
va = 3/v3a = (1/v3l +2/v3t)
Debye characteristic temperature θD = θD = planckover2pi1ωD/kB
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5.3.3,Elastic Waves in Periodic Composites
Phononic Crystals
Elastic or acoustic waves in inhomogeneous media
Geophysics and Solid State Physics
Gaps,stop-bands for elastic wave propagating in periodic media
Transversal and a longitudinal modes
Elastic wave equations
ρ(r)?
2uα
t2 =
xα
parenleftbigg
λ(r)?uβ?x
β
parenrightbigg
+x
β
bracketleftbigg
μ(r)
parenleftbigg?u
α
xβ +
uβ
xα
parenrightbiggbracketrightbigg
,α,β = 1,2,3
(5.3.25)
Density ρ(r),Displacement vector u(r),Lam′e coefficients,λ(r),μ(r)
Periodic conditions
λ(r +l) = λ(r),μ(r+l) = μ(r),ρ(r+l) = ρ(r) (5.3.26)
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Bloch function
u(r) = Uk(r)exp(k·r)
Secular equations
ω2cuα,k+G =
summationdisplay
Gprime
summationdisplay
β,Gprimeprime
ρ?1G?Gprimeprime [λGprimeprime?Gprime(k+Gprime)β(k+Gprimeprime)α
+ μGprimeprime?Gprime(k+Gprime)α(k+Gprimeprime)β]uβ,k+Gprime
+
summationdisplay
Gprimeprime
ρ?1G?GprimeprimeμGprimeprime?Gprimesummationdisplay
β
(k+Gprime)β(k+Gprimeprime)β
uα,k+Gprime

,
(5.3.27)
c,acoustic velocity; take N reciprocal vectors,3N ×3N matrix eigen-
value equations
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Theoretically,Economou and Sigalas (1994),Au spherical inclusions in
a Si matrix,fcc lattice,choose N = 400,volume fraction is 10%,the
convergence is better than 1%,Fig,5.3.2
Experimentally,F,R,Montero de Espinosa,E,Jimenez,andM,Torres,
Phys,Rev,Lett,(1998),Fig,5.3.3
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W KXLU0.0
1.4
2.7
4.1
5.4
Au / Si
ω/2pi
(kHz)
Γ
Figure 5.3.2 The dispersion relation of a periodic composite.
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1.2
0.8
0.4
0.0
[110] k (mm-1) [100]
01 1
ω /2
pi (MHz
)
Figure 5.3.3 Experimentally observed dispersion relation of ultrasonic wave in a
two-dimensional periodic composite.
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5.4,Electromagnetic Waves in Periodic
Structures
Spatially dependent epsilon1(r),μ(r)
Assumed to be scalar
For nonmagnetic media,μ = 1
5.4.1,Photonic Bandgaps in Layered Periodic
Media
Two kinds of slabs,d1 and d2,epsilon11 and epsilon12
Fig,5.4.1(a)
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d1 d2
n1 n2
d
(a)
l
Figure 5.4.1 (a) Layered periodic medium composed with two kinds of slabs with
thicknesses d1 and d2.
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Equation of electric field
d2
dz2E(z) +
ω2
c2 epsilon1(z)E(z) = 0 (5.4.1)
Periodic potential
epsilon1(z +d) = epsilon1(z) (5.4.2)
d = d1 + d2
Bloch function
E(z +d) = E(z)eikd (5.4.3)
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Trial solutions,amplitudes of electric fields Al,Bl Cl,Dl
E(1)l (z) = Aleiq1z +Ble?iq1z
E(2)l (z) = Cleiq2z +Dle?iq2z (5.4.4)
E(1)l+1(z) = Al+1eiq1z + Bl+1e?iq1z
Implied frequency
q1 = √epsilon11ω/c = n1ω/c,q2 = √epsilon12ω/c = n2ω/c
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Relation for Al+1,Bl+1 Al,Bl
parenleftBigg
Al+1
Bl+1
parenrightBigg
= T
parenleftBigg
Al
Bl
parenrightBigg
(5.4.5)
Matrix elements
T11 = eiq1d1
bracketleftbigg
cos(q2d2) + i2
parenleftbiggn
1
n2 +
n2
n1
parenrightbigg
sin(q2d2)
bracketrightbigg
T12 = eiq1d1 i2
parenleftbiggn
1
n2?
n2
n1
parenrightbigg
sin(q2d2)
T21 = T?12 (5.4.6)
T22 = T?11
Bloch theorem parenleftBigg
Al+1
Bl+1
parenrightBigg
= eikd
parenleftBigg
Al
Bl
parenrightBigg
(5.4.7)
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(T?eikdI)
parenleftBigg
Al
Bl
parenrightBigg
= 0 (5.4.8)
(T?1?e?ikdI)
parenleftBigg
Al
Bl
parenrightBigg
= 0 (5.4.9)
coskd = 12(T +T?1) = 12TrT (5.4.10)
Dispersion relation
cosk(d1+d2) = cos n1ωd1c cos n2ωd2c?12
parenleftbiggn
1
n2 +
n2
n1
parenrightbigg
sin n1ωd1c sin n2ωd2c
(5.4.11)
Compare it with the one for electron in semiconductor superlattices
Fig,5.4.1(b)
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kpi/d 2pi/d
(b)
ω
Figure 5.4.1 (b) The dispersion relation for a one-dimensional periodic dielectric
structure.
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5.4.2,Dynamical Theory of X-Ray Diffraction
X-ray diffraction in three dimensional crystals
kinematical theory for small or imperfect crystals
dynamical theory for large and perfect crystals
Rewritten (5.1.13),Stationary equation
2D +K2D +?×?×[χ(r)D] = 0 (5.4.12)
Free wavevector
K = |K| = ω/c
Electric susceptibility
χ(r) =? e
2
mω2ρ(r) (5.4.13)
Electron density ρ(r) ≈ 1023-1025 cm?3 → χ ≈ 10?6-10?4 → epsilon1 ≈ 1
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Fourier transform
χ(r) =
summationdisplay
G
χGe?iG·r (5.4.14)
and
χG = 1?
c
integraldisplay
c
χ(r)eiG·rdr =? 4pie
2
mω2?cFG (5.4.15)
Structure factor
FG =
summationdisplay
j
fjeiG·rj (5.4.16)
Atomic scattering factor fj at rj
Bloch function
D(r) =
summationdisplay
G
DGeikG·r (5.4.17)
Incident wavevector k0 in crystal,scattering wavevector kG
kG = k0 +G (5.4.18)
(5.4.12) → General secular equation
parenleftbigK2?k2
G
parenrightbigD
G?
summationdisplay
Gprime
χG?Gprime [kG ×(kG ×DGprime)] = 0 (5.4.19)
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Two-wave approximation,only kG near K,the amplitude DG is large
Vectorial secular equations
(k20?K2)D0 = k20(χ0D0 +χ?GDG)
(k2G?K2)DG = k2G(χGD0 + χ0DG) (5.4.20)
Two polarization states,σ (C = 1),pi (C = cos2θ)
Scalar secular equations
(k20?K2)D0 = k20(χ0D0 +Cχ?GDG)
(k2G?K2)DG = k2G(CχGD0 + χ0DG) (5.4.21)
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Further approximation
n = kK = 1 + 12χ0 =
parenleftbigg
1? e
2〈ρ(r)〉
mω2
parenrightbigg
< 1 (5.4.22)
k20?K2?k20χ0 similarequal k20?K2
parenleftbigg
1 + 12χ0
parenrightbigg2
= k20?k2
k2G?K2?k2Gχ0 similarequal k2G?k2 (5.4.23)
k0 + k similarequal kG + k similarequal 2K
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Secular determinant
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
k0?k?12CKχ?G
12CKχG kG?k
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= 0 (5.4.24)
Dispersion relation in X-ray dynamical diffraction theory
(k0?k)(kG?k)? 14C2K2χ?GχG = 0 (5.4.25)
Fig,5.4.2,dispersion surface in reciprocal space
If the gap can be tested from experiments?
exp(?iK ·τ) = exp(?ik0 ·τ) = exp(?ikG ·τ) (5.4.26)
K?k0 = K?kG = δkn (5.4.27)
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(a) (b)
K
k
A
BC
D
L'
L
O G
L'
L
A
C
B
D
P
P' P''I
k0 kG
n
O G
A'
C'
B'
D'
E F
S(2)
S(1)
Figure 5.4.2 Dispersion surface in reciprocal space in two-wave approximation.
(a) Globle picture,parenleftbigK?k = Kvextendsinglevextendsingle12χ0vextendsinglevextendsingleparenrightbig; (b) Local magnification.
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5.4.3,Bandgaps in Three-Dimensional
Photonic Crystals
Difficult for X rays to show bandgaps
Three-dimensional artificial periodic structures show
Bands and gaps for electromagnetic waves
Frequency-wavevector relation for electromagnetic wave
→ photonic band structure
Relevant quantities,(1) lattice constant,(2) shape of the embedded
dielectric object,(3) dielectric constants,(4) filling fraction
First experiment,E,Yablonovitch and T,J,Gmitter,Phys,Rev,Lett.
(1989)
Fig,5.4.3
Microwaves,photonic crystal
Measurement of phases and amplitudes → ω ～k
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L
UX
WK
X U L X W K
X U L X W K
ω
k
Γ
Γ
Figure 5.4.3 The experimentally observed photonic band structure in reciprocal
space of spherical-air-atom crystal.
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Theoretically,first scalar wave theory,then vector wave theory
Bloch function
Dk(r) = uk(r)eik·r (5.4.28)
Dispersion relation ω2(k) → band structure
Fourier expansion
Dk(r) =
summationdisplay
G
DGei(k+G)·r (5.4.29)
·D = 0 gives
DG ·(k+G) = 0 (5.4.30)
Secular equations
QGDG +
summationdisplay
Gprime
χG?Gprimebracketleftbig(k+G)·DGprime(k+G)?|k+G|2DGprimebracketrightbig= 0
(5.4.31)
with
QG = |k+G|2?ω2/c2 (5.4.32)
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Secular determinant
vextendsinglevextendsingleQ
GδG,Gprimeδα,βχGprime?G
bracketleftbig(k+Gprime)
α(k+Gprime)β?|k+Gprime|2δα,β
bracketrightbigvextendsinglevextendsingle= 0
(5.4.33)
Numerical solution
Choose epsilon1a,epsilon1b,N plane waves,N ≥ 300
3N ×3N order determinant,3N eigenmodes
2N real modes in total
Fig,5.4.4,epsilon1a = 1,epsilon1b = 12.25
K,M,Leung and Y,E,Liu,Phys,Rev,Lett,(1990)
Symmetry points W and U,Quasigap
Decreasingsymmetryoftheunitcellsinfcclattice,nonspherical‘atoms’,
complete photonic bandgap
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fcc
20
15
10
5
0
X U L X W K
ω
k
Γ
Figure 5.4.4 Theoretical photonic band structure of fcc dielectric structure.
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Special advantages by using magnetic field H
Wave equation
×
bracketleftbigg 1
epsilon1(r)?×H(r)
bracketrightbigg
=
parenleftBigω
c
parenrightBig2
H(r) (5.4.34)
Operator
Θ ≡?×
bracketleftbigg 1
epsilon1(r)?×
bracketrightbigg
Operator equation
ΘH(r) =
parenleftBigω
c
parenrightBig2
H(r) (5.4.35)
Hamilton-like
H ≡? planckover2pi1
2
2m?
2 + V(r)? Θ ≡?×
bracketleftbigg 1
epsilon1(r)?×
bracketrightbigg
Electric field
E(r) =
bracketleftbigg
icωepsilon1(r)
bracketrightbigg
×H(r) (5.4.36)
Hk(r) = H0eik·r (5.4.37)
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Experimentally
At the beginning,microwave frequencies
Tuning condition
Longer wavelengths of microwaves →
Larger unit cell size,easy to fabricate
In the later 1990s,the investigation were concentrated on
Infrared and visible range
Possibility of photonic crystals to manipulate the light propagation
Its applicability in laser technology,optical communications
Theoretically
Extended to various possible cases for structures and materials
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5.4.4,Quasi Phase Matching in Nonlinear
Optical Crystals
Strong electric field,nonlinearity
Electric polarization vector
P = χE + χ(2)EE +χ(3)EEE +··· (5.4.38)
Nonlinear polarization → frequency conversion of laser lights:
Frequency doubling,Sum frequency,Difference frequency
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Frequency doubling from ω to 2ω
A plane wave along x
E1 = e1E0 sin(ω1t?k1x)
Wavevector
k1 = n1ω1/c = 2pin1/λ1
n1 for ω1
Polarization wave
P(2ω1) =?12χ(2)e1e1E21 cos(2ω1t?2k1x)
Velocity of polarization wave
vP = 2ω1/2k1 = ω1/k1 = v1
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The second harmonic,superposition of two waves:
(1) forced wave
Eprime2ω = eprime2Eprime2 cos(2ω1t?2k1x)
(2) free wave
E2ω = e2E2 cos(2ω1t?2k2x)
k2 = 2ω1n2/c,ω2 = 2ω1,n2 for ω2,and n2 negationslash= n1
Forced wave and free wave propagate with different velocities,although
same frequency
Phase mismatching
k = k2?2k1
Spatial period of polarization
2lc = 2pi/?k
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To realize phase-matching,?k = 0
Artificial periodic structures
Alternating positive and negative nonlinear susceptibilities
G to compensate phase-mismatching?k,so
k = k2?2k1 = G = pil
c
or more generally in vector form
k = G
2lc is the spatial period for positive and negative layers,
lc = λ4(n
2?n1)
about one micron for normal nonlinear crystals
Fig,5.4.5
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+deff +deff +deff +deff-deff -deff -deff -deff
QPM
NPM
lc
l
E2ω
ω 2ω
Figure 5.4.5 Amplitude of second harmonic waves versus optical path for a mod-
ulated structure satisfying the quasi phase matching.
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5.5,Quasiperiodic Structures
5.5.1,One-Dimensional Quasiperiodic Structure
5.5.2,Two-Dimensional Quasiperiodic Structures
5.5.3,Three-Dimensional Quasicrystals
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5.6,Waves in Quasiperiodic Structures
5.6.1,Electronic Spectra in a One-Dimensional
Quasilattice
5.6.2,Wave Transmission through Artificial
Fibonacci Structures
5.6.3,Pseudogaps in Real Quasicrystals
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Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
February 18,2004
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Contents
5 Wave Propagation in Periodic andQuasiperiodic Structures 3
5.1 Unity of the Concept for WavePropagation,,,,,,,4
5.1.1 Wave Equations and Periodic Potentials,,,,4
5.1.2 Bloch Waves,,,,,,,,,,,,,,,,,,,,4
5.1.3 Revival of the Study on Classical Waves,,,,4
5.2 Electrons in Crystals,,,,,,,,,,,,,,,,,,,5
5.2.1 Free Electron Gas Model,,,,,,,,,,,,,5
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5.2.2 Nearly-Free Electron Model,,,,,,,,,,,5
5.2.3 Tight-Binding Electron Model,,,,,,,,,,5
5.2.4 Kronig-Penney Model for Superlattices,,,,,5
5.2.5 Density of States and Dimensionality,,,,,,5
5.3 Lattice Waves and Elastic Waves,,,,,,,,,,,,6
5.3.1 Dispersion Relation of Lattice Waves,,,,,,6
5.3.2 Frequency Spectrum of Lattice Waves,,,,,,6
5.3.3 Elastic Waves in Periodic Composites,,,,,,6
5.4 Electromagnetic Waves in PeriodicStructures,,,,,,7
5.4.1 Photonic Bandgaps in Layered PeriodicMedia,7
5.4.2 Dynamical Theory of X-Ray Diffraction,,,,7
5.4.3 Bandgaps in Three-DimensionalPhotonic Crystals 7
5.4.4 Quasi Phase Matching in NonlinearOptical Crystals 7
5.5 Quasiperiodic Structures,,,,,,,,,,,,,,,,,8
5.5.1 Three-Dimensional Quasicrystals,,,,,,,,9
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5.5.2 Two-Dimensional Quasiperiodic Structures,,,16
5.5.3 One-Dimensional Quasiperiodic Structure,,,21
5.6 Waves in Quasiperiodic Structures,,,,,,,,,,,,26
5.6.1 Electronic Spectra in a One-DimensionalQuasilattice 26
5.6.2 Wave Transmission through ArtificialFibonacci Structures 35
5.6.3 Pseudogaps in Real Quasicrystals,,,,,,,,40
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Chapter 5
Wave Propagation in Periodic and
Quasiperiodic Structures
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5.1,Unity of the Concept for Wave
Propagation
5.1.1,Wave Equations and Periodic Potentials
5.1.2,Bloch Waves
5.1.3,Revival of the Study on Classical Waves
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5.2,Electrons in Crystals
5.2.1,Free Electron Gas Model
5.2.2,Nearly-Free Electron Model
5.2.3,Tight-Binding Electron Model
5.2.4,Kronig-Penney Model for Superlattices
5.2.5,Density of States and Dimensionality
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5.3,Lattice Waves and Elastic Waves
5.3.1,Dispersion Relation of Lattice Waves
5.3.2,Frequency Spectrum of Lattice Waves
5.3.3,Elastic Waves in Periodic Composites
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5.4,Electromagnetic Waves in Periodic
Structures
5.4.1,Photonic Bandgaps in Layered Periodic
Media
5.4.2,Dynamical Theory of X-Ray Diffraction
5.4.3,Bandgaps in Three-Dimensional
Photonic Crystals
5.4.4,Quasi Phase Matching in Nonlinear
Optical Crystals
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5.5,Quasiperiodic Structures
Two types of atomic structures for solids
(1) Crystals,highly ordered
Long-range translational order? Periodic repetition of unit cells
Long-range orientational order? Discrete rotations
(2) Glasses,only short-range correlation
Metallic glasses,randomly densely-packed spheres
Amorphous semiconductors,random networks
What may be between them?
(3) Quasicrystals,another class of long-range order
Long-range quasiperiodic translational order
Long-range orientational order with
Forbidden rotation symmetry
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5.5.1,Three-Dimensional Quasicrystals
Shechtman et al,1984,Electron diffraction,Phys,Rev,Lett.
Crystallographically forbidden icosahedral symmetry
Levine and Steinhardt,1984,quasicrystal
A new state of matter,between crystals and glasses
Generalized crystallography
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{3,3} {3,4} {4,3}
{5,3} {3,5}
Figure 5.5.1 Five kinds of regular polyhedra.
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79.2o 58.29o 37.38o
31.72o
63.43o
36o
Figure 5.5.2 Electron diffraction patterns of Al-Mn quasicrystal.
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79.2
ringoperator
58.29
ringoperator
37.37
ringoperator
63.43ringoperator
31.72ringoperator
Figure 5.5.3 The stereogram showing the relation of the axial (face) angles between
the cube and icosahedron symmetry groups.
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A 3D quasicrystal with icosahedral symmetry may be deduced
From the origin,six 5-fold symmetry axes of a regular icosahedron
Unit vectors ei (i = 1,2,3,4,5,6)
A series of planes perpendicular to these vectors
Intersections of these planes → a quasilattice
Two unit cells,the oblate and prolate rhombohedra
3D quasilattice may be formed by the cut and projection from
A periodic lattice in 6D hyperspace
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a6
a2
a3a
1
a4
a5
b3
b1 b2
Figure 5.5.4 Regular icosahedron with Cartesian coordinates and six vertices which
determine the directions of 5-fold axes.
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(0,0,0) (0,1,τ)
(0,-1,τ)
(τ,0,1)
(-τ,0,1)
(0,0,0)
(τ,0,-1)
(0,1,τ)
Figure 5.5.5 Unit cells for 3D Penrose tiling,(a) Oblate; (b) Prolate.
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5.5.2,Two-Dimensional Quasiperiodic Structures
Nonperiodic tiling of plane
Penrose tiling,1974,by two rhombs,matching rule
Deflation rules for rescaling are shown in Fig,2.3.9
From this,the ratio of the numbers of the fat and the thin rhombs in
a Penrose tiling is just τ
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(a) (b)
Figure 5.5.6 Pentagrids,(a) Periodic; (b) Quasiperiodic.
Penrose tilings,quasiperiodic geometric figures
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36o72o
Figure 5.5.7 Penrose tiling.
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Figure 5.5.9 Inflation and deflation rules for Penrose tiles.
Marckay,1962,optical diffraction
After ordered icosahedral phase in 3D,several 2D quasicrystals were
discovered
2D quasicrystals,atoms or ions are packed to form
With 8,10 and 12-fold symmetries
Decagonal phase is very like the Penrose tiling
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5.5.3,One-Dimensional Quasiperiodic Structure
Fibonacci,1202,rabbit breeding
Substitution rule
A→AB,B →A (5.5.1)
Matrix form
Mij =
1,1
1,0
,Mij
A
B
=
AB
A
→ ABA (5.5.2)
Next generation
Mij
AB
A
=
ABA
AB
→ ABAAB,.,(5.5.3)
A sequence B → A → AB → ABA → ABAAB → ABAABABA
→ ABAABABAABAAB ···
Total numbers of As and Bs for a generation →
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Fibonacci numbers,1,1,2,3,5,8,13,21,34,...
Recursion relation
un+1 = un +un?1 (5.5.4)
so
un+1
un =
un +un?1
un = 1 +
un?1
un = 1 +
1
un
un?1
= 1 + 1
1 + un?2u
n?1
= 1 + 1
1 + 1
1 + 11 +···
(5.5.5)
A irrational number=a infinite sequence of the rational number
The limit form
τ = limn→∞un+1u
n
= 1 + 11 +τ (5.5.6)
τ2?τ?1 = 0
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L L L L L
LLLLLLLL
S S S
S S S S S
Figure 5.5.10 The self-similarity of a Fibonacci lattice.
Golden number
τ = 1 +
√5
2 = 1.618···,(5.5.7)
Self-similarity of Fibonacci lattice,deflation and inflation rules
To generate 1D quasiperiodic lattice from 2D periodic lattice
From a 2D square lattice with lattice spacing a
ρ(x,y) =
summationdisplay
n,m
δ(x?na)δ(y?ma) (5.5.8)
Fig,2.3.2
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α
a
R
R
S
L
Figure 5.5.11 Cut and projection of 2D square lattice (projection line tanα = 1/τ,
stipe width?).
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(3,3)
(5,1)
(2,3)
(3,1)
(1,2)
(1,1)
(2,6)
(0,12)
(3,2)
(4,1)
(0,4)
(2,1)
(0,1)0
Figure 5.5.12 Electron micrograph and electron diffraction pattern of an artificial
Fibonacci superlattice of a-SiH/SiN:H,taken from Chen Kunji et al,J,Noncrys.
Solid,97 & 98,341 (1987).
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5.6,Waves in Quasiperiodic Structures
Structural feature → physical properties
Periodic structures → Bloch waves,extended states
Disordered structures → exponentially decreased,localized states
Quasicrystalstructures→intermediatebetweenboth,neitherextended
nor strictly localized.
5.6.1,Electronic Spectra in a One-Dimensional
Quasilattice
An electron in a Fibonacci lattice
M,Kohmoto,B,Sutherland,and C,Tang,Phys,Rev,B (1987)
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Tight-binding model
tiψi?1 +εiψi +ti+1ψi+1 = Eψi (5.6.1)
Transfer model,εi = 0 for all i; two transfer energies tA,tB
To (5.6.1),adopt periodic boundary condition
l-th generation with sites N = Fl.
Define transfer matrix
M(ti+1,ti) =
E/ti+1?ti/ti+1
1 0
(5.6.2)
Column vector
Ψi =
ψi
ψi?1
(5.6.3)
(5.6.1) → a matrix equation
Ψi+1 = M(ti+1,ti)Ψi (5.6.4)
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Sequential products of the matrices →
Ψi+1 = M(i)Ψ1 (5.6.5)
where
M(i) =
iproductdisplay
j=1
M(tj+1,tj) (5.6.6)
M(i) can be obtained recursively
Define Ml ≡M(Fl),then
Ml+1 = Ml?1Ml (5.6.7)
Initial conditions
M1 = M(tA,tA),M2 = M(tA,tB)M(tB,tA)
The transfer matrix for a general value i
M(i) = Mlj ···Ml2Ml1 (5.6.8)
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i = Fl1 +Fl2 +···+Flj,l1 >l2 >···>lj
A nonlinear dynamical map for trace
χl = (1/2)trMl
Trace map
χl+1 = 2χlχl?1?χl?2 (5.6.9)
Invariance for Fibonacci lattice
I = χ2l+1 +χ2l +χ2l?1?2χl+1χlχl?1?1 (5.6.10)
Invariance for transfer model
I = 14
parenleftbiggt
B
tA?
tA
tB
parenrightbigg2
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Electronic energy spectra are like Cantor set with three branches
Integrated density of states in Fig,5.6.1(a)
Wavefunctions display self-similar amplitude distributions
A critical state in Fig,5.6.1(b)
Three different types of energy spectrum:
(1) Absolute continuous:
A smooth density of states g(E)
(2) Point-like:
A set of δ-functions-like countable {Ei}
(3) Singular continuous:
Energies E continuously increases
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-3 0 3
E
0
1
(a)
0.5
IDOS
Figure 5.6.1 One-dimensional Fibonacci lattice,(a) IDOS.
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(b)
i
Figure 5.6.1 One-dimensional Fibonacci lattice (b) a self-similar wave function.
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But non-differentiable at any value,just like the Cantor set
DOS is not well-defined,only IDOS is well-defined
Three kinds of wavefunctions
(1) Extended state
with asymptotical uniform amplitude as
integraldisplay
|r|<L
|ψ(r)|2dr ～Ld
L is the size of the sample and d the spatial dimension
(2) Localized state
characterized by a square integrable wavefunction
integraldisplay
|r|<∞
|ψ(r)|2dr ～L0
(3) Critical state
A typical example is a power-law function
ψ(r) ～|r|?ν,with ν ≤d/2
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integraldisplay
|r|<L
|ψ(r)|2dr ～L?2ν+d,(0 < 2ν <d)
Three kinds of wavefunctions may correspond to three energy spectra
The method of transfer matrix for electronic structure,can also be
applied to lattice vibration in one-dimensional quasiperiodic structures
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5.6.2,Wave Transmission through Artificial
Fibonacci Structures
Molecular-Beam Epitaxy and Lithography techniques
Artificial Fibonacci structures
To study the wave behavior experimentally and theoretically
For example,electromagnetic waves
Fibonacci superlattice composed of two types of dielectric layers A and
B
Electric field for the light in layer A (similar for layer B)
EA = E(1)A exp[i(k(1)A ·r?ωt)] +E(2)A exp[i(k(2)A ·r?ωt)] (5.6.11)
Boundary condition at the interface gives
E(1)A +E(2)A = E(1)B +E(2)B
nA cosθA
parenleftBig
E(1)A?E(2)A
parenrightBig
= nB cosθB
parenleftBig
E(1)B?E(2)B
parenrightBig
(5.6.12)
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Indices of refraction of layer A and layer B,nA and nB; Incident and
reflective angles,θA and θB
Choose independent variables as
E+ = E(1) +E(2),E? = (E(1)?E(2))/i,(5.6.13)
Matrix equation?
E+
E?
B
= TBA
E+
E?
A
(5.6.14)
TBA represents light propagation across interfaces from A to B
TBA =
1 0
0 nA cosθA/nB cosθB
(5.6.15)
TAB = T?1BA
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Propagation within layer A (similar for layer B)
TA =
cosδA?sinδA
sinδA cosδA
(5.6.16)
with
δA = nAkdA/cosθA,δB = nBkdB/cosθB (5.6.17)
k is the wave number in vacuum,dA and dB are the thicknesses of the
layers
For one layer A,and two layers BA,the light propagation are given by
M1 = TA,M2 = TABTBTBATA (5.6.18)
Recursion relation (5.6.7) is applicable
Invariance
I = 14 sin2δA sin2δB
parenleftbiggn
A cosθA
nB cosθB?
nB cosθB
nA cosθA
parenrightbigg2
(5.6.19)
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For the case nA = nB,I = 0,no quasiperiodicity
Transmission coefficient T
T = 4/(|Ml|2 +2) (5.6.20)
|Ml|2 is the sum of the squares of the four elements of Ml.
Simplest experimental setting,incident light is normal,θA = θB = 0,
Let nAdA = nBdB → δA = δB = δ
For δ = mpi,1/2 wavelength layer,I = 0,the transmission is perfect;
forδ = (m+12)pi,1/4 wavelength layer,I is maximum,quasiperiodicity
is most effective
Fig,5.5.5.
These characteristics were verified experimentally,W,Gellermann et
al.,Phys,Rev,Lett,(1994)
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1.5pi 2pi
1.398pi 1.602pi1.5pi
pi
T
T
δ
Figure 5.6.2 The transmission coefficient T vs the optical phase length of a layer
δ for a Fibonacci multilayer F9 (55 layers),The indices of refraction are chosen as
nA = 2 and nB = 3.
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5.6.3,Pseudogaps in Real Quasicrystals
The efficient approach in 1D quasiperiodic lattices
Not applicable to 2D or 3D quasicrystals
Tight-binding approximation also used to 2D or 3D Penrose lattice
Only numerical calculations,no analytic solution
Real quasicrystals are all alloys,nearly-free electron model may help us
to elucidate the electronic structures and properties of quasicrystals.
Electrons in a perfect crystal,strong Bragg reflections → energy gaps,
and the peaks,and valleys in g(E) versus E
In 3D quasicrystal,no reciprocal lattices and BZ
Diffraction pattern shows sharp diffraction spots with icosahedral sym-
metry → a series of quasi-reciprocal vectors → QBZ
Fig,5.5.3 shows QBZs for two icosahedral phases.
Stability of quasiperiodic order is intimately related to their electronic
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(a) (b)
Figure 5.6.3 Quasi BZ for two i phases,(a) Al-Mn alloys; (b)Al-Cu-Li alloys.
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structure
Icosahedral structure may be established through energetic considera-
tions
When Fermi surface intersects the quasi Brillouin zone boundaries →
a pseudogap in DOS
Fermi level situated near the minimum of pseudogap → low energy
Theoretical calculation in Fig,5.5.4 suggests the stability of the qua-
sicrystal relative to various competing crystalline phases
The existence of pseudogap at Fermi energy is generic in quasicrystals.
The origin of pseudogap is attributed to strong electron scattering by
the quasi-lattice and suggests the gap-opening mechanism by a touch-
ing of the Fermi surface at the effective Brillouin zone → an enhance-
ment of cohesive energies
Hume-Rothery mechanism for stability of metals and alloy
It works more efficiently in quasicrystals because the effective Brillouin
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0 5 10 15
0
0.5
1.0
g (E
)
E (eV)
periodic
quasiperiodic
Figure 5.6.4 DOS curves calculated from the NFE model for Al quasi i phase
(solid curve) and crystalline phase (dashed curve),From A,P,Smith and N,W.
Ashcroft,Phys,Rev,Lett,59,1365 (1987).
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zone is almost spherical
TheelectronicstructuresofquasicrystalsAl-Cu-LiisshowninFig.5.5.5.
Low values of DOS at the Fermi level are observed in the crystalline
as well as in the quasicrystalline phases,Hence,although the pseudo-
gap is a generic property of quasicrystals,it is not a specific property
distinguishing the quasiperiodic from the periodic or aperiodic phases.
Crystalline,quasicrystalline,and amorphous alloys have to be con-
sidered as Hume-Rothery phases with a varying degree of band-gap
stabilization
Unusual transport properties
Experimental measurements shows semimetallic transport properties:
high resistivity,negative temperature coeffcient for the resistivity
strong temperature and composition dependence of the Hall coefficient
and thermopower
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-10 -5 0 50
0.5
1.0
1.5
(E?EF ) (eV)
g(E
)
Figure 5.6.5 Pseudoenergy gap in the calculated DOS curve of i-Al-Cu-Li phase.
From M,Windish et al.,J Phys,Condens,Matter 6,6977 (1994).
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
March 6,2004
Contents
Chapter 6,Dynamics of Bloch Electrons 3
§6.1 Basic Properties of Electrons in Bands,,,,,,,,,,,,6
6.1.1 Electronic Velocity and Effective Mass,,,,,,,,,7
6.1.2 Metals and Nonmetals,,,,,,,,,,,,,,,,,15
6.1.3 Hole,,,,,,,,,,,,,,,,,,,,,,,,,,,26
6.1.4 Electronic Specific Heat in Metals,,,,,,,,,,,30
§6.2 Electronic Motion in Electric Fields,,,,,,,,,,,,,35
6.2.1 Bloch Oscillations,,,,,,,,,,,,,,,,,,,,36
6.2.2 Negative Differential Resistance,,,,,,,,,,,,45
6.2.3 Wannier-Stark Ladders,,,,,,,,,,,,,,,,,49
§6.3 Electronic Motion in Magnetic Fields,,,,,,,,,,,,60
6.3.1 Cyclotron Resonance,,,,,,,,,,,,,,,,,,60
6.3.2 Landau Quantization,,,,,,,,,,,,,,,,,,60
6.3.3 de Haas-van Alphen Effect,,,,,,,,,,,,,,,60
6.3.4 Susceptibility of Conduction Electrons,,,,,,,,,60
Chapter 6
Dynamics of Bloch Electrons
Bloch electrons in periodic structures
Existence of electric field or magnetic field,or both
Dynamical problem
Technical applications
Wave-particle duality,T <T0
Semiclassical approach,λF lessmuchL
Correspondence of Quantum and Classical Physics
E = planckover2pi1ω
p = planckover2pi1k
v =?kω = 1planckover2pi1?kE(k)
dp
dt = planckover2pi1
dk
dt = F
§6.1 Basic Properties of Electrons in Bands
An electron has energy,momentum and mass
In a crystal,it behaves as Bloch wave
Energy → band structure
Momentum → crystal momentum,or quasi-momentum
Mass → effective mass
6.1.1 Electronic Velocity and Effective Mass
An electron in a Bloch state ψk(r)
Group velocity of a wave packet is given by v =?kω(k)
de Broglie relation ω = E/planckover2pi1 → for a Bloch electron
v = 1planckover2pi1?kE(k) (6.1.1)
Velocity of an electron ∝ the gradient of energy in k-space
Figure 6.1.1 Fermi surface of a metal.
In general,v is not necessary bardbl k
Except a free electron satisfying v = planckover2pi1k/m
Near the center of Brillouin zone,E = planckover2pi12k2/2m?,thus
v = planckover2pi1km? (6.1.2)
effective mass m?
However,near the zone boundaries,distorted energy contours → gaps
(6.1.1) must be used,result is different from a free electron
Figure 6.1.2 Energy contour and velocity.
An applied force F → energy absorption by a Bloch electron
dE(k)
dt = F ·v (6.1.3)
To rewritten the left-hand side of (6.1.3)
dE(k)
dt =?kE(k)·
dk
dt = planckover2pi1v ·
dk
dt
a simple relation
planckover2pi1dkdt = F (6.1.4)
Newton’s second law,the rate of momentum change is equal to the force
Electron undergoes an acceleration
a = dvdt (6.1.5)
Combined with (6.1.1)
a =?kv ·dkdt = 1planckover2pi12?k?kE(k)·F (6.1.6)
Its component
ai =
summationdisplay
j
1
planckover2pi12
2E
ki?kjFj,i,j = x,y,z (6.1.7)
Definition of dynamic effective mass
parenleftbigg 1
m?
parenrightbigg
i,j
= 1planckover2pi12?
2E
ki?kj,i,j = x,y,z (6.1.8)
a second-order tensor with nine components
In semiconductors Si and Ge,the dispersion relation is
E(k) = α1k2x +α2k2y +α3k2z (6.1.9)
The effective mass with three components:
m?xx = planckover2pi12/2α1,m?yy = planckover2pi12/2α2,m?zz = planckover2pi12/2α3
The mass of electron is anisotropic and
depends on the direction of external force
Strictly m? is not a constant,but a function of k
There may exist an inflection point kc,mass → becomes negative
This effect comes from a large retarding force from the lattice
6.1.2 Metals and Nonmetals
Difference in conducting behaviors of materials
Most successful application of band theory
A criterion for metals and nonmetals
A band is partially or fully occupied by carriers
Wilson’s contribution,1931
(1) Metal
take sodium Na as example,11 electrons per atom [1s22s22p6]3s1
10 inner electrons form closed shells in an isolated atom
and form very narrow bands in the solid
Inner bands 1s,2s,2p fully occupied,not contribute to current
The uppermost occupied band,the 3s band
N unit cells form bcc structure,each cell has one atom
contributing one valence electron,N valence electrons in total
while 3s band can occupy 2N electrons,half of it is filled
Na behaves like a metal,Fig,6.1.1(a)
E EE E
k k k k
Eg
(a) (d)(c)(b)
Figure 6.1.1 Distribution of electrons in the bands of (a) a metal,(b) an insulator,
(c) a semiconductor,and (d) a semimetal.
(2) Insulator
take diamond C as example,6 electrons for a atom 1s22s22p2
For diamond structure,sp3 hybridization for 2s and 2p electrons
gives rise to two bands split by an energy gap as shown in Fig,6.1.1(b)
Each cell contains two atoms,these bands can accomodate 8N electrons
There are 8 valence electrons per unit cell
Thus valence band is completely full,conduction band is empty,with gap
about 7 eV.
(3) Semiconductor
Intermediate between metals and insulators
gap is small as shown in Fig,6.1.1(c)
The typical examples are Si and Ge also in diamond structure
but their gaps are narrower,about 1 and 0.7 eV,respectively
To distinguish semiconductor from insulator:
The gap at room temperature is less than 2 eV
(4) Semimetal
Example are Bi,As,Sb,and white Sn
gap vanishes,as shown in Fig,6.1.1(d)
Divalent elements,such as Be,Mg,Ca,Zn,are interesting
Magnesium atom,[1s2,2s2,2p6]3s2
Mg crystallizes in the hcp structure with one atom per cell
Two valence electrons per cell,the 3s band should be fully filled up
However,Mg is a metal,3s and 3p bands overlap
resulting in incompletely filled bands
The same condition accounts for the metallicity of Be,Ca,Zn
and other divalent metals
A conclusion:
(1) The number of valence electrons per cell is odd → a metal
(2) Even number of valence electrons,a metal,bands being overlapping
(3) Even number of valence electrons,an insulator,bands being disparate
In transport process,a fully filled band carries no electric current
but unfully filled band does carry electric current
Electronic velocity with opposite Bloch wavevectors k and?k satisfy
v(?k) =?v(k) (6.1.10)
according to (6.1.1) and the symmetry relation E(?k) = E(k)
Current density due to all electrons in the band
j =?e?
summationdisplay
k
v(k) (6.1.11)
Sum is over all states in the band
For a fully filled band,the sum over a whole band → j = 0
For a partially filled band,when an electric field is applied
The correspondence of wavevectors k and?k destroyed
the velocities of electrons cannot be cancelled out each other
the sum over this band gives a finite value,thus j negationslash= 0
Metal-Nonmetal transitions in band picture
Wilson transition,band overlap by pressure
For example,I2,layered molecular crystal,insulator
The electronic configuration of an atom is 53I [ ] 5p5
Apply 160 k bar in layer or 220 k bar ⊥ layer
to make it a metal
Another important example,solid hydrogen H2
Figure 6.1.2 Metal-nonmetal transition.
6.1.3 Hole
For a semiconductor,thermal excitation
electrons transit from valence band to conduction band
vacant states in the valence band,holes
it is convenient to focus on the motion of holes
The concept of hole is important in band theory
Consider only a hole in valence band
Fig,6.1.2 shows a hole in an external field
ε
E
Hole
k1 0 kpi/a-pi/a
Figure 6.1.2 The hole and its motion in the presence of an electric field.
The current density is
jh =?e?
summationdisplay
k
primev
e(k) (6.1.12)
The sum over the filled band is zero,the current density is
jh = e?ve(k1) (6.1.13)
It is the same as an electron of positive charge +e located at k1
The hole current in δt
δjh = e?
parenleftbiggdv
e
dk
parenrightbigg
k1
dk
dtδt =
e
1
m?(k1)Fδt =
e2
1
m?(k1)Eδt (6.1.14)
effective mass m?(k1)
The mass of a hole as
m?h =?m?(k1) (6.1.15)
(6.1.14) is now rewritten as
δjh = 1? e
2
m?hEδt (6.1.16)
Hole current is in the same direction as the electric field
From (6.1.13) and (6.1.16),the motion of the hole like a particle
with a positive charge e and a positive mass m?h
6.1.4 Electronic Specific Heat in Metals
All electrons in a metal are accommodated
from the lowest level up to the Fermi level
The distribution function,f(E),is defined as
the probability that the level E is occupied by an electron
At T = 0 K,it has the form
f(E) =

1,E <EF
0,E >EF
(6.1.17)
Fig,6.1.3 shows the discontinuity at the Fermi energy
f (E)
T = 0 K
T > 0 K
EEF0
1
Figure 6.1.3 Fermi distribution function.
Above 0 K,thermal energy excites electrons to higher energy states
An electron may absorb kBT ≈ 0.025 eV at room temperature
much smaller than EF ≈ 5 eV
To use EF = kBTF to define a Fermi temperature,TF ≈ 60000 K
Only those electrons close to the Fermi level can be excited
This explains the low electronic specific heat
The distribution function f(E) at finite temperature
f(E) = 1e(E?E
F)/kBT + 1
(6.1.18)
Fermi-Dirac distribution plotted in Fig,6.1.3
Only a fraction kBT/EF of electrons contributes to specific heat
The number of electrons excited per mole is about N(kBT/EF)
each electron absorbs an energy kBT on the average
The thermal energy per mole is given approximately by
U = N(kBT)
2
EF (6.1.19)
Specific heat is
Ce =?U/?T = 2RkBTE
F
(6.1.20)
R = NkB
It is reduced from its classical value R by the factor kBT/EF
For EF = 5 eV and T = 300K,this factor is equal to 1/200
This great reduction is in agreement with experiment
An more exact evaluation
Ce = pi
2
2 R
kBT
EF (6.1.21)
§6.2 Electronic Motion in Electric Fields
A applied electric field → electronic states changed
in momentum space as well as in position space
Impurities or defects affect the electronic transport substantially
one-band model
6.2.1 Bloch Oscillations
In Fig,6.2.1(a) and (b),a 1D band structure and corresponding velocity
Dispersion curve is distorted a lot near BZ boundaries
(6.1.1) is reduced to
v = 1planckover2pi1?E?k (6.2.1)
As k varies,the velocity increases at first
reaches a maximum,and then decreases to zero at boundaries
An inflection point corresponding to maximum velocity
E A
B
C
A'
B'
C'
kpi/a-pi/a 0
Electron
(a) F
Figure 6.2.1 Energy-wavevector relation of an electron moving in an electric field.
v
kpi/a-pi/a 0
(b)
Figure 6.2.1 Velocity-wavevector relation of an electron moving in an electric.
A static electric field E → acceleration of electrons
In absence of any scattering for a perfect periodic structure
planckover2pi1dkdt = F =?eE (6.2.2)
An electron,starting from k = 0 → A → equivalent point Aprime
→ AprimeBprimeCprime continuously
The motion in k-space is periodic with period
τB = 2piplanckover2pi1eEa (6.2.3)
Corresponding frequency of oscillation
ωB = eEa/planckover2pi1 (6.2.4)
This is well-known Bloch oscillation
In Fig,6.2.1(b) the velocity starts from k = 0,increases as time passes
reaches a maximum,decreases and vanishes at the zone edge
The electron turns around and acquires a negative velocity,and so on
A Bloch electron,in the presence of a static electric field
also executes an oscillatory periodic motion in real space
Net displacement
z =
integraldisplay t
0
v(t)dt = 1planckover2pi1
integraldisplay t
0
E
kdt
= 1planckover2pi1
integraldisplay t
0
E
k
dt
dkdk =?
1
eE {E[k(t)]?E(0))} (6.2.5)
The maximum displacement is
zmax = B/eE
B is the bandwidth
After one cycle t = τB,the net displacement is z = 0
Theory
Experiment
ε=16.5 (kV/cm)
Time (ps)
Peak Shift
(meV
)Displacement (nm)
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
-8
-4
0
4
8
Figure 6.2.2 Spatially electronic oscillation.
The oscillatory motion is hard to be observed in real crystals
The period τB ～ 10?5 s
a typical electron collision time is τ = 10?14 s at room temperatures
About 109 scatterings in the time of one cycle
Semiconductor superlattices with high purity makes it possible
τB ≤τ
Bloch oscillations were observed in the time domain
by using four-wave mixing method
V,G,Lyssenko et al.,1997,Phys,Rev,Lett,79
Fig,6.2.2 shows peak shift (right scale) with delay time
for heavy-hole transition under an applied field
The peak shift can be related to the displacement shown as the left scale
The electron wave packet performs a sinusoidal oscillation with a total am-
plitude of about 140?A
6.2.2 Negative Differential Resistance
Early successful application of superlattice is to NDR
In Fig,6.2.1,under E,according to (6.1.6)
dv = eEplanckover2pi12?
2E
k2dt (6.2.6)
The average drift velocity,taking into account the scattering time τ
vd =
integraldisplay ∞
0
e?t/τdv = eEplanckover2pi12
integraldisplay ∞
0
2E
k2 e
t/τdt (6.2.7)
exp(?t/τ) represents the probability of free acceleration by E
A sinusoidal approximation E = α?2βcoskd
vd = eEτm?(0)
bracketleftBigg
1 +
parenleftbiggeEτd
planckover2pi1
parenrightbigg2bracketrightBigg?1
(6.2.8)
The current-field curves is
j = envd (6.2.9)
m?(0) is determined by the curvature of E(k) at k = 0 and
n the electron concentration
j with E in Fig,6.2.3(a) shows the maximum at eEτd/planckover2pi1 = 1
Problem,To prove formula (6.2.8) and account for negative
differential resistance
A negative differential resistance indicates under E
conduction electrons may gain enough energies to crossover inflection point
and will be decelerated rather accelerated by an electric field
The obstacle for observing NDR in crystals is τ lessmuchτB
electrons have no chance to surmount the inflection point of velocity
Esaki et al,(1972) found superlattice to exhibit NDR
In Fig,6.2.3(b) current-voltage characteristic at room temperature
for a superlattice having 100 periods
with each period consisting of a GaAs well 60?A thick
and Ga0.5Al0.5As barrier 10?A thick
Vo l t
CurrentCurrent
Vo l t
(a) (b)
Figure 6.2.3 Current-voltage characteristic for negative differential resistance,(a)
theoretical curve; (b) experimental result.
6.2.3 Wannier-Stark Ladders
This is another effect from electron motion in periodic structures
in the presence of an applied electric field
A periodic structure is actually destroyed by the static electric field
Especially,eEd> planckover2pi1/τ
Electronic states are no longer given by Bloch-like solutions
but are given by levels localized in space within a few periods of the lattice
and the energy spectrum is discrete with a level separation of eEd
These equally spaced energy levels are called Wannier-Stark ladders
A wavefunction displaced by d still satisfies the same wave equation
but the energy displacement is eEd
The effect may be observable only in superlattice
because larger lattice modulated period leads to larger energy splitting
A semiconductor with d～ 2?A at E ～ 104 V/cm → eEd～ 0.2 meV
this splitting is extremely small
A superlattice with dsimilarequal 100?A,eEd = 10 meV
WSL has been observed by optical experiments
E,E,Mendez et al.,1988,Phys,Rev,Lett,60
(a) (b)
Figure 6.2.4 A Wannier-Starkladder in a superlattice,(a) the perpendicular energies
corresponding to the resonance transmission miniband states; (b) the miniband breaks
up into discrete Wannier-Stark levels when a large electric field is applied.
A simple theory for Wannier-Stark ladders in superlattices
G,Bastard et al.,1988,Phys,Rev,Lett,60; 1989,Superl & Micros,6
For a superlattice composed of 2N + 1 quantum wells and 2N barriers
Consider a tight-binding picture of localized states in wells
The level and wavefunction of each isolated quantum well is E0 and φ(z)
Interaction between neighboring quantum wells is?λ
When there is no external field
ψk(z) =
Nsummationdisplay
l=?N
ck(l)φ(z?ld),ck(l) = 1√2N + 1eikld (6.2.10)
l labels the quantum wells and k is the wavevector along z direction
The dispersion relation is
E(k) = E0?2λcoskd (6.2.11)
Coupling between neighboring wells → minibands,and extended state
in Fig,6.2.4(a)
When E is applied perpendicular to superlattice
Static electric energy is?eEz
Energy of lth quantum well is changed by?eEld
If eEd>λ,the energy of the superlattice is determined by
the levels of all quantum wells
El = E0?eEld (6.2.12)
a Wannier-Stark ladder appears in Fig,6.2.4(b)
To obtain the wavefunctions and energies of the system
To write the coefficient equations as
λc(l?1) + (E0?eEld?E)c(l)?λc(l+ 1) = 0 (6.2.13)
with boundary conditionse
c(?N?1) = c(N + 1) = 0
The eigenenergies
E = E0?νeEd,?N ≤ν <N (6.2.14)
For any eigenenergy,(6.2.13) is transformed to
c(l?1) +c(l+ 1) = 2(l?ν)(2λ/eEd)c(l) (6.2.15)
This is the recursion relation of Bessel function,so
cν(l) = Jl?ν
parenleftbigg 2λ
eEd
parenrightbigg
(6.2.16)
An example,take eEd = 2λ,then
Jl?ν(1) =?0.0199,0.1150,?0.4401,0.7652,0.1150 and 0.0199
for l?ν =?3,?2,?1,0,1,2,3 schematically shown in Fig,6.2.5
Figure 6.2.5 Schematic probability density of the Wannier-Stark ladder states in
semiconductor superlattice.
Theoretical and experimental studies on
the quantum motion of ultracold atoms
in an accelerating optical lattice have exhibit the same quantum behaviors
as for electrons,such as Bloch oscillations and Wannier-Stark ladders
The optical potential is spatially periodic
yielding an energy spectrum of Bloch bands for the atoms
The acceleration provides an inertial force
The theoretical studies was provided by Q,Niu,X,G,Zhao et al.,1996
Phys,Rev,Lett,76
Experimental verifications can be seen from M,B,Daham,E,Peik et al.,
1996,Phys,Rev,Lett,76,and also S,R,Wilkinson,C,F,Bharucha et al.,
1996,Phys,Rev Lett.
§6.3 Electronic Motion in Magnetic Fields
6.3.1 Cyclotron Resonance
6.3.2 Landau Quantization
6.3.3 de Haas-van Alphen Effect
6.3.4 Susceptibility of Conduction Electrons
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
March 11,2004
Contents
Chapter 6,Dynamics of Bloch Electrons 3
§6.1 Basic Properties of Electrons in Bands,,,,,,,,,,,,4
6.1.1 Electronic Velocity and Effective Mass,,,,,,,,,4
6.1.2 Metals and Nonmetals,,,,,,,,,,,,,,,,,4
6.1.3 Hole,,,,,,,,,,,,,,,,,,,,,,,,,,,4
6.1.4 Electronic Specific Heat in Metals,,,,,,,,,,,4
§6.2 Electronic Motion in Electric Fields,,,,,,,,,,,,,5
6.2.1 Bloch Oscillations,,,,,,,,,,,,,,,,,,,,5
6.2.2 Negative Differential Resistance,,,,,,,,,,,,5
6.2.3 Wannier-Stark Ladders,,,,,,,,,,,,,,,,,5
§6.3 Electronic Motion in Magnetic Fields,,,,,,,,,,,,6
6.3.1 Cyclotron Resonance,,,,,,,,,,,,,,,,,,7
6.3.2 Landau Quantization,,,,,,,,,,,,,,,,,,17
6.3.3 de Haas-van Alphen Effect,,,,,,,,,,,,,,,33
6.3.4 Susceptibility of Conduction Electrons,,,,,,,,,42
Chapter 6
Dynamics of Bloch Electrons
§6.1 Basic Properties of Electrons in Bands
6.1.1 Electronic Velocity and Effective Mass
6.1.2 Metals and Nonmetals
6.1.3 Hole
6.1.4 Electronic Specific Heat in Metals
§6.2 Electronic Motion in Electric Fields
6.2.1 Bloch Oscillations
6.2.2 Negative Differential Resistance
6.2.3 Wannier-Stark Ladders
§6.3 Electronic Motion in Magnetic Fields
Magnetic fields → many results:
carrier masses at bandedges
symmetry of electronic states in Brillouin zone
shape of Fermi surface
6.3.1 Cyclotron Resonance
For nonmagnetic solids,an electron in a magnetic field
planckover2pi1dkdt =?ecv×H (6.3.1)
The right side is the Lorentz force
Change of k in dt
dk =? ecplanckover2pi1v×Hdt (6.3.2)
In k-space,dk ⊥ the plane defined by v and H
The electron rotates along an energy contour ⊥ H
shown in Fig,6.3.1
Take k⊥ and v⊥ perpendicular to H
dk⊥ =? ecplanckover2pi1v⊥ ×Hdt (6.3.3)
dk is equivalent to dk⊥
vbardbl is a constant
ky
Electron
Trajectory υdk
H
kx
Figure 6.3.1 Trajectory of an electron in k-space in a magnetic field H.
To examine two orbits in k⊥-space with E and E +dE
shown in Fig,6.3.2
The separation of the orbits
δk⊥ = dE|?
k⊥E|
= dEplanckover2pi1|v
⊥|
(6.3.4)
The rate of area change along one orbit
vextendsinglevextendsingle
vextendsinglevextendsingledk⊥
dt
vextendsinglevextendsingle
vextendsinglevextendsingleδk⊥ = e
cplanckover2pi12|v⊥ ×H|
dE
|v⊥| =
eH
cplanckover2pi12dE (6.3.5)
dk
dA
υ
E
E+dE
δk⊥
Figure 6.3.2 Two orbits of an electron at energiesE and E+dE in a magnetic field.
The annulus area
dS = Tc ·eHcplanckover2pi12dE (6.3.6)
S is the area with energy less than E
The period of the orbit
Tc = cplanckover2pi1
2
eH
dS
dE (6.3.7)
The cyclotron resonance frequency
ωc = 2piT
c
= 2pieHcplanckover2pi12 1dS/dE = eHm
cc
(6.3.8)
The cyclotron resonance mass
mc = planckover2pi1
2
2pi
dS
dE (6.3.9)
It is not the effective mass in general,only for a parabolic band
E = planckover2pi12k2/2m?
It is the same
S = pik2 = 2m?piE/planckover2pi12 →mc = m?
For Ge and Si with an ellipsoidal constant energy surface (6.1.9)
E(k) = planckover2pi12
parenleftBigg
k2x +k2y
2mt +
k2z
2ml
parenrightBigg
(6.3.10)
transverse and longitudinal effective mass mt and ml
The velocity components are
vx = planckover2pi1
2kx
mt,vy =
planckover2pi12ky
mt,vz =
planckover2pi12kz
ml (6.3.11)
(1) If H bardblkx and lies in the equatorial plane of the spheroid
dkx
dt = 0,
dky
dt =?ωlkz,
dkz
dt = ωtky (6.3.12)
ωl = eH/cml and ωt = eH/cmt
From (6.3.12)
d2ky
dt2 +ωlωtky = 0 (6.3.13)
The frequency
ωc = (ωlωt)1/2 = eHc(m
lmt)1/2
(6.3.14)
(2) If H bardblkz,the frequency is simply
ωc = ωt = eHcm
t
(6.3.15)
(3) In general,an angle θ between H and kz
The cyclotron resonance mass
parenleftbigg 1
mc
parenrightbigg2
= cos
2θ
m2t +
sin2θ
mtml (6.3.16)
Altering the magnetic field direction →
probe various combinations of ml and mt
Cyclotron resonance experiment → carrier masses
Conditions for measurement,ωcτ ≥ 1
(i) very pure samples
(ii) low temperature
(iii) very strong magnetic field
Problem,Prove expression (6.3.16)
6.3.2 Landau Quantization
Quantum mechanical description for an electron in a magnetic field
Hamiltonian
H = 12m
parenleftBig
p+ ecA
parenrightBig2
+V(r) (6.3.17)
vector potential A,periodic potential V(r)
Effective mass approximation to absorb crystal potential
Schr¨odinger equation
1
2m?
parenleftbiggplanckover2pi1
i?+
e
cA
parenrightbigg2
ψ = Eψ (6.3.18)
The interaction between the spin and the magnetic field is ignored here
gLμBσ·H
spin operator σ,Bohr magneton μB = eplanckover2pi1/2cm
Lande factor gL,for free electron gL = 2.
In the gauge A = (0,Hx,0) → H = Hz
planckover2pi1
2
2m?
bracketleftBigg
2
x2 +
parenleftbigg?
y +
ieHx
planckover2pi1c
parenrightbigg2
+?
2
z2
bracketrightBigg
ψ = Eψ (6.3.19)
All energies are to be measured from the bandedges
The wavefunction can be written
ψ(x,y,z) = ei(kyy+kzz)φ(x) (6.3.20)
Denoting
Eprime = E? planckover2pi1
2
2m?k
2
z (6.3.21)
The equation for φ(x)
bracketleftbigg
planckover2pi1
2
2m?
d2
dx2 +
1
2m
ω2
c(x?l
2
cky)
2
bracketrightbigg
φ(x) = Eprimeφ(x) (6.3.22)
From (6.3.20) and (6.3.22)
the motion along the magnetic field is unaffected
the motion in the xy plane is given by harmonic equation
with frequency ωc = eH/cm? and centered around the point
x0 =?l2cky (6.3.23)
where
lc =
parenleftbiggcplanckover2pi1
eH
parenrightbigg1/2
(6.3.24)
is the cyclotron radius or magnetic length,about 100?A for H = 105 Oe
The eigenfunctions of (6.3.22)
φ(x) ∝Hν(x?x0)e?(x?x0)2/2l2c (6.3.25)
Hν(x) are the Hermitian polynomials,and the eigenvalues
Eprimeν =
parenleftbigg
ν + 12
parenrightbigg
planckover2pi1ωc,ν = 0,1,..,(6.3.26)
These are Landau levels
ψ(r) are extended in the y- and z-directions,but localized in x-direction
Total energy
Eν =
parenleftbigg
ν + 12
parenrightbigg
planckover2pi1ωc + planckover2pi1
2
2m?k
2
z (6.3.27)
quantized in the xy plane and continuous along the magnetic field
Consider DOS,for a box of sides Lx,Ly,and Lz
Both kz and ky are quantized in units of 2pi/Lz and 2pi/Ly
The center x0 in (6.3.23) must be inside the dimension of the system
0 ≤x0 ≤Lx (6.3.28)
By using?ky = 2pi/Ly
x0 = 2pil2c/Ly (6.3.29)
The degeneracy of a level in 2D
D = Lx?x
0
= LxLy2pil2
c
(6.3.30)
comes from the linear oscillations with same energy
but different central positions
Equivalently,the total magnetic flux and the flux quantum
Φ = HLxLy,φ0 = hc/e
(6.3.30) can also be expressed in the form
D = Φ/φ0
Distribution of states in kbardbl-space drawn in Fig,6.3.3(a)
(kx,ky) points condense into a series of circles
representing constant energy surfaces with energies planckover2pi1ωc/2,3planckover2pi1ωc/2,···
2D DOS
g2D(E) = 1L
xLy
D
planckover2pi1ωc =
m?
2piplanckover2pi12 (6.3.31)
Extended to 3D,kz is continuous,Fig,6.3.3(b)
3D DOS is given by 1D DOS weighted by the degeneracy factor D
kz is still a good quantum number
E-kz gives Landau subbands
(a)
Figure 6.3.3 Quantization scheme for electrons when a magnetic field is applied in
a kx-ky plane.
H
kx
ky
kz (b)
Figure 6.3.3 Quantization scheme for electrons when a magnetic field is applied in
whole k-space.
Landau levels are shown in Fig,6.3.4(a)
1D DOS for a particular Landau level with energy Eν
g1D(E) = 14pi
parenleftbigg2m?
planckover2pi12
parenrightbigg1/2
(E?Eν)?1/2 (6.3.32)
By taking into account 2D DOS (6.3.31),the total density of states
g3D(E) = 1(4pi)2
parenleftbigg2m?
planckover2pi12
parenrightbigg3/2
planckover2pi1ωc
summationdisplay
ν
bracketleftbigg
E?
parenleftbigg
ν + 12
parenrightbigg
planckover2pi1ωc
bracketrightbigg?1/2
(6.3.33)
Fig,6.3.4(b)
Periodic variation of DOS with magnetic field
This has important effects:
de Haas-van Alphen effect and Shubnikov-de Haas effect
H=0
E
kz
n=0
n=1
n=2
n=3
(a)
Figure 6.3.4 Effects of magnetic field in a three-dimensional electronic system on
the band structure.
E
g0(E)
n=0
1 2 3g(E)
(b)
g3d(E)
Figure 6.3.4 Effects of magnetic field in a three-dimensional electronic system on
the density of states.
Consider 2DEG confined in z-direction and
a magnetic field is along the z-axis
not only x-y energies,but also the z energies,are quantized
DOS becomes a series of δ-functions shown in Fig,6.3.5(a)
electronic motion can be described as a series of circles
with cyclotron radius lc shown in Fig,6.3.5(b)
If an electric field is applied along x-direction
Landau quantization still exists and a more interesting effect
the quantum Hall effect may appear under strong magnetic field and at low
temperature
g(E)
1
2 planckover2piωc
3
2 planckover2piωc
5
2 planckover2piωc
7
2 planckover2piωc
E
(a)
Figure 6.3.5 In a two dimensional electronic system with a magnetic field perpen-
dicular to it,(a) DOS.
l
(b)
Figure 6.3.5 In a two dimensional electronic system with a magnetic field perpen-
dicular to it,(b) Schematic trajectories of electronic cyclotron motion.
6.3.3 de Haas-van Alphen Effect
A kind of magnetic oscillatory behavior related to the Landau quantization
The key point is the position of Landau levels with respect to EF
3D electron gas,kF = (3EF/4pi)1/3 if no magnetic field
applied field,all the states → cylinders in Fig,6.3.3(b)
H increases → the separation between levels increases →
the highest filled Landau level moves up above EF
and will be emptied into lower energy levels
1
H
E
E'F
λ=0
2
345678910
Figure 6.3.6 The spectrum of the Landau levels as a function of magnetic field H.
This process repeats itself whenever
EprimeF/planckover2pi1ωc = m?cEprimeF/planckover2pi1eH = ν + 1/2 (6.3.34)
where EprimeF = EF?planckover2pi12k2z/2m?
Any physical quantities sensing this oscillation should have
a constant period in H?1
(H?1) = eplanckover2pi1m?cEprime
F
(6.3.35)
de Haas-van Alphen (dHvA) effect is the oscillation of magnetic moment
pure specimens,low temperatures,strong magnetic fields
satisfying planckover2pi1ωc >kBT,and ωcτ > 1
For clearness,consider zero temperature
a slice in k-space around kz and thickness δkz in Fig,6.3.4(a)
Lzδkz/2pi states in the kz-space,the number of electronic states in this slice
for a particular Landau level is
δN = DLzδkz2pi =?βH (6.3.36)
= LxLyLz,and β = eδkz/4pi2cplanckover2pi1
δn = (ν + 1)βH (6.3.37)
ν+3 ν+2 ν+1 ν∝1/H
δM
δE
δn
H increasing
Figure 6.3.7 Variation of excess number δn,excess energy δE,and excess magnetic
moment δM of slice δkz,as the magnetic field is increased.
δn?δn0 = 2pikprimeFδkprimeδkz/(2pi)3 (6.3.38)
shown in the upper part of Fig,6.3.7
δE = μβ(δn?δn0)2 (6.3.39)
effective Bohr magneton μ≡eplanckover2pi1/2m?c,the middle part of Fig,6.3.7
dE/dH =?M at 0 K,from (6.3.39) and (6.3.37)
δM =? ddH
bracketleftbiggμ
β(δn?δn0)
2
bracketrightbigg
=?E
prime
F
H (δn?δn0) (6.3.40)
(δn?δn0) oscillates between ±βH/2
δM oscillates between?βEprimeF/2 shown in the down part of Fig,6.3.7
To determine the response M of the entire system
δM = β
∞summationdisplay
p=1
Apsinpx (6.3.41)
where
x = piE
prime
F
μH (6.3.42)
For?pi<x<pi
δM =?βE
prime
F
2pi x (6.3.43)
therefore
Ap = (?1)pE
prime
F
ppi (6.3.44)
To sum over all slices,remembering
EprimeF = EF?planckover2pi12k2z/2m?
and
β = eδkz/4pi2cplanckover2pi1
the total magnetization is
M = e4pi3cplanckover2pi1
∞summationdisplay
p=1
(?1)p
p
integraldisplay kF
kF
EprimeF sin
bracketleftbiggppi
μH
parenleftbigg
EF? planckover2pi1
2k2z
2m?
parenrightbiggbracketrightbigg
dkz (6.3.45)
M = eEF(2mμH)
1/2
4pi3cplanckover2pi1
∞summationdisplay
p=1
(?1)p
p3/2 sin
parenleftbiggppiE
F
μH?
pi
4
parenrightbigg
(6.3.46)
with the approximation kF → ∞
In (6.3.46),the p = 1 is dominant → periodic variation of M with 1/H
Except this de Haas-van Alphen effect,many other physical properties
such as specific heat and thermoelectric power associated with DOS
also display the de Haas-van Alphen effect
The oscillation period in (6.3.46) is related to EF → determine EF →
topological structure of complex Fermi surface
6.3.4 Susceptibility of Conduction Electrons
Many metals,Li,Na,K,Rb and Cs,being weakly paramagnetic and
showing a susceptibility varying little with temperature
A simple explanation based upon the free-electron model
At T = 0 K,a uniform external field H → the energy of electrons
with spin direction bardbl H decreases by μBH
while the energy of electrons with antiparallel spin
increases by the same amount,Fig,6.3.8
E
EF
1
2 g(EF) μBH
g↑(E) g↓(E)
Figure 6.3.8 Energy distribution of electrons in the presence of a magnetic field.
The magnetization is
M = μB
integraldisplay EF
0
[g(E +μBH)?g(E?μBH)]dE = μB
integraldisplay EF+μBH
EF?μBH
g(E)dE
(6.3.47)
Even in very strong field μBH/EF ≈ 10?3
The magnetization becomes
M = 2μ2Bg(EF)H (6.3.48)
Pauli paramagnetic susceptibility
χp(0) = 2μ2Bg(EF) (6.3.49)
At T negationslash= 0,the magnetization is
M(T) = μB
summationdisplay
k
[f(Ek↑)?f(Ek↓)] (6.3.50)
Ek± = EF ±μBH
The summation → integral and expand the Fermi function for weak field
M(T) = 2μ2BH
integraldisplay ∞
0
bracketleftbigg
f(E)?E
bracketrightbigg
g(E)dE (6.3.51)
The Pauli susceptibility at finite temperatures
χp(T) = 2μ2B
integraldisplay ∞
0
bracketleftbigg
f(E)?E
bracketrightbigg
g(E)dE (6.3.52)
Furthermore
χp(T) = χp(0)
bracketleftBigg
1? pi
2
12
parenleftbiggk
BT
EF
parenrightbigg2
+···
bracketrightBigg
(6.3.53)
kBT lessmuch EF → the susceptibility is little decreased with T
Landau diamagnetic susceptibility
In addition a conduction electron system has a diamagnetic susceptibility
originating from changing the orbital states by applied magnetic field
For free electron gas the value is given by (6.3.52) multiplied by?(1/3)
Spin polarization of electrons in Fig,6.3.8 and
Magnetization in (6.3.48) and (6.3.51) are all dependent on applied fields
Imagine there exist interactions between electrons with spins
these interactions are equivalent to an internal field
electronic spins will be affected by the internal filed
even no external field exists,spin polarization and
magnetization will appear
Consider a spatially varying external field H(r)
decompose the field into Fourier components
H(r) = 1?
summationdisplay
q
Hqeiq·r (6.3.54)
with Hq = Hq
Define the susceptibility,χ(q),magnetization is obtained as
M(r) = 1?
summationdisplay
q
χqHqeiq·r,(6.3.55)
Assume the magnetic field is applied along the z-axis
siz is the z-component of the spin of the i-electron
The Zeeman energy
Hprime =?μB
summationdisplay
i
siz(Hqeiq·ri +H?qe?iq·ri) (6.3.56)
taken as perturbation
In first order,the electron state described by a plane wave
ψkq± = 1√?eik·r
bracketleftBigg
1± 12μB
parenleftBigg
Hqeiq·r
εk+q?εk +
H?qe?iq·r
εk?q?εk
parenrightBiggbracketrightBigg
(6.3.57)
Its square gives the number density of electrons
ρkq± = 1?
bracketleftbigg
1± 12μB
parenleftbiggf
k(1?fk+q)
εk+q?εk +
fk(1?fk?q)
εk?q?εk
parenrightbigg
(Hqeiq·r +H?qe?iq·r)
bracketrightbigg
(6.3.58)
Spatial density of spin magnetic moment is obtained from (6.3.58)
Mq(r) = χq12(Hqeiq·r +H?qe?iq·r) (6.3.59)
where
χq = μ2B
summationdisplay
k
fk?fk+q
εk+q?εk (6.3.60)
Define the important function
F(q) =
summationdisplay
k
fk?fk+q
εk+q?εk (6.3.61)
which is in fact dimension-dependent
Substitute the energy spectrum of free electron gas and finish the summation
F1(q) = 2mpiplanckover2pi12q ln
vextendsinglevextendsingle
vextendsinglevextendsingle2kF +q
2kF?q
vextendsinglevextendsingle
vextendsinglevextendsingle (6.3.62)
F3(q) = 3N4ε
F
bracketleftbigg
1 + 4k
2
F?q
2
4kFq ln
vextendsinglevextendsingle
vextendsinglevextendsingle2kF +q
2kF?q
vextendsinglevextendsingle
vextendsinglevextendsingle
bracketrightbigg
(6.3.63)
F2(q) =

