6. Free Electron Fermi Gas Energy Levels in One Dimension Effect of Temperature on the Fermi-Dirac...
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Transcript of 6. Free Electron Fermi Gas Energy Levels in One Dimension Effect of Temperature on the Fermi-Dirac...
6. Free Electron Fermi Gas
• Energy Levels in One Dimension
• Effect of Temperature on the Fermi-Dirac Distribution
• Free Electron Gas in Three Dimensions
• Heat Capacity of the Electron Gas
• Electrical Conductivity and Ohm’s Law
• Motion in Magnetic Fields
• Thermal Conductivity of Metals
• Nanostructures
Introduction
Free electron model:Works best for alkali metals (Group I: Li, Na, K, Cs, Rb)Na: ionic radius ~ .98A, n.n. dist ~ 1.83A.
Successes of classical model:Ohm’s law.σ / κ
Failures of classical model:Heat capacity.Magnetic susceptibility.Mean free path.
Quantum model ~ Drude model
Energy Levels in One Dimension
2 2
22n n n
dH
m dx
Orbital: solution of a 1-e Schrodinger equation
Boundary conditions: 0 0n n L
sinn
nA x
L
2n L
n
Particle in a box
2sin
n
A x
1,2,n
22
2n
n
m L
Pauli-exclusion principle: No two electrons can occupy the same quantum state.
Quantum numbers for free electrons: (n, ms ) ,sm
Degeneracy: number of orbitals having the same energy.
Fermi energy εF = energy of topmost filled orbital when system is in ground state.
N free electrons:22
2F
F
n
m L
2F
Nn
Effect of Temperature on the Fermi-Dirac Distribution
Fermi-Dirac distribution : 1
1f
e
1
Bk T
Chemical potential μ = μ(T) is determined by N d g f g = density of states
At T = 0: 1for
0f
→ 0 F
1
2f For all
T :
For ε >> μ : f e
(Boltzmann distribution)
3D e-gas
Free Electron Gas in Three Dimensions
2 2 2 2
2 2 22
d d dH
m dx dy dz
r r
Particle in a box (fixed) boundary conditions:
0, , , , ,0, , , , ,0 , , 0y z L y z x z x L z x y x y L n n n n n n
sin sin sinyx znn n
A x y zL L L
n
Periodic boundary conditions:
Standing waves
, , , , , , , ,x y z x L y z x y L z x y z L k k k k
→
→iA e k r
k2 i
i
nk
L
0, 1, 2,in
2 2
2
k
m k
Traveling waves
1, 2,in
i k kp
kk → ψk is a momentum eigenstate with eigenvalue k.
p km
k
v
N free electrons: 33
42
8 3 F
VN k
1/323F
Nk
V
2 2
2F
F
k
m
2/32 23
2
N
m V
FF
kv
m
1/323 N
m V
Density of states:
32
8
dSVD
k
k
k k
2
3 2
4
4 /
V k
k m
2 2
V mk
3/2
2 2
2
2
V m
323 F
VN k
3/2
2 2
2
3FmV
→
3
2FF
ND
3
2 F F
ND
Heat Capacity of the Electron Gas
(Classical) partition theorem: kinetic energy per particle = (3/2) kBT.
