The free electron theory of metals

The Drude theory of metals

Paul Drude (1900): theory of electrical and thermal conduction in a metal
application of the kinetic theory of gases to a metal,
which is considered as a gas of electrons
mobile negatively charged electrons are confined in a
metal by attraction to immobile positively charged ions

isolated atom in a metal

nucleus charge eZa

Z valence electrons are weakly bound to the nucleus (participate in chemical reactions)
Za – Z core electrons are tightly bound to the nucleus (play much less of a role in chemical
reactions)
in a metal – the core electrons remain bound to the nucleus to form the metallic ion
the valence electrons wander far away from their parent atoms
called conduction electrons or electrons
 Doping of Semiconductors

C, Si, Ge, are valence IV , Diamond fcc structure. Valence

band is full
Substitute a Si (Ge) with P. One extra electron donated
to conduction band
N-type semiconductor
 density of conduction electrons in metals ~1022 – 1023 cm-3
rs – measure of electronic density
rs is radius of a sphere whose volume is equal to the volume per electron
3 1/ 3
4rs V 1  3  1
  rs    ~
3 N n  4n  n1 / 3
mean inter-electron spacing

in metals rs ~ 1 – 3 Å (1 Å= 10-8 cm) rs/a0 ~ 2 – 6

2
a0  2
 0.529 Å – Bohr radius
me

● electron densities are thousands times greater than those of a gas at normal conditions
● there are strong electron-electron and electron-ion electromagnetic interactions

in spite of this the Drude theory treats the electron gas

by the methods of the kinetic theory of a neutral dilute gas
 The basic assumptions of the Drude model

1. between collisions the interaction of a given electron

with the other electrons is neglected independent electron approximation
and with the ions is neglected free electron approximation

2. collisions are instantaneous events

Drude considered electron scattering off
the impenetrable ion cores

the specific mechanism of the electron scattering is not considered below

3. an electron experiences a collision with a probability per unit time 1/τ

dt/τ – probability to undergo a collision within small time dt
randomly picked electron travels for a time τ before the next collision
τ is known as the relaxation time, the collision time, or the mean free time
τ is independent of an electron position and velocity

4. after each collision an electron emerges with a velocity that is randomly directed and
with a speed appropriate to the local temperature
 DC electrical conductivity of a metal

V = RI Ohm’s low
the Drude model provides an estimate for the resistance
introduce characteristics of the metal which are independent on the shape of the wire
E  rj j  sE
j=I/A – the current density
r – the resistivity L
R=rL/A – the resistance A
s = 1/r - the conductivity
j  -env
v is the average electron velocity

eE  ne 2 
v-  j   E
m  m 
ne 2
j  sE s
m
 m

rne 2
at room temperatures
resistivities of metals are typically of the order of microohm centimeters (mohm-cm)
and  is typically 10-14 – 10-15 s

mean free path l=v0

v0 – the average electron speed
l measures the average distance an electron travels between collisions
estimate for v0 at Drude’s time 1 2 mv0 2  3 2 k BT → v0~107 cm/s → l ~ 1 – 10 Å
consistent with Drude’s view that collisions are due to electron bumping into ions

at low temperatures very long mean free path can be achieved

l > 1 cm ~ 108 interatomic spacings!
the electrons do not simply bump off the ions!

the Drude model can be applied where

a precise understanding of the scattering mechanism is not required

particular cases: electric conductivity in spatially uniform static magnetic field

and in spatially uniform time-dependent electric field

Very disordered metals and semiconductors

motion under the influence of the force f(t) due to spatially uniform
electric and/or magnetic fields average average
momentum velocity
equation of motion dp(t ) p( t )
-  f (t ) p ( t )  mv ( t )
for the momentum per electron dt 

electron collisions introduce a frictional damping term for the momentum per electron