m/piplanckover2pi12,for q < 2kF
(m/piplanckover2pi12){1?[1?(2kF/q)2]1/2},for q > 2kF
(6.3.64)
The two parts are joined at q = 2kF
q/2kF
0 0.5 1.0 1.5 2.0
0.5
1.0
1.5
2.0
'
'
'
χ (q
) /
χ (0)
Figure 6.3.9 Normalized χ(q) versus q in one,two,and three dimensions,χ(0) is
identified as the Pauli paramagnetic susceptibility at q = 0.
χ(q) versus q in one,two,and three dimensions given in Fig,6.3.9
Different singularities at q = 2kF for χ(q) in 1D,2D,3D
reflecting the existence of Fermi surface and
leading to some peculiar behaviors
in electron gases,especially in lower dimensions
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
February 23,2004
Contents
Chapter 7,Surface and Impurity Effects 3
§7.1 Electronic Surface States,,,,,,,,,,,,,,,,,,,7
7.1.1 Metal Surface,,,,,,,,,,,,,,,,,,,,,,8
7.1.2 Semiconductor Surface States,,,,,,,,,,,,,28
§7.2 Electronic Impurity States,,,,,,,,,,,,,,,,,,38
7.2.1 Shielding Effect of Charged Center,,,,,,,,,,,39
7.2.2 Localized Modes of Electrons,,,,,,,,,,,,,,47
7.2.3 Electron Spin Density Oscillation,,,,,,,,,,,,55
§7.3 Vibrations Related to Surface and Impurity,,,,,,,,,56
7.3.1 Surface Vibrations,,,,,,,,,,,,,,,,,,,56
7.3.2 Impurity Vibration Modes,,,,,,,,,,,,,,,56
7.3.3 Elastic Wave Defect Modes,,,,,,,,,,,,,,,56
§7.4 Defect Modes in Photonic Crystals,,,,,,,,,,,,,57
7.4.1 Electromagnetic Surface Modes,,,,,,,,,,,,,57
7.4.2 Point Defect,,,,,,,,,,,,,,,,,,,,,,,57
7.4.3 Line Defects,,,,,,,,,,,,,,,,,,,,,,,57
Chapter 7
Surface and Impurity Effects
Ideal perfect lattices → easy to study wave behavior
Real crystals,imperfections,surface and impurity
Weak imperfection,band model is effective
Effects of surface and impurity on the band model
Surfaces are not just here or there,
For surfaces are everywhere,
Splitting nature into phases,
Solids,liquids and the gases
—— S,G,Davison and M,Ste′slika
Basic Theory of Surface States,1996
Figure 7.0.1 Cui Hu,Tang Dynasty.
§7.1 Electronic Surface States
Surface → deviation from perfect periodicity
Born-von Karmen cyclic boundary condition abandoned
Electronic structure is modified from infinite crystal
Real structure of a solid surface is complex
Ideal surface as a model
7.1.1 Metal Surface
Two fundamental parameters for a metallic surface
(1) Work function,energy difference for an electron
between Fermi level and vacuum level
(2) Surface energy,energy increase of a unit surface area
when all bonds across a planar surface are truncated
Both are involved in the electronic states related to a surface
Nearly-free electron approximation
V0 a
V(z)
W= -EF+V0
EF
z
Figure 7.1.1 Schematic potential of an electron in a semi-infinite lattice.
Simplified to a one-dimensional problem
Surface represents an abrupt transition at z = 0
Between vacuum and periodic lattice
Schr¨odinger equation of a semi-infinite periodic chain
bracketleftbigg
planckover2pi1
2
2m
d2
dz2 +V(z)
bracketrightbigg
ψ(z) = Eψ(z) (7.1.1)
with the potential
V(z) =
V0,for z < 0
V(z +la),for z > 0
a,lattice constant; l,unit cell
Wavefunctions in two half semi-spaces
ψ(z) =

Aexp
bracketleftBig
1
planckover2pi1
radicalbig2m(V
0?E)z
bracketrightBig
,for z < 0
Bukeikz +Cu?ke?ikz,for z > 0
(7.1.2)
A,B,C,to be determined by continuity conditions at z = 0
More interested is extended states in energy bands not fully filled
As in Fig,7.1.2
Wavefunction is a Bloch wave in crystal,
and attenuate exponentially outside the surface
ψ
z0
Figure 7.1.2 Extended wavefunction near metallic surface.
Dipolar double layer across the surface
(1) layer out of the surface is negatively charged due to
the electrons spilled out
(2) layer of one or two unit cells near the surface in crystal is
positively charged due to the deficiency of electrons
Leads potential profile across the surface modified
shown by the dashed line in Fig,7.1.1
Work function for an electron at metallic surface
W = V(?∞)?V(0)?EF ≈V0?EF
It is important in technology to
thermionic emission and field emission
To calculate the surface energy and electronic density variation near a surface
NFE → Jellium model
Interior of a crystal,homogeneity and charge neutrality
ρ+ +ρ? = 0
Variation near the surface,dipolar double layer and Friedel oscillation
Problem → solve Schr¨odinger equation in a potential well,width L
Let L→ ∞ → electronic surface state
Comparison of two cases:
(1) confined case with infinite barriers
Wavefunction
ψk(z) =
parenleftbigg2
L
parenrightbigg1/2
sinkz,k = npiL,n = 1,2,···,nF (7.1.3)
Eigenenergies
En = planckover2pi1
2
2m
parenleftBignpi
L
parenrightBig2
,n = 1,2,···,nF
k is always positive
k = pi/L
one pair of electrons with spin up and down in the state |k|
(2) unconfined case,such as periodic potential
Wavefunction
ψprimekprime(z) =
parenleftbigg1
L
parenrightbigg1/2
eikprimez,kprime = 2n
primepi
L,n
prime = 0,±1,±2,···,±nprime
F (7.1.4)
Eigenenergies
Eprimenprime = planckover2pi1
2
2m
parenleftbigg2nprimepi
L
parenrightbigg2
,nprime = 0,±1,±2,···,±nprimeF (7.1.5)
kprime can be positive or negative
kprime = 2pi/L
two pairs of electrons in the state |kprime|
Using N,taking it even,electrons to fill the energy levels
as in Fig,7.1.3
nF?2nprimeF = 1 or 0
Correspondingly
kF?kprimeF =?kF = pi/2L
An energy difference
E =
nFsummationdisplay
n=1
En?
nprimeFsummationdisplay
|nprime|=0
Eprimenprime (7.1.6)
7
3
4
5
6
1
2
0
n
2
3
1
0
|n'|
Surface
Confinement
No Surface
Confinement
4
6
2
0
|n|=2|n'|
Figure 7.1.3 Filled energy levels of one-dimensional potential wells,The left is with
surface confinement and the right without surface confinement.
Energy difference arises from surface confinement
Surface confinement → (1/2)(N/2) levels to rise an energy
(planckover2pi12/2m)(pi/L)2
Total increment,principal part of metallic surface energy
E = planckover2pi1
2k2
F
2m (7.1.7)
corresponding an electron from the level k = 0 to the Fermi level k = kF
Problem,Please prove formula (7.1.7)
More exact calculation for surface of a 3D solid
Surface energy from cutting a solid to form two surfaces
2L2?E∞s =
integraldisplay
k≤kF+δkF,kz≥pi/2L
E(k)2
parenleftbiggL
2pi
parenrightbigg2 L
pidk?
integraldisplay
k≤kF
E(k)2
parenleftbiggL
2pi
parenrightbigg3
dk
(7.1.8)
δkF = pi/4L is the increment due to the surface in 3D jellium model
To introduce kbardbl
2L2?E∞s =
integraldisplay
k≤kF+δkF,kz≥0
E(k)4
parenleftbiggL
2pi
parenrightbigg3
dk?
integraldisplay
kbardbl≤kF+δkF
d2kbardbl
integraldisplay pi/2L
0
E(k)4
parenleftbiggL
2pi
parenrightbigg3
dkz
2×
integraldisplay
k≤kF,kz≥0
E(k)2
parenleftbiggL
2pi
parenrightbigg3
dk
≈EF L
3
2pi32pik
2
FδkF?
integraldisplay kF
0
planckover2pi12k2bardbl
2m
L2
4pi22pikbardbldkbardbl
=
kFintegraldisplay
0
parenleftBigg
EF? planckover2pi1
2k2
bardbl
2m
parenrightBigg
L2
4pi22pikbardbldkbardbl (7.1.9)
Surface energy is the energy increase of a half of electrons
moved from the level k = 0 to the Fermi surface
The result is
E∞s = planckover2pi1
2k4
F
32pim (7.1.10)
In 1D jellium model,with two confined surfaces,from (7.1.3)
Electron density distribution
ρ(z) =
Nsummationdisplay
n=1
ψ?n(z)ψn(z) = 2L
Nsummationdisplay
n=1
sin2npizL (7.1.11)
Carry out the summation
ρ(z) = N + 1/2L? sin[2pi(N + 1/2)z/L]2Lsin(piz/L) (7.1.12)
To examine the effect of surface on the electron density,let L→ ∞
ρ(z) = 2L
Nintegraldisplay
0
sin2 npizL dn (7.1.13)
Friedel oscillation
ρ(z) = ρ0?ρ0sin(2kFz)2k
Fz
,ρ0 = N/L (7.1.14)
Fig,7.1.4(a) shows density oscillation with wavelength pi/kF
1
z0 10 2k
F
ρ-(
z)/ρ
0
(a)
+
Figure 7.1.4 Electronic density oscillation near metal surface,(a) Infinite surface
barriers.
More real result should use 3D jellium model with a finite barriers
Fig,7.1.4(b)
Experimental tests in recent years on Be surfaces due to steps and defects
observed by STM
P,H,Hofmann et al.,Phys,Rev,Lett,(1997)
P,T,Sprunger et al.,Science (1997)
-10 -5 0 0.5 1
rs=2
rs=5
ρ-(
z)
ρ0-(b)
+
z
Figure 7.1.4 Electronic density oscillation near metal surface,(b) Finite surface
barriers.
7.1.2 Semiconductor Surface States
Return to nearly-free electron approximation
Half-infinite periodic chain with weak lattice potential
Band structure is almost the same as in Fig,5.2.2
BZ extends from k =?pi/a to +pi/a
0 pi/a 2pi/a 3pi/a-pi/a-2pi/a-3pi/a k 0 pi/a-pi/a k
E E
(a) (b)
Figure 7.1.5 Bands and gaps in one-dimensional nearly-free electron model for (a)
extended zone scheme,and (b) reduced zone scheme.
Study the solutions in the vicinity of k = ±pi/a
ψk(z) = αeikz +βei(k?2pi/a)z (7.1.15)
α and β determined from the following equations
bracketleftbiggplanckover2pi12
2mk
2?E(k)
bracketrightbigg
α+V1β = 0 (7.1.16)
V?1 α+
bracketleftBigg
planckover2pi12
2m
parenleftbigg
k? 2pia
parenrightbigg2
E(k)
bracketrightBigg
β = 0 (7.1.17)
Fourier coefficient of the periodic potential V1 = V?2pi/a
V?2pi/a = 1L
integraldisplay
V(x)ei2pix/adx
Defining k = pi/a+ε,γ = (planckover2pi12pi/ma|V1|)ε
(1) ε is real
Energies
E± = planckover2pi1
2
2m
parenleftBigpi
a +ε
parenrightBig2
+|V1|
parenleftBig
γ±
radicalbig
1 +γ2
parenrightBig
(7.1.18)
Wavefunctions
ψ±(z) = B
bracketleftbigg
eipiz/a + |V1|V
1
parenleftBig
γ±
radicalbig
1 +γ2
parenrightBig
e?ipiz/a
bracketrightbigg
eiεz,for z > 0
(7.1.19)
with B being an another constant
Bands are shown in Fig,7.1.5(a)
For each energy,free parameters A and B determined by
Connecting the solutions in vacuum and periodic lattice at z = 0
The result is an extended state in Fig,7.1.2.
The energy bands of an infinite lattice is only slightly modified.
(2) ε = iμ with real positive μ
Solutions decrease exponentially into the crystal or
localized to the surface
k μ|ε|
(a) (b)
E E
pi/a
Figure 7.1.5 Dispersion relations when surface exists,(a) A little modified energy
spectra for a nearly free electron model; (b) In the energy gap between the two bands,
solutions with imaginary k can appear.
Take
γ = isin(2δ) = i(planckover2pi12pi/ma|V1|)μ
Energies are
E± = planckover2pi1
2
2m
bracketleftbiggparenleftBigpi
a
parenrightBig2
μ2
bracketrightbigg
±|V1|
bracketleftBigg
1?
parenleftbigg planckover2pi12piμ
ma|V1|
parenrightbigg2bracketrightBigg1/2
(7.1.20)
The energies in (7.1.20) are always real for
0 ≤μ≤μmax = ma|V1|/planckover2pi12pi
as shown in Fig,7.1.5(b)
Wavefunctions are
ψ±(z) = C
bracketleftbigg
e(ipiz/a±δ) ± |V1|V
1
e(?i(piz/a±δ)
bracketrightbigg
e?μz (7.1.21)
C is an another constant
There is a surface state to appear
Fig,7.1.6
z
ψ
0
Figure 7.1.6 A localized wavefunction near semiconductor surface.
In 1D model,a surface state has a discrete energy level
denoted as E0,for fixed μ in the energy gap
Extending to 3D,the wavefunctions
ψk = ψ0(z)exp[i(kxx+kyy)] (7.1.22)
Energies
E(k) = E0 +
parenleftbiggplanckover2pi12
2m
parenrightbigg
(k2x +k2y) (7.1.23)
For transition metals,or covalent semiconductors like Si,Ge,and C
Tight-binding model is used to study their surface states
§7.2 Electronic Impurity States
Impurities → departure from perfect periodicity
Dilute impurities → a single impurity in metals or semiconductors
To study the effect of impurity center
on electronic states and energy band structure
What potential is adopted?
7.2.1 Shielding Effect of Charged Center
Simplest case,a static impurity charge Ze embedded in
A metal as free electron gas
Plane wave solution
1/2eik·r
Impurity charge → a spherically symmetric scattering potential U(r)
The plane waves of conduction electrons → ψk(r)
k labels the unperturbed state before Ze is introduced
Schr¨odinger equation
planckover2pi1
2
2m?
2ψk +U(r)ψk = Ekψk (7.2.1)
Energy of free electrons
Ek = planckover2pi12k2/2m
(7.2.1) is rewritten as
2ψk +k2ψk = 2mplanckover2pi12 U(r)ψk (7.2.2)
To introduce Green’s function G(rrprime)
2G(rrprime) +k2G(rrprime) =?4piδ(r?rprime) (7.2.3)
Green’s function is used as the propagator
G(rrprime) in (7.2.3) is so-called preliminary solution
It can be found simply by taking Fourier transform,the solution is
G(rrprime) = e
ik|r?rprime|
|r?rprime| (7.2.4)
Formal solution
ψk(r) =1/2eik·r? m2piplanckover2pi12
integraldisplay
drprimeG(rrprime)U(rprime)ψk(rprime) (7.2.5)
An integral equation related to wavefunction
Recursion method,in the first-order Born approximation
ψk(r) =1/2
bracketleftbigg
eik·r? m2piplanckover2pi12
integraldisplay
drprimeG(rrprime)U(rprime)eik·rprime
bracketrightbigg
(7.2.6)
The electron density,summing ψ?k(r)ψk(r) over all k to the Fermi surface
ρ(r) =
summationdisplay
|k|<kF
ψ?k(r)ψk(r) =1

summationdisplay
|k|<kF
summationdisplay
|k|<kF
m
2piplanckover2pi12
integraldisplay
drprimeU(rprime)
×
bracketleftBig
G(rrprime)eik·(rprime?r) +G?(rrprime)e?ik·(rprime?r)
bracketrightBigbracerightBig
(7.2.7)
Summation → integration
ρ(r) = ρ0? 2m(2pi)4planckover2pi12
integraldisplay
drprimeU(rprime)
integraldisplay
|k|<kF
dk
bracketleftBig
G(rrprime)eik·(rprime?r)
+ G?(rrprime)e?ik·(rprime?r)
bracketrightBig
(7.2.8)
ρ0 = N/? is the average density
Fulfill the integral for angles in k space
ρ(r) = ρ0? 2m(2pi)4planckover2pi12
integraldisplay
drprimeU(rprime)
integraldisplay kF
0
dk4pik2
bracketleftbiggsink|r?rprime|
k|r?rprime| ·
2cosk|r?rprime|
|r?rprime|
bracketrightbigg
(7.2.9)
After integrating over k
ρ(r)?ρ0 =? mk
2
F
2pi3planckover2pi12
integraldisplay
drprimeU(rprime)j1(2kF|r?r
prime|)
|r?rprime|2 (7.2.10)
j1(x) = (sinx?xcosx)/x2,first-order spherical Bessel function
Finally,ρ(r) depends on the impurity potential chosen
For a simple example,U(r) = U0δ(r)
ρ(r)?ρ0 =?mk
2
FU0
2pi3planckover2pi12 ·
j1(2kFr)
r2 ～
sin2kFr?2kFrcos2kFr
r4 (7.2.11)
For long range
ρ(r)?ρ0 ～ cos2kFrr3 (7.2.12)
This is the Friedel oscillation
Reversely,Friedel oscillation of potential,from Poisson equation,(7.2.10)
transformed into
2U = 2me
2k2
F
pi2planckover2pi12
integraldisplay
drprimeU(rprime)j1(2kF|r?r
prime|)
|r?rprime|2 (7.2.13)
The asymptotic behavior of the potential is
U(r) ～ cos2kFrr3 (7.2.14)
Fig,7.2.1 shows the oscillation of screened potential,experimentally can be
confirmed by Knight shifts in dilute alloys
U
r
Figure 7.2.1 Screened potentials round a point charge in a sea of free electron gas.
7.2.2 Localized Modes of Electrons
Single impurity charge in a semiconductor,impurity potential U(r)
From (5.1.7) to (7.2.15)
H =? planckover2pi1
2
2m?
2 +V(r) → H =? planckover2pi1
2
2m?
2 +V(r) +U(r)
Schr¨odinger equation
bracketleftbigg
planckover2pi1
2
2m?
2 +V(r) +U(r)
bracketrightbigg
ψ = Eψ (7.2.15)
Periodic potential V(r) → Perturbated potential V(r) +U(r)
Bloch state ψk(r) → new state ψ
Bloch energy E(k) → new eigenenergy E
A covalent-bonded elemental semiconductor of Z-valent atoms,like Si
One of the original atoms is substituted by a (Z + 1)-valent atom,like P
The donor atom introduces one electron and one additional positive charge
Crystal lattice screens the Coulomb potential like a homogeneous medium
with dielectric constant epsilon1
U(r) =?e
2
epsilon1r (7.2.16)
Problem ≡ a hydrogen atom with dielectric constant epsilon1
Wavefunction is expanded in terms of the Bloch functions
ψ =
summationdisplay
k
c(k)ψk(r) (7.2.17)
To substitute it into (7.2.15)
[E(?i?) +U(r)]ψ = Eψ (7.2.18)
E(k) → E(?i?)
E(k) is a periodic function in k-space
E(k) =
summationdisplay
l
Elexp(il·k)
E(?i?) =
summationdisplay
l
Elel·? (7.2.19)
exp(l·?) is a translational operator
To make any f(r) → f(r + l)
Shallow impurity with relatively weak potential to bind an electron
The orbital traverses many lattice cells → its extent in k-space is small →
uk(r) = u0(r) = ψ0(r)
ψ = u0(r)
summationdisplay
k
c(k)eik·r = ψ0(r)F(r) (7.2.20)
where
F(r) =
summationdisplay
k
c(k)exp(ik ·r)
Note that
E(?i?)ψ0(r)F(r) = ψ0(r)E(?i?)F(r) (7.2.21)
Puts (7.2.20) into (7.2.18)
[E(?i?) +U(r)]F(r) = EF(r) (7.2.22)
Expanded operator into second order
E(?i?) = Ec? planckover2pi1
2
2m
2 (7.2.23)
where Ec is lower edge of the conduction band
Hydrogen atom-like equation
parenleftbigg
planckover2pi1
2
2m
2? e
2
epsilon1r
parenrightbigg
F(r) = (E?Ec)F(r) (7.2.24)
Eigenenergies
En = Ec? e
4m?
2planckover2pi12epsilon12n2,n = 1,2,..,(7.2.25)
Discrete levels below the conduction band as shown in Fig,7.2.2.
Ground state,a bound state
F(r) = 1(pia?3
0 )1/2
exp
parenleftbigg
ra?
0
parenrightbigg
,a?0 = planckover2pi1
2
me2
m
m?epsilon1 (7.2.26)
n=1
n=2n=3
n=4
k
E
0
Figure 7.2.2 TheE-kdiagramfor the localizedimpuritylevels lie below the minimum
of the conduction band.
7.2.3 Electron Spin Density Oscillation
§7.3 Vibrations Related to Surface and Impurity
7.3.1 Surface Vibrations
7.3.2 Impurity Vibration Modes
7.3.3 Elastic Wave Defect Modes
§7.4 Defect Modes in Photonic Crystals
7.4.1 Electromagnetic Surface Modes
7.4.2 Point Defect
7.4.3 Line Defects
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
February 25,2004
Contents
Chapter 7,Surface and Impurity Effects 3
§7.1 Electronic Surface States,,,,,,,,,,,,,,,,,,,4
7.1.1 Metal Surface,,,,,,,,,,,,,,,,,,,,,,4
7.1.2 Semiconductor Surface States,,,,,,,,,,,,,4
§7.2 Electronic Impurity States,,,,,,,,,,,,,,,,,,5
7.2.1 Shielding Effect of Charged Center,,,,,,,,,,,5
7.2.2 Localized Modes of Electrons,,,,,,,,,,,,,,5
7.2.3 Electron Spin Density Oscillation,,,,,,,,,,,,6
§7.3 Vibrations Related to Surface and Impurity,,,,,,,,,11
7.3.1 Surface Vibrations,,,,,,,,,,,,,,,,,,,11
7.3.2 Impurity Vibration Modes,,,,,,,,,,,,,,,19
7.3.3 Elastic Wave Defect Modes,,,,,,,,,,,,,,,28
§7.4 Defect Modes in Photonic Crystals,,,,,,,,,,,,,29
7.4.1 Electromagnetic Surface Modes ina layered dielectric structure
7.4.2 Point Defect,,,,,,,,,,,,,,,,,,,,,,,35
7.4.3 Line Defects,,,,,,,,,,,,,,,,,,,,,,,43
Chapter 7
Surface and Impurity Effects
§7.1 Electronic Surface States
7.1.1 Metal Surface
7.1.2 Semiconductor Surface States
§7.2 Electronic Impurity States
7.2.1 Shielding Effect of Charged Center
7.2.2 Localized Modes of Electrons
7.2.3 Electron Spin Density Oscillation
A magnetic impurity S locates at site R in a metal
Free electron approximation
Scattering of conduction electrons by the impurity moment
Spins of conduction electrons sj
Eigenstate of a conduction electron is spin-dependent
ψkσ(r) =1/2eik·r|σ〉,σ =↑ or ↓ (7.2.27)
Contact interaction
Hprime =?J
summationdisplay
j
sj ·Sδ(r?R) (7.2.28)
Impurity moment provides an effective field
Heff(r) =? Jg
LμB
Sδ(r?R) (7.2.29)
Lande factor gL and Bohr magneton μB
This field is nonhomogeneous
Its Fourier transform is important
Heff(q) =? Jg
LμB
S (7.2.30)
Electron gas in metal responses to this field determined by
Its susceptibility χ(q)
By assuming R = 0,the spin density at r
s(r) = Jg
L2μ2B?
summationdisplay
q
χ(q)eiq·rS (7.2.31)
Spin susceptibility of a free electron gas in 3D is derived in §6.3.4
χ(q) = 3gL
2μ2
B
16EF
N
bracketleftbigg
1 + kFq
parenleftbigg
1? q
2
4k2F
parenrightbigg
ln
vextendsinglevextendsingle
vextendsinglevextendsingle2kF +q
2kF?q
vextendsinglevextendsingle
vextendsinglevextendsingle
bracketrightbigg
(7.2.32)
Sum over q in (7.2.31) → an integral
1
summationdisplay
q
χ(q)eiq·r = 3g
2μ2
BkF
128piEF
N
sin2kFr?2kF cos2kFr
(kFr)4 (7.2.33)
Substitute it into (7.2.31),spin density for kFr greatermuch 1
s(r) =?3pin
2
64EFJS
cos2kFr
(kFr)3 (7.2.34)
A localized impurity moment in a metal →
Spin density of conduction electrons oscillates,shown in Fig,7.2.3
This is called the Rudermann-Kittel-Kasuya-Yosida RKKY oscillation
for electron spin density
Analogous to the Friedel oscillation of electron charge density
A
B
C
Figure 7.2.1 Electron spin density oscillation arising from an impurity moment at
A.
§7.3 Vibrations Related to Surface and Impurity
Like electrons in real crystals
Surface or impurity → slight modification to states related to
Lattice waves or elastic waves and
Localized modes to appear
7.3.1 Surface Vibrations
Semi-infinite crystal in the z > 0 half space
Vacuum in the z < 0 half space
Surface at z = 0
0 z
Figure 7.3.1 A semi-infinite atomic chain used to surface vibrations.
2D periodicity in the x-y directions,the Bloch theorem still applied
Lattice vibrations formally → determination of the modes of a semi-infinite
linear chain
From the equations of motion,two types of solutions:
(1) Bulk modes propagating to z → ∞
(2) Surface modes vanishes rapidly when z increases.
A semi-infinite linear chain shown in Fig,7.3.1
Identical atoms of mass M,spacing a,nearest-neighbor coupling constant β
β0 negationslash= β between the first two atoms coming from effect of surface
Equations of motion
M¨u0 =?β0(u0?u1),(7.3.1)
M¨u1 =?β0(u1?u0)?β(u1?u2),(7.3.2)
M¨ul =?β(ul?ul?1)?β(ul?ul+1),for l ≥ 2 (7.3.3)
Two boundary conditions (7.3.1) and (7.3.2)
Solutions can be written as
u0 = U0e?iωt (7.3.4)
and
ul = Ueikla?iωt,for l ≥ 1 (7.3.5)
U0 and U are two amplitude variables
(1) ka is real,bulk modes
Substitute (7.3.5) into (7.3.3),dispersion relation of infinite periodic chain
ωb =
parenleftbigg4β
M
parenrightbigg1/2vextendsinglevextendsingle
vextendsinglevextendsinglesinka
2
vextendsinglevextendsingle
vextendsinglevextendsingle (7.3.6)
(2) ka is complex due to surface,surface mode
Substitute (7.3.4),(7.3.5) into (7.3.1),(7.3.2),and using (7.3.6)
[β0?2β(1?coska)]U0?β0eikaU = 0,
β0e?ikaU0 + [β0?β(1?e?ika)]U = 0
Coefficient determinant is zero →
[β0?2β(1?coska)]bracketleftbigβ0?βparenleftbig1?e?ikaparenrightbigbracketrightbig?β20 = 0 (7.3.7)
Several non-trivial solutions:
One is exp(ika) = 1 corresponding to
a rigid translation of the chain and has no physical interest
The other two solutions are
eika = 1±
radicalbigg
1 + 1ε (7.3.8)
with
ε = β0?ββ (7.3.9)
Consider the solutions in (7.3.8)
β0 and β are positive,so ε>?1
There are two possibilities:
(1) β0 <β,i.e.,?1 <ε< 0,(1 + 1/ε) < 0 and
eika = 1±i
radicalbigg
1? 1ε
these are not exponentially decaying solutions
(2) β0 >β,i.e.,ε> 0
There are two solutions of (7.3.8):
(i) The solution with plus sign is larger than 1 and corresponds to a
negative imaginary k which is contrary to the assumption as can
be seen from (7.3.5)
(ii) The solution with minus sign corresponds to ka = pi+iμ
μ = ln(ε+√ε2 +ε)
If μ is positive,then
ε+√ε2 +ε> 1
or by using (7.3.9)
β0
β >
4
3 (7.3.10)
When β0 > 4β/3,there is a localized mode
ωs =
parenleftbigg2β
M
parenrightbigg1/2
(1 + coshμ) (7.3.11)
above the bulk mode
The displacement of lth atom is
ul = U(?1)le?lμ,for l≥ 1 (7.3.12)
an oscillatory damped solution in z direction very similar to (7.1.21)
If (7.1.10) is not satisfied,there is no surface mode because μ is not a
positive real number
Alternatively,if M0 also deviates from M,the condition for surface mode
β0
β >
4M0
2M0 +M (7.3.13)
7.3.2 Impurity Vibration Modes
Two kinds of impurities to influence lattice vibrations:
One is the mass defect and the other is the force constant defect
Most important results,slight influences on the phonon states and
appearance of localized states between the acoustic and optical branches
and above the optical branches
A perfect crystal with a single atom per primitive cell
Equations of motion (7.3.1) →
ω2ul? 1M
summationdisplay
lprime
Φllprime ·ulprime = 0 (7.3.14)
Displacements ul and force constant Φllprime
(7.3.2) is transformed into
ul = Ukeik·l (7.3.15)
Uk is normalized,l runs from 1 to N
Define a 3N ×3N matrix
D(ω2) = ω2I?(1/M)?Φ (7.3.16)
Φ,3N ×3N force constant matrix
(7.3.14) is rewritten as
D(ω2)u = 0 (7.3.17)
u is a 1×3N column matrix standing for the set of displacement ul
For simplicity,consider mass defect of single atom
The mass at site l = 0 is changed to M0,mass difference δM = M0?M
Dynamic equations (7.3.14) is changed into
ω2ul? 1M
summationdisplay
lprime
Φllprime ·ulprime + δMM ω2u0δl0 = 0 (7.3.18)
and can be symbolized as
D(ω2)u+δD(ω2)u = 0 (7.3.19)
Mass defect → a perturbation of the dynamic matrix of perfect crystal
Rewrite (7.3.18)
(1 +GδD)u = 0 (7.3.20)
by Classical Green’s function G satisfying
G(ω2)D(ω2) = 1 (7.3.21)
Solve (7.3.17) to get the eigenfrequencies ωk and eigenvectors uk satisfying
ω2kuk? 1MΦuk = 0 (7.3.22)
Then
G?1uk = Duk = (ω2?ω2k)uk (7.3.23)
or
Guk = 1ω2?ω2
k
uk (7.3.24)
× u?k on both sides,and summationtextk
summationdisplay
k
Guku?k =
summationdisplay
k
uku?k
ω2?ω2k (7.3.25)
Using u?kuk = I →
Gllprime(ω2) = 1N
summationdisplay
k
UkU?k
ω2?ω2ke
ik·(l?lprime) (7.3.26)
Mass defect at l = 0 → a highly localized perturbation
Substitute (7.3.26) into (7.3.20) → an equation about u0
Then only G00(ω2) is considered,i.e.
δM
NM
summationdisplay
k
UkU?k ω
2
ω2k?ω2 = I (7.3.27)
A 3×3 matrix expression
Uk is a unit vector of mode k,Ukα is the α component,UkU?k is a matrix
with elements of UkαU?kβ
(7.3.27) in components
δM
NM
summationdisplay
k
UkαU?kβ ω
2
ω2k?ω2 = δαβ (7.3.28)
If frequencies and polarizations of the normal modes of a perfect lattice are
known
All frequencies of the normal modes of the crystal with the defect can be
found from (7.3.28)
In the case of a defect with cubic symmetry in a cubic crystal,for all ω
A simplified relation is
UkαU?kβ = (1/3)δαβ
(7.3.28) →
δM
3NM
summationdisplay
k
ω2
ω2k?ω2 = 1 (7.3.29)
Its roots are the normal mode frequencies,graphical solution in Fig,7.3.2
f(ω2) = 13N
summationdisplay
k
ω2
ω2?ω2k (7.3.30)
intersects the horizontal line at (?M/δM)
(1) δM > 0 for a heavy impurity
Each root lie below a pole ω2k of f(ω2)
all of the new solutions are indistinguishable from the old
(2) δM < 0 for a light impurity
Each root lie above a pole ω2k of f(ω2)
Almost all of the new solutions are indistinguishable from the old
A localized mode is found away from the top of the band,Fig,7.3.2
Its square amplitudes can be obtained and they decay exponentially with
distance
1
0 ωmax ω
Localized Mode
-M/δM
f (ω2)
Figure 7.3.2 Graphical solution of lattice modes with a single mass defect.
Localized lattice vibrations are often infrared active and
can be detected in the absorption spectrum of crystal
7.3.3 Elastic Wave Defect Modes
Elastic wave propagation in artificial periodic composites in §5.3.3
Artificial structures are always finite and imperfect
There must be surface and impurity modes for elastic waves
§7.4 Defect Modes in Photonic Crystals
In §5 the artificial periodic structures,photonic crystals,photonic bandgaps
Unified picture of wave propagation → surface and impurity modes in pho-
tonic crystals
7.4.1 Electromagnetic Surface Modes in
a layered dielectric structure
In §7.4.1,the dispersion relation and band structure of
electromagnetic waves in layered periodic dielectric structures
Consider a semi-infinite periodic dielectric medium in z ≥ 0 half space
Homogeneous medium with index of refraction n0 in z < 0 half space
Fig,7.4.1
There are evanescent electromagnetic surface waves guided by the boundary
of a semi-infinite periodic dielectric layered medium
They are are propagating modes in xy plane
but confined in the z-direction in the vicinity of the interface
One surface wave corresponds to an eigenfrequency in forbidden band
Consider a confined wave propagates in the positive y direction,and
is transverse electric (TE) modes with the electric field polarized in the x
direction
Electric field equation
d2E(y,z)
dy2 +
d2E(y,z)
dz2 +
ω2
c2epsilon1(z)E(y,z) = 0 (7.4.1)
Its solution can be separated
E(y,z) = E(z)e?ikyy
then in z-direction
d2E(z)
dz2 +
parenleftbiggω2
c2epsilon1(z)?k
2
y
parenrightbigg
E(z) = 0 (7.4.2)
Its solution can be divided into two parts
E(z) =
Cek0z,z ≤ 0
A1eiq1z +B1e?iq1z,0 ≤z <d1
(7.4.3)
C,A1,B1 are constants,and q1 = n1ω/c
k0 is given by
k0 =
bracketleftbigg
k2y?
parenleftBign0ω
c
parenrightBig2bracketrightbigg1/2
(7.4.4)
Solution for z > 0 approximated as Bloch wave E(z)exp(ikz) as in §5.4.1.
For a localized mode,k must be complex → field decays as z deviates from
z = 0
Continuity conditions of E(z) and dE(z)/dz at z = 0
C = A1 +B1,
k0C = iq1(A1?B1) (7.4.5)
Eliminating C and replacing A1/B1 by (5.4.6)
Mode condition for surface waves
k0 = iq1T11 +T12?e
ikd
T11?T12?eikd (7.4.6)
Combined with (7.4.4),this is a complicated transcendental equation
to determine the dispersion relation ω(ky,k)
A lot of propagating wave solutions for z < 0 or z > 0 half-space
To find a surface mode,first take ky >n0ω/c to assure real and positive k0
k may become a complex,k = η +iμ,corresponds to surface waves
CalculatedelectromagneticfieldforatypicalsurfacewaveisshowninFig.7.4.1(a)
Electromagnetic surface waves observed experimentally by measuring its in-
tensity distribution
(b)(a)
Figure 7.4.1 Transverse field distribution for typical surface modes guided by the
surface of a semi-infinite periodic stratified media,(a) Theoretical result; (b) Experi-
mental measurement.
as shown in Fig,7.4.1(b)
7.4.2 Point Defect
Adielectricstructurewithdefectscanbedescribedbytheposition-dependent
dielectric function epsilon1(r) as
epsilon1(r) = epsilon10(r) +δepsilon1(r),(7.4.7)
whereepsilon10(r) is a periodic function in the real space,whileδepsilon1(r) represents the
deviation at each site with epsilon10(r),For a real defect,the deviation is actually
concentrated in a certain region,for example,a unit cell,corresponding to a
point defect; or a tube constructed with a sequence of unit cells,correspond-
ing to a line defect,If defects are known,then the dielectric function (7.1.1)
is determined,we can substitute it into the following equation
×
bracketleftbigg 1
epsilon1(r)?×H(r)
bracketrightbigg
= ω
2
c2 H(r),(7.4.8)
to obtain the eigenvalues and the localized eigenstates,The general method
to solve the equation is to perform numerical calculations for supercells.
We take a two-dimensional photonic crystal with a point defect as an exam-
ple,Consider a periodic structure composed of circular rods with same radius
R and dielectric constant epsilon1,and the lattice constant is a and R = 0.2a,We
choose a 7 × 7 supercell,The simplest point defect is obtained by inflat-
ing or deflating the radius of a dielectric rod in the middle of the structure.
Certainly the real case for defects can be more sophisticated,such as filling
materials with different epsilon1,and involved with more than one unit cell,We
are concerned here about only the simplest case,Let the magnitude of R
decreases from 0.2a,At the beginning,disturbance is not large enough to
induce even one localized state,When the radius is decreased to R = 0.15a,
there is a localized mode to appear not far from the top of valence band.
Afterwards it sweeps over the band gap until the rod disappears at R = 0,
ω = 0.38c/a,Reversely,we let the radius increases gradually,When R
arrives at 0.25a,there appears a pair of doubly degenerated localized modes,
dipolar modes with nodes on the midplane,near the bottom of the conduc-
tion band,Continuously increasing the radius leads the localized states with
different symmetries sequentially to appear,and their eigenfrequencies sweep
over the gap,as shown in Fig,7.4.2,These localized states due to the point
defect correspond to the acceptor and donor states in doped semiconductors.
If there is an atom placed in an otherwise perfect photonic crystal,and the
atomic transition frequency is just located in the gap of the photonic crystal,
then its spontaneous transition for radiation will be inhibited,But if this
atom is placed in the point defect of a photonic crystal,the situation is quite
different,When the atomic transition frequency matches with the energy
level of local modes of the point defect,the probability of the atomic spon-
0.40
0.45
0.25
0.30
0.35
0 0.2 0.4 0.6 0.8
Double Degenerate Modes
ω
a/2
pic
Defect Radius
Figure 7.4.2 The relation of frequency and stick radius for local state in two di-
mensional photonic crystals,From Villeneuve,P,R,et al.,Phys,Rev,B 54,7837
taneous emission of radiation will be enhanced,Therefore,a void (R = 0)
in a photonic crystal is like a optical resonant cavity,called microcavity,
surrounded by perfectly reflecting walls,The resonant frequency of the mi-
crocavity corresponds to the local mode of the point defect.
Usually the qualityfactorQis used to characterize the magnitude of resonant
loss,i.e.,
Q = ω0EdE/dt similarequal ω0?ω,(7.4.9)
where E is the energy stored by resonant cavity,ω0 is the eigenfrequency
of resonant cavity,and?ω is the half width of the Lorentz profile of this
frequency,The physical meaning of Q denotes the oscillating cycles expe-
rienced by a resonant cavity with its energy declined to e?2pi (～ 0.2%),It
is obvious that Q is closely related to the size of a photonic crystal,the
more the number of unit cells,the larger the Q value,for example,when the
number of unit cells is 9 × 9 for a two-dimensional rod periodic structure,
photonic crystal composed of parallel Qsimilarequal 104.
Assume that the atom is coupled with light field,then the Einstein coefficient
Af,characterizing the possibility of the atomic spontaneous radiation in free
space,is proportional to the photonic density of states in unit volume,that
is
Af similarequal 1ω
0λ3
,(7.4.10)
while the corresponding coefficient in a microcavity is
Ac = 1?ω?,(7.4.11)
where? is the volume of a microcavity,Now the enhancement factor in a
microcavity for the spontaneous radiation is about
Ac
Af similarequal
ω0
ω
λ3
=
Q
(?/λ3),(7.4.12)
Due to the size of a microcavity is? ～ λ3,so the enhancement factor of a
microcavity is almost equal toQvalue,Larger probability of spontaneous ra-
diation is beneficial for the fabrication the highly efficient luminescent diodes
and lasers,It is expected that there is a large potential in application of
point defects in photonic crystals as microcavities for lasers.
7.4.3 Line Defects
A point defect in a photonic crystal can be used to restrict electromagnetic
waves in a local region; and a line defect can be used as a waveguide for
electromagnetic waves,that is,it can guide electromagnetic waves from one
place to other.
To illustrate this problem,we can still use the two-dimensional square array
of dielectric circular rods as in the case of point defects,The corresponding
defect modes form a band of conduction waves,as shown in Fig,7.4.3,just
as the impurity band in a semiconductor,This band of conduction waves
allows electromagnetic waves propagate freely along a narrow channel of a
waveguide,Because it is impermissible for electromagnetic waves to pene-
trate the wall of a waveguide,so even if the waveguide bends 90?,there are
nearly no electromagnetic wave leaking out,Theoretical calculations have
verified this conclusion.
In the traditional waveguide techniques,at the microwave frequency range
the metallic walls and coaxial cables are used to guide electromagnetic waves;
while at the optical frequency range the dielectric waveguide and optical fi-
bres are used,The latter are based on the effects due to the gradient of
indices of refraction and total reflection at boundary,From the viewpoint
of applications,there are some shortage for optical fibre waveguides and di-
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5
Conduction Modes
ka/2pi
ω
a/2
pic
Figure 7.4.3 Energy strucutures for in a two dimensional photonic crystal with a
line defect,From Joannopoulos,J,D,et al.,Photonic Crystals,Princeton Univ,Press
(1995).
electric waveguides,Especially the index of refraction has dispersion,i.e.,
there are differences for the velocities of light with different frequencies,The
original very-short optical pulse (according to the uncertainty relation,very
short in time regime means very wide in frequency regime) propagating in
the dispersive media will be widen,thus restricted the quality of informa-
tion to be transmitted,On the other hand,if optical fibres bend with large
angles,the energy loss will be considerable,The existence of these disad-
vantages provides the opportunities for line defects of photonic crystals as
optical waveguides in applications to optical techniques of information trans-
mission,1?.
1?For taking line defects in photonic crystals as waveguides,see a review by Knight,J,C.,Nature
428,847 (2003).
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
March 15,2004
Contents
Chapter 8,Transport Properties 3
§8.1 Normal Transport,,,,,,,,,,,,,,,,,,,,,,5
8.1.1 Boltzmann Equation,,,,,,,,,,,,,,,,,,6
8.1.2 DC and AC Conductivities,,,,,,,,,,,,,,,9
8.1.3 Microscopic Mechanism of Metallic Conductivity,,,18
8.1.4 Electric Transport in Semiconductors,,,,,,,,,24
8.1.5 Other Transport Coefficients,,,,,,,,,,,,,,28
§8.2 Charge Transport and Spin Transport in Magnetic Fields,29
8.2.1 Classical Hall Effect,,,,,,,,,,,,,,,,,,,30
8.2.2 Shubnikov-de Haas Effect,,,,,,,,,,,,,,,,36
8.2.3 Ordinary Magnetoresistance and Its Anisotropy,,,,41
8.2.4 Spin Polarization and Spin Transport,,,,,,,,,52
8.2.5 Resistivity and Magnetoresistance ofFerromagnetic Metals 57
§8.3 Tunneling Phenomena,,,,,,,,,,,,,,,,,,,,63
8.3.1 Barrier Transmission,,,,,,,,,,,,,,,,,,63
8.3.2 Resonant Tunneling through Semiconductor Superlattices 63
8.3.3 Zener Electric Breakdown and Magnetic Breakdown,63
8.3.4 Tunneling Magnetoresistance,,,,,,,,,,,,,,63
8.3.5 Scanning Tunneling Microscope,,,,,,,,,,,,63
Chapter 8
Transport Properties
Two competing factors to carriers
(1) driven by electric field and temperature gradient
(2) scattered by impurities,defects and lattice vibrations
at equilibrium,stationary transport
First,semiclassical Boltzmann equation and relaxation time approximation
Then discuss electronic transport in magnetic field
Involving charge transport as well as spin transport
Finally electronic transport based on the tunnelling effect
§8.1 Normal Transport
The theoretical basis for transport properties in solids
Boltzmann equation and its relaxation time approximation
The normal transport,i.e,without magnetic field
mainly concentrate on the transport under electric field
or under temperature gradient
8.1.1 Boltzmann Equation
An electron from k → kprime
Inhomogeneous on a macroscopic scale,and the scattering is weak
a semiclassical distribution function f(k,r,t)
Boltzmann equation
df
dt =
f
t +?kf ·
dk
dt +?f ·
dr
dt =
f
t
vextendsinglevextendsingle
vextendsinglevextendsingle
coll
(8.1.1)
f
t +
1
planckover2pi1F ·?kf +v·?f =
f
t
vextendsinglevextendsingle
vextendsinglevextendsingle
coll
(8.1.2)
Relaxation time approximation
f
t
vextendsinglevextendsingle
vextendsinglevextendsingle
coll
=?f?f0τ (8.1.3)
f
t +
1
planckover2pi1F ·?kf +v·?f =?
f?f0
τ (8.1.4)
Linear response,f = f0 + f1
f1 is the deviation from the equilibrium distribution function f0
f1
t +
1
planckover2pi1F ·?kf0 +v ·?f0 =?
f1
τ (8.1.5)
The linearized Boltzmann equation can be used to study
electric as well as heat transport,and the external fields
includes electric field or temperature gradient
R,Kubo’s quantum transport theory
σμν = 1k
BT
integraldisplay ∞
0
〈jμ(t)jν(0)〉dt (8.1.6)
G,D,Mahan,Many-Particle Physics,3rd ed.,Plenum Press,
New York,(1995)
8.1.2 DC and AC Conductivities
In the stationary case?f1/?t = 0
1
planckover2pi1?kf0 =
1
planckover2pi1
f0
E?kE = v
f0
E
and
f0 =?f0?E?EF
From (8.1.5),first-order term in the distribution function
f1 = eτ?f0?Ev·Eprime (8.1.7)
Electromotive force
Eprime = E? (1/e)?EF
j =? 2e(2pi)3
integraldisplay
vf1dk = σEprime (8.1.8)
σ =? 2e
2τ
(2pi)3
integraldisplay
vv?f0?Edk (8.1.9)
In principle,σ is a tensor in crystals
For convenience,assume isotropic bands,and
let the electric field E lie in the z direction
replace v by vz,and averaging over angle to get 〈v2z〉 = v2/3
Conductivity in the scalar form
σ =? 2e
2τ
3(2pi)3
integraldisplay
dkv2?f0?E (8.1.10)
For a simplest case,E = planckover2pi12k2/2m?,and then
v(f0/?E) =?(1/planckover2pi1)?f0/?k
σ =? 2e
2τ
3(2pi)3m?
integraldisplay ∞
0
dk4pik3?f0?k
= 2e
2τ
3(2pi)3m?
parenleftbigg
4pik3f0vextendsinglevextendsingle∞0 + 3
integraldisplay ∞
0
dk4pik2f0
parenrightbigg
Drude conductivity
σ = ne
2τ
m? (8.1.11)
n = [2/(2pi)3]integraltext f0dk is the number density of electrons
An understanding intuitively,eE/m? →eτE/m? →e2τE/m? →ne2τE/m?
In (8.1.8)
j = σE? 1eσ?EF
first term is the current density due to electric field
second term is the current density due to diffusion?eD?n
n =?EF(dn/dEF)
Einstein relation
σ = e2D dndE
F
= e2DgF (8.1.12)
Alternating electric field
E = E0e?iωt (8.1.13)
The equation of motion for a quasi-free electron
m?˙v =?eE?m?vτ (8.1.14)
Consider that v and E have the same alternating frequency
iωm?v =?eE?m?vτ (8.1.15)
j =?nev = ne
2τ
m?(1?iωτ)E (8.1.16)
Alternating conductivity
σ = ne
2τ
m?(1? iωτ) =
ne2τ(1 + iωτ)
m?(1 + ω2τ2) (8.1.17)
Electric polarization is
P =?nex =? ω
2p
ω2 + iω/τE (8.1.18)
Plasma oscillation frequency
ω2p = ne
2
epsilon10m? (8.1.19)
Frequency-dependent dielectric function
epsilon1(ω) = 1? ω
2p
ω(ω + i/τ) = 1?
ω2p(ω?i/τ)
ω(ω2 + 1/τ2) (8.1.20)
which implies a specific physical meaning
Negative dielectric constant in the optical frequency range
In 1968,Vesalago theoretically studied the propagation behavior of elec-
tromagnetic waves in materials with different combinations of negative and
positive epsilon1 and μ,and pointed out that epsilon1 and μ are all positive for normal
materials,which are called right-handed materials,Its Poynting vector is in
the same direction as the wavevector; if epsilon1 and μ are all negative,the mate-
rial will be called left-handed,the Poynting vector and the wavevector are in
opposite directions,so their indices of refraction are negative
Early theoretical study
V.G,Vesalago,Sov,Phys,Uspeshi,10,509 (1968)
A detailed discussion
J,Pendry et al.,IEEE 47,11 (1999)
Some experimental results
R,A,Shelby et al.,Science 292,77 (2001)
C,G,Parazzoli et al.,Phys,Rev,Lett,90,107401 (2003)
A,A,Houck et al.,Phys,Rev,Lett,90,137401 (2003)
8.1.3 Microscopic Mechanism of Metallic Conductivity
The relaxation times and transition probabilities
τ?1(k) = τ?1d (k) + τ?1L (k) (8.1.21)
If τd(k) and τL(k) are all isotropic,then due to ρ = σ?1
Matthiesen rule for resistivity
ρ = ρd + ρL (8.1.22)
The influence of thermal vibrations of ions to resistivity
k +q = kprime
and energy conservation
E(k) + E(q) = E(kprime)
This is normal scattering process,called N process
It is possible to exist a scattering
k +q = kprime +G
it is Umklapp scattering process,called U process
Fig,8.1.1
k
q
(a)
k
G
q k'
P Q
(b)
k'
k'
Figure 8.1.1 Electron-phonon scattering,(a) Normal process k +q = kprime; (b) Ultra
process k+q = kprime +G.
τ?1 = 2pik2
integraldisplay
P(θ)(1?cosθ)sinθdθ (8.1.23)
〈P(θ)〉 ∝ kBTθ2
D
(8.1.24)
integraldisplay x
0
(1?cosθ)sinθdθ =
integraldisplay x
0
8sin3 θ2d
parenleftbigg
sin θ2
parenrightbigg
(8.1.25)
At high temperatures (T greatermuch θD),Einstein model
ρ ∝ τ?1 ∝ T (8.1.26)
At low temperatures (T < θD),Debye model
ρL ∝ τ?1 ∝ T5
ρL = 4.225
parenleftbiggT
θD
parenrightbigg5
J5
parenleftbiggθ
D
T
parenrightbigg
ρ(θD) (8.1.27)
J5(x) =
integraldisplay x
0
z5dz
(ez? 1)(1?e?z)
Fig,8.1.2
Au θD=175
Na θD=202
Cu θD=333
Al θD=395
Ni θD=472
0 0.1 0.2 0.3 0.4
0.1
0.2
0.3
T /θD
ρ(
T)
/ρ(
θ D
)
Figure 8.1.2 The relation of reduced resistivity ρ(T)/ρ(θD) and reduced temperature
T/θD of several metals,Curves are the theoretical calculations.
8.1.4 Electric Transport in Semiconductors
In semiconductors,as shown in Fig,8.1.3
Ec?EF greatermuch kBT,EF?Ev greatermuch kBT
f(E) = 1e(E?E
F)/kBT + 1
E?EF > Ec?EF greatermuch kBT
f(E) ≈ e?(E?EF)/kBT
1?f(E) ≈ e?(EF?E)/kBT
σ = n|e|μ = ne|e|μe + nh|e|μh
τI ∝ T3/2,τL ∝ T?3/2
so
τ?1 = aT?3/2 + bT3/2
Fig,8.1.4
1?f (E) lessmuch1
f (E) lessmuch1
EF
Ev
Ec
f (E)
E
Figure 8.1.3 The distribution of energy levels of conduction band and valence band
in semiconductors.
Sample104H
Slope1.5
Sample 98C
Sample 99B
Crystal Scattering

