Transcript Part VII

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
4rs
V 1
 
3
N n
3
1/ 3
 3 
rs  

 4n 
~
1
n1 / 3
mean inter-electron spacing
in metals rs ~ 1 – 3 Å (1 Å= 10-8 cm)
rs/a0 ~ 2 – 6
2
a0 
 0.529 Å – Bohr radius
me2
● 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=I/A – the current density
r – the resistivity
R=rL/A – the resistance
s = 1/r - the conductivity
j  sE
L
A
j  - env
v is the average electron velocity
eE
v- 
m
j  sE
 ne2 
E
j  
 m 
ne2
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
2
estimate for v0 at Drude’s time 1 2 mv0  3 2 kBT → 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
momentum
equation of motion
for the momentum per electron
dp ( t )
p( t )
 f (t )
dt

average
velocity
p ( t )  mv ( t )
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
 


scattered
part
fraction of electrons that
does not experience scattering
p(t  dt )  p(t )  f (t )dt -
p( t )

total loss of momentum
after scattering
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:
magnetoresistance
R ( H )  Rxx 
(transverse magnetoresistance)
Hall (off-diagonal) resistance R yx 
Vy
Ix
Vx
Ix
resistivity
r ( H )  r xx 
Hall resistivity
r yx 
Ey
the Hall coefficient RH 
Ey
RH → measurement of the sign of the carrier charge
RH is positive for positive charges and negative for negative charges
jx
jx H
Ex
jx
1


f  - e E  v  H 
c


dp
1

 p
equation of motion
 - e E 
p  H mc
for the momentum per electron dt

 
p
0  - eE x - c p y - x
in the steady state px and py

satisfy
p
0  - eE y  c p x - y
force acting on electron

c 
eH
mc
cyclotron frequency
frequency of revolution
of a free electron in the
magnetic field H
e
2
mc r  c r H
c
multiply by - ne / m
the Drude
model DC
conductivity
at H=0
j  - ne
ne 
m
jy  0
s0 
jx  s 0 E x
p
m
2
s 0 E x  cj y  jx
s 0 E y  -cjx  j y
  
 H 
E y  - c  jx  -
 jx
s
nec


 0 
the resistance does not
depend on H
RH  -
1
nec
c 
c
~ 1GHz
2
at H = 0.1 T
RH → measurement of the density
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
c  1
j is at a small angle f to E
f is the Hall angle
tan f  c
measurable quantity – Hall resistance r H  RH H
Vy
H
rH   Ix
n2 D ec
E  rj
j  sE
in the presence of magnetic field the
resistivity and conductivity becomes tensors
 r xx r xy 
 E x   r xx r xy  jx 

for 2D: r  
E   r
 j 
r
r
r
yx
yy
y
yx
yy


  
 y 
s 0 E x  cj y  jx
s 0 E y  -cjx  j y
Ex 
1
jx 
c
jy
s0
s0

1
E y  - c jx 
jy
s0
s0
c s 0 
 1 s0

r  


s
1
s
0
0 
 c
c s 0   jx 
 Ex   1 s 0
E   
 j 


s
1
s
y
0
0  y 
   c
1
m
r xx 
 2
s 0 ne 

H
r xy  c 
s 0 nec
for 3D systems n2 D  nLz
for 2D systems n2D=n
 jx   s xx s xy  Ex 
 j   s
 E 
s
yy  y 
 y   yx
 s xx s xy   r xx
  
s  
s
s
yy 
 yx
 r yx
r xy 

r yy 
s0
1  (c ) 2
- s 0c
s xy  -s yx 
1  (c ) 2
s xx  s yy 
s xx 
r xx
r xx 2  r xy 2
s xy  -
r xy
r xx 2  r xy 2
-1
H
nec
the Drude
model
Hall resistance r H  RH H  -
1
m
weak
r xx 
 2
s 0 ne 
magnetic
fields
c
H
r


c  1
xy
s 0 nec
the classical
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
both in metals and dielectric
historically
used mainly for
electric current j
conductivity s  j/E
polarization P
polarizability c  P/E
dielectrics
dielectric function e 14c
electric field leads to
dr
j   qi i
dt
i
P   qiri
i
e ( )  1  4
mainly
described by
P( )
E ( )
E( , t )  E( )e - it
dP
 -iω P
dt
P 1 j
σ
χ 
i
E E -iω
ω
4π
ε(ω)  1  i σ(ω)
ω
j
D  E  4P  εE
e(,0) describes the collective
div D  4rext
excitations of the electron
gas – the plasmons
e(0,k) describes the electrostatic
screening
div E  4r  4 ( r ext  rind )
metals
j( , t ) ~ j( )e - it
P( , t ) ~ P( )e - it
AC electrical conductivity of a metal
dp( , t )
p( , t )
- eE( , t )
dt

