Post on 11-Jan-2016
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The free electron theory of metalsThe Drude theory of metals
Paul Drude (1900): theory of electrical and thermal conduction in a metalapplication 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
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
in a metal
Doping of Semiconductors
C, Si, Ge, are valence IV , Diamond fcc structure. Valence band is fullSubstitute a Si (Ge) with P. One extra electron donated to conduction bandN-type semiconductor
density of conduction electrons in metals ~1022 – 1023 cm-3
rs – measure of electronic densityrs is radius of a sphere whose volume is equal to the volume per electron
nN
Vrs 1
3
4 3
3/1
3/11
~4
3
nnrs
mean inter-electron spacing
in metals rs ~ 1 – 3 Å (1 Å= 10-8 cm) rs/a0 ~ 2 – 6
529.02
2
0 me
a
Å – Bohr radius
● 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 electronwith the other electrons is neglectedand with the ions is neglected
2. collisions are instantaneous eventsDrude 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/τdt/τ – probability to undergo a collision within small time dtrandomly 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 timeτ is independent of an electron position and velocity
4. after each collision an electron emerges with a velocity that is randomly directed andwith a speed appropriate to the local temperature
independent electron approximationfree electron approximation
DC electrical conductivity of a metal
V = RI Ohm’s lowthe Drude model provides an estimate for the resistance
introduce characteristics of the metal which are independent on the shape of the wire
j=I/A – the current density – the resistivityR=L/A – the resistance= 1/ the conductivity
v is the average electron velocity
enj v
e
m
Ev Ej
m
ne 2
jE Ej
m
ne 2
Ej
L
A
at room temperatures resistivities of metals are typically of the order of microohm centimeters (ohm-cm)and is typically 10-14 – 10-15 s
2ne
m
mean free path l=v0v0 – the average electron speed l measures the average distance an electron travels between collisionsestimate for v0 at Drude’s time → v0~107 cm/s → l ~ 1 – 10 Åconsistent with Drude’s view that collisions are due to electron bumping into ions
at low temperatures very long mean free path can be achievedl > 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 fieldand in spatially uniform time-dependent electric field
Very disordered metals and semiconductors
Tkmv B2321 20
motion under the influence of the force f(t) due to spatially uniform electric and/or magnetic fields
equation of motionfor the momentum per electron
electron collisions introduce a frictional damping term for the momentum per electron
)()()(
tt
dt
tdf
pp
Derivation:
2
( ) 1 ( ) ( ) 0
( )( ) ( ) ( )
( ) ( ) ( )( )
( ) ( )( )
dt dtt dt t t dt
tt dt t t dt dt O dt
t dt t tt O dt
dtd t t
tdt
p p f
pp p f
p p pf
p pf
fraction of electrons that does not experience scattering
scatteredpart
total loss of momentumafter scattering
( ) ( )t m tp v
average momentum
average velocity
Hall effect and magnetoresistanceEdwin Herbert Hall (1879): discovery of the Hall effect
HvF c
eLthe Lorentz force
in equilibrium jy = 0 → the transverse field (the Hall field) Ey due to the accumulated charges balances the Lorentz force
quantities of interest:magnetoresistance (transverse magnetoresistance)
Hall (off-diagonal) resistance
( ) xxx
x
yyx
x
yH
x
EH
j
E
j
ER
j H
RH → measurement of the sign of the carrier charge RH is positive for positive charges and negative for negative charges
the Hall effect is the electric field developed across twofaces of a conductorin the direction j×Hwhen a current j flows across a magnetic field H
( ) xxx
x
yyx
x
VR H R
I
VR
I
resistivity
Hall resistivity
the Hall coefficient
mc
eH
ppeE
ppeE
mce
dt
d
ce
c
yxcy
xycx
0
0
1
1
pHpE
p
HvEfforce acting on electron
equation of motionfor the momentum per electron
in the steady state px and py satisfy