2
3
4
T (K)
μ
(cm
2,V
-1,s
-1 )
Figure 8.1.4 The relation of mobility and temperature in GaAS.
8.1.5 Other Transport Coefficients
A brief discussion on the other transport coefficients:
thermoelectric coefficient,thermal conductivity etc.
§8.2 Charge Transportand Spin Transportin Magnetic Fields
The influence of cyclic motion on electronic transport properties:
classical Hall effect,Shubnikov-de Haas effect,normal magnetoresistance
spin polarization and spin transport
resistivity and magnetoresistivity in transition metals
8.2.1 Classical Hall Effect
Langevin equation
dv
dt =?
e
m?
parenleftbigg
E + 1cv ×H
parenrightbigg
vτ (8.2.1)
v is the average velocity and τ is the relaxation time
Current is given by j =?nev
Magnetic field is applied H = Hz
Two examples in Fig,8.2.1(a) and in Fig,8.2.1(b)
H
E
H
E
(a)
(b)
Figure 8.2.1 Classical motion of an electron in electric and magnetic fields,(a)
E bardbl H; (b) E ⊥ H.
In general case E = (Ex,Ey,Ez)
j = σ ·E (8.2.2)
E = ρ·j (8.2.3)
In the steady state,dv/dt = 0
σ0Ex = jx + ωcτjy,σ0Ey =?ωcτjx + jy,σ0Ez = jz (8.2.4)
σ0 = ne2τ/m? and ωc = eH/m?c
σ =
σxx σxy
σyx σyy
,ρ =
ρxx ρxy
ρyx ρyy
(8.2.5)
σxx and σxy are the longitudinal and the transverse components
σxx = σyy = σ01 + (ω
cτ)2
,σxy =?σyx =? σ0ωcτ1 + (ω
cτ)2
(8.2.6)
Resistivity formulas
ρxx = ρyy = 1σ
0
,ρxy =?ρyx = ωcτσ
0
(8.2.7)
σ·ρ = I (8.2.8)
Relationship between the resistivity and conductivity
ρxx = σxxσ2
xx + σ2xy
,ρxy =? σxyσ2
xx + σ2xy
(8.2.9)
σxy =? σ0ω
cτ
+ 1ω
cτ
σxx (8.2.10)
When the magnetic field is strong and temperature is low → ωcτ greatermuch 1
then σxx = 0,the Hall conductivity
σH = σxy =?necH (8.2.11)
the Hall resistivity
ρH = ρxy =? Hnec (8.2.12)
1/nec is the Hall coefficient
The Hall resistivity changes continuously as the magnetic field and carrier
density vary,This is purely a classical result.
In fact,at low temperatures and under strong magnetic field,quantum Hall
effect can be observed
1980,von Klitzing
8.2.2 Shubnikov-de Haas Effect
Shubnikov-de Haas (SdH) effect is the oscillationof longitudinal conductivity,
or resistivity,with magnetic fields
The conditions planckover2pi1ωc > kBT and ωcτ > 1
This effect,just like the de Haas-van Alphen effect,also comes from the Lan-
dau quantization in magnetic fields,Although this effect has been observed
in metals for many years,it has become a very powerful tool to characterize
the transport in heterostructures
AlGaAs-Si
GaAs jH⊥
Hparallelto Dark
H⊥ Light
Hparallelto Light
22.5
22.5
(a)
(b)
(c)
(d)
0 10 20 30 40 50 60 70
H (kG)
d 2
R/d
H 2
(Arbitrary Units)
H⊥
Hparallelto
Dark
Figure 8.2.2 Angulardependence of the Shubnikov-deHaas oscillation,FromSt¨omer
H,Dingle R,et al.,Solid State Commun,29,705 (1979).
n = e/ch?(1/H) (8.2.13)
These measured values are usually in excellent agreement with those deter-
mined by Hall measurements
σxx = ne
2τ
m?
1
1 + (ωcτ)2
bracketleftbigg
1? 2(ωcτc)
2
1 + (ωcτc)2
× 2pi
2kBT/planckover2pi1ωc
sinh(2pi2kBT/planckover2pi1ωc) exp
parenleftbigg
piω
cτc
parenrightbigg
cos
parenleftbigg2piE
F
planckover2pi1ωc
parenrightbiggbracketrightbigg
(8.2.14)
ρxx = ρxx(H = 0)ne
2τ
m?
bracketleftbigg
1? 2(ωcτc)
2
1 + (ωcτc)2
× 2pi
2kBT/planckover2pi1ωc
sinh(2pi2kBT/planckover2pi1ωc) exp
parenleftbigg
piω
cτc
parenrightbigg
cos
parenleftbigg2piE
F
planckover2pi1ωc
parenrightbiggbracketrightbigg
(8.2.15)
ρxx ∝ 2pi
2m?ckBT/planckover2pi1eH
sinh(2pi2m?ckBT/planckover2pi1eH) exp
parenleftbigg
piω
cτc
parenrightbigg
cos
parenleftbigg2pi2cplanckover2pi1n
eH
parenrightbigg
(8.2.16)
H (10kOe)
1.66 2.10 2.54 2.98 3.42
1.00
0.50
0.00
-0.50
-1.00
ρ (
k?
,s-
1 )
Figure 8.2.3 The oscillatory portion of the magnetoresistivity of a quantum well at
six different temperature between 1.7 K and 3.7 K,As the temperature increases,the
amplitude of the oscillations decreases,From Singh J.,Physics of Semiconductors and
Their Heterostructures,McGraw-Hill,New York (1993).
8.2.3 Ordinary Magnetoresistance and Its Anisotropy
Ordinary magnetoresistance(OMR) arisesfrom the cyclicmotionofelectrons
in magnetic field
All metals have positive ordinary magnetoresistance,i.e.,ρH > ρ0
The longitudinal magnetoresistance of magnetic field H parallel with current
j does not vary obviously
the transverse magnetoresistance of H perpendicular to j varies remarkably
with H
Kohler’s rule
ρ/ρ0 = ρH?ρ0 = F(H/ρ0) (8.2.17)
F is a function related to metallic properties,the relative orientations of
current,magnetic field and crystalline axes,but H/ρ0 appears as a combined
quantity
From Drude conductivity
l = mv/e2nρ0 (8.2.18)
and cyclotron frequency
r = mv/eH (8.2.19)
l
r =
H
ρ0
1
ne (8.2.20)
It is obvious that H/ρ0 in the Kohler rule is actually the measurement of
l/r,It should be noted that the Kohler rule has its restriction.
The transverse magnetoresistance of crystals usually behaves as one of the
three types
(1) resistance is saturated,several times of the value at zero field
saturation appears along all directions
(2) resistance increases continuously for all crystalline axes
(3) resistance is saturated along some crystalline axes,but not saturated for
other directions,anisotropy
Fig,8.2.4
0 9 18 27 36
300
600
BA
6
3
ρ
/ρ
Figure 8.2.4 Variation of resistivity in two directions of magnetic field of Sn single
crystal,Curves A and B correspond the minimum and maximum of resistivity varia-
tion,From Olsen J L.,Electron Transport in Metals,John Wiley & Sons,New York
(1962).
Magnetoresistance is a tool to investigate Fermi surface
whether it is closed,or includes open orbits,and
the directions of the open orbits
Fig,8.2.5 gives the schematic diagrams for several two-dimensional Fermi
surfaces
Fig,8.2.6 gives the anisotropic magnetoresistace for Cu single crystal
Fig,8.2.7
1
2
4 3
3
1
2
4
pi/a
kx
ky
-pi/a
Figure 8.2.5 Schematic Fermi surfaces in two-dimensional Brillouin zone,1,Closed
orbit; 2,Self intersectional orbit; 3.Open orbit; 4,Hole orbit.
0 200200 400 600400600
ρ⊥(H)/ρ(0)
[010]
[011][001]
Figure 8.2.6 Anisotropicmagnetoresistancein Cu single crystals,From R,G,Cham-
bers,Electrons in Metals and Semiconductors,Chapman and Hall,London (1990).
[001]
0
2
4
6
θ
ρH /ρ0
100
110
010001
112
111
I
II
(a) (b)
-100o -50o +50o +100o
III
[110][111]
Figure 8.2.7 High field magnetoresistance in Au single crystals,(a) The relation
of magnetoresistivity and angle of magnetic field; (b) Stereogram of distribution of
magnetoresistance related to magnetic field direction,From J,M,Ziman,Electron in
the Metals,Taylor & Francis,London (1963).
Free electron model can be used to classical Hall effect
but not magnetoresistance,because the magnetoresistance will be zero
ρxx is independent of H,its reason is that the Hall electric field Ey cancels
the Lorentz force from magnetic field
One simple but important model for drift velocity is to introduce two kind
of carriers,electrons and holes,or s electrons and d electrons,or open orbits
and closed orbits
This is the two-band model for illustrating magnetoresistance
ρ
ρ0 =
σ1σ2(σ1/n1 + σ2/n2)2(H/e)2
(σ1 + σ2)2 + σ21σ22(1/n1?1/n2)2(H/e)2 (8.2.21)
where σ1 and σ2 are the conductivities for each band,while n1 and n2 are
correspondingly the carrier densities for each band
In general,this formula is not consistent with the Kohler rule,but when
σ1 = λσ2,and λ is a constant,the above formula is reduced to
ρ
ρ0 =
A(H/ρ0)2
1 + C(H/ρ0)2 (8.2.22)
Figure 8.2.8 shows the experimental data of several polycrystalline materials
for their transverse magnetoresistance with magnetic field
Bi Ga W
Zn
Cd
Re
Mo
Mg
Fe
Sn
Pb
Pd
Pt
Cu
Al In
Au Nb
K Ni
ρ
⊥(
B)
/ ρ(0)
102 103 104 105
10-1
1.0
10
102
103
106
107
(H ρRT(0)/ρ(0)) (104 Oe)
1.0
10
102
103
104
105
10-1
Figure 8.2.8 Transverse magnetoresistance of several metals under high magnetic
fields,From E,Fawcett,Adv,Phys,13,139 (1964).
8.2.4 Spin Polarization and Spin Transport
An electron has not only the charge,but also the spin
a new discipline,spintronics
Fig,8.2.9 schematically gives the densities of states of
electronic spin subbands in ferromagnetic metals Fe,Co and Ni
g↑(EF) and g↓(EF)
n↑ negationslash= n↓ → the spin susceptibility
fracn↑?n↓n↑ + n↓ negationslash= 0
4s 3d 4s 3d 4s 3d
Fe Co Ni
E E E E E E
0.3 0.3 0.3 0.30.350.35
2.6 3.3 4.44.8
5
5
Figure 8.2.9 Schematic DOS curves in ferromagnetic metals.
N,F,Mott,1936
proposed a two-current model,i,e,electrons with different spins contributed
to resistance correspond to two channels connected in parallel
Basic assumption of this model:
the electrons with different spins have different distribution functions and
relaxation times
Consider there exists spin-flip scattering process,another relaxation time τ↑↓
Boltzmann equation should be extended
eE ·v?f0?E =?f↑?f0τ
↑
f↑?f↓τ
↑↓
eE ·v?f0?E =?f↓?f0τ
↓
f↓?f↑τ
↑↓
(8.2.23)
Cambell and Fert solved these coupled equations
ρ = ρ↑ρ↓ + ρ↑↓(ρ↑ + ρ↓)ρ
↑ + ρ↓ + 4ρ↑↓
(8.2.24)
where
ρ↑ = m?ne2τ
↑
,ρ↓ = m?ne2τ
↓
,ρ↑↓ = m?ne2τ
↑↓
If the effect of spin-flip scattering can be neglected,(8.2.4) → Mott’s result
ρ = ρ↑ρ↓ρ
↑ + ρ↓
(8.2.25)
Problem,to prove the formula (8.2.24)
8.2.5 Resistivity and Magnetoresistance of
Ferromagnetic Metals
The resistance and magnetoresitance of ferromagnetic metals
There are three sources for resistivity of ferromagnetic metals
ρ(T) = ρd + ρL(T) + ρm(T) (8.2.26)
Fig,8.2.10.
In lower temperatures,ρ↓ > ρ↑ and ρ↑↓ can be neglected,why ρM is so small,
due to the short-circuit role played by ρ↑
As temperature is raised,the spin-flip scattering gives rise spin mixing effect,
Ni
Tc
ρtot
ρL
ρM
200 400 600 800 1000 1200
0
10
20
30
40
50
T (K)
ρ (10
-6?
,cm)
Figure 8.2.10 The relation of resistivity-temperature in Ni.
ρ
ρ0
Tc
T (K)
ρ⊥
ρparallelto
Troom
Figure 8.2.11 Schematic anisotropic resistivity in ferromagnets.
the short-circuit role of low resistance reduced,so ρM rises,
in accordance with the modified expression in two-current model
At high temperatures,ρ↑ρ↓ greatermuch ρ↑ (or ρ↓)
ρM approaches the saturation value
ρsM = 14(ρ↑ + ρ↓)
In addition,below Tc the resistivity anisotropy expressed as
ρbardbl(I bardbl Ms) negationslash= ρ⊥(I ⊥ Ms),
Fig,8.2.11
Anisotropic magnetorisistance (AMR)
Fig,8.2.12
AMR = ρbardbl?ρ0ρ
0
,or ρ⊥?ρ0ρ
0
(8.2.27)
8.20
8.15
8.10
8.05
8.00
0 5 10 15 20 25
ρav
B
A
294K
Ni.9942Co.0058
H (kG)
0 5 10 15 20 25
0.59
0.60
0.61
0.62
0.63
0.64
ρ (
μ?
,cm)
H (kG)
B
A
4.2K
Ni.9942Co.0058ρ
⊥
ρparallelto ρ
parallelto
ρ⊥
(a) (b)
Figure 8.2.12 Dependence ofresistivitywithmagneticfield in columnarCo-Nialloys.
(a) Room temperature; (b) Low temperature (4.2K)
§8.3 Tunneling Phenomena
8.3.1 Barrier Transmission
8.3.2 Resonant Tunneling through Semiconductor Superlat-
tices
8.3.3 Zener Electric Breakdown and Magnetic Breakdown
8.3.4 Tunneling Magnetoresistance
8.3.5 Scanning Tunneling Microscope
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
March 17,2004
Contents
Chapter 8,Transport Properties 3
§8.1 Normal Transport,,,,,,,,,,,,,,,,,,,,,,4
8.1.1 Boltzmann Equation,,,,,,,,,,,,,,,,,,4
8.1.2 DC and AC Conductivities,,,,,,,,,,,,,,,4
8.1.3 Microscopic Mechanism of Metallic Conductivity,,,4
8.1.4 Electric Transport in Semiconductors,,,,,,,,,4
8.1.5 Other Transport Coefficients,,,,,,,,,,,,,,4
§8.2 Charge transport and Spin Transport in Magnetic Fields,,5
8.2.1 Classical Hall Effect,,,,,,,,,,,,,,,,,,,5
8.2.2 Shubnikov-de Haas Effect,,,,,,,,,,,,,,,,5
8.2.3 Ordinary Magnetoresistance and Its Anisotropy,,,,5
8.2.4 Spin Polarization and Spin Transport,,,,,,,,,5
8.2.5 Resistivity and Magnetoresistance of Ferromagnetic Metals 5
§8.3 Tunneling Phenomena,,,,,,,,,,,,,,,,,,,,6
8.3.1 Barrier Transmission,,,,,,,,,,,,,,,,,,8
8.3.2 Resonant Tunneling through SemiconductorSuperlattices 18
8.3.3 Electric Breakdown and Magnetic Breakdown,,,,,25
8.3.4 Tunneling Magnetoresistance,,,,,,,,,,,,,,34
8.3.5 Scanning Tunneling Microscope,,,,,,,,,,,,43
Chapter 8
Transport Properties
§8.1 Normal Transport
8.1.1 Boltzmann Equation
8.1.2 DC and AC Conductivities
8.1.3 Microscopic Mechanism of Metallic Conductivity
8.1.4 Electric Transport in Semiconductors
8.1.5 Other Transport Coefficients
§8.2 Charge transport and Spin Transport in Magnetic Fields
8.2.1 Classical Hall Effect
8.2.2 Shubnikov-de Haas Effect
8.2.3 Ordinary Magnetoresistance and Its Anisotropy
8.2.4 Spin Polarization and Spin Transport
8.2.5 Resistivityand Magnetoresistanceof FerromagneticMet-
als
§8.3 Tunneling Phenomena
Began in the initiative period of quantum mechanics
Based upon the nature of waves of quantum particles
a particle can tunnel through an energy barrier into other region
Two same or different materials intervened with a barrier layer
form a tunneling junction
materials may be metals,semiconductors,or superconductors
may be magnetic or nonmagnetic
Nobel Prize to tunneling phenomena,1973
L,Esaki (1925),Japan,Tunneling diode
I,Giaever (1929),Norway,Superconducting gap
B,D,Josephson (1940),England,DC and AC effects
Nobel Prize to STM,1986
H,Rohrer (1933),Swiss
G,Binnig (1947),German
8.3.1 Barrier Transmission
A typical tunneling structure shown in Fig,8.3.1
A potential barrier V(z)
Electronic wavefunction with energy E = planckover2pi12k2/2m
Linear combination of forward travelling wave and backward travelling wave
ψ1(z) = A1eikz +B1e?ikz,ψ2(z) = A2eikz +B2e?ikz (8.3.1)
I II
V(x)
x
Figure 8.3.1 A particle with wave property and energyE tunneling from left to right
across a barrier.
Write the electronic wavefunction in the barrier
using the continuity conditions,obtain a transfer matrix T
A2
B2
= T
A1
B1
=
T11 T12
T21 T22
A1
B1
(8.3.2)
or a scattering matrix S
A2
B1
= S
A1
B2
=
t1 r2
r1 t2
A1
B2
(8.3.3)
T = |t1|2 represents the transmission probability
R = |r1|2 is the reflection probability
There are relations for the matrix elements of T matrix
T11 = T?22,T12 = T?21 (8.3.4)
and S matrix
t1t?2 = 1?|r1|2,|r1|2 = |r2|2 = |?r|2,t1r?2 =?r1t?2,t2r?1 =?r2t?1 (8.3.5)
T =
t
1
2 r2t
1
2
r?2t12 t?12
(8.3.6)
||T|| = t1/t2 = t?1/t?2 = 1
By using conservation condition for probability flow
|t1|2 + |?r|2 = |t2|2 + |?r|2 = 1 (8.3.7)
For a square barrier,height V0,between z1 =?a/2 and z2 = a/2
a particle with the energy E <V0,its transmission amplitude
t = |?t|eiφ (8.3.8)
where φ satisfies
tan(φ+ka) = 12
parenleftbiggk
κ?
κ
k
parenrightbigg
tanhκa (8.3.9)
in which E = planckover2pi12k2/2m and V0?E = planckover2pi12κ2/2m
The transmission coefficient is
T = |?t|2 = 4k2κ2
4k2κ2 + (k2 +κ2)2 sinh2κa =
bracketleftbigg
1 + V
20
4E(V0?E) sinh
2κa
bracketrightbigg?1
(8.3.10)
Figure 8.3.2,V0 = 0.3 eV,a = 10 nm
When κa is very large
T = 16E
V0 exp(?2κa) (8.3.11)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
Barrier
δfunction
Classic
E (eV)
T (
E)
Figure 8.3.2 The relation of transmission coefficient and particle energy for a square
barrier,From Davis,J,H.,The Physics of Low-Dimensional Semiconductors,Cam-
bridge University Press,Cambridge (1998).
If in (8.3.1) z is substituted by (z?d)
Td =
e
ikd 0
0 eikd
T
e
ikd 0
0 e?ikd
(8.3.12)
Double barrier structure at z = 0 and z = d,total transfer matrix
Tt = TdT (8.3.13)
Total transmission amplitude
tt =?t21 +|?r|2e2i(kd+φ) (8.3.14)
Total transmission probability
Tt = |?tt|2 =?T2
|1 +|?r|2e2i(kd+φ)|2 =
(1?|?r|2)2
|1 +|?r|2e2i(kd+φ)|2 (8.3.15)
For some specific values satisfying 2(kd+φ) = (2n+1)pi → resonant trans-
mission
corresponds to a filter → double barrier diode
J(V) ∝
integraldisplay
|M|2g1(E?eV)g2(E){f(E?eV)[1?f(E)]
[1?f(E?eV)]f(E)}dE
∝
integraldisplay
|M|2g1(E?eV)g2(E)[f(E?eV)?f(E)]dE(8.3.16)
g1 and g2 are the densities of states at Fermi surface
f(E) is the Fermi distribution function
8.3.2 Resonant Tunneling through Semiconductor
Superlattices
Extended (8.3.13) to periodically repeated structure
a crystal or superlattice
resonant tunneling → energy bands
Consider the multiple barrier resonance tunneling shown in Fig,8.3.3
R,Tsu and L,Esaki,Appl,Phys,Lett,22,562 (1973)
l
eV
EF
I T
R
(a)
(b)
tildenosp
tildenosp
Figure 8.3.3 A finite superlattice of lengthl,(a) in equilibrium state with incidence,
reflection,and transmission amplitudes; (b) after applying a voltage V.
The energy of incident electron
E = El + planckover2pi12k2t/2m? (8.3.17)
longitudinal energy El and transverse energy planckover2pi12k2t/2m?
Wavefunction
ψ = ψlψt (8.3.18)
Left and right end wavefunctions
ψ1 = ψt ·(eik1z +re?ik1z) (8.3.19)
and
ψN = ψt ·teikNz (8.3.20)
Continuity condition → the reflection and transmission amplitudes
r,?t,and the transmission coefficient |?t|2
Fig,8.3.4,double,a triple,and a quintuple barrier structures
The parameters are m? = 0.067me,V0 = 0.5 eV,d1 = 20?A,d2 = 50?A
Tunneling current J
J = e4pi3planckover2pi1
integraldisplay ∞
0
dkl
integraldisplay ∞
0
dkt[f(E)?f(Eprime)]?tt?E?k
l
(8.3.21)
incident energy E and transmitted energy Eprime
0
-2
-4
-6
0
-2
-4
0
-2
-4
-6
-8
-10
-12 0.04 0.12 0.20 0.28 0.36
5
Barriers
E (eV)
3
Barriers
2
Barriers
ln (
t t*
)
tildenosptildenosp
Figure 8.3.4 Plot of logarithmic transmission coefficient ln?tt versus electron energy
showing peaks at the energies of the bound states in the potential well.
Integrated over the transverse direction
J = em
kBT
2pi2planckover2pi13
integraldisplay ∞
0
ttln
parenleftbigg 1 + exp[(E
F?El)/kBT]
1 + exp[(EF?El?eV)/kBT]
parenrightbigg
dEl (8.3.22)
Low temperature limit T → 0,for V ≥EF
J = em
2pi2planckover2pi13
integraldisplay EF
0
(EF?El)t?tdEl (8.3.23)
and for V <EF
J = em
2pi2planckover2pi13
bracketleftbigg
eV
integraldisplay EF?eV
0
ttdEl +
integraldisplay EF
EF?eV
(EF?El)?ttdEl
bracketrightbigg
(8.3.24)
Resonant tunneling in double barriers in Fig,8.3.5
Well width of GaAs is 50?A,barrier width of Ga0.3Al0.7As is 80?A
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.9
0.6
0.3
0
-0.3
-0.6
-0.9
-1.2
-0.8 -0.4 0 0.4 0.8 1.2
V (V)
(a)
(b)
(c)
(a)
(b)
(c)dI / dV
I (mA)
σ (
10
-3 e
2 h
-1 )
Figure 8.3.5 Current-voltage and conductance-voltage characteristics of a double-
barrier structure at 77 K,Condition at resonance (a) and (c) and off-resonance (b)
are indicated.
8.3.3 Electric Breakdown and Magnetic Breakdown
From Bloch oscillation to Zener breakdown
From Fig,6.2.1 to Figs,8.3.6 and 8.3.7
Increase electric field to very large value
edE >Eg
E A
B
C
A'
B'
C'
k
u
k
pi/a
pi/a
-pi/a
-pi/a
0
0
Electron
(a)
(b)
F
Figure 8.3.6 Bloch oscillation of an electron moving in Brillouin zone under electric
fields.
Eg
A
A''
B
E(k)
O k
Figure 8.3.6 Electric breakdown for an electron moving in Brillouin zone under
electric fields.
O
d
A''A
eE
Figure 8.3.7 Tilted energy bands in real space under strong electric fields.
From (8.3.11),the probability of electric breakdown
T = exp
bracketleftBigg
pi
2
4
E2g
E0eEa
bracketrightBigg
(8.3.25)
lattice constant a,kinetic energy E0 = (planckover2pi12/2m)(G/2)2 ≈EF,G/2 = pi/a
Zener breakdown condition
eEaE0
E2g > 1 (8.3.26)
Similarly,a magnetic field strong enough → electrons to transit
between different energy bands,the magnetic breakdown
Introduce a periodic potential for a crystal
V(r) =
summationdisplay
G
VGeiG·r (8.3.27)
as a perturbation
Fig,8.3.8,the electronic orbit may be transferred from AB to AC
This case comes from the possibility of orbit reconnection at
the boundary of Brillouin zone in a periodic lattice
A
B
C
Z.B.
A
B
C
Z.B.
(a) (b)
Figure 8.3.8 Magnetic breakdown,(a) Free-electron orbit in magnetic field,(b)
Orbit connected at the boundary of a Brillouin zone when there is a periodic potential.
Increase perturbation → energy split at A,path AC become preferential
the electron moves in open orbits
The section B is connected to form a separating branch
In very strong magnetic field,the orbit may jump back to
the circular movement
It does not move along AC,but the energy gap breakdown takes place
a tunneling between a region in reciprocal space to separate two orbits
From the electric breakdown formula (8.3.26) to magnetic breakdown
E similarequalvH/c,v ≈ planckover2pi1kF/m →
eplanckover2pi1H
m
kFaE0
E2g > 1 (8.3.28)
kFa≈ 1,E0 ≈EF
magnetic breakdown condition simplified to
planckover2pi1ωcEF
E2g > 1 (8.3.29)
8.3.4 Tunneling Magnetoresistance
Two ferromagnetic metals isolated by a nonmagnetic insulating thin layer
Electronic transport depends on tunneling process through the barrier
and the relative orientation of magnetizations of two ferromagnetic metals
mainly on their parallel or antiparallel alignment
Spin-polarized transport
Brief history
(1) The earliest experimental verification of electronic spin polarization
P,M,Tedrow and R,Meservery,Phys,Rev,Lett,26,192 (1971)
measuring the tunneling current of S-I-FM tunneling junction
under applied magnetic fields,confirmed that currents of magnetic metals
like Fe,Co,and Ni under applied electric fields are spin-polarized
(2) Afterwards,Slonczewski suggested (1975) FM-I-FM tunneling junction
J,C,Slonczewski (1989) Phys,Rev,B 39,6995
(3) Julliere,1975,studied the transport properties of Fe-Ge-Co tunneling
junction
M,Julliere,Phys,Lett,54A,225 (1975)
a very short paper cited many times in recent years
experimentally verified that resistance is related to
the relative magnetization orientations of two ferromagnetic layers
Julliere,1975,proposed a simple model for tunneling magnetoresistance
the electrons close to the Fermi level participate in the transport process
The density of states of majority spins n↑ is higher than
that of minority spins n↓
parallel magnetization orientations → low resistance
antiparallel magnetization orientations → high resistance
Total conductivity σ ∝ 1/R is the sum of the two spin channels
Parallel orientation
σ↑↑ ～n↑n↑ +n↓n↓
Antiparallel orientation
σ↑↓ ∝n↑n↓ +n↓n↑
Spin polarization
P = n↑?n↓n
↑ +n↓
(8.3.30)
Two electrodes with different spin polarizations P1 and P2
TMR?R/R,two definitions
parenleftbigg?R
R
parenrightbigg
c
= R↑↓?R↑↑R
↑↓
(8.3.31)
always less than 100%
parenleftbigg?R
R
parenrightbigg
i
= R↑↓?R↑↑R
↑↑
(8.3.32)
can even be infinite
Their relation is
parenleftbigg?R
R
parenrightbigg
i
=
parenleftbigg?R
R
parenrightbigg
c
slashbiggbracketleftbigg
1?
parenleftbigg?R
R
parenrightbigg
c
bracketrightbigg
(8.3.33)
The resulting magnetoresistance
parenleftbigg?R
R
parenrightbigg
c
= 2P1P21 +P
1P2
(8.3.34)
and parenleftbigg
R
R
parenrightbigg
i
= 2P1P21?P
1P2
(8.3.35)
The spin polarizations are 0.44 and 0.34 for Fe and Co,respectively
the calculating value?R/RA is 26%
Recent experimental progress
T,Miyazaki and N,Tezuka,J,Mag,Mag,Mat,139,L231 (1995)
J,S,Moodera,L,R,Kinder et al.,Phys,Rev,Lett,74,3273 (1995)
Recent theoretical work,Z,Z,Li et al.,Phys,Rev,B,2004
Problem,Prove the formulas from (8.3.30) to (8.3.35)
Co Film
CoFe Film
CoFe/Al2O3/Co
Junction
0-600 -400 -200 0 200 400 600
2.5
5.0
7.5
10.0
0
0.06
0.12
-0.50
-0.25
0
H (kG)
R/R
(%)
Figure 8.3.9 The relation of tunneling magnetoresistance and magnetic field for
CoFe-Al2O3-Co three layer structure,Arrows represent the magnetization directions
in two magnetic layers,From J,S,Moodera,L,R,Kinder and T,M,Wong,Phys.
Rev,Lett.,74,3273 (1995).
8.3.5 Scanning Tunneling Microscope
Transmission probability through rectangular barrier
T ∝ exp(?2κa) (8.3.36)
κ = radicalbig2m(U?E)/planckover2pi12
A barrier structure can be two metals separated by a thin insulating layer
A bias voltage is applied,the current density j
j = 2eh
summationdisplay
kz
integraldisplay
Ez
T(Ez,V)[f(E)?f(E +eV)]dEz (8.3.37)
E
0 z
EF
E
z0
eV
(a) (b)
Figure 8.3.10 Potential as a function of position in a metal-insulator-metal tunnel
junction,(a) the equilibrium state; (b) an applied potential V.
Finishing the integral over k space,the total current density at 0 K
j(V) = 4pime
2V
h3
integraldisplay EF?eV
0
T(Ez,V)dEz
+4pimeh3
integraldisplay EF
EF?eV
(EF?Ez)?T(Ez,V)dEz (8.3.38)
Tunneling effect in M-I-M → an important modern experimental technique
scanning tunneling microscope (STM)
with vacuum as the insulating layer
G,Binnig and H,Rohrer et al.,1982,Helv,Phys,Acta,55,726;
Appl,Phys,Lett,40,178; Phys,Rev,Lett,49,57
STM was designed by Binnig and Rohrer (1982),and
finished by Binnig,Rohrer,Gerber and Weibel (1982)
a metal tip sharpened to a point with atomic dimensions
Fig,8.3.11,the tip is controlled by three piezoelectric elements
x,y and z perpendicular to each other
the distance between the tip and sample below 1 nm
the electronic wavefunctions in the tip and at the surface overlaps
LyL
zL
x
LyLz
Lx
Sample
Sample
nA
I
I
Vb
Vb
Vx Vx
lgI
0 z
z0
(a)
(b) nA
Figure 8.3.11 Two work modes for scanning tunneling microscope,(a) shows the
constant height mode; (b) shows the constant current mode.
Take the tip and the surface of a sample as two electrodes
applied a bias voltage,current by the tunneling effect
extreme sensitivity of the tunneling current to tip-surface separation
and electron density,STM images routinely achieve atomic resolution
Assume the work functions of the tip and sample φ1 and φ2
and the separation a between both
Tunneling current density is approximated
j = 2e
2
h
parenleftBig κ
4pi2a
parenrightBig
V exp(?2κa) (8.3.39)
κ = planckover2pi1?1[m(φ1 +φ2)]1/2
More detailed STM theories based on Bardeen’s tunneling current theory
J = 2pieplanckover2pi1
summationdisplay
μν
f(Eμ)[1?f(Eν +eV)]|Mμν|2δ(Eμ?Eν) (8.3.40)
f(E) is Fermi distribution function,V applied voltage
Mμν transition matrix element
Mμν = planckover2pi1
2
2m
integraldisplay
dS ·(ψ?μ?ψν?ψν?ψ?μ) (8.3.41)
The key point is to determine the eigenstates ψμ and ψν
for the tip and sample separately,to assume a reasonable barrier
and also to consider the influences of temperature and bias voltage
STM is a very useful tool to ascertain local electronic structures
at the atomic scale on solid surfaces,to infer their local atomic structures
Based on the principles of STM,there are a series of related techniques
including
atomic force microscope (AFM)
the laser force microscope
electrostatic force microscope
magnetic force microscope
scanning thermal microscope
ballistic electron emitting microscope
scanning ionic conduction microscope
photon scanning tunneling microscope
and so on
These are powerful tools to study magnetic,electric,mechanical and thermal
properties on surfaces or interfaces of matter
Figure 8.3.12 Waves in an elliptical dish of mercury,1825.
Figure 8.3.13 Quantum mirage,2000.
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
March 1,2004
Contents
Chapter 9,Wave Localization in Disordered Systems 3
§9.1 Physical Picture of Localization,,,,,,,,,,,,,,,8
9.1.1 A Simple Demonstration of Wave Localization,,,,9
9.1.2 Characteristic Lengths and Characteristic Times,,,16
9.1.3 Particle Diffusion and Localization,,,,,,,,,,,22
§9.2 Weak Localization,,,,,,,,,,,,,,,,,,,,,,29
9.2.1 Enhanced Backscattering,,,,,,,,,,,,,,,,30
9.2.2 Size-Dependent Diffusion Coefficient,,,,,,,,,,43
9.2.3 Interference Correction to Conductivity,,,,,,,,50
§9.3 Electron Localization,,,,,,,,,,,,,,,,,,,,,59
9.3.1 Continuum Percolation Model,,,,,,,,,,,,,59
9.3.2 Anderson Model,,,,,,,,,,,,,,,,,,,,,59
9.3.3 Mobility Edges,,,,,,,,,,,,,,,,,,,,,59
9.3.4 Edwards Model,,,,,,,,,,,,,,,,,,,,,59
9.3.5 Hopping Conductivity,,,,,,,,,,,,,,,,,59
§9.4 Strong Localization of Light,,,,,,,,,,,,,,,,,60
9.4.1 Propagation of Optical Waves in Disordered Media,,60
9.4.2 Independent Scatterers,,,,,,,,,,,,,,,,,60
9.4.3 Coherent Scatterers,,,,,,,,,,,,,,,,,,,60
Chapter 9
Wave Localization in Disordered Systems
Formal analogy for wave propagation in the structures
With perfect or nearly perfect periodicity in Chapters 5 and 7
Both de Broglie waves and classical waves including
Electromagnetic waves and lattice waves or elastic waves
Perfect periodic structures,band theory
Weak imperfection,impurity and surface
Almost all states are extended,but some localized modes
Now consider disordered structures
Compositional disorder and structural disorder
band theory breaks down,almost all states localized
Wave characteristics of periodic,weak disorder and strong disorder
Formal analogy for three kinds of waves can still be demonstrated
Figure 9.0.1
§9.1 Physical Picture of Localization
P,W,Anderson,1958,Phys,Rev,109
Absence of diffusion in certain random lattices
Proposed concept of disorder-induced electron localization
This concept can also be applied to classical waves
S,John’s extension to elastic waves and optical waves
1983,Phys,Rev,B 27,28; 1984,Phys,Rev,Lett,53
9.1.1 A Simple Demonstration of Wave Localization
In Chapter 5,wave equations introduced for three kinds of waves
With the space-dependent potential functions
In disordered structures,potential functions randomly distributed
For strong disorder,waves will be localized
Exponentially localized functions in space as
ψ(r) ∝ exp(?|r?r0|/ξ)
A central position r0 and localization length ξ
For simplicity,all wave equations are expressed in the same scalar form
2ψ+ [k2?V(r)]ψ = 0 (9.1.1)
where
k2 =

2mE/planckover2pi12,for electrons
ω2/c2,for classical waves
Potential V(r) normalized with 2m/planckover2pi12 for electrons
with combination of stiffness coefficient and mass density for elastic waves
with dielectric constant for electromagnetic waves
A pedagogical example
S,He and J,D,Maynard,1987,Phys,Rev,Lett,57
One-dimensional acoustic system made by a long steel wire
Tension K,small lead masses along the wire with equidistance or not
provide a periodic or disordered potential field V
Small masses → potential a series of δ functions with strength mω2/K
Transverse field ψ generated with electromechanical actuator
at one end of the wire
(1) Comparison of frequency responses of periodic and disordered systems
in Fig,9.1.1
Distinct edges separating pass band from forbidden bands on either sides
These features were lost in the case of random distribution
(2) Comparison of amplitude distributions of some eigenstates
in Fig,9.1.2
First two correspond to Bloch-wave-like eigenstates
and the other two localized eigenstates
(a)
(b)
Figure 9.1.1 Frequency response of the wire as a one-dimensional acoustic system
for (a) periodic potential and (b) random potential.
(a)
(b)
(c)
(d)
Figure 9.1.2 Eigenstate amplitude as a function of position along the wire,(a) and
(b) are Bloch states corresponding to two eigenfrequencies in Fig,9.1.1(a); (c) and
(d) are eigenstates of the disordered system with frequencies corresponding two peaks
in Fig,9.1.1(b).
Two main conclusions from the acoustic simulation:
(1) Sufficient disorder is introduced into a system
band and gap structures disappear
(2) Wavefunctions are localized
9.1.2 Characteristic Lengths and Characteristic Times
From above discussion,introduction of strong disorder → localized states
Localization comes from multiple scattering
among the randomly distributed scatterers
Non-dissipative disordered medium,three characteristic lengths:
(1) wavelength λ related to eigenwavevector or eigenenergy
(2) elastic mean free path l characteristic of disorder ～ (nσ?)?1
number of scatterers per unit volume n,elastic scattering cross section σ?
(3) size of the sample L
Their relative values are important,to specify three regimes:
(1) Propagative case l>L
Homogeneous medium,incident wave,reflected wave,transmitted wave
(2) Diffusive case λ<l<L
Multiple scattering → waves lose the memory of initial direction
An additional coherent effect,the weak localization
(3) Strongly disordered case l≤λ<L
Ioffe-Regel criterion l≤λ,coherent effect becomes so important
Diffusive constant vanishes,nothing is transmitted through the system
Localized state,Anderson localization
Mean free path deserves further discussion
A collision time τ is defined by
l = vτ (9.1.2)
Characteristic velocity v,e.g.,Fermi velocity vF for electron
τ denotes the average time interval of two consecutive elastic scatterings
Involved in the transition between eigenstates of different momentums but
with degenerate energy
Temperature → thermally excited → inelastic scatterings →
Transition between eigenstates of different energies
Another characteristic time τin
Average time interval of two consecutive inelastic scatterings
In general,τin >τ
Elastic scattering keeps phase coherence
Inelastic scattering breaks phase coherence
Phase coherent length defined as lin ≡vτin
Inelastic scattering rate 1/τin ～Tp,p is a constant
An estimation for electrons at room temperature:
Inelastic scattering by phonons at very high rate
1/τin ～kBT/planckover2pi1 ～ 1013s?1 →τin similarequal 10?13 s
vF ～ 108cm/s = 1016?A/s and lin = vFτin ≈ 1000?A
Typical distance for an electron keeping phase coherent →
Elastic mean free path ～ 100?A
Phase information is destroyed for macroscopic samples
Two approaches to display phase coherence for electrons
(1) to decrease the size of sample and also temperature →
mesoscopic electronic systems
(2) to analyze the physical effects to macroscopic samples
coming from the multiple scattering at small size →δσ
9.1.3 Particle Diffusion and Localization
Diffusion,a concept of central importance in localization
For simplicity,first consider the classical diffusion of
A particle in d-dimensional disordered system
At t = 0 the particle is at r = 0,begins its random walk
After t = τ,it experiences an elastic scattering →
Its direction of motion is changed
This process continues →
Diffusion in disordered medium
A possible diffusion path in Fig,9.1.3
A particle is localized ≡
a nonzero probability to be around r = 0 when t→∞
Otherwise it is delocalized
r2
r1
r4
r3
Figure 9.1.3 Diffusion path of a moving particle in the disordered system.
Classical diffusion equation in d-dimension
The probability density distribution p(r,t)
p
t?D?
2p = 0 (9.1.3)
Diffusion coefficient D
Its solution
p(r,t) = (4piDt)?d/2 exp(?r2/4piDt) (9.1.4)
A diffusive volume Vd can be defined
Vd ≈ (Dt)d/2 (9.1.5)
Fig,9.1.4
Figure 9.1.4
The chance to return r = 0 at t
p(0,t) = (4piDt)?d/2 (9.1.6)
Integrated probability for diffusive particle to return
P(0,t) =
parenleftbigg 1
4piD
parenrightbiggd/2integraldisplay t
τ
dt
td/2 ∝

(t/τ)1/2,d = 1
ln(t/τ),d = 2
(t/τ)?1/2,d = 3
(9.1.7)
Dimensionality is important from (9.1.7)
Scale theory is more reliable to discuss localization with dimensionality
The interference of scattered waves may modify
this classical picture of diffusion →
to enhanced backscattering as well as
size-dependent diffusion coefficient
§9.2 Weak Localization
Weak disorder is intermediate between
the scattering from a single impurity and strong localization
due to the scatterings by a large amount of random scatterers
In the case of weak disorder,λlessmuchl<L
A great many energy eigenstates being extended although not periodic
Weak localization is found as the precursor for strong localization
9.2.1 Enhanced Backscattering
For classical diffusion
A particle propagating on same path in opposite directions
with identical probabilities
Total probability is by adding up two probabilities
For microscopic particle with wave-like character
two partial waves propagate in opposite directions on same path
Returned to the origin,total probability is twice as large as
in the classical diffusion problem expressed in (9.1.6)
It is instructive to study the wave behavior of classical waves
in disordered media
To examine the wave diffusion in disordered systems
Wave diffusing from origin O to some point Oprime shown in Fig,9.2.1
Different trajectories with a probability amplitude Ai
connected to every path i
λF
O
O'
Figure 9.2.1 Various possible paths for wave diffusing from O to Oprime.
The total intensity
I = |
summationdisplay
i
Ai|2 =
summationdisplay
i
|Ai|2 +
summationdisplay
inegationslash=j
AiA?j (9.2.1)
There are noninterference contribution and interference contribution
In most cases interference contribution can be ignored
One particular exception:
Points O and O’ coincide,i.e.,the path crosses itself
Fig,9.2.2
1
2
O=O'
Figure 9.2.2 Self-crossing path of a diffusing particle.
Amplitudes A1 and A2 are coherent in phase →
constructive interference
From (9.2.1) for A1 = A2 = A,the classical return probability 2|A|2
Wave character yields 2|A|2 + 2A1A?2 = 4|A|2
Wave diffusion is slower than classical diffusion
due to the existence of a more effective backscattering effect
Backscattering interference effect of optical waves
experimentally demonstrated in disordered dielectric media
In Fig,9.2.3,incident light ki = k0
scattered at point r1,r2,...,rN
final state kN = kf
The path γ and its time reversed path?γ
Relative phase factor is given by relative optical path length
A～ exp[i(ki + kf)·(rN?r1)] (9.2.2)
l2
l1
γ
r2
r1
rN
rN-1
-γk
i
ki
kf
kf
θ
Figure 9.2.3 Schematic diagram of light backscattering.
In the exact backscattering direction
q = ki + kf = 0
Constructive interference → intensity doubling
Take θ as the angle between?ki and kf
Coherent condition for small θ is
q ·(rN?r1) = 2piθ|rN?r1|/λ< 1 (9.2.3)
In the diffusion limit
|rN?r1|2 ≈D(tN?t1) ≈lL?/3
photon diffusion coefficient D = lveff/3,total length of path L?
θm = λ/(2pi
radicalbig
lL?/3) (9.2.4)
Problem,To prove the formulas (9.2.3) and (9.2.4)
Width and enhancement factor of the contribution to the backscattering cone
as a function of the depth in the sample
From M,B,van der Mark,M,P,van der Albada,and A,Lagendijk,1988,
Phys,Rev,B 37
Fig,9.2.4(a) and (b),experimental and theoretical results
θ ～L?1/2
0 4 8 12 16
1.6
1.8
2.0
5
-16 -12 -8 -4 0 4 8 -16 -12 -8 -4
1.0
1.2
1.4 1
2
3 4
12 16
1.0
1.2
1.4
1.6
1.8
2.0
1
2
3 4
5
(a) (b)
Figure 9.2.4 Light backscattering pattern,(a) experimental results; (b) calculated
results.
Multi-scattering of lights by fine particles can trap lights
Random laser can be manufactured
Theoretical suggestion
V,S,Letokhov,Sov,Phys,JETP 26,835 (1968)
Experimental test
C,Gouedard et al.,1993,J,Opt,Soc,Am,B 10
N,M,Lawandy et al.,1994,Nature 368
D,S,Wiersma and A,Lagendijk,1996,Phys,Rev,E 54
9.2.2 Size-Dependent Diffusion Coefficient
Diffusion used to describe the motion of electron and photon
When l<L,a particle randomly walks with D = vl/3
However,classical random walk is not enough
Because of the wave character for electron and photon
Diffusion process must be described by amplitude rather than probability
Interference between all possible diffusion paths must be considered
in evaluating the transport of waves
A precise definition of diffusion coefficient for wave propagation needed
Diffusion coefficient is no longer a local variable
but determined by wave interference throughout disordered medium
In the case of weak scattering,reduced to the familiar diffusion coefficient
veffl/3 with effective medium speed veff
In the vicinity of incipient localization,Ddepends on the size of entire sample
A renormalized picture of transport shown in Fig,9.2.5
L
ξc
Dielectric Slab
Mean Free Path
Incident Light Transmission
l
Figure 9.2.5 Physical picture of optical transport at incipient localization.
Wave interference plays an important role
In a random medium,scatterers far apart
do not cause large interference corrections
to the classical diffusion picture
Define a finite coherent length ξcoh >～l represents a scale
Interference effect must be taken into account
to determine the effective diffusion coefficient
Effective diffusion coefficient of photon
is renormalized by wave interference
For a finite size sample of linear size L
Effective diffusion coefficient D(L)
D(L) ≈ vl3
parenleftbigg l
ξcoh +
l
L
parenrightbigg
(9.2.5)
Renormalized diffusion coefficient
as a function of sample size for three cases
(1) weak scattering limit,D is independent of sample
(2) strong scattering,D is renormalized downward
to an asymptotic value less than classical value
(3) strong localization,the renormalized diffusion coefficient vanishes
Fig,9.2.6,P,Sheng,1995,Introduction to Wave Scattering,Localization,
and Mesoscopic Phenomena
Weak Scattering
Strong Scattering
LocalizationRenormalized Dif
fusion Coef
ficient
DB
Coherent Diffusion
Regime
l L
Regime
Figure 9.2.6 Schematic variation of renormalized diffusion coefficient with sample
size,DB denotes the classical Boltzmann value of the diffusion coefficient.
9.2.3 Interference Correction to Conductivity
Conductivity σ is related to diffusion coefficient D
Einstein relation (8.1.12) for independent scattering
σ = e2Dg(EF) (9.2.6)
In conventional Boltzmann theory
Multiple scatterings → breakdown of the validity (9.2.6)
Interference between the scattered waves must be considered
Enhanced backscattering,special coherent superposition of
scattered waves → decrease of conductance
In Fig,9.2.7,an electron with k and exp(ik·r) at t = 0
penetrates into the medium and be scattered by scatterers 1,...,N
Two time reversal complementary processes
time-dependent phase changes (Et/planckover2pi1) are identical
k
k
k1'' k2'' k3'' -k (+q)
g3g2g1
-k (+q)k2'k1' k3'
g4
Figure 9.2.7 Diffusion path of a conduction electron.
Final amplitudes of Aprime and Aprimeprime are phase coherent
take |Aprime| = |Aprimeprime| = A
Backscattering intensity is
I = |Aprime +Aprimeprime|2 = |Aprime|2 +|Aprimeprime|2 +Aprime?Aprimeprime +AprimeAprimeprime?
= 4|A|2 = 2×2|A|2 (9.2.7)
leads to the decrease of the conductance σ
To calculate the correction of conductivity
Characteristic wavelength λF = 2pi/kF
Weak disorder corresponds lgreatermuchλF
Consider a d-dimensional tube with diameter λF
and its cross section λd?1F,shown in Fig,9.2.8
Particle moves a distance vFdt
Corresponding volume element of the tube dV = vFdtλd?1F
Maximum possible volume for the diffusing particle is Vdiff in (9.1.5)
}
dl
Cross Section ~ λFd-1
Figure 9.2.8 Enlarged cross section of a d-dimensional quantum mechanical trajec-
tory of diameter λF = h/mvF.
Probability for a particle to be in a closed tube
P =
integraldisplay τin
τ
dV
Vdiff = vFλ
d?1
F
integraldisplay τin
τ
dt
(Dt)d/2 (9.2.8)
Integral time over a single elastic collision τ to
the inelastic relaxation time τin
Correction for conductivity
δσ
σ ∝?κ×

(τin/τ)1/2,d = 1
planckover2pi1ln(τin/τ),d = 2
planckover2pi12(τin/τ)?1/2,d = 3
(9.2.9)
Disorder parameter κ
Assuming 1/τin ∝Tp,p is a positive constant,(9.2.9) gives
temperature-dependent conductivity
δσ
σ ∝?κ×

