1 Characteristics of Solids Bonding Electrons in Solids Band Theory Defects Semiconductor Classification of Solids There are several forms solid state materials can adapt §Single Crystal -Preferred for characterization of structure and properties. §Polycrystalline Powder (Highly crystalline) -Used for characterization when single crystal can not be easily obtained, preferred for industrial production and certain applications. §Polycrystalline Powder (Large Surface Area) -Desirable for further reactivity and certain applications such as catalysis and electrode materials §Amorphous (Glass) -No long range translationalorder. §Thin Film -Widespread use in microelectronics, telecommunications, optical applications, coatings, etc. Molecular solid Ionic Covalent Metallic Types of Crystals (a)ion crystal(NaCl) (b)metallic crystal (c)covalent crystal(InSb) (d)molecule crystal (solid state Ar) (e)hydrogen bond crystal(H3BO3) (f)mixed bond crystals (graphite) According to the ionic bonding in solids Bonding Electrons transfer between atoms Between many atoms Electrons sharing two atoms Bonding Primary bonds ? strong attractions between atoms Ionic ? Metal ion(+) & Nonmetallic ion (?) Covalent ? local sharing of electrons between atoms Metallic ? global sharing of electrons by all atoms Secondary bonds ? attraction forces between molecules 2 Primary Bonding e + Na+ e e e e ee e e e e + e e e e ee e e e F- Ionic + + e e Covalent e + + + +++ + + + e e ee e e e e Metallic Solid State ? Strength The strength of the solid depends on the molecular forces that hold the solid together. Ionic interactions are strong because the opposite charges resist breaking of the intermolecular bonds. For example, the melting temperature of salts is very high, typically 400 to 800°C. Some molecular interactions are strong because of the 3- D arrangement of the atoms. For example, diamonds are exceptionally hard because the solid state forms, a 3-D network such that each carbon is held close to 4 other atoms. The bonds are molecular, not ionic. Types of Non-Bonded (intermolecular, van der Waals) Interactions Dipole-dipole Ion-dipole Induced dipole Dispersion or London Hydrogen bonding H Cl + ? H + Cl ? Dipole-dipole interactions: when two polar molecules with net dipoles come closer, one end of the dipole in one molecule will be attracted to the opposite end in the second molecule. (a). These forces are strong, but are weaker than ion-ion interactions in ionic compounds. (b). These forces decrease with distance as 3r 1 Ion-dipole interactions: When polar molecules encounter ions, the positive end of the dipole is attracted to negative ions and vice versa. The ion-dipole interactions require two species to be present, one to provide ions and another to provide dipoles. O H Hδ+ δ+ δ?Na + Cl- Induced dipole interactions: In non-polar molecules, electrons are distributed symmetrically. This symmetry can be distorted by an ion/dipole, by inducing dipole in non-polar molecule. These forces are weak and are of short range. e e e ee e e+ e e e ee e+ e Dispersion or London forces: Noble gases are atomic gases and do not have dipole moments or net charges. The fact that they can be liquefied suggests that forces of attraction exist between atoms of a noble gas. These forces are called Dispersion or London forces. The distribution of electrons in an atom/molecule fluctuate over time. These fluctuations set up temporary dipoles which induce dipoles in others. The attraction between temporary dipoles is responsible for dispersion forces. These forces exist between all atoms and molecules. Strength of dispersion forces increases with number of electrons in atoms/molecules. “Dispersion force increases down a group”. Thus, for rare gases, dispersion forces increase as Xe > Kr > Ar > Ne > He. 3 Hydrogen bonding: A hydrogen atom covalently bonded to N, O, or F is attracted to the lone pair of a different atom nearby forming a hydrogen bond. Hydrogen bonding is stronger than any other non- bonded interaction, yet weaker than covalent bonds/ionic bonds O HH .. H O H Hydrogen bond Types of Non-Bonded Interactions Interaction Energy Ion-ion interaction ~250 kJ/mol H-bonding ~20 kJ/mol Ion-dipole Dipole-dipole ~2 kJ/mol Dispersion/London <2 kJ/mol Induced dipole Examples: Compound/element Type Dominant Interaction NaH ionic Ion-ion ClBr covalent Dipole-dipole Rn noble gas Dispersion NH3 covalent Hydrogen bonding NH4Cl ionic Ion-ion HBr(g) covalent Dipole-dipole HF(g) covalent Hydrogen bonding Variation of Ionic Radii With Coordination Number The radius of one ion has to be fixed to a reasonable value (r(O2-) = 1.40?) → Linus Pauling. That value is then used to compile a set of self consistent values for all other ions. 1. Ionic radii increase on going down a group. (Lanthanide contraction restricts the increase of heavy ions !!) 