Solid state physics 04-free electrons in metals

Solid State PhysicsUNIST, Jungwoo Yoo

1. What holds atoms together - interatomic forces (Ch. 1.6)2. Arrangement of atoms in solid - crystal structure (Ch. 1.1-4) - Elementary crystallography - Typical crystal structures - X-ray Crystallography3. Atomic vibration in solid - lattice vibration (Ch. 2) - Sound waves - Lattice vibrations - Heat capacity from lattice vibration - Thermal conductivity4. Free electron gas - an early look at metals (Ch. 3) - The free electron model, Transport properties of the conduction electrons---------------------------------------------------------------------------------------------------------(Midterm I) 5. Free electron in crystal - the effect of periodic potential (Ch. 4) - Nearly free electron theory - Block's theorem (Ch. 11.3) - The tight binding approach - Insulator, semiconductor, or metal - Band structure and optical properties6. Waves in crystal (Ch. 11) - Elastic scattering of waves by a crystal - Wavelike normal modes - Block's theorem - Normal modes, reciprocal lattice, brillouin zone7. Semiconductors (Ch. 5) - Electrons and holes - Methods of providing electrons and holes - Transport properties - Non-equilibrium carrier densities8. Semiconductor devices (Ch. 6) - The p-n junction - Other devices based on p-n junction - Metal-oxide-semiconductor field-effect transistor (MOSFET)---------------------------------------------------------------------------------------------------------------(Final)

All about atoms

backstage

All about electrons

Main character

Main applications


Free Electrons in Metals

1. The electron as a wave

2. Quantum mechanical description

3. Introduction

4. The free electron model

5. Transport properties of the conduction electrons

A number of electrical properties of metals can be well described with the model of free electrons, in which we ignored the attractive interaction between electrons and ions. In the free electron model, the valence electron move freely in the specimen of size L solid.


The electron as a wave

Electron also have wavelike properties too!

A good example is the interference of electron waves in the experiment of Davisson and Germer in 1927

A

B

Screen with holes

Target Screen

Gun

A

B

Source of waves

Source of particles

nx

2

1 nx

Interference:Constructive interference

Destructive interference



De Broglie’s matter wave:According to de Broglie’s hypothesis, particles of matter (such as electrons) have wave properties.

mv

h

Davisson and Germer’s experiment

Electron gunIncident beam Reflected beam

Detector

Path difference a sin2d

The reflected beam displayed an interference pattern a The wavelike nature of the electron

was conclusively demonstrated

Condition for constructive inteference

n (Bragg condition)



sin2d n (Bragg condition)

The difference in angle between two successive maxima is of the order of

d/~

For good resolution, d~ (lattice spacing)

16111031

34

ms1025.7mJskg10101.9

106.6

m

hv

Then, the accelerating voltage in e-gun to produce electron with sufficient energy is

eVmv 2

2

1

V150Cskgm106.12

)1025.7(101.9

2122

19

26312

e

mvVa

Electron ? Proton ? Neutron ?

How about bullet ? ,kg10 3m ,ms10 13 v a m10 34Extremely difficult to observe !


The electron microscope

The usage of wave nature of electron

The resolving power of an optical microscope is fundamentally limited by the wave-length of the light. For greater resolution, we need a shorter wavelength.

Electron can have very short wavelength and can be easily focused by electric and magnetic field


Some properties of waves

A wave of frequency, w, and wave number, k, can be described as )()( tkziaezu

Then, the phase velocity can be defined as

fkt

zvp

constant

For a single frequency wave this is fairly obvious.

But, what happens when several waves are superimposed ?

The resultant wave is given by )()( tzki

nn

nneazu For the continuum case,

dkekazu tkkzi ))(()()(

Here, a(k) and w(k) are functions of k



Let’s consider for the time being, t = 0, than

dkekazu ikz)()(

Now, let’s investigate the relationship between a(k) and u(z)

Consider the simplest possible case of

1)( ka

0)( ka

for22 0

kkk

k

k

)(ka

1

k

2/

2/

0

0

)(kk

kk

ikzdkezu a

2/

2/sin)( 0

kz

kzkezu zik

z

z

)(zu

Wave packetWidth of packet is determined by the point where the amplitude drops to 0.63 of its maximum value

22

kz

a 2 zk



Let’s consider for the time-varying case, and a(k) is zero beyond Dk.

