Notes:
• Reminder, Exam: Tues, Oct 13.
§
§
§
In class.
Covers through ch. 5
Open book = Kittel only.
• HW 4: Can turn in Thursday, Oct. 8. (I will have solutions ready.)
• Note on HW grades: averages 38/50 (HW1), 31/40 (HW2), 37/45 (HW3). Important to understand what you missed! Fermi gases, and Transport properties (ch. 6):
• “Fermi gas” means non-interacting (or weakly-interacting)
gas of electrons, which are Fermions.
• Particles of this type will exhibit a Fermi surface due to the
Pauli exclusion principle.
• Electron states are plane waves. (Later, with crystal
potential: Bloch waves.) State density in k-space for a
crystal volume L3: Dk = 2×(2π/L)3, as before except
factor 2× for electron spin.
2 2
!
k
εF = F
ε
TF = F
kB
2m
Fermi
energy
vF = kF / m
Fermi temperature & velocity
Free-­‐electron gas, and Transport properties (ch. 6):
• “Fermi gas” = non-interacting (or weakly-interacting) gas
of electrons.
Notes:
v Fermi gas concept works surprisingly well for simple
metals (sodium or copper); Fermi sphere replace by
sometimes convoluted Fermi surface.
v Electrons do interact strongly (Coulomb interaction). We
will see later, screening & other effects can reduce this
effect.
v “Fermi liquid” refers to situations with stronger
interactions, e.g. with magnetic interactions at low T.
v Other more exotic situations of current interest, strong
effects of interactions: “non-Fermi liquids”; “Luttinger
liquids” …
Fermi gases, and Transport properties (ch. 6):
• State density in k-space for a crystal volume L3:
Dk = V /(4π 3 ) , factor 2× for electron spin.
with N states inside sphere, kF = 3 3π 2 n
εF =
 2 kF2
2
⎡
⎤ (3π 2 n)2/3

=
2m ⎣ 2m ⎦
Fermi gases, and Transport properties (ch. 6):
• State density in k-space for a crystal volume L3:
Dk = V /(4π 3 ) , factor 2× for electron spin.
with N states inside sphere, kF = 3 3π 2 n
εF =
 2 kF2
2
⎡
⎤ (3π 2 n)2/3

=
2m ⎣ 2m ⎦
Metal, unit cell ~1 nm or less; suppose (e.g. Cu) 4 atoms/cell, each with
1 valence electron:
7
n ~ 4×1021 cm-3, find kF = 4.9 ×10
cm-1,
εF = 0.9 eV, TF ~ 10,000 K; vF ~ 108 cm/s.
► Note, Metals typically strongly degenerate; Semiconductors (much
smaller n) may be in classical regime.
Thermal properties:
• Chemical potential:
ε=µ
• Occupation function [Fermi-Dirac]:
f (ε ) = [ exp(ε − µ ) / kBT +1]
−1
Symmetric about µ
Equivalent to classical in low-density limit.
Density of states :
Dk = V /(4π 3 ) , have seen, with

k = (2π / L)(n1iˆ + n2 ˆj + n3 kˆ )
kF = 3 3π 2 n
Same method as for phonons gives:
V 2⎛ m⎞
D(ε ) = 2 ⎜ 2 ⎟
π ⎝ ⎠
3/2
ε
ε = 2 k 2 2m
Density of states :
Dk = V /(4π 3 ) , have seen, with

k = (2π / L)(n1iˆ + n2 ˆj + n3 kˆ )
kF = 3 3π 2 n
Same method as for phonons gives:
V 2⎛ m⎞
D(ε ) = 2 ⎜ 2 ⎟
π ⎝ ⎠
3/2
ε
Average over occupied states:
∞
U = ∫ εD(ε ) f (ε )dε
Energy avg.
−∞
Avg. # occupied states in dε
ε = 2 k 2 2m
U =
Specific heat etc.:
∂U
∞
∫ ε D(ε ) f (ε )d ε
−∞
∞
1
∂
C=
= ∫ εD(ε )
f (ε )dε , and
∂T
V −∞
∂T
∞
∞
U − Nε F = ∫ (ε − ε F ) D(ε ) f (ε )dε
∫ D(ε ) f (ε )d ε = N
−∞
−∞
“delta function”.
∞
1
∂
C = ∫ (ε − ε F )D(ε )
f (ε )d ε
V −∞
∂T
∂
∂x ∂
ex
2
(ε − ε F )
f (ε ) = (ε − ε F )
f (x) = kB x x
∂T
∂T ∂x
(e +1)2
∞
x
2
⎛
⎞
1
e
1
π
2
C ≅ kBTD(ε F ) ∫ x x
dx = kBTD(ε F ) ⎜ ⎟ ≡ γ T
2
V
(e +1)
V
⎝ 3⎠
−∞
Specific heat etc.:
⎛π 2 ⎞
1
C ≅ k BTD (ε F )⎜⎜ ⎟⎟ ≡ γT
V
⎝ 3 ⎠
( )
π 2 nkB T
Same as C ≅
TF
2
Free electrons
Also can show
⎛T
π2
µ ≅ ε F − k BT ⎜⎜
12
⎝ TF
⎞
π 2 ⎛ D′(ε F ) ⎞
⎟⎟ = ε F − ⎜⎜
⎟⎟(k BT )2
6 ⎝ D(ε F ) ⎠
⎠
Real metals with crystalline periodic potentials:
• Derivation of relationships containing D(ε) unchanged.
• However generally D(ε) is modified due to crystal-potential
(especially at B.Z. boundaries).
Transport properties:
Electrical conductivity: The standard relationship is,
ne τ
σ=
m
ne 2τ
j = −ne v = +
m
2
• Velocity is fermi velocity (good metals).
• Mobility defined as:
eτ
µ=
m
 = vF τ
σ = neµ
Metal, n is constant, mean free path changes. Conductivity
decrease vs. T. Semiconductor n and σ increase as T increases.