2. Radii of equal charge ions decrease across a period 3. Ionic radii increase with increasing coordination number (the higher its CN the bigger the ions seems to be !!) 4. The ionic radius of a given atom decreases with increasing charge (r(Fe2+) > r(Fe3+)) 5. Cations are usually the smaller ions in a cation/anion combination (exceptions: r(Cs+) > r(F-) ...!!!) 6. Frequently used for rationalization of structures: “radius ratio” r(cation)/r(anion) (< 1) Some General Trends for Ionic Radii Ion Bond and Ion Crystal Some Properties of Ion Crystal Lattice energy of ion crystals Ionic radii in crystals Pauling’s Rules Ion bonds with part covalent bond Some Properties of Ionic Crystals Relative stable and hard crystals Poor electrical conductors (lack of free electrons) High melting and vaporization temperatures Transparent to visible light but absorb strongly infrared light Soluble in water and polar liquids! 4 Two definitions: The lattice enthalpy change is the standard molar enthalpy change for the following process: M+(gas) + X-(gas) → MX(solid) If entropy considerations are neglected the most stable crystal structure of a given compound is the one with the highest lattice enthalpy. Lattice Energy (U) of Ionic Compounds: disassemble one mole of a crystalline ionic compound at 0K into free components o LH? 0HoL <? Lattice Enthalpy o LHU ??= Equilibrium Distance & Cohesive Energy )n11(r eZZ EEE 0 2 repulsiveattractive ??= += ?+ Ep r nr B r eZZ 2?+? total At equilibrium: (Erepulsive) (Eattractive) 0 2 attractive r eZZE ?+?= Find B and n at equilibrium: nrepulsive r BE = n 2 repulsiveattractiveTotal r B r eZZEEE +?=+= ?+ 0drdE Total = 0rr = 1n 0 2 1n 0 2 0 2 rn eZZB0rnBr eZZ ? ?+ + ?+ =?=+?? Total energy at r0: )n11(r eZZE 0 2 rr 0 ??= ?+ = What is n? Compressibility ? n ≈ 9 Bohr-Madelung equation Lattice Energy of Ionic Compounds: Bohr-Madelung equation： N = 6.02x1023 mol-1 The Madelung constant is independent of the ionic charges and the lattice dimensions, but is only valid for one specific structure type If know the crystal structure, you can choose a suitable Madelung constant, and the distance between the ions ro , you can estimate the lattice energy of ion compound. )n11(r eZANZU 0 2 ?= ?+ Structure Type Madelung Constant CsCl 1.763 NaCl 1.748 ZnS (Wurtzite) 1.641 ZnS (Zinc Blende) 1.638 thermal stabilities of ionic solids stabilities of oxidation states of cations solubility of salts in water calculations of electron affinity data stabilities of “non existent”compounds Applications of Lattice Enthalpy Calculations The Kapustinskii Equation Kapustinskii noticed that A /ν, is almost constant for all structures ν is the number of ions in the formula unit ro = r++r?, unit: pm Variation in A /ν with structure is partially canceled by change in ionic radii with coordination number )r 5.341(r ZZ125200U 00 ?υ= ?+ 5 BONDS Coord. No. Length (?) C-O 3 1.32 Si-O 4 1.66 Si-O 6 1.80 Ge-O 4 1.79 Ge-O 6 1.94 SnIV-O 6 2.09 PbIV-O 6 2.18 PbII-O 6 2.59 Notes: Ion radii for given element increase with coordination number (CN) Ion radii for given element decrease with increasing oxidation state/positive charge Radii increase going down a group Anions often bigger thancations not “in touch” in touch Limiting and Optimal Radius Ratios for Specific Coordinations Radius Ratio Rules Rationalization for octahedral coordination: R= radius of large ion, r=radius of small ion 414.0Rr rR)12( rRR2 2 145cos rR R =? =?? +=? ==+ o If r/R < 0.414, the cation is too small and can “rattle”inside the octahedral site If r/R > 0.414, the anions are pushed apart If r/R ≤ or ≥ 0.414, coordination changes: Coordination Minimum r/R Linear, 2 ? Trigonal, 3 0.155 Tetrahedral, 4 0.225 Octahedral, 6 0.414 Cubic, 8 0.732 Close packed, 12 1.000 A simple prediction tool, but beware ? it doesn’t always work! Limiting Radius Ratios - anions in the coordination polyhedron of cation are in contact with the cation and with each other Radius Ratio Coordination no. Binary (AB) Structure-type r+/r- = 1 12 none known 1 > r+/r- > 0.732 8 CsCl 0.732 > r+/r- > 0.414 6 NaCl 0.414 > r+/r- > 0.225 4 ZnS 6 The critical ratio determined by geometrical analysis 2 <0.155 3 0.155?0225 4 0.225 ? 0.414 6 0.414 ? 0.732 8 0.732 ? 