2/

2/

)(0

0

)()(kk

kk

tkzi dkekazu

Rewrite the formula as the following form

)( 00),()( tzkietzAzu

Then,

2/

2/

))()((0

0

00)(),(kk

kk

tzkki dkekatzA

Now, we can define two velocity,

I. Phase velocity: the velocity with which the central components propagate

00 / kvp

II. Group velocity: the velocity with which the envelop of the wave packet propagate

constant.)()( 00 zkkt

a ,0

0

kkt

zvg

a0kk

g kv


Applications to electrons

Now, let’s try to apply the properties of waves to the particular case of the electrons

I) Position of electron as a wave,

If wave ripples are uniformly distributed in space, as is the case for a single fre-quency wave, the electron can be everywhere

If wave ripples are concentrated in space in the form of a wave packet, the posi-tion of an electron is defined with certain uncertainty. We also can identify the velocity of the wave packet with the electron velocity.

I) Energy of the electron

A photon of frequency w has an energy

hfEAnalogously, the energy of an electron in a wave packet cnetred at the fre-quency w is given by the same formula.

hfETaking the potential energy as a zero,

2

2

1mvE a

k

vmv

kg

g

ak

vm g

a gmvk

agmv

h


Applications to electrons

Uncertainty principle

The electron of a wave packet have a width of position Dz, an uncer-tainty about the position of the electron

From the previous relationship in the wave packet

2 zk and gmvk

We get

hzp

Ex) if we know the position of the electron with an accuracy of 10-9m then the uncertainty in momentum is

125 kgms106.6 p a 15 ms107 vThe uncertainty in velocity is quite appre-ciable

For Dx ~ 10-9m for the uncertainty in position, and a bullet with a mass of 10-3kg, the uncertainty in velocity decreases to

122 ms106.6 v


Brief Review of Quantum Mechanics

Important consequences from Quantum mechanical description

I) Discreet energy level

II) Tunneling

III) Spin & Magnetism

III) Theory of Conventional Superconductor (BCS theory)

Ex) particle in 1D box, potential well, harmonic oscillation, hydrogen atom, etcand new particles with fundamental unit of energy, photon, phonon, magnon, etc

II) Band theory




2

22

2ˆ

zmH

H

In a free space, V=0, the Schröndinger eqn. is

The general solution for the time dependent Schröndinger eqn. is

),(ˆ),( trHtrt

i

The general solution for the Schröndinger eqn. is plane wave propagating left and right

)exp()exp( ikzBikzA m

kE

2

22

)exp()exp()exp( ikzBikzAti E

The momentum of the electron is equal tok

Then, how about the position of the electron ?

For simplicity, take B=0 (the case for the forward traveling wave)

a The probability of finding the electron at any particular point is unity

.const)(2 z



The electron as a particle

02 2

22

Ezm

The Schröndinger eqn.

is the linear differential equation, hence the sum of the solutions is still a solution.

Therefore, a wave packet (a sum of many waves) is also solution of Schröndinger eqn.

A wave packet represents an electron as a particle because is appre-ciably different from zero only within the packet.

2)(z

With the choice in the interval , the probability of finding the elec-tron is given by

22

)2/(

)2/sin(K)(

kz

kzz

1)( ka k


Particle in 1D box

An electron of mass m is confined to a length L by infinite barriers

From the schrödinger equation H 2

22

2ˆ

xmH

From the fixed boundary condition, 0)()0( L

integral number of half wavelength should fit in size L

22

2

L

n

mn

Ln )2/(L

nk

a

Wavefunction can be given by the form kxAx sin)(




Harmonic Oscillator )

2

1( nn

,2

ˆ

2

ˆˆ22 x

Km

pH

m

K

,2

ˆ

2

ˆ 22

2 xm

m

p

)2

1ˆˆ( aa

N

m

pix

ma

2

ˆˆ

2ˆ

m

pix

ma

2

ˆˆ

2ˆ



The electron meeting a potential barrier

For finite potential barrier

z

V

Region I Region II

,ˆ H )(2

ˆ2

2

xVxm

H

VI=0 VII=V

,11 xikxikI BeAe E

m

k

2

21

2For region I,

,2xikII Ce VE

m

k

2

22

2For region II,

From boundary condition (or matching cond.), At z=0, both and should be continuous

z

CBA

,)( 21 CikBAik

a ,21

21

kk

kk

A

B

.2

21

1

kk

k

A

C



The electron meeting a potential barrier


z

V

Region I Region II

For E > V

,ˆ H )(2

ˆ2

2

xVxm

H

VI=0 VII=V

Oscillatory solution in region II, and finite amount of re-flection

022 k a k is real

For E < V

022 k a k is imaginary

The amplitude of wave de-cline exponentially in region II (there is finite, though de-clining, probability of finding the electron at z > 0)