E( , t )  E( )e -it
equation of motion
for the momentum per electron
time-dependent electric field
p( , t )  p( )e -it
seek a steady-state solution in the form
eE( )
1 /  - i
nep( ) ( ne 2 / m )E( )
j( )  
m
1 /  - i
p( ) 
j( )  s ( )E( )
s0
s ( ) 
1 - i
4
e ( )  1  i s ( )

s0
1 - i
ne 2
AC conductivity
s ( ) 
DC conductivity
s0 
  1
e ( )  1 -
4ne
m 2
2
p
2
Re s ( ) 
s0
1   2 2
m
4ne2

m
the plasma frequency
p
e ( )  1 - 2

2
a plasma is a medium with positive
and negative charges, of which at
least one charge type is mobile
even more simplified:
no electron collisions (no frictional damping term,   1)
equation of motion of a free electron
if x and E have the time dependence e-iwt
the polarization as the dipole moment per unit volume
d 2x
m 2  -eE
dt
eE
x
m 2
ne2
E
P  -exn  2
m
4 ne 2
P( )
 1e ( )  1  4
m 2
E ( )
p
2
4 ne 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
look for a solution with
dispersion relation for
electromagnetic waves
e (, K ) 2E / t 2  c 22E
E  exp( -it  iK  r )
e (, K ) 2  c 2 K 2
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
-Kr
with a characteristic length 1/|K|: E  e
(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
Dispersion relation for
electromagnetic waves
p
e (, K ) 2  c2 K 2
(1) For  > p → K2 > 0, K is real,
waves with  > p propagate in the media
with the dispersion relation  2   p 2  c 2 K 2
an electron gas is transparent
e ( )
(2) for  < p → K2 < 0, K is imaginary, E  e
waves with  < p incident on the medium
do not propagate, but are totally reflected
-Kr
vgroup  d dK  c
/p
 = cK
4ne2

m
p2
e ( )  1 - 2

 2 - p 2  c2 K 2
v ph   K  c
2
Metals are shiny
due to the reflection
of light
/p
(2)
(1)
E&M waves propagate
with no damping when
e is positive & real
E&M waves are totally reflected
from the medium when e is negative
forbidden
frequency gap
cK/p
vph > c → vph
This does not correspond to the velocity of
the propagation of any quantity!!
Ultraviolet transparency of metals
plasma frequency p and free space wavelength lp = 2c/p
range
metals
semiconductors
ionosphere
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
spectral range
UV
IF
radio
electron gas is transparent when  > p i.e. l < lp
p
2
4ne2

m
p2
e ( )  1 - 2

the reflection of
light from a metal
is similar to the
reflection of radio
waves from the
ionosphere
plasma frequency
ionosphere
semiconductors
metals
metal
ionosphere
reflects
visible
radio
transparent for
UV
visible
Skin effect
when  < p electromagnetic wave is reflected
the wave is damped with a characteristic length d = 1/|K|: E  e-r d  e- K r
the wave penetration – the skin effect
the penetration depth d – the skin depth
2
  4
K  2 e  2 1  i
si 2
s
c
c 
 
c 
2
2
2 
4
12

2s
K
(1  i )
d cl 
c
2s1 2
the classical skin depth
c
 2s1 2
E  exp( -it  iKr )  exp  c


r 

d >> l – the classical skin effect
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
d'
c
c

2s '  1 2  d ' 1 2
 2 s 
l


13
 lc 

2
s


d '  
2
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
motion
 2d
Nm 2  - NeE  - Ne( 4nde)
t
 2d
oscillation at the
2


d

0
p
plasma frequency
t 2
for longitudinal plasma oscillation L=p
/p
transverse
electromagnetic waves
forbidden
frequency gap
cK/p
longitudinal plasma
oscillations
p
2
4ne2

m
L 2
e L   1 - 2  0
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
jq  -T
Fourier’s law
 – thermal conductivity
the thermal energy per electron
n
n
vET (T [ x - v ]) - vET (T [ x  v ])
2
2
dE  dT 
j q  nv 2 T  
dT  dx 
jq  -T
to
1 2
2
2
2
3D: v x  v y  v z  v
high T
low T
1 2
1
3


v

c

lvcv
v
after each collision an electron emerges
3
3
dE
T
with a speed appropriate to the local T
n
 cv
dT
→ electrons moving along the T
gradient are less energetic
the electronic specific heat
1D: j q 
ne2
s
m
 cv mv 2 3  k B 