cyclotron frequencyfrequency of revolutionof a free electron in the magnetic field H
at H = 0.1 T0yj xx
cy j
nec
HjE
0
multiply by
yxcy
xycx
jjE
jjE
0
0
m
ne
m
pnej
mne
2
0
/
the Drudemodel DCconductivityat H=0
Hrc
erm cc 2
weak magnetic fields – electrons can complete only a small part of revolution between collisionsstrong magnetic fields – electrons can complete many revolutions between collisions1
1
c
c
1cj is at a small angle to E is the Hall angle tan c
RH → measurement of the density nec
RH
1xx Ej 0 the resistance does not
depend on H
GHzcc 1~
2
measurable quantity – Hall resistance HRHH
ecn
H
I
V
Dx
yH
2
zD nLn 2for 3D systems for 2D systems n2D=n
jE Ej in the presence of magnetic field the resistivity and conductivity becomes tensors
yxcy
xycx
jjE
jjE
0
0
yyyx
xyxx
0 0
0 0
1
1
cx x y
cy x y
E j j
E j j
for 2D:
00
00
1
1
c
c
nec
H
ne
m
cxy
xx
0
20
1
22
22
xyxx
xyxy
xyxx
xxxx
20
20
)(1
)(1
c
cyxxy
cyyxx
1
yyyx
xyxx
yyyx
xyxx
x xx xy x
y yx yy y
E j
E j
0 0
0 0
1
1x xc
y yc
E j
E j
x xx xy x
y yx yy y
j E
j E
from D.C. Tsui, RMP (1999) and from H.L. Stormer, RMP (1999)
nec
H
ne
m
cxy
xx
0
20
1
nec
HHRHH Hall resistance
strong magnetic fieldsquantization of Hall resistance
at integer and fractional
the integer quantum Hall effectand the fractional quantum Hall effect
2e
hxy
hceH
n
the Drude model
the classicalHall effect
1c
1c
weak magnetic fields
consider the case >> l wavelength of the field is large compared to the electronic mean free pathelectrons “see” homogeneous field when moving between collisions
AC electrical conductivity of a metalApplication to the propagation of electromagnetic radiation in a metal
electric field leads toelectric current j conductivity j/E metals
polarization P polarizability P/E dielectricsdielectric function
response to electric field mainly historicallyboth in metals and dielectric described by used mainly for
ii
i
i
ii
q
dt
dq
rP
rj
EPED ε 4
)(44div
4div
indext
ext
E
D
(,0) describes the collective excitations of the electron gas – the plasmons (0,k) describes the electrostatic screening
1
41
dPj iω
dtP j σ
χ iE E iω ω
πε(ω) i σ(ω)
ω
)(
)(41)(
E
P
P
( , ) ( )
( , ) ~ ( )
( , ) ~ ( )
i t
i t
i t
t e
t e
t e
E E
j j
P P
AC electrical conductivity of a metal
i
mne
m
ne
i
e
et
et
tet
dt
td
ti
ti
/1
)()/()()(
/1
)()(
)(),(
)(),(
),(),(),(
2 Epj
Ep
pp
EE
Eppequation of motion
for the momentum per electron
seek a steady-state solution in the form
time-dependent electric field
m
ne
i
2
0
0
1)(
)()()(
Ej
AC conductivity
DC conductivity
)(4
1)( i
2
2
22
1)(
4
p
p m
ne
i
1
)( 0
2
241)(
m
ne
the plasma frequencya plasma is a medium with positive and negative charges, of which at least one charge type is mobile
1
02 2
Re ( )1
even more simplified:
2
2
2
2
2
2
2
22
2
2
( ) 4( ) 1 4 1
( )
4
( ) 1
p
p
d xm eE
dteE
xm
neP exn E
m
P ne
E m
ne
m
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
no electron collisions (no frictional damping term, )1
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 forelectromagnetic waves
222
2222
),(
)exp(
/),(
Kc
iti
ct
K
rKE
EEK
real and > 0 → for is real, K is real and the transverse electromagnetic wave propagates with the phase velocity vph= c/
real and < 0 → for is real, K is imaginaryand the wave is damped with a characteristic length 1/|K|:
(3) complex → for is real, K is complexand the waves are damped in space
(4) = → The system has a final response in the absence of an applied force (at E=0); the polesof (,K) define the frequencies of the free oscillations of the medium
(5) = 0 longitudinally polarized waves are possible
rKeE
Application to the propagation of electromagnetic radiation in a metal
(1) for > p → K2 > 0, K is real,
waves with > p propagate in the media
with the dispersion relationan electron gas is transparent
(2) for < p → K2 < 0, K is imaginary,
waves with < p incident on the medium
do not propagate, but are totally reflected
Transverse optical modes in a plasma222),( Kc K
dispersion relation forelectromagnetic waves
2
