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
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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.
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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