Notes:
• Exams  will  be  graded  by  next  Tuesday!

• Recall , “Fermi gas” concept applies to typical metals, with
T f very large.
• But semiconductors (small n ) may be in classical regime.
f (
ε
)
=
[ exp(
ε − µ
) / k
B
T
+
1
] −
1
Fermi function = occupation probability
F
Fermi energy

Transport  properties:
Electrical  conductivity: Classic  relationship    (Drude): j = − ne v = + ne 2 τ m
σ = ne 2 τ m
• Instantaneous v = Fermi velocity (good metals).
 = v
F
• Collision time = τ .
• Mobility defined:
µ = e
τ m
= ne
Metal, n is constant, mean free path changes. Conductivity decrease vs. T. Semiconductor n and σ increase as T increases.

Transport  properties:
Electrical  conductivity: Classic  relationship    (Drude): j = − ne v = + ne 2 τ m
σ = ne 2 τ m
• Electron wave-packets: in field, can show,
& see  text  ch. 8,  p.  192 d k
dt
e
E
• Scattering; allowed for electrons near Fermi surface.
• But as k changes, τ = scattering time applied to all electrons.
dn dt
= −
τ n
⇒ τ = lifetime
• So classical formula often applies to Fermi gas situation; scattering due to defects or phonons.
or  other  electrons,  etc.
Not from  regular  crystal  positions

Transport  properties:
Electrical  conductivity: Classic  relationship    (Drude): j = − ne v = + ne 2 τ m
σ = ne 2 τ m k
F
= 3 3 π 2 n
F
=
D ( ε
F
)
!
2 k
F
2
=
V
π 2
2
2 m
⎝
⎜
⎛ m
!
2
= !
2 ( 3
⎠
⎟
⎞ 3 / 2
ε
F
2
= n ) 2 / 3 2 m
Vm
( )
!
2 π 2 n
2 / 3
σ = ne m
2 τ
=
2
3
D ( ε
V
F
) ε
F e 2 τ m
=
D ( ε
F
) v
F
2 e 2 τ
3 V
(Expressed   in  terms  of
Fermi  surface   properties   only)
• Classical formula often applies to Fermi gas situation; scattering due to defects or phonons.

Transport  properties:
Matthiessen’s rule: Scattering  rates  add.
τ
1
=
1
τ impurity
+
τ
1 phonons
+ ...
• General rule; uncorrelated processes, metal resistivities add.
• Impurities always decrease mobility (increase resistivity)
• Alloy resistivity may be large & constant vs. T .
ρ
µ = e
τ m
= ρ impurity
= ne
≡ 1 /
+ ρ phonons
+ ...
1
High T, n phonon
Good metal

Transport  properties:
Matthiessen’s rule: Scattering  rates  add.
τ
1
=
1
τ impurity
+
τ
1 phonons
+ ...
• General rule; uncorrelated processes, metal resistivities add.
• Impurities always decrease mobility (increase resistivity)
• Alloy resistivity may be large & constant vs. T .
With  impurities
ρ
µ = e
τ m
= ρ impurity
= ne
≡ 1 /
+ ρ phonons
+ ...
1
High T, n phonon
Good metal

Transport  properties:
Mobilities:
µ = e
τ m
Measures   scattering  rate
• Mass also modified for states in crystal potential, we will see
[example, graphene “ m * = 0’].
• Copper,
• GaAs,
~ 100 cm 2 V .
s
~ 1000 cm 2 V .
s
• “world record”
~ 10 7
RT cm 2 V .
s

Transport  properties:
Mobilities:
µ = e
τ m
Measures   scattering  rate
• Mass also modified for states in crystal potential, we will see
[example, graphene “ m * = 0’].
τ ~ 10 − 13 s , !
~ 10 − 5 cm
• Copper,
• GaAs,
~ 100 cm 2 V .
s
~ 1000 cm 2 V .
s
• “world record”
~ 10 7
RT cm 2 V .
s quantum hall effect – type systems; electrons “ballistic”
“high electron mobility transitor”

Transport  properties:
Hall  effect ,
R
H
=
E y j x
B z
=
1 nec
Cgs units!
• Standard means to measure carrier densities.
(Especially for semiconductors.)
• Actually can be either sign; positive signifies holes.
• Works for classical or Fermi gas; with more than one carrier type, relationships much more complicated.

Transport  properties:
Thermal  conductivity,
el
=
1
3
C !
v
• We saw this before for phonons.
• Works for degenerate system too
• v (mean speed) is Fermi velocity for good metals.
• Specific heat we have seen: C = γ T
• Generally electron + phonon contributions to κ add.
el
el =
2
3 ⎝
⎜
⎛ k
B e ⎠
⎟
⎞ 2
T
Wiedemann-Franz law degenerate metals only
C ≅
1
V k
B
TD ( ε
F
)
⎛
⎜⎜⎝
π
3
2 ⎞
⎟⎟⎠
≡ γ T σ =
D ( ε
F
) v
F
2 e
3 V
2 τ