1.0 C.N. rC/rA Geometry Coordination Number vs Geometrical Shapes C.N. = 3 C.N. = 4 C.N. = 6 Cubic hole Cuboctahedral hole Cuboctahedral and Anti-cuboctahedral Structure Cuboctahedron ?? Cubic close packing Anti-cuboctahedron ?? Hexagonal close packing Common Coordination Polyhedra Ceramic Crystal Structures The ratio of ionic radii (rcation/ranion ) dictates the coordination number of anions around each cation. As the ratio gets larger (i.e. as rcation/ranion→1), the coordination number gets larger and larger. Crystal Structure of AB vs Ion Ratio (r+/r-) 7 Structure Maps ?? Plots of rA versus rB with structure-type indicated Separation of structure types is achieved on structure diagrams ? but, the boundaries are complex Conclusion ? Size does matter, but not necessarily in any simple way! Structure Maps ?? Plots of rA versus rB with structure-type indicated Separation of structure types is achieved on structure diagrams ? but, the boundaries are complex Conclusion ? Size does matter, but not necessarily in any simple way! Pauling’s Rules Cation environment in a polyhedron (cation- anion distance and Coordination Number) Relationship between bond valence and oxidation number Corner, edge and face sharing polyhedra Large valence and small Coordination Number cations tend not to share polyhedra elements Rule of parsimony Pauling’s Rules for Ionic Crystals Deal with the energy state of the crystal structure 1st Rule The cation-anion distance = Σ radii Can use r+/r? to determine the coordination number of the cation Pauling’s Rules for Ionic Crystals 2nd Rule First note that the strength of an electrostatic bond = valence / CN Cl Cl Cl Cl Na Na+ in NaCl is in VI coordination For Na+ the strength = +1 divided by 6 = + 1/6 Pauling’s Rules for Ionic Crystals 2nd Rule ? the electrostatic valence principle + 1/6 + 1/6 + 1/6 + 1/6 Na NaNa Na Na Cl- An ionic structure will be stable to the extent that the sum of the strengths of electrostatic bonds that reach an anion from adjacent cations = the charge of that anion 6( + 1/6 ) = +1 (sum from Na ’s) charge of Cl = ?1 These charges are equal in magnitude so the structure is stable 8 In [SiO4], strengths of Si-O=1, ∑Strength=2,stable If 1 Al replace 1 Si, ∑Strength = 1+3/4=1.75, unstable If 2 Al replace 2 Si, ∑Strength =3/4+3/4=1.5 very unstable Pauling’s Rules for Ionic Crystals ?2nd Rule ? the electrostatic valence principle 3rd Rule: The sharing of edges, and particularly of faces, of adjacent polyhedra tend to decrease the stability of an ionic structure Pauling’s Rules for Ionic Crystals Polyhedral Linking The stability of structures with different types of polyhedral linking is vertex-sharing > edge-sharing > face-sharing effect is largest for cations with high charge and low coordination number especially large when r+/r- approaches the lower limit of the polyhedral stability 4th Rule: In a crystal with different cations, those of high valence and small CN tend not to share polyhedral elements An extension of Rule 3 Si4+ in IV coordination is very unlikely to share edges or faces Pauling’s Rules for Ionic Crystals 5th Rule ? Rule of Parsimony The number of different kinds of constituents in a crystal tends to be small Pauling’s Rules for Ionic Crystals Polarization of Ion Polarization of an ion is the distortion of the electron cloud of the anion due to the influence of the nearby cation. + – + – Perfect model of ionic compound Electron cloud of anion is attracted towards the cation and result in higher electron density between the ion, stronger bond is resulted. Ionic compound with polarization of ion ?? Not a purely ionic compound 9 Polarizing Power of Cation The ability to distort the electron distribution of adjacent ions or atom. The polarizing power of cation is favored by higher charge and smaller size ? to have a higher charge density Al3+ > Mg2+ > Na+ Li+ > Na+ Polarizability of Anion The ease of the electron cloud being distorted by the influence of adjacent ion or atom. The more polarizable anion would have higher charge and larger size S2- > O2- S2- > Cl- Polarization Effect on AB2 Structures (A = Transition Metal) Same r+/r?, larger size of anion induced more polarizable anion → part of electrons are active in the whole crystal → property of semiconductor and metal, such as FeS2 smaller cation has more polarizing power → ion crystals to molecule crystals along vertical axis. Polarization Increasing Polarization in bonding low-dimensionality ?? layers/chains Covalent Crystals ? Held Together by Covalent Bonds Share electrons lead to strongest bonds Some Properties: - Very hard. - High melting points. - Insulators/semiconductors. Covalent Bonding Review some important features of covalent bonding: §Basic Concepts of Molecular Orbital Theory half-filled orbital hybrid orbital atomic orbital & molecular orbital