T?p/2,d = 1
planckover2pi1pln(1/T),d = 2
planckover2pi12Tp/2,d = 3
(9.2.10)
Fig,9.2.9 confirms the theoretical result in two dimension experimentally,
by G,J,Dolan and D,D,Osheroff,1979,Phys,Rev,Lett,43
0 0.1 0.4-0.2-0.4
0
2
4
6
-2
-4
-6
log10 (T/K)
[(V/I
-R
0)/
R 0
]×100
Figure 9.2.9 The dependence of resistance and temperature in disordered Au-Pd
film.
§9.3 Electron Localization
9.3.1 Continuum Percolation Model
9.3.2 Anderson Model
9.3.3 Mobility Edges
9.3.4 Edwards Model
9.3.5 Hopping Conductivity
§9.4 Strong Localization of Light
9.4.1 Propagation of Optical Waves in Disordered Media
9.4.2 Independent Scatterers
9.4.3 Coherent Scatterers
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
March 4,2004
Contents
Chapter 9,Wave Localization in Disordered Systems 3
§9.1 Physical Picture of Localization,,,,,,,,,,,,,,,4
9.1.1 A Simple Demonstration of Wave Localization,,,,4
9.1.2 Characteristic Lengths and Characteristic Times,,,4
9.1.3 Particle Diffusion and Localization,,,,,,,,,,,4
§9.2 Weak Localization,,,,,,,,,,,,,,,,,,,,,,5
9.2.1 Enhanced Backscattering,,,,,,,,,,,,,,,,5
9.2.2 Size-Dependent Diffusion Coefficient,,,,,,,,,,5
9.2.3 Interference Correction to Conductivity,,,,,,,,5
§9.3 Electron Localization,,,,,,,,,,,,,,,,,,,,,6
9.3.1 Continuum Percolation Model,,,,,,,,,,,,,9
9.3.2 Anderson Model,,,,,,,,,,,,,,,,,,,,,17
9.3.3 Mobility Edges,,,,,,,,,,,,,,,,,,,,,27
9.3.4 Edwards Model,,,,,,,,,,,,,,,,,,,,,32
9.3.5 Hopping Conductivity,,,,,,,,,,,,,,,,,37
§9.4 Strong Localization of Light,,,,,,,,,,,,,,,,,43
9.4.1 Propagation of Optical Waves in Disordered Media,,44
9.4.2 Independent Scatterers,,,,,,,,,,,,,,,,,47
9.4.3 Coherent Scatterers,,,,,,,,,,,,,,,,,,,53
Chapter 9
Wave Localization in Disordered Systems
§9.1 Physical Picture of Localization
9.1.1 A Simple Demonstration of Wave Localization
9.1.2 Characteristic Lengths and Characteristic Times
9.1.3 Particle Diffusion and Localization
§9.2 Weak Localization
9.2.1 Enhanced Backscattering
9.2.2 Size-Dependent Diffusion Coefficient
9.2.3 Interference Correction to Conductivity
§9.3 Electron Localization
Single impurity → localized electronic modes,in Chapter 7
Strongly disorder → localization of one-electron states
P,W,Anderson,1958
Studied the diffusion of electrons in a random potential
Found the localization of electron waves in strong randomness
Mott,1960s
Investigated impurity conductance in doped semiconductors
Suggested disordered-induced metal-nonmetal transition
(a) (b)
Figure 9.3.1 Bond and site percolation.
Glass Al
Figure 9.3.2 Percolation of the random close-packed two-phase mixture of Al balls
and glass ones.
9.3.1 Continuum Percolation Model
Percolation model is a simple method to
demonstrate the consequences of disorder
In §4.2,bond percolation and site percolation
Continuum percolation simulates the situation of an electron
moving in a random mixture of conductors and insulators
A classical particle with energy E moving in an irregular potential V(r)
E >V(r) is the allowed region
E <V(r) is the forbidden region
Fig,9.3.1
Fraction of space allowed to particles of energy E
p(E) ≡
integraldisplay
E>V(r)
dr
slashbiggintegraldisplay
dr (9.3.1)
p(E) ≤ 1,increases monotonically with E
P
P
P
P
PA
A
A
A
E < Ec
A
A
AA
A P
P
P
P
P
E greatermuch Ec
P
P
P
P P
P
A A
A A
E > Ec
A
(a) (b) (c)
A
Figure 9.3.1 The motion of a classical particle in a random potential.
A percolation threshold pc
When p(E) >pc,an infinitely extended region appears
A threshold energy Ec satisfies
p(Ec) = pc (9.3.2)
In two dimensions,pc = 1/2
In one dimension,pc = 1
In three dimensions,a reasonable estimate is pc ≈ 0.16
Strictly,electrons can tunnel through the regions E <V(r)
Some differences between classical continuum percolation and
quantum mechanical approach
(1) E >Ec:
(i) in classical case,localized states coexist with extended states
(ii) in quantum sense,a localized state mixes with extended states
(2) Dimension-dependence:
(i) all states localized by disorder in one dimension
(ii) percolation transition is present in two or higher dimensions
(iii) quantized localized-delocalized transition occur in dimensions >2
(3) Difference between Bloch states in periodic structure and
extended states within the percolation model
Continuum percolation model →
intuitive picture about extended states
localized states,and localization-delocalization transition
in disordered systems
Percolation may be applicable if the characteristic length related V(r)
is much larger than a typical electron de Broglie wavelength
Classical percolation can be extended to quantum percolation
C,M,Soukoulis et al.,1987,Phys,Rev,B 36
Weak localization effect of waves be added to percolation processes
Thresholds of quantum percolation pq and classical percolation pc
pq >pc
Giant Hall effect in metallic grainy films can be explained by the
quantum percolation model
X,X,Zhang et al.,2001,Phys,Rev,Lett,86
9.3.2 Anderson Model
Now turn to quantum model
In 1958,Anderson studied the electronic diffusion in random potentials
found electron waves localized when random potentials are strong enough
Anderson’s quantum theory is
a tight-binding disordered one-electron model
The Hamiltonian in diagonal disorder model
H =
summationdisplay
i
εi|i〉〈i|+
summationdisplay
inegationslash=j
tij|i〉〈j| (9.3.3)
|i〉,εi,tij,only for nearest neighboring sites,tij = t,εi is randomly chosen,
as in Fig,9.3.2
W/2 ≤ε≤W/2
Probability distribution of energy
p(E) = 1W (9.3.4)
E
W/2
-W/2
0
Figure 9.3.2 Disordered distribution of energy in the Anderson model.
Tight-binding expansions of wavefunction
|ψ〉 =
summationdisplay
i
ai|i〉 (9.3.5)
|i〉 is the atomic orbital centered at site i
From
Hψ = Eψ
a stationary matrix equation for the amplitudes ai
Eai = εiai +
summationdisplay
j
tijaj (9.3.6)
The time-dependent equation is
planckover2pi1
i
dai
dt = εiai +
summationdisplay
j
tijaj (9.3.7)
A definition for localized states:
at t = 0,an electron is placed at site i,ai(t = 0) = 1,aj = 0 for j negationslash= i
If ai(t→ ∞) = 0 → the electron is in extended state
If ai(t→ ∞) is finite,the electron is in localized state
In the nearest neighbor approximation,(9.3.6) is transformed into
Eai = εiai +t
zsummationdisplay
α=1
ai+α (9.3.8)
To examine some limiting cases:
(1) W = 0 corresponds to all sites with same energy,no disorder
bandwidth is
B = 2zt (9.3.9)
(2) εi distribution with W,but t = 0,bandwidth B = 0
The solution is simply atomic orbitals at each site
Initial situation will be kept invariant,i.e.,ai = 1,aj = 0 for j negationslash= i
(3) Intermediate between these two limits,W and B are all finite values
The magnitude of the ratio
δ = W/B (9.3.10)
shows the competition between disorder and order
A critical value δc for localization-delocalization transition
For determining δc,Anderson’s full treatment with Green’s function
A qualitative discussion
From localized limit B = 0,turn on t as a perturbation
At first ai = 1,aj = 0 for all j negationslash= i
First order perturbation by an amplitude of order
t
εi?εj
Higher orders of perturbation add terms containing
higher powers of this quantity,then
|ψ〉 = |i〉+ tε
i?εj
|j〉+··· (9.3.11)
How big t/(εi?εj) can be before localization is destroyed
and extended state arises?
Suppose εi at the center of the energy distribution
and assume εjs of the z nearest-neighbor sites are
uniformly spaced over W/z
Smallest energy denominator is |εi?εj| = W/2z
So the largest of perturbation parameter is
t
εi?εj =
2zt
W =
B
W (9.3.12)
Perturbation expansion convergence requires (B/W) < 1
When W >B,localization occurs at the center of band
Localization-delocalization transition occurs at B = W
Anderson localization criterion
δc = 1 (9.3.13)
smaller than Anderson’value,but consistent with computer simulation
9.3.3 Mobility Edges
Based on Anderson,Mott put forward a useful idea of mobility edge
shown in Fig,9.3.3 which is a critical energy value Ec
to distinguish localized state from delocalized state
This concept can be used to study the conductance in doped semiconductors
metal-insulator transition takes place
g(E)
B/t < (W/t)c
-Ec Ec E
Figure 9.3.3 Mobility edges.
(1) For a small degree of disorder,the states in the band tails are localized
The tails stretch into the energy gap through
disordered potential distribution
Within the main body of each band,the states are extended
(2) For an intermediate degree of disorder less than the critical value
There will be two energies separating localized from extended states
shown in Fig,9.3.3
Mott,1974,emphasized the possibility of Anderson transition
By changing the degree of disorder by,doping,pressure,electric field
Fermi level EF can cross a mobility edge
Fig,9.3.4 shows the effect of disorder to produce localized states
as the disorder increases,the tails become longer,until eventually
the mobility edges move inward and coalesce at the center
all states are localized
Band Width
Localized States Extended States
δ
Figure 9.3.4 Localized states increase as disorder parameter δ increases.
9.3.4 Edwards Model
For electrons in disordered metals,it cannot be satisfactorily described
in terms of tight-binding wavefunctions
In contrast to the Anderson model,Edwards,1958,considered
the disordered distribution of scatterers in a nearly-free electron model
S,F,Edwards,1958,Phil,Mag,3
The one-electron Schr¨odinger equation
parenleftbigg
planckover2pi1
2
2m?
2 +V(r)
parenrightbigg
ψ(r) = Eψ (9.3.14)
random potential V(r)
The random potential V(r) satisfies
〈V(r)〉 = 0 (9.3.15)
and
〈V(r)V(0)〉 = V2rmse?r2/ξ2 (9.3.16)
ξ a length scale characterizing the random fluctuations
Correlation length ξ defines an energy scale εξ = planckover2pi12/2mξ2
One-electron DOS in 3D is depicted in Fig,9.3.5
Weak disorder corresponds to Vrms lessmuchεξ
For weak disorder there is a tail of strongly localized states,Urbach tail
for E < 0 separated by a mobility edge Eprimec similarequal?V2rms/εξ
Electrons of energy lower thanEprimec are trapped and do not conduct electricity
As the disorder parameter Vrms increased
the mobility edge eventually moves into the positive Ec
An electron with E >Ec → successive tunnelings
to transverse the entire solid by a slow diffusive process →
conduct electricity
Electrons with E <Ec are trapped,do not conduct electricity
Only in the limit of very strong disorder Vrms greatermuchεξ
The states with E > 0 exhibit localization
EcEc' E
g(E)
0
Figure 9.3.5 One electron DOS in a correlated Gaussian random potential,As the
disorder is increased,the mobility edge moves into the positive energy regime denoted
by Ec.
9.3.5 Hopping Conductivity
Conduction properties of semiconductors is more interesting
whose conductivity vanishes at zero temperature
DOS in Fig,9.3.3,EF locates between Ev and Ec
For crystal,EF lies at an energy devoid of states
For glass,it lies within the region of localized states
In both cases,conductivity is zero unless thermal energy is supplied
E
R
Figure 9.3.6 Variable-range hopping.
An applied field can bias the motion and create a current
This is called hopping conduction,temperature-dependent hopping
phonon-assisted quantum mechanical tunneling
If T is lower enough,variable-range hopping is prominent
N,F,Mott,1968,Phil,Mag,17
The tail states are especially susceptible to localization
Thouless,1974,provided a percolation argument
a polychromatic percolation process
Transition probability including two contributions
p～ exp[?2αr?(W/kBT)] (9.3.17)
hopping distancer,energy separationW,temperatureT,inverse localization
length α,
electron wavefunction
ψ(r) ～ exp(?αr)
phonon assistance in overcoming the energy mismatch W
exp(?W/kBT)
Competition between hopping distance r and mismatch energy W
A sphere of radius r surrounding the initial site
(4pi/3)r3W(r)g(EF) = 1 (9.3.18)
gives
W(r) ∝r?3
Combining (9.3.17),using?p/?r = 0 to maximize p yields
the most probably hopping distance
r = [αkBTg(EF)]?1/4 (9.3.19)
Assume that p ∝ σ,then
σ ～ exp(?A/T1/4) (9.3.20)
with
A = [α3/kBg(EF)]1/4 (9.3.21)
Well-known T1/4-law in a number of amorphous semiconductors
verified by experiments
In d space dimension,1/4 is replaced by 1/(d+ 1)
Problem,For variable-range hoppingind-dimensions,to verify
the T1/d+1 law for conductivity
§9.4 Strong Localization of Light
In disordered dielectric microstructures,light can also be localized
S,John’s work in 1980s
Electron localization is complicated by electron-electron
and electron-phonon interactions
Optical waves in nondissipative dielectric media is ideal
to study localization even at room temperature
9.4.1 Propagation of Optical Waves in Disordered Media
Light localization arises entirely from coherent multiple scattering
and interference
Light localization occurs when the scale of coherent multiple scattering
is reduced to wavelength itself
Wave equation for electric field
2E +?(?·E)?ω
2
c2epsilon1f(r)E = epsilon10
ω2
c2E (9.4.1)
Dielectric constant
epsilon1(r) = epsilon10 +epsilon1f(r) (9.4.2)
average value epsilon10,and random fluctuation satisfying
〈epsilon1f(r)〉 = 0 (9.4.3)
Several important observations:
(1) epsilon10ω2/c2 always positive
unlike electrons,lowering photon energy,i.e.,let ω → 0 leads to
complete disappearance of scattering
(2) ω → high value,geometric optics becomes available
In both limits the normal electromagnetic modes are extended
(3) localized light must be in intermediate frequency window
within the positive energy continuum,in agreement with Ioffe-Regel rule
depending on the filling ratio and arrangement of scatterers
9.4.2 Independent Scatterers
Wave interference → large spatial fluctuations in light intensity
Wavelength λ,mean free path l
l plays a central role in localization
Three different regimes
(1) λgreatermuchl,long wavelength Rayleigh scattering limit
l～λ4
Consider incident wave with wavelength λ and intensity I0,
scattered by a sphere with the size a and dielectric constant epsilon1a,
background dielectric constant epsilon1b
Scattered intensity Is at a distance R from the sphere
could be written as
Is = I0f1(λ,epsilon1a,epsilon1b) a
2d
Rd?1 (9.4.4)
(ad)2 is proportional to the square of the induced dipole moment of the sphere
and 1/Rd?1 expresses the conservation of energy
Since Is/I0 is dimensionless,it is follows that
f1(λ,epsilon1a,epsilon1b) = f2(epsilon1aepsilon1b)/λd+1 (9.4.5)
f2 is another dimensionless function of the dielectric constants
Purely dimensional argument → the scattering cross section
for long wavelengths ∝λ?(d+1)
This is a familiar result for explaining
why the sky is blue in three dimensions
(2) λlessmuchl,all states are extended
noninterfering,multiscattering paths
transport of wave energy is diffusive on long length scale
(3) λ～l in strongly disordered medium,localized states
Ioffe-Regel rule
2pil/λ≈ 1 (9.4.6)
is satisfied
Interference between multiply scattered paths drastically modifies
the average transport properties
S,John,1984,Phys,Rev,Lett,53
Fig,9.4.1,a is the correlation length
Independent scatterers,free photon states,multiple scattering
microscopic resonance
Dashed curve in Fig,9.4.1 for dilute scatterers would not induce localization
a
a
λ/2pi
l l ~ λ4
λ/2pi
Figure 9.4.1 Behavior of the elastic mean free path l as a function of wavelength λ.
9.4.3 Coherent Scatterers
Free photon scattering among independent scatterers
satisfying Ioffe-Regel condition → strong localization
completely overlooks the possibility of gap
Free photon criterion for localization is difficult to achieve
Gap in photon density of states may be important in determining
the transport properties and localization of light
Another approach used to discuss the light localization
Coherently arranged scatterers,like photonic crystals
Dielectric constant epsilon1fluc(r)
epsilon1f(r) = epsilon11(r) +V(r) (9.4.7)
Spatially periodic function epsilon11(r)
Fourier expansion
epsilon11(r) = epsilon11
summationdisplay
G
UGeiG·r (9.4.8)
V(r) is a perturbation to the originally periodic dielectric constant
Bragg condition
k·G = (1/2)G
for a photonic wavevector k
Electric field amplitude below gap is a linear superposition of
free-photon field of wavevector k and
its Bragg reflected partner at k?G
Disordered V(r) → localized states in gap region depicted in Fig,9.4.2
Bandgap → strongly localized photonic band tail states
analogous to the Urbach tail in the electron systems
The two extreme limits discussed:
(1) a structureless random medium,2pil/λ≈ 1 yields localization
(2) a medium with nearly sharp Bragg peaks and a band gap
2pil/λenv ≈ 1 yields localization
In real disordered system,both band picture and single scattering approach
may provide a complete description
An experiment performed in a random 3D sample by
randomly mixing of metallic aluminium and dielectric teflon spheres
A,Z,Genack,N,Garcia,1991,Phys,Rev,Lett,66
Localization of microwave radiation was found
in a narrow window of frequency
Fig,9.3.8 shows the variation of D with the filling fraction and frequency
D → 0 at a metallic filling fraction f = 0.35
and frequencies around ν = 19 GHz
The filling fraction is associated with the sample length L
The results are consistent with the theoretical formula (9.2.5) for localization
f (GHz)
16 18 20 22 24
D (10
-9 cm
2 s
-1 )
1
10
Figure 9.4.2 The experimental results for frequency dependence of diffusion co-
efficient D,The triangle,dot and circle are,respectively,for the filling fraction
f = 0.20,0.25,0.35.
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
March 29,2004
Part II
Wave Behavior in
Various Structures
5,Wave Propagation in Periodic and
Quasiperiodic Structures
6,Dynamics of Bloch Electrons
7,Surface and Impurity Effects
8,Transport Properties
9,Wave Localization in Disordered Systems
10,Mesoscopic Quantum Transport
Part III
Bond,Band and Beyond
Only connect ···
— E,M,Forster
A genuine symbiosis may also emerge from comple-
mentaryapproaches,Thetypicalchemistwantsabove
all to understand why one substance behaves differ-
ently from another; the physicist usually wants to find
principles that transcend any specific substance.
— Dudley Herschbach (1997)
This part concerned with the electronic structure of matter
11,Bond Approach
12,Band Approach
13,Correlated Electronic States
14,Quantum Confined Nanostructures
Contents
Part II Wave Behavior inVarious Structures 3
Part III Bond,Band and Beyond 7
Chapter 11,Bond Approach 3
§11.1 Atoms and Ions,,,,,,,,,,,,,,,,,,,,,,,,5
11.1.1 A Hydrogen Atom,,,,,,,,,,,,,,,,,,,5
11.1.2 Single-Electron Approximation for Many-ElectronAtoms 10
11.1.3 Intraatomic Exchange,,,,,,,,,,,,,,,,,,14
11.1.4 Hund’s Rules and Magnetic Moments in Ions,,,,,17
§11.2 Diatomic Molecules,,,,,,,,,,,,,,,,,,,,,,20
11.2.1 The exact solution for hydrogen molecular ion H+2,,,20
11.2.2 Molecular Orbital Method,,,,,,,,,,,,,,,26
11.2.3 Heitler and London’s Treatment of Hydrogen Molecule 36
11.2.4 The Spin Hamiltonian and the Heisenberg Model,,,40
§11.3 Polyatomic Molecules,,,,,,,,,,,,,,,,,,,,,42
11.3.1 Molecular Orbital Method for Polyatomic Molecules,42
11.3.2 Valence Bond Orbitals,,,,,,,,,,,,,,,,,43
11.3.3 H¨uckel Approximation for Molecular Orbital Method,46
11.3.4 Electronic Structure of Some Molecules,,,,,,,,52
§11.4 Ions in Anisotropic Environment,,,,,,,,,,,,,,,61
11.4.1 Three Types of Crystal Fields,,,,,,,,,,,,,61
11.4.2 Transition metal ions in crystal fields,,,,,,,,,64
11.4.3 Jahn-Teller Effect,,,,,,,,,,,,,,,,,,,,67
11.4.4 Ions in Ligand Fields,,,,,,,,,,,,,,,,,,73
Chapter 11
Bond Approach
Development of CMP leads to include:
(1) Different objects and construction blocks
(2) Interaction,local environment and external fields
This chapter is devoted to quantum chemistry approach
More concern on atomic configuration,local potential
Effective theoretical methods,HF equations,LDA,SCF
Dirac’s comment on chemistry,1930
§11.1 Atoms and Ions
11.1.1 A Hydrogen Atom
bracketleftbigg?
r
parenleftbigg
r2r
parenrightbigg
+ 1sinθθ
parenleftbigg
sinθθ
parenrightbigg
+ 1sin2θ?
2
φ2 +
2mr2
planckover2pi12
parenleftbiggZe2
r +E
parenrightbiggbracketrightbigg
ψ = 0
(11.1.1)
ψnlm(r,θ,φ) = Rnl(r)Ylm(θ,φ) (11.1.2)
En =? e
4
2planckover2pi12
1
n2,n = 1,2,..,(11.1.3)
n?1summationdisplay
l=0
lsummationdisplay
m=?l
1 =
n?1summationdisplay
l=0
(2l+ 1) = n2 (11.1.4)
x
y
z
r
θ
φ
Figure 11.1.1 The coordinate system of a hydrogen atom.
s
px pz py
x y
z
p
dz2 dx2-y2 dzx dxy dyz
d
Figure 11.1.2 The three-dimensional graphs for s,p,d orbitals.
Figure 11.1.2 The three-dimensional graphs for f orbitals.
11.1.2 Single-Electron Approximation for Many-Electron
Atoms
planckover2pi122m
Zsummationdisplay
i=1
2i?
Zsummationdisplay
i=1
Ze2
ri +
1
2
summationdisplay
inegationslash=j
e2
|ri?rj|
Ψ = EΨ (11.1.5)
one-electron Schr¨odinger equation
bracketleftbigg
planckover2pi1
2
2m?
2 +v(r)
bracketrightbigg
ψ = εψ (11.1.6)
3
1
6
4
5
7
2
s p d f
1s
2s
3s
4s
5s
6s
7s
2p
3p
3d
4p
5p
4d
5f
6p 5d4f
6d
7p
E
1s
2s
3s
4s
5s
6s
7s
2p
3p
3d
4p
5p
4d
5f
6p
5d
4f
6d7p
Figure 11.1.3 Energy levels of electron shells in many-electron atoms.
E
Z
0 20 40 60 80
1s
2s
2p
3s
3d 3p
4f
4p 4d4s
5d
5s
5p
6s6s
5d
5p
5s
4d
4f
4p
3d
3s
3p
4s
2p
2s
1s
Figure 11.1.4 Ordering of the energy sub-shell in atoms versus atomic number.
HeBeC O
3s
2p
2s
M Shell
L Shell
1s
K Shell
10
102
103
Z
H Li B N FNe P Ca Mn Zn Br
4s3d
3p
N Shell100
5 10 15 20 25 30
4s 4p
10-1
E
(R
y)
Figure 11.1.5 Electron binding energies versus atomic number for different shells.
11.1.3 Intraatomic Exchange
ψ(x,y,z,sz) = φ(x,y,z)χ(sz) (11.1.7)
Ψss =

χ11 = α(1)α(2),
χ10 = 1√2[α(1)β(2) +α(2)β(1)]
χ1?1 = β(1)β(2)
(11.1.8)
Ψsa = χ00 = 1√2[α(1)β(2)?α(2)β(1)] (11.1.9)
total antisymmetric wave function
Ψ = [φ0(r1)φnl(r2)?φ0(r2)φnl(r1)]

χ11
χ10
χ1?1

(11.1.10)
Ψ = [φ0(r1)φnl(r2) +φ0(r2)φnl(r1)]χ00 (11.1.11)
χ1,1
χ0,0
χ1,0
χ1,-1
Sz
1
0
0
-1
Triplet
Singlet g40g32g32g32g32g32?g32g32g32g32g32g4121
g40g32g32g32g32g32 +g32g32g32g32g32g4121{
Figure 11.1.6 Combined wave functions for spins of two electrons.
11.1.4 Hund’s Rules and Magnetic Moments in Ions
effective magnetic moments for an atom or ion
μeff = gL[J(J + 1)]1/2μB = pμB (11.1.12)
μB = eplanckover2pi12mc (11.1.13)
Table 11.1.1 Calculated and measured effective Bohr magneton numbers p of triva-
lent rare earth ions
Element Electronic State to Hund’s
Rule
f-shell(l = 3)nd = d-
electron number
S L = |Σ| J Ground State Term
nd ml =3,2,1,0,-1,-2,-3
Ce3+ 1 ↑ 1/2 3 5/2
J = |L?S|
2F5/2
Pr3+ 2 ↑ ↑ 1 5 4 2H4
Nd3+ 3 ↑ ↑ ↑ 3/2 6 9/2 4I9/2
Pm3+ 4 ↑ ↑ ↑ ↑ 2 6 4 5I4
Sm3+ 5 ↑ ↑ ↑ ↑ ↑ 5/2 5 5/2 6H5/2
Eu3+ 6 ↑ ↑ ↑ ↑ ↑ ↑ 3 3 0 7F0
Gd3+ 7 ↑ ↑ ↑ ↑ ↑ ↑ ↑ 7/2 0 7/2 8S7/2
Tb3+ 8 ↓↑ ↑ ↑ ↑ ↑ ↑ ↑ 3 3 6 7F6
Dy3+ 9 ↓↑ ↓↑ ↑ ↑ ↑ ↑ ↑ 5/2 5 5/2 6H15/2
Table 11.1.2 Calculated and measured effective number of Bohr magnetons p for 3d
transition metal ions
element Electronic State to Hund’s Rule Calculated
d-electron number S L = |Σ| J Ground State Term J = S
n ml =2,1,0,-1,-2
Ti3+ 1 ↑ 1/2 2 3/2
J = |L?S|
2D3/2 1.73
V4+ 2D3/2 1.73
V3+ 2 ↑ ↑ 1 3 2 3F2 2.83
V2+
3 ↑ ↑ ↑ 3/2 3 3/2
4F3/2 3.87
Cr3+ 4F3/2 3.87
Mn4+ 4F3/2 3.87
Cr2+ 4 ↑ ↑ ↑ ↑ 2 2 0 5D0 4.90
Mn3+ 5D0 4.90
Mn2+ 5 ↑ ↑ ↑ ↑ ↑ 5/2 0 5/2 6S5/2 5.92
Fe3+ 6S5/2 5.92
Fe2+ 6 ↓↑ ↑ ↑ ↑ ↑ 2 2 4 5D4 4.90
§11.2 Diatomic Molecules
11.2.1 The exact solution for hydrogen molecular ion H+2
planckover2pi1
2
2m?
2ψ?
parenleftbigge2
rA +
e2
rB
parenrightbigg
ψ = E(R)ψ (11.2.1)
ξ = rA +rBR (1 lessorequalslantξ <∞)
η = rA?rBR (?1 lessorequalslantηlessorequalslant +1) (11.2.2)
A B
R
P φ
Figure 11.2.1 The coordinate system for hydrogen molecular ion.
2planckover2pi1
2
mR2
bracketleftbigg 1
ξ2?η2
ξ
braceleftbigg
(ξ2?1)?ψ?ξ
bracerightbigg
+ 1ξ2?η2η
braceleftbigg
(1?η)2?ψ?ξ
bracerightbigg
+ 1(ξ2?1)(1?η2)?
2ψ
φ2
bracketrightbigg
4e
2ξ
R(ξ2?η2)ψ = E(R)ψ
(11.2.3)
ψ(ξ,η) = X(ξ)Y(η) (11.2.4)
d
dξ
braceleftbigg
(ξ2?1)dXdξ
bracerightbigg
+
parenleftbiggmR2E
2planckover2pi12 ξ
2 + mRe
2
planckover2pi12 ξ +A
parenrightbigg
X = 0 (11.2.5)
d
dη
braceleftbigg
(η2? 1)dYdη
bracerightbigg
+
parenleftbiggmR2E
2planckover2pi12 η
2 +A
parenrightbigg
Y = 0 (11.2.6)
U(R) = E(R) + e
2
R (11.2.7)
0 5 10 15-2.0
-1.5
-1.0
-0.5
0 4
3
2sσ
g50ppi
2pσ
g49sσ
E (a.u.)
R (a.u.)
Figure 11.2.2 Electronic energy levels versus internuclear distance.
0 2 4 6 8 10 12 14 16-
0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
2ppi
3dσ
2pσ
1sσ
R (a.u.)
U
(a.u.)
Figure 11.2.3 Total energy versus internuclear distance.
11.2.2 Molecular Orbital Method
Bonding is an important concept in quantum chemistry
Molecular Orbital Method,Hund and M¨ulliken,1931
ψ(?x,?y,?z) = ±ψ(x,y,z) (11.2.8)
Acommonlyadopted approximationprocedure islinearcombination ofatomic
orbitals (LCAO) to treat molecular problems,i.e.,
ψ(ri) = ψA(ri) ±ψB(ri) (11.2.9)
-2 -1 0 1 2
-2 -1 0 1 2
-2 -1 0 1 2
(a)
(b)
(c)
Figure 11.2.4 Ground-state wave functions of H+2 along the central line,(a) exact
result; (b) bonding LCAO; (c) antibonding LCAO.
0 1 2 3 4 5 6
-0.60
-0.55
-0.50
-0.45
-0.40
R (a.u.)
U
(a.u.)
Figure 11.2.5 Bonding energy versus internuclear distance for H+2,(a) exact result;
(b) bonding atomic orbital; (c) antibonding atomic orbital.
single-electron Schr¨odinger equation
Hiψi =
parenleftbigg
planckover2pi1
2
2m?
2
i?
e2
rAi?
e2
rBi
parenrightbigg
= εiψi (11.2.10)
ψi = Ni[cAψA(ri) +cBψB(ri)] (11.2.11)
E(ψ) =
integraltext ψ?Hψdr
iintegraltext
ψ?ψdri (11.2.12)
E
ci = 0 (i = A,B) (11.2.13)
(HAA?ESAA)cA + (HAB?ESAB) = 0 (11.2.14)
(HAA?ESAA)cA + (HAB?ESAB) = 0 (11.2.15)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
HAA?ESAA HAB?ESAB
HAB?ESAB HBB?ESBB
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle= 0 (11.2.16)
Coulomb integral
α =
integraldisplay
ψAHiψAdri =
integraldisplay
ψBHiψBdri (11.2.17)
Let HAB = HBA = β
β =
integraldisplay
ψBHiψAdri =
integraldisplay
ψAHiψBdri (11.2.18)
SAB = SBA =
integraldisplay
ψAψBdri (11.2.19)
ψ± = 1radicalbig2(1±S)(ψA ±ψB) (11.2.20)
E± = α±β1±S
AB
(11.2.21)
Bond MO
AntibondMO
1sB1sA
E
1sA 1sB
(a) (b)
Figure 11.2.6 (a) Energy diagram showing MO formation with (b) corresponding
pictorial representation of orbitals.
(a) σ Orbital (b) pi Orbital (c) δ Orbital
Figure 11.2.7 Symmetry of σ-,pi- and δ- orbitals viewing perpendicular to bonding
axis.
3σu
1pig
3pig
1piu
2p2p
2σu
2σg
2s 2s
1s 1s1σu
1σg
Figure 11.2.8 The occupancy of MOs in O2.
Figure 11.2.9 Orbitals and Bonding.
11.2.3 Heitlerand London’sTreatmentof Hydrogen Molecule
Valence bond method,Heitler and London,1927
For the first time to consider many-body interactions
Difficult to be extended to polyatomic molecules
H =? planckover2pi1
2
2m?
2
1?
planckover2pi12
2m?
2
2?
e2
rA1?
e2
rA2?
e2
rB1?
e2
rB2 +
e2
r12 +
e2
rAB (11.2.22)
The triplet (S = 1)
|Ψ1〉 = 1radicalbig2(1?S2
AB)
α(s1)α(s2)[ψA(r1)ψB(r2) +ψA(r2)ψB(r1)],Sz = 1
(11.2.23)
|Ψ2〉 = 1radicalbig2(1?S2
AB)
[α(s1)β(s2) +β(s1)α(s2)][ψA(r1)ψB(r2)?ψA(r2)ψB(r1)]
(11.2.24)
|Ψ3〉 = 1radicalbig2(1?S2
AB)
β(s1)β(s2)[ψA(r1)ψB(r2)?ψA(r2)ψB(r1)],Sz =?1
(11.2.25)
The singlet (S = 0)
|Ψ4〉 = 12radicalbig(1 +l2) [α(s1)β(s2)?β(s1)α(s2)][ψA(r1)ψB(r2) +ψA(r2)ψB(r1)]
(11.2.26)
SAB =
integraldisplay
ψ?A(r)ψB(r)dr (11.2.27)
Et = 〈Ψ1|H|Ψ1〉 = 2Eat + CAB?IAB1?S2
AB
(11.2.28)
Es = 〈Ψ4|H|Ψ4〉 = 2Eat + CAB +IAB1 +S2
AB
(11.2.29)
CAB =
integraldisplay
dr1
integraldisplay
dr2|ψA (r1)|2 e
2
|r1?r2||ψB (r2)|
2 (11.2.30)
integraldisplay
dr1 e
2
|r1?RB||ψA (r1)|
2?
integraldisplay
dr2 e
2
|r2?RA||ψB (r2)|
2
IAB =
integraldisplay
dr1
integraldisplay
dr2ψ?A (r1)ψB (r1) e
2
|r1?r2|ψ
B (r1)ψA (r2)
SAB
integraldisplay
dr1 e
2
|r1?RB|ψ
A (r1)ψB (r1)
SAB
integraldisplay
dr2 e
2
|r2?RA|ψ
B (r2)ψB (r2) (11.2.31)
J12 = Es?Et =?2S
2
ABCAB?IAB
1?S4AB (11.2.32)
11.2.4 The Spin Hamiltonian and the Heisenberg Model
Combining electronic spins and local interactions with bond method
magnetic interactions between neighboring ions obtained
S2i = 12(12 + 1) = 34 (11.2.33)
S2 = (S1 +S2)2 = 32 + 2S1 ·S2 (11.2.34)
Hs = 14(Es + 3Et)? (Es?Et)S1 ·S2 (11.2.35)
Hs =?J12S1 ·S2 (11.2.36)
Direct exchange interaction
Hs =?
summationdisplay
i<j
JijSi ·Sj (11.2.37)
Problem,Combining §11.2.3 and §11.2.4 to derive
the Heisenberg Hamiltonian (11.2.37) detailedly.
Except the direct exchange,there are superexchange interaction
and double exchange interaction
§11.3 Polyatomic Molecules
11.3.1 Molecular Orbital Method for Polyatomic Molecules
Ψ = 1√N!
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
ψ1(1) ψ1(2),.,ψ1(N)
ψ2(1) ψ2(2),.,ψ2(N)
...,..,..,..
ψN(1) ψN(2),.,ψN(N)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(11.3.1)
11.3.2 Valence Bond Orbitals
Hybrid orbitals
(1) sp hybridization
ψ1 = 1√2(s+px),ψ1 = 1√2(s?px)
(2) sp2 hybridization
ψ1 = 1√3s+
radicalbigg2
3px,ψ2 =
1√
3s?
1√
6px+
1√
2py,ψ3 =
1√
3s?
1√
6px?
1√
2py
(3) sp3 hybridization
ψ1 = 12(s+px +py +pz),ψ2 = 12(s+px?py?pz)
ψ3 = 12(s?px +py?pz),ψ4 = 12(s?px?py +pz) (11.3.2)
E = EA +EB +EC + ···+GAB +GAC +GBC + ··· (11.3.3)
GAB ≈ e2
integraldisplay ρ
A(1)ρB(2)
r12 dr1dr2 (11.3.4)
x xx
yy
z
(a) (b) (c)
Figure 11.3.1 Hybridized orbitals for s,p electrons (a) sp orbitals,(b) sp2 orbitals,
(c) sp3 orbitals.
11.3.3 H¨uckel Approximation for Molecular Orbital Method
parenleftBigg
h
2
2m?
2?summationdisplay
i
e2
ri
parenrightBigg
ψ = Eψ (11.3.5)
ψ =
Nsummationdisplay
i=1
ciφi (11.3.6)
(H11?E)C1 + (H12?ES12)C2 +··· + (H1N?ES12)CN = 0
............
(HN1?ESN1)C1 + (HN2?ESN2)C2 +··· + (HNN?E)CN = 0
(11.3.7)
H¨uckel,1931
In H¨uckel’s treatment,further simplifications
(1) The Coulomb integral Hrr = α
(2) The resonance integral Hrs = β
(3) The overlap between neighboring atoms is ignored Srs = 0 when r negationslash= s
α?E
β =?ε,or E = α+βε (11.3.8)
εc1 +c2 +cN = 0
............
cm?1?εcm +cm+1 = 0
............
c1 +cN?1?εcN = 0
(11.3.9)
cm?1?εcm +cm+1 = 0
c0 = cN,cN+1 = c1 (11.3.10)
ckm = ck exp(2piimkN ) (11.3.11)
εk = 2cos
parenleftbigg2kpi
N
parenrightbigg
(11.3.12)
Ek = α+ 2βcos
parenleftbigg2pik
N
parenrightbigg
(11.3.13)
N=1 N=∞N=3 N=4 N=5 N=6
ε=α 4β
Antibonding
Bonding
Nonbonding}
}
Figure 11.3.2 Conjugated rings and their energy level diagrams,Note when N
increases,discrete levels are transformed into a continuous band.
k = 0,±1,...,

±(N? 1)/2,for N = odd
±N/2,for N = even
(11.3.14)
11.3.4 Electronic Structure of Some Molecules
(1) Water (H2O)
Fig,11.3.3
(2) Aromatics
Fig,11.3.4-6
(3) Fullerene C
Kroto,Smalley and Curl,1985
Fig,11.3.7
p
s
b1,b2
h,h'
2b2
4a1
1b1
3a1
1b2
2a1
φa1
φb1
a2b(H1s)
Ha,HbH2OO
Figure 11.3.3 Energy level diagram of H2O.
H
HH
H
H H
C
C
C
C
C
C
H
H
H
H
H
H
C
C
C
C
C
C
H
H
H
H
H
H
(a) (b)
Figure 11.3.4 Structure of Benzene (a) Kekul′e structures; (b) Electron density for
delocalized pi electrons of Benzene.
255 nm
315 nm
380 nm
480 nm
580 nm
Benzene
Naphthalin
Anthracene
Tetracene
Pentacene
Figure 11.3.5 Structure and optical absorption of Benzene and other molecules with
planar region of fused rings (polyacenes).
E=α
HOMO
LUMO
(a) (b)
Figure 11.3.6 The molecular orbital energy levels of naphthalene,(a) calculated
by H¨uckel approximation; (b) calculated by H¨uckel approximation with additional
overlap correctional (l = 0.25).
(a) (b)
Figure 11.3.7 The structures of a C60 and C70 molecules.
E
LUMO
HOMO
Figure 11.3.7 Energy levels of C60 from HMO calculation.
Figure 11.3.8 Nano carbon tube.
A
a1^ a2^
ch A' θ
h
O
a2^
a1^(1.0) (2.0) (3.0) (4.0) (5.0) (6.0)
(7.2)
(2.1) (3.1) (4.1) (5.1) (6.1) (7.1)
(4.2) (5.2) (6.2)
(7.0)
(8.2)
(6.3) (7.3) (8.3)
(8.4) (9.4a)
(a) (b)
Figure 11.3.9 Nano carbon tube.
§11.4 Ions in Anisotropic Environment
Crystal field,also called ligand field
Effect from electrostatic field of symmetrically arranged neighboring atoms
on a substitutional atom or an atom of the host lattice
mainly on 3d or 4f electrons in orbitals not fully occupied
11.4.1 Three Types of Crystal Fields
Total hamiltonian of the electrons in a magnetic ion
H = H0 + Hee + HLS +HCF (11.4.1)
Principal contribution
H0 =
summationdisplay
i
bracketleftbigg
planckover2pi1
2
2m?
2? Ze
2
ri
bracketrightbigg
(11.4.2)
Electron-electron interaction
Hee = 12
summationdisplay
inegationslash=j
e2
rij (11.4.3)
Spin-orbital interaction
HLS =
summationdisplay
i
HLS(i) (11.4.4)
Crystal contribution
HCF =
summationdisplay
i
eV(ri) (11.4.5)
with
VCF = C(x4 +y4 +z4? 3r4/5) (11.4.6)
C is a numerical constant
(1) strong crystal fields,HCF >Hee >HLS
(2) medium crystal fields,Hee >HCF > HLS
(3) weak crystal fields,Hee > HLS >HCF
11.4.2 Transition metal ions in crystal fields
Fig,11.4.1
H3d(r) = H(Ti)(r) +
6summationdisplay
j=1
V(O)(r?Rj) (11.4.7)
Fig,11.4.2
Figure 11.4.1 Local environment of a transition metal ion at the center of the oxygen
octahedron.
dxy,dyz,dzx dx2-y2 dz2 (a) (b) (c)
Figure 11.4.2 3d orbitals of a center ion in the octahedral environment of O anions.
11.4.3 Jahn-Teller Effect
Jahn and Teller,1937
Jahn-Teller distortion originally described
the instabilityofsymmetricmolecules inthe presence ofelectronic degeneracy
Extended to solids,the cooperative Jahn-Teller effect is a phase transition
involving the simultaneous splitting of the electronic states and
a symmetry-lowering distortion of the lattice
Assume a correction to the Born-Oppenheimer energy in distortion Q
If a term linear in Q exists,then an instability will result
a Jahn-Teller contribution to the hamiltonian
HJT = AQSz (11.4.8)
Sz is an electronic operator have eigenvalues ±1
The constant A measures the strength of the coupling.
Elastic energy
(1/2)kQ2
There are two minima for the total energy
Two minima has the depth
EJT =?A2/2k (11.4.9)
at Q = ±Q0,where
Q0 = A/k (11.4.10)
Fig,11.4.3
Fig,11.4.4
Figure 11.4.3 Energy diagram for Jahn-Teller effect.
δz
Figure 11.4.4 Jahn-Teller distortion of an oxide octahedron.
d
t2g
eg
E0
dx2-y2
dx2
dxy
dxx,dyz
Figure 11.4.5 Energy levels in octahedral crystal field.
11.4.4 Ions in Ligand Fields
Fig,11.4.5
Table 11.4.1
Fig,11.4.6
Fig,11.4.7-8
ML6
t1u
a1g?
eg?
t2g
eg
t1u
a1g
σ
6LM
4p
4s
3d D0
Figure 11.4.6 The energy levels for the MOs of octahedral transition metal com-
plexes.
t2g
Lower Spin State
Higher Spin State
1a
1g
S=0
5t
2g
S=2
t2gTemperaturePressure
Under Light
eg
eg
Figure 11.4.7 The electronic spin transition of Fe2+ ion in octahedral conjugated
molecules.
Table 11.4.1 High-spin and low-spin complexes with octahedral structures
number of d- electrons high-spin state low-spin state
t2g eg n p t2g eg n p
1 —↑ — — — — 1 1.73
2 —↑ —↑ — — — 2 2.83
3 —↑ —↑ —↑ — — 3 3.87
4 —↑ —↑ —↑ —↑ — 4 4.90 —↑↓ —↑ —↑ — — 2 2.83
5 —↑ —↑ —↑ —↑ —↑ 5 5.92 —↑↓ —↑↓ —↑ — — 1 1.73
6 —↑↓ —↑ —↑ —↑ —↑ 4 4.90 —↑↓ —↑↓ —↑↓ — — 0 0
7 —↑↓ —↑↓ —↑ —↑ —↑ 3 3.87 —↑↓ —↑↓ —↑↓ —↑ — 1 1.73
8 —↑↓ —↑↓ —↑↓ —↑ —↑ 2 2.83
9 —↑↓ —↑↓ —↑↓ —↑↓ —↑ 1 1.73
Fe
Figure 11.4.8 The structure of ferrocene.
e1g,e1u
a1g,a2u
e1g*
e1u*
a1g*
a2u*
e2g
a1g
e1g
e1u
a1g
a2u
4p
4s
3d
Figure 11.4.9 The energy levels of ferrocene.
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
April 1,2004
Contents
Chapter 12,Band Approach 3
§12.1 Different Ways to Calculate the Energy Bands,,,,,,,5
12.1.1 Orthogonized Plane Waves,,,,,,,,,,,,,,,6
12.1.2 Pseudopotential,,,,,,,,,,,,,,,,,,,,,9
12.1.3 The Muffin-Tin Potential and Augmented Plane Waves 12
12.1.4 Symmetry of Energy Bands and k·p Method,,,,,14
§12.2 From Many-Particle Hamiltonian to Self-Consistent Field Approach
12.2.1 Many-Particle Hamiltonians,,,,,,,,,,,,,,17
12.2.2 Valence Electron and Adiabatic Approximations,,,19
12.2.3 Hartree Approximation,,,,,,,,,,,,,,,,,22
12.2.4 Hartree-Fock Approximation,,,,,,,,,,,,,,25
§12.3 Electronic Structure via Density Functionals,,,,,,,,28
12.3.1 From Wavefunctions to Density Functionals,,,,,,29
12.3.2 Hohenberg-Kohn Theorems,,,,,,,,,,,,,,,31
12.3.3 Self-Consistent Kohn-Sham Equations,,,,,,,,,37
12.3.4 Local Density Approximation and Beyond,,,,,,,41
12.3.5 Car-Parinello Method,,,,,,,,,,,,,,,,,,45
§12.4 Electronic Structure of Selected Materials,,,,,,,,,48
12.4.1 Metals,,,,,,,,,,,,,,,,,,,,,,,,,,48
12.4.2 Semiconductors,,,,,,,,,,,,,,,,,,,,,58
12.4.3 Semimetals,,,,,,,,,,,,,,,,,,,,,,,64
12.4.4 Molecular Crystals,,,,,,,,,,,,,,,,,,,64
12.4.5 Surfaces and Interfaces,,,,,,,,,,,,,,,,,70
Chapter 12
Band Approach
One of the most important aspects in condensed matter physics
is the electronic properties in many-particle systems
Energy band theory based on single electron approximation
§12.1 Different Ways to Calculate the Energy Bands
Free electron model
Near-free electron model
Tight-binding electron model
Fundamental problems are what potential and base functions chosen
12.1.1 Orthogonized Plane Waves
Plane waves
|k+G〉 =1/2ei(k+G)·r (12.1.1)
ψk(r) =1/2
summationdisplay
G
c(G)ei(k+G)·r (12.1.2)
Secular equations
summationdisplay
Gprime
braceleftbiggbracketleftbigg planckover2pi12
2m(k+G)
2?E(k)
bracketrightbigg
δGGprime +V(G?Gprime)
bracerightbigg
c(Gprime) = 0 (12.1.3)
Fourier coefficients
V(G?Gprime) = 1?
integraldisplay
drV(r)e?i(G?Gprime)·r (12.1.4)
(a)
(b)
(c)
Figure 12.1.1 (a) Plane wave; (b) Core state; and (c) Orthogonal plane wave state.
Determinant equation
vextendsinglevextendsingle
vextendsinglevextendsingle
bracketleftbigg planckover2pi12
2m(k+G)
2?E(k)
bracketrightbigg
δGGprime +V(G?Gprime)
vextendsinglevextendsingle
vextendsinglevextendsingle = 0 (12.1.5)
Diagonalizing → eigenvalues E(k) and coefficients c(G)
Orthogonalized plane wave (OPW) corresponding to k
φk(r) =1/2eik·r?
summationdisplay
j
μjkφjk(r) (12.1.6)
φjk(r) = 1√N
summationdisplay
l
eik·Rlχj(r?Rl) (12.1.7)
ψk(r) =
summationdisplay
G
cGφk+G(r) (12.1.8)
12.1.2 Pseudopotential
Eigenfunction of the Schr¨odinger equation as
ψk(r) = ψpsk?
summationdisplay
j
μjφj (12.1.9)
Fig,12.1.2(a)
parenleftbigg
planckover2pi1
2
2m?
2 +V
parenrightbigg
ψk = E(k)ψk (12.1.10)
parenleftbigg
planckover2pi1
2
2m?
2 +V
parenrightbigg
φj = E(j)φj (12.1.11)
(a)
(b)
Figure 12.1.2 The pseudopotential concept,(a) The actual potential and corre-
sponding wavefunction; (b) The pseudopotential and corresponding pseudofunction.
parenleftbigg
planckover2pi1
2
2m?
2 +Vps
parenrightbigg
ψpsk = E(k)ψpsk (12.1.12)
Vps = V +
summationdisplay
j
(E(k)?Ej)|φj〉〈φj| (12.1.13)
Fig,12.1.2(b)
ψps(r) =
summationdisplay
c(G)ei(k+G)·r (12.1.14)
12.1.3 TheMuffin-Tin Potentialand AugmentedPlane Waves
V (r) =

Va (r),r<rc
0,rgreaterorequalslantrc
(12.1.15)
wk =

atomic function,r<rc
1/2eik·r,rgreaterorequalslantrc
(12.1.16)
φk =
summationdisplay
G
a(k+G)wk+G (12.1.17)
ψk(r) =
summationdisplay
G
c(G)φk+G(r) (12.1.18)
Figure 12.1.3 A schematic diagram for the muffin-tin potential.
12.1.4 Symmetry of Energy Bands and k·p Method
bracketleftbiggp2
2m +
planckover2pi1
mk·p+
planckover2pi12k2
2m +V (r)
bracketrightbigg
unk (r) = E(k)unk (r) (12.1.19)
unk (r) =
summationdisplay
nprime
Cnprimenunprime0 (r) (12.1.20)
summationdisplay
nprime
braceleftbiggbracketleftbigg
En(0) + planckover2pi1
2k2
2m
bracketrightbigg
δnprimen + planckover2pi1mk·pnprimen
bracerightbigg
Cnprimen = En(k)Cnn (12.1.21)
pnprimen =
integraldisplay
u?n0 (r)punprime0 (r)dr (12.1.22)
R
ST
Z
M
XΓ
Λ
Σ
Σ
Σ
Γ
Λ