Derivation:
 dt  dt
p(t  dt )  1 -   p(t )  f (t )dt    0
   

fraction of electrons that scattered total loss of momentum

does not experience scattering part after scattering

p( t )
p(t  dt )  p(t )  f (t )dt - dt  O dt 2 

p(t  dt ) - p(t ) p( t )
 f (t ) -  O  dt 
dt 
dp ( t ) p(t )
 f (t ) -
dt 
 Hall effect and magnetoresistance
Edwin Herbert Hall (1879): discovery of the Hall effect

the Hall effect is the electric

field developed across two
faces of a conductor
in the direction j×H
when a current j flows across
a magnetic field H
e
the Lorentz force FL  - v  H
c
in equilibrium jy = 0 → the transverse field (the Hall field) Ey due to the accumulated charges
balances the Lorentz force
quantities of interest:
Vx Ex
magnetoresistance R ( H )  R xx  resistivity r ( H )  r xx 
(transverse magnetoresistance) Ix jx
Vy Ey
Hall (off-diagonal) resistance R yx  Hall resistivity r yx 
Ix jx
Ey
the Hall coefficient RH 
jx H
RH → measurement of the sign of the carrier charge
RH is positive for positive charges and negative for negative charges
  1 
force acting on electron f  - e E  v  H 
 c 
equation of motion dp  1  p
 - e E  p  H -
for the momentum per electron dt  mc  
px
in the steady state px and py 0  - eE x -  p
c y -

satisfy p
0  - eE y   c p x - y

eH
c 
mc cyclotron frequency
multiply by - ne / m frequency of revolution
p of a free electron in the
j  - ne s 0 E x  cj y  jx
the Drude magnetic field H
m s 0 E y  -cjx  j y e
mc r  c r H
2
model DC
conductivity ne 2
c
s0  c
at H=0 m    c  ~ 1GHz
 H  2
jy  0 E y  - c  jx  -  jx
s  nec  at H = 0.1 T
 0 
the resistance does not 1
jx  s 0 E x RH  - RH → measurement of the density
depend on H nec
c  1 weak magnetic fields – electrons can complete only a small part of revolution between collisions
c  1 strong magnetic fields – electrons can complete many revolutions between collisions
j is at a small angle f to E f is the Hall angle tan f  c c  1
 measurable quantity – Hall resistance r H  RH H
Vy H for 3D systems n2 D  nLz
rH   -
Ix n2 D ec for 2D systems n2D=n

E  rj j  sE  jx   s xx s xy   E x 
in the presence of magnetic field the  j   s s  E 
resistivity and conductivity becomes tensors  y   yx yy   y 

-1
 r xx r xy   E x   r xx r xy   jx   s xx s xy   r xx r xy 
for 2D: r    
  E   r  j  s      
 r yx r yy    
y yx r yy  y  s
 yx s yy   r yx r yy 
s 0 E x  cj y  jx
s0
s 0 E y  -cjx  j y s xx  s yy 
1  (c )2
1 c - s 0c
Ex  jx  jy s xy  -s yx 
s0 s0  Ex   1 s 0 c s 0   jx  1  (c )2
 1 E     j 
   -  s 1 s  y 
E y  - c jx  jy y c 0 0
s0 s0 r xx
1 m s xx 
r xx   2 r xx 2  r xy 2
s 0 ne 
 1 s0 c s 0  r xy
r 
   H
s xy  -
r xy  c 
 - c s 0 1 s0 
s 0 nec r xx 2  r xy 2
 H
Hall resistance r H  RH H  -
nec
weak 1 m the Drude
r xx   2
magnetic s 0 ne  model
fields  H
c  1 r xy  c  the classical
s 0 nec Hall effect

strong magnetic fields c  1 h

quantization of Hall resistance r xy  2
e
 eH 
at integer and fractional   n  
 hc 
the integer quantum Hall effect
and the fractional quantum Hall effect
from D.C. Tsui, RMP (1999) and from H.L. Stormer, RMP (1999)
 AC electrical conductivity of a metal
Application to the propagation of electromagnetic radiation in a metal

consider the case

l >> l wavelength of the field is large compared to the electronic mean free path
electrons “see” homogeneous field when moving between collisions
 response to electric field mainly historically
both in metals and dielectric described by used mainly for

electric current j conductivity s  j/E metals

electric field leads to
polarization P polarizability c  P/E dielectrics
dielectric function e 14c

dr E(, t )  E( ) e - it

j   qi i
dt dP
i j  -iω P j(, t ) ~ j( ) e - it
P   qiri dt
P 1 j σ P(, t ) ~ P( ) e - it
i
χ  i
E E - iω ω
P ( ) 4π
e ( )  1  4 ε(ω)  1  i σ(ω)
E ( ) ω