  T
2
s
3ne
2 e 
Drude:
application of
classical ideal
gas laws
cv 
2
3
nk B
2
1 2 3
mv  k BT
2
2
Wiedemann-Franz law (1853)
Lorenz number ~ 2×10-8 watt-ohm/K2
success of the Drude model is due to the cancellation
of two errors: at room T the actual electronic cv is 100
times smaller than the classical prediction, but v2 is 100
times larger
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
 dv 2
2
dx
dx  2 
vQ  T
to
6 dT
1 2
2
2
2
3D: v x  v y  vz  v
3
eE
mean electronic velocity due to electric field: v E  m
2
d
mv
2

1
c
1 mdv 2
 
- v
T  Q  -  
in equilibrium vQ + vE = 0 → E  dT
3en
6e dT
 3e 
Drude:
no cancellation of two errors:
kB
3
application of
observed metallic thermopowers at room T are
Qcv  nk B
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
the Drude model: electronic velocity distribution
is given by the classical
Maxwell-Boltzmann distribution
 m 
f MB ( v )  n 

2

k
T
B


the Sommerfeld model: electronic velocity distribution f
FD
is given by the quantum
Fermi-Dirac distribution
m / 
( v) 
4 3
3/ 2
 mv 2 
exp  
2
k
T
B


3
1
1 2

mv
k
T
B
0


exp  2
1

k BT




Pauli exclusion principle: at most one electron
can occupy any single electron level
n   dv f ( v )
normalization
condition
T0
consider noninteracting electrons
electron wave function
associated with a level of energy E
satisfies the Schrodinger equation
2   2
2
2 
 2  2  2  ( r )  E ( r )
2m  x
y
z 
  x, y , z  L     x, y , z 
periodic
boundary
conditions
  x, y  L, z     x, y , z 
  x  L, y , z     x, y , z 
a solution neglecting
the boundary conditions
 k (r ) 
1 ikr
e
V
3D:
L
1D:
normalization constant: probability
2
of finding the electron somewhere 1  dr  (r )
in the whole volume V is unity

energy
momentum
velocity
wave vector
de Broglie
wavelength
 2k 2
E (k ) 
2m
p  k
k
v
m
k
2
l
k
p2 1 2
E
 mv
2m 2
  x, y , z  L     x, y , z 
 k (r ) 
1 ikr
e
V
  x, y  L, z     x, y , z 
  x  L, y , z     x, y , z 
apply the boundary conditions eik x L  e
components of k must be
 eik z L  1
2
2
2
kx 
nx , k y 
ny , kz 
nz
L
L
L
nx, ny, nz integers
a region of k-space of
volume  contains
the number of states
per unit volume of k-space,
k-space density of states
ik y L

V

3
2 / L  2 3
V
2 3
the area
per point
 2 
 
 L 
the volume
per point
 2  2 
  
V
 L 
states
i.e. allowed
values of k
k-space
2
3
3
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
density
of states
volume
state of the lowest energy
the number of allowed values of
k within the sphere of radius kF
 4 k F 3  V
kF 3
 2V


3
3
6

  2 
to accommodate N electrons
2 electrons per k-level due to spin
kF 3
N 2 2V
6
kF 3
n 2
3
~108 cm-1
Fermi wave vector k F
Fermi energy
E F   2 k F / 2m
2
p F  k F
Fermi velocity
compare to the
classical
thermal velocity
v F  k F / m
Fermi sphere
1/ 2
~108 cm/s
~ 107 cm/s at T=300K
0
kx
k F  3 2n 
13
~104-105 K
vthermal  3k BT / m 
kF
Fermi surface
at energy EF
~1-10 eV
Fermi temperature TF  E F k B
Fermi momentum
ky
at T=0
23
2