M
L
U
S
XZW
K
Q
Λ
Γ
P
F
H
N T K
P
HSA
I
L
x
y
z
x
y
z
x
y
z
z
x
(a) (b)
(c) (d)
Figure 12.1.4 Symmetry points in the Brillouin zones for different real lattices,(a)
sc; (b) fcc; (c) bcc; (d) sh.
§12.2 FromMany-ParticleHamiltonian toSelf-ConsistentField
Approach
Condensed matter composed of a large number of
electrons and ions
From many particles to single particle
From individual particles to many particles
Interaction between particles
Simplification approaches,Effective potential and Perturbation
12.2.1 Many-Particle Hamiltonians
Schr¨odinger equation for many particles
iplanckover2pi1tΨ = HΨ (12.2.1)
H =
summationdisplay
i
p2i
2m+
summationdisplay
α
P2α
2Mα+
1
2
summationdisplay
inegationslash=j
e2
|ri?rj|+
1
2
summationdisplay
αnegationslash=β
ZαZβe2
|Rα?Rβ|?
summationdisplay
iα
Zαe2
|ri?Rα|
(12.2.2)
Ψ = Ψ(r1,r2,···,rN)
For a set of noninteracting particles
H =
summationdisplay
i
Hi =
summationdisplay
i
bracketleftbigg
planckover2pi1
2
2m?
2
i +v(ri)
bracketrightbigg
(12.2.3)
bracketleftbigg
planckover2pi1
2
2m?
2 +v(r)
bracketrightbigg
ψn = εnψn (12.2.4)
12.2.2 Valence Electron and Adiabatic Approximations
Core and valence electrons
Valence electrons are more important to physical properties
Valence electron approximation
H =? planckover2pi1
2
2m
summationdisplay
i
2i? planckover2pi1
2
2M
summationdisplay
α
2α + 12
summationdisplay
inegationslash=j
e2
|ri?rj| +
summationdisplay
iα
v(ri,Rα)
+ 12
summationdisplay
αnegationslash=β
v(Rα,Rβ) (12.2.5)
Screening potential v(ri,Rα),and short range potential v(Rα,Rβ)
This approximation may fail for transition metals and rear earth materials
Adiabatic approximation also called Born-Oppenheimer approximation
Separation of electronic system and ionic system
Quantum mechanical theory was developed by Born and Oppenheimer,1927
an error less than a small parameter (m/M)1/4
Identical interacting ionic system
H =? planckover2pi1
2
2M
summationdisplay
α
2α + 12
summationdisplay
αnegationslash=β
v(Rα,Rβ) +
summationdisplay
α
ve(Rα) (12.2.6)
Identical interacting electronic system
H =? planckover2pi1
2
2m
summationdisplay
i
2i + 12
summationdisplay
inegationslash=j
e2
|ri?rj| +
summationdisplay
i
v(ri) (12.2.7)
Schr¨odinger equation for electrons
planckover2pi122msummationdisplay
i
2i +
summationdisplay
i
v(ri) + 12
summationdisplay
inegationslash=j
e2
|ri?rj|
Ψ(r1,···,rN)
= EΨ(r1,···,rN) (12.2.8)
Difficulty comes from Coulomb interactions,needs further approximation
12.2.3 Hartree Approximation
Neglecting antisymmetry for wavefunction
Ψ(r1,···,rN) =
Nproductdisplay
i=1
ψi(ri) (12.2.9)
Hartree variational calculation,energy minimum → ground state
E = 〈Ψ|H|Ψ〉〈Ψ|Ψ〉 (12.2.10)
From integraltext |ψi|2dri = 1,use Lagrange multiple εi as variation parameter
A set of Hartree equations?
planckover2pi122m?2 +v(r) +summationdisplay
j
primee2integraldisplay ψ?j(rprime)ψj(rprime)drprime
|r?rprime|
ψi(r) = εiψi(r) (12.2.11)
Effective potential
veff = v(r) +
summationdisplay
j
primee2integraldisplay ψ?j(rprime)ψj(rprime)drprime
|r?rprime| (12.2.12)
Self-consistent solution for (12.2.11-12)
Ground-state energy
E = 〈Ψ|H|Ψ〉〈Ψ|Ψ〉 =
summationdisplay
i
εi? 12
summationdisplay
inegationslash=j
e2
integraldisplayintegraldisplay ψ?
j(rprime)ψj(rprime)ψ?i(r)ψi(r)
|r?rprime| drdr
prime
(12.2.13)
ψi(r) = 1√?eik·r (12.2.14)
veff(r) = 2e
2
kFsummationdisplay
k=0
integraldisplay drprime
|r?rprime|?
e2
integraldisplay drprime
|r?rprime| (12.2.15)
e2
integraldisplay drprime
rprime =
4pie2
integraldisplay
rprimedrprime (12.2.16)
This term → 0 when r → ∞
bracketleftbigg planckover2pi12
2m?
2 +E(k)
bracketrightbigg
ψk(r) = 0 (12.2.17)
12.2.4 Hartree-Fock Approximation
Fermions,Pauli principle,a Slater determinant
Ψ ({ri}) = 1√N!
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
ψ1(r1) ··· ψ1(rN)
··· ··· ···
ψN(r1) ··· ψN(rN)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
(12.2.18)
Variational calculation → a set of Hartree-Fock equations?
planckover2pi122m?2 +v(r) +summationdisplay
j
primee2integraldisplay ψ?j(rprime)ψj(rprime)
|r?rprime| dr
prime
ψi(r)
summationdisplay
j
prime
bracketleftbigg
e2
integraldisplay ψ?
j(rprime)ψi(rprime)
|r?rprime| dr
prime
bracketrightbigg
ψj(r) = εiψi(r) (12.2.19)
Self-consistent calculations can be going on
Ground-state energy is like(12.2.13),but subtracted the exchange interaction
from parallel spins
e2
2
summationdisplay
inegationslash=j
integraldisplayintegraldisplay ψ?
i(r)ψj(r)ψ?j(rprime)ψi(rprime)
|r?rprime| drdr
prime (12.2.20)
The drawback of HF approximation,Correlation energy was not considered
Recently,extending version is nonconstraint SCF method
Problem:
To verify the Hartree-Fock equations (12.2.19)
by variational calculation
and get the expression of ground-state energy.
Koopmans theorem,1934
To confirm single-electron property,we should know the meaning of εi
If an electron was emitted from a N electron system
By assuming the one-electron wavefunctions are the same for N and N?1
states
The energy change is
= E(N)?E(N?1) = εi
and the ionization energy is?εi
When an electron transfers from one state to another
Eij = εi?εj
§12.3 Electronic Structure via Density Functionals
The theoretical basis for modern calculation of electronic structure
First for ground state,then extended to finite temperature
and also for magnetic systems,and even relativistic correction
12.3.1 From Wavefunctions to Density Functionals
In 1927,Thomas-Fermi (TF) theory
Total energy functional
E[n] =
integraldisplay
drv(r)n(r)+e
2
2
integraldisplay
drdrprimen(r)n(r
prime)
|r?rprime| +
3planckover2pi12
10m(3pi
2)2/3
integraldisplay
drn5/3(r)
For local homogeneity
EK = 35EF = 35 planckover2pi1
2
2m(3pi
2n)2/3
Variational treatment
δ
braceleftbigg
E[n]?μ
integraldisplay
drn(r)
bracerightbigg
= 0
Integral equation
v(r) +e2
integraldisplay
drprime n(r
prime)
|r?rprime| +
planckover2pi12
2m(3pi
2)2/3n2/3(r) = μ
To define an effective potential
veff (r) = v(r) +e2
integraldisplay n(rprime)
|r?rprime|dr
prime (12.3.1)
n(r) = γ[μ?veff (r)]3/2 (12.3.2)
γ = 13pi2
parenleftbigg2m
planckover2pi12
parenrightbigg3/2
12.3.2 Hohenberg-Kohn Theorems
H = T +U +V (12.3.3)
n(r) = N
integraldisplay
|ΨG(r,r2,···,rN)|2dr2···rN (12.3.4)
V =
Nsummationdisplay
i=1
v(ri) =
integraldisplay
Ψ?(r)v(r)Ψ(r)dr =
integraldisplay
v(r)n(r)dr (12.3.5)
n(r) = 〈G|Ψ?(r)Ψ(r)|G〉 (12.3.6)
P,Hohenberg and W,Kohn,1964,Phys,Rev,136,B 864
Theorem 1,The ground-state density n(r) of a bound system of inter-
acting electrons in some external potential v(r) determines this potential
uniquely,The term ‘uniquely’ means here,up to an uninteresting additive
constant.
That is to say,In the ground state,the external potential v(r) is
the unique functional of the electronic density n(r)
If v(r) is known
v(r) → V → solve Schr¨odiger equation → |G〉 → n(r)
If there is a different vprime(r) → |Gprime〉 → nprime(r)
but n = nprime,then it must be v(r) = vprime(r)
The proof,letn(r) be the nondegenerate ground-state density ofN electrons
in the potential v1(r),the ground state wavefunction is Ψ1 with the energy
E1,then
E1 = 〈Ψ1|H1|Ψ1〉
=
integraldisplay
v1(r)n(r)dr+〈Ψ1|T +U|Ψ1〉 (12.3.7)
similarly
E2 =
integraldisplay
v2(r)n(r)dr+〈Ψ2|T +U|Ψ2〉 (12.3.8)
General variational principle of quantum mechanics (Rayleigh-Ritz variation
principle)
Ground-state energy is the functional minimum of wavefunction ΨG
E1 < 〈Ψ2|H1|Ψ2〉
=
integraldisplay
v1(r)n(r)dr+〈Ψ2|T +U|Ψ2〉
= E2 +
integraldisplay
[v1(r)?v2(r)]n(r)dr (12.3.9)
E2 <〈Ψ1|H2|Ψ1〉 = E1 +
integraldisplay
[v2(r)?v1(r)]n(r)dr (12.3.10)
E1 +E2 <E1 +E2 (12.3.11)
Theorem 2,The ground-state energy E can be obtained through the
variation of trial densities n(r) instead of trial wavefunctions Ψ
This is called Hohenberg-Kohn variational principle
also from general variational principle
To consider the ground state and write its energy
EG[n] = F[n(r)] +
integraldisplay
n(r)v(r)dr (12.3.12)
with
F[n(r)] = 〈G|T +U|G〉 (12.3.13)
It is to show that EG[n] has its minimum when n(r) is the correct ground
state density,To assume that the density is varied but satisfies
N =
integraldisplay
n(r)dr (12.3.14)
Suppose Ψ is the correct ground state with v and corresponding n
and Ψprime is a different wavefunction with vprime and corresponding nprime
EG[Ψprime] = 〈Gprime|T +U|Gprime〉+ 〈Gprime|V|Gprime〉
= F[nprime] +
integraldisplay
vprime(r)nprime(r)dr
> EG[Ψ] = F[n] +
integraldisplay
v(r)n(r)dr (12.3.15)
So,EG[n] is a minimum relative to other density function associated with
some different vprime
12.3.3 Self-Consistent Kohn-Sham Equations
W,Kohn and L,J,Sham,1965,Phys,Rev,140,A 1133
They are Effective Schr¨odinger equations for single-particle functions
Write
F[n] = T [n] + e
2
2
integraldisplayintegraldisplay n(r)n(rprime)
|r?rprime| drdr
prime +Exc[n] (12.3.16)
Exc[n] is the exchange and correlation energy,its exact form is unknown
Take variation of (12.3.12) with μ as the Lagrange multiple
integraldisplay
δn(r)
bracketleftbiggδT [n]
δn(r) +v(r) +e
2
integraldisplay n(rprime)
|r?rprime|dr
prime + δExc[n]
δn(r)?μ
bracketrightbigg
dr = 0
(12.3.17)
δT [n]
δn(r) +v(r) +e
2
integraldisplay n(rprime)
|r?rprime|dr
prime + δExc[n]
δn(r) = μ (12.3.18)
veff(r) = v(r) +e2
integraldisplay n(rprime)
|r?rprime|dr
prime +vxc(r) (12.3.19)
vxc(r) = δExc[n]/δn(r) (12.3.20)
Assume a set of N single electron function satisfying
n(r) =
Nsummationdisplay
i=1
|ψi(r)|2 (12.3.21)
The kinetic energy functional can be written as
Ts[n] = planckover2pi1
2
2m
Nsummationdisplay
i=1
integraldisplay
ψ?i(r)·?ψi(r)dr = planckover2pi1
2
2m
Nsummationdisplay
i=1
integraldisplay
ψ?i(r)· (2)ψi(r)dr
(12.3.22)
The difference between Ts[n] and T can be absorbed into Exc[n]
Use the variation to ψi instead of the variation to n
also the Lagrange multiple μ is replaced by εi
δ
δψi
braceleftBigg
E[n(r)]?
Nsummationdisplay
i=1
εi
bracketleftbiggintegraldisplay
drψ?i(r)ψi(r)?1
bracketrightbiggbracerightBigg
= 0
A set of eigen equations
bracketleftbigg
planckover2pi1
2
2m?
2 +veff(r)
bracketrightbigg
ψi(r) = εiψi(r) (12.3.23)
The ground-state energy may be constructed
EG =
Nsummationdisplay
i=1
εi?e
2
2
integraldisplay n(r)n(rprime)
|r?rprime| drdr
prime+Exc[n]?
integraldisplay
n(r)vxc(r)dr (12.3.24)
12.3.4 Local Density Approximation and Beyond
Generally the exchange and correlation energy functional Exc[n] is nonlocal
Kohn and Sham adopted a local density approximation (LDA)
W,Kohn,1996,Phys,Rev,Lett,76,3168
proved ‘nearsightedness’ principle:
The local static physical properties of a many-electron system at r are de-
pendent on the particles in neighborhood region of r,for example,within
the sphere with radius about λF (r),the local Fermi wavelength λF (r) ≡
bracketleftbig3pi2n(r)bracketrightbig?1/3,and are insensitive to the change of potential outside this
region
When the density varies slowly
Exc[n] ≈
integraldisplay
εxc[n(r)]dr (12.3.25)
vxc[n(r)] ≈ dεxc[n(r)]dn(r) ≡ μxc[n(r)] (12.3.26)
μxc[n(r)] is called the exchange and correlation chemical potential
A simplest form is for the homogeneous electron gas,correlation is neglected
εxc[n(r)] ≈?3e
2
2pi(3pi
2n(r))1/3n(r) (12.3.27)
vxc[n(r)] ≈?2e2
parenleftbigg3
pi
parenrightbigg1/3
n1/3(r) (12.3.28)
For considering the correlation,electronic pair correlation function
g(r,rprime) = 1n(r)n(rprime)
angbracketleftBigg
G
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
summationdisplay
inegationslash=j
δ(rprime?ri)δ(r?rj)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingleG
angbracketrightBigg
(12.3.29)
Exc(n) = e
2
2
integraldisplayintegraldisplay
drdrprimen(r)?g(r,r
prime)?1
|r?rprime| n(r
prime) (12.3.30)
g(r,rprime) =
integraldisplay 1
0
dλg(r,rprime,λ) (12.3.31)
e2
|r?rprime| →
λe2
|r?rprime|,v(r) →vλ(r) (12.3.32)
Extended to consider the case with spins
n =
summationdisplay
σ
nσ
Exc[nσ] and vxc[nσ]
Local spin density approximation (LSDA)
12.3.5 Car-Parinello Method
Molecular dynamics computer simulation (MD) based on Newton equation
to get equilibrium or nonequilibrium structures on atomic scale
Sometimes needs to consider quantum effects from electrons
by combining MD and DFT
L = T?V (12.3.33)
T = 12
summationdisplay
l
Ml ˙R2l + 12
summationdisplay
μi| ˙?i|2 (12.3.34)
˙?i = d?idt (12.3.35)
Equation of motion associated with Lagrangian
d
dt
L
i?
L
i = 0 (12.3.36)
σij = 1?
integraldisplay
i(r)?j (r)dr?δij = 0 (12.3.37)
L = 12
summationdisplay
i
μi| ˙?i(r)|2 + 12
summationdisplay
l
Ml ˙R2l +V [{?i},{Rl}]
+2
summationdisplay
ij
λij
parenleftbigg 1
integraldisplay
i(r)?j(r)dr?δij
parenrightbigg
(12.3.38)
μ¨?i =V
i
summationdisplay
λij?j (12.3.39)
Ml ¨Rl =V?R
l
(12.3.40)
§12.4 Electronic Structure of Selected Materials
12.4.1 Metals
(1) Simple Metals
(2) Noble Metals
(3) Transition Metals
(4) Lanthanides and Actinides
Fig,12.4.1-9
Figure 12.4.1 The Fermi surface of Cu.
10
5
0
-5
Γ X W L Γ K X
EF
E?
E F
(eV)
k
Figure 12.4.2 The band structure of Cu(full lines) compared with NFE approxima-
tion (broken lines).
Ti(hcp)
K
V(bcc)
HΓ
Cr(bcc)
Γ H
Fe(bcc)
Γ H
Co(fee)
Γ X
Ni(fee)
Γ X
Cu(fcc)
XΓ Γ
Figure 12.4.3 Parts of band structure of transition metals.
0.1 0.2 0.3 0.4 0.5 0.6 0.7
11
9
7
5
3
E (Ry)
g(E
)
Figure 12.4.4 Density of states of transition metal for fcc structure.
IA
IIA
IIIB
IVB
VB
VIB
VIIB VIIIB IB
IIB
Cs Rb
K SrCa
Ba
Se
Y
La Ti
Zr
Hf
Ta
W
Nb
Mo
V
Re Os
Ru Ir
Tc
Cr Fe
Rh
Co
Pb
Pt
Ni
Mn Ag
Cu
Au
Zn
Cd
Hg
40
80
120
160
200
240
U ( 4.19
,10
-3 J
,g-
1 )
Figure 12.4.5 Cohesive energy of transition metals.
-0.60
-0.40
-0.20
-0.00
3.00
4.00
5.00
E c (R
y)
R ws
(a.u.)
K
Ca
Se
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
Ga
Figure 12.4.6 Wigner-Seitz radiuses and cohesive energies of the transition metals
(curves are calculated with the LDA and dots are experimental values).
25'α
25'β
12α
12β
4'β
4'α
5β
2β5α
2α
32β
1β3α
1α
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Ec (Ry)
Γ Χ?
εF V4
α Spin
β Spin
1α
1β
1β
2'β
3β
3α
3α
2'α
1α1α
3α
4α
2α
1β 1β
3β
4β
2β
L K SΛ Γ Σ Χ
Figure 12.4.7 Calculated bands and densities of states for up and down spins (α
and β) for Ni.
0 1 2 3 4 5
5d 6s
5p
4f
α γ
r / a0
ψ
Figure 12.4.8 Schematic diagram for the radical distribution of orbits in Ce atom
(the arrows point at the Wigner-Seitz radii of α- and γ-phase).
Ac Th Pa U Np PuAm CmBk
α
δ
0 1 2 3 4 5 6 7 8
Wi
gner
-Seitz
Vo
lume
Figure 12.4.9 The variation of the Wigner-Seitz volumes of solid phases vs,the
number of f-electrons (from the data of α- and δ-phase of Pu).
12.4.2 Semiconductors
(1) Direct-Gap and Indirect-Gap
(2) Band Gaps
(3) Effective Masses
(4) Electron Density Distribution in Real Space
kk
E (eV)
E (eV)
4
0
-4
-8
-12
Si Ge
4
0
-4
-8
-12
L U,KΛ Γ? Χ Σ Γ
GaAs
L U,KΛ Γ? Χ Σ Γ
AlAs
Figure 12.4.10 Band structure of four common semiconductors Si,Ge,GaAs and
AlAs,The calculation excludes the spin-orbit interaction.
Table 12.4.1 Bandgaps of semiconductors,experimental values vs theoret-
ical ones
crystal the type of the Eg(eV) (experimental values) Eg(eV) (theoretical
energy gap? 300 K 0 K LDA? QP
diamond i 5.4 3.9 5.6
Si i 1.11 1.17 0.5 1.29
Ge i 0.66 0.744 < 0 0.75
αSn d 0.00 0.00
InSb d 0.17 0.23
InAs d 0.36 0.43
InP d 1.27 1.42
Ek
'
Figure 12.4.11 Schematic diagram for the band edges in direct-gap semiconductor.
24
16 12
8 4
2420
16 12
8
4
(Si)
Figure 12.4.12 Valence electron density distribution in Si.
GaAs
ZnSe
82 20 20 14
As
18
10
2
Ga
(a)
2 4 Zn 2
30
Se20
22 24
20
10
2
(b)
Figure 12.4.13 Valence electron density distribution,(a) GaAs; (b) ZnSe.
12.4.3 Semimetals
Graphite,As,Sb and Bi are semimetals
12.4.4 Molecular Crystals
(1) Monoatomic Crystals
(2) Diatomic Molecular Crystals
(3) H2 Crystal and Metallic Hydrogen
(4) C60 Solids
κ0 pi/α
E
EF
Figure 12.4.14 Schematic energy band picture for a semimetal.
Γ
E E
ky
M kx
K
K Γ Mk
(a) (b)
Figure 12.4.15 The band structure of single graphite layer,(a) Band structure
for the triangle MK in 3D representation; (b) Band structure for line KM in 2D
representation.
g(E)
E
EF
g(E) g(E)
E E
EF
EF
(a) (b) (c)
Figure 12.4.16 (a) The density of states vs,energy of a single layer of graphite; and
(b),(c) its modification by intercalation.
Melting Line
Insulated Liquid Phase
PPT?
Insulated Solid Phase
Metallic Phase
Infrared Active
Livermore
Experiment
Boundary of
Insulator-Metal
Thermalized Ionization
P
I
II III
T
Figure 12.4.17 Schematic phase diagram of H2.
Γ? Σ ΧΧ W L Γ
g(E
)
-0.5
0
0.5
1
1.5
2
E (eV)
4.0?3.0?2.0?1.0 0.0 1.0 2.0
g(E
)
(a) (b)
k
Figure 12.4.18 (a) Band Structure of Solid C60 showing the top of the valence band
and the bottom of the conduction band; (b) Densities of states for solid C60 and some
potassium fullerites.
12.4.5 Surfaces and Interfaces
(1) 7× 7 Cell of Reconstructed Si(111) Surface
(2) Metal-Semiconductor Interfaces
(3) Insulator-Semiconductor Interfaces
ψ(z) = Aexp(?qz)cos
parenleftBigpiz
a +?
parenrightBig
,(z > 0) (12.4.1)
Figure 12.4.19 STM image of 7×7 cell of reconstructed Si(111) surface.
16 17
1 2 3 4
5 6
7
8 9 10 11
12 13
141518 19 20 21 22 23 242526 27 30 31
32 33 34 35 36 37
(a)
(b)
Figure 12.4.20 DAS model for 7×7 cell of reconstructed Si (111) surface,(a) The
view perpendicular to surface; (b) The view of cross-section along the long diagonal.
ψ(z)
E
Ec
EF
Ev
z
Figure 12.4.21 Schematic diagram for the metal induced gap states (MIGS).
EF
EF EFV
g>V0Vg=V0Vg= 0
Semiconductor
E
(a) (b) (c)
Figure 12.4.22 Metal-Insulator-Semiconductor interfaces in a MOS device,(a)Vg =
0; (b)Vg = V0,depletion layer → inversion lager; (c)Vg > V0,degenerate electrons in
triangular-shaped potential well.
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
April 5,2004
Contents
Chapter 13,Correlated Electronic States 3
§13.1 Mott Insulators,,,,,,,,,,,,,,,,,,,,,,,,6
13.1.1 Idealized Mott Transition,,,,,,,,,,,,,,,,7
13.1.2 Hubbard Model,,,,,,,,,,,,,,,,,,,,,16
13.1.3 Kinetic Exchange and Superexchange,,,,,,,,,22
13.1.4 Orbital Ordering versus Spin Ordering,,,,,,,,,30
13.1.5 Classification of Mott Insulators,,,,,,,,,,,,40
§13.2 Doped Mott Insulators,,,,,,,,,,,,,,,,,,,,45
13.2.1 Doping of Mott Insulators,,,,,,,,,,,,,,,45
13.2.2 Cuprates,,,,,,,,,,,,,,,,,,,,,,,,,50
13.2.3 Manganites and Double Exchange,,,,,,,,,,,59
13.2.4 Charge-Ordering and Electronic Phase Separation,,,59
§13.3 Magnetic Impurities,Kondo Effect and Related Problems,60
13.3.1 Anderson Model and Local Magnetic Moment,,,,,60
13.3.2 Indirect Exchange,,,,,,,,,,,,,,,,,,,,60
13.3.3 Kondo Effect,,,,,,,,,,,,,,,,,,,,,,60
13.3.4 Heavy-Electron Metals and Related Materials,,,,,60
§13.4 Outlook,,,,,,,,,,,,,,,,,,,,,,,,,,,,61
13.4.1 Some Empirical Rules,,,,,,,,,,,,,,,,,,61
13.4.2 Theoretical Methods,,,,,,,,,,,,,,,,,,61
Chapter 13
Correlated Electronic States
Challenge to study of electronic structures
Anomalous phenomena often occur
Narrow band phenomena,between bond and band
Conventional solids,weak interactions,single electron,effective masses
There are little differences between wide-band solids
but large differences between narrow-band materials
Figure 13.0.1 Potentials and wavefunction overlaps for conventional solids (a) and
narrow-band materials.
§13.1 Mott Insulators
de Bohr and Verway’s experiments,1937
NiO,MnO,CoO
Peierls and Mott’s argument,1937
d-d Coulomb interaction leads to correlation gap
Transition metal monoxides,3d-electrons
Lanthanum compounds,4f-electrons
Atomic correlation and mixed valence
Mott,1949,1952 and later
13.1.1 Idealized Mott Transition
Outer shells of Co atom 3d4s and O atom 2s2p
Number of electrons per unit cell is 9 + 6 = 15,an odd number
According to band theory,CoO was predicted to be metallic state
However,experimentalists found it an insulator with large gap
These transition metal oxides are called Mott insulators
High Tc superconductors in cuprates and
colossal magneto-resistors in manganites
Mott considered metal-insulator transition for Na crystal
with changing interatomic spacing and
predicted a critical interatomic spacing ac
Overlap of wavefunctions and Bandwidth B
increases when interatomic spacing decreases
Fig,13.1.1
(a)
(b)
(c)ψ
1/a
U
B
Eσ
Figure 13.1.1 Idealized Mott transition,(a) Electrical conductivity and (b) band-
width vesus inverse interatomic spacing; (c) Orbitals on both sides of Mott transition.
Na + Na → Na+ + Na?
At Na? site,the double occupancy of an orbital
Energy cost is called Hubbard energy U
In the limit of isolated atoms
U = I?A (13.1.1)
An approximate estimate of Mott transition
Band width B,half-filled,average kinetic energy for each electron B/4
Probability for a site vacant,singly occupied,and doubly occupied is
1/4,1/2,and 1/4,average correlation energy for each electron U/4
Metallic state for U <B; insulating state U >B
The condition for Mott transition
U = B (13.1.2)
Mott transition is an electron correlation induced collective localization
In general,the ground state of most Mott insulators is antiferromagnet
Figure 13.1.1 Three kind of metal-insulator transitions.
Figure 13.1.1 Wilson transition.
Figure 13.1.1 Anderson transition.
Figure 13.1.1 Mott transition.
13.1.2 Hubbard Model
Hubbard Hamiltonian,1962-5
H =
summationdisplay
i,j,σ
tijc?jσcjσ +U
summationdisplay
i
niσniσ (13.1.3)
Three parameters,t0 = tii,t = tij (i,j = n.n) and U
t0 is the mean energy,t is equal to the half width of the band
(1) U = 0
c?iσ = 1√N
summationdisplay
k
eik·Ric?kσ,ciσ = 1√N
summationdisplay
k
e?ik·Rickσ (13.1.4)
H =
summationdisplay
kσ
Ekσc?kσckσ,Ek =
summationdisplay
ij
tijeik·(Ri?Rj) (13.1.5)
Bloch state Ek
(2) t = 0
E =
summationdisplay
i
[t0(niσ +niσ) +Uniσniσ] = N1t0 +N2(2t0 +U) (13.1.6)
A more precise definition of U
U =
integraldisplay
dr1
integraldisplay
dr2|ψ(r1?R)|2 e
2
|r1?r2||ψ(r1?R)|
2 (13.1.7)
The ground states of electrons are
itinerant states when tgreatermuchU or localized when tlessmuchU
The spin direction is antiferromagnetic
If an electron with a given spin enters,it can be accommodated at
one of the N/2 atoms occupied by an electron with opposite spin
The energy amounts to t0 + U at any isolated atom,and splits up into a
band centering around t0 +U
Figure 13.1.2 Hubbard model.
1.15
B/U
O
t0
t0+U
E
Figure 13.1.2 Transition from localized states to delocalized states in a half-filled
energy band in Hubbard model.
1.4142
1.1547
0.5
W/U
t0 t0+U E
0
0.2
0.4
0.6
Ug(E)
Figure 13.1.3 Density of states g(E) showing band splitting by electron-electron
interactions in Hubbard model.
13.1.3 Kinetic Exchange and Superexchange
At half-filling,for tlessmuchU,effective Hamiltonian is
Antiferromagnetic Heisenberg model
H =?J
summationdisplay
ij
Si ·Sj (13.1.8)
with J =?4t2/U
Kinetic exchange process for a pair of neighboring sites
Assume spins are antiparallel,a virtual hopping process
Associated energy by second order perturbation →?t2/U
Kinetic exchange → antiferromagnetism of insulators,Fig,13.1.4
TiO
PM
Structures
Rock Salt
MnO
118
AF
FeO
198
AF AF AF AF
CoO
293
NiO
520
CuO
230
Mn2O4
41
119
(Verwey)
850
Fe3O4 Co3O4
40
AF
Ti2O3
500 150
AF AF AF AF
α-Cr2O3
308
α-Mn2O3 α-Fe2O3
963
Slanted
TiO2
'0
VO
340 84
AF
β-MnO2
VF2 CrF2 MnF2 FeF2 CoF2 NiF2 CuF2
AF AF AF AF AF AF AF
27-42 53 67 78 37 73 69
K2MnF4
42Rb
2MnF438
AF
Rb2FeF4
56AF
K2CoF4
107Rb
2CoF4101
AF
K2NiF4
97Rb
2NiF490
AF
K2CuF4
6
F
Spinel
Corundum
Rutile
Rutile
Insulator (or semiconductor) Metal
80
VO
PM
383
F
CrO2
V2O3
2
Monoclinic
30
PM
30
Ferri
Ferri
Two Dimension
Figure 13.1.4 Summary of the magnetic states of various transition metal oxides
and fluorides,
Superexchange
For example,in MnO,Mn++ interact over 0.4 nm apart,Mn25,3d54s2
The antiferromagnetic coupling is between second nearest-neighbor cations
such as A=Mn++ and C=Mn++,via intervening nearest neighbor anion
Fig,13.1.5
Extended to periodic structure,atomic orbitals replaced
by Wannier functions
the transfer integral between Ri and Rj sites becomes
bij = 〈w(r?Ri)|H|w(r?Rj)〉 (13.1.9)
A B C
3d 3dp
Figure 13.1.5 The spin configurations of four electrons in the 3d and p orbitals on
sites Mn-O-Mn.
Superexchange regarded as a generalization of kinetic exchange
Electrons of intervening oxygen atom participated in virtual hopping pro-
cesses.
Goodenough and Kanamori,1950s
Exchange interactions between half-filled orbitals described by virtual hop-
ping process shown in Fig,13.1.6(a),antiferromagnetic coupling is derived
Jkinij =?2b2ij/4S2U (13.1.10)
where S is the net spin of a magnetic atom and U is the Hubbard energy
Furthermore,an electron with up-spin transferred from a full orbital
to an empty orbital,shown in Fig,13.1.6(b),intraatomic exchange energy,
Hund’s rule coupling,?ex/U2 favors ferromagnetic exchange coupling
Jkinij = +2b2ij?ex/4S2U (13.1.11)
3d 3dp 3d 3dp
(a) (b)
Figure 13.1.6 Virtual processes for the Goodenough-Kanamori rule of the superex-
change,(a) cationd-shell half filled (antiferromagnetic); (b) cationd-shell<half filled
(ferromagnetic).
+
+
+
y
x
dxy px
Figure 13.1.7 Possibility for ferromagnetic alignment through the superexchange.
13.1.4 Orbital Ordering versus Spin Ordering
Orbital degeneracy of d-electrons introduces further richness
Hubbard model generalized to n-fold degenerate orbitals
Simplest case in Fig,13.1.8
Two electrons on two sites A and B,both have two-fold degeneratedorbitals
As example,two eg orbitals in octahedral site
ψa = d3z2?r2 and ψb = dx2?y2
a b
A B
a b
d3z2-r2 dx2-y2
(a) (b)
a b
Figure 13.1.8 Schematic diagram of a eg molecule for two orbitals a and b on two
sites A and B.
Like Heitler-London H2 problem,generalized to consider a diatomic molecule
each atom has two orbitals
4 different low-energy states of H2 classified into 1 singlet and 1 triplet
Now 16 states classified into 4 singlets and 4 triplets
To distinguish the Hubbard energy on different orbitals Ua,Ub and Uab
(1) virtual process I in Fig,13.1.9(a),like kinetic exchange in §13.1.3
Two electrons occupy the same orbital ψa at two sites
gives the energy for a singlet in a-orbital
Es ≈?4t
2a
Ua (13.1.12)
a b
A B A B A B
A B A B A B
a b a b a b a b a b
?
?
(a)
(b)
Figure 13.1.9 Virtual hopping processes for electrons between two sites,(a) Process
I,i.e.,virtual hopping process in a single orbital (a or b); (b) process II,i.e.,virtual
hopping process in mixed orbitals (a and b).
(2) virtual process II in Fig,13.1.9(b),the energy gain for the triplet
Et ≈? tatbU
ab?J
,(13.1.13)
J is the energy gain due to Hund’s rule coupling (intraatomic exchange)
Roughly Ua = Ub = U >Uab = U?2J
The triplets have lower energy than the singlets
the ground state is the triplet with the lowest energy
For degenerate orbitals,a new kind of order,i.e.,orbital order
Degenerate orbitals may have different shapes or orientations
Orbital order is related with the arrangement of orbital shapes on sites
Staggered orbitals have antisymmetric orbital wave functions
accompanied by symmetric spin wavefunctions,i.e.,parallel spins
Regular orbitals accompanied by anti-parallel spins
Results in a two-electron system extended to a lattice
Introduce a pseudospin T variable to describe the orbital ordering
Its component Tz = ±1/2,for dx2?y2 or d3z2?r2 occupied
This pseudospin has three components satisfying
Similar commutation relation as those of spin operators
There are interactions between spins S and pseudospins T of different ions
Generalized Heisenberg Hamiltonian
H =?
summationdisplay
i<j
[JS (Si ·Sj) +JT (Ti ·Tj) +JST (Si ·Sj)(Ti ·Tj)] (13.1.14)
When more than two orbitals involved,many different situations can be
realized
Orbital order is defined by its pseudospins:
Staggered orbital order as orbital antiferromagnetism,Fig,13.1.10
Regular orbital order as orbital ferromagnetism
In general,the spin ferromagnetism favors the pseudospin antiferromag-
netism,staggered orbital order
Spin antiferromagnetism favors the pseudospin ferromagnetism,regular or-
bital order
Surely orbital order together with spin order play conspicuous roles
in physics of Mott insulators and doped Mott insulators
Figure 13.1.10 Staggered orbital order,i.e.,pseudospin antiferromagnetism,alter-
nation of dx2?y2 and d3z2?r2 orbitals in a simple cubic lattice.
Figure 13.1.11 Orbital order and spin order of LaMnO3.
13.1.5 Classification of Mott Insulators
Hubbard subbands in Mott insulator sandwiched with 2p bands
Their mutual arrangements will determine their insulating behavior
A classification proposed by J,Zaanen,G,A,Sawatzty,and J,W,Allen
1985,Phys,Phys,Lett,55,418
Mott insulators are classified into two types
Hubbard energy U and charge transfer energy?
Fig,13.1.12
￠
U
Charge Transfer Insulator
(￠<U)
Lower Hubbard Band
Upper Hubbard Band
p BandU
￠
Mott-Hubbard Insulator
(U<￠)
Upper Hubbard Band
Lower Hubbard Band
p Band
g(E)
E
g(E)
Figure 13.1.12 The relative positions of Hubbard subbands and oxygen 2p band.
The smaller gap is of crucial importance for the transport behavior
(1) Mott-Hubbard type,? >U
(2) charge transfer type,U >?
In MH insulators,the charge for transport are d-electrons
In CT insulators,d-like quasi-particle and p-hole
Fig,13.1.13
Fig,13.1.14
CT
MH
LaMO3
YMO3
Sc Ti V Cr Mn Fe Co Ni Cu
0 2 4 6 8
6
4
2
0
E g
(eV)
3d Elctron Number
Figure 13.1.13 Optically determined gaps in Mott insulators in LaMnO3 and YMO3.
Full symbols are MH gaps,open symbols are CT gaps.
Covalence Insulator
A
Charger Transfer Insulator
B
Mott-Hubbard
Insulator
tpd =
0.5eV1.010
8
6
4
2
0-2 0 2 4 6
U dd
/t pd
C
Metal
D
￠/tpd
Figure 13.1.14 The Zaanen,Sawatsky and Allen diagram with modification.
§13.2 Doped Mott Insulators
13.2.1 Doping of Mott Insulators
Doping is important to modify physical properties of
band insulators and semiconductors as well as Mott insulators
Electron doping or hole doping
Usual chemical doping is the substitution of ions
with different valence for cations or anions in the parent compound
For example,La1?xSrxTiO3,La2?xSrxCuO4
Both parent compounds are antiferromagnetic insulators with S = 1/2
After doping both become metals with anomalous properties
the former is a nonmagnetic heavy Fermi liquid
the latter is a high Tc superconductor
Majority of high Tc superconductors are hole-doped
only Nd2?xCexCuO4 is electron-doped
Doped Mott insulators introduce extra charge carriers as well as extra spins
into the original insulator with antiferromagnetic order
d0 d1 d2 d3 d4 d5 d6 d7 d8 d9
SrTiO3 LaTiO3
SrVO3 LaVO3
SrCrO3 LaCrO3
SrMnO3 LaMnO3
SrFeO3 LaFeO3
SrCoO3 LaCoO3
SrNiO3 LaNiO3
SrCuO3 LaCuO3
Sr2TiO4 LaSrTiO4
Sr2VO4 LaSrVO4
Sr2CrO4 LaSrCrO4
Sr2MnO4LaSrMnO4 La2MnO4
Sr2FeO4 LaSrFeO4 La2FeO4
Sr2CoO4 LaSrCoO4 La2CoO4
Sr2NiO4 LaSrNiO4 La2NiO4
LaSrCuO4 La2CuO4
Figure 13.2.1 A guide map for the synthesis of filling-controlled 3d transition-metal
oxides with perovskite and layered perovskite structures.
Substitution also brings distortions into structures,so may indirectly reduce
the bandwidth B of compounds
In Fig,13.2.2 YMO3 shows stronger electron correlation than LaMO3 due to
reduced B introduced by distorted oxygen octahedra
Band
Insulator
Mott-
Hubard
Insulator
(Y,Ca)T
iO 3
(La,Y)T
iO
3
(Y,Ca)VO
3
(La,Sr)T
iO 3
(La,Sr)VO
3
B=Ti V Cr
(La,Sr)CrO
3
Fe
Sr(Fe,Co)O3 (La,Sr)CoO
3Co
Ni Cu
RNiO
3
RCuO
3
ABO3
ReO3 Metal
d0 d1 d2 d3 d4 d5 d6 d7 d8
UWU/W
A=
Y
B3+
La
Ca
Sr
B4+
￠￠/W
Charge Transfer
Insulator
Insulator
(La,Sr)MnO
3
(Sr
,Ca)FeO
3
(La,Sr)FeO
3
(La,Ca)CoO3
(Y,Ca)FeO
3
(La,Ca)MnO
3
Figure 13.2.2 A metal-insulator phase diagram for 3d transition-metal oxides with
perovskite structure.
13.2.2 Cuprates
Cuprates high Tc superconductors are doped charge transfer insulators
Crystal structures are mostly perovskite-like displayed in Fig,13.2.3
Fig,13.2.4shows the CuO2 layers playing the crucial partin carryingcurrents
In Y-Ba-Cu-O (YBCO),in addition,Cu-O chains also important
All parent compounds are Mott insulators with antiferromagnetic order
For La compound,TN = 330 K; for Y compound,TN = 500 K
O
O
O
O
O
O
Cu OO
O
O
O
Cu OO
O
O
Cu
(a) (b) (c)
Figure 13.2.3 Crystal structures of (a) La2CuO4; (b) YBa2 CuO7; (c) NdCuO4.
Cu2+
O2-
Figure 13.2.4 CuO2 in cuprates with arrows indicating the spin orientation and
shaded area as the chemical bond.
Take La2?xSrxCuO4(0 <x< 0.3) to analyze electronic structure
Cu atoms are situated in the center of distorted oxygen octahedra
Cu-O distances are 0.23 nm and 0.19 nm for O atoms
outside and in the CuO2 plane
Stability of CuO2 plane ascribed to σ bonds formed by
Cu 3dx2?y2 and O 2px,y,see Fig,13.2.5
much weaker pi-bonds between Cu 3dx2?y2 and O 2pz
Γ? Λ Γ ΧU Z S Z
Z xΓ?
y Χ S
U
A
A
B
B
-8
-6
-4
-2
0
2
4
E / eV
EF
O2-2p5
Cu2+3d9
dx2-y2 (1-x)
d3z2-r2 (2)
dxy,dxz,dyx (6)
Figure 13.2.5 Dispersion curves of CuO2 plane in cuprates.
Energy band calculation → highest antibonding pdσ? orbital is half filled
→ metallic character for La2CuO4
This is just in contradiction to the fact that La2CuO4 is an insulator
Even LDA calculations gave wrong answer for the ground state
Actually La2CuO4 without doping is an excellent insulator with antiferro-
magnetic long range order
Only doped with sufficient Sr can destroy this antiferromagnetic long range
order and turn it into a metallic superconductor
Strong correlation→electronic properties of cuprates surprising and extraor-
dinary
If the occupation of a single sd hole is expressed as
ciσ = ciσ(1?niˉσ) (13.2.1)
Hubbard model may be simplified to the t-J model
H = t
summationdisplay
ijσ
c?iσ?cjσ?J
summationdisplay
i<j
Si ·Sj (13.2.2)
The first term describes the electron moving in the CuO2 plane
The second term is the Heisenberg Hamiltonian describing
the effective exchange interaction of neighboring spins of 3d holes
This exchange interaction is realized through O 2porbitals between Cu atoms
by the superexchange mechanism
This highly correlated state of electrons making the normal properties of
high Tc superconductors extremely anomalous and difficult to account for.
Fig,13.2.6 is a typical electronic phase diagram for cuprates
Fermi Liquid
Non Fermi Liquid
Te
mperature
Charge Density
SC
AFM
Figure 13.2.6 Electronic phase diagram for cuprates.
13.2.3 Manganites and Double Exchange
13.2.4 Charge-Ordering and Electronic Phase Separation
§13.3 Magnetic Impurities,Kondo Effect and Related Prob-
lems
13.3.1 Anderson Model and Local Magnetic Moment
13.3.2 Indirect Exchange
13.3.3 Kondo Effect
13.3.4 Heavy-Electron Metals and Related Materials
§13.4 Outlook
13.4.1 Some Empirical Rules
13.4.2 Theoretical Methods
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
April 8,2004
Contents
Chapter 13,Correlated Electronic States 3
§13.1 Mott Insulators,,,,,,,,,,,,,,,,,,,,,,,,4
13.1.1 Idealized Mott Transition,,,,,,,,,,,,,,,,4
13.1.2 Hubbard Model,,,,,,,,,,,,,,,,,,,,,4
13.1.3 Kinetic Exchange and Superexchange,,,,,,,,,4
13.1.4 Orbital Ordering versus Spin Ordering,,,,,,,,,4
13.1.5 Classification of Mott Insulators,,,,,,,,,,,,4
§13.2 Doped Mott Insulators,,,,,,,,,,,,,,,,,,,,5
13.2.1 Doping of Mott Insulators,,,,,,,,,,,,,,,5
13.2.2 Cuprates,,,,,,,,,,,,,,,,,,,,,,,,,10
13.2.3 Manganites and Double Exchange,,,,,,,,,,,19
13.2.4 Charge-Ordering and Electronic Phase Separation,,,31
§13.3 Magnetic Impurities,Kondo Effect andRelated Problems,,40
13.3.1 Anderson Model and Local Magnetic Moment,,,,,41
13.3.2 Indirect Exchange,,,,,,,,,,,,,,,,,,,,48
13.3.3 Kondo Effect,,,,,,,,,,,,,,,,,,,,,,50
13.3.4 Heavy-Electron Metals and Related Materials,,,,,60
§13.4 Outlook,,,,,,,,,,,,,,,,,,,,,,,,,,,,65
13.4.1 Some Empirical Rules,,,,,,,,,,,,,,,,,,66
13.4.2 Theoretical Methods,,,,,,,,,,,,,,,,,,69
Chapter 13
Correlated Electronic States
§13.1 Mott Insulators
13.1.1 Idealized Mott Transition
13.1.2 Hubbard Model
13.1.3 Kinetic Exchange and Superexchange
13.1.4 Orbital Ordering versus Spin Ordering
13.1.5 Classification of Mott Insulators
§13.2 Doped Mott Insulators
13.2.1 Doping of Mott Insulators
Doping is important to modify physical properties of
band insulators and semiconductors as well as Mott insulators
Electron doping or hole doping
Usual chemical doping is the substitution of ions
with different valence for cations or anions in the parent compound
For example,La1?xSrxTiO3,La2?xSrxCuO4
Both parent compounds are antiferromagnetic insulators with S = 1/2
After doping both become metals with anomalous properties
the former is a nonmagnetic heavy Fermi liquid
the latter is a high Tc superconductor
Majority of high Tc superconductors are hole-doped
only Nd2?xCexCuO4 is electron-doped
Doped Mott insulators introduce extra charge carriers as well as extra spins
into the original insulator with antiferromagnetic order
d0 d1 d2 d3 d4 d5 d6 d7 d8 d9
SrTiO3 LaTiO3
SrVO3 LaVO3
SrCrO3 LaCrO3
SrMnO3 LaMnO3
SrFeO3 LaFeO3
SrCoO3 LaCoO3
SrNiO3 LaNiO3
SrCuO3 LaCuO3
Sr2TiO4 LaSrTiO4
Sr2VO4 LaSrVO4
Sr2CrO4 LaSrCrO4
Sr2MnO4LaSrMnO4 La2MnO4
Sr2FeO4 LaSrFeO4 La2FeO4
Sr2CoO4 LaSrCoO4 La2CoO4
Sr2NiO4 LaSrNiO4 La2NiO4
LaSrCuO4 La2CuO4
Figure 13.2.1 A guide map for the synthesis of filling-controlled 3d transition-metal
oxides with perovskite and layered perovskite structures.
Substitution also brings distortions into structures,so may indirectly reduce
the bandwidth B of compounds
In Fig,13.2.2 YMO3 shows stronger electron correlation than LaMO3 due to
reduced B introduced by distorted oxygen octahedra
Band
Insulator
Mott-
Hubard
Insulator
(Y,Ca)T
iO 3
(La,Y)T
iO
3
(Y,Ca)VO
3
(La,Sr)T
iO 3
(La,Sr)VO
3
B=Ti V Cr
(La,Sr)CrO
3
Fe
Sr(Fe,Co)O3 (La,Sr)CoO
3Co
Ni Cu
RNiO
3
RCuO
3
ABO3
ReO3 Metal
d0 d1 d2 d3 d4 d5 d6 d7 d8
UBU/B
A=
Y
B3+
La
Ca
Sr
B4+
￠￠/B
Charge Transfer
Insulator
Insulator
(La,Sr)MnO
3
(Sr
,Ca)FeO
3
(La,Sr)FeO
3
(La,Ca)CoO3
(Y,Ca)Fe
O 3
(La,Ca)MnO
3
Figure 13.2.2 A metal-insulator phase diagram for 3d transition-metal oxides with
perovskite structure.
13.2.2 Cuprates
Cuprates high Tc superconductors are doped charge transfer insulators
Crystal structures are mostly perovskite-like displayed in Fig,13.2.3
Fig,13.2.4shows the CuO2 layers playing the crucial partin carryingcurrents
In Y-Ba-Cu-O (YBCO),in addition,Cu-O chains also important
All parent compounds are Mott insulators with antiferromagnetic order
For La compound,TN = 330 K; for Y compound,TN = 500 K
O
O
O
O
O
O
Cu OO
O
O
O
Cu OO
O
O
Cu
(a) (b) (c)
Figure 13.2.3 Crystal structures of (a) La2CuO4; (b) YBa2 CuO7; (c) NdCuO4.
Cu2+
O2-
Figure 13.2.4 CuO2 in cuprates with arrows indicating the spin orientation and
shaded area as the chemical bond.
Take La2?xSrxCuO4 (0 < x < 0.3) to analyze electronic structure
Cu atoms are situated in the center of distorted oxygen octahedra
Cu-O distances are 0.23 nm and 0.19 nm for O atoms
outside and in the CuO2 plane
Stability of CuO2 plane ascribed to σ bonds formed by
Cu 3dx2?y2 and O 2px,y,see Fig,13.2.4
much weaker pi-bonds between Cu 3dx2?y2 and O 2pz
Energy band calculation → highest antibonding pdσ? orbital is half filled
→ metallic character for La2CuO4
This is just in contradiction to the fact that La2CuO4 is an insulator
Even LDA calculations gave wrong answer for the ground state
Actually La2CuO4 without doping is an excellent insulator
with antiferromagnetic long range order
Only doped with sufficient Sr can destroy
this antiferromagnetic long range order
and turn it into a metallic superconductor
Γ? Λ Γ ΧU Z S Z
Z xΓ?
y Χ S
U
A
A
B
B
-8
-6
-4
-2
0
2
4
E / eV
EF
O2-2p5
Cu2+3d9
dx2-y2 (1-x)
d3z2-r2 (2)
dxy,dxz,dyx (6)
Figure 13.2.5 Dispersion curves of CuO2 plane in cuprates.
Strong correlation → electronic properties of cuprates
surprising and extraordinary
If the occupation of a single sd hole is expressed as
ciσ = ciσ(1?niˉσ) (13.2.1)
Hubbard model may be simplified to the t-J model
H = t
summationdisplay
ijσ
c?iσ?cjσ?J
summationdisplay
i<j
Si ·Sj (13.2.2)
The first term describes the electron moving in the CuO2 plane
The second term is the Heisenberg Hamiltonian describing
the effective exchange interaction of neighboring spins of 3d holes
This exchange interaction is realized through O 2porbitals between Cu atoms
by the superexchange mechanism
This highly correlated state of electrons making the normal properties of
high Tc superconductors extremely anomalous and difficult to account for
Fig,13.2.6 is a typical electronic phase diagram for cuprates
Fermi Liquid
Non Fermi Liquid
Te
mperature
Charge Density
SC
AFM
Figure 13.2.6 Electronic phase diagram for cuprates.
13.2.3 Manganites and Double Exchange
Manganite LaMnO3 has a distorted perovskite structure
Mott insulator of charge transfer type
Antiferromagnet with in-plane ferromagnetism
Doping → La1?xDxMnO3 (D= Ca2+,Sr2+,or Ba2+)
Colossal magnetoresistance (CMR) effect since 1993
Hole-doping with Sr x ≈ 0.2-0.4,balanced by creating Mn4+ ions (d3+)
(1?x) manganese ions are in Mn3+ (d4+) state
Fig,13.2.7,resistivity drastically decreases when temperature falls below Tc
T > Tc,resistivity behaves like an semiconductor with (dρ/dT) < 0
T < Tc,metallic behavior with (dρ/dT) > 0
Onset of ferromagnetism is coincident with the insulator-metal transition
explained by double exchange mechanism for mixed valent oxides
C,Zener,1952,Phys,Rev,82,403
P,W,Anderson and H,Hasegawa,1955,Phys,Rev,118,675
Tc
La1-xSrxMnO3
x=0
x=0.05
x=0.1x=0.15
x=0.175
x=0.2
x=0.3
x=0.4
0 200 400
10-4
10-2
100
102
T (K)
ρ (
.
cm)
Figure 13.2.7 Resistivity versus temperature for La1?xSrxMnO3 (0 < x < 0.4).
Before doping,x = 0,all Mn3+,ionic state t32g e1g
three d- electrons in t2g orbitals plus one electron in eg orbital
Hund’s rule total spin is S = 2
Real hopping motion of electrons is forbidden by Mott correlation effect
After doping,x- fraction of Mn3+ ions transformed into Mn4+ ions
Now ionic state t2g and Hund’s rule total spin S = 3/2
Making eg electrons free to move and giving a large conductivity
To describe eg electrons moving on the background with t2g cores
Double exchange Hamiltonian
H =?t
summationdisplay
ij
(c?iσcjσ + h.c.)?JH
summationdisplay
i
Si ·si (13.2.3)
First term is one-electron band term,t is the hopping integral
Second term is the exchange interaction between si of itinerant electrons
and Si of t2g orbitals,JH is intraatomic exchange (Hund’s rule coupling)
If t lessmuch JH,the energy simplifies as
E =?JHS ±tcos(θ/2) (13.2.4)
θ is the angle between neighboring spins,Fig 13.2.8
θ
M4+M3+
Figure 13.2.8 The angle between neighboring spins.
Bandwidth is related to direction of neighboring spins:
wide for parallel spins and narrow for antiparallel spins
In contrast to superexchange due to virtual hopping processes of electrons
Double exchange related to real electron hopping processes
between neighboring atoms
Fig,13.2.9
Mn4+Mn3+ O2-
Figure 13.2.9 Mechanism of double exchange for a manganite.
Figure 13.2.9 Mechanism of double exchange for a manganite.
LaMnO3,insulator,planar ferromagneticspins coupled antiferromagnetically
La1?xSrxMnO3(x > 0.2),ferromagnetic metal with double exchange
What is the physical nature of this transition?
To account for this transition,de Gennes’s pure spin model
P,G,de Gennes,1960,Phys,Rev,118,114
Spins on neighboring planes depend on the doping level x
When x = 0,θ = pi; as x reaches some critical value xc,θ = 0
Energy density is composed of two terms:
superexchange between fixed t2g cores and double exchange,i.e.,
the spin-orientation dependent hopping energy of eg electrons
E(θ) = JH cosθ?x[4t + 2tcos(θ/2)] (13.2.5)
Minimizing E(θ) with respect to θ to find
cos(θ/2) = tx2J
H
(13.2.6)
A reasonable physical picture from this solution for transition
from AFM insulator to FM metal with doping level x:
at x = 0,θ = pi,AFM state
at small x,x lessmuch 1,θ ≈ pi?(tx/JH),the canted AFM state
at critical value xc = 2JH/t,θ = 0,FM state
So a gradual transition,from the canted phase for 0 < x < xc to
the full ferromagnetic alignment for x > xc,is predicted
13.2.4 Charge-Ordering and Electronic Phase Separation
Competition between kinetic energy and Coulomb repulsion
Electron concentration is important
Wigner crystallization
In the jellium model,let r0 be the mean radius defined by (4pi/3)r30 = n?1
Average kinetic energy due to uncertainty principle
δεkin = (?p)2/2m ≈ h2/2mr20
Average potential energy due to the Coulomb repulsion δεpot ≈ e2/r0
For very low concentration,δεpot > δεkin
Electrons crystallize into a lattice at r0 about 40-100 Bohr radius
Here consider the charge ordering of doped manganites
Take La1?xCaxMnO3 as an example
Phase diagram in Fig,13.2.10
At x = 0,there is no conduction electron for each Mn site
At x = 1,there is one conduction electron
between these limits,there is x electrons for each Mn site
La1-xCaxMnO3T
N
TC
PC
0 10 20 30 40 50 60 70 80 90 100100
120
140
160
180
200
220
240
260
280
300
x (% Ca)
T (K)
AFM
insulator
FM
metal AFMinsulator
Figure 13.2.10 Phase diagram of La1?xCaxMnO3.
Figure 13.2.10 Magnetic structure of La-Ca-Mn-O.
The problem is how to distribute these electrons in space?
It is seen that spin-ordering and orbital-ordering in manganites
now charge ordering added to make physics of manganites more rich
At x = 0.5,a charge-ordered AFM state below Tc = 160 K is found
J,Goodenough’s model,1955,Phys,Rev.,100,564
Fig,13.2.11
Mn3+ and Mn4+ arranged like a checkerboard
exhibiting charge ordering,spin ordering and orbital ordering altogether
This model verified by X-ray and neutron diffraction measurements
Mn3+ Mn
4+
Mn3+
b a
d3y2-r2
d3x2-r2
(a) (b)
Figure 13.2.11 Spin,charge and orbital ordering in La0.5Ca0.5MnO3 (a) schematic
diagram (b) checkerboard pattern with dark and light shaded squares show Mn3+ and
Mn4+ sites,Taken from T,Mizokawa and A,Fujimori,Phys,Rev,B,56 R493 (1997).
Manganites showing large CMR effect are those exhibiting
double exchange and charge ordering simultaneously
Fig,13.2.12,magneto-resistances versus temperature for La0.7?xPrxCaO3
(ρ0?ρH)/ρH may reach 104-105
Theory for CMR is still unsettled
Mechanism involved in Jahn-Teller effect and Anderson localization
Experimental evidences are increasing to show that
electronic phase separation into nanoclusters of FM metals and
charge-ordered insulators may play important role in CMR
La0.7-xPrxCa0.3MnO3x=0.7 x=0.5
x=0.525
x=0.35
x=0.175
x=0
0 50 100 150 200 250 300-2
0
2
4
log (
ρ 0T
-ρ g53
g84 )/
ρ g53
g84
Tg32g40g75g41
Figure 13.2.12 Magnetoresistance at 5 T versus temperature for La0.7?xPrxCaO3.
Curie temperatures Tc are marked by arrows.
Electronic phase separation phenomenon
competitions involving various types of ordered states
also interactions such as Coulombic,exchange and Jahn Teller
Clusters or stripes of various electronic phases with nanometer sizes or larger
Stripe phases in cuprates involving charge and spin ordering
§13.3 Magnetic Impurities,Kondo Effect and
Related Problems
Another important area of correlated electronic states
Magnetic impurities in metals
Magnetic moment formation and their influence on resistivity of metals
Kondo effect → Kondo problem → heavy-electron metals,
Kondo insulators and superconductors
13.3.1 Anderson Model and Local Magnetic Moment
In §7.2,impurity atoms in metals were treated in a free electron gas
Here transition metal impurity with unfilled d-band
Anderson model is important to handle many-body effect
In Anderson model,localized description for impurity and
delocalized description for conduction electrons
Fig,13.3.1
(a) (b) (c)
U
Ed-
Ed+
V
V
ρd+(E)
EF
w
ρd-(E)
Figure 13.3.1 PictorialrepresentationofAndersonmodel,(a)originalatomicd-level;
(b) it is split and polarized by on-site Coulomb repulsion; (c) it is further broadened
by hybridization with s-electrons in the conduction band of the host.
Hamiltonian for d-electrons on an impurity site
Hd =
summationdisplay
σ
εdndσ + Und↑nd↓ (13.3.1)
Hamiltonian for s-electrons in conduction band of the host
Hs =
summationdisplay
kσ
εknkσ (13.3.2)
s-d hybridization interaction with strength Vkd
Hsd =
summationdisplay
kσ
Vkσ[c?kσcdσ + c?dσckσ] (13.3.3)
Anderson Hamiltonian is the sum of three terms
H = Hd +Hs +Hsd (13.3.4)
It requires εd < εF and εd + U > εF
Transition rate between d-level and s-band described by Fermi Golden Rule
in which the density of states in the s-band at the d-level g(εdσ) makes d-level
extend to some resonance with the width
Γσ = piV 2g(εdσ)
Sharp level at εdσ replaced by spectral density function
ρdσ(ε) = 1pi Γσ(ε?ε
dσ)2 + Γ2σ
(13.3.5)
The consequence of broadening d-levels by resonance
〈nd↑〉 < 1,〈nd↓〉 > 0
Whether the local moment retained on the impurity site demands evaluation
〈nd↑〉 = 1pi
integraldisplay εF
∞
Γdε
(ε?εd)2 + Γ2 =
1
piarccot
εd?εF +〈nd↓〉U
Γ (13.3.6)
〈nd↓〉 = 1piarccotεd?εF +〈nd↑〉UΓ (13.3.7)
Self-consistent calculation → graphical solutions,in Fig,13.3.2
Larger U is needed to retain the local moment
〈nd↑〉?〈nd↓〉 = ±0.644μB
Anderson explained why the iron group elements from V to Co show
magnetic moments when dissolved in Cu,Ag and Cu,but show
no magnetic moment when dissolved in Al
Numerically solveEqs,(13.3.6)and (13.3.7)to verify Fig.13.3.2,
and to find a critical U = Uc for the transition from no local
moment to a local moment.
0 0.2 0.4 0.6 0.8 1.00
0.2
0.4
0.6
0.8
1.0
0 0.2 0.4 0.6 0.8 1.0
(a) (b)
< nd↑ >
< n
d↓
>
< nd↑ >
Figure 13.3.2 Self-consistency plot of 〈nd↓〉 against 〈nd↑〉,(a) U = Γ and εF?εd =
0.5Γ,no local moment is formed,(b) U = 5Γ and εF?εd = 2.5Γ,local moment.
13.3.2 Indirect Exchange
A local magnetic moment → spin polarization of host electron gas
As in §7.2.3,the local magnetic moment brings the oscillation to spin density
This has been verified in Cu-Mn and Ag-Mn by NMR measurements
Every impurity atom introduces spin density oscillation (Fig,7.2.3)
Magnetic interaction may be set up between different impurity atoms
This is an indirect interaction called
Ruderman-Kittel-Kasaya-Yoshida (RKKY) interaction
E12 ≈ JS2 ·s(r) ≈ 3pin
2
64EF
cos(2kFr)
(kFr)3 J
2S1 ·S2 (13.3.8)
This long-range oscillation of positive and negative couplings used
to explain the mechanism of ferromagnetism,antiferromagnetism and
many exotic magnetic configurations of the rare earth solids
and also the behavior in spin glasses
Problem,give a summary to various exchange interactions,
their physical origins and applications.
13.3.3 Kondo Effect
Scattering of conduction electrons classified into two types:
Ordinary potential scattering and spin-flip scattering by magnetic impurity
H =?
parenleftbiggJ
N
parenrightbigg
s·Sjδ(r?Rj) (13.3.9)
J < 0 → the coupling is antiferromagnetic
Fig,13.3.3
Spin scattering may produce an extra sharp resonance
in density of states at the Fermi energy and a broader resonance
due to the coupling of the impurity level to the conduction band
k'k
Figure 13.3.3 Spin flip scattering of a localized impurity moment by a conduction
electron (the base line indicates the spin state of the impurity).
Resistivity of a material is dramatically influenced
by this extra resonance → Kondo effect
Variation of electrical resistance with temperature is
an important characteristic for materials,shown in Fig,13.3.4
The resistance minimum of the noble metals was discovered in the1930s
identified as the effect of transition metal impurities in the 1950s
explained by J Kondo theoretically in 1964
(b)
(a)
(c)
~10 K
ρ
T
Figure 13.3.4 Resistance vs,temperature curves for metals,curve (a) shows the
residual resistance independent of temperature due to impurities and defects in spec-
imen; curve (b) shows the normal state to superconductor phase transition; curve (c)
shows Kondo effect.
Kondo effect is a many-electron effect in which localized electrons with spins
interact with conduction electrons in the band
The resistivities due to impurity and lattice vibrations are
ρ = aT5 + ρimp = aT5 + cimp
parenleftbigg
ρ0?4|J|gF ln 2kBTB
parenrightbigg
(13.3.10)
Tmin =
parenleftbigg4|J|g
F
5a
parenrightbigg15
c15imp (13.3.11)
In Kondo’s original paper a full comparison with experimental results of Fe
in Au,as shown in Fig,13.3.5
Specific heat and magnetic susceptibility show similar behavior
0.030
0.032
0.034
0.074
0.076
0.078
0.080
0.082
0.084
0.086
0.088
0.090 0.200
0.198
0.196
0.194
0.192
0.190
0.188
0.186
0.184
0 1 2 3 4
0.002at,%
0.006at,%
0.02at,% Fe
T (K)
ρ tmp
(μ?
,cm)
AuFe
Figure 13.3.5 Comparison of experimental results for the resistivity of dilute Fe in
Au at very low temperatures with logarithmic form,reproduced from the paper of
Kondo in 1964.
Kondo’s perturbation theory is not good when T → 0 K
A,A,Abrikosov et al,dealt this problem by introducing higher order
perturbation terms,so-called the most divergent sum method
The magnetic susceptibility
χimp = (gLμB)
2S(S + 1)
3kBT
braceleftbigg
1 + 2JgF1?2Jg
F ln(2kBT/B)
+ c2 (2JgF)2
bracerightbigg
(13.3.12)
By introducing Kondo temperature
kBTK = B2 exp
parenleftbigg
12|J|g
F
parenrightbigg
(13.3.13)
χ → ∞ → near TK,perturbation calculations become invalid
Kondo gave a correct physical explanation of resistance minimum
and other anomalous physical properties due to transition metal impurities
Now new question is the physics for systems near or below TK
so-called Kondo problem
Scaling theory (Anderson,late 1960s) and
renormalization group method (K,G,Wilson,early 1970s)
were used to Kondo problem
The results from RG method
Low-temperature magnetic susceptibility
χimp(T) = ω(gLμB)
2
4kBTK
braceleftBigg
1?ax
parenleftbigg T
TK
parenrightbigg2
+ O
parenleftbigg T
TK
parenrightbigg4bracerightBigg
(13.3.14)
Wilson number ω = 0.5772
Low-temperature electrical resistance
ρimp(T) = ρ0
braceleftBigg
1?aR
parenleftbigg T
TK
parenrightbigg2
+ O
parenleftbigg T
TK
parenrightbigg4bracerightBigg
(13.3.15)
Whole picture for the Kondo problem:
(1) at T greatermuch TK,the d-spin is free
its contribution to magnetic susceptibility is Curie-like
(2) as temperature is lowered,in a wide crossover regime around TK
spin-flip scattering events becomes more frequent → resistance minimum
(3) at low temperature,T lessmuch TK,spin-compensation is nearly complete
impurity spin with its compensating electron spin cloud is strongly binding
together forming a Kondo singlet and appear to be non-magnetic
Electrons are scattered by the big singlet with strong potential scattering
but the spin-flip scattering disappears
13.3.4 Heavy-Electron Metals and Related Materials
Heavy-electron metals are rare earth or actinide compounds
showing metallic behavior with a variety of
anomalous thermodynamic,magnetic and transport properties
Theprominentexamplesofheavy-electronmetalsareCeAl3,CeCuSi2,CeCu6,
UBe13,UPt3,UCd11,U2 Zn17,and NpBe13,containing Ce,U and Np
At low temperatures,f-electron magnetic moments are strongly coupled
to conduction electrons whole effective mass becomes heavy
typically about 10 to 100 times the bare electron mass
Fig,13.3.6 and Fig,13.3.7
CeAl3
0.005 0.01
T 2 / K2
0.7
0.9
1.1
0
0 100 200 300
T (K)
0
50
100
150
200
250
ρ (
μ?
,cm)
ρ (
μ?
,cm)
U2Zn17
CeCu2Si2
UBl12 CeAl3
Figure 13.3.6 Temperature dependence of electrical resistivity R(T) for CeAl3,
UBe13,CeCu2Si2 and U2Zn11,The inset shows the T2 behavior for CeAl3 for T < 0.1
K.
T / K0 1
0 100 200
T (K)
CeAl3
27
28
UBe12
CeAl3
0
100
200
300
χ-1
(mol/10J
,T-2
)
χ-1
Figure 13.3.7 Inverse susceptibility 1/χ versus T for CeAl3 and UBe13,The Curie-
Weiss form above 100K corresponds to local moment behavior of f electrons,The
inset shows saturation indicating itinerant f electron behavior at low T.)
For explanation of anomalous properties of heavy-electrons
to generalize Kondo Hamiltonian to Kondo lattice model (KLM)
HKLM =?t
summationdisplay
ijσ
(c?iσcjσ + h.c.)?J
summationdisplay
i
si ·Si (13.3.16)
Kondo lattice model is one of the standard models for heavy electrons
Mean-field phase diagram for the Kondo coupling J/t versus
the conduction-electron number nc is shown in Fig,13.3.8.
There are many possible ground states for Kondo-like materials,
except metallic state,one is Kondo insulator,and other is superconductor
0.5 1.00.0
2.0
1.0
0.0
J/6
t Kondo Singlet
FM AFM
nc
Figure 13.3.8 Schematic phase diagram,Kondo coupling versus electron density,
obtained by the mean-field treatment for the three-dimensional Kondo lattice model.
§13.4 Outlook
Bad metals is important for high-tech
Control conductivity
Semiconductors as doped band insulators
High Tc superconductors and colossal magnetoresistors
as doped Mott insulators
The study of compounds with 4 or 5 elements just in the beginning
Potential for further research in materials with strongly correlated electrons
13.4.1 Some Empirical Rules
Some empirical rules used as maps to guide the course
Elements with unfilled d and f shells are closely related with narrow bands
Fig,13.4.1 and Fig,13.4.2
Intensive studies of physical properties of compounds with many components
began with the discovery of high Tc superconductors
Only a few have been studied in detail
Explore and study the fertile field of complex compounds
with superior or peculiar physical properties is
one of important topics of condensed matter physics
Tl
In
Ga
Lu
Y
Sc
Th
Cf LuLa
f14 d10
0.0
0.1
0.2
0.3
0.4
0.5
(r/r
WS
)3
s1 d2f1 p1d1
Figure 13.4.1 Ratio of l-shell volume to Wigner-Seitz volume in elements with un-
filled f or d shells.
4f
5 f
3d
4d
5d
La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr
Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn
Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd
Ba Lu Hf Ta W Re Os Ir Pt Au Hg
Localization
Shell FullShell Partly FilledShell Empty
Delocalization
Figure 13.4.2 Modified periodic table with vertical column (4f-5f-3d-4d-5d).
13.4.2 Theoretical Methods
Strongly correlated electrons belong to a fertile field for theorists
To introduce new ideas,new models and new methods
How to reconcile density functional theory with
the model Hamiltonians,such as Hubbard model and Anderson model
An important direction of recent progress consists of
combination of these two different types of theoretical approaches
LDA+U method,add a mean field Hubbard term to LDA functional
V,I,Anisimov,E,Aryasetiaswan,and A,I,Lichtenstein,1997,J,Phys.
Condens,Matter,9,767
Further,LDA++ method and LDA-DMFT (dynamical mean field theory)
were developed
Combination of two approaches can solve difficult problems of
strongly correlated electrons
S,Y,Sarasov,G,Kotliar,and Abrahams,2001,Nature 410,793
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
April 22,2004
Part II
Wave Behavior in Various Structures
5,Wave Propagation in Periodic and
Quasiperiodic Structures
6,Dynamics of Bloch Electrons
7,Surface and Impurity Effects
8,Transport Properties
9,Wave Localization in Disordered Systems
10,Mesoscopic Quantum Transport
Part III
Bond,Band and Beyond
11,Bond Approach
12,Band Approach
13,Correlated Electronic States
14,Quantum Confined Nanostructures
Part IV
Broken Symmetry and Ordered Phases
Order is heav’n’s first law···
— Alexander Pope
Symmetry cannot change continuously:
what I have called the first theorem
of condensed matter physics.
— P,W,Anderson (1981)
This part concerned with various kinds of phase transitions
15,Landau Theory of Phase Transitions
16,Crystals,Quasicrystals and Liquid Crystals
17,Ferromagnets,Antiferromagnets and
Ferrimagnets
18,Superconductors and Superfluids
19,Broken Ergodicity
Contents
Chapter 15,Landau Theory of Phase Transitions 3
§15.1 Two Important Concepts,,,,,,,,,,,,,,,,,,,8
15.1.1 Broken Symmetry,,,,,,,,,,,,,,,,,,,,9
15.1.2 Order Parameter,,,,,,,,,,,,,,,,,,,,24
15.1.3 Statistical Models,,,,,,,,,,,,,,,,,,,,31
§15.2 Second-Order Phase Transitions,,,,,,,,,,,,,,,39
15.2.1 Series Expansion of Free Energy,,,,,,,,,,,,40
15.2.2 Thermodynamic Quantities,,,,,,,,,,,,,,,45
15.2.3 System with a Complex Order Parameter,,,,,,,48
§15.3 Weak First-Order Phase Transitions,,,,,,,,,,,,,53
15.3.1 Influence of External Field,,,,,,,,,,,,,,,53
15.3.2 Landau-Devonshire Model,,,,,,,,,,,,,,,53
15.3.3 Landau-de Gennes Model,,,,,,,,,,,,,,,,53
15.3.4 Coupling of Order Parameter with Strain,,,,,,,53
§15.4 Change of Symmetry in Structural Phase Transitions,,,,54
15.4.1 Density Function and Representation Theory,,,,,54
15.4.2 Free Energy Functional,,,,,,,,,,,,,,,,,54
15.4.3 Landau Criteria,,,,,,,,,,,,,,,,,,,,,54
15.4.4 Lifshitz Criterion,,,,,,,,,,,,,,,,,,,,54
Chapter 15
Landau Theory of Phase Transitions
ìe??