D  E  4P  εE e(,0) describes the collective

divD  4r ext excitations of the electron
divE  4r  4 ( r ext  r ind ) gas – the plasmons
e(0,k) describes the electrostatic
screening
 AC electrical conductivity of a metal
equation of motion dp(, t ) p(, t )
- - eE(, t )
for the momentum per electron dt 
time-dependent electric field E(, t )  E( )e -it
seek a steady-state solution in the form p(, t )  p( )e -it
eE( )
p( ) 
1 /  - i
nep( ) (ne2 / m)E( )
j( )  - 
m 1 /  - i

j( )  s ( )E( )
s0 s0
AC conductivity s ( )  Re s ( ) 
1 - i 1   2 2
ne2
DC conductivity s0 
m

s0 2 4ne 2
s ( )  p  the plasma frequency
1 - i   1 4ne2
m a plasma is a medium with positive
e ( )  1 -
4 m 2 p 2
and negative charges, of which at
e ( )  1  i s ( ) e ( )  1 - 2 least one charge type is mobile
 
 even more simplified:

no electron collisions (no frictional damping term,   1)

d 2x
equation of motion of a free electron m 2  -eE
dt
eE
if x and E have the time dependence e-iwt x
m 2
ne 2
the polarization as the dipole moment per unit volume P  -exn  - E
m 2

P ( ) 4 ne 2
e ( )  1  4 1-
E ( ) m 2
4 ne 2
p 2

m
p2
e ( )  1 - 2

Application to the propagation of electromagnetic radiation in a metal

transverse
electromagnetic
wave
Application to the propagation of electromagnetic radiation in a metal
electromagnetic wave equation
in nonmagnetic isotropic medium e ( , K ) 2E / t 2  c 2 2E
look for a solution with E  exp( -it  iK  r )
dispersion relation for
e ( , K ) 2  c 2 K 2
electromagnetic waves

1 e real and > 0 → for  is real, K is real

and the transverse electromagnetic wave propagates
with the phase velocity vph= c/e1/2
2 e real and < 0 → for  is real, K is imaginary
and the wave is damped
with a characteristic length 1/|K|: E  e - K r
(3) e complex → for  is real, K is complex
and the waves are damped in space
(4) e =  → The system has a final response in the
absence of an applied force (at E=0); the poles
of e(,K) define the frequencies of the free
oscillations of the medium
(5) e = 0 longitudinally polarized waves are possible
 Transverse optical modes in a plasma p 2

4ne 2
dispersion relation for m
electromagnetic waves e ( , K ) 2  c 2 K 2
p2
e ( )  1 - 2
 2 - p2  c2 K 2 
(1) for  > p → K2 > 0, K is real,
waves with  > p propagate in the media
2
with the dispersion relation  2   p  c 2 K 2 e ( )
an electron gas is transparent
-Kr
(2) for  < p → K2 < 0, K is imaginary, E  e
waves with  < p incident on the medium
do not propagate, but are totally reflected
/p
vgroup  d dK  c
v ph   K  c (2) (1)
metals are shiny
/ p  = cK due to the reflection
of light
electromagnetic waves are electromagnetic waves
totally reflected from the propagate without damping
forbidden medium when e is negative when e is positive and real
frequency gap
cK/p

vph > c → vph does not correspond to the velocity of the real physical propagation of any quantity
 Ultraviolet transparency of metals 2 4ne 2
p 
m
plasma frequency p and free space wavelength lp = 2c/p
range metals semiconductors ionosphere p2
e ( )  1 - 2
n, cm-3 1022 1018 1010 
p, Hz 5.7×1015 5.7×1013 5.7×109
lp, cm 3.3×10-5 3.3×10-3 33
the reflection of
spectral range UV IF radio
light from a metal
electron gas is transparent when  > p i.e. l < lp is similar to the
reflection of radio
waves from the
ionosphere