EF 
3 2n 
2m
13

vF  3 2n 
m
density of states
V 3
k
3 2
total number of states with wave vector < k
N
total number of states with energy < E
V  2mE 
N 2 2 
3   
the density of states – number of states per unit energy
dN
V  2m 
D( E ) 
 2 2 
dE 2   
2
k2
E
2m
32
32
dn
1  2m 
the density of states per unit volume or the density of states D ( E ) 



dE 2 2   2 
k-space density of states – the number of states per unit volume of k-space
E
32
V
2 3
E
Ground state energy of N electrons
2 2
add up the energies of all electron
E  2 k
states inside the Fermi sphere
2m
k k F
volume of k-space per state
k  8 3 V
 smooth F(k)
 F (k ) 
k
V
V
k 0 i . e.V 
F
(
k
)

k






F ( k )dk
3 
3 
8 k
8
5
the energy density
the energy per electron
in the ground state
E
1
 2k 2
1  2k F
 3  dk
 2
V 4 k k F
2m  10m
2
E
3  2k F
3

 EF
N 10 m
5
2k 2
F (k ) 
2m
dk  4k 2dk
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 
ia



,


e
 2 , 1 
1
2
of the system wavefunction by a phase factor only
 1, 2    2 , 1 
repeated particle exchange → e2ia  1
system of N=2 particles
1, 2 - coordinates and
spins for each of the
particles
antisymmetric wavefunction with respect
to the exchange of particles
1
 1, 2  
 p1 1  p2 2  - p1 2  p2 1 
2
symmetric wavefunction with respect
to the exchange of particles
1
 1, 2  
 p1 1  p2 2   p1 2  p2 1 
2
fermions are particles which have half-integer spin
the wavefunction which describes a collection
of fermions must be antisymmetric with respect
to the exchange of identical particles
bosons are particles which have integer spin
the wavefunction which describes a collection
of bosons must be symmetric with respect
to the exchange of identical particles
fermions: electron, proton, neutron
bosons: photon, Cooper pair, H atom, exciton




p1, p2 – single particle states
if p1 = p2   0
→ at most one fermion can occupy
any single particle state – Pauli principle
unlimited number of bosons can occupy
a single particle state
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
particles with
half-integer spins
Fermi-Dirac
distribution
function
f FD ( E ) 
1
E-m
exp 
 1
k
T
 B 
degenerate
Fermi gas
fFD(k) < 1
bosons
particles with
integer spins
Bose-Einstein
distribution
function
f BE ( E ) 
1
E-m
exp 
 -1
k
T
 B 
degenerate
Bose gas
fBE(k) can be any
both fermions and
bosons at high T
when E - m  kBT
Maxwell-Boltzmann
distribution
function
m-E
f MB ( E )  exp 

 k BT 
classical
gas
fMB(k) << 1
n   dEn( E )   dED ( E ) f ( E )
m=m(n,T) – chemical potential
remarks on statistics II
m v
BE and FD distributions differ from the classical MB distribution
because the particles they describe are indistinguishable.
Particles are considered to be indistinguishable if their wave packets
overlap significantly.
Two particles can be considered to be distinguishable
if their separation is large compared to their de Broglie wavelength.
(x)
vg=v
x
x
k
g(k’)
12
thermal de Broglie
wavelength
 2 
ldB  

mk
T
 B 
particles become
indistinguishable when
ldB ~ d  n
2
~
h
p
-1 3
2 2 2 3
n
i.e. at temperatures below TdB 
mk B
at T < TdB fBE and fFD are strongly different from fMB
at T >> TdB fBE ≈ fFD ≈ fMB
electron gas in metals:
n = 1022 cm-3, m = me → TdB ~ 3×104 K
gas of Rb atoms:
n = 1015 cm-3, matom = 105me → TdB ~ 5×10-6 K
excitons in GaAs QW
n = 1010 cm-2, mexciton= 0.2 me → TdB ~ 1 K
x
k’
k0
 
k '2  
 (r, t )   g ( k ') exp i  k ' r t 
2m  
k'
 
A particle is represented by a
wave group or wave packets
of limited spatial extent,
which is a superposition of many matter
waves with a spread of wavelengths
centered on l0=h/p
The wave group moves
with a speed vg – the group speed,
which is identical to the classical
particle speed
Heisenberg uncertainty principle, 1927:
If a measurement of position is made with
precision x and a simultaneous
measurement of momentum in the x
direction is made with precision px,
then
px x 
2
T≠0
the Fermi-Dirac distribution
3D
dn
1  2m 
density of states
D( E ) 
 2 2 
dE 2 

