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