plasma frequency
ionosphere
semiconductors
metals
reflects transparent for
metal visible UV
ionosphere radio visible
 Skin effect
when  < p electromagnetic wave is reflected
-Kr
the wave is damped with a characteristic length d = 1/|K|: E  e -r d  e
the wave penetration – the skin effect
the penetration depth d – the skin depth
2 2  4
  4
2
K  2 e  2 1  i
2
si 2 s c
c c    c  d cl 
2s 1 2
K
2s 
12
(1  i ) the classical skin depth
c
 2s 1 2  d >> l – the classical skin effect
E  exp( -it  iKr )  exp - r 
 c 
d << l – the anomalous skin effect (for pure metals at low temperatures)
the ordinary theory of the electrical conductivity is no longer valid; electric field varies rapidly over l
not all electrons are participating in the absorption and reflection of the electromagnetic wave
only electrons that are running inside the skin depth for most of the mean
d’ l free path l are capable of picking up much energy from the electric field
only a fraction of the electrons d’/l are effective in the conductivity
13
c c  lc 2 
d'  d '   
2s '  1 2  d ' 1 2  2s 
 2 s 
 l 
 Longitudinal plasma oscillations

a charge density oscillation, or

a longitudinal plasma oscillation, or plasmon

nature of plasma oscillations:

displacement of entire electron gas through d
with respect to positive ion background creates
surface charges s = nde and an electric field E = 4nde
equation of  2d
Nm 2  - NeE  - Ne( 4nde )
motion t
 2d oscillation at the 2 4ne 2
  2
d  0 p 
t 2
p plasma frequency m
L 2
for longitudinal plasma oscillation L=p e L   1 - 2  0
p
/ p transverse
electromagnetic waves

forbidden
frequency gap longitudinal plasma
oscillations
cK/p
 Thermal conductivity of a metal
assumption from empirical observation - thermal current in metals is mainly carried by electrons
thermal current density jq – a vector parallel to the direction of heat flow
whose magnitude gives the thermal energy per unit time
crossing a unite area perpendicular to the flow
Fourier’s law the thermal energy per electron
jq  -T
 – thermal conductivity
n n
1D: j q  vET (T [ x - v ]) - vET (T [ x  v ])
2 2
dE  dT 
j q  nv 2 T  - 
dT  dx 
to jq  -T
2 2 2 1
high T low T 3D: v x  v y  vz  v 2 1 2 1
3   v c  lvcv
after each collision an electron emerges 3
v
3
with a speed appropriate to the local T dE
n T
 cv
→ electrons moving along the T dT
gradient are less energetic the electronic specific heat
2
ne2  cv mv 2 3  k B  Wiedemann-Franz law (1853)
s    T
m s 3ne 2
2 e  Lorenz number ~ 2×10-8 watt-ohm/K2
Drude: 3
cv  nk B success of the Drude model is due to the cancellation
application of 2 of two errors: at room T the actual electronic cv is 100
classical ideal 1 2 3 times smaller than the classical prediction, but v2 is 100
gas laws mv  k BT times larger
2 2
 Thermopower
Seebeck effect: a T gradient in a long, thin bar should be accompanied by an electric field
directed opposite to the T gradient
high T low T thermoelectric field
E  QT
gradT
E thermopower
mean electronic velocity due to T gradient:
1 dv d  v2 
1D: vQ  [v ( x - v ) - v ( x  v )]  -v  -  
2 dx dx  2  vQ  -
 dv 2
T
to
2 2 2 1 2 6 dT
3D: v x  v y  vz  v
3 e E
mean electronic velocity due to electric field: v E  -
m

in equilibrium vQ + vE = 0 → E  -
1 mdv 2
T  Q  -  
1
   d mv 
2
2
- v
c
6e dT  3e  dT 3en
Drude:
kB no cancellation of two errors:
application of 3
observed metallic thermopowers at room T are
cv  nk B Q-
classical ideal 2 2e 100 times smaller than the classical prediction
gas laws
inadequacy of classical statistical mechanics in describing the metallic electron gas
 The Sommerfeld theory of metals
3/ 2
 m   mv 2 
the Drude model: electronic velocity distribution f MB ( v )  n   exp  - 
is given by the classical  2 k T
B   2 k T
B 

Maxwell-Boltzmann distribution

m / 
3
the Sommerfeld model: electronic velocity distribution f 1
( v) 
1 2
FD
is given by the quantum 4 3 
 mv - k T
B 0
Fermi-Dirac distribution exp  2 1
k BT 
 