Tunneling


x

V

Region I Region II Region III

For E > V

,11 xikxikI BeAe E

m

k

2

21

2

,22 xikxikII DeCe VE

m

k

2

22

2

xikIII Fe 1 E

m

k

2

21

2

-a a

)2(sin)(4

11

1

22

2

akVEE

VT

,ˆ H )(2

ˆ2

2

xVxm

H

,2

A

FT

2

A

BR 1RT



From boundary condition at x = a and x = -a

aikaikaikaik eA

De

A

Ce

A

Be 2211

aikaikaikaik e

A

De

A

Cke

A

Bek 2211

21

aikaikaik eA

Fe

A

De

A

C122

aikaikaik eA

Fke

A

De

A

Ck 122

12

1

221

22

21

22 )2sin(

2)2cos(2

akkk

kkiake

A

F aik

)2sin(2 221

21

22 ak

kk

kk

A

Fi

A

B

122

A

B

A

FRT

)2(sin4

11

12

2

21

22

21

2

akkk

kk

F

A

T

a )2(sin

)(4

11

12

22

akVEE

V

T




x

V


For E < V

,11 xikxikI BeAe E

m

k

2

21

2

,xxII DeCe 0

2

22

EVm

xikIII Fe 1 E

m

k

2

21

2

-a a

)2(sinh)(4

11

1

22

aEVE

VT

,ˆ H )(2

ˆ2

2

xVxm

H

,2

A

FT

2

A

BR 1RT

Tunneling

2ik



Tunneling


x

V

Region I

Region II Region III

-a a

For E < V



The Ramsauer effect

x

V


,11 xikxikI BeAe E

m

k

2

21

2

,22 xikxikII DeCe VEEV

m

k

2

22

2

xikIII Fe 1 E

m

k

2

21

2

-a a

)2(sin)(4

11

1

22

2

akVEE

VT

,ˆ H )(2

ˆ2

2

xVxm

H

,2

A

FT

2

A

BR 1RT

For E > 0



Potential well

,xI Ae 0

2

22

Em

,ikxikxII CeBe 0

2

22

EVm

k

,xIII De 0

2

22

Em

,ˆ H )(2

ˆ2

2

xVxm

H

x

V


-a a

For E < 0

From boundary condition (or matching cond.), At x=a,-a both and should be continuous

z

1E2E

3E



The uncertainty principle

V

V

dV

dVAA 2

ˆ

The average value of a physically measurable quantity is

The rms value of position and momentum are

2/12xxx 2/12

ppp

The uncertainty relationship is

hpx

htE


Introduction

Electrical resistivity of materials

Materials fall into 3 main classes:

Metals: resistivities between 10-8 to 10-5 Wm

Semiconductors: resistivities between 10-5 to 10 Wm

Insulators: resistivities above 10 Wm

Resistivity increases by addition of small amount impurities

Resistivity decreases by addition of small amount impurities


Introduction

For most metals: T For most semiconductors:TkBe /

tend to become insulator at low T


The Free Electron Model

Nucleus and many electrons

Most electrons are boundedElectrons at outer shell mainly address properties of atomsElectrons at Fermi energy mainly ad-dress physical properties of materi-als electrical, magnetic, optical

Ions and electrons

Solid



Solid

Drude model: an early look at metal (proposed by Paul Drude in 1900)

It is based on the elemental kinetic theory of gases

eliminate all the electron ion interactions and replace them by a single pa-rameter t.

t is collision time

Ch 3.3 transport properties of conduction electrons is based on Drude model.

How is solid so transparent to conduction electrons ? i) The electron matter wave can propagate freely in a periodic structure ii) The conduction electron is scattered very rarely



Solid

Sommerfeld model: impose Fermi-Dirac distribution on Drude model

treat electrons as Fermions

a Free electron model



• The valance electrons are responsible for the conduction of electricity, and for this reason these electrons are termed conduction electrons.

• Na11 ￫ 1s2 2s2 2p6 3s1

• This valance electron, which occupies the third atomic shell, is the electron which is responsible chemical properties of Na.