bonding (symmetric) molecular orbital antibonding (antisymmetric) molecular orbital 10 What is the Largest Molecule in the world? Not polymer aggregates The largest molecules that have ever been found are Diamonds. The largest of all was the so-called Cullinan diamond, found on Jan. 25, 1905, in South Africa and sporting a weight of 621.6 g (3106 carats). The largest man-made synthetic molecule is an artificial diamond of 38.4 carats, the growth of which required 25 days. The hardest element with the highest thermal conductivity of >2000 W/mK (the best metallic thermal conductor is Ag (429 W/mK). Molecular Crystals ?held together by van der Waals bonds weak … but everywhere. Polar molecules ? electric dipole moment Polar molecules attract each other: Van der Waals Attraction between Non-polar molecules: On the average, non-polar molecules are symmetric distributions, but at any moment the distributions are asymmetric. The fluctuations in the charge distributions of nearby molecules lead to an attractive force, given also by: 6attractive r 1~U ? Metallic Bond and Metallic Crystal Free Electrons Gas Model in Metals The Nearly-Free-Electron Model in Metals Distribution of Electrons in Metals Band Theory Like copper and gold, most of the known elements are metals. Metals are solids which require extra discussion to explain their special properties: Ductility Shiny surface High electrical conductivity high heat conductivity 11 Superductibility of Nanoscaled Copper A metallic solid “free range”electrons Metals are solids which require extra discussion to explain their special properties: Ductility Shiny surface High electrical conductivity High heat conductivity Metallic Crystals – Held Together by Metallic Bonds A gas of negatively charged free electrons holds metal ions together. Some Properties: Weaker than covalent or ionic crystals. High melting points. Electrical and thermal conductors. Free Electron Gas Model Drude and Lorentz: The valence electrons in metals are loosely bound to their atoms and form a gas of particles that may wander through the crystal and conduct electricity. Many properties of metals can be understood by treating electrons in metals as free- electron gas. Metallic Sea of Electrons High electrical conductivity High thermal conductivity High reflectivity of visible light High malleability and ductility + + + + + + + + + + + + + + + + + + + + + + + + + + + + Valence electrons are not bonded to any particular atom and are free to move about in the solid. Hall Effect If an electric current flows through a conductor in a magnetic field, the magnetic field exerts a transverse force on the moving charge carriers which tends to push them to one side of the conductor. This is most evident in a thin flat conductor as illustrated. A buildup of charge at the sides of the conductors will balance this magnetic influence, producing a measurable voltage between the two sides of the conductor. The presence of this measurable transverse voltage is called the Hall effect after E. H. Hall who discovered it in 1879. 12 When electrons flow without magnetic field... t d conductor slice + _ I I When the magnetic field is turned on ... B-field I qBv As time goes by... I qE qBv = qE low potential high potential Finally... B-field I VH Hall Effect Lorentz force likes to deflect Ix However, E-field is set up which balances Lorentz force Balance occurs when Ey = Bzvx is defined as Hall coefficient Nq1R H = xzHyxzy x xxx IBREIBNq 1E Nq IvNqvI =?=?=?= xz y H IB ER = the Nearly Free Electron Model Sommerfeld modified free electron gas model: the Nearly- Free-Electron Model To better account for the electrical properties of different materials, we need to consider the interaction that arises between the electrons and the crystal structure. One model that attempts to do this in a simple way is the nearly-free electron model. To understand the key features of this we need to recall the diffraction of waves that is generated by a crystal structure. In the nearly-free electron model it is therefore considered that the main effect of the electron-crystal interaction is to diffract electrons whenever the Bragg condition is satisfied Electrons can exhibit a variety of wave effects including diffraction， it accords with De Broglie relation: λ =h/mυ 13 Basic Concepts of Band Theory Electrons in crystals are arranged in energy bands. These bands are separated by regions in which no electron states exist. These regions are called energy gaps or band gaps. The lowest empty band is the conduction band.The highest band with occupied electron levels is the valence band. The highest