 
Pauli exclusion principle: at most one electron
can occupy any single electron level
normalization
n   dvf ( v ) condition T0
consider noninteracting electrons
electron wave function
associated with a level of energy E 2   2 2 2 
-  2  2  2  ( r )  E ( r )
satisfies the Schrodinger equation 2m  x y z 

  x, y , z  L     x, y , z 
periodic
boundary   x , y  L, z     x , y , z 
conditions   x  L, y , z     x , y , z  3D:
a solution neglecting 1 ikr
 k (r )  e L
the boundary conditions 1D:
V
normalization constant: probability 2

of finding the electron somewhere 1  dr  (r )
in the whole volume V is unity
 2k 2
energy E (k ) 
2m
momentum p  k
k p2 1 2
velocity v E  mv
m 2m 2
wave vector k
de Broglie 2
wavelength l
k
   x, y , z  L     x, y , z 
1 ikr
  x , y  L, z     x , y , z 
2
 k (r )  e the area  2 
per point  
V  L 
  x  L, y , z     x , y , z 
the volume  2  2 
3 3

apply the boundary conditions eik x L  e

ik y L
 e ik z L  1 per point   
 L  V
2 2 2
components of k must be kx  nx , k y  ny , kz  nz
L L L
nx, ny, nz integers

a region of k-space of  V states

 i.e. allowed
volume  contains 2 / L  2 3
3

values of k
the number of states V
per unit volume of k-space, 2 3
k-space density of states k-space
 consider T=0
the Pauli exclusion principle postulates that only one electron can occupy a single state
therefore, as electrons are added to a system, they will fill the states in a system
like water fills a bucket – first the lower energy states and then the higher energy states

the ground state of the N-electron system is formed by occupying all single-particle levels with k < kF
volume density
state of the lowest energy of states
the number of allowed values of  4 k F 3  V kF 3
k within the sphere of radius kF    2V
  2 
3
 3 6
ky
to accommodate N electrons kF 3
N  2 2V Fermi sphere
2 electrons per k-level due to spin 6 kF
kF 3
n 2
3 kx
Fermi surface
Fermi wave vector k F ~108 cm-1 at energy EF

Fermi energy 2
E F   2 k F / 2m ~1-10 eV
k F  3 2 n 
13

Fermi temperature TF  EF k B ~104-105 K

2
Fermi momentum pF  k F EF 
2m
3 2 n 
23

Fermi velocity vF  k F / m ~108 cm/s

vF  3 2 n 
 13

compare to the vthermal  3k BT / m  ~ 107 cm/s at T=300K

1/ 2
m
classical
thermal velocity 0 at T=0
 density of states
V 3
total number of states with wave vector < k N k  2k 2
3 2 E
32 2m
total number of states with energy < E V  2mE 
N 2 2 
3   
32
the density of states – number of states per unit energy dN V  2m 
D( E )     E
dE 2 2   2 

32
dn 1  2m 
the density of states per unit volume or the density of states D ( E )     E
dE 2 2   2 

V
k-space density of states – the number of states per unit volume of k-space
2 3
 Ground state energy of N electrons

add up the energies of all electron 2 2

states inside the Fermi sphere E  2 k
2m
k k F

volume of k-space per state k  8 3 V

V V
 smooth F(k)  F (k ) 
k
3 
8 k
F ( k ) k k 0 i . e .V 
    
8 3 
F ( k )dk

5
E 1  2k 2 1  2k F
3 
the energy density  dk  2
V 4 k k F 2m  10m
 2k 2
F (k ) 
2 2m
the energy per electron E 3  2k F 3
in the ground state   EF dk  4k 2 dk
N 10 m 5

kF 3
N  2V
3
 remarks on statistics I
in quantum mechanics particles are indistinguishable
systems where particles are exchanged are identical
exchange of identical particles can lead to changing   ,    eia  ,   system of N=2 particles
of the system wavefunction by a phase factor only 1 2 2 1
1, 2 - coordinates and
spins for each of the
repeated particle exchange → e2ia  1  1 , 2    2 , 1  particles

antisymmetric wavefunction with respect symmetric wavefunction with respect

to the exchange of particles to the exchange of particles
1
 1 , 2    
 p1 1  p2 2  -  p1 2  p2 1   1 , 2  
1
 
 p1 1  p2 2    p1 2  p2 1 
2 2
p1, p2 – single particle states
fermions are particles which have half-integer spin bosons are particles which have integer spin
the wavefunction which describes a collection the wavefunction which describes a collection
of fermions must be antisymmetric with respect of bosons must be symmetric with respect
to the exchange of identical particles to the exchange of identical particles
fermions: electron, proton, neutron bosons: photon, Cooper pair, H atom, exciton