• When Na atoms come closer to form a Na metal, Solid state of Na atoms overlap slightly. Then, a valance electron is no longer attached to a particular ion, but belongs to the whole crystal, since it can move readily from one ion to its neighbour, and then the neighbour’s neighbour, and so on.

• This mobile electron becomes a conduction electron in a solid.

Free electron model

Valance electron (loosely bound)

Core electrons

+

+ + +

+ + The removal of the valance elec-trons leaves a positively charged ion.



Consider 1D case

One-dimentional periodic potential associated with a chain of identical atoms



In free electron model: the positive ion cores is spread uniformly throughout the metal so that the electrons move in a constant electrostatic potential. No details of crystal structure No el-ions interaction, No el-el interaction



Particle in 1D box

An electron of mass m is confined to a length L by infinite barriers

From the schrödinger equation H 2

22

2ˆ

xmH

From the fixed boundary condition, 0)()0( L

integral number of half wavelength should fit in size L

Ln )2/(L

nk

22

2

L

n

mn

a

Wavefunction can be given by the form kxAx sin)(


For 3D


H 22

2ˆ

mH

From the periodic boundary condition,

),,(),,( zyxLzLyLx

Solution of the schrödinger equation

)(

2/12/1

11),,( zkykxkirki zyxe

Ve

Vzyx

pNa pNa

k2

a

2xk p

L

2yk q

L

2zk r

L

m

k

2

22 )(

2222

2

zyx kkkm



kdkdkkg R

3)()( dkk

Vkd

L 23

33

482

2

the number of allowed k values inside a spherical shell of k-space of radius k

,2

2

dkVk

For spin up and down

dkkgdg )()(

d

dkkgg )()(

m

k

2

22

m

k

dk

d 2

2

2

m

k

2/12/33222222

2

)2(2

2

22)(2)(

m

VmVm

k

mVk

d

dkkgg

2/3320

2/12/3320

)2(3

)2(2

)( FmV

dmV

dgNFF


When all states are filled up to a certain energy, this upper limit energy is being called Fermi energy EF. The Fermi energy is obtained by integrating density of states between 0 and EF, should equal N. Hence


2/12/332

)2(2

)(

mV

Eg

m

k

V

N

mF

F 2

3

2

223/222

3/222 3

V

NkF

kz

ky

kx

Fermi surfaceE=EF

kF

2/3320

2/12/3320

)2(3

)2(2

)( FmV

dmV

dgNFF

)(3

2FF g

2/12/332

)2(2

)( FF mV

g


The Free Electron ModelEx) Monovalent potassium metal as an example; the atomic density (the same as valence electron density) N/V is 1.402x1028 m-3

Only at a temperature of the order of TF that the particles in a classical gas can attain (gain) kinetic energies as high as EF

At T > TF , the free electron gas behave like a classical gasAt T < TF , behaviour of the free electron gas is dominated by Pauli exclusion principle

Fermi energy:

Fermi wavenumber:

Fermi temperature:

Fermi momentum:

Fermi velocity:

eV12.2J1040.33

219

3/222

V

N

mF

746.03

3/12

V

NkF

Å-1

16 ms1086.0 e

F

e

FF m

k

m

Pv

FF kP

Kk

TB

FF

41046.2



Free electron gas at finite T

At a temperature T , the probability of occupation of an electron state of energy E is given by the Fermi distribution function

1

1),(

/)( TkBe

Tf

EFE<EF E>EF

0.5

fFD(E,T)

E

At absolute zero Temp. T=0K

11

1)( /)(

TkBe

f For

01

1)( /)(

TkBe

f For

At T = 0K, m is eF



Free electron gas at finite T

The number of electrons per unit energy range in thermal equilibrium is given by

The # of electrons per unit energy =

Density of state g(e) ⅹ probability of occupation f(e, T)

),()(),( TfgTn

T > 0

n(e,T)

e

g(e)

eF

T = 0

TkB~The effect of finite temperature



Heat capacity of the free electron gas

T > 0

n(e,T)

e

g(e)

eF

T = 0

TkB~

The amount of energy absorbed by electrons at finite T = shaded area ⅹ kBT

½ ⅹ height (½ g(eF)) ⅹ base (2kBT)