occupied band can be completely or partly filled depending on the ratio between the number of electron levels available in the band and the number of valence electrons given by each atom in the crystal. The occupancy of the valence band and the size of the gap between the valence and the conduction band will determine the conductive, semiconductive or insulator character of a crystal. If a particle is confined into a rectangular volume, the same kind of process can be applied to a three-dimensional "particle in a box", and the same kind of energy contribution is made from each dimension. The energies for a three-dimensional box are Particle in a Box The idealized situation of a particle in a box with infinitely high walls is an application of the Schrodinger equation which yields some insights into particle confinement. The wave function must be zero at the walls and the solution for the wave function yields just sine waves. 2 2 2 2 m8 hm 2 1E κ pi=υ= 0 Lx 2 2 1 mL8 hE = ??? ? ??? ?= 2 2 2 2 mL8 h2E ??? ? ??? ?= 2 2 2 3 mL8 h3E ??? ? ??? ?= 2 2 2 4 mL8 h4E 0 Lx 2 2 1 2 2 2 2 2 2 2 3 2 2 2 4 mL8 hE L 2L2 mL8 h2E L 22L mL8 h3E L 32L 3 2 mL8 h4E L 42L 2 1 =?pi=λpi=κ?=λ ??? ? ??? ?=?pi= λ pi=κ?=λ ??? ? ??? ?=?pi= λ pi=κ?=λ ??? ? ??? ?=?pi= λ pi=κ?=λ De Broglie Relation 22 2 m8 hE κ pi= 2m 2 1E m h 2 υ= υ=λ λ pi=κ Fermi Energy (Fermi Level) "Fermi level" is the term used to describe the top of the collection of electron energy levels at absolute zero temperature. Fermi energy (or Fermi level): highest occupied energy level in the ground state (T=0K) of the N electron system. At T=0K, the N electron system is in the ground state: The electrons occupied all the energy states up to the Fermi level. Enrico Fermi (1901-1954) Fermi Energy Distribution Function The distribution function f(E) is the probability that a particle is in energy state E. The Fermi function f(E) gives the probability that a given available electron energy state will be occupied at a given temperature. Some electrons excited above Fermi level at T > 0K, ? Fermi distribution. The basic nature of this function dictates that at ordinary temperatures, most of the levels up to the Fermi level EF are filled, and relatively few electrons have energies above the Fermi level. 1e 1)E(f kT/)EE( F += ? vThe possibility that a particle will have the energy E is: Origin and Temperature Dependence of Electronic Conductivity in Metals Electrons with energies close to the Fermi level can easily be promoted to nearby empty levels. they are mobile and can easily move through the solid in an electric potential gradient (free electrons) At elevated temperatures thermal vibrations of the atoms reduce the mobility of the free electrons. As a consequence the electrical conductivity decreases with increasing temperature 14 Fermi Energy Distribution Function of Semiconductor The illustration below shows the implications of the Fermi function for the electrical conductivity of a semiconductor. The band theory of solids gives the picture that there is a sizable gap between the Fermi level and the conduction band of the semiconductor. At higher temperatures, a larger fraction of the electrons can bridge this gap and participate in electrical conduction. Classification of Solids Depending on Their Structures Metal: partly filled valence band Semiconductor: completely filled valence band, small energy gap (Eg) between valence and conduction band Insulator: completely filled valence band, large energy gap (Eg) between valence and conduction band Semi-metal: partly filled valence band due to overlapping of conduction and valence band Two different pictures of the same problem Bands in Metals Example: Aluminum Consider Al(1s22s22p63s23p1) Core atomic orbitals : contracted, poor overlap, narrow bands, filled Valence atomic orbitals: diffuse, partially occupied, good overlap, wide bands Partially filled VB, electrons responsible for electrical and thermal conductivity, optical (reflectivity) and magnetic properties Metals: Free Electron Model Many physical properties of metals can be explained in terms of a free electron model. It assumes a lattice of positive ion cores surrounded by a cloud of electrons that can move freely through the solid. In this approximation, we neglect the electron-electron and electron-core interaction. The free electron model allows us to understand electrical and thermal conductivity. Parameters Affecting the Conductivity of Metals §Electrons are scattered by the thermal vibration of ionic cores, impurities and defects. This scattering reduces the conductivity (enhances resistivity). In a very perfect (no defects) and pure (no impurities) copper crystal conductivity is 105 times larger at T=4K than at room temperature. 