if p1 = p2   0 unlimited number of bosons can occupy

→ at most one fermion can occupy a single particle state
any single particle state – Pauli principle

obey Fermi-Dirac statistics obey Bose-Einstein statistics

 distribution function f(E) → probability that a state at energy E
will be occupied at thermal equilibrium

fermions Fermi-Dirac 1 degenerate

f FD ( E ) 
particles with distribution E-m Fermi gas
half-integer spins function exp   1 fFD(k) < 1
 k BT 
bosons Bose-Einstein 1 degenerate
particles with distribution f BE ( E ) 
E-m Bose gas
integer spins function exp   -1 fBE(k) can be any
k T
 B 
both fermions and Maxwell-Boltzmann m-E classical
bosons at high T distribution f MB ( E )  exp   gas
when E - m  k BT function  k BT  fMB(k) << 1

n   dEn ( E )   dED ( E ) f ( E )

m=m(n,T) – chemical potential

 remarks on statistics II m v
x
BE and FD distributions differ from the classical MB distribution
because the particles they describe are indistinguishable. (x)
vg=v
Particles are considered to be indistinguishable if their wave packets
x
overlap significantly.
Two particles can be considered to be distinguishable
if their separation is large compared to their de Broglie wavelength. x
g(k’) k
12 k’
thermal de Broglie  2  2
h k0
  k ' 2  
wavelength ldB    ~  (r, t )   g ( k ') exp i  k ' r - t 
mk
 B  T p k'   2 m 
A particle is represented by a
particles become
ldB ~ d  n -1 3
wave group or wave packets
indistinguishable when of limited spatial extent,
2  2 2 3 which is a superposition of many matter
i.e. at temperatures below TdB  n waves with a spread of wavelengths
mk B
centered on l0=h/p
at T < TdB fBE and fFD are strongly different from fMB The wave group moves
at T >> TdB fBE ≈ fFD ≈ fMB with a speed vg – the group speed,
which is identical to the classical
electron gas in metals: particle speed
n = 1022 cm-3, m = me → TdB ~ 3×104 K Heisenberg uncertainty principle, 1927:
If a measurement of position is made with
gas of Rb atoms: precision x and a simultaneous
n = 1015 cm-3, matom = 105me → TdB ~ 5×10-6 K measurement of momentum in the x
direction is made with precision px,
excitons in GaAs QW then
p x x   2
n = 1010 cm-2, mexciton= 0.2 me → TdB ~ 1 K
 T≠0 the Fermi-Dirac distribution
3D
32
dn 1  2m 
density of states D( E )   2 2  E
dE 2   
1
distribution function f ( E )  lim f ( E )  1, E  m
E-m T 0
exp   1 0 Em
k
 B T
n   dED ( E ) f ( E )  m lim m  EF
T 0
1
D ( E )dE   [the number of states in the energy range from E to E + dE]
V
1
D ( E ) f ( E )dE   [the number of filled states in the energy range from E to E + dE]
V
density of
states
D(E)

per unit volume

density of
filled states
D(E)f(E,T)
shaded area – filled
states at T=0
EF E
 specific heat of the degenerate electron gas, estimate

T ~ 300 K for typical metallic densities

1  U   u  T=0
specific heat cv     
V  T V  T V
U – thermal U
u
kinetic energy V f(E) at T ≠ 0 differs from f(E) at T=0
1 3 only in a region of order kBT about m
classical gas u  n mv 2  nk BT because electrons just below
2 2 EF have been excited to levels just above EF
3
cv  nk B the observed electronic
2
contribution at room T is
usually 0.01 of this value

classical gas: with increasing T all electron gain an energy ~ kBT

Fermi gas: with increasing T only those electrons in states within
an energy range kBT of the Fermi level gain an energy ~ kBT
k T
number of electrons which gain energy with increasing temperature ~ N B
EF
 k BT 
the total electronic thermal kinetic energy U ~  N  k BT
 E F 
EF/kB ~ 104 – 105 K
1 U  k BT
the electronic specific heat cv    ~ nk B
kBTroom / EF ~ 0.01
V  T V EF
 specific heat of the degenerate electron gas