2
22
1)(
4
p
p m
ne
2222 Kcp
rKeE
metals are shiny due to the reflection of light
/p = cK
cK/p
cKv
cdKdv
ph
group
forbidden frequency gap
vph > c → vph does not correspond to the velocity of the real physical propagation of any quantity
(2) (1)
/p
electromagnetic waves are totally reflected from the
medium when is negative
electromagnetic waves propagate without dampingwhen is positive and real
( ) 2222 Kcp
Ultraviolet transparency of metals
2
2
22
1)(
4
p
p m
ne
plasma frequency p and free space wavelength p = 2c/p
range metals semiconductors ionospheren, cm-3 1022 1018 1010
p, Hz 5.7×1015 5.7×1013 5.7×109
p, cm 3.3×10-5 3.3×10-3 33 spectral range UV IF radio
electron gas is transparent when > p i.e. < p
plasma frequencyionospheresemiconductorsmetals
the reflection of light from a metal is similar to the reflection of radio waves from the ionosphere
reflects transparent formetal visible UVionosphere radio visible
Skin effectwhen < p electromagnetic wave is reflectedthe wave is damped with a characteristic length = 1/|K|:the wave penetration – the skin effectthe penetration depth – the skin depth
rKr eeE
rc
iKrtiE
ic
K
cii
ccK
21
21
2
2
2
2
2
22
2exp)exp(
)1(2
441
the classical skin depth
212 c
cl
>> l – the classical skin effect
<< 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 waveonly electrons that are running inside the skin depth for most of the mean free path l are capable of picking up much energy from the electric fieldonly a fraction of the electrons ’/l are effective in the conductivity
’ l
2121'
2'2
'
l
cc312
2'
lc
Longitudinal plasma oscillations
0
)4(
2
2
2
2
2
dt
d
ndeNeNeEt
dNm
p
a charge density oscillation, ora longitudinal plasma oscillation, or plasmon
nature of plasma oscillations:displacement of entire electron gas through d with respect to positive ion background createssurface charges = nde and an electric field E = 4nde
m
nep
22 4 oscillation at the
plasma frequency
equation ofmotion
01 2
2
p
LL
/p
cK/p
for longitudinal plasma oscillation L=p
transverse electromagnetic waves
longitudinal plasma oscillations
forbidden frequency gap
Thermal conductivity of a metalassumption 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
Tq j Fourier’s law – thermal conductivity
high T low Tafter each collision an electron emergeswith a speed appropriate to the local T→ electrons moving along the T gradient are less energetic
the thermal energy per electron
vT
zyx
Tq
TTq
cdT
dEn
vvvv
dx
dT
dT
dEnvj
vxTvEn
vxTvEn
j
2222
2
3
1
])[(2
])[(2
the electronic specific heat
1D:
to3D:
vv
q
lvccv
T
3
1
3
1 2
j
Tkmv
nkc
Te
k
ne
mvc
B
Bv
Bv
2
3
2
12
3
2
3
3
2
2
2
2
Wiedemann-Franz law (1853)Lorenz number ~ 2×10-8 watt-ohm/K2
Drude:application of classical ideal gas laws
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
m
ne 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
thermoelectric fieldhigh T low T
gradTE
TQE
thermopower
2222
2
3
1
2)]()([
2
1
vvvv
v
dx
d
dx
dvvvxvvxvv
zyx
Q
1D:
to3D:
TdT
dvQ
2
6
v
mean electronic velocity due to T gradient:
mean electronic velocity due to electric field:m
eE
Ev
in equilibrium vQ + vE = 0 → 2 21
3 3v
d mv cQ
e dT en
Drude:application of classical ideal gas laws
Bv nkc2
3
e
kQ B
2
no cancellation of two errors: observed metallic thermopowers at room T are100 times smaller than the classical prediction
inadequacy of classical statistical mechanics in describing the metallic electron gas
21
6
mdvT
e dT E
The Sommerfeld theory of metals
the Drude model: electronic velocity distribution is given by the classical Maxwell-Boltzmann distribution
the Sommerfeld model: electronic velocity distribution is given by the quantum Fermi-Dirac distribution
3/ 2 2
3
32
0
( ) exp2 2
/ 1( )
142exp 1
( )
MBB B
FD
B
B
m mvf n
k T k T
mf
mv k T
k T
n d f
v
v
v v
normalizationcondition T0
Pauli exclusion principle: at most one electron can occupy any single electron level
)()(2 2
2
2
2
2
22
rErzyxm
, , , ,
, , , ,
, , , ,
x y z L x y z
x y L z x y z