Thermal energy =

2))((2

1~)0()( TkgETE BF

TkgT

EC BFv

2)(

)(

3

2FF gN

FBFF Tk

NNg

2

3

2

3)(

FBv T

TNkC

2

3




FBv T

TNkC

2

3

For exact calculation,

0

),( dTnN

0

),()( dTnTE

FBBFv T

TNkTkgC

2)(

3

22

2

The lattice heat capa-city,

at room T, 3NkB and falls off T3 below Debye temperature

The total heat capacity at low temperat-ure,

ElectronicHeat capacity

Lattice HeatCapacity

3TTC




Intercept gives g

slope gives bFor metal:

For insulator:

Heat capacity of KCl

Heat capacity of K

21KmolmJ08.2 212

KmolmJ67.12

F

B

T

Nk

KTF41046.2

No contribution from conduction electrons


The Free Electron ModelHeat capacity of the free electron gas

The discrepancy between predicted values and experimental values can often be corrected by introducing effective mass of electron m*, which dif-fers from their bare mass

Effective mass correction

For change by a factor,F ,FT ,Fvm

m

For change by a factor),( Fg ,vCm

m

For a theoretical calculation m*, electron-phonon interaction and electron-electron interaction need to be considered

The effective mass m*, associated with any physical property provides a use-ful way of quantifying departures of that property from the free electron prediction

For Potassium, 25.1

m

m


The Free Electron ModelMetallic binding

22

2

L

n

mn

2

22

1 2ma

a

5a

2

22

1 50ma

2

22

5 2ma

Consider 4 electrons in each 5 atoms

Delocalization of electrons reduction of kinetic energy metallic bonding


Transport properties of the conduction electronsThe equation of motion of the electrons

In the presence of electric field (E) and magnetic field (B), the force on an electron of charge e and mass me

BveEedt

vdmF e

In free electron model, the electron is being described as a plane wave states, which extend through the crystal.

To describe with classical equation of motion for a plane wave states of elec-tronsNeed to introduce wave packet with well defined position and momentum to give a particle like entity for a electrons.

The velocity of wave packet is the group velocity of the waves

ee m

p

m

k

kd

d

kd

dv

1

The size of wave packet should be much larger than atomic spacing with well defined positionto approximate electron-ion interaction

to use equation of motion~ 10 a

For 10a, k ~ kF/10 P ~ PF/10



In the presence of electric field (E) and magnetic field (B), the force on an electron of charge e and mass me

BveEedt

vdmF e

In the absence of magnetic field, the applied E results a constant acceleration but this will not cause a continuous increase in current. Since electrons suffer collisions withI) phonons II)electrons

a impose collision by adding v

me

BveEev

dt

vdme

In the absence of electric and magnetic field, v decay exponentially to zero, a v is drift velocity, is departure from the thermal equilibrium state given by Fermi distribution

Drude model



In the presence of dc electric field (E) only

The steady state solution (no acceleration) is

Em

ev

e

eMobility:

The Ohm’s law for the electric current density J

EEm

nevnqj

e

2

where n=N/V

ee

nem

ne 2

Electrical Resistivity and Resistance

A

LR

,

1

Collistions: 1) electron-phonon a as T g 0, tph(T) infinity 2) electron with impurity atoms gives a finite scattering time t0 even at T g 0


Transport properties of the conduction electrons

At finite temperature, the electron scattering rate

0

1

)(

11

Tph

term for perfect crystal(phonon scattering)

term for scattering with impurity

Valid if two scattering mechanism is independent, for low impurity material

The electrical resistivity (Mattheisen’s rule):

00

222)(

1

)(

111

Tne

m

Tne

m

ne

mI

e

ph

ee

Ideal resistivity Residual resistivity

Residual resistivity ratio: 0T

RT

can be as high as 106


Temperature (K)

pureimpure

Rel

ativ

e re

sist

ance

104

R/R

29

0K


The resistivity vs T graphs for different spec-imens of the same mate-rial differ only by a dis-placement, This dis-placement is associated with the variation in r0 due to different imper-fection densities

The r vs T curves for sodium specimens of differing purity

For sodium: 117 m100.2 RT 114 m103.5 residual

Taking mmn e ,m107.2 328s106.2~ 14

2

ne

ms100.7~ 11

at room temperature

at T = 0

Fermi velocity: 16

3/22

ms101.13

V

N

mm

kv

ee

FF

a electron mean free path: 29 nm at RT 77 mm at T g 0


Transport properties of the conduction electronsThe thermal conductivity

Electrons coming from a hotter region of the metal carry more thermal energy than those from a cooler region, resulting in a net flow of heat. The thermal conductivity

