3.155J/6.152J,Fall Term,2003
Some useful equations for Quiz # 2
Constants,k
B
= 1.38 × 10
-23
J/K = 8.62 × 10
-5
eV/K,e = 1.6 × 10
-19
Coul,
1 atm = 760 Torr = 10
5
Pa
x
Ideal gas,PV = Nk
B
T,N/V = n = C (concentration),J =
nv
.
x
2
k
B
T
Gas kinetics,v =
2k
B
T
,Mean free path,λ=
2πd
2
P
,d = molecular diameter,
x
πm
P
J =
2πk
B
Tm
,m = species mass,
E = E
1
4 M
1
M
2
)
2
cos
Energy transfer in collision of M
1
and M
2
,
(M
1
+ M
2
2
θ
∞
∝ J
n
e
+ E
a
/ k
B
T
Reliability,mean time to failure,MTTF =
∫
t? f ( t ) dt or
0
Electromigration,
J
D
A
qZ
A
Jρ
A
= c
A
v
A
= c
A
D
A
F
= c
A
*
(number/m
2
x sec)
RT RT
J
A
is flux of species A,having a concentration of c
A
,where flux is due to current J in a
medium of electrical resistivity is ρ.,at temperature,T,by grain-boundary diffusion
obeying,
D
A
= D
A
o
exp[? E
a
/(k
B
T)]
Change in concentration due to electromigration and temperature gradients,
J
A
dTdc
A
=?
J
A
dt?x?T dx
v
σ eτ
= =
Electrical,Resistivity,R =ρ l/A (Ohm),mobility,
μ=
E ne m *
Capacitance,C = κε
0
A/d (Farad),ε
0
= 8.85 x 10
-12
(F/m)
Crystal growth,vacancy concentration,n = n
0
exp
[
E /k
B
T
] E
g
= 2.6 eV
vac g
n
0
= 5 x 10
22
Si/cm
3
,
Oxygen concentration,
C
oxy
= 2 × 10
22
exp[?1.03eV /k
B
T ]
Maximum crystal growth velocity,
v
max
=
k
s
T
z
Lρ
m

solid
1