x L y z x y z
k
m
m
kE
d
eV
i
2
2)(
)(1
1)(
22
2
k
kv
kp
k
rr
r rkk
22
2
1
2mv
m
pE
electron wave functionassociated with a level of energy Esatisfies the Schrodinger equation
consider noninteracting electrons
L
3D:
1D:
periodicboundary conditions
a solution neglecting the boundary conditions
normalization constant: probability of finding the electron somewhere in the whole volume V is unity
energy
momentum
velocity
wave vector
de Broglie wavelength
1
2 2 2, ,
yx zik Lik L ik L
x x y y z z
e e e
k n k n k nL L L
rkk r ie
V
1)(
, , , ,
, , , ,
, , , ,
x y z L x y z
x y L z x y z
x L y z x y z
apply the boundary conditions
components of k must be
nx, ny, nz integers
3
33
2
2/2
V
V
L
a region of k-space of volume contains
states i.e. allowed values of kthe number of states
per unit volume of k-space, k-space density of states
VL
L33
2
22
2
the area
per point
the volume per point
k-space
compare to the ~ 107 cm/s at T=300Kclassicalthermal velocity 0 at T=0
2/1
22
/3
/
2/
mTkv
mkv
kp
kET
mkE
k
Bthermal
FF
FF
BFF
FF
F
Fermi wave vector ~108 cm-1
Fermi energy ~1-10 eV
Fermi temperature ~104-105 K
Fermi momentum
Fermi velocity ~108 cm/s
consider T=0
3 3
3 2
3
2
3
2
4
3 62
26
3
F F
F
F
k V kV
kN V
kn
the number of allowed values of k within the sphere of radius kF
to accommodate N electrons2 electrons per k-level due to spin
the ground state of the N-electron system is formed by occupying all single-particle levels with k < kF
312
3222
312
3
32
3
nm
v
nm
E
nk
F
F
F
kx
ky
kFFermi sphere
Fermi surfaceat energy EF
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
state of the lowest energy density of states
volume
total number of states with energy < E
the density of states – number of states per unit energy EmV
dE
dNED
mEVN
23
22
23
22
2
2)(
2
3
density of states
32V
k-space density of states – the number of states per unit volume of k-space
the density of states per unit volume or the density of states Em
dE
dnED
23
22
2
2
1)(
323
VN k
total number of states with wave vector < k 2 2
2
kE
m
FF
F
kk
Vei
kk
Em
k
N
E
m
k
m
kd
V
E
dFV
FV
F
V
km
E
F
F
5
3
10
3
10
1
24
1
)(8
)(8
)(
8
22
22
52
2
22
3
3..0
3
3
22
k
kkkkk
k
k
kk
dkkd
m
kF
2
22
4
2)(
k
k
3
23Fk
N V
Ground state energy of N electrons
add up the energies of all electron states inside the Fermi sphere
volume of k-space per state
smooth F(k)
the energy density
the energy per electronin the ground state
in quantum mechanics particles are indistinguishable
systems where particles are exchanged are identical
exchange of identical particles can lead to changing of the system wavefunction by a phase factor only
repeated particle exchange → e2i
1221 ,, ie
1221 ,,
122121 21212
1, pppp 122121 21212
1, pppp
antisymmetric wavefunction with respect to the exchange of particles
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
fermions: electron, proton, neutron
if p1 = p2 → at most one fermion can occupy any single particle state – Pauli principle
unlimited number of bosons can occupy a single particle state
p1, p2 – single particle states
obey Fermi-Dirac statistics obey Bose-Einstein statistics
system of N=2 particles - coordinates and spins for each of the particles
remarks on statistics I
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
bosons: photon, Cooper pair, H atom, exciton
symmetric wavefunction with respect to the exchange of particles
1( )
exp 1
1( )
exp 1
( ) exp
( ) ( ) ( )
FD
B
BE
B
MBB
f EEk T
f EEk T
Ef E
k T
n dEn E dED E f E
Fermi-Diracdistribution function
Bose-Einsteindistribution function
Maxwell-Boltzmanndistribution function
distribution function f(E) → probability that a state at energy E will be occupied at thermal equilibrium
fermionsparticles with half-integer spins
bosonsparticles with integer spins
both fermions and bosons at high T when TkE B
degenerate Fermi gasfFD(k) < 1
degenerate Bose gasfBE(k) can be any
classicalgasfMB(k) << 1
=(n,T) – chemical potential
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.