dk
u   3 E (k ) f E (k )    dED( E ) Ef ( E )
4 0  u 
 m and u  c v   
dk  T V
n   3 f E (k )    dED( E ) f ( E )
4 0
 EF
the way in which integrals of the form - H ( E ) f ( E )dE differ from their zero T values - H ( E )dE
is determined by the form of H(E) near E=m 
( E - m )n d n
replace H(E) by its Taylor expansion about E=m H (E)   n
H ( E ) E m
n 0 n! dE
 m 
d 2 n -1
the Sommerfeld expansion
 H ( E ) f ( E )dE   H ( E )dE   k T 
2n
B an 2 n -1
H ( E ) E m
- - n 1 dE
m 6
2 7 4
  H (E )E  k BT  H ( m ) 
2
k BT 4 H ( m )  O k BT 
-
6 360  m 
successive terms are m
2
smaller by O(kBT/m)2 u   ED( E )dE  k BT 2 mD( m )  D( m )  O(T 4 )
0
6
m
2
for kBT/m << 1 n   D( E )dE  k BT 2 D( m )  O(T 4 ) replace m
6
0
by mT0 = EF
m EF

 H ( E )dE   H ( E )dE  ( m - E
0 0
F ) H ( EF ) correctly to order T2
 specific heat of the degenerate electron gas

 1   k T 2
m  E F 1 -  B  
 3  2 E F  
2
k BT 
2
u  u0  D (E F )
6
u  2 2
cv   k B TD (E F )
1  2m 
32 T 3
D( E )  E
 
2 2  2  3 n
D( EF ) 
EF 
2
 3 2 n 
23 2 EF
2m
 2 k BT
cv  nk B
2 EF
3
(1) cclassical  nk B
2
FD statistics depress  k BT
2

cv by a factor of 3 EF
(2) cv  T
 thermal conductivity
thermal current density jq – a vector parallel to the direction of heat flow
whose magnitude gives the thermal energy per unit time
j  -T
q
crossing a unite area perpendicular to the flow
1 1
  v 2cv  lvcv
3 3
2
ne2  cv mv 2 3  k B  Wiedemann-Franz law (1853)
s    T
m s 3ne 2 2 e  Lorenz number ~ 2×10-8 watt-ohm/K2
Drude: 3
cv  nk B success of the Drude model is due to the cancellation
application of 2 of two errors: at room T the actual electronic cv is 100
classical ideal 1 2 3 times smaller than the classical prediction, but v is 100
gas laws mv  k BT times larger
2 2

for  2 k BT
degenerate
the correct cv  nk B cv cv -classical ~ k BT / EF ~ 0.01 at room T
2 EF
Fermi gas of 2 2
the correct estimate of v2 is vF2 v v classical ~ E F / k BT ~ 100
at room T
electrons F

2
  2  kB 
  
sT 3  e 
 thermopower
Seebeck effect: a T gradient in a long, thin bar should be accompanied by an electric field
directed opposite to the T gradient
high T low T E  QT thermoelectric field

cv
gradT thermopower Q-
3ne
E
Drude:
application of 3 kB
cv  nk B Q-
classical ideal 2 2e
gas laws

for  2 k BT
degenerate the correct cv  nk B cv cv -classical ~ k BT / EF ~ 0.01 at room T
2 EF
Fermi gas of
electrons Q/Qclassical ~ 0.01 at room T

 2 k B  k BT 
Q-  
6 e  EF 
 Electrical conductivity and Ohm’s law
equation of motion dv dk
Newton’s law m   - eE dp(t ) p( t )
dt dt -  f (t )  0
in the absence of collisions the Fermi sphere in dt 
k-space is displaced as a whole at a uniform rate k (t ) - k (0)  - eE t p  f  -eE
by a constant applied electric field 
because of collisions the displaced Fermi sphere eE
is maintained in a steady state in an electric field k avg  - 

ky Ohm’s law
F k avg eE
v avg  -   ne 2 
Fermi sphere m m j   E
 m 
kx
j  -nev avg
ne 2
s
kavg m
1 m
r  2
s ne 
the mean free path l = vF
because all collisions involve only electrons near the Fermi surface
vF ~ 108 cm/s
for pure Cu:
at T=300 K  ~ 10-14 sl ~ 10-6 cm = 100 Å
at T=4 K  ~ 10-9 s l ~ 0.1 cm
kavg << kF for n = 1022 cm-3 and j = 1 A/mm2 vavg = j/ne ~ 0.1 cm/s << vF ~ 108 cm/s