0D, Point defects vacancies interstitials impurities 1D, Dislocations edge screw 2D, Stacking Faults and Grain Boundaries mosaic structure high angle grain boundary tilt grain boundary twist grain boundary 3D, Bulk or Volume defects precipitates second phase particles voids Defects Types Real crystals are never perfect, there are always defects Defects in Solids 15 Defects have a profound impact on the macroscopic properties of materials Bonding + Structure + Defects Properties “Crystals are like people, it is the defects in them which tend to make them interesting!” ? Colin Humphreys Defects in Solids § A defect is a break in the pattern § Defects make things work and make them beautiful –Pentium chips –Rubies Point Defects Intrinsic defects: interstitials and vacancies Interstitials: Self-interstitial (host atom in interstitial position) ? complexes of interstitial: di-interstitial and tri- interstitial Vacancies: lack of an atom ? complexes of vacancies: di-vacancy, tri-vacancy etc Extrinsic defects: chemical impurities Substitutional and interstitial Point Defects vacancy: the site of the missing atom Substitutional atom interstitial atom self-interstitial atom fi disturbances in a crystal ~ a few interatomic distances Vacancy- a lattice position that is vacant because the atom is missing. Interstitial - an atom that occupies a place outside the normal lattice position. It may be the same type of atom as the others (self interstitial) or an impurity interstitial atom. Point defects in ionic crystals are charged. Coulombic forces are large and any charge imbalance has wants to be balanced. Charge neutrality ? several point defects created: Frenkel defect: pair of cation (positive ion) vacancy and a cation interstitial or an anion (negative ion) vacancy and anion interstitial. (Anions are larger so it is not easy for an anion interstitial to form). Schottky defect: pair of anion and cation vacancies Frenkel and Schottky Defects Frenkel defect Schottky defect Frenkel or Schottky Defects: no change in cation to anion ratio → compound is stoichiometric Non-stoichiometry (composition deviates from the one predicted by chemical formula) may occur when one ion type can exist in two valence states, (e.g. Fe2+, Fe3+). In FeO, usual Fe valence state is 2+. If two Fe ions are in 3+ state, then a Fe vacancy is required to maintain charge neutrality → fewer Fe ions → non-stoichiometry Imperfections in Ceramics 16 Methods of Introducing Point Defects Nonintentional during growth (host lattice defects, impurities coming from contamination) during processing (ion implantation) as a result of radiation damage Intentional by changing crystal growth parameters by annealing by irradiation by implantation by diffusion the Processing Determines the Defects Composition Bonding Crystal Structure Thermomechanical Processing Microstructure defects introduction and manipulation Defect Equilibrium Concentration (Intrinsic Defects) The change in free energy ?H associated with the introduction of n vacancies or interstitial ?G= ?nEf ? T?S Ef: the formation energy of one defect ?S: the change in entropy n: the number of defects Equilibrium condition ?G/?n=0 Defect concentration: N: the total number of atoms kT Ef Nen ?= Defect Equilibrium Concentration Interstitial concentration: Vacancy concentration: Vacancies in ionic crystals Schottky defects: E+: formation energy of a cation vacancy E?: formation energy of an anion vacancy Frenkel defects: N: number of lattice sites; Ni: number of interstitial site kT E i i Nen ?= kT E V V Nen ?= kT2 EE V Nen ?+ + ?= kT E iv i NeNn ?= Point Defects in Metals Intrinsic defects?? Vacancies are predominant Ev ~ 1 eV for Cu, Ag, Au Concentration at T=1000°C n/N~10-5 Ei ~ 3 eV at T=1000°C n/N~10-16 Extrinsic defects ??Small atomic radius form the interstitial solid solutions with metals, such as H, B, C, N, O Defects in Semiconductors Concentration of intrinsic electrons and holes n(E) = N(E) F(E) N(E): density of states (the volume density of electron levels of energy E) F(E): Fermi Energy Distribution Function , the probability that the energy level E are occupied kT2 E vche kT EE vh kT EE ce q vf fc eNNnn eNn eNn ? ?? ?? == = = 17 Defects in Ionic Crystals Kr?ger-Vink Notation: a standard notation used to describe the point defects x yM Charge Position siteDefect type { Main body M : V ? Vacancy; M? central ion Superscript : the effective charge or the relative charge of the defect with respect to the original species . ? positive effective charge , ? negative effective charge x ? neutrality, zero charge x x Defect Notations Vacancies: the effective charge of a vacancy is the opposite sign of a missing ion charge NaCl: MgO: Interstitials: the effective charge of an interstitial ion is the same sign of the ion charge NaCl: MgO: ? Cl ' NaVV ?? O " MgVV ' iiClNa ? '' iiOMg ?? Kr?ger-vink Notation for MX Crystals Frenkel defect: O fi VM + Mi Schottky defect: O fi VM + Vx Where VM: void at the site of M Mi: interstitial atom, M Vx: void at the site of X Xi: interstitial atom, X Point defects thermodynamic equilibrium (concentration) ? (temperature) line defects interface defects fl Non-equilibrium Color Centers Electrons trapped in vacant sites give rise to colored materials color centers color arises due to transitions between electron in a box levels Trapped electrons can be produced by irradiation of the sample treatment with an electron donor like sodium or potassium vapor Color Centers Exposure to radiation can induce defects Useful for imaging Useful for dating KBr KCl NaCl F, H and V Centers F Center –electron trapped in anion vacancy H Center –interstitial Clatom bonds to lattice Cl- V Center –electron removed from lattice anion site, resulting Clatom pairs with neighboring Cl- 18 § Alkali-halides made from Groups I and VII § F-center is an electron in place of a halogen –Long studied –Not completely understood The F-center in Alkali-Halides e- F-center Applications Tunable solid-state lasers optical performance Ways of Creating Color Centers: Heating the crystal in the vapor of the metal ion Introduction of impurities (extrinsic defects) X-ray, γ-ray, neutron or electron beam irradiation Electrolysis Main Features of Color Centers: Color characteristic of host crystal Color shifts to red as the anion size increases Color does not shift if Na or K is added to NaCl ESR (Electron Spin Resonance) indicates F-center is an electron trapped in an anion vacancy: electron in an octahedral box problem Number of F-centers ～1 in 10,000 halide ions Electron in an octahedral box (halide ion vacancy) of size L Difference between energy levels of the box proportional to 1/L2 Absorption takes place between energy levels of the box, corresponds to the energy of UV-Vis-NIR photons The larger L is, the lower the absorption energy. i.e. Absorption red shifts with the lattice parameter Pressure causes box to shrink and absorption to blue shift Color of crystals results from the light they reflect or transmit. Reflection + Transmission + Absorption=1 Example: KBr absorbs red, it looks blue-green 2 22 mL8 hnE = Optical Absorption of F-centers Absorption spectra obtained from color centers in halide salts exhibit a clear trend in the variation of the wavelength with the size of the halide vacancy (as estimated by the lattice parameter, the length of an edge of the cubic unit cell). 2 22 mL8 hnE = Solid Solutions Solid solutions are made of a host (the solvent or matrix) which dissolves the minor component (solute). The ability to dissolve is called solubility. Solvent: in an alloy, the element or compound present in greater amount Solute: in an alloy, the element or compound present in lesser amount Solid Solution: homogeneous maintain crystal structure Contain randomly dispersed impurities (substitutional or interstitial) Second Phase: as solute atoms are added, new compounds/ structures are formed, or solute forms local precipitates Whether the addition of impurities results in formation of solid solution or second phase depends the nature of the impurities, their concentration and temperature, pressure… 19 Substitutional Solid Solutions Factors for high solubility: Atomic size factor: atoms need to “fit”? solute and solvent atomic radii should be within ~ 15% Crystal structures of solute and solvent should be the same Electronegativities of solute and solvent should be comparable (otherwise new inter-metallic phases are encouraged) Generally more solute goes into solution when it has higher valence than solvent Ni Cu Substitutional Solid Solutions A1.0Z0.0 A0.8Z0.2 A0.6Z0.4 A0.4Z0.6 A0.2Z0.8 A0.0Z1.0 Random and Ordered Solid Solutions a: random solid solution b and c: partly ordered solid solution d:ordered solid solution 1. Temperature Cation disorder in a solid solution increases the configurational entropy: solid solution is stabilized at high temperature Cation-size mismatch increases the enthalpy (structure must strain to accommodate cations of different size): solid solution is destabilized at low temperatures Extent of solid solution tolerated is greater at higher temperatures Factors Controlling the Extent of Solid