N free electrons:3
2e BC N k ( 2 orders of magnitude too large at room temp)
Pauli exclusion principle → ~e BF
TC N k
T TF ~ 104 K for metal
U d D f
1
1f
e
1
Bk T 3
2 F F
ND
free electronsUsing the Sommerfeld expansion formula
2 1
22 1
2 11
2 2 2n
nnB n
n
d Hd H f d H n k T
d
2 42 B
dDU d D k T D O T
d
2 42 B
dDN d D k T O T
d
2 42 B
dDU d D k T D O T
d
2 42F
F
F F F B F F
dDd D D k T D O T
d
2 42 B
dDN d D k T O T
d
2 42F
F
F F B
dDd D D k T O T
d
→ 22 0
F
F F B
dDD k T
d
22
F
F B
dDk T
D d
2 42F
B Fd D k T D O T
F
N d D
2
2
3V F B
N
UC D k T
T
2
26
2
2B
V BF
k TC N k
3-D e-gas
1
2
d D
D d
for 3-D e-gas
22
F
F B
dDk T
D d
Bk T
1
2
d D
D d
for 3-D e-gas
1
2
d D
D d
for 1-D e-gas
Experimental Heat Capacity of Metals
For T << and T << TF : 3C T A T el + ph
2CA T
T
thobsm
m e gas
2
2
3 F BC D k T 3
2FF
ND
2 2
3 2
2 F
N m
k
1/323F
Nk
V
Deviation from e-gas value is described by mth :
thobsm
m e gas
Possible causes: e-ph interaction e-e interaction
Heavy fermion: mth ~ 1000 m
UBe3 , CeAl3, CeCu2Si2.
Electrical Conductivity and Ohm’s Law
d
dt
pFLorentz force on free electron:
1e
c E v B d
dt
k
No collision: 0e t
t E
k k t k
Collision time :
nqj v n e
m
k 2ne
m
E
Ohm’s law
2 1ne
m
Heisenberg picture: ,d
i Hdt
p
p , q p E r q i E d
qdt
p
EFree particle in constant E field
Experimental Electrical Resistivity of Metals
Dominant mechanismshigh T: e-ph collision.low T: e-impurity collision.
phonon impurity1 1 1
ph imp
ph imp
Matthiessen’s rule:
0 imp Sample dependent
ph impT T Sample independent
Residual resistivity:
Resistivity ratio: room
imp
T
imp ~ 1 ohm-cm per atomic percent of impurity
K
imp indep of T
(collision freq
additive)
Consider Cu with resistivity ratio of 1000:
32951.7 10 ohm-cm
resistivity ratioimp
K
3 21.7 10 10ic Impurity concentration: = 17 ppm
Very pure Cu sample: 54 10 300K K
94 2 10K s 4 4 0.3Fl K v K cm 8 11.57 10Fv cm s
For T > : T See App.J
From Table 3, we have 295 1.7 ohm-cmL K
imp ~ 1 ohm-cm per atomic percent of impurity
Umklapp Scattering
Normal: k k q
Umklapp:
k k q G
Large scattering angle ( ~ ) possible
Number of phonon available for U-process exp(U /T )
For Fermi sphere completely inside BZ, U-processes are possible only for q > q0
q0 = 0.267 kF for 1e /atom Fermi sphere inside a bcc BZ.
For K, U = 23K, = 91K U-process negligible for T < 2K
Motion in Magnetic Fields
1d
dt
k FEquation of motion with relaxation time : 1q
c E v B
1 1dm q
dt c
v E v B
/ / / /
1dm q
dt
v E
1 1 2
1 1dm q B
dt c
v E v
be a right-handed orthogonal basis 1 2 / /ˆ, , e e e BLet
2 2 1
1 1dm q B
dt c
v E v
Steady state:
1 1 2c
q q
m q
v E v
2 2 1c
q q
m q
v E v
/ / / /
q
m
v E
c
q B
m c = cyclotron frequency
q = –e for electrons
Hall Effect
0yj →
0y c x
q qE v
m q
x x
qv E
m
0z z
qv E
m
y c x
qE E
q x
qBE
mc
Hall coefficient:
yH
x
ER
j B 2
x
x
qBE
mcnq
E Bm
1
nqc
electrons
Thermal Conductivity of Metals
From Chap 5:1
3K C v l
Fermi gas:21
3 2B
el B F FF
k TK N k v v
2
3B
B
k TN k
m
In pure metal, Kel >> Kph for all T.
Wiedemann-Franz Law:2
23
BB
k TN kK m
nqm
22
3Bk
Tq
T
Lorenz number:K
LT
22
3Bk
q
8 22.45 10 watt-ohm/deg
for free electrons
8 22.45 10 watt-ohm/degL for free electrons