u|b
8èD÷>%
Phase transitions are cooperative phenomena
Involved in a large amount of particles
Macroscopic variable T or P changed→
Global change of structure and physical properties
A successful theory is base on mean-field method
Detailed description needs microscopic models
Phenomenon,rule and theory
Mean-field theory has been discovered for many times
and used in many kinds of phase transitions
van der Waals for vapor-liquid transition (1873)
Weiss for paramagnetism-ferromagnetism transition (1907)
Bragg-Williams for order-disorder transition in alloys (1934)
Bardeen-Cooper-Schrieffer for superconductivity theory (1957)
Landau (1908-1968),a Russian physicist
Landau’s second-order phase transition theory (1937)
A phenomenological theory based on thermodynamic principles
Unifies various mean-field theories
Its simplicity of formalism and universality of applications
§15.1 Two Important Concepts
In Landau theory,two important and general concepts
Broken symmetry and Order parameter
They are closely related to each other
15.1.1 Broken Symmetry
Phase transition is usually accompanied by change of symmetry
What is symmetry?
Concrete and Abstract
Symmetry H,Weyl,1951
Gè
?
ús
/?2N
'?24A8
ê?????
òà"Oà
,&-?
Symmetry←invariance of physical quantities under operations
All these operations form a closed set called symmetric group
Table 15.1.1 enumerates some continuous symmetric operations
Take liquid state as an example
Its physical properties are invariant to arbitrary translation and rotation
composed of the Euclidean group E(3)
A physical system is described byH,so
the symmetry is related to invariance ofH
For liquid state,His invariant under group E(3)
TakeG0 to denote the symmetric group ofH
Let g be an element of the group,g∈G0,then
Hg = gH,or g?1Hg =H (15.1.1)
At high T,all microscopic states are equally accessible
Hand Φ have same symmetry
Table 15.1.1 Transformations with various continuous symmetries
symmetric infinitesimal finite characteristics of
group generator transformation transformation
translation k eik·r translation r in wavevector
k direction
rotation L eiφn·L rotation φ along n axis
with angular momentum L
spin precession S eiχn·S rotation χ along n axis
when spin S precesses
gauge N eiθN phase change θ caused by the
action of the number operator N
When macroscopic conditions changed,T ↓,or P ↑,or E (H)negationslash= 0
one or more symmetric elements disappear→broken symmetry
The system does not have the full symmetry possessed byH
Magnetic system is a well-known example
T >Tc and H = 0,M = 0,high symmetric phase
T <Tc,spontaneous magnetization M negationslash= 0,symmetry decreases
Phase transitions are the many-body problem
Different kind of interactions lead to different ordered phases
through symmetry breaking,when T ↓,or P ↑,or E (H)negationslash= 0
Table 15.1.2
As T ↓,a system may display many different ordered phases
At zero temperature,it is the ground state
For structural phase transition as T ↓
from liquid state to a crystalline state
continuous translation and rotation symmetries are broken
Crystal with symmetry groupG,one of 230 possible space groups
In many cases,especially of second-order phase transitionsG?G0
Assume|G〉is the ground state
h|G〉=|G〉for h∈G; g|G〉negationslash=|G〉for g /∈G
even though g∈G0
Table 15.1.2 Broken Symmetry and Ordered Phases
phase broken symmetry order parameter
crystal translation and rotation ρ = summationtextGρGeiG·r
nematic rotation ηij = 12(3ηiηj?δij)
smetic rotation and 1D translation ηij = A|ψ|cos(qz?φ)
ferroelastic inversion P
antiferroelastic inversion summationtextp (sublattice)
ferromagnetic time reversal M
antiferromagnetic time reversal summationtextm (sublattice)
superfluid 4He gauge (U(1) group) ψ =|ψ|eiθ
superconductivity gauge (U(1) group) ψ =|ψ|eiθ
In the cases with degenerate ground states
spontaneous broken symmetry→to choose one of them
Two types of broken symmetries:
one is spontaneous breaking keepingHinvariant
the other is externally disturbed asH→H+Hprime
Hprime is an additional perturbation to originalH
Landau emphasized the importance of broken symmetry
A given symmetry element is either there or it is not
In each state there is either one symmetry or the other
the situation is never ambiguous
Like liquid and crystal or different crystalline states
It is impossible to change symmetry gradually
When symmetry is broken,there is concurrent ordering
Most of phase transitions?sudden change of symmetry
If there are phase transitions without change of symmetry?
The answer is yes !
Vapor-liquid transition,liquid-glass transition,
paramagnetic-spin glass transition,metal-insulator transition
are all unrelated to broken symmetry
They are involved in more general concept broken ergodicity
Table 15.1.3 Broken Symmetry and Ordered Phases
phase broken symmetry order parameter
crystal translation and rotation ρ = summationtextGρGeiG·r
nematic rotation ηij = 12(3ηiηj?δij)
smetic rotation and 1D translation ηij = A|ψ|cos(qz?φ)
ferroelastic inversion P
antiferroelastic inversion summationtextp (sublattice)
ferromagnetic time reversal M
antiferromagnetic time reversal summationtextm (sublattice)
superfluid 4He gauge (U(1) group) ψ =|ψ|eiθ
superconductivity gauge (U(1) group) ψ =|ψ|eiθ
15.1.2 Order Parameter
Phase transitions = loss or gaining of some symmetry elements
A quantitative description of phase transition in a system
Order parameter,a physical quantity
η = 0 in the high symmetry phase
ηnegationslash= 0 in the low symmetry phase
In structural phase transition,the amount of atomic displacement
For magnetic transition,macroscopic magnetization appears
High symmetry phase is disordered phase
Low symmetry phase is ordered phase
Order parameter is a macroscopic quantity to quantify ordered phase
a thermodynamic variable as the ensemble average of
some microscopic variables σi(t)
Two kinds of phase transitions:
(1) first-order transition,or discontinuous phase transition
order parameter appears suddenly below Tc
Figure 15.1.1,BaTiO3,Tc = 120? C
cubic-tetragonal transition with electric polarization
(2) second-order transition,or continuous phase transition
order parameter appears gradually below Tc
Figure 15.1.2,SrTiO3,Tc = 110 K
cubic-tetragonal by a tilt of neighboring oxygen octahedra
Order parameter can be a scalar,or vector,or tensor,or multicomponents
Scalar order parametern = 1,like (c/a?1) and?/?0 in BaTiO3 and SrTiO3
A vector order parameter n = 3,macroscopic magnetization M
For two-dimensional isotropic ferromagnet,the magnetization n = 2
Similarly,for superfluids and superconductors,n = 2
macroscopic wavefunction,ψ = ψ0 exp(iθ) as order parameter
For vapor-liquid phase transition,no change of symmetry
Density difference ρl?ρg taken as order parameter,Fig,15.1.3
O
Ba
Ti
0.5 1.0
0.5
1.0
T/Tc
0
(a) (b)
(c/a
-1) / 10
2
Figure 15.1.1 First-order phase transition in BaTiO3,c/a is the ratio of lattice
constants.
(a)
0 0.5 1.0
0.5
1.0
T/Tc
/
0
(b)
O
Ti
Sr
Figure 15.1.2 Second-order phase transition of SrTiO3,? is the tilting angle of the
oxygen octahedra,?0 = 1.3? is the maximum value of the tilting angle.
Order Parameter
ρl(T)
Tc T
ρc
ρ
ρg(T)
Figure 15.1.3 The ρ-T phase diagram of gas-liquid.
15.1.3 Statistical Models
Physical realization of order parameters at microscopic level
In most cases,internal interaction is the main reason of
spontaneous symmetry breaking when T <Tc
Internal interactions suppress thermal fluctuation and
leads to an internal field conjugate to the order parameter
driving the entire system into an ordered state
Phase transitions are cooperative phenomena in nature
It is necessary to employ microscopic theories
beyond simple thermodynamic theories→statistical models
Can statistical model be used to describe phase transition?
In 1935,international conference on statistical physics,50% - 50%
They are simple but include enough information of many-body interactions
It is conventional and convenient to use magnetic language
write the model Hamiltonian in terms of spin variables
and are applicable to many non-magnetic systems
Heisenberg Hamiltonian
H=?
summationdisplay
ij
JijSi·Sj?H·
summationdisplay
i
Si (15.1.2)
exchange energy J and applied field H
Rewritten in the form
H=?Jz
summationdisplay
ij
SziSzj?J⊥
summationdisplay
ij
(SxiSxj +SyiSyj)?H
summationdisplay
i
Szi (15.1.3)
For J⊥ = 0→Ising model,Jz = 0→XY model
In all three cases,ordered state characterized by
occurrence of a mean magnetic moment vectors on the sites
Ising Hamiltonian
H=?J
summationdisplay
ij
SiSj?H
summationdisplay
i
Si (15.1.4)
positive J favors parallel and negative J antiparallel alignment
MnF2 is a good approximation using Ising model
Ising model is widely applicable to interacting two-state systems
such as order-disorder transitions in binary alloys in§20.2.1
For H = 0 Ising spin system involves in Z2 group
Z2 ={E,I}
identity transformation E and inversion transformation I
At T >Tc,η = 0,so Eη = η,and Iη = η,satisfying Z2 symmetry
At T <Tc,ηnegationslash= 0,so Eη = η,Iη =?η,Z2 symmetry is broken
The thermodynamic state below Tc lacks the full symmetry ofH
For the XY model
Si = iSix +jSiy = S(icosθi +jsinθi)
Take the average of Si as the order parameter
η =〈Si〉
T >Tc,spin vectors are randomly distributed satisfyingO(2) group
Orientational angle in two-dimensional plane can take any value
η = 0
T <Tc,the symmetry ofO(2) group is broken
If J⊥> 0,average θi takes a definite value
η =〈S〉(icosθ+jsinθ)
Broken symmetry comes from interactions of spins at different sites i and j
Si·Sj = S2 cosθij (15.1.5)
〈θij〉→0 when symmetry is broken
For an isotropic XY system,the ordered states are infinitely degenerate
Symmetry groupsO(2) and U(1) are isomorphic
Order parameter can also expressed as a complex number
η =〈S〉eiθ
Complex parameter such as macroscopic wavefunction ψ =|ψ|eiθ
used to describe superfluid and superconducting phase transition
with the breaking of gauge symmetry
§15.2 Second-Order Phase Transitions
Order of a phase transition according to Ehrenfest scheme
Landau’s phenomenological theory of second-order phase transition
based on the idea of spontaneous symmetry breaking at phase transition
Take scalar order parameter to give an illustration in principle
15.2.1 Series Expansion of Free Energy
Gibbs free energy Φ is a function of P,T and η
Continuity of change of state in a second-order phase transition
η takes arbitrarily small values near the transition point
Φ(P,T,η) = Φ0 +αη+Aη2 +Cη3 +Bη4 +··· (15.2.1)
Stability condition→Φ minimum
parenleftbigg?Φ
η
parenrightbigg
= 0,
parenleftbigg?2Φ
η2
parenrightbigg
> 0 (15.2.2)
T >Tc,η = 0→A> 0
T <Tc,ηnegationslash= 0→A< 0
At T = Tc,A = 0
A(P,T) = a(P)(T?Tc) (15.2.3)
with a(P) > 0
Phase transition point,T = Tc,itself is stable→the conditions
parenleftbigg?2Φ
η2
parenrightbigg
η=0
= 0,
parenleftbigg?3Φ
η3
parenrightbigg
η=0
= 0,
parenleftbigg?4Φ
η4
parenrightbigg
η=0
> 0 (15.2.4)
then
A(P,Tc) = 0,C(P,Tc) = 0,B(P,Tc) > 0 (15.2.5)
Assume η and?η are equivalent→C = 0 and B > 0
Φ(P,T,η) = Φ0 +A(P,T)η2 +Bη4 (15.2.6)
From?Φ/?η = 0,the equation of state
η(A+ 2Bη2) = 0 (15.2.7)
There are two solutions
η = 0 (15.2.8)
and
η =±
parenleftbigg
A2B
parenrightbigg1/2
=±
bracketleftbigga(T
c?T)
2B
bracketrightbigg1/2
(15.2.9)
Fig,15.2.1
η

T>Tc T<T
c
(a) (b)
η
Figure 15.2.1 Free energy as the function of scalar order parameter in the vicinity
of the second-order phase transition,(a) T >Tc; (b) T <Tc.
15.2.2 Thermodynamic Quantities
Phase transitions→extraordinary physical properties
Thermodynamic quantities may change drastically
Anomalies are thermal expansion coefficients,elastic constants
refractive indices,transport coefficients
For second-order phase transition,entropy,volume,etc.,vary continuously
specific heat,thermal expansion coefficient,compressibility are discontinuous
Temperature dependence of entropy S =Φ/?T
When T >Tc,η = 0
S =Φ0?T = S0 (15.2.10)
When T <Tc,ηnegationslash= 0
S = S0 + a
2
2B(T?Tc) (15.2.11)
At T = Tc,S = S0
Entropy is continuous at Tc
Specific heat CP = T(?S/?T)P
For high symmetry phase
CP = T
parenleftbigg?S
0
T
parenrightbigg
P
(15.2.12)
For low symmetry phase
CP = T
parenleftbigg?S
0
T
parenrightbigg
P
+ a
2Tc
2B (15.2.13)
At Tc,there is a discontinuous jump
CP = a
2Tc
2B (15.2.14)
Other quantities,thermal expansion coefficient,compressibility
are also discontinuous
15.2.3 System with a Complex Order Parameter
Gauge symmetry breaking→superconducting or superfluid
A macroscopic wavefunction denotes broken gauge symmetry
Complex order parameter
η = η0eiθ
Expand the free energy
Φ = Φ0 +A|η|2 +B|η|4 (15.2.15)
A = a(T?Tc) and B > 0
Minimum of free energy given by?Φ/?η0 = 0
(A+ 2Bη20)η0 = 0 (15.2.16)
(1) Normal state,T >Tc,η0 = 0,Φ = Φ0
θ takes any value,gauge symmetry is intact
(2) Superflow state,T <Tc,η0 = [a(Tc?T)/2B]1/2
θ takes a definite value,Gauge symmetry is broken
Fig,15.2.2
T>Tc T<Tc
(a) (b)
η η
Figure 15.2.2 Free energy surfaces for complex order parameter corresponding to
two cases for (a) T >Tc,and (b) T <Tc.
Transformation
η?→ηprime = eiφη (15.2.17)
corresponds to a rotation by the angle φ in the complex plane
all rotations constitute the continuous group U(1)
Φ is invariant under group U(1)
Ordered states are degenerate,the minimum of Φ lies on a circle
Symmetry is spontaneously broken,Fig,15.2.2(b)
Phase of macroscopic wavefunction arisen in condensation process
Number N of condensed particles↑as T ↓below Tc
θ and N are a couple of conjugate variables
analysis showed uncertainty relation
N?θ～1 (15.2.18)
More intricate broken gauge symmetries in
liquid crystals and superfluid 3He
§15.3 Weak First-Order Phase Transitions
15.3.1 Influence of External Field
15.3.2 Landau-Devonshire Model
15.3.3 Landau-de Gennes Model
15.3.4 Coupling of Order Parameter with Strain
§15.4 Change of Symmetry in Structural Phase Transitions
15.4.1 Density Function and Representation Theory
15.4.2 Free Energy Functional
15.4.3 Landau Criteria
15.4.4 Lifshitz Criterion
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
April 26,2004
Contents
Chapter 15,Landau Theory of Phase Transitions 3
§15.1 Two Important Concepts,,,,,,,,,,,,,,,,,,,4
15.1.1 Broken Symmetry,,,,,,,,,,,,,,,,,,,,4
15.1.2 Order Parameter,,,,,,,,,,,,,,,,,,,,4
15.1.3 Statistical Models,,,,,,,,,,,,,,,,,,,,4
§15.2 Second-Order Phase Transitions,,,,,,,,,,,,,,,5
15.2.1 Series Expansion of Free Energy,,,,,,,,,,,,5
15.2.2 Thermodynamic Quantities,,,,,,,,,,,,,,,5
15.2.3 System with a Complex Order Parameter,,,,,,,5
§15.3 Weak First-Order Phase Transitions,,,,,,,,,,,,,6
15.3.1 Influence of External Field,,,,,,,,,,,,,,,8
15.3.2 Landau-Devonshire Model,,,,,,,,,,,,,,,15
15.3.3 Landau-de Gennes Model,,,,,,,,,,,,,,,,23
15.3.4 Coupling of Order Parameter with Strain,,,,,,,28
§15.4 Change of Symmetry in Structural Phase Transitions,,,,34
15.4.1 Density Function and Representation Theory,,,,,35
15.4.2 Free Energy Functional,,,,,,,,,,,,,,,,,40
15.4.3 Landau Criteria,,,,,,,,,,,,,,,,,,,,,43
15.4.4 Lifshitz Criterion,,,,,,,,,,,,,,,,,,,,47
Chapter 15
Landau Theory of Phase Transitions
§15.1 Two Important Concepts
15.1.1 Broken Symmetry
15.1.2 Order Parameter
15.1.3 Statistical Models
§15.2 Second-Order Phase Transitions
15.2.1 Series Expansion of Free Energy
15.2.2 Thermodynamic Quantities
15.2.3 System with a Complex Order Parameter
§15.3 Weak First-Order Phase Transitions
Landau theory
successful to
second-order phase transitions
extended to
weak first-order phase transitions
order parameter still effective
￡?é x?éó
,?Q?ü

;?R?
15.3.1 Influence of External Field
Conjugated variables:
P and V in vapor-liquid transition
H and M in paramagnetic-ferromagnetic transition
E and P in paraelectric-ferroelectric transition
σ and ε in paraelastic-ferroelastic transition
Consider a conjugate field h of the scalar order parameter η
Φh(P,T,η) = Φ0 +a(T?Tc)η2 +Bη4?ηh (15.3.1)
Figure 15.3.1
Equilibrium condition?Φh/?η = 0
→ Equation of state
2a(T?Tc)η+ 4Bη3?h = 0 (15.3.2)
Figure 15.3.1
h
T>Tc T>Tc T<T
cT<Tc
η
Figure 15.3.1 Asymmetric free energy under external field.
Tc T
η
Figure 15.3.2 Phase diagram of η vs T under the fixed external field h,dot line
corresponds to h = 0.
Susceptibility χ = (?η/?h)T,h→0
χ = 12a(T?T
c) + 12Bη2
(15.3.3)
T >Tc
χ = 12a(T?T
c)
(15.3.4)
T <Tc,Curie-Weiss law
χ = 14a(T
c?T)
(15.3.5)
T →Tc,χ→ ∞
Figure 15.3.3,η-h
T<Tc
T=Tc
T>Tc
A
B
B'
A'
D'
h
η
O
D
Figure 15.3.3 Phase diagram of η vs T under the fixed external field h,dotted line
corresponds to h = 0.
Solid lines,stable states; dashed lines,unstable states
Segments A-B and Aprime-Bprime,metastable states
Segments B-O and Bprime-O,unstable states with?2Φ/?h2 < 0
or with an inverse susceptibility χ?1
χ?1 =
parenleftbigg?h
η
parenrightbigg
η=0
=
parenleftbigg?2Φ
η2
parenrightbigg
η=0
(15.3.6)
Discontinuities between the states B-Dprime and D-Bprime
Hysteresis loop D-A-B-Dprime-Aprime-Bprime
Ccoercive field = (hBprime?hB)/2
First-order phase transition appears when T <Tc
15.3.2 Landau-Devonshire Model
Devonshire,1949,found in many ferroelectrics
Spontaneous polarization,weak first-order phase transitions
Polarization is taken as scalar order parameter
Free energy should be expanded to higher terms
Suppose B < 0,expand the free energy in 6th power
Φ(P,T,η) = Φ0 +a(T?Tc)η2 +Bη4 +Dη6 (15.3.7)
D> 0
O
Ba
Ti
0.5 1.0
0.5
1.0
T/Tc
0
(a) (b)
(c/a
-1) / 10
2
Coefficient A = a(T?Tc) is kept invariant
Tc is not a transition temperature
Equilibrium condition?Φ/?η = 0 → Equation of state
2a(T?Tc)η + 4Bη3 + 6Dη5 = 0 (15.3.8)
Solutions are
η = 0 (15.3.9)
η2 =?B + [B
2?3aD(T?Tc)]1/2
3D (15.3.10)
and
η2 =?B?[B
2?3aD(T?Tc)]1/2
3D (15.3.11)
The condition for real roots
T+ = Tc + B
2
3aD >Tc (15.3.12)
For T <T+,(15.3.10) is a solution to minimize the free energy
(15.3.11) is meaningless for ordered phase
T+ is not a transition temperature
Real transition temperature T = Tt is determined by Φ?Φ0 = 0
a(T?Tc)η2 +Bη4 +Dη6 = 0 (15.3.13)
From the condition of real root
Tt = Tc + B
2
4aD (15.3.14)
Tc <Tt <T+
At T = Tt,three minimums of Φ,η = 0,and η = ±(?B/2D)1/2
Figure 15.3.4
T >T+,η = 0 → minimum of Φ,disordered phase is stable
T+ >T >Tt,η = 0 and η negationslash= 0 for Φ taking equilibrium values
still the disordered phase is stable,ordered phases are metastable
T
η
Figure 15.3.4 Free energy vs order parameter in the Landau-Devonshire theory.
At T = Tt,Φ?Φ0 = 0,first-order phase transition
The polarization is changed discontinuously from zero to a finite value
η2 = B2D (15.3.15)
The change of entropy
S =?Φ?TΦ0?T = aB2D (15.3.16)
T <Tt,disordered phase unstable,ordered phase stable
At T = Tc,?Φ/?η = 0 and?2Φ/?η2 = 0,so η = 0 is spinodal point
Tc corresponds to absolutely unstable limit of disordered phase
η = ±(?2B/3D)1/2 are perfectly stable
n
15.3.3 Landau-de Gennes Model
The microscopic theory of isotropic-nematic phase transition
of liquid crystals in §16.3
η denotes the orientational order parameter
This phase transition is first-order
de Gennes,1971,gave a phenomenological description
Φ(P,T,η) = Φ0 +a(T?Tc)η2 +Cη3 +Bη4 (15.3.17)
C < 0,still B > 0,Tc is for C = 0
Fig,15.3.5
T>T
c T=Tc T<Tc
η
Figure 15.3.5 Free energy vs order parameter in Landau-de Gennes theory.
Equilibrium condition?Φ/?η = 0 → Equation of state
2a(T?Tc)η+ 3Cη2 + 4Bη3 = 0 (15.3.18)
The solutions are
η = 0 (15.3.19)
η =?3C + [9C
2?32aB(T?Tc)]1/2
8B (15.3.20)
η =?3C?[9C
2?32aB(T?Tc)]1/2
8B (15.3.21)
Real root condition
T+ = Tc + 9C
2
32aB (15.3.22)
T >T+,η = 0 stable; T <T+,a metastable minimum for η negationslash= 0
First-order phase transition point from Φ?Φ0 = 0
Tt = Tc + C
2
4aB <T+ (15.3.23)
Two stable minima at T = Tt,η = 0 and η negationslash= 0
A jump of order parameter at Tt
η =? C2B (15.3.24)
Absolute unstability appears at T ≤Tc
T = Tc is spinodal point for η = 0,?2Φ/?η2 = 0
From equilibrium condition,η =?3C/4B
Problem,To verify and illuminate by yourself that
Landau-Devonshire model and Landau-de Gennes model
are all first-order transition
15.3.4 Coupling of Order Parameter with Strain
Structural phase transitions,interplay of ε and η
An interaction of η2ε is reasonable choice
Φ = Φ0 +a(T?Tc)η2 +Bη4 +Jη2ε+ 12Kε2 (15.3.25)
coupling constant J and elastic constant K
Φ/?η = 0 gives
a(T?Tc) + 2Bη2 +Jε = 0 (15.3.26)
Equation of state for the variable ε
σ =
parenleftbigg?Φ
ε
parenrightbigg
η,T
= Jη2 +Kε (15.3.27)
In the case of no external stress σ = 0
ε =?Jη
2
K (15.3.28)
η2 =?a(T?Tc)2B? (15.3.29)
B? = B? J
2
2K (15.3.30)
The equilibrium value ε depends on temperature linearly
The inverse susceptibility
χ?1 =
parenleftbigg?σ
ε
parenrightbigg
σ=0
= K + 2Jη
parenleftbigg?η
ε
parenrightbigg
σ=0
(15.3.31)
Discontinuous change at the transition point
χ?1 = K,for T >Tc (15.3.32)
and
χ?1 = K? J
2
2B,for T <Tc (15.3.33)
Fig,15.3.6
Tc T
1
K
χ
Figure 15.3.6 Susceptibility versus temperature for the model with coupling between
stain and order parameter.
A conjugate field h to order parameter η
h =
parenleftbigg?Φ
η
parenrightbigg
ε,T
= 2a(T?Tc)η+ 4Bη3 + 2Jηε (15.3.34)
The inverse susceptibility
χ?1η = 2a(T?Tc) + 12Bη2 + 2Jε (15.3.35)
T >Tc,η = 0,ε = 0
χ?1η = 2a(T?Tc) (15.3.36)
T <Tc,η and ε take equilibrium values
χ?1η = 4aBB?(Tc?T) (15.3.37)
Curie-Weiss law
Substitute (15.3.28) into (15.3.25),the free energy
Φ = Φ0 +a(T?Tc)η2 +B?η4 (15.3.38)
B >B? > 0,the phase transition is second-order
If the coupling is strong,B? <B,Dη6 needs to be included
The coupling of η-ε may drive phase transition
from second-order to first-order
§15.4 Change of Symmetry in Structural Phase Transitions
Landau theory is most significant in structural phase transitions in crystals
A prerequisite is the theory of space group of crystals
T ↓ → crystal undergo a series of structural phase transitions
A generalized order parameter with multicomponents
15.4.1 Density Function and Representation Theory
Crystal symmetry ↓ → a structural phase transition →
a new crystal structure
Begin from ρ(r) to describe crystal structures
To ascertain the symmetry of crystals
Initial high symmetry group G0? ρ0(r)
Below Tc,for the low symmetry phase
ρ(r) = ρ0(r) +δρ(r) (15.4.1)
described by G? G0
Landau’s theory is to expand ρ(r) or δρ(r)
in a complete set ψνi (r) of IR of G0
δρ(r) =
summationdisplay
ν
primesummationdisplay
i
ηνiψνi (r) (15.4.2)
In general,second-order phase transition is only related to one IR
δρ(r) =
dsummationdisplay
i=1
ηiψi(r) (15.4.3)
expansion coefficient ηi,{η1,···,ηd}
as a vector order parameter,η,transforming according to IR
T >Tc,all ηi = 0; T <Tc,at least one ηi negationslash= 0
To study structural phase transitions in crystals →
to find IR of space group
A successful method is to adopt the modulated wave vector
Space group IRs specified by wavevectors
related to reciprocal lattice vectors with their symmetry
To classify all the points belong to a Brillouin zone
Lifshitz points with particularly high symmetry
IRs of G associated with its elements g on a given wavevector k
Point symmetry operation (rotation and reflection)
gk = k (15.4.4)
nonsymmorphic operation (glide-reflection or screw rotation)
gk = k+G (15.4.5)
All such elements forms the wavevector group Gk? G0
IRs of Gk characterized by k and ν
IRs of G0 characterized by a wavevector star {k}
star arms kL
kL = gLk (15.4.6)
Coset decomposition of G0
G0 =
lksummationdisplay
L=1
gLGk (15.4.7)
The basis functions of Gk IR are Bloch functions
ψνkλ(r) = uνkλ(r)eik·r (15.4.8)
λ = 1,2,...,dν
gψνkλ(r) =
dνsummationdisplay
μ=1
Γνkμλ(g)ψνkμ(r) (15.4.9)
A basis of IR of G is generated by the set of Bloch functions
{ψνk1λ},{ψνk2λ},...,{ψνkl
kλ
} prescribed on all star arms k1,k2,...,klk
15.4.2 Free Energy Functional
Φ = Φ(P,T,ρ(r)) (15.4.10)
transformed according to IR in (15.4.3)
Fix ψi and let {ηi} transform
Φ = Φ(P,T,{ηi}) (15.4.11)
ηi found by equilibrium condition and form multicomponent order parameter
High symmetry phase T ≥Tc,δρ = 0,all ηi = 0
Low symmetry phase T <Tc,δρnegationslash= 0,at least one ηi negationslash= 0
As T →Tc,δρ→ 0,ηi → 0
Φ expanded with {ηi} near Tc
Polynomial expansions in powers of a multicomponent order parameter
Introduce normalization definition
ηi = ηγi,
summationdisplay
i
γ2i = 1 (15.4.12)
η2 =
summationdisplay
i
η2i (15.4.13)
T >Tc,η = 0
T <Tc,η increases continuously from zero
Φ = Φ0(P,T)+η2A(P,T)+η3
summationdisplay
α
Cα(P,T)I(3)α (γi)+η4
summationdisplay
α
Bα(P,T)I(4)α (γi)
(15.4.14)
Φ = Φ0 +A(P,T)
summationdisplay
i
η2i = Φ0 +A(P,T)η2 (15.4.15)
Φ/?η = 0,and?2Φ/?η2 > 0
T >Tc,A> 0 → ηi = 0
T <Tc,A< 0,ordered state occurs,at least one ηi negationslash= 0
15.4.3 Landau Criteria
Landau proposed and solved,in general form,which of initial group IRs
cannot give rise to a second-order phase transition
No third order invariants constituted by ηi for IR
Landau criterion
I(3)α (γi) = 0 (15.4.16)
At T = T?c,A(P,Tc) = 0
Φ = Φ0 +η3Cα(P,T)I(3)α (γi) +η4Bα(P,T)I(4)α (γi) (15.4.17)
For brevity,only one third order term and one fourth order term,?Φ/?η = 0
3I(3)α (γi)Cα(P,T)η2 + 4I(4)α (γi)Bα(P,T)η3 = 0 (15.4.18)
High symmetry phase
η = 0,→Φ = Φ0 (15.4.19)
Low symmetry phase
η =?3I
(3)
α (γi)Cα(P,T)
4I(4)α (γi)Bα(P,T)
→Φ = Φ0? 3
2
43
[I(3)α (γi)Cα(P,T)]4
[I(4)α (γi)Bα(P,T)]3
(15.4.20)
It must be assumed
Bα(P,T)I(4)α (γi) > 0 (15.4.21)
otherwise for
Bα(P,T)I(4)α (γi) < 0 →Φ>Φ0
or
Bα(P,T)I(4)α (γi) = 0 →Φ→?∞
unreasonable,there is no stable solution
It is clear at T = T?c,η negationslash= 0,the order parameter
jumps from 0 to?3I(3)α Cα/4I(4)α Bα discontinuously
This is not a second-order phase transition,except 3I(3)α Cα = 0
Φ = Φ0 +η2A(P,T) +η4
summationdisplay
α
Bα(P,T)I(4)α (γi),(15.4.22)
Two Landau criteria for second-order phase transitions
(1) G for low symmetry phase is subgroup of G0 for high symmetric phase
(2) No third order invariant in free energy functional
15.4.4 Lifshitz Criterion
Original Landau theory,homogeneous ordered phase
η has the same value everywhere
Lifshitz demonstrated
there may be spatially inhomogeneous phases to occur
Free energy involves order parameter derivatives with coordinates
Linear invariants in derivatives → Lifshitz invariants
Density of free energy?
=?(P,T,ηi(r),?ηi(r)) (15.4.23)
Free energy
Φ =
integraldisplay
(P,T,ηi(r),?ηi(r))dr,(15.4.24)
Still use modulated wavevector
For homogeneous phase transition,at Tc
only one characteristic wavevector,k = k0
to satisfy Φ minimum for stability of new phase
If inhomogeneity to appear
k = k0 +κ (15.4.25)
κ is a small quantity,1/κ offers macroscopic inhomogeneity
In first-order approximation
parenleftbigg
P,T,ηi,?ηi?x
p
parenrightbigg
=?0(P,T,ηi) +
summationdisplay
ip
Ui,p(P,T)?ηi?x
p
+ 12
summationdisplay
ijp
Vijp(P,T)
bracketleftbigg
ηi?ηj?x
p
+ηj?ηi?x
p
bracketrightbigg
+ 12
summationdisplay
ijp
Vijp(P,T)
bracketleftbigg
ηi?ηj?x
p
ηj?ηi?x
p
bracketrightbigg
+···
(15.4.26)
The expansion coefficients
Uip(P,T) =?Φ?(?η
i/?xp)
(15.4.27)
equals zero due to the equilibrium condition,and
Vijp(P,T) =?
2Φ
ηi?(?ηi/?xp) (15.4.28)
It is clear integraldisplay parenleftbigg
ηi?ηj?x
p
+ηj?ηi?x
p
parenrightbigg
dxp ～ηiηj
can be included into the first term in (15.4.26)
The total free energy is
Φ =
integraldisplay
dr =
integraldisplay
0dr+ 12
summationdisplay
ijp
Vijp(P,T)
integraldisplay parenleftbigg
ηi?ηj?x
p
ηj?ηi?x
p
parenrightbigg
dr
(15.4.29)
From stability condition of?ηi/?xp as independent variable
δΦ
δ(?ηj/?xp) =
summationdisplay
i
Vijpηi(r) = 0 (i,j = 1,...,d) (15.4.30)
This is a set of linear equations
Low symmetry phase,ηi(r) are not all zero
Coefficient matrix V(p) = {Vijp} should satisfy
det[V(p)] = 0 (15.4.31)
Vijp is the function of P and T,it is accidental for det[V(p)] = 0
In general,we may require
summationdisplay
ijp
Vijp
integraldisplay
dxp
bracketleftbigg
ηi?ηj?x
p
ηj?ηi?x
p
bracketrightbigg
= 0 (15.4.32)
This is the Lifshitz criterion (homogeneous condition):
No Lifshitz invariant is the condition for a phase transition
from homogeneous to homogeneous phase
Lifshitz also proved
only Γ point and the end points with high symmetry of k
at boundary of Brillouin zone related to second-order phase transition
So wavevector k of the new phase is simple fraction of
the wavevector of initial phase
aprimei =
summationdisplay
ij
lijaj (i,j = 1,2,3) (15.4.33)
ai and aprimej are basic vectors of old and new phase,lij is integer
R
ST
Z
M
XΓ
Λ
Σ
Σ
Σ
Γ
Λ
M
L
U
S
XZW
K
Q
Λ
Γ
P
F
H
N
T K
P
HSA
L
x
y
z
x
y
z
x
y
z
z
x

