Transcript ASU Talk 5/99

N. Newman, [email protected] Handout 2
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Reciprocal Lattice
t3
R = n1 t 1 + n 2 t 2 + n3 t 3
t2
G = 2p (h1b1 + h2b2 + h3b3)
t2 x t3
where b1 =
t 1 . t 2 x t3
t 3 x t1
b2 = t . t x t
1 2
3
0
t1
t 1 x t2
and b3 =
t 1 . t 2 x t3
i.e. b1.t1 = 1 ; b1.t2 = 0; and b1.t3 = 0



Cubic t1 = ax ; t2 = ay and t3 = az



b1 = (2p/a)x; b2 = (2p/a) y ; & b3 = (2p/a) z
Volume of unit cell t1. t2 x t3 = a3
Volume of first Brillion zone (2p/a)3
Primitive Translation for FCC
 
 
 
t1 = (a/2)(y + z) ; t2 = (a/2)(x + z); t3 = (a/2)(x + y)
Volume V = a3/4
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Sommerfield: Quantum Mechanics
p2
E
ħ2k2
E = ½ mv2 = 2m =
2m
-ħ2
2
+V =E
2m
Free electron (V = 0)
1
  = v eik.r
2 2
ħ k
2m
0
k
constant such that v2d3v = 1
ħ2k2
 = E  i.e 2m
ħk
=E
momentum
If impose boundary conditions
(x, y, z) = (x, y, z+L) = (x, y+L, z) = (x+L, y, z)
1
1
1
ik.x
ik.(x+L) =
ik.x i.kL
v e e
v e =v e
i.e eik.L = 1
 kx = 2pnx/L Similarly ky = 2pny/L and kz = 2pnz/L
y
Therefore, one electron occupies a
volume of (2p/L)3 i.e 8p3/L3 in k-space
# of electrons in k-space = V/8p3
where  is the area in k-space
2p/L
0
x
2p/L
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Density of States
 = Ceik.r with periodic boundary conditions at L
 is eigenstate iff kx = 2pnx/L; ky = 2pny/L and kz=2pnz/L


Each k-cell is (2p/L)3 in volume.
Each holds 2 electrons.
number of k-points between k
and k+dk, with 2 electrons per
k points (spin up & down)
k
2
k+dk
4pk2dK
(2p/L)3
2L3
k
=
2 dK
p
Define g(k) as number of states/unit volume = (k2/p2)dk
Convert to E using E =
ħ2k2
2m
ħ2k
dE =
dk
m
ħ2 2mE
dE = m
dk
ħ
dk
m1/2 dE
i.e
dk
=
volume of shell is 4pk2
2E ħ
g(E)dE = k2/p2dk
= 2mE/ħ2
p2
=
m3/2 E1/2
2p2 ħ3
m1/2
dE
2E ħ
dE
i.e g(E) =
N. Newman, [email protected] Handout 2
Density of
states
m3/2E1/2
2p2 ħ3
p. 13
Fermions
n =  N(E) dE =
N(E)
Ef =
Ef
ħ2
2m
Ef3/2
3p2ħ3
(3p2n)2/3
kf = (3p2n)1/3
k
2 2 m3/2
Vf =
ħkf
m
Vf of 108 cm/sec for most metals, mfp of 10 nm at R.T
Specific Heat of Metal’s Electron Gas
E (#) * (excitation energy)
= [N(Ef)  dE]  kBT
N(E)
E~kT
Ef
k
= [n(kB T /Ef)] 3/2 kBT  3/2 nkB2T2 /Ef
cv = E/T = 3nkB2T/Ef
Very different from classical
/T(3/2nkBT) =~ 3/2nkB

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Photoemission Spectroscopy
h
Solid
e- kinetic energy
measured
Kinetic energy
primary
electron
s
primary
electron
s
secondary
electrons
vacuum level
h
valence
band
VBM
Core level
Core
level
(1) Excitation
Binding energy
conductio
n
band
(2) Transport to surface(3) Escape from surface
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Escape Depth (Angstroms)
Surface Sensitivity
15
10
5
Maximum surface sensitivity
0
25
50
100
1000
Kinetic Energy of Photoelectrons
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From Ashcroft and Mermin, Solid-State Physics
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R=
rL/
A
L
= / (s A)
For a 1 mm x 1mm x 10 mm sample
(A = 10-2 cm2, L = 1 cm)
I
V
metal = 10-3 W
intrinsic Si = 107 W
doped Si (1017/cm3) = 10 W
doped Si (1020/cm3) = 10-1 W
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R=
rL/
A
L
= / (s A)
where
metal
r = 10 mW cm
intrinsic Si
r = 1011 mW cm
doped Si
(1017/cm3)
r = 105 mW cm
doped Si
(1020/cm3)
r = 103 mW cm
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Conductivity
J (A/cm2)
= q n v e + q p vh
= q n me E + q p mp E
= (se + sp) E
vdrift = m E
me [cm2/(V sec)]
mh [cm2/(V sec)]
where m is the mobility (cm2/V.sec)
Si
GaAs
Ge
Cu
Al
1500
8500
3900
32
12
480
400
1900
-
-
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What is mobility?
Vd
F = m a = -q E
if the electron accelerates between
collisions in an average time, t E
from zero velocity to a
terminal velocity, vd
then m (vd/2) / t = - q E
and vd = -(2qt/ m) E
vd = -m E
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E
n=N
Eo
ECBM
E
Ef
e-E/ kT
EVBM
n = Nc e-Eg/ 2kT
= 2(2pmekT/h2)3/2 e-Eg/ 2kT
For a semiconductor Eo is the Fermi level (Ef) and
Nc is called the density of states. Because the
electronic states of a semiconductor are distributed
throughout the valence/conduction bands, N is not
the total number of electrons in the band. Instead
it is a statistically
weighted number which turns out to be the number
of states which are very close (within ~kT) of the
band edge.
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