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 QWn = 1010 cm-2, mexciton= 0.2 me → TdB ~ 1 K
at T < TdB fBE and fFD are strongly different from fMB
at T >> TdB fBE ≈ fFD ≈ fMB
1 22
1 3
22 3
2~
~
2
dBB
dB
dBB
h
mk T p
d n
T nmk
thermal de Broglie wavelength
particles become indistinguishable when
i.e. at temperatures below
remarks on statistics II
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 0=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
2xp x
g(k’)
k0
kk’
m v
vg=v
x
x
(x)
2
'
'( , ) ( ') exp '
2
kt g i t
m
k
r k k r
x
3 2
2 2
1 2( )
2
dn mD E E
dE
density of states
distribution function1
( )
exp 1
( ) ( )
B
f EEk T
n dED E f E
3D
T≠0 the Fermi-Dirac distribution
E
EEfT
0
,1)(lim0
FT
E
0
lim
VdEEfED
VdEED
1)()(
1)( [the number of states in the energy range from E to E + dE]
[the number of filled states in the energy range from E to E + dE]
EF E
density of filled statesD(E)f(E,T)
shaded area – filled states at T=0
density of states D(E)
per unit volume
specific heat of the degenerate electron gas, estimate
Bv
B
nkc
Tnkmvnu
2
32
3
2
1 2
1v
V V
U uc
V T T
Uu
V
specific heat
U – thermal kinetic energy
classical gas
the observed electronic 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
number of electrons which gain energy with increasing temperature ~
the total electronic thermal kinetic energy
the electronic specific heat1
~ Bv B
V F
U k Tc nk
V T E
EF/kB ~ 104 – 105 KkBTroom / EF ~ 0.01
f(E) at T ≠ 0 differs from f(E) at T=0only in a region of order kBT about because electrons just below EF have been excited to levels just above EF
T ~ 300 K for typical metallic densities T = 0
B
F
k TN
E~ B
BF
k TU N k T
E
03
03
)()()(4
)()()()(4
EfEdEDEfd
n
EEfEdEDEfEd
u
kk
kkk
)()()()(
)()(6
)(
)()()(6
)(
00
422
0
422
0
FF
E
B
B
EHEdEEHdEEH
TODTkdEEDn
TODDTkdEEEDu
F
correctly to order T2
specific heat of the degenerate electron gas
the way in which integrals of the form differ from their zero T values is determined by the form of H(E) near E=
E
nn
nn
EHdE
d
n
EEH )(
!
)()(
0
replace H(E) by its Taylor expansion about E=
the Sommerfeld expansion
6
44
22
12
12
1
2
)(360
7)(
6)(
)()()()(
TkOHTkHTkEEH
EHdE
daTkdEEHdEEfEH
BBB
En
n
nn
nB
dEEfEH )()(
FEdEEH )(
successive terms are smaller by O(kBT/)2
for kBT/<< 1 replace by = EF
and vV
uu c
T
FD statistics depress cv by a factor of
2
22
0
22
2
2
11
3 2
( )6
( )3
3( )
2
2
3
2
3
BF
F
B F
v B F
FF
Bv B
F
classical B
B
F
v
k TE
E
u u k T D E
uc k TD E
Tn
D EE
k Tc nk
E
c nk
k T
E
c T
(1)
(2)
2
2 3232FE n
m
3 2
2 2
1 2( )
2
mD E E
specific heat of the degenerate electron gas
thermal conductivity
vv
q
lvccv
T
3
1
3
1 2
j
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
Tkmv
nkc
Te
k
ne
mvc
B
Bv
Bv
2
3
2
12
3
2
3
3
2
2
2
2
Wiedemann-Franz law (1853)Lorenz number ~ 2×10-8 watt-ohm/K2
Drude:application of classical ideal gas laws
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
m
ne 2
the correct at room T
the correct estimate of v2 is vF2 at room T
01.0~/~2
2
FBclassicalvvBF
Bv ETkccnk
E
Tkc
100~/~22 TkEvv BFclassicalF
22
3
e
k
TB
for degenerateFermi gas of electrons
thermopower
ne
cQ v
3
Drude:application of classical ideal gas laws
Bv nkc2
3
e
kQ B
2
Seebeck effect: a T gradient in a long, thin bar should be accompanied by an electric field directed opposite to the T gradient
TQE
thermopower
for degenerateFermi gas of electrons
the correct at room T
Q/Qclassical ~ 0.01 at room T
01.0~/~2
2
FBclassicalvvBF
Bv ETkccnk
E
Tkc
F
BB
E
Tk
e
kQ
6
2
high T low T
gradTE
thermoelectric field
Electrical conductivity and Ohm’s law
2
2
1
ne
mm
ne
( ) (0)
avg
avgavg
d dm e
dt dte
t t
e
e
m m
v kE
Ek k
Ek
k Ev
Ej
m
ne 2
avgnevj
equation of motionNewton’s law
in the absence of collisions the Fermi sphere in k-space is displaced as a whole at a uniform rate by a constant applied electric field
because of collisions the displaced Fermi sphere is maintained in a steady state in an electric field
Ohm’s law
the mean free path l = vFbecause all collisions involve only electrons near the Fermi surfacevF ~ 108 cm/s for pure Cu:at T=300 K ~ 10-14 s l ~ 10-6 cm = 100 Åat 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
kavg
( ) ( )( ) 0
d t tt
dte
p pf
p f E
FFermi sphere
ky
kx