C

D

E

F
Γ
When Lifshitz condition is unfulfilled
inhomogeneous phase may have lower free energy
incommensurate phase may appear,as in §16.2.3
In (15.4.29) Φ is only expanded to first-order derivatives?η/?xp
Lifshitz condition is not very strict
Higher order expansion may lead to the domain structure
One of the examples is the Ginzburg-Landau free energy density
(P,T,η,?η) =?0(P,T,η) +K(?η)2 (15.4.34)
K > 0
Total free energy is
Φ =
integraldisplay
(P,T,η,?η)dr (15.4.35)
From its minimum,the distribution of order parameter
and domain structure can be determined
Landau theory of phase transitions can be extended to
anothertypeof structuralphasetransitions—reconstructivephasetransitions
in which the group-subgroup relationship is entirely missing
only the concept of order parameter is kept
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
April 29,2004
Contents
Chapter 16,Crystals,Quasicrystals and Liquid Crystals 3
§16.1 Liquid-Solid Transitions,,,,,,,,,,,,,,,,,,,5
16.1.1 Free Energy Expansion Based on Density Waves,,,6
16.1.2 Crystallization,,,,,,,,,,,,,,,,,,,,,,14
16.1.3 Quasicrystals,,,,,,,,,,,,,,,,,,,,,,21
§16.2 Phase Transitions in Solids,,,,,,,,,,,,,,,,,,27
16.2.1 Order-Disorder Transition,,,,,,,,,,,,,,,28
16.2.2 Paraelectric-Ferroelectric Transition,,,,,,,,,,38
16.2.3 Incommensurate-Commensurate Transition,,,,,,46
§16.3 Phase Transitions in Soft Matter,,,,,,,,,,,,,,,47
16.3.1 Maier-Saupe Theory for Isotropic-Nematic Transition,47
16.3.2 Onsager Theory for Isotropic-Nematic Transition,,,47
16.3.3 Phase Separation in Hard-Sphere Systems,,,,,,,47
Chapter 16
Crystals,Quasicrystals and Liquid
Crystals
In Part I,to consider from geometry
Structures of crystals,quasicrystals,and liquid crystals
Broken spatial translational and orientational symmetries
In this chapter,to clarity the physical reason
Landau theory may be the first step
Furthermore microscopic theories are needed
§16.1 Liquid-Solid Transitions
A homogeneous and isotropic liquid,T ↓
mass or compositional density waves
T < Tc,some modes are locked in → ordered solid
From continuous symmetry to discrete symmetry
16.1.1 Free Energy Expansion Based on Density Waves
2D or 3D liquid with full translational and rotational symmetries
corresponding to the Euclidean group
Question,when condense into solid phase at low temperature
which is the possible ordered structure to arise
In liquid phase,ρ0 is average density,a constant
T < Tc,ρ0 → ρ0 + δρ(r) = ρ(r)
Condensed phase described by a symmetry-breaking order parameter
transforms as an IR of symmetry group of liquid phase labelled by q
Density of low-temperature ordered phase
ρ(r) = ρ0 +
summationdisplay
q
ρqeiq·r (16.1.1)
ρq are taken as the order parameters
ρq = ρq (16.1.2)
Free-energy is expanded in terms of ρq
Due to the rotational symmetry,the free energy depends only on |q|
It is reasonable to fix q to G and then ρG
Free energy of the solid Φ = Φ(P,T,ρ(r))
Φ = Φ0 + Φ1 + Φ2 + Φ3 + Φ4 +··· =
summationdisplay
n=0
Φn (16.1.3)
From the invariance of translation of Φ,r → r +R
ρG1ρG2 ···ρGn = ρG1ρG2 ···ρGn exp{i(G1 +G2 +···+Gn)·R}
R is an arbitrary constant vector
Permissible Φn can only include the terms satisfying
G1 +G2 +···+Gn = 0 (16.1.4)
Constraint relation of possible wavevectors
n = 1,G1 = 0,→ Φ1 = 0
n = 2,G1 =?G2,Φ2 satisfies
Φ2 =
summationdisplay
G
AG|ρG|2 (16.1.5)
Designating coefficient AG simply by A
Φ2 = A
summationdisplay
G
|ρG|2 (16.1.6)
Third order terms have the form
Φ3 =
summationdisplay
G1G2G3
CG1G2G3ρG1ρG2ρG3 (16.1.7)
Every term
G1 +G2 +G3 = 0 (16.1.8)
G1,G2,G3 form an equilateral triangle,all CG1G2G3 are equal
Φ3 = C
summationdisplay
G1G2G3
ρG1ρG2ρG3 (16.1.9)
In the same way Φ4 and Φ5··· can also be written out
Free energy expanding to fifth order
Φ = Φ0 + A(P,T)
summationdisplay
G
|ρG|2 + C(P,T)
summationdisplay
|Gi|=G
ρG1ρG2ρG3
+ B(P,T)
summationdisplay
|Gi|=G
ρG1ρG2ρG3ρG4
+ E(P,T)
summationdisplay
|Gi|=G
ρG1ρG2ρG3ρG4ρG5 (16.1.10)
From which to discuss the stability of a variety of structures
Wavevector combinations of some possible structures in Fig,16.1.1
(a) smectic structure,(b) rodlike triangular structures or triangular atomic
monolayers,(c) bcc structures,(d) two-dimensional Penrose structures or
three-dimensional rodlike lyotropic structures,and (e) icosahedral quasicrys-
tals,From P,Bak,Phys,Rev,Lett,54,1517 (1985)
Third-order terms in (16.1.10) are essential to liquid-solid transitions
Liquid-solid transitions are first order
Landau theory are still valid
G1
G1
G2
G1
G2
G3G4
G5
G1
G2
G3
G4
G5
(a) (b) (c)
(d) (e)
G3
G3
G2
G1
Figure 16.1.1 Several representative wavevector combinations.
16.1.2 Crystallization
A density wave in 1D
ρ(r) = 1√2ρcos(G·r) (16.1.11)
a smectic liquid crystal with G and?G in Fig,16.1.1(a)
Translational invariance is broken in one direction
Minimum of free energy satisfied by Φ2 in (16.1.5)
(a) (b) (c)
Density waves in 2D
Three waves form an equilateral triangle in Fig,16.1.1(b)
To lock triple-G,represents triangular (or honeycomb) crystal
on the surface of graphite,xenon atoms can form triangular lattice
Fig,16.1.2
In 3D,rodlike structures with 2D periodicity and
liquid translational symmetry in third direction
as observed for lyotropic mesophases
G1
G2
G3
G1
G2
G3
(a) (b)
Figure 16.1.2 Density waves for two-dimensional crystals.
From (16.1.8),G1 +G2 +G3 = 0
ρGi = 12√3ρeiθi
ρ(r) =
3summationdisplay
i=1
ρ√
3 cos(Gi ·r + θi)
Take advantage of third-order terms in (16.1.10)
Φ3 = C3√3ρ3 cos(θ1 + θ2 + θ3)
Free energy minimum
(Φtria3 )min =? C3√3ρ3
For metallic elements of IA,IIA,IIIB-VIB groups,except Mg
and almost all elements in lanthanum and actinium systems
T lessorsimilar Tm,all bcc structure → What is the general factor to control it?
S,Alexander and J,Mctaque’ s theory,Phys,Rev,Lett,41,702 (1978)
n = 12 → 6 pairs of ±Gi form an octahedron shown in Fig,16.1.1(c)
Free energy minumum → 3D bcc structure
ρGi = (1/2√6)ρexp(iθi)
ρ(r) =
summationdisplay
octa
ρ√
6 cos(Gi ·r + θi) (16.1.12)
Not all six pairs of vectors Gi are linearly independent
Free energy takes the form
Φ3 =
summationdisplay C
6√6ρ
3 cos(θi + θj + θk) (16.1.13)
can be minimized by choosing p as an integer satisfying
θi + θj + θk = pip
(Φocta3 )min =? 2C3√6ρ3 (16.1.14)
This theory answer why metallic elements under and near melting point
have bcc structure
16.1.3 Quasicrystals
In Chap,2,Al86Mn14 alloy with icosahedral point symmetry
Icosahedral ordering as a 6-G structure
a 6D space group formed by compositional density waves
Diffraction pattern is spanned by six linearly independent
reciprocal lattice vectors G1,...,G6
Landau theory confirms the stability of icosahedral symmetry
Alexander and McTague,1978
predicted the existence of icosahedral structure
Generalized Penrose structure
A slightly simpler 2D structure formed by five density waves
with wavevectors G1,...,G5 forming a regular pentagon
Five-folded rotational symmetry,no translational invariance
in Fig,16.1.1(d)
Characterized by 5D space group
Example for Penrose structure is decagonal phase of 2D quasicrystals
P,Bak,Phys,Rev,Lett,54,1517 (1985); Phys,Rev,B 32,5764 (1985)
Fifth-order term in (16.1.10) favors it,ρi = (1/2√5)ρexp(iθi)
Density becomes
ρ(r) =
5summationdisplay
i=1
ρ√
5 cos(Gi ·r + θi) (16.1.15)
Free energy
Φ5 = E25√5ρ5 cos(θ1 + θ2 + θ3 + θ4 + θ5) (16.1.16)
If E is positive,minimum of free energy is
(Φpenr5 )min =? E25√5ρ5 (16.1.17)
The structure has actually tenfold symmetry
Fig,2.4.6 shows the symmetry
AlMn quasicrystals were produced by rapid quenching
quasicrystalline state is stable metastable state
In (16.1.10) fifth order terms combining with third order terms
favor wavevectors forming regular icosahedra in Fig,16.1.1(e)
n = 30 → 15 pairs of edge vectors ±Gi define a structure
ρ(r) =
summationdisplay
i
ρ√
15 cos(Gi ·ri + θi) (16.1.18)
Φ3 = ρ
3C
15√15
summationdisplay
10triangles
cos(θi + θj + θk) (16.1.19)
Φ5 = ρ
5E
225√15
summationdisplay
6pentagons
cos(θi + θj + θk + θl + θm) (16.1.20)
If C and E are all positive,the resulting free energy
(Φ3 + Φ5)min =?2ρ
3C
3√15?
2ρ5E
75√15 (16.1.21)
Landau theory allows that the icosahedral structures may be stable
Over a certain temperature interval the quasicrystalline state
may have a lower free energy as compared with the usual crystalline state
with a bcc structure
Problem,To deduce all the formulas in §16.1.2 and §16.1.3
§16.2 Phase Transitions in Solids
Ordered structures are unstable as T ↓ continuously
Symmetry will be broken further
Phase transitions take place from solid to solid
16.2.1 Order-Disorder Transition
Order-disorder transition in substitutional binary alloys
Two metals form an alloy with interactions between n.n,sites
εAB,εAA and εBB
For T = 0 K,two possible ordered states
(a) if εAB > (εAA + εBB)/2
all atoms A are separated from atoms B
(b) if εAB < (εAA + εBB)/2
A atoms mixed with B atoms
At higher temperatures,disordered state,entropy is dominant
Φ = U?TS
in mixing two types of atoms on lattice sites
At Tc,a order-disorder phase transition will take place
Bragg and Williams’s theory,1934
for order-disorder transition in alloys
A simple case of β-brass (CuZn) in Fig,16.2.1
Second-order phase transition
T < Tc atomic order arises from a diffusive rearrangement on sites
Zn,Cu
Cu,Zn
Zn
Cu
T>Tc T<Tc
(a) (b)
Figure 16.2.1 Unit cell of CuZn alloy in the ordered and disordered phases.
Rearrangement described by a local order parameter σi
σi = pi(A)?pi(B) (16.2.1)
Local probabilities pi(A) and pi(B) satisfies
pi(A) + pi(B) = 1 (16.2.2)
Macroscopic order parameter η
η = 〈σi〉 = 1N
summationdisplay
i
σi (16.2.3)
Summation is taken over the whole subsystem
Pseudospin model for ordering in binary systems
Correlation energy
H =?
summationdisplay
ij
Jijσiσj (16.2.4)
Short-range correlation energy Ei between i and n.n,j
Ei =
summationdisplay
j
pi(A)pj(A)εAA +pi(B)pj(B)εBB+pj(A)pi(B)εAB +pj(B)pi(A)εBA
(16.2.5)
According to (16.2.1) and (16.2.2)
pi(A) = 12(1 + σi),pi(B) = 12(1?σi) (16.2.6)
Substituting into (16.2.5)
Ei =
summationdisplay
j
const.?K(σi + σj)?Jσiσj (16.2.7)
const,= 12(2εAB+εAA+εBB),K = 14(εBB?εAA),J = 14(εAA+εBB?2εAB)
Considering z nearest neighbors i = 1,...,z
Ei =?Jσi
summationdisplay
j
σj (16.2.8)
Jsummationtextj σj may be interpreted as the local field Fi
Average 〈summationtextj σj〉 replaced by zη
F = 〈Fi〉 = Jzη
A binary system consists of two subsystems characterized by ±η
invariant Gibbs free energy under inversion η →?η
An alternative description
Local probabilities averaged over all lattice sites in the subsystems
p(A) = 〈pi(A)〉,p(B) = 〈pi(B)〉
p(A) + p(B) = 1
Order parameter can be defined as
η = p(A)?p(B) (16.2.9)
p(A) = 12(1 + η),p(B) = 12(1?η)
For complete disorder,p(A) = p(B) = 1/2 and hence η = 0
For ordered states η = ±1 correspond to p(A) = 1,p(B) = 0
or p(B) = 1,p(A) = 0
Self-consistent equation for the order parameter
η = tanh zJ2k
BT
η (16.2.10)
Solution of (16.2.10) can be obtained graphically
y = (zJ/2kBT)η
and
η = tanhy
Fig,16.2.2
For 2kBT/zJ ≥ 1 the intersection is only at η = 0
For 2kBT/zJ < 1 another intersection with nonzero η
transition temperature
Tc = zJ2k
B
(16.2.11)
0
1
y
T>Tc T=Tc T<Tc
A
tanh y
η
Figure 16.2.2 Graphically solutions for the order parameter η.
16.2.2 Paraelectric-Ferroelectric Transition
Ferroelectricity is related to piezoelectricity and pyroelectricity
A ferroelectric crystal possesses a spontaneous electric polarization
reversible under the action of an external electric field
PE-FE phase transition is in general a structural transition
P versus E hysteresis loop in Fig,16.2.3
E
P
Figure 16.2.3 Electric hysteresis loop.
PE-FE phase transitions can be realized in two ways
(1) Displacive type
FE phase is obtained by minute displacement of atoms or molecules
(2) Order-disorder transition type
FE phase results because of ordering of atoms or molecules
It is possible for a transition to have both characteristics
FE phase transitions can be either continuous or discontinuous
TTc
P
(a)
P
Tc T
(b)
Figure 16.2.4 Temperaturedependence ofspontaneouspolarizationin (a)continuous
and (b) discontinuous ferroelectric phase transitions.
A simple case of a uniaxial rigid elementary dipoles
reorientale in one of two opposite directions
An effective field approach to ferroelectric transitions
analogous to the Weiss theory for ferromagnets
Effective field
Eeff = E + γP (16.2.12)
External field E and cooperative field γP
Latter is due to partially ordered system of dipoles
Energies associated with two possible orientations
w = ±(E + γP)μ
elementary dipole moment μ
Number of dipoles pointing in ± directions to Eeff
N+ = NZew/kBT,N? = NZe?w/kBT (16.2.13)
N is the total number of the dipoles
Partition function Z
Z = ew/kBT + e?w/kBT (16.2.14)
A self-consistent equation for polarization
P = (N+?N?)μ = Nμtanh (E + γP)μk
BT
(16.2.15)
For spontaneous polarization,E = 0,T lessorsimilar Tc,P → 0
Critical temperature
Tc = γNμ2/kB (16.2.16)
Spontaneous polarization
Ps = Nμ√3
bracketleftbigg
1? TT
c
bracketrightbigg1/2
(16.2.17)
Dielectric constant defined as epsilon1 = 4pidP/dE
epsilon1(T) = CT?T
c
,T ≥ Tc (16.2.18)
epsilon1(T) = C2(T?T
c)
,T ≤ Tc (16.2.19)
Curie-Weiss law with Curie constant C = 4piTc/γ
Thermal properties can be calculated from
U =?12EeffPs =?12γP2s (16.2.20)
Transition heat
U = U(Tc)?U(0) = 12γP2s = 12NkBTc (16.2.21)
Transition entropy
S =
integraldisplay Tc
0
1
Td
bracketleftbigg
12γP2s(T)
bracketrightbigg
(16.2.22)
Specific heat jump
Cp = TcdSdT = ddT
parenleftbigg
12γP2s
parenrightbigg
Tc
= 32NkB (16.2.23)
FE transitions? anomalies near Tc
in structural,thermal,elastic,optical properties
making ferroelectric crystals useful in applications
16.2.3 Incommensurate-Commensurate Transition
§16.3 Phase Transitions in Soft Matter
16.3.1 Maier-Saupe Theory for Isotropic-Nematic Transition
16.3.2 Onsager Theory for Isotropic-Nematic Transition
16.3.3 Phase Separation in Hard-Sphere Systems
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
May 8,2004
Contents
Chapter 16,Crystals,Quasicrystals and Liquid Crystals 3
§16.1 Liquid-Solid Transitions,,,,,,,,,,,,,,,,,,,4
16.1.1 Free Energy Expansion Based on Density Waves,,,4
16.1.2 Crystallization,,,,,,,,,,,,,,,,,,,,,,4
16.1.3 Quasicrystals,,,,,,,,,,,,,,,,,,,,,,4
§16.2 Phase Transitions in Solids,,,,,,,,,,,,,,,,,,5
16.2.1 Order-Disorder Transition,,,,,,,,,,,,,,,5
16.2.2 Paraelectric-Ferroelectric Transition,,,,,,,,,,5
16.2.3 Incommensurate-Commensurate Transition,,,,,,6
§16.3 Phase Transitions in Soft Matter,,,,,,,,,,,,,,,23
16.3.1 Maier-Saupe Theory for Isotropic-Nematic Transition,28
16.3.2 Onsager Theory for Isotropic-Nematic Transition,,,38
16.3.3 Phase Separation in Hard-Sphere Systems,,,,,,,47
Chapter 16
Crystals,Quasicrystals and Liquid
Crystals
§16.1 Liquid-Solid Transitions
16.1.1 Free Energy Expansion Based on Density Waves
16.1.2 Crystallization
16.1.3 Quasicrystals
§16.2 Phase Transitions in Solids
16.2.1 Order-Disorder Transition
16.2.2 Paraelectric-Ferroelectric Transition
16.2.3 Incommensurate-Commensurate Transition
Discovered in 1960s,before the discovery of quasicrystals
Another kind of quasiperiodic structure
Incommensurate phases occur in various materials
Structure,or composition,or charges,or spins
Modulation are mostly 1D,NaNO3,Na2CO3,Ba2NaNb5O15
Sometimes it may be 2D,quartz,or 3D,wustite Fe1?xO
2D periodic structure with 1D incommensurate modulation
Unmodulated atomic sites determined by a vector
r = a(nxx+nyy)
x,y are orthogonal unit vectors
A modulation along x-axis is introduced
rprime = a[(nx +epsilon1sinqnxa)x+nyy]
a
a
λ
Stable only in a limited temperature range
experimentally observed with decreasing temperature
Prototypic-Incommensurate-Commensurate phase transitions
molecular crystal thiourea,CO(NH2)2
T > 202 K,paraelectric; T < 169 K,ferroelectric
169 K <T < 202 K,incommensurate phase
a polarization wave of dipole moments in the crystal
its wavelength is incommensurate with the underlying lattice
Density wave description
δρk(r) =
summationdisplay
i
ηikψik(r) (16.2.24)
Modulated wavevector k changes with T
Two transition temperatures T = TI and T = TL
Prototypic TI?→ Intermediate TL?→ Wavevector lock-in k = kc
A simple example for incommensurate transition
Two component order parameter η1 and η2
2D IR of symmetry group of prototypic phase
This model is realized in numerous materials
Displacement type structural phase transition
Normal coordinates of the soft mode Q,Q? as complex order parameter
Equivalently Q = η1 +iη2 and Q? = η1?iη2
Order parameter depends on spatial coordinates x
Free energy density taking into account two fourth-order invariants
and gradient terms including a Lifshitz invariant and a Ginzburg term
=?0 + A(η21 +η22) +B1(η21 +η22)2 +B2η21η22
+ δ
bracketleftbigg
η1?η2?x?η2?η1?x
bracketrightbigg
+ κ2
bracketleftBiggparenleftbigg
η1
x
parenrightbigg2
+
parenleftbigg?η
2
x
parenrightbigg2bracketrightBigg
(16.2.25)
To introduce the transformation
η1 = ηsinθ,η2 = ηcosθ (16.2.26)
α = 2A,β1 = 4[B1 +B2/8],β2 =?B2/2
Free energy density
=?0 + α2η2 + β14 η4 + β24 η4 cos4θ?δη2?θ?x + κ2
bracketleftBiggparenleftbigg
η
x
parenrightbigg2
+η2
parenleftbigg?θ
x
parenrightbigg2bracketrightBigg
(16.2.27)
η(x) and θ(x) are modulated along the x-direction
Stability of commensurate phase requires β1 >β2
a positive wavenumber k implies δ> 0 and κ> 0
Free energy
Φ =
integraldisplay
L
parenleftbigg
η,θ,?η?x,?θ?x
parenrightbigg
dx (16.2.28)
L is the length of the crystal in x-direction
Φ/?η = 0,?Φ/?θ = 0 → a coupling nonlinear differential equations
αη+β1η3 +β2η3 cos4θ?2δη?θ?x +κη
parenleftbigg?θ
x
parenrightbigg2
κ?
2η
x2 = 0 (16.2.29)
β2η4 sin4θ+ 2κη?η?x
parenleftbigg?θ
x?
δ
κ
parenrightbigg
+κη2?
2θ
x2 = 0 (16.2.30)
For general value of the coefficients,only numerical solutions
Analytic treatments can be get on for constant η,spatial modulated θ(x)
mathematical analysis is still complicated
Here discuss some special cases from (16.2.29) and (16.2.30)
The solutions for commensurate phases from?η/?x = 0,?θ/?x = 0
(16.2.29) and (16.2.30) are simplified to
η(α+β1η2) +β2η3 cos4θ = 0 (16.2.31)
β2η4 sin4θ = 0 (16.2.32)
There are two sets of solutions
(1) η = 0,θ arbitrary value,high-T prototypic phase
(2) η negationslash= 0,ordered structure,sin4θ = 0 → eight θ values
Only four directions? minimum of Φ depending on the sign of β2
For β2 > 0,θ = ±pi/4,±3pi/4
For β2 < 0,θ = 0,±pi/2,pi
The amplitude of order parameter
η2 =? αβ
1?|β2|
(16.2.33)
As usual,set α = α0(T?TL)
T <TL,η2 > 0,low T commensurate and ordered phase
denoted by some isolated dots in Fig,16.2.5(a)
on the (η1,η2) plane with ηe and θe
η2(a) (b) (c)
η1
η2
η1
η2
η1
η1,η2 η1,η2 η1,η2
x x x
Figure 16.2.5 The thermodynamic stable solutions in the order parameter plane
with β2 > 0,(a) Low-temperature commensurate phases; (b) Ignoring anisotropic
energy; (c) Numerical results.
Lifshitz invariant → prohibiting second-order transition
directly to the commensurate phase from the prototypic phase
T <TI,prototypic phase to incommensurate phase
order parameter is a small quantity
η1 = ηsin(kIx),η2 = ηcos(kIx) (16.2.34)
i.e.,η similarequal 0 and θ = kIx
From (16.2.30),ignoring higher-order terms
κη?η?x
parenleftbigg?θ
x?
δ
κ
parenrightbigg
= 0 (16.2.35)
Modulated wavevector
kI = δ/κ (16.2.36)
related to the coefficients of Lifshitz and Ginzburg terms
If β1 greatermuchβ2 and neglecting anisotropic terms,(16.2.29) →
α0
parenleftbigg
T?TL? δ
2
α0κ
parenrightbigg
+β1η2 = 0 (16.2.37)
Transition temperature for prototypic phase to incommensurate phase
TI = TL + δ
2
α0κ (16.2.38)
When TL <T <TI
η2I =?α0(T?TI)β
1
> 0,(16.2.39)
As T →TI,η2I → 0,transition is continuous
Near TI,λ ～ 1/kI = κ/δ is irrational
On η1η2 plane,stable phase is any point on the circle with radius ηI
η1 and η2 changes along x-axis sinusoidally
with amplitude ηI and wavenumber kI in Fig,16.2.5(b)
This is the single plane wave form of incommensurate phase
Fig,16.2.6(a)
T ↓,anisotropic energy increases,representative points on η1η2 plane
will be dense near the low-T commensurate phases
Numerical solutions of (16.2.29) and (16.2.30) show that
near TL the incommensurate modulated wave becomes a square wave
in Fig,16.2.5(c)
This square wave is composed with a lot of domain structures
Fig,16.2.6(b)
T ↓,domain walls diminish,at TL,domain walls vanish
only a commensurate phase
Problem,Begin from (16.2.25),derive equations (16.2.27-39)
(a)
(b)
Figure 16.2.6 Modulation wave in incommensurate phase,(a) Single plane modu-
lation for T near TI,(b) Domain and wall structure for Tc <T <TI.
§16.3 Phase Transitions in Soft Matter
In §3.3-5,soft matter,includes thermotropic and lyotropic liquid crystals,
polymers,biopolymers,colloids,···
In general,Phase transition driven by entropy as well as energy
Polymorphic configurations → interesting phenomena
Introduction to thermotropic liquid crystals
Nematic Phase,Cholesteric Phase,Smectic Phase
Its brief history,de Gennes’ contribution
Discovery of liquid crystal,1888,Austria botanist,Reinitzer
C6H5CO2C27H45 cholesteric crystal
Heated to 145.5? C,turbid liquid; increased to 178.5? C,clear liquid
Afterwards,German physicist O,Lehmann did systematic researches,found
many organic materials have the same features,In the mesophases,between
melting point and clear point,the mechanical property is like isotropic liquid,
but the optical property is anisotropic,Actually,in the process of heating,
these materials lose positional order at first; but there are correlations among
molecular orientations,i.e.,orientational order
I like liquid crystal,because its beauty and secret
—– de Gennes
Liquid crystal was discovered by German scholar 100 years ago;
G,Friedel (1922) of France established the basic scheme of
structural classification;
Americans notices the potential applications on display devices;
now the application technique are grasped by Japanese
—— de Gennes
L O N N C
O
O C2H5
O
C
O
H5C2
H3C O N N CH3
O
R R
R
R
R
R
C
C
C
C
C
C
C
R
C
R
C RCR
C
R
C
R
(a)
(b)
D
D
h
(a) (b) (c)
16.3.1 Maier-Saupe Theory for Isotropic-Nematic Transition
Rod-like molecules,T ↓,isotropic-nematic transition
Symmetry is cylindrical,a symmetry axis ˉn,director
T >Tc,isotropic normal liquid phase
T <Tc,ordering in the polar angle θ,nematic phase
x
y
z
ψ
φ
θ
n
n
Figure 16.3.1 Schematic diagrams of the structure of a nematic liquid crystal and
single rod-like molecule.
Definition of long-range orientational order parameter
Is it possible to use cosθ? No
Because ˉn and -ˉn are fully equivalent
The preferred axis is non-polar,it is suitable to consider cos2θ
If all molecules fully aligned with ˉn,〈cos2θ〉 = 1
If molecules randomly distributed in direction,〈cos2θ〉 = 1/3
To choose scalar order parameter
η = 〈P2〉 = 12〈3cos2θ?1〉,(16.3.1)
second-order Legendre function P2
r
2
θ1 θ2
1
Figure 16.3.2 Schematic diagram of the interaction between two rod-like molecules.
Pair potential between two rod-like molecules
V12 = V12(r,θ1,φ1,θ2,φ2) (16.3.2)
It is difficult to get its exact form
Maier-Saupe molecular field method,1958
This theory can be extended to isotropic-smectic phase transition
W,L,McMillan,Phys,Rev,A 4,1238 (1971)
A molecule is in the mean field of all other molecules
V(cosθ) =?vP2(cosθ)〈P2〉 (16.3.3)
v is strength of intermolecular interaction,v> 0
Orientational distribution function
f(cosθ) = Z?1 exp[?βV(cosθ)] (16.3.4)
Single molecule partition function
Z =
integraldisplay 1
0
exp[?βV(cosθ)]d(cosθ) (16.3.5)
β = 1/kBT
A self-consistent integral equation to calculate the order parameter
〈P2〉 = η =
integraldisplay 1
0
P2(cosθ)f(cosθ)d(cosθ)
=
integraltext1
0 P2(cosθ)exp[βvP2(cosθ)·η]d(cosθ)integraltext
1
0 exp[βvP2(cosθ)·η]d(cosθ)
(16.3.6)
To determine the temperature dependence of order parameter
Choosing one value of kBT/v → 〈P2〉
Numerical results in Fig,16.3.3
T <Tc = 0.22019v/kB,three solutions appear
To judge which one is stable by minimizing the free energy
Internal energy
U = 12N〈V〉 = 12N
integraldisplay 1
0
V(cosθ)f(cosθ)d(cosθ) (16.3.7)
Number of molecules N
Entropy
S =?NkB〈lnf〉 = NT 〈V〉+NkB lnZ (16.3.8)
Tc=0.22019
0
4
8
12
-4
0 4 8 12 16 20 24
kBT/v
P 2
Figure 16.3.3 Phase diagram of Maier-Saupe transition.
Combining (16.3.7) and (16.3.8),Helmholtz free energy
F =?NkBT lnZ? 12N〈V〉 (16.3.9)
The second term is duo to pair interactions
From minimum of F
T <Tc,nematic phase is stable
T >Tc,isotropic phase is stable
〈P2〉 discontinuously changes from 0 to 0.4289
Phase transition is first-order
16.3.2 Onsager Theory for Isotropic-Nematic Transition
Besides the anisotropic attractive interaction
there must also be an anisotropic steric interaction
due to the impenetrability of the molecules
Onsager’s theory for hard rods as the density is increased
from isotropic phase to anisotropic phase
Onsager’s theory was also successfully applied to smectic phase
A,Stroobants et al.,Phys,Rev,A 36,2929 (1987);
X,Wen et al.,Phys,Rev,Lett,63,2760 (1989)
Two kinds of entropy in a gas of hard rods
Translational entropy and orientational entropy
Both are coupling through the effect of excluded volume
due to the impenetrability of molecules
Excluded volume is smallest when two hard rods parallel
parallel alignment favors translational entropy
but represents low orientational entropy
A competition exists between translational and orientational entropies
Zero density limit → orientational entropy
Increasing density excluded volume becomes important
Tight-packing density,hard rods must be parallel
At some intermediate density,there is a transition
Onsager’s approach uses density expansion of free energy
G,J,Vroege and H,N,W,Lekkerkerker,Rep,Prog,Phys,55,1241 (1992)
Consider a fluid of long thin hard-rod molecules
Length L and diameter D,LgreatermuchD
Steric repulsion is only interactions
volume fraction ν = (1/4)ρpiLD2,density of rods ρ
Angular distribution f(?),the number of rods per unit volume
Condition of normalization
integraldisplay
f(?)d? = 1 (16.3.10)
Free energy expanded to first order in density
F = F0+kBT
braceleftbiggintegraldisplay
f(?)ln[4pif(?)]d? + 12ρ
integraldisplay integraldisplay
f(?)f(?prime)u(prime)d?d?prime
bracerightbigg
(16.3.11)
First term is a constant; Second term describes the entropy contribution;
Third term describes the excluded volume effects,u(prime)
u = 2L2D|sinγ| (16.3.12)
γ is the angle between? and?prime
γ
'
Figure 16.3.4 Excluded volume of two hard rods with angle γ.
Minimize free energy (16.3.11) by all variations of f(?)
with the constraint (16.3.10)
δF = kBTλ
integraldisplay
δf(?)d? (16.3.13)
λ is Lagrange multiplier
Self-consistent equation for f(?)
ln[4pif(?)] = λ?1?ρ
integraldisplay
u(prime)f(?prime)d?prime (16.3.14)
λ is determined by the normalized condition (16.3.10)
An isotropic solution,f(?) = 1/4pi,independent of?
If νL/D is large enough,anisotropic solutions,a nematic phase
To solve the nonlinear integral equation (16.3.14)
Onsager’s variational method based on a trial function
f(?) = Acosh(αcosθ) (16.3.15)
variational parameter α,constant A to normalize f
α turns out to be large (～ 20)
f is strongly peaked around θ = 0 and θ = pi
Order parameter for αgreatermuch 1
η = 12
integraldisplay
f(?)(3cos2θ?1)sinθdθ =similarequal 1?3/α (16.3.16)
First-order transition from isotropic (α = 0) to nematic (α≥ 18.6)
T lessorsimilarTc,νnc = 4.5D/L
greaterorsimilarTc,νic = 3.3D/L.
νnc and νic independent of T
In this model,hard rods are an athermal system
Phase transition is driven by entropy
ηc similarequal 0.84 → an abrupt transition
between an ordered nematic and disordered isotropic phase
16.3.3 Phase Separation in Hard-Sphere Systems
Entropy-driven phase transitions in soft matter
Phase separation of hard sphere systems
A concise introduction,
T,C,Lubensky,Solid St,Commun.,102,187 (1997)
Phases of condensed matter in equilibrium,free energy
F = U?TS (16.3.17)
should be minimum
Energy-driven phase transition
Conventional solids,hard matter,U is important than S
To determines the structure of phase in equilibrium
The crystallization discussed in §16.1.2 as example
T ↓,a phase transition from disordered liquid to ordered crystal
S ↓→F ↑,so ordered phase is due to U ↓
Entropy-driven phase transition
In the case of soft matter,TS changes large,U small
Decrease of free energy mainly due to increase of entropy
Equilibrium state is determined by entropy maximum
The key point is that increase of microscopic disorder is
beneficial to the appearance of macroscopic order
Formally,entropy deviated from a equilibrium value
will give rise to an entropic force
just like the gradient of a potential
A hard sphere system with single size for radius
Internal energy is always zero for different configurations
Entropy is prominent,which is related to the total volume fraction ν
When ν is small,a ideal gas
As ν increases,restriction of movement increased
In the case of close packing,all particles are trapped
There are two close packing densities
hexagonal close packing density νh = 0.7405
random close packing density νr = 0.638
It is imagined to magnify hcp lattice and keep crystalline structure invariant
ν ↓,entropy increases
For rcp,a lot of configurations cannot access each other
Magnified lattice with ν = νr has higher entropy
This means volume fraction increases,
system is in favor to form a periodic crystal
Entropy-driven first-order liquid-solid phase transition
For glass transition,it is necessary to adopt the quenching
a nonequilibrium process
Figure 16.3.5 Schematic diagram of exclusive volume between large sphere,small
sphere,and wall of container.
Mixtures of hard spheres with different sizes
Consider a dispersion system composed of colloidal spheres
with two different diameters dL and dS,dL greatermuchdS
If both the volume fractions are about the same,nS greatermuchnL
Entropy of small spheres play principal role
in determining the structure of the system
Fig,16.3.5
There are attractive forces between large spheres,and large spheres and wall
A phase separation will appear between large and small particles
by increasing volume fractions
Three parameters,the volume fractions νL and νS
The ratio of radii of large and small spheres α = rL/rS
Fig,16.3.6 shows the experimental result and theoretical analysis
+
+
+
+
+ + +
**
*
*
*
*
*
*
*
0.00 0.05 0.10 0.15 0.20 0.25
0.1
0.2
0.3
0.0
fL
α = 0.825 μm / 0.069 μm =12.0
*
f s
Figure 16.3.6 Phase diagram of mixture with two components,+,no phase separa-
tion; square,solid phase appears on the surface of container; triangle,phase separations appear
on surface as well as in bulk;?,liquid-solid phase separation appears only in bulk.
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
May 10,2004
Contents
Chapter 17,Ferromagnets,Antiferromagnets and Ferrimagnets
§17.1 Basic Features of Magnetism,,,,,,,,,,,,,,,,,5
17.1.1 Main Types of Magnetism,,,,,,,,,,,,,,,6
17.1.2 Spatial Pictures of Magnetic Structures,,,,,,,,10
17.1.3 Band Pictures of Magnetic Structures,,,,,,,,,18
17.1.4 Hamiltonians with Time Reversal Symmetry,,,,,23
§17.2 Theory based on Local Magnetic Moments,,,,,,,,,31
17.2.1 Mean Field Approximation for HeisenbergHamiltonian 32
17.2.2 Ferromagnetic Transition,,,,,,,,,,,,,,,,38
17.2.3 Antiferromagnetic Transition,,,,,,,,,,,,,,43
17.2.4 Ferrimagnetic Transition,,,,,,,,,,,,,,,,49
17.2.5 Ferromagnetic and Antiferromagnetic Ground States,49
§17.3 Theory based on Itinerant Electrons,,,,,,,,,,,,50
17.3.1 Mean-Field Approximation of Hubbard Hamiltonian,50
17.3.2 Stoner Theory of Ferromagnetism,,,,,,,,,,,50
17.3.3 Very Weak Itinerant Ferromagnetism,,,,,,,,,50
17.3.4 Spin Density Waves and Antiferromagnetism,,,,,50
Chapter 17
Ferromagnets,Antiferromagnets and
Ferrimagnets
Magnetic ordering:
ferromagnetism,antiferromagnetism and ferrimagnetism
Broken symmetry of time reversal or spin rotation
Two physical models to describe the magnetism
Localized to magnetic insulators
Itinerant to magnetic metals
Opposed but complementary models to each other
§17.1 Basic Features of Magnetism
Magnetism of materials←spins of microscopic particles
Spin in quantum mechanical sense,sometimes as classical vector
Langevin’s treatment for paramagnetism
Classical Ising model,XY model,and Heisenberg model
H=?Jz
summationdisplay
ij
SziSzj?J⊥
summationdisplay
ij
(SxiSxj +SyiSyj)?H
summationdisplay
i
Szi
17.1.1 Main Types of Magnetism
Magnetic behavior in solids←orientations of magnetic dipoles,
or magnetic moments composed of electronic spins,
electronic orbitals,and nuclear magnetic moments
Figure 17.1.1,M; χ = M/H
(1) Diamagnetism
(2) Paramagnetism
(3) Ferromagnetism
(4) Antiferromagnetism
(5) Ferrimagnetism
T
χ-1
0 T0 T
c
χ-1
M
χ-1
,M
χ-1
0 TN Tθ T0 T
c
χ-1
M
χ-1
,M
(a) (b)
(c) (d)
Figure 17.1.1 Several kind of magnetic behavior,(a) Paramagnetism; (b) Ferro-
magnetism; (c) Antiferromagnetism; and (d) Ferrimagnetism.
Several kinds of magnetic interactions:
Direct exchange (kinetic exchange)
Superexchange
Double exchange
Indirect (RKKY) exchange
Exchange interactions between itinerant electrons
Figure 17.1.2
RKKY
Itinerant
Electrons
Direct
Exchange
Super-
exchange
Nonlocal
moment
Nonmetal
Local
Moment
Metal
Double
Exchange
Figure 17.1.2 The relationship among five exchange interactions.
17.1.2 Spatial Pictures of Magnetic Structures
Fig,17.1.3
Fig,17.1.4
Table 17.1.1.
Fig,17.1.5
Table 17.1.2.
Fig,17.1.6
Table 17.1.3
Figure 17.1.3 Magnetic structure of MnO.
Table 17.1.1 Intrinsic magnetic properties and crystalline structures of
several antiferromagnets
compounds crystalline structures TN(?K) θ(?K) θ/TN Cmole χp(0)/χp(T
MnO fcc 122 610 5.0 4.40 0.69
FeO fcc 185 570 3.1 6.24 0.77
CoO fcc 291 280 0.96 3.0 —
NiO fcc 515 — — — 0.67
MnS fcc 165 528 3.2 4.30 0.82
MnF2 rutile 74 113 1.5 4.08 0.75
FeF2 rutile 85 117 1.4 3.9 0.72
CoF2 rutile 40 53 1.3 3.3 —
Fe
O
1
1
23
2 x y
z
Figure 17.1.4 Magnetic structure of αFe2O3.
[111]
O2-
Me2+
S-Me2-Fe4O8
Fe3+
2
Figure 17.1.5 Magnetic structure of Fe3O4.
Table 17.1.2 Intrinsic magnetic properties and crystalline structures of
several ferrimagnets
(a) Spinel Ferrimagnets
Structures Curie Ionic moments (μB) Saturation
Compounds types point A B Net value magnetization
Tc/K position position Theoretical Experimental μ
MnFe2O4 I 575?(1+4) 1+9 5 46～5
Fe3O4 I 860?5 4+5 4 4.1
CoFe2O4 I 790?5 3+5 3 3.7
NiFe2O4 I 865?5 2+5 2 2.3
CuFe2O4 I 728?5 1+5 1 1.3
Li Fe O I 943?5 0+7.5 2.5 2.5～3
(a) (b) (c) (d) (e) (f) (g)
Figure 17.1.6 Different types of magnetic ordered structures for rare earth metals.
Table 17.1.3 Magnetic structures of several rare earth metals and alloys
Element Gd Tb Dy Ho Er Tm
293K
230K
220K
176K
88K
130K
20K
85K
52K
20K
57K
32K
0K
100K
200K
300K
χ=18?20o (e)
PM PM PM PM PM
χ=26?43o(e)
helical AFM
χ=30?50o(e)
FM moment
along base
plane is
μB (f)
FM
(along c axis,
orientantional
angles of
moments are
variable)
μ
μ
=9.5 μB (d)
=1.7 μB =7.6 μBμ
cone helical
μ=4.3 μB (d)
c axis modulated
AFM 7
fFM moment
along base
plane is
μB (f)
helical AFM
helical
cone helical
17.1.3 Band Pictures of Magnetic Structures
Figure 17.1.7
Table 17.1.4
m = (n↑?n↓)μB = [5?(n+ 1.35?5)]μB = (8.65?n)μB (17.1.1)
Fig,17.1.8
m/μB = (1?χ)mprimeA +χmprimeB (17.1.2)
Fig,17.1.9
3
2
1
0
1
2
3-10 -5 0
2
1
0
1
2
-10 -5 0
sp-g(E)
EF
fcc-Cobcc-Fe
EF
E (eV)
g(E
) (eV)
-1
(a) (b)
Figure 17.1.7 DOS curves,(a) Fe; (b) Co,From J,K¨ubler,Theory of Itinerant
Electron Magnetism,Oxford University Press,Oxford (2000).
Table 17.1.4 Electron distribution in energy bands and intrinsic magnetic
properties of ferromagnetic metals
electronic distribution of hole E value of momen
element configuration band electrons number spin magnetic neutron
of isolated 3d↑3d↓ 4s↑ 4s↓ 3d↑3d↓number measure 3d
atoms
Cr 3d44s2 2.7 2.7 0.3 0.3 2.3 2.3 0 0
Mn 3d54s2 3.2 3.2 0.3 0.3 1.8 1.8 0 0
Fe 3d64s2 4.8 2.6 0.3 0.3 0.2 2.4 2.2 2.216 2.39?
Co 3d74s2 5.0 3.3 0.35 0.35 0 1.7 1.7 1.715 1.99?
Ni 3d84s2 5.0 4.4 0.3 0.3 0 0.6 0.6 0.616 0.620?
bcc Fe-Co
fcc
Fe-Ni
fcc
Ni-CubccCr-Fe
7 8 9 10
0
1
2
3
m at (
μ B)
Number of 3d+4s Electrons
Figure 17.1.8 Slater-Pauling curve for two component 3d magnetic alloys.
E (eV)
0 2 4-2-4-6-88
6
4
2
0
2
4
6
EF
CrO2
a=0.4410 nm
c/a=0.6599
g(E
) (eV
-1 )
Figure 17.1.9 DOS of spin subbands of CrO2,From J,K¨ubler,Theory of Itinerant
Electron Magnetism,Oxford University Press,Oxford (2000).
17.1.4 Hamiltonians with Time Reversal Symmetry
Time reversal,an important symmetric operation,T
Related to the complex conjugation,proved from
Time-dependent Schr¨odinger equation
Hψ = iplanckover2pi1?ψ?t (17.1.3)
Performing operationT
Hψ? =?iplanckover2pi1?ψ
t = iplanckover2pi1
ψ?
(?t) (17.1.4)
Time inversion,the equation of motion is invariant
Commutativity between some physical quantities andT
TrT?1 = r (17.1.5)
For p,p = dr/dt or p = (planckover2pi1/i)?
TpT?1 =?p (17.1.6)
TLT?1 =?L (17.1.7)
TST?1 =?S (17.1.8)
T reverses a magnetic moment
Alternatively,T reverses the direction of an electric current
I = dQ/dt,?I = dQ/(?dt)
Fig,17.1.10.
SupposeH(r,p,S)
TH(r,p,S)T?1 =H(r,?p,?S) (17.1.9)
If no applied field
TH(r,p,S)T?1 =H(r,p,S) (17.1.10)
Hamiltonian has time reversal symmetry
Figure 17.1.10 Schematic diagram for magnetic moments given by currents.
Examples with time reversal symmetry
(1) Single electron in periodic potential
H= p
2
2m +V(r) (17.1.11)
Time reversal symmetry→Kramers degeneracy
ψk(r) and ψ?k(r),E(k) = E(?k)
(2) Single-electron Hamiltonian containing SO coupling
H= 12mp2 +V(r) + planckover2pi14m2c2σ·(?V ×p) (17.1.12)
(3) Heisenberg Hamiltonian
H=?
summationdisplay
i>j
JijSi·Sj (17.1.13)
Jij = Jij(|Ri?Rj|) (17.1.14)
In the n.n,approximation
H=?J
summationdisplay
i>j
Si·Sj (17.1.15)
It is invariant under time reversal transformation
However,H→?H·S,destroys invariance
Even though no H,T↓→spontaneously symmetry broken
J > 0 for ferromagnetic order; J < 0 for antiferromagnetic order
(4) Hubbard Hamiltonian
H=
summationdisplay
ijσ
Tijc?iσciσ + U2
summationdisplay
iσ
niσniˉσ (17.1.16)
Tij is a matrix element between sites i and j and the interaction is
U =
integraldisplay integraldisplay
|ψ(r1)|2 e
2
|r1?r2|ψ(r2|
2dr1dr2 (17.1.17)
It is invariant under time inversion
To apply H or decrease T for breaking time reversal symmetry
Paramagnetic substance is invariant under time reversal operation
Broken time reversal invariance→
a rich variety of magnetic structures
1191 magnetic space groups,instead of 230 colorless space groups (§1.5.2)
§17.2 Theory based on Local Magnetic Moments
Molecular field
First introduced by Weiss (1907) to study ferromagnetism
Later used by Neel (1936,1948) to study antiferromagnetism
and ferrimagnetism
Its essential idea is an effective magnetic field
proportional to magnetization
Heisenberg Hamiltonian can give Weiss’s effective field
17.2.1 Mean Field Approximation for Heisenberg
Hamiltonian
From Heisenberg Hamiltonian (17.1.15)
Hi =?JSi·
zsummationdisplay
j=1
Sj (17.2.1)
An effective field Heff
gLμBHeff = J
zsummationdisplay
j=1
Sj = J
zsummationdisplay
j=1
〈Sj〉= zJ〈Sj〉 (17.2.2)
Land′e factor gL and Bohr magneton μB = eplanckover2pi1/2mc
Single atom Hamiltonian
Hi =?gLμBSi·Heff (17.2.3)
Magnetization
M = NgLμB〈Sj〉 (17.2.4)
Effective field
Heff = zJg
LμB
〈Sj〉= zJNg2
Lμ
2
B
M = γM (17.2.5)
γ = zJ/Ng2Lμ2B
Apply an field H,total field
Ht = H +Heff = H +γM (17.2.6)
Assume H along z axis→Heff along z axis
Instead of (17.2.1),single atom Hamiltonian
Hi =?gLμBSizHt (17.2.7)
Its eigenvalues
Eν =?gLμBνHT,ν =?S,···,S (17.2.8)
Partition function
Z =
Ssummationdisplay
ν=?S
e?Eν/kBT =
Ssummationdisplay
ν=?S
eνgLμBHt/kBT (17.2.9)
after summation
Z = sinh[gLμBHt(2S + 1)/2kBT]sinh[g
LμBHt/2kBT]
(17.2.10)
Magnetization
M = NkBT?lnZ?H
t
= NgLμBSBS(x) (17.2.11)
Brillouin function defined as
BS(x) = 2S + 12S coth
parenleftbigg2S + 1
2S x
parenrightbigg
12S coth
parenleftbigg 1
2Sx
parenrightbigg
,(17.2.12)
and
x = gLμBSHtk
BT
(17.2.13)
Define the maximum of magnetization
M0 = NgLμBS (17.2.14)
The reduced magnetization
m = M/M0 = BS(x),(17.2.15)
BS(x) takes value from 0→1 as x from∞→0
Let x0 = gμBSH/kBT,then
x = x0 + (zJS2/kBT)m (17.2.16)
(17.2.15) and (17.2.16) are two coupling equations
Its graphical solution is in Fig,17.2.1
Derive coupling equations (17.2.15) and (17.2.16),Find
the numerical solutions and plot figures forS = 3/2 andS = 5/2.
T>Tc
H=0
T=Tc
H=0
T<Tc
H=0
T>Tc
H0=0
B7/2
0.0 1.00.0 2.0 3.0 4.0 5.0
0.2
0.4
0.6
0.8
1.0
x
m
Figure 17.2.1 graphical solution of magnetization.
17.2.2 Ferromagnetic Transition
Consider external field H = 0,when T↓
Spontaneous broken symmetry of time reversal
paramagnetic phase→ferromagnetic phase
For small x,(17.2.12) is expanded
BS(x) = S + 13S x?[(S + 1)
2 +S2](S + 1)
90S3 x
3 +··· (17.2.17)
A spontaneous internal magnetic field develops at Tc
Substitute (17.2.17) to (17.2.15)
Tc = 2JzS(S + 1)3k
B
(17.2.18)
T >Tc,a magnetic solution exists only if H is present
T <Tc,the spontaneous magnetization in zero field satisfies
m = BS
parenleftbigg 3S
S + 1
Tc
Tm
parenrightbigg
(17.2.19)
T lessorsimilarTc
m2 = 103 (S + 1)
2
(S + 1)2 +S2
Tc?T
Tc (17.2.20)
m～(Tc?T)1/2
Magnetization M is not unique in its direction
due to Hamiltonian (17.1.16) is isotropic
Heating and recooling,magnetization will occur with different direction
If considering crystal structures,an anisotropic term added into
the Hamiltonian,the degeneracy will be lifted
From (17.2.16) using the small argument expansion
x = x0 + TcTx (17.2.21)
This gives Curie-Weiss law for the susceptibility
χ = MH = CT?T
c
(17.2.22)
Curie constant
C = Ng
2
Lμ
2
BS(S + 1)
3kB (17.2.23)
Internal energy
U =?M
parenleftbigg1
2γM +H
parenrightbigg
=?12γM20m
parenleftbigg
m+ 2HγM
0
parenrightbigg
Specific heat
CM =?U?T =?γM20
parenleftbigg
m+ HγM
0
parenrightbigg?m
T (17.2.24)
When H = 0,for T >Tc,m = 0
CM = 0 (17.2.25)
for T <Tc,mnegationslash= 0
CM =?12γM0dm
2
dT = 5NkB
S(S + 1)
(S + 1)2 +S2 (17.2.26)
Specific heat indicates a second-order phase transition
These are consistent with Landau theory,if take m = M/M0
as the order parameter to construct free energy
Φ = Φ0 +A(T)m2 +Bm4?Hm (17.2.27)
17.2.3 Antiferromagnetic Transition
J < 0 in (17.1.16)→antiferromagnetism
Taking two sublattice system as an example in n.n.n,model
Two interaction constants between two kinds of spins
J12 = J21 and J11 = J22
Also effective field method but for two sublattice i,j (i,j = 1,2)
Two effective fields
Hi = H +
summationdisplay
j
γijMj (17.2.28)
Coupling constants
γij = 2zijJijNg2μ2
B
(17.2.29)
zij is the number of neighbors
z12 = z1 is the total number of nearest neighbors
z11 = z22 = z2 is the total number of next nearest neighbors
Defining the exchange parameter for nearest neighbors J1 = J12
and the exchange parameter for next nearest neighbors J2 = J11
γ12 = γ21 = 2z1J1Ng2
Lμ
2
B
,γ11 = γ22 = 2z2J2Ng2
Lμ
2
B
Repeating the arguments in the ferromagnetic case
Reduced magnetization on sublattice i
mi = BS(xi) (17.2.30)
xi = gLμBHiS/kBT and M0i = NgLμBS/2
T >Tc,paramagnetic phase,magnetized only along H
Expanding Brillouin function (17.2.30) to first term
mi = BS(xi) = S + 13S xi = (S + 1)gLμBHi3k
BT
Substitute it into (17.2.28)
Mi? C2T
summationdisplay
j
γijMj = C2TH (17.2.31)
Curie constant C is defined by (17.2.23)
Summing (17.2.31) with respect to i and using γij = γji,for inegationslash= j,
and γii = γjj
M
1? C2T summationdisplay
j
γij
= CTH (17.2.32)
Susceptibility from (17.2.32)
χ = MH = CT +Θ
A
(17.2.33)
ΘA =?C2
summationdisplay
γij =?2S(S + 1)3k
B
summationdisplay
i
ziJi (17.2.34)
ΘA is a positive temperature
Curie-Weiss law with a negative Curie temperature Tc =?ΘA
Set H = 0 in (17.2.31) and look for a nontrivial solution
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
2T/C?γ11 γ12
γ12 2T/C?γ11
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle = 0 (17.2.35)
The solutions for transition temperatures
T = C2 (γ11±γ12) = 2S(S + 1)3k
B
(z2J2±z1J1) (17.2.36)
The + solution→T =?ΘA,a transition at negative temperature
the eigenvector (M1 = M2),unstable,not observable
The - solution gives M1 =?M2,antiferromagnetic state,
the transition temperature called N′eel temperature
TN = 2S(S + 1)3k
B
(z2J2?z1J1) (17.2.37)
If only n.n,interaction,then J1 =?|J|,J2 = 0
TN = ΘA = 2S(S + 1)3k
B
z1|J|
17.2.4 Ferrimagnetic Transition
17.2.5 Ferromagnetic and Antiferromagnetic Ground States
§17.3 Theory based on Itinerant Electrons
17.3.1 Mean-Field Approximation of Hubbard Hamiltonian
17.3.2 Stoner Theory of Ferromagnetism
17.3.3 Very Weak Itinerant Ferromagnetism
17.3.4 Spin Density Waves and Antiferromagnetism
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
May 13,2004
Contents
Chapter 17,Ferromagnets,Antiferromagnets and Ferrimagnets
§17.1 Basic Features of Magnetism,,,,,,,,,,,,,,,,,4
17.1.1 Main Types of Magnetism,,,,,,,,,,,,,,,4
17.1.2 Spatial Pictures of Magnetic Structures,,,,,,,,4
17.1.3 Band Pictures of Magnetic Structures,,,,,,,,,4
17.1.4 Hamiltonians with Time Reversal Symmetry,,,,,4
§17.2 Theory based on Local Magnetic Moments,,,,,,,,,5
17.2.1 Mean Field Approximation for HeisenbergHamiltonian 5
17.2.2 Ferromagnetic Transition,,,,,,,,,,,,,,,,11
17.2.3 Antiferromagnetic Transition,,,,,,,,,,,,,,16
17.2.4 Ferrimagnetic Transition,,,,,,,,,,,,,,,,22
17.2.5 Ferromagnetic and Antiferromagnetic Ground States,30
§17.3 Theory based on Itinerant Electrons,,,,,,,,,,,,40
17.3.1 Mean-Field Approximation of Hubbard Hamiltonian,41
17.3.2 Stoner Theory of Ferromagnetism,,,,,,,,,,,46
17.3.3 Very Weak Itinerant Ferromagnetism,,,,,,,,,55
17.3.4 Spin Density Waves and Antiferromagnetism,,,,,64
Chapter 17
Ferromagnets,Antiferromagnets and
Ferrimagnets
§17.1 Basic Features of Magnetism
17.1.1 Main Types of Magnetism
17.1.2 Spatial Pictures of Magnetic Structures
17.1.3 Band Pictures of Magnetic Structures
17.1.4 Hamiltonians with Time Reversal Symmetry
§17.2 Theory based on Local Magnetic Moments
17.2.1 Mean Field Approximation for Heisenberg
Hamiltonian
From Heisenberg Hamiltonian (17.1.15)
Hi =?JSi ·
zsummationdisplay
j=1
Sj (17.2.1)
An effective field Heff
gLμBHeff = J
zsummationdisplay
j=1
Sj = J
zsummationdisplay
j=1
〈Sj〉 = zJ〈Sj〉 (17.2.2)
Land′e factor gL and Bohr magneton μB = eplanckover2pi1/2mc
Single atom Hamiltonian
Hi =?gLμBSi ·Heff (17.2.3)
Magnetization
M = NgLμB〈Sj〉 (17.2.4)
Effective field
Heff = zJg
LμB
〈Sj〉 = zJNg2
Lμ
2
B
M = γM (17.2.5)
γ = zJ/Ng2Lμ2B
Apply an field H,total field
Ht = H +Heff = H +γM (17.2.6)
Assume H along z axis → Heff along z axis
Instead of (17.2.1),single atom Hamiltonian
Hi =?gLμBSizHt (17.2.7)
Its eigenvalues
Eν =?gLμBνHt,ν =?S,···,S (17.2.8)
Partition function
Z =
Ssummationdisplay
ν=?S
e?Eν/kBT =
Ssummationdisplay
ν=?S
eνgLμBHt/kBT (17.2.9)
after summation
Z = sinh[gLμBHt(2S + 1)/2kBT]sinh[g
LμBHt/2kBT]
(17.2.10)
Magnetization
M = NkBT?lnZ?H
t
= NgLμBSBS(x) (17.2.11)
Brillouin function defined as
BS(x) = 2S + 12S coth
parenleftbigg2S + 1
2S x
parenrightbigg
12S coth
parenleftbigg 1
2Sx
parenrightbigg
,(17.2.12)
and
x = gLμBSHtk
BT
(17.2.13)
Define the maximum of magnetization
M0 = NgLμBS (17.2.14)
The reduced magnetization
m = M/M0 = BS(x),(17.2.15)
BS(x) takes value from 0 → 1 as x from ∞ → 0
Let x0 = gμBSH/kBT,then
x = x0 + (zJS2/kBT)m (17.2.16)
(17.2.15) and (17.2.16) are two coupling equations
Its graphical solution is in Fig,17.2.1
Derive coupling equations (17.2.15) and (17.2.16),Find
the numerical solutions and plot figures forS = 3/2 andS = 5/2.
T>Tc
H=0
T=Tc
H=0
T<Tc
H=0
T>Tc
H0=0
B7/2
0.0 1.00.0 2.0 3.0 4.0 5.0
0.2
0.4
0.6
0.8
1.0
x
m
Figure 17.2.1 graphical solution of magnetization.
17.2.2 Ferromagnetic Transition
Consider external field H = 0,when T ↓
Spontaneous broken symmetry of time reversal
paramagnetic phase → ferromagnetic phase
For small x,(17.2.12) is expanded
BS(x) = S + 13S x? [(S + 1)
2 +S2](S + 1)
90S3 x
3 +··· (17.2.17)
A spontaneous internal magnetic field develops at Tc
Substitute (17.2.17) to (17.2.15)
Tc = 2JzS(S + 1)3k
B
(17.2.18)
T >Tc,a magnetic solution exists only if H is present
T <Tc,the spontaneous magnetization in zero field satisfies
m = BS
parenleftbigg 3S
S + 1
Tc
Tm
parenrightbigg
(17.2.19)
T lessorsimilarTc
m2 = 103 (S + 1)
2
(S + 1)2 +S2
Tc?T
Tc (17.2.20)
m～ (Tc?T)1/2
Magnetization M is not unique in its direction
due to Hamiltonian (17.1.16) is isotropic
Heating and recooling,magnetization will occur with different direction
If considering crystal structures,an anisotropic term added into
the Hamiltonian,the degeneracy will be lifted
From (17.2.16) using the small argument expansion
x = x0 + TcTx (17.2.21)
This gives Curie-Weiss law for the susceptibility
χ = MH = CT?T
c
(17.2.22)
Curie constant
C = Ng
2
Lμ
2
BS(S + 1)
3kB (17.2.23)
Internal energy
U =?M
parenleftbigg1
2γM +H
parenrightbigg
=?12γM20m
parenleftbigg
m+ 2HγM
0
parenrightbigg
Specific heat
CM =?U?T =?γM20
parenleftbigg
m+ HγM
0
parenrightbigg?m
T (17.2.24)
When H = 0,for T >Tc,m = 0
CM = 0 (17.2.25)
for T <Tc,mnegationslash= 0
CM =?12γM0dm
2
dT = 5NkB
S(S + 1)
(S + 1)2 +S2 (17.2.26)
Specific heat indicates a second-order phase transition
These are consistent with Landau theory,if take m = M/M0
as the order parameter to construct free energy
Φ = Φ0 +A(T)m2 +Bm4?Hm (17.2.27)
17.2.3 Antiferromagnetic Transition
J < 0 in (17.1.16) → antiferromagnetism
Taking two sublattice system as an example in n.n.n,model
Two interaction constants between two kinds of spins
J12 = J21 and J11 = J22
Also effective field method but for two sublattice i,j (i,j = 1,2)
Two effective fields
Hi = H +
summationdisplay
j
γijMj (17.2.28)
Coupling constants
γij = 2zijJijNg2μ2
B
(17.2.29)
zij is the number of neighbors
z12 = z1 is the total number of nearest neighbors
z11 = z22 = z2 is the total number of next nearest neighbors
Defining the exchange parameter for nearest neighbors J1 = J12
and the exchange parameter for next nearest neighbors J2 = J11
γ12 = γ21 = 2z1J1Ng2
Lμ
2
B
,γ11 = γ22 = 2z2J2Ng2
Lμ
2
B
Repeating the arguments in the ferromagnetic case
Reduced magnetization on sublattice i
mi = BS(xi) (17.2.30)
xi = gLμBHiS/kBT and M0i = NgLμBS/2
T >Tc,paramagnetic phase,magnetized only along H
Expanding Brillouin function (17.2.30) to first term
mi = BS(xi) = S + 13S xi = (S + 1)gLμBHi3k
BT
Substitute it into (17.2.28)
Mi? C2T
summationdisplay
j
γijMj = C2TH (17.2.31)
Curie constant C is defined by (17.2.23)
Summing (17.2.31) with respect to i and using γij = γji,for inegationslash= j,
and γii = γjj
M
1? C2T summationdisplay
j
γij
= CTH (17.2.32)
Susceptibility from (17.2.32)
χ = MH = CT +Θ
A
(17.2.33)
ΘA =?C2
summationdisplay
γij =?2S(S + 1)3k
B
summationdisplay
i
ziJi (17.2.34)
ΘA is a positive temperature
Curie-Weiss law with a negative Curie temperature Tc =?ΘA
Set H = 0 in (17.2.31) and look for a nontrivial solution
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
2T/C?γ11 γ12
γ12 2T/C?γ11
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle = 0 (17.2.35)
The solutions for transition temperatures
T = C2 (γ11 ±γ12) = 2S(S + 1)3k
B
(z2J2 ±z1J1) (17.2.36)
The + solution → T =?ΘA,a transition at negative temperature
the eigenvector (M1 = M2),unstable,not observable
The - solution gives M1 =?M2,antiferromagnetic state
the transition temperature called N′eel temperature
TN = 2S(S + 1)3k
B
(z2J2?z1J1) (17.2.37)
If only n.n,interaction,then J1 =?|J|,J2 = 0
TN = ΘA = 2S(S + 1)3k
B
z1|J|
17.2.4 Ferrimagnetic Transition
Ferrimagnets are the extension of antiferromagnets
but M1 negationslash= M2
N′eel’s mean-field treatment to ferrimagnetic transition
Using the similar reasoning for antiferromagnets in §17.2.3
Mi = NigLμBSiBSi(xi) (17.2.38)
Near Tc,expand Brillouin function to first order
Mi = CiT Hi (17.2.39)
Curie constant Ci
(T?C1γ11)M1?C1γ12M2 = C1H
C2γ12M1 + (T?C2γ22M2) = C2H (17.2.40)
From these two equations
M1 = C1(T?C2γ22) +C1C2γ12(T?C
1γ11)(T?C2γ22)?C1C2γ212
H
M2 = C2(T?C1γ11) +C1C2γ12(T?C
1γ11)(T?C2γ22)?C1C2γ212
H (17.2.41)
T
χ-1
0 T0 T
c
χ-1
M
χ-1
,M
χ-1
0 TN Tθ T0 T
c
χ-1
M
χ-1
,M
(a) (b)
(c) (d)
Total magnetization
M = M1 +M2
Susceptibility is determined by χ = M/H
1
χ =
T
C?
1
χ0?
σ
T?θ (17.2.42)
C = C1 +C2,χ0,σ and θ determined by Ci and γij
1/χ-T shown in Fig,17.1.1(d)
Problem,Combined the expressions in §17.2.3,to derive
the formulas (17.2.38-17.2.42),especially to give the relations
χ0,σ and θ in Eq,(17.2.42) with Ci and γij
Asymptotic behavior T → ∞
1
χ =
T
C?
1
χ0 (17.2.43)
For general ferrimagnets
Θ =?C/χ0 > 0 (17.2.44)
χ = CT +Θ (17.2.45)
similar to (17.2.33) antiferromagnetic case
For H = 0 from (17.2.40),N′eel temperature
TN = 12{C1γ11 +C2γ22 + [(C1γ11?C2γ22)2 + 4C1C2γ212]1/2} (17.2.46)
T <TN,two sublattices → antiparallel
M(T) = M1(T) +M2(T) (17.2.47)
For different material parameters,interesting magnetic properties
Unconventional curves can often appear as shown in Fig,17.2.2
According to N′eel,all M(T)-T curves in Fig,17.2.3
M1(T)
M1+M2
M2(T)
|M1+M2|
Tc T
M
Figure 17.2.2 Reversal of magnetization direction in ferrimagnets.
T
M
M
T
M
L
T
M
N
T
M
Q
T
M
P
Figure 17.2.3 Relation of magnetization-temperature in ferromagnets.
17.2.5 Ferromagnetic and Antiferromagnetic Ground States
Ferromagnetic and antiferromagnetic ordered phases in §17.2.2 and §17.2.3
What are their ground states?
Heisenberg Hamiltonian
H =?
summationdisplay
i>j
JijSi ·Sj =?
summationdisplay
i>j
Jij(SziSzj + 12S+i S?j + 12S?i S+j ) (17.2.48)
Raising and lowering operators
S±i = Sxi ±iSyi (17.2.49)
Commutation relations
[Szi,S±j ] = ±δijS±i,[S+i,S?j ] = 2δijSzi (17.2.50)
Difference of the operators
1
2S
i S
+
j?
1
2S
+
i S
j = i(S
x
iS
y
j?S
y
iS
x
j) (17.2.51)
antisymmetric about i and j,→ 0 on summation
Heisenberg Hamiltonian
H =?
summationdisplay
i,j
Jij(SziSzj +S?i S+j ) (17.2.52)
When each Szi = S,the completely ferromagnetic state
|0〉 =
productdisplay
i
|S〉i (17.2.53)
This state satisfies
Szi|S〉i = S|S〉i,S+i |S〉i = 0 (17.2.54)
The eigenenergy
H|0〉 = E0|0〉,E0 =?S2
summationdisplay
ij
Jij
In n.n.a.
E0 =?NzS2J (17.2.55)
|0〉 is the ground state for J > 0
AFM ordered state in MFA is not the ground state of
the Heisenberg Hamiltonian,and not even an eigenstates
Anderson,1951,Phys,Rev,86,694,pointed out
the N′eel state energy
E = NzS2J (17.2.56)
is an upper bound for the ground state energy
Consider the Hamiltonian of a cluster in two sublattice AFM
Si and its z neighboring spins Si+δ
Hi =?2JSi ·
summationdisplay
δ
Si+δ (17.2.57)
Define St = summationtextδSj+δ
Hi =?2JSi ·St =?J[(Si +St)2?S2i?S2t]
=?J[(Si +St)2?S(S + 1)?St(St + 1)]
For J < 0,Si +St = St?S → the lowest energy
J[(St?S)(St?S + 1)?S(S + 1)?St(St + 1)] = 2JS(St + 1)
(St)min = zS leads to
Emin = 2JS(zS + 1) (17.2.58)
Sum over i,the ground state energy satisfies the inequalities
NzJS2 >EG >NzJS2
parenleftbigg
1 + 1zS
parenrightbigg
(17.2.59)
AFM ground state is an unsolved problem except 1D S = 1/2 n.n,chain
based on Bethe ansatz
A simplified analysis by Anderson (1973),Mat,Rev,Bull,8,153
N′eel state and its energy
ψNeel = α(1)β(2)α(3)β(4)··· (17.2.60)
E = 〈ψNeel|H|ψNeel〉 = NJzS2/2 = NJ/4 (17.2.61)
The state of an alternating chain of paired atoms and its energy
ψ = α(1)β(2)?α(2)β(1)2 (34)(56)··· (17.2.62)
E = NJ2 S(S + 1) = 0.75NJ (17.2.63)
<ENeel,closer to the correct energy E0 = 0.886NJ
Correct wavefunction obtained by linear combination of (17.2.62)
Anderson called it the RVB,Fig,17.2.4
Anderson extended this treatment to 2D triangular lattice
three sublattices of spins at 120? to each other
The energy of this state
EtriangleNeel = NJ
parenleftbigg
3× 12S2
parenrightbigg
= 0.75NJ (17.2.64)
This energy may be improved as
EtriangleNeel = 2×(0.463±0.007)NJ (17.2.65)
1 2
3 4
(a) (b)
Figure 17.2.4 Two-dimensional resonating valence bonds.
The 2D ground state may be linearly combined pair-bond trial wavefunction
Further result
Etriangle = 2×(0.54±0.01)NJ (17.2.66)
At 0 K,the N′eel state can be a locally stable minimum in the total energy,
but not the absolute minimum
Two states are so far apart in phase space
LiNiO2 has triangular lattice,a possible candidate for RVB
as AFM ground state
§17.3 Theory based on Itinerant Electrons
Conduction electrons with exchange interaction →
Certain metals are
FM like Fe,Co,Ni
AFM like Cr,Mn,γ-Fe or
Nonmagnetic Sc,V,Ti,···
17.3.1 Mean-Field Approximation of Hubbard Hamiltonian
d-orbitals in transition metals localized than s-electrons
the overlap of d- wavefunctions less than s- wavefunctions
Consider a conduction-electron system in one band
with short-range interaction → Hubbard Hamiltonian,its MFA
H =
summationdisplay
ijσ
Tijc?iσciσ +U
summationdisplay
iσ
niσ〈niˉσ〉 (17.3.1)
In a homogeneous system,〈niˉσ〉 = 〈nˉσ〉
H =
summationdisplay
ijσ
Tijc?iσciσ +U
summationdisplay
iσ
〈nˉσ〉c?iσciσ (17.3.2)
Bloch representation
H =
summationdisplay
kσ
εkσc?kσckσ +
summationdisplay
kσ
U〈nˉσ〉c?kσckσ (17.3.3)
The first term is minimum for nσ = nˉσ
The second term is lower when nσ?nˉσ is larger
Competition of these two terms → splitting of energy band
This is the Stoner model,To be clear,(17.3.3) is rewritten as
H =
summationdisplay
kσ
Ekσc?kσckσ (17.3.4)
with
Ekσ = εkσ +U〈nˉσ〉 (17.3.5)
Ekσ is correlated to antiparallel electrons
Define for each atom
n = 〈n↑〉+〈n↓〉,m = 〈n↑〉?〈n↓〉 (17.3.6)
magnetization
M = NμBm (17.3.7)
From (17.3.6) and (17.3.7)
〈nσ〉 = 12(n+σm),σ = ±1 (17.3.8)
Ekσ =
parenleftbigg
εkσ + 12nU
parenrightbigg
σμB
parenleftbigg U
2Nμ2BM
parenrightbigg
(17.3.9)
The second term corresponds to a molecular field
If 〈n↑〉 negationslash= 〈n↓〉,the energy band splitting
Ek↓?Ek↑ = U(〈n↑〉?〈n↓〉) ≡ 2? (17.3.10)
shown in Fig,17.3.1,there is spontaneous magnetization in the system
g↓(E)g↑(E)
E
EF
g↓(E)g↑(E)
E
EF
(a) (b)
Figure 17.3.1 Band splitting of ferromagnetic phases in transition metals,g↑(E) is
the density of states with spin up,and g↓(E) is the density of states with spin down.
17.3.2 Stoner Theory of Ferromagnetism
Fermi distribution of itinerant electrons (kσ)
〈c?kσckσ〉 = 〈nkσ〉 = f(Ekσ)
Under an applied field H
Ekσ = εkσ?σμB
parenleftbigg
H + U2Nμ2
B
M
parenrightbigg
(17.3.11)
Magnetization
M(T) = NμB(〈n↑〉?〈n↓〉) = μB
summationdisplay
k
[f(Ek↑)?f(Ek↓)] (17.3.12)
These are two self-consistent equations
For weak H and M,f(Ekσ) can be expanded
M(T) ∝ 2μ2B
parenleftbigg
H + U2Nμ2
B
M
parenrightbiggintegraldisplay ∞
0
bracketleftbigg
f(E)?E
bracketrightbigg
g(E)dE (17.3.13)
the generalization of (6.3.51) by adding the molecular field to external field
For U negationslash= 0,by using (6.3.52)
M(T) = χp(T)1?2(U/N)[χ
p(T)/4μ2B]
H ≡χ(T)H (17.3.14)
χp(T) is the paramagnetic susceptibility of itinerant electrons
2(U/N)χp(T) = 1 → paramagnetic phase unstable
The condition of stability for ferromagnetic phase
2UNχp(T) > 1 (17.3.15)
In terms of (6.3.52),χp(T = 0)/4μ2B = g(EF)/2
U
Ng(EF) > 1 (17.3.16)
This is called Stoner criterion,FM ground state is more favorable
For paramagnetic phase at 0 K
χ(T = 0) = χp(T = 0)1?(U/N)g(E
F)
(17.3.17)
Stoner factor
S = 11?(U/N)g(E
F)
(17.3.18)
The product of interaction constant and the DOS
Fig,17.3.2,exchange integral J and DOS g(EF)
Z
-0.04 10 20 30 40 50
20
40
60
80
100
0
-0.02
0.0
0.02
0.04
00.6
0.08
0.10
0
g (
E F
)
J (R
y)
Figure 17.3.2 Exchange integral J and DOS at Fermi surface g(EF) with atomic
number.
To study spontaneous magnetization
Ekσ = εkσ? 12σUm (17.3.19)
Fermi distribution function
f(Ekσ) =
bracketleftbigg
expεkσ?
1
2σUm?μ
kBT + 1
bracketrightbigg?1
(17.3.20)
Total number of electrons in spin-up band N+
N+ =
integraldisplay ∞
0
bracketleftbigg
expE?
1
2Um?μ
kBT + 1
bracketrightbigg?1
g(E)dE (17.3.21)
Take the free electron form for further calculation
g(E) = 34NE?3/2F E1/2 (17.3.22)
The following abbreviations
x = Ek
BT
,η = μk
BT
,β = Um2k
BT
(17.3.23)
Define the function
F1/2(α) =
integraldisplay ∞
0
x1/2dx
ex?α + 1 (17.3.24)
N+ = 34N
parenleftbiggk
BT
EF
parenrightbigg3/2
F1/2(η+β) (17.3.25)
Total number of electrons in spin-down band
N? = 34N
parenleftbiggk
BT
EF
parenrightbigg3/2
F1/2(η?β) (17.3.26)
The total number of electrons N = N+ +N? and
magnetization M = μB(N+?N?)
N = 34N
parenleftbiggk
BT
EF
parenrightbigg3/2bracketleftbig
F1/2(η+β) +F1/2(η?β)bracketrightbig (17.3.27)
and
M = 34NμB
parenleftbiggk
BT
EF
parenrightbigg3/2bracketleftbig
F1/2(η+β)?F1/2(η?β)bracketrightbig (17.3.28)
From (17.2.27) and (17.2.28),a little mathematics
U
EF =
1
m[(1 +m)
2/3?(1?m)2/3]
bracketleftBigg
1 + pi
2
12
parenleftbiggk
BT
EF
parenrightbigg2
(1?m2)?2/3
bracketrightBigg
(17.3.29)
At T = 0 K
U
EF =
1
m
bracketleftBig
(1 +m)2/3?(1?m)2/3
bracketrightBig
(17.3.30)
The condition for spontaneous magnetization
4
3 <
U
EF < 2
2/3 (17.3.31)
Curie temperature obtained from (17.3.29)
pi2
12
parenleftbiggk
BTc
EF
parenrightbigg2
= 34 UE
F
1 (17.3.32)
Near Currie point,mlessmuch 1 and EF greatermuchkBT
m2 = 9pi
2
8
parenleftbiggk
BTc
EF
parenrightbigg2bracketleftBigg
1?
parenleftbiggT
Tc
parenrightbigg2bracketrightBigg
(17.3.33)
M = M0 3pi2√2
parenleftbiggk
BTc
EF
parenrightbiggbracketleftBigg
1?
parenleftbiggT
Tc
parenrightbigg2bracketrightBigg1/2
(17.3.34)
χ(T) ～ T
2
F
T2?T2c (17.3.35)
Stoner band model is a mean-field theory,gives a satisfactory explanation
for many properties of metals and alloys,There are still some discrepancies
between experiments and theory,It may be necessary to treat the interaction
between the electrons in a less simplistic way
17.3.3 Very Weak Itinerant Ferromagnetism
We have seen above that the itinerant electron model for describing ferro-
magnetism of transition metals is very successful,however,it shortcomings
are also obvious,especially the temperature dependence of magnetization as
well as paramagnetic susceptibility is inconsistent with Curie-Weiss law,On
the other side,the Tc values estimated from this model for transition metals
are often too high,The reason for these is that spin fluctuation has not
been taken into account sufficiently,this is the common fault for mean-field
theory,However,it is interesting to note that there are a kind of materials,
called very weak itinerant ferromagnets,their Curie temperatures are very
low,i.e.,～ 10 K,and the energy splitting between the up and down spin
subbands is very small,the temperature dependence of magnetization and
susceptibility are in consistent with (17.3.34) and (17.3.35).
Theoretical consideration can similarly begin from (17.3.21),but assume
there is an applied magnetic field,so the number of electrons with positive
or negative spin can be written as
N± =
integraldisplay ∞
0
bracketleftbigg
expE?μ?
1
2Um?μBH
kBT + 1
bracketrightbigg?1
g(E)dE,(17.3.36)
The reduced magnetization is a function of temperature as well as magnetic
field,i.e.,m = m(H,T),Very weak itinerant ferromagnetism means m0 =
m(0,0) lessmuch 1,If the applied field is not very strong,it is reasonable to
assumem(H,T) lessmuch 1,Therefore,(17.3.36)can be expanded for temperature
satisfying kBT/EF lessmuch 1,we have
2
Ng(EF)
parenleftbiggmU
2 +μBH
parenrightbigg
= m
bracketleftBigg
1 +α
parenleftbiggT
Tc
parenrightbigg2bracketrightBigg
+γm3···,(17.3.37)
with
α = 16pi2(kBTc)2(D21?D2),
γ = 18
bracketleftbigg N
g(EF)
bracketrightbigg2parenleftbigg
D21?D23
parenrightbigg
,
Dν = g(ν)(EF)/g(EF),
where Tc is the Curie temperature and g(ν)(EF) is ν-order derivatives of g.
It is found from (17.3.37) that at zero temperature and under no applied
field,there is an expression
1
Ng(EF)U = 1 +γm
2
0,(17.3.38)
which returns to Stoner criterion when its righthand side larger than or equal
to 1,So γ greatermuch 0 is required,This is the necessary but not sufficient condition
for the appearance of the very weak ferromagnetism.
Defining a zero-field dynamic susceptibility as
χ0 = χ(0,0) =
parenleftbigg?M(H,T)
H
parenrightbigg
0,0
,(17.3.39)
it can be deduced
χ0 = g(EF)μ2Bμ0/γm20,(17.3.40)
This formula tells us that susceptibility increases with decreasingm0 and this
has physical meaning,From (17.3.37) and (17.3.38),and Tc as the divergent
temperature for (?m/?H)H=0,i.e.,(?H/?m)m=m(0,Tc) = 0,we have
α = γm20,(17.3.41)
BecauseαisproportionaltoT2c,soaboveexpressioncanbeused todetermine
Tc.
From (17.3.38) to (17.3.41),(17.3.37) can be written into
bracketleftbiggM(H,T)
M(0,0)
bracketrightbigg3
bracketleftbiggM(H,T)
M(0,0)
bracketrightbiggbracketleftBigg
1?
parenleftbiggT
Tc
parenrightbigg2bracketrightBigg
= 2χ0HM(0,0),(17.3.42)
or further into
M2(H,T) = M2(0,0)[1?(T/Tc)2 + 2χ0H/M(H,T)],(17.3.43)
So the theory predicts that at different temperatures,the relation between
M2 andH/M gives a series of parallel straight lines,and the line forT = Tc
passes through the origin,This type of plot is called ‘Arrott chart’,The
parallel straight lines in Fig,17.3.3 describe this variation,and are the results
of one typical weak itinerant ferromagnet ZrZn2.
From (17.3.42),the temperature dependence of zero-field dynamic suscepti-
H / M (kOe/ emu?g-1)
0 5 10 15
10
20
30
M
2 (emu/g)
2
T (K)=4.27
1013
1619
22
25
Figure 17.3.3 Square magnetization of ZrZn2 compounds M2(H,T) versus
H/M(H,T),From B,Barbara,D,Gignoux,and C,Vettier,Lectures on Modern
Magnetism,Science Press and Springer-Verlag,Beijing (1988).
T 2 (K2)
0 200 4000
5
10
M
2 (emu/g)
2
Figure 17.3.4 Square spontaneous magnetization M2(0,T) versus square tempera-
ture,From B,Barbara et al.,ibid.
bility χ = χ(0,T) can be given above or below Tc
χ =