Solution G = H – TS 2. Structural flexibility Cation size alone is not enough to determine the extent of solid solution, it also depends on the ability of the structural framework to flex and accommodate differently-sized cations e.g. there is extensive solid solution between MgCO3 and CaCO3 at high temperature 3. Cation charge Complete solid solution is usually only possible if the substituting cations differ by a maximum of ± 1. Heterovalent substitutions often lead to complex behaviour at low temperatures due to the need to maintain local charge balance. Factors Controlling the Extent of Solid Solution Deformation of Solids Motion of a dislocation (line of missing particles) in a crystalline solid results in a permanent change in the shape of the solid. 20 Dislocations— Linear Defects Dislocations are linear defects: the interatomic bonds are significantly distorted only in the immediate vicinity of the dislocation line. This area is called the dislocation core. Dislocations also create small elastic deformations of the lattice at large distances. Dislocations are very important in mechanical properties of material. Introduction/discovery of dislocations in 1934 by Taylor, Orowan and Polyani marked the beginning of our understanding of mechanical properties of materials. Linear Defects ?? Dislocation Two kinds of dislocations: Edge dislocation Screw dislocation GaN Pd Existence of dislocation even without deformation 106/cm3 in usual Edge dislocation E E’ Extra half plane Edge Dislocation Slip System Preferred planes for dislocation movement (slip planes) Preferred crystallographic directions (slip directions) Slip planes + directions (slip systems) àhighest packing density. Distance between atoms shorter than average; distance perpendicular to plane longer than average. Far apart planes can slip more easily. BCC and FCC have more slip systems compared to HCP: more ways for dislocation to propagate ? FCC and BCC are more ductile than HCP. Slip in a Single Crystal Each step (shear band) results from the generation of a large number of dislocations and their propagation in the slip system Zn Stacking Faults and Grain Boundaries mosaic structure high angle grain boundary tilt grain boundary twist grain boundary 2D Defects in Solids 21 Stacking Faults Stacking faults occur in wide variety of materials not just simple metals. Consider a structure to be built up from successive layers of atoms or other units, if the regular stacking of these units is interrupted, we have a stacking fault. Close packed metals provide simple examples Perfect FCC has a ABCABCABCABCABC sequence The sequence ABCABCBCABCABC has a stacking fault Perfect HCP is ABABABABABABAB ABABABCABABABABAB has a stacking fault Faults that put two of the same layers together AA BB or CC are unlikely due to their very high energy Grain Boundaries Mosaic structure High angle grain boundary Low angle grain boundary Left: tilt Right: twist Tilt Grain Boundaries Low angle grain boundary is an array of aligned edge dislocations. This type of grain boundary is called tilt boundary (consider joint of two wedges) Transmission electron microscope image of a small angle tilt boundary in Si. The red lines mark the edge dislocations, the blue lines indicate the tilt angle Tilt boundary Low angle symmetrical tilt boundary θ < 10~15° 22sinD 2b θ≈θ= Low Angle Symmetrical Tilt Boundary The number of dislocations per unit length, 1/D bD 1 θ≈ Interfacial energy, γ θ∝∝γ D1 Twist Grain Boundaries Twist boundary - the boundary region consisting of arrays of screw dislocations (consider joint of two halves of a cube and twist an angle around the cross section normal) Chemistry in Two Dimensions: Surfaces Model of a heterogeneous solid surface, depicting different surface sites. These sites are distinguishable by their number of nearest neighbors 22 Surfaces & Grain Boundaries External Surfaces Surface atoms have unsatisfied atomic bonds, and higher energies than the bulk atoms ? Surface energy, γ(J/m2) Surface areas tend to minimize (e.g. liquid drop) Solid surfaces can “reconstruct”to satisfy atomic bonds at surfaces. Grain Boundaries Polycrystalline material comprised of many small crystals or grains. The grains have different crystallographic orientation. There exist atomic mismatch within the regions where grains meet. These regions are called grain boundaries. Surfaces and interfaces are reactive and impurities tend to segregate there. Since energy is associated with interfaces, grains tend to grow in size at the expense of smaller grains to minimize energy. This occurs by diffusion, which is accelerated at high temperatures.