1
distribution function f ( E ) 
E-m
exp 
 1
k
T
 B 
n   dED ( E ) f ( E )  m
32
E
lim f ( E )  1, E  m
T 0
0 Em
lim m  EF
T 0
1
 [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
D ( E )dE 
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
1  U 
 u 





V  T V  T V
U
U – thermal
u
kinetic energy
V
specific heat
T ~ 300 K for typical metallic densities
T=0
cv 
f(E) at T ≠ 0 differs from f(E) at T=0
only in a region of order kBT about m
because electrons just below
EF have been excited to levels just above EF
1
3
classical gas u  n mv2  nk BT
2
2
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 
kBT
kBTroom / EF ~ 0.01
the electronic specific heat cv  
~
nk
B

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

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)  
H ( E ) E m
n
n
!
dE
n 0
EF
m

the Sommerfeld expansion

 H ( E ) f ( E )dE   H ( E )dE   k T 
-
n 1
-
m
2n
B
d 2 n -1
an
H ( E ) E m
2 n -1
dE
7 4
2
k BT  H ( m ) 
k BT 4 H ( m )  O  k BT 
  H (E )E 
6
360
 m 
-
successive terms are
smaller by O(kBT/m)2
2
m
u   ED( E )dE 
2
0
for kBT/m << 1
m
n   D ( E )dE 
0
m
6
2
6
k BT 2 mD( m )  D( m )  O (T 4 )
k BT 2 D( m )  O(T 4 )
EF
 H ( E )dE   H ( E )dE  ( m - E
0
0
F
) H ( EF )
replace m
by mT0 = EF
correctly to order T2
6
specific heat of the degenerate electron gas
 1   k T 2 
m  EF 1 -  B  
 3  2 E F  
u  u0 
1  2m 
D( E )  2  2 
2 

EF 
2
 3 n 
2m
2
23
32
E
 2 k BT
2 EF
cclassical 
FD statistics depress  k BT
cv by a factor of
3 EF
2
(2)
6
 k BT 
2
D( EF )
u  2 2
cv 

k B TD ( E F )
T
3
3 n
D( EF ) 
2 EF
cv 
(1)
2
cv  T
nk B
3
nk B
2
thermal conductivity
thermal current density jq – a vector parallel to the direction of heat flow
whose magnitude gives the thermal energy per unit time
q
j  -T
crossing a unite area perpendicular to the flow
1
1
  v 2cv  lvcv
3
3
ne2
s
m
 cv mv 2 3  k B 

  T
s
3ne 2
2 e 
Drude:
application of
classical ideal
gas laws
cv 
2
3
nk B
2
1 2 3
mv  k BT
2
2
Wiedemann-Franz law (1853)
Lorenz number ~ 2×10-8 watt-ohm/K2
success of the Drude model is due to the cancellation
of two errors: at room T the actual electronic cv is 100
times smaller than the classical prediction, but v is 100
times larger
 2 k BT
for
cv 
nk B cv cv -classical ~ k BT / EF ~ 0.01 at room T
the correct
degenerate
2 EF
Fermi gas of
2
2
the correct estimate of v2 is vF2
at room T
v
v
F
classical ~ EF / kBT ~ 100
electrons
  2  kB 

 
sT
3  e 
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
E  QT
high T
low T
thermoelectric field
gradT
E
Drude:
application of
classical ideal
gas laws
for
degenerate
Fermi gas of
electrons
the correct
thermopower
cv 
3
nk B
2
cv 
 2 k BT
2 EF
Q-
cv
3ne
Q-
kB
2e
nk B
cv cv -classical ~ k BT / EF ~ 0.01 at room T
Q/Qclassical ~ 0.01 at room T
Q-
 2 k B  k BT 


6 e  EF 
Electrical conductivity and Ohm’s law
equation of motion
dv
dk
Newton’s law
m

 - eE
dt
dt
in the absence of collisions the Fermi sphere in
k-space is displaced as a whole at a uniform rate k (t ) - k (0)  - eE t
by a constant applied electric field
eE
because of collisions the displaced Fermi sphere
k


avg
is maintained in a steady state in an electric field
ky
k avg
eE
F
v avg 
- 
Fermi sphere
m
m
kx
kavg
j  - nev avg
dp(t )
p( t )
 f (t )  0
dt

p  f  - eE
Ohm’s law
 ne2 
E
j  
 m 
ne2
s
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 s
l ~ 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