χ0
bracketleftbigg
1?
parenleftBig
T
Tc
parenrightBig2bracketrightbigg?1
,T <Tc,
2χ0
bracketleftbiggparenleftBig
T
Tc
parenrightBig2
1
bracketrightbigg?1
,T >Tc.
From (17.3.43),we find that the temperature dependence of spontaneous
magnetization is
M2(0,T) = M2(0,0)
bracketleftBigg
1?
parenleftbiggT
Tc
parenrightbigg2bracketrightBigg
,(17.3.44)
that is to say,if the relation of M2 and T2 is plotted,it should be a straight
line,This was verified in the investigation of the compound ZrZn2,as shown
in Fig,17.3.4.
17.3.4 Spin Density Waves and Antiferromagnetism
Antiferromagtism in metals can also be investigated in the band approach
when the electron-electron interaction is considered,Its ground state is char-
acterized by a periodic modulation of spin density,This type of ground state
was first proposed by Overhauser (1960,1962) for isotropic metals,1? By
common consensus that the antiferromagnetism of chromium is due to spin
density waves,2? On the other hand,it is confirmed that there are spin
density waves apper in highly anisotropic,so-called quasi-one-dimensional
1?A,W,Overhauser,Phys,Rev,Lett,4,462 (1960); Phys,Rev,128,1437 (1962); Phys,Rev.
167,691 (1968).
2?E,Fawcett,Rev,Mod,Phys,60,209 (1988).
metals,1?
Just like in §6.3.4,we introduce a spatially varying external magnetic field
along z-axis
H(r) =
summationdisplay
q
Hqeiq·r,(17.3.45)
The coupling of electron system to this field is described by an extra term
Hprime =?
summationdisplay
q
MqH?q,(17.3.46)
whereMq is theq-th component of magnetization alongz and can be related
to spin operator as
Mq = gLμBSq,(17.3.47)
1?G,Gr¨uner,Rev,Mod,Phys,66 1 (1994).
Now we can write the total Hamitonian as
H =
summationdisplay
kσ
εkc?kσckσ +U
summationdisplay
i
ni↑ni↓?
summationdisplay
q
MqH?q,(17.3.48)
This Hamiltonian can be rewritten as
H =
summationdisplay
kσ
Ekc?kσckσ? 2Ug2
Lμ
2
B
summationdisplay
q
MqM?q?
summationdisplay
q
MqH?q,(17.3.49)
where Ek = εk+nU/2,The first term describes the nonmagnetic behavior,
and the last two terms are related to magnetism,We can write the last two
terms as
summationdisplay
q
M?q
parenleftbigg
Hq + 2UNg2
Lμ
2
B
Mq
parenrightbigg
.
The term (2U/Ng2Lμ2B)Mq acts exactly as a mean field,By definition,
Mq = χ(q)Hq = χ0(q)
parenleftbigg
Hq + 2UNg2
Lμ
2
B
Mq
parenrightbigg
,(17.3.50)
from which
χ(q) = χ0(q)
parenleftbigg
1 + 2UNg2
Lμ
2
B
χ(q)
parenrightbigg
,(17.3.51)
We can find in the normalized susceptibility,in unit of g2Lμ2B,that
χ(q) = χ0(q)1?(2U/N)χ
0(q)
,(17.3.52)
The normalized susceptibility of independent particles was given in §6.2,i.e.,
χ0(q) =
summationdisplay
k
f(Ek+q)?f(Ek)
Ek?Ek+q,(17.3.53)
For a given q,χ(q) may become infinite at the temperature Tc(q) when
2U
N χ0(q) = 1,(17.3.54)
If,at T = 0,this condition is not satisfied for any value of the q the system
is nonmagnetic at all temperatures,If it is satisfied for various values of q,
the system which is nonmagnetic at high temperature will be ordered at the
temperature Tc = max [Tc(q)].
We obtain ferromagnetic order if q0 = 0 and AFM order if q0 = pi/a for a
simple cubic,for example,If q0 has an arbitrary value,there may be two
types of order (the z-axis being arbitrary),i.e.,one is sinusoidal spin density
wave with
M(Ri) = M0 cos(q0 ·Ri),(17.3.55)
and the other is helicoidal spin density wave with
Mx(Ri) = M0 cos(q0 ·Ri),My(Ri) = ±M0 sin(q0 ·Ri),(17.3.56)
The choice of plus or minus corresponds to two possible rotations of the
magnetization when q0·Ri increases,Starting with these three spin density
waves one may construct spin density waves of any polarization,At Tc,one
of the three spin density waves builds up with an infinitesimal amplitudes.
The existence of static spin density waves depends crucially on the zero order
susceptibility χ0(q),We have seen the form of χ0(q) for free particles,In
this case q0 = 0 and the system becomes ferromagnetic,In general χ0(q)
depends in a detailed fashion on the band structure,but one can give some
general guidelines,χ0(q) is important if,for a large number of k values,
(Ek?Ek+q) is small,This situation occurs if two portions of the Fermi
surface coincide (nearly) over a large area for a translation of wavevector q0
or q0 +K,This condition is just the one to obtain Kohn anomalies.
In one dimension χ0(2kF) is infinite and there is an instability for any in-
finitesimal value of the interaction for q0 = 2kF,In three dimension,χ0(q)
is finite,But with a Coulomb interaction,the Hartree-Fock theory gives an
instability for q0 = 2kF whatever the strength of the interaction may be.
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
May 17,2004
Contents
Chapter 18,Superconductors and Superfluids 3
§18.1 Macroscopic Quantum Phenomena,,,,,,,,,,,,,4
18.1.1 Concept of Bose-Einstein Condensation,,,,,,,,5
18.1.2 Bose-Einstein Condensation of Dilute Gases,,,,,,12
18.1.3 Superfluidity of Liquid Heliums,,,,,,,,,,,,,18
18.1.4 Superconductivity of Various Substances,,,,,,,,28
§18.2 Ginzburg-Landau Theory,,,,,,,,,,,,,,,,,,45
18.2.1 Ginzburg-Landau Equations andBroken Gauge Symmetry 46
18.2.2 Penetration Depth and Coherence Length,,,,,,,53
18.2.3 Magnetic Properties of Vortex States,,,,,,,,,,60
18.2.4 Anisotropic Behavior of Superconductor,,,,,,,,67
§18.3 Pairing States,,,,,,,,,,,,,,,,,,,,,,,,,76
18.3.1 Generalized Cooper Pairs,,,,,,,,,,,,,,,,76
18.3.2 Conventional Pairing of Spin-Singlet s-Wave,,,,,,76
18.3.3 Exotic Pairing for Singlet d-Wave,,,,,,,,,,,76
18.3.4 Pseudogaps and Associated Symmetry,,,,,,,,,76
18.3.5 Exotic Pairing for Spin-Triplet p-Wave,,,,,,,,,76
§18.4 Josephson Effects,,,,,,,,,,,,,,,,,,,,,,,77
18.4.1 Josephson Equations,,,,,,,,,,,,,,,,,,77
18.4.2 Josephson Effects in Superconductors,,,,,,,,,77
18.4.3 Phase-Sensitive Tests of Pairing Symmetry,,,,,,77
18.4.4 Josephson Effect in Superfluids,,,,,,,,,,,,,77
Chapter 18
Superconductors and Superfluids
§18.1 Macroscopic Quantum Phenomena
Microscopic,Mesoscopic,Macroscopic quantum phenomena
Quantum degeneracy temperature T0 = planckover2pi12/3mkBa2
Einstein (1924) predicted Bose-Einstein condensation (BEC)
Ideal gas composed of identical bosons
First experimental confirmation of BEC of dilute gases (1995)
Postulate for BEC has facilitated the study on
superconductivity of metals and alloys and
superfluidity of He liquids
18.1.1 Concept of Bose-Einstein Condensation
A phase transition in an ideal gas of identical bosons
when λT = (2piplanckover2pi1/kBT)1/2 >d
bosons are stimulated by the presence of other bosons
macroscopic occupation of a single quantum state i
ni(T) = 1exp[β(ε
i?μ)]?1
(18.1.1)
N =
summationdisplay
i
1
exp[β(εi?μ)]?1 (18.1.2)
Chemical potential μ is determined from (18.1.2)
When T is low enough
n0(T) ≡N0(T) = 1exp[β(ε
0?μ)]?1
≈N (18.1.3)
N ～ 1023 →kBT/(ε0?μ) ≈N → μlessorsimilarε0 → n1(T) lessmuchN → ε1 greatermuchε0
Under the cyclic boundary condition,εq = planckover2pi12q2/2m,DOS
g(ε) = V4pi2
parenleftbigg2m
planckover2pi12
parenrightbigg3/2
ε1/2 (18.1.4)
g(0) = 0 →
N = N0(T) +Nprime(T) (18.1.5)
Nprime(T) = V4pi2
parenleftbigg2m
planckover2pi12
parenrightbigg3/2integraldisplay ∞
0
ε1/2dε
exp[β(ε?μ)]?1 (18.1.6)
μlessorequalslant 0,μ = 0 → upper bound of Nprime(T),With x = βε
Nprime(T) lessorequalslant V4pi2
parenleftbigg2mk
BT
planckover2pi12
parenrightbigg3/2integraldisplay ∞
0
x1/2dx
ex?1 (18.1.7)
Nprimem(T) = 2.612V
parenleftbiggmk
BT
2piplanckover2pi12
parenrightbigg3/2
(18.1.8)
T <Tc,Nprimem(Tc) <N,which yields
Tc = 2piplanckover2pi1
2
mkB
parenleftbigg N
2.612V
parenrightbigg2/3
(18.1.9)
Nprime(T) = N
parenleftbiggT
Tc
parenrightbigg3/2
(18.1.10)
N0(T) = N
bracketleftBigg
1?
parenleftbiggT
Tc
parenrightbigg3/2bracketrightBigg
(18.1.11)
Fig,18.1.1
BEC can be regarded as a separation in momentum space
Original BEC theory for ideal gas
Bogoliubov’s theory of superfluidity (1947) for weakly repulsive bosons
Penrose and Onsager (1956) gave a more general theory
A precise term introduced by C,N,Yang (1962)
characterize by an off-diagonal component of a density matrix
ρ(r,rprime) =
angbracketleftBig?
Ψ(r)?Ψ?(rprime)
angbracketrightBig
= N0cΨ0(r)Ψ?0(rprime) +G(r?rprime) (18.1.12)
Due to interactions the condensate fraction may be much smaller
Fig,18.1.2
(a) (b)
E E
Figure 18.1.1 Occupation of energy levels for a BE condensate of ideal gas,(a)
T = 0; (b) 0 <T <Tc.
T=0
(a) (b)
E
Superfluid Superfluid Normal Fluid
Thermal
Excitation
Figure 18.1.2 Occupation of energy levels for a BE condensate of interacting bosons:
(a) T = 0; (b) 0 <T <Tc.
18.1.2 Bose-Einstein Condensation of Dilute Gases
Dilute gas with weak interaction,1995,E,Cornell and C,Wieman
First successful BEC of trapped 87Rb atoms
Ultra low temperature with laser cooling plus evaporation cooling
Afterwards BEC for dilute gases,23Na,7Li,1H
A new family of condensed matter (condensed in momentum space)
T < 1 μK,ρ< 1012 atoms cm?3,a trap with about 103-107
Quantum statistics of particles,rather than interactions,dominates
Confining potential for alkali atoms in a trap
V(r) = m2 (ω2xx2 +ω2yy2 +ω2zz2) (18.1.13)
For an ideal Bose atomic gase,the Hamiltonian is the sum of
single particle Hamiltonians with eigenvalues
εnxnynz =
parenleftbigg
nx + 12
parenrightbigg
planckover2pi1ωx +
parenleftbigg
ny + 12
parenrightbigg
planckover2pi1ωy +
parenleftbigg
nz + 12
parenrightbigg
planckover2pi1ωz (18.1.14)
nx,ny,nz are non-negative integers
Ground state wavefunction
Ψ(r1,...,rN) =
productdisplay
i
ψ0(ri) (18.1.15)
ψ0(r) =
parenleftBigmω0
piplanckover2pi1
parenrightBig1/3
exp
bracketleftBig
m2planckover2pi1(ωxx2 +ωyy2 +ωzz2)
bracketrightBig
(18.1.16)
ω0 = (ωxωyωz)1/2 (18.1.17)
ρ(r) = N|ψ0(r)|2 The size of the cloud,or harmonic oscillator length
a0 =
parenleftbigg planckover2pi1
mω0
parenrightbigg1/2
～ 1μ m (18.1.18)
corresponding to the average width in (18.1.16)
At finite temperatures,for kBT greatermuch planckover2pi1ω0
the radius of the thermal cloud aT ～a0(kBT/planckover2pi1ω0)1/2
Condensate fraction of an ideal Bose gas in a harmonic trap
N0
N = 1?
parenleftbiggT
Tc
parenrightbigg3
(18.1.19)
Fig,18.1.3,theoretical and experimental comparison
In Fig,18.1.4,condensate as a narrow peak in both real space and
momentum space is a peculiar feature of trapped Bose gases
Comparison of photon laser and atom laser
0.0 0.5 1.0 1.5
4
8
12
N
/ 10
4
T/Tc
0.0 0.5 1.0 1.5
0.0
0.2
0.4
0.6
0.8
1.0
N 0
/N
Figure 18.1.3 The condensate fraction number N versus the scaled temperature for
BE condensate of Rb gas of 4000 atoms.
0 20 40 60-20-40-60
0
200
400
600
800
1000
z (μm)
ρ l (
μm
-2 )
Figure 18.1.4 The condensate column density versus the radial coordinate.
18.1.3 Superfluidity of Liquid Heliums
An aggregate of helium atoms remains in the liquid
high zero point energy and small atomic mass
Two kinds of stable isotopes,3He and 4He
3He,spin 1/2,Fermi liquid; 4He,spin 0,Bose liquid
Fig,18.1.5,sharp specific heat peak,λ transition
from the normal to superfluid state
Fig,18.1.6,the phase diagram for 4He
Fig,18.1.7,two-component model
-1.5 -0.5 0 0.5 1.5
2
6
10
14
18
22
C p
(J
,g-
1 K
-1 )
T-Tg108 (K)
Figure 18.1.5 Specific heat anomaly in the superfluid transition of 4He.
1 2 3 4 5 6
10
20
30
40
T (K)
P (0.1MPa)
Normal Liquid
Superfluid
(He II)
(He I)
Solid
(bcc)
Melting Line
Boiling Curve
Critical Point
λ Curve
Gas
Figure 18.1.6 Phase diagram of 4He.
0 0.5 1.0 1.5 2.0 2.5
0.2
0.4
0.6
0.8
1.0
nn /n
ns /n
T (K)
λ
Figure 18.1.7 Superfluid fraction ns/n versus T of liquid 4He (inset,schematic
experimental set-up).
London conjectured that superfluidity of liquid 4He is BEC
In Fig,18.1.8,the curve may be fitted to an empirical relation
n0(T) = n0
bracketleftbigg
1?
parenleftbiggT
Tλ
parenrightbiggαbracketrightbigg
,(18.1.20)
α = 3.6 not 3/2,n0(0) = 13.9%,Interaction !
0.0 0.5 1.0 1.5 2.0 2.5
0.00
0.05
0.10
0.15
0.20
T (K)
n 0
(T)
Figure 18.1.8 Condensation fraction versus temperature of liquid 4He.
Macroscopic quantum state and macroscopic wavefunction
ψ(r) = ψ0(r)eiθ(r) (18.1.21)
Average number of superflow particles
ψ?(r)ψ(r) = ψ20(r) = ns (18.1.22)
Condensation momentum p satisfies
pψ = hi?ψ (18.1.23)
For a homogeneous system,ψ0 is independent of r
p = planckover2pi1?θ (18.1.24)
To interpret the canonical momentum of one particle in superfluid 4He
p = m4vs (18.1.25)
vs = planckover2pi1m
4
θ,(18.1.26)
‘rigidity’ exists in the momentum space
Fig,18.1.9,quantization of circulation
Γ =
contintegraldisplay
L
vs ·dl (18.1.27)
Li
Figure 18.1.9 Superfluid in annular (multiply connected) region with a closed con-
tour L.
Due to (18.1.26),single-valued for a trip along a closed loop
Γ = ν hm
4
,ν = 0,±1,±2,··· (18.1.28)
By taking curl of (18.1.27)
×vs = 0 (18.1.29)
irrotationality of superfluid flow
In Fig,18.1.10,the cores of vortex lines are normal regions?×vs negationslash= 0
3He liquid composed of fermions,T <2.7 mK,superfluid state
BEC of bosons consisting of pairs of 3He atoms
18.1.4 Superconductivity of Various Substances
Two fundamental features below Tc:
(1) zero resistivity,persistent currents,Fig,18.1.11
(2) perfect diamagnetism,Meissner effect
Thermodynamic critical field
H2c(T)
8pi = fn(T)?fs(T) (18.1.30)
Empirically Hc(T) described by a parabolic law (Fig,18.1.12)
Hc(T) ≈Hc(0)
bracketleftBigg
1?
parenleftbiggT
Tc
parenrightbigg2bracketrightBigg
(18.1.31)
A specific heat anomaly in the superconducting transition,Fig,18.1.13
Figure 18.1.10 Array of vortex lines in rotating superfluid 4He.
TTc
ρ0
ρ(T)
0
Figure 18.1.11 Temperature dependence of the resistivity of a superconductor.
0 Tc T
Hc
Hc(0)
S
N
Figure 18.1.12 Phase diagram of a type I superconductor.
0 Tc T
C
Figure 18.1.13 Specific heat anomaly in the superconducting transition.
Fig,18.1.14,type I superconductors; Fig,18.1.15,type II superconductors
In a superconductor the condensate particle are Cooper pair
In an applied magnetic field B =?×A
the canonical momentum of a Cooper pair
p = 2mvs + 2eA (18.1.32)
vs = planckover2pi12m?θ?eAmc (18.1.33)
corresponding to (18.1.26) of a superfluid,means a macroscopic wavefunction
0 Hc H
-M
Figure 18.1.14 Meissner effect of type I superconductors.
-M
0
Hc Hc2Hc1 H
Figure 18.1.15 Meissner effect of type II superconductors.
Supercurrent density
js = nsevs (18.1.34)
Combining with (18.1.33)
js = nseplanckover2pi12m?θ? nse
2
mcA (18.1.35)
Taking curl of (18.1.35),assuming nonmagnetic,H =?×A
×js + nse
2
mcH = 0 (18.1.36)
In the stationary case,Maxwell equations give
×H = 4pic js (18.1.37)
Substituting it into (18.1.36),the London equation
H +λ2L?×?×H = 0 (18.1.38)
London penetration depth about 50-200 nm
λL = (mc2/4pinse2)1/2 (18.1.39)
London equation can be used to illustrates Meissner effect
For a semi-infinite specimen,the region z < 0 being empty
From (18.1.38) and with?·H = 0
2H? 1λ2
L
H = 0 (18.1.40)
0 z
B
B(0) λL
Figure 18.1.16 Penetration of the magnetic field inside the surface layer of a super-
conductor according to the London equation.
d2H
dz2 =
H
λ2L (18.1.41)
Final result
H(z) = H(0)e?z/λL,js = c4pidHdz = js(0)e?z/λL (18.1.42)
Fig,18.1.17,Meissner effect
Figure 18.1.17 An illustrationof the Meissner effect withan additionalconsideration
on the penetration of a magnetic field into the surface layer.
Fig,18.1.18,multiply-connected superconductors,flux quantization
contintegraldisplay
L
θ·dl =
contintegraldisplay
L
2e
planckover2pi1cA·dl (18.1.43)
Quantization of magnetic flux
Φ = 2piνplanckover2pi1c2e = νhc2e (18.1.44)
Flux quantum or fluxoid defined
φ0 = hc2e = 2.07×10?7 Gs·cm2 (18.1.45)
observed experimentally and its value gives direct confirmation of
the pairing of electrons in superconductors
L
Figure 18.1.18 A ring-shaped superconductor.
Table 18.1.1 Periodic table of superconducting elements
H
Li
Na
K
Rb
Cs
(1.5)
Fr
Be
0.026
(9)
Mg
Ca
Sr
Ba
(5.4)
Ra
Sc
Y
(2.5)
La
(6.0)
Ac
Ti
Zr
0.61
Hf
0.12
Th
1.4
V
5.4
Nb
9.25
Ta
4.47
Cr
Mo
0.92
W
0.01
Mn
Tc
7.7
Re
1.7
Fe
(2)
Ru
0.49
Ir
0.11
Co
Rh
.0003
Os
0.66
Ni
Pd
Pt
Cu
Ag
Au
Zn
0.85
Cd
0.52
Hg
4.15
B
(11)
Al
1.18
Ga
1.08
In
3.41
Ti
2.38
C
Si
(7.1)
Ge
(5.3)
Sn
3.72
Pb
7.2
N
P
(5.5)
As
(0.5)
Sb
(3.5)
Bi
(8.5)
O
S
(17)
Se
(5.9)
Te
(4.3)
Po
F
Cl
Br
I
At
He
Ne
Ar
Kr
Xe
Rn
e/a=1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8
s s-d s-p
Elements
Tc (K)
(Tc) Metastable
Ce
(1.8)
Pr-
Yb
Pa
(1.4)
Lu
(2.8)
U
(2.4)
Table 18.1.2 Some superconducting alloys and compounds
typematerial Tc (K)Hc2
N2H)remark
alloys
Mo0.5Re0.5 12.5 115
(solid
solutions)
Nb0.4Ti0.6 9.3 124 practical superconducting
materials for cablesNb0.26Ti0.7Ta0.04 9.9 91
Nb0.75Zr0.25 11 ~26
Pb0.75Bi0.25 8.7
alloys
V3Ga 16.8 240 A15 structures,practical supercon-
ducting materials for cables
(com-
pounds)
dittoNb3Al 18.8 300
dittoNb3Sn 18.1 245
dittoNb3Al0.75Ge0.25 21.0 420
A15 structure,the highest TcNb3Ge 23.2
B1 structureNbN 17
Chevrel-phasePb3Mo6S 14.7 600
Chevrel-phase with highest Hc2 losingGa0.25Eu0.3Pb0.7Mo6S8 14.3 700
superconducting at 0.9K,changing intoErRh4B4 8.5
ferromagnet <0.2K,superconducting
coexisted with antiferromagnet
ErMo6S8 2.2
39 The highest Tc for alloysMgB2 16~18
coexistence of SC and WFMZrZn2 0.29
heavy CeCu2Si2 0.5~0.6
having three superconducting phases,
p wave pairing(?)
electron UPt3 0.5
intermetallic
compounds
UBe3
coexistence of SC and WFMUGe2 <(1)
oxides
perovskite type structureSiTiO3-x 0.05~0.5
perovskite type structureLiTiO4 13
x~0.25,Ba(Pb1-xBix)O3 13
Ba1-2KxBiO3 >30 x>0.35 isolator x~0.4
layered perovskite type structure,
p-wave paring(?)
SrRuO4 1.2
ocuprates
HTSC,La2-xBaxCuO4 >30
layered perovskite type structure
in liguid nitrogen
discovered first superconductor with TcYBa2Cu3O7-x 93 150(77K)
Bi2Sr2CaCu2O8 110 500(4.2K)
Tl2Ba2Ca2Cu3O8 125
very unstableHgBa2Ca2Cu3O8-x 135
164 coexistence of SC and WFM in
different layers
RuSr2GdCu2O8 15~40
organic 1D charge transfer salt (TMTSF)2ClO4 1.2~1.4
compounds [BEDT-TTF]2Cu(MCS)2 11.4 1D charge transfer salt,
having the highest Tc for organicsand K3C60 ~18
molecular
materials
Rb3C60 ~28~15
abnormal Meissner effect,single wall nanocarbon-
tube and 1D SC uctuations
(at high pressure)
§18.2 Ginzburg-Landau Theory
A phenomenological theory near Tc
applied to both the homogeneous and inhomogeneous superconductors
Order parameter ψ of superconductors like the macroscopic wavefunction
GL theory was developed before the microscopic BCS theory
Later Gor’kov proved that it is a rigorous consequence of microscopic theory
It is better to be used to intermediate state,mixing state and spatial inho-
mogeneity
18.2.1 Ginzburg-Landau Equations and
Broken Gauge Symmetry
For a spin-singlet s-wave superconductor,the pair wavefunction
ψ(r) = 〈ψ↑(r)ψ↓(r)〉
The N-S phase transition is broken gauge symmetry
Condensed phase corresponds to a macroscopically occupied quantum state
described by a macroscopic wavefunction
|ψ|2 = ψ?ψ →ns,the local pair density
Density functional of free energy
fs(r) = fn +α|ψ(r)|2 + β2|ψ(r)|4 + 12m?
vextendsinglevextendsingle
vextendsinglevextendsingle
parenleftbigg
iplanckover2pi1?+ e
cA
parenrightbigg
ψ
vextendsinglevextendsingle
vextendsinglevextendsingle
2
+ H
2(r)
8pi
(18.2.1)
e? = 2e,and m? = 2m
Taking the minimum of free energy with ψ
δF = δ
integraldisplay
f(r)dr = 0 (18.2.2)
and gauge condition?·A = 0 → GL equation
1
2m?
parenleftbigg
iplanckover2pi1?+ e
cA
parenrightbigg2
ψ+αψ +β|ψ|2ψ = 0 (18.2.3)
The boundary condition is the normal component being zero
n·
parenleftbigg
iplanckover2pi1?+ e
cA
parenrightbigg
ψ = 0 (18.2.4)
Taking minimum of free energy with A,the supercurrent equation
js =?ie
planckover2pi1
2m?(ψ
ψ?ψ?ψ?)? e
2
m?cψ
ψA (18.2.5)
In general,ψ = |ψ|exp(iθ)
js = e
m?|ψ|
2
parenleftbigg
planckover2pi1?θ?e
cA
parenrightbigg
= e?|ψ|2vs (18.2.6)
consistent with (18.1.34),just the formula for current in quantum mechanics
GL equation is nonlinear for the term β|ψ(r)|2ψ(r)
ψ(r) → exp(iθ)ψ(r),global gauge transformation
When θ takes different constants,f in (18.2.1) is invariant
a gauge symmetry group,in Fig,18.2.1
GL equations give a simple illustration for superconductors
For simplicity,assume size of the superconductor infinite
and A = 0
Minimum of free energy in (18.2.1)
αψ0 +β|ψ0|2ψ0 = 0 (18.2.7)
|ψ0| =

0,T >Tc
[α(T?Tc)/β]1/2,T <Tc
(18.2.8)
In the normal state T >Tc,ψ0 = 0,ψ0 exp(iθ) = 0
In the superconducting state T <Tc,ψ0 →ψ0 exp(iθ) negationslash= 0
T >Tc,the global gauge symmetry is intact
T <Tc,it is spontaneously broken
eiθψ
Re(ψ)
ψ'=eiθψ=0 θ
Im(ψ)
θ0
Figure 18.2.1 Global gauge transformation of the GL order parameter.
Ginzburg-Pitaevskii equation for 4He
parenleftbigg
planckover2pi12m
4
2 +α+β|ψ(r)|2
parenrightbigg
ψ(r) = 0 (18.2.9)
An extension to the case of a normal fluid
f(r) = 12m4|(?iplanckover2pi1vn))ψ(r)|2 +f0(|ψ(r)|2) (18.2.10)
Gross-Pitaevskii equation for BE condensate in the trap
parenleftbigg
planckover2pi1
2
2m?
2 +V(r) +g|ψ(r)|2
parenrightbigg
ψ(r) = μψ(r) (18.2.11)
g and μ denote the interaction strength and chemical potential
18.2.2 Penetration Depth and Coherence Length
In the interior of a superconductor,no magnetic field exists and the gradient
term vanishes,the free energy density in (18.2.1) is reduced to
fs?fn = α|ψ|2 + 12β|ψ|4,(18.2.12)
wherethetemperaturedependence ofparametersareassumedα = a(T?Tc),
and β being a constant larger than zero as determined from free energy
minimum,In this case,the first term in (18.2.3) vanishes,and we can write
|ψ|2 = |ψ∞|2 =?αβ ∝ (1?t),(18.2.13)
where t = T/Tc,Combined with 18.1.30,the thermodynamic critical field is
determine from
fs?fn = H
2c
8pi =?
α2
2β,(18.2.14)
i.e.,
Hc =
parenleftbigg4pi
β
parenrightbigg1/2
|α| ∝ (1?t),(18.2.15)
from which when T →Tc,Hc → 0.
From (18.2.6),the current is simply given by
js = em|α|β
parenleftbigg
planckover2pi1?θ? 2ec A
parenrightbigg
,(18.2.16)
If we notice that B =?×A and js = (c/4pi)?×B,we find the usual
London penetration depth is temperature dependent
λL =
parenleftbigg mc2β
8pie2|α|
parenrightbigg1/2
～ (1?t)?1/2,(18.2.17)
The supercurrent is confined within this distance λ of the surface.
To study the variation of the wavefunction near surface,we consider GL
equations (18.2.3) in the absence of field,i.e.,A = 0,Since all of the
coefficients are real,ψ must be real,and there is no supercurrent,js = 0.
Introducing a dimensionless wavefunction G = ψ/ψ∞,and assuming that it
varies only in z-direction,(18.2.3) becomes
ξ2(T)d
2Gprime
dz2 +G
prime?Gprime3 = 0 (18.2.18)
where a coherence length ξ(T) is defined as
ξ(T) =
parenleftbigg planckover2pi12
2m?|α|
parenrightbigg1/2
∝ (1?t)?1/2,(18.2.19)
ξ is the characteristic length for variation of the order parameter and di-
verges at T = Tc,To see the significance of ξ(T) clearer,we should get an
approximate solution for (18.2.18),Let Gprime = 1 +gprime,with gprime lessmuch 1,then the
first order of (18.2.18) becomes
ξ2 d
2
dz2g
prime + (1 +gprime)?(1 + 3gprime +...) = 0,(18.2.20)
or
d2
dz2g
prime = 2
ξ2g
prime,(18.2.21)
and its solution is
gprime(z) ～ e±
√2z/ξ(T)
,(18.2.22)
This means that ψ will decay to ψ∞ within the characteristic length ξ(T).
There is an important GL parameter defined as a ratio of above two length
parameters
κ = λξ,(18.2.23)
In type I superconductors,λ～ 50 nm,ξ ～ 400 nm,so κlessmuch 1 for these ma-
terials,Fig,18.2.2 shows the crossover region of the normal-superconductor
boundary,how the magnetic field penetrates the superconductor to a depth
λ,and how ψ increases in the superconductor to its value at infinity ψ∞ in a
distance ξ,Abrikosov (1957) shows the importance of the κgreatermuch 1 case which
leads to type II superconductors,Essentially all superconducting compounds
and all high-Tc materials have κ greatermuch 1,In general,the value κ = 1/√2 is
used as a criterion to separate superconductors of type I and II.
x x
λ
ξξ
λ
fw fw
H H
Hc Hc
h
ψ
ψ h
κ<< 1 κ>> 1
(a) (b)
Figure 18.2.2 Free energy of the wall,local magnetic field and order parameter
versus the distance from a normal-superconductor interface.
18.2.3 Magnetic Properties of Vortex States
As stated in §18.1.4,the flux quantization is an important concept related to
macroscopic wavefunction,It should be noted that there is a profound differ-
ence in magnetic behavior of type I and type II superconductors as deduced
from L,V,Shubnikov’s experimental results in the 1930s,In contrast to type
I,the magnetization of type II material begins to decrease atH >Hc1,then
decreases more gradually,and goes to zero at H >Hc2,This peculiar mag-
netic behavior is instrumental in making the applications of superconducting
magnets and cables possible,Abrikosov in 1957 first explained the magnetic
properties of type II superconductors theoretically from the solutions of GL
equations,The essential point is that in type II material,magnetic flux can
penetrate into bulk specimen as an array of vortex lines between Hc1 and
Hc2,So Meissner effect is not so complete as in type I materials,this mixture
of a superconductor and with normal state in the cores of a vortex lattice is
called the vortex state or the mixed state,This bold hypothesis was experi-
mentally confirmed with neutron diffraction in 1965 and direct observations
later.
With the GL theory,we can determine the value ofHc2 even without explicit
knowledge of the structure of a vortex state,Since H →Hc2,ψ → 0; then
at magnetic field slightly less than Hc2,ψ is very small,so the nonlinear
term in GL equation may be dropped and instead the simplified linear GL
equation
1
4m
parenleftbigg
iplanckover2pi1?+ 2ec A
parenrightbigg2
ψ = |α|ψ (18.2.24)
may be used,Here A may be regarded as the vector potential for the homo-
geneous field H at ψ = 0,in this situation,the magnetic field permeates the
normal state,Formally (18.2.24) is just a Schr¨odinger equation to describe
the motion of a particle with charge 2e and mass 2m in the magnetic field
leading to Landau levels (discussed in §7.3.2),and |α|is the spacing of en-
ergy levels,Furthermore the boundary condition thatψ = 0 at infinity is the
same for both problems,We have already solved the problem of a charged
particle in magnetic field,the minimum value for energy is E0 = planckover2pi1ωH/2,
where ωH = eH/2mc,Due to similarity of these two problems,so we may
conclude that superconducting phase appears only when the condition
|α|> eplanckover2pi12mcH (18.2.25)
is satisfied,Then we arrive at
Hc2 = 2mc|α|eplanckover2pi1 = φ02piξ(T)2 = √2κHc,(18.2.26)
This is the highest field at which superconductivity can nucleate in a bulk
sample in a decreasing external magnetic field.
Hc1 marks the magnetic field at which the first vortex in the form of flux
quantum penetrates into the specimen,By definition,when H = Hc1,the
Gibbs free energy must have the same value whether the first vortex is in or
out of the sample,Thus,Hc1 is determined by the condition
Φ(0)s = Φ(1)s,(18.2.27)
Φ(0)s and Φ(1)s are the Gibbs free energies with no magnetic flux trapped and
with the first isolated vortex line entering the sample,Since
Φ = F? H4pi ·
integraldisplay
hdr,Φs = fs,(18.2.28)
introduce εi (or in other words,the line tension of the vortex) and L the
length of vortex line,the condition becomes
fs = fs +εiL?Hc1φ0L4pi,(18.2.29)
Thus we get
Hc1 = 4piεiφ
0
,(18.2.30)
In the extreme type II limit κ = λ/ξ greatermuch 1,at the center of an isolated
vortex,ψ = 0,then gradually rises to a limiting value at radius ξ,This
defines a vortex core region within which the behavior is like the normal
metal,while outside the core region the behavior just like an ordinary London
superconductor (see Fig,18.2.3).
h(r)
ψ∞
ψ(r)
0ξ λ r
Figure 18.2.3 Magnetic field and superconducting order parameter of an isolated
vortex line.
18.2.4 Anisotropic Behavior of Superconductor
After the discovery of high Tc superconductors,it was found that phenome-
nally the GL equation is still valid,however,their parameters become very
anisotropic,Now 18.2.3 can be rewritten as
planckover2pi1
2
2
parenleftbigg
i2eplanckover2pi1cA
parenrightbigg
·
parenleftbigg1
m
parenrightbigg
·
parenleftbigg
i2eplanckover2pi1cA
parenrightbigg
ψ+αψ+β|ψ|2ψ = 0,(18.2.31)
here (1/m) is the reciprocal mass tensor with principal values 1/mab and
1/mc,Due to the fact that the interlayer coupling is weak,then mc greatermuchmab.
The anisotropy of mass causes the coherence length ξ to be very anisotropic.
We may generalize (18.2.19) to the anisotropic case and get
ξ2i(T) = planckover2pi1
2
2mi|α(T)|,(18.2.32)
where the subscriptirefers a particular principal axis,Sinceα(T) is isotropic
andproportionalto(T?Tc),ξi scaleswith1/√mi anddivergesas|T?Tc|?1/2
when T →Tc,The penetration depth λ is also anisotropic for the relation
2√2piHc(T)ξi(T)λi(T) = φ0,(18.2.33)
It shows that the anisotropy of the penetration depth λi will be the inverse
to that ofξi sinceHc is isotropic,Since whatλi describes is the screening by
supercurrents flowing along theith axis,not the screening of a magnetic field
along the ith axis,We may take an Abrikosov vortex line in a sample with
the magnetic field along a-axis,In an isotropic superconductor,the vortex
will be circularly symmetric; however,in an anisotropic superconductor,the
core radius along plane direction will be ξab,but the core radius along c-
direction will be ξc lessmuch ξab,On the other hand,the flux-penetration depth
will be λc along the plane direction,and will be much smaller,i.e.,λab along
c-direction,Thus,both the core and the current streamlines confining the
flux are flattened into ellipses with long axes parallel to the planes (b-axis),
and aspect ratio (mc/mab)1/2,as shown in Fig,18.2.4.
Furthermore,the anisotropy of the upper critical field along the two distinct
λc
λab b
c
ξab ξ
c
Core
Flux
Figure 18.2.4 Cross section of a vortex line along the a-axis in an anisotropic su-
perconductor.
principal axes may be derived,
Hc2bardblc = φ02piξ2ab,Hc2bardblab = φ02piξabξc (18.2.34)
Since ξab greatermuch ξc,so that Hc2bardblab greatermuch Hc2bardblc,Because Hc1 ～ 1/λ2,which is
inversely related to Hc2,the anisotropy in Hc1 will be inverse to that for
Hc2,i.e.,Hc1bardblab lessmuchHc1bardblc.
Now we introduce the dimensionless anisotropy parameter
γ =
parenleftbiggm
c
mab
parenrightbigg1/2
= ξabξ
c
= λcλ
ab
=
parenleftbiggH
c2bardblab
Hc2bardblc
parenrightbigg
=
parenleftbiggH
c1||c
Hc1||ab
parenrightbigg
,(18.2.35)
The mass ratio mc/mab and γ for YBCO are about 50 and 7 respectively,
whileforBSCCO,theseare20000and150respectively,Thislargeanisotropy
is one of the decisive factors to make the high Tc superconductors acting so
differently from the conventional ones,Near Tc,ξc(T) ≈ ξc(0)(1?t)?1/2
will always be large enough to justify the GL approximation discussed above.
But when the temperature is lowered,ξc(T) shrinks toward a limiting value.
If this value is smaller than the interplanar spacing,it is obvious that the
smooth variation assumed in GL equations will break down at some inter-
mediate temperature T,At temperatures below Tc,it is expected that the
3-D continuum approximation will be replaced by 2D behavior of a stack of
individual layers,This may be described by a model proposed by Lawrence
and Doniach(LD) 1? In this model,the free energy may be expressed as
F =
summationdisplay
n
integraldisplay bracketleftBigg
α|ψn|2 + 12β|ψn|4 + planckover2pi1
2
2mab
parenleftBiggvextendsinglevextendsingle
vextendsinglevextendsingle?ψn
x
vextendsinglevextendsingle
vextendsinglevextendsingle
2
+
vextendsinglevextendsingle
vextendsinglevextendsingle?ψn
y
vextendsinglevextendsingle
vextendsinglevextendsingle
2parenrightBigg
+ planckover2pi1
2
2mcs2 |ψn?ψ
(18.2.36)
where the z is along c-axis,x,y are the coordinates in the plane,s is the
distance between the layers,the sum runs over layers and the integral is over
the area of each layer,Note that if we write ψn = |ψn|eiθn,and assume that
all |ψn| are equal,the last term of (18.2.36) becomes
planckover2pi12
mcs2|ψn|
2[1?cos(ψn?ψn?1)],(18.2.37)
1?W,E,Lawrence and S,Doniach,in Proc 12th Int,Cont,Low Temp,Phys,(Kyoto,1970); E.
Kanda (ed.),Keigaku,Tokyo,1971.
This term is equivalent to a Josephson coupling energy (1/mc) between adja-
cent planes (see §18.4),This crossover from 3D behavior [Hc2 ∝ (Tc?T)] to
2D one [Hc2 ∝ (Tc?T)1/2] in superconductors has been verified experimen-
tally by the artificial layered composites Nb/Ge with certain thickness range
(Fig,18.2.5),We can expect that high Tc superconductors have analogous
behavior.
2-D
4.5nm/5nm
DNb/DGe=6.5nm/3.5nm
3-D
4.5nm/0.7nm
Hc2parallelto
Hc2parallelto
1.00.90.80.0
2.5
5.0
7.5
10.0
T/Tc
μ 0H
c 2 (T)
Hc2⊥
Figure 18.2.5 Upper critical fields of Nb/Ge composites with layer thickness DNb
for Nb layer and DGe for Ge layer.
§18.3 Pairing States
18.3.1 Generalized Cooper Pairs
18.3.2 Conventional Pairing of Spin-Singlet s-Wave
18.3.3 Exotic Pairing for Singlet d-Wave
18.3.4 Pseudogaps and Associated Symmetry
18.3.5 Exotic Pairing for Spin-Triplet p-Wave
§18.4 Josephson Effects
18.4.1 Josephson Equations
18.4.2 Josephson Effects in Superconductors
18.4.3 Phase-Sensitive Tests of Pairing Symmetry
18.4.4 Josephson Effect in Superfluids
Introduction to
Condensed Matter Physics
Jin Guojun
Nanjing University
May 20,2004
Contents
Chapter 18,Superconductors and Superfluids 3
§18.1 Macroscopic Quantum Phenomena,,,,,,,,,,,,,4
18.1.1 Concept of Bose-Einstein Condensation,,,,,,,,4
18.1.2 Bose-Einstein Condensation of Dilute Gases,,,,,,4
18.1.3 Superfluidity of Liquid Heliums,,,,,,,,,,,,,4
18.1.4 Superconductivity of Various Substances,,,,,,,,4
§18.2 Ginzburg-Landau Theory,,,,,,,,,,,,,,,,,,5
18.2.1 Ginzburg-Landau Equations andBroken Gauge Symmetry 6
18.2.2 Penetration Depth and Coherence Length,,,,,,,13
18.2.3 Magnetic Properties of Vortex States,,,,,,,,,,13
18.2.4 Anisotropic Behavior of Superconductor,,,,,,,,13
§18.3 Pairing States,,,,,,,,,,,,,,,,,,,,,,,,,14
18.3.1 Generalized Cooper Pairs,,,,,,,,,,,,,,,,15
18.3.2 Conventional Pairing of Spin-Singlet s-Wave,,,,,,28
18.3.3 Exotic Pairing for Singlet d-Wave,,,,,,,,,,,28
18.3.4 Pseudogaps and Associated Symmetry,,,,,,,,,28
18.3.5 Exotic Pairing for Spin-Triplet p-Wave,,,,,,,,,28
§18.4 Josephson Effects,,,,,,,,,,,,,,,,,,,,,,,29
18.4.1 Josephson Equations,,,,,,,,,,,,,,,,,,30
18.4.2 Josephson Effects in Superconductors,,,,,,,,,36
18.4.3 Phase-Sensitive Tests of Pairing Symmetry,,,,,,45
18.4.4 Josephson Effect in Superfluids,,,,,,,,,,,,,46
Chapter 18
Superconductors and Superfluids
§18.1 Macroscopic Quantum Phenomena
18.1.1 Concept of Bose-Einstein Condensation
18.1.2 Bose-Einstein Condensation of Dilute Gases
18.1.3 Superfluidity of Liquid Heliums
18.1.4 Superconductivity of Various Substances
§18.2 Ginzburg-Landau Theory
A phenomenological theory near Tc,1950
GL theory was developed before the microscopic BCS theory (1957)
Gor’kov,1959,derived GL equations from microscopic theory
Applied to both the homogeneous and inhomogeneous superconductors
Especially to intermediate state,mixing state
BCS can be used to obtain ground state,excited states,energy gap
zero resistivity,Meissner effect
18.2.1 Ginzburg-Landau Equations and
Broken Gauge Symmetry
N-S phase transition is involved in gauge symmetry breaking
Condensed phase corresponds to a macroscopically occupied quantum state
Order parameter ψ described by a macroscopic wavefunction
For a spin-singlet s-wave superconductor,the pair wavefunction
ψ(r) =〈ψ↑(r)ψ↓(r)〉
|ψ|2 = ψ?ψ→ns,the local pair density
Density functional of free energy
fs(r) = fn +α|ψ(r)|2 + β2|ψ(r)|4 + 12m?
vextendsinglevextendsingle
vextendsinglevextendsingle
parenleftbigg
iplanckover2pi1?+ e
c A
parenrightbigg
ψ
vextendsinglevextendsingle
vextendsinglevextendsingle
2
+ H
2(r)
8pi
(18.2.1)
e? = 2e,and m? = 2m
ψ(r)→exp(iθ)ψ(r),global gauge transformation
When θ takes different constants,f in (18.2.1) is invariant
Gauge symmetry groupU(1),in Fig,18.2.1
eiθψ
Re(ψ)
ψ'=eiθψ=0 θ
Im(ψ)
θ0
Figure 18.2.1 Global gauge transformation of the GL order parameter.
Taking the minimum of free energy with ψ
δF = δ
integraldisplay
f(r)dr = 0 (18.2.2)
and gauge condition?·A = 0→GL equation
1
2m?
parenleftbigg
iplanckover2pi1?+ e
cA
parenrightbigg2
ψ+αψ +β|ψ|2ψ = 0 (18.2.3)
GL equation is nonlinear for the term β|ψ(r)|2ψ(r)
The boundary condition is the normal component being zero
n·
parenleftbigg
iplanckover2pi1?+ e
cA
parenrightbigg
ψ = 0 (18.2.4)
Taking minimum of free energy with A,the supercurrent equation
js =?ie
planckover2pi1
2m?(ψ
ψ?ψ?ψ?)? e
2
m?cψ
ψA (18.2.5)
just the formula for current in quantum mechanics
In superconducting phase,ψ =|ψ|exp(iθ)
js = e
m?|ψ|
2
parenleftbigg
planckover2pi1?θ?e
cA
parenrightbigg
= e?|ψ|2vs (18.2.6)
consistent with the London equation in§18.1.4
Derive GL equations (18.2.3) and (18.2.6) from (18.2.1)
GL equations give a simple illustration for superconductors
For simplicity,assume size of the superconductor infinite and A = 0
Minimum of free energy,or from (18.2.2) and (18.2.3)
αψ0 +β|ψ0|2ψ0 = 0 (18.2.7)
|ψ0|=

0,T >Tc
[α(T?Tc)/β]1/2,T <Tc
(18.2.8)
In the normal state T >Tc,ψ0 = 0,ψ0 exp(iθ) = 0
Global gauge symmetry is intact
In the superconducting state T <Tc,ψ0→ψ0 exp(iθ)negationslash= 0
Global gauge symmetry is spontaneously broken
Ginzburg-Pitaevskii equation for 4He
parenleftbigg
planckover2pi12m
4
2 +α+β|ψ(r)|2
parenrightbigg
ψ(r) = 0 (18.2.9)
An extension to the case of a normal fluid
f(r) = 12m4|(?iplanckover2pi1vn)ψ(r)|2 +f0(|ψ(r)|2) (18.2.10)
Gross-Pitaevskii equation for BE condensate in the trap
parenleftbigg
planckover2pi1
2
2m?
2 +V(r) +g|ψ(r)|2
parenrightbigg
ψ(r) = μψ(r) (18.2.11)
g and μ denote the interaction strength and chemical potential
18.2.2 Penetration Depth and Coherence Length
18.2.3 Magnetic Properties of Vortex States
18.2.4 Anisotropic Behavior of Superconductor
§18.3 Pairing States
Key points of the microscopic theory for superconductivity
and 3H superfluidity
(1) Symmetry of the paired electrons or atoms
(2) Mechanism of corresponding pair formation
Spin-singlet s-wave pairing
Spin-singlet d-wave pairing of cuprate superconductors
Spin-triplet p-wave pairing pairing of 3He superfluid
and some exotic superconductors
18.3.1 Generalized Cooper Pairs
Cooper pair,one of the fundamental ideas from BCS theory
Normal state is unstable with respect to the Fermi sea
in the presence of an arbitrarily small attractive interaction
→electron pairing→Cooper pairs
Consider a simple model of two electrons r1 and r2
added to a Fermi sea of electrons at T = 0,in Fig,18.3.1
Interact via the potential V(r1?r2)
k
kF
-k
Figure 18.3.1 Two electrons with oppositely directed momenta just outside the
Fermi sphere.
Schr¨odinger equation
planckover2pi1
2
2m(?
2
1 +?
2
2)ψ(r1,r2) +V(r1,r2)ψ(r1,r2) = (E + 2EF)ψ(r1,r2)
(18.3.1)
E measured from Fermi level
Coordination transformation,center of mass coordination R = (r1 + r2)/2
and relative coordination r = r1?r2
Ground state requires zero momentum→k and?k
A simplified Schr¨odinger equation
planckover2pi1
2
m?
2ψ(r) +V(r)ψ(r) = (E + 2EF)ψ(r) (18.3.2)
Momentum representation of the wavefunction
φ(k) =
integraldisplay
dre?ik·rψ(r) (18.3.3)
Wavefunction has a definite angular momentum quantum number l
and can be expanded by spherical harmonics Ylm
in terms of magnetic quantum number m
Define κ = k/kF
φl(k) =
lsummationdisplay
m=?l
alm(k)Ylm(κ) (18.3.4)
Anisotropy of wavefunction←anisotropy of interaction potential
which is also l-dependent and a function of the direction of k
the potential can be written in the reciprocal space
Vl(k) =
integraldisplay
dre?ik·rVl(r) (18.3.5)
expanded by spherical harmonics
Vl(k) =
lsummationdisplay
m=?l
Vlm(k)Ylm(κ) (18.3.6)
The equation for φl(k) from (18.3.1)
planckover2pi12k2
m φl(k) +
summationdisplay
kprime
φl(kprime)Vl(k?kprime) = (E + 2EF)φl(k) (18.3.7)
When k<kF,all states below EF are occupied
φ(k) = 0 (18.3.8)
There is a continuous spectrum of solution of two electrons±planckover2pi1k for E > 0
Cooper pointed out that the bound states exist for E < 0 if V < 0
A drastic simplification for the interaction
Vl(k?kprime) =
Vl,EF < planckover2pi12k2/2m,planckover2pi12kprime2/2m<EF +εl
0,other cases
(18.3.9)
Then (18.3.7) becomes
parenleftbiggplanckover2pi12k2
m?E?2EF
parenrightbigg
φl(k) = Vl
summationdisplay
kprime
φl(kprime) (18.3.10)
Define
ξk = planckover2pi1
2k2
2m?EF (18.3.11)
(18.3.10) can be written
1 = Vl
summationdisplay
k
1
2ξk?E (18.3.12)
Summation over k is replaced by an integration
1 = Vl
integraldisplay εl
0
N(ξ) dξ2ξ?E (18.3.13)
For εllessmuchE,N(ξ)→N(0)
1 = 12N(0)VllnE?2εlE (18.3.14)
From it,bound energy E,and energy gap? =?E/2
For N(0)V lessmuch1,the weak-coupling approximation gives
El =?2εlexp
parenleftbigg
2N(0)V
l
parenrightbigg
,?l = εlexp
parenleftbigg
2N(0)V
l
parenrightbigg
(18.3.15)
An allowed energy state exists with E < 0 for a pair of bound electrons
Generalized Cooper pair with finite bound energy 2?l
Several remarks:
1) Weak attractive interaction Vl→pairing
2) Total wavefunction of the Cooper pair must be antisymmetrical
with exchanging r1 and r2
Orbital wavefunction φl(k) is even or odd
according to l being even or odd,φl(?k) = (?1)lφl(k)
On the other hand,spin function for a paired fermions is
odd for a spin-singlet (α1β2?α2β1) with S = 0
even for the spin-triplets (α1α2,α1β2 +α2β1,β1β2) with S = 1
α and β represent↑and↓states respectively
Important conclusion:
For even l,l = 0 (s-wave),l = 2 (d-wave),l = 4 (g-wave),→spin-singlets
while odd l,l = 1 (p-wave),l = 3 (f-wave),→spin-triplets
3) Particular form of E in (18.3.15) for Vl→
perturbation technique fails in microscopic theories of superconductivity
4) Uncertainty principle→ξ0δkgreaterorequalslant 1
2?～δE～planckover2pi1
2
mkFδk～planckover2pi1vFδk (18.3.16)
Coherence length of Cooper pairs
ξ0～planckover2pi1vF2? (18.3.17)
Critical temperature
kBTc～2?
Fermi velocity can be estimated from the electronic density of metals
In conventional superconductors,Tc is about several Kelvins
Electron mass is about the free electron mass,ξ0～10?4-10?5 cm
For 3He,its mass is larger,Tc is lower,→ξ0～10?6 cm
In heavy fermion superconductors and cuprate superconductors
ξ0～10?6-10?7 cm
In conventional superconductors,ξ0greatermuch10?8 cm,
there are many electron-pairs distributed crisscrossing and overlapping
with each other in the same spatial range
This is the original picture of Cooper pairs BCS theory
Schafroth’s theory for pairing of two local electrons,bipolarons
may useful for cuprates and heavy fermions
5) The Cooper problem is decisive to microscopic theory of superconductivity
For realistic calculation of physical properties,it should go outside
3He atoms are Fermions,BCS type condensation
Anderson and Morel,1961
p-wave pairing,l=1,spin triplet
indirect paramagnetic exchange interaction from van der Waals force
18.3.2 Conventional Pairing of Spin-Singlet s-Wave
18.3.3 Exotic Pairing for Singlet d-Wave
18.3.4 Pseudogaps and Associated Symmetry
18.3.5 Exotic Pairing for Spin-Triplet p-Wave
§18.4 Josephson Effects
Typical example of broken gauge symmetry
Importance of phase in superconductors
Heisenburg uncertainty relation for phase and number of particles
θ?Nsimilarequal1
P,W,Anderson,1961,Cambridge,Lectures
Josephson,Pippard’s graduate
Phases of two weakly coupled superconductors,Jesephson junctions
18.4.1 Josephson Equations
Two superconductors separated by an insulating layer shown in Fig,18.4.1
Feynman,1962,Quantum Mechanics
ψ1 and ψ2 represent the wavefunctions of electron pairs
Wave equations
iplanckover2pi1?ψ1?t = E1ψ1 +Kψ2,iplanckover2pi1?ψ2?t = E2ψ2 +Kψ1 (18.4.1)
S1 S2
ψ1 ψ2
θ1 θ2
ψ1 ψ2
I
Figure 18.4.1 Two superconductors separated by a thin insulator.
Two superconductors are connected to a battery
a potential difference V across the junction
E1?E2 = 2 eV
Define the zero of energy at (E1 +E2)/2
iplanckover2pi1?ψ1?t = eVψ1 +Kψ2,iplanckover2pi1?ψ2?t =?eVψ2 +Kψ1 (18.4.2)
Actually,two coupling superconductors become one system
described by a single condensate wavefunction
|ψ〉= ψ1|1〉+ψ2|2〉
which obeys the wave equation
iplanckover2pi1?|ψ〉?t =H|ψ〉
The hamiltonian includes three terms
H=H1 +H2 +HT,
among them
H1 = E1|1〉〈1|,H2 = E2|2〉〈2|,HT = K(|1〉〈2|+|2〉〈1|)
Densities of Cooper pairs in the two superconductors
|ψ1|2 = ρ1,|ψ2|2 = ρ2 (18.4.3)
Two macroscopic wavefunctions can be written as
ψ1 =√ρ1eiθ1,ψ2 =√ρ2eiθ2 (18.4.4)
Substituting them into (18.4.2),four equations
˙ρ1 = 2Kplanckover2pi1 √ρ2ρ1 sin(θ2?θ1),˙ρ2 =?2Kplanckover2pi1 √ρ2ρ1 sin(θ2?θ1)
˙θ1 = K
planckover2pi1
radicalbiggρ
2
ρ1 cos(θ2?θ1)?
eV
planckover2pi1,
˙θ2 = K
planckover2pi1
radicalbiggρ
1
ρ2 cos(θ2?θ1) +
eV
planckover2pi1 (18.4.5)
From 2e ˙ρ1(or?2e ˙ρ2)
J = 4eKplanckover2pi1 √ρ2ρ1 sinθ = J0 sinθ (18.4.6)
Phase difference θ = θ2?θ1,maximum current density J0 = 4eK√ρ2ρ1/planckover2pi1
From other pair equations in (18.4.5)
˙θ = ˙θ2? ˙θ1 = 2eV
planckover2pi1 (18.4.7)
or
θ(t) = θ0 + 2eplanckover2pi1
integraldisplay t
0
V(t)dt (18.4.8)
(18.4.6) and (18.4.8) are called the Josephson equations
18.4.2 Josephson Effects in Superconductors
Theoretical prediction for (18.4.6) and (18.4.8)
Experimental confirmation by Anderson and Rowell,1963
Suitable weak links can be made in a large variety of ways
tunneling junction,point contacts,microbridge,etc.
A lot of quantum effects related to weak links
To show the importance of the phases of superconductors
Derive Josephson equations (18.4.6) and (18.4.8) from (18.4.1)
(1) dc Josephson effect,V = V0 = 0
J = J0 sinθ0 (18.4.9)
No voltage,the current between J0 and?J0,depending θ0
(2) ac Josephson effect,V = V0negationslash= 0,2eV0 <? to avoid pair breaking
J = J0 sin
parenleftbigg
θ0 + 2eplanckover2pi1V0t
parenrightbigg
(18.4.10)
dc voltage leads to an ac current
alternating frequency ω = 2piν = 2eV0/planckover2pi1
detected experimentally through electromagnetic radiation
a constant defined by ν/V0 = 2e/h = 4.8×1014 HzV?1
(3) Shappiro steps,an ac voltage at a microwave frequency + dc voltage
V = V0 +vcosωt,vlessmuchV0
J = J0 sin
parenleftbigg
θ0 + 2eplanckover2pi1V0t+ 2evplanckover2pi1ω sinωt
parenrightbigg
(18.4.11)
Using sin(x+?x)≈sinx+?xcosx
J = J0
bracketleftbigg
sin
parenleftbigg
θ0 + 2eplanckover2pi1V0t
parenrightbigg
+ 2evplanckover2pi1ω sinωtcos
parenleftbigg
θ0 + 2eplanckover2pi1V0t
parenrightbiggbracketrightbigg
(18.4.12)
After time average,first term=0,second term= dc current if V0 = planckover2pi1ω/2e
More exactly,using Fourier-Bessel expansion to higher ranks
dc components in the frequency modulated current,if
V0 = planckover2pi12enω (18.4.13)
n is any integer,and its amplitude
J = J0Jn
parenleftbigg2ev
planckover2pi1ω
parenrightbigg
sinθ0 (18.4.14)
Jn(2ev/planckover2pi1ω) is n-rank Bessel function
Fig,18.4.2,experimental result of Shapiro steps,1963
The physical implication,Cooper pair tunneling is facilitated
by applied radiation
1
2
3
Nb-Nb
T=4.2 K
-600
-450
-300
-150
0
150
300
450
600
750
J (μA)
V (
μV)
Figure 18.4.2 Current vs,voltage of Josephson junction,showing Shapiro steps.
Effect of magnetic field on the Josephson junctions
Consider a magnetic field through a loop
Vector potential A will influence phase of macro-wavefunction
θ (18.4.6) and (18.4.8) is changed
γ = θ?2piφ
0
integraldisplay
A·dl (18.4.15)
φ0 is the magnetic flux quantum
A special case for a pair of Josephson junctions in Fig,18.4.3
Take the integration contour in the region vs = 0
From (18.1.33),A = (φ0/2pi)?θ
Φ =
contintegraldisplay
A·dl = φ02pi
contintegraldisplay
θ·dl (18.4.16)
contintegraltext?θ·dl = 2piν,the maximum supercurrent
Imax = 2Ic cos(2piΦ/φ0) (18.4.17)
Fig,18.4.3,reminiscent of two slit interference pattern in optics
Physical basis of dc-SQUID (superconducting quantum interference device)
magnetometer,the most sensitive device for the measurement of magnetic
flux
Fig,18.4.4,rf-SQUID
1
2
A B
-3 -2 -1 0 1 2 3
/φ0
Imax
(a) (b)
Figure 18.4.3 (a) Schematic diagram for two Josephson junctions in magnetic field;
(b) corresponding quantum interference pattern.
IT
L
M
LT VT
I1
I1?VT
=nφ0
=(n+1/2)φ0
I1
VT
0
(a) (b)
Figure 18.4.4 (a) Schematic diagram for the circuit of rf-SQUID; (b) corresponding
VT vs,Il relations for integral and half-integral numbers of flux quanta.
An array of 1D,2D or 3D Josephson junctions used to study
Quantum phase transitions
Granular superconductivity
Quantum computer
18.4.3 Phase-Sensitive Tests of Pairing Symmetry
18.4.4 Josephson Effect in Superfluids
It has been believed since Richard and Anderson,1965
Analog,Josephson equations are still valid in superfluids
Although it failed for a long time
Electric current replaced by mass transport of neutral atoms
ac Josephson effects in superfluid helium are hydrodynamic effects
A superfluid weak link in Fig,18.4.7
weak link is a very small aperture
E
1 2
Figure 18.4.5 Two reservoir of liquid helium connected by a small aperture.
A level difference?z between two reservoirs
gives chemical potential difference
μ2?μ1 = mg?z (18.4.18)
m is the mass of one 4He atom or two 3He atoms
˙θ = ˙θ2? ˙θ1 =?1
planckover2pi1(mg?z) (18.4.19)
Define a characteristic frequency
ω = mg?z/planckover2pi1 (18.4.20)
Difficulties for experimental observation
Coherence lengths,or healing lengths,are very small:
for 4He,ξc～0.1 nm; for 3He,ξc～50 nm
First,successful realizations of Josephson effect in superfluid 3He
E,Varoquax and O,Avenel,1988,Phys,Rev,Lett,60,416;
S,V,Pereversev et al.,1997,Nature 388,449;
R,W,Simmonds et al.,2001,Nature 412,55
Crucial technique is to fabricate an array of apertures
with diameters less than the coherence length
This mass oscillation frequency can be heard by using loudspeaker
The interference effect of dc SQUID-like device of superfluid 3He
has been reported,the rotation of earth acting like magnetic field
Return to 4He problem,Noting coherence length diverges near Tc
ξ = ξ0
parenleftbigg
1? TT
λ
parenrightbigg?γ
,γ = 0.672 (18.4.21)
An ingenious experiment using Tλ?T = 3.72 mK,ξc≈10 nm
accessible to modern microfabrication,ac Josephson signal is found
K,Sikhatme et al.,2001,Nature 411,280
BE condensates of dilute gases→interference patterns observed
M,R,Andrews et al.,1997,Science 275,637

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课件名称：	南京大学：凝聚态物理学导论讲义英文版PartII（金国均）
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