Typically, due to the contribution of conduction electrons metalnonmetal KK

From elementary kinetic theorylvCK FV3

1

Mean free path of conduction electrons: Fvl

2

3

1FV vCK a

FBV T

Tk

V

NC

2

2 2

2

1FF mv

ae

B

m

TnkK

3

22



Wiedemann-Franz law

e

B

m

TnkK

3

22 e

e

nem

ne 2

The ratio of the electrical and thermal conductivities

a independent of the electron gas parameters;

2822

KW1045.23

e

k

T

K B

Lorentz number

28 KW1023.2 T

KL

For copper at 0 C

The collision time limiting the flow of electric and heat currents are the same

Electrical and Thermal Conductivity



The group velocity of the electronic waves

ee m

p

m

k

kd

d

kd

dv

1 BveEe

v

dt

vdme

The drift velocity associ-ated with electric current

Em

ev

e

The change in the wavevector of each elec-tron

Ee

vm

k e

The electric current carrying state corresponds to a shift by of the whole Fermi sphere

k

For 2D

E

For a current density of 107 Am-2

Fv

ne

jv

8

1

11928

7

10~

mms10~

ms1010

10




E

k

At finite T, T≠0

xk

yk

0 Tx HotCold




At finite T, T≠0

xk

yk

0 Tx HotCold




Scattering associated with electrical and thermal conductivity by electrons

fk

ik

q

fi kqk

Phonon absorption

fk

ik

q

qkk fi

Phonon emission

Electron-phonon scattering

Need to satisfy conservation of momentum and energy



E

k

xk

yk

HotCold


A typical relaxation processes associated with electric-current carrying state

Momentum of phonon is importnat



At finite T, T≠0

xk

yk

0 Tx HotCold


A typical relaxation processes associated with heat-current carrying state



xk

yk

At very high T , maximum phonon energy a to satisfy conservation of energy only states near the Fermi energy involve scattering with phonon

But, maximum phonon momentum

a both large and small momentum transfer can occur by phonon scatter-

ing

)( DT


q

Fk

~

DBk FBTk

Since wavelength for is scale of atomic spacingand Fermi wavelength is alsoscale of atomic spacing

DBk



xk

yk

At low T , typical phonon energy

a phonon energy has momentum of order

a allow scattering between the electron states close to each other

in the vicinity of Fermi energy

)( DT


TkB

FD pT )/( TkB



Temperature Dependence of Electrical and Thermal Conductivity

At high T ( ), the typical phonon energy is . and electron mean free path lph is

inversely proportional to the phonon number.

Lattice vibration energy is and each phonon energy ( ) is nearly const.

a phonon # a the electron scattering time T 1Tph

,)(22

Tlne

vm

ne

mT

ph

Fe

ph

eI

T

TNkB3

Bk

Bk

T

D

TI

T

Temperature Dependence of Electrical Conductivity




At high T ( ), the typical phonon energy is . and electron mean free path lph is

inversely proportional to the phonon number.

Lattice vibration energy is and each phonon energy ( ) is nearly const.

a phonon # a the electron scattering time T 1TphTNkB3

Bk

Bk

T

K

D T

Temperature Dependence of Thermal Conductivity

0TK

022

3T

m

TnkK

e

B




At very low T ( ), scattering dominated by impurity

D

0

T


0T const.ph

.)(2

constne

mT

ph

eI

a




K

D T


a Tm

TnkK

e

B 3

22

TK

At very low T ( ), scattering dominated by impurity0T const.ph




At low T ( ), the average phonon energy is .

Lattice vibration energy is and

phonon energy only allow low momentum transfer scattering.

a mean free path and scattering time are inversly proportional to the phonon #

K

D T

,3th

T

TkB

TkB


T4T

a 222

3 T

m

TnkK

e

B

2TK




At low T ( ), the average phonon energy is .

phonon energy has momentum of order

TkB

TkBT

FD pT )/(

D

5TI

T


Too small to introduce Large momentum change

Fk

Fk q

k

22

2T

k

q

F

For small theta

5

21

~1

TT

thDel

Therefore,




D

0 5TI

TI

T

K

D T

2TK0TK



TK

Impurity scattering dominant

Phonon scattering dominant



Wiedemann-Franz law fails

Wiedemann-Franz law22

3

e

k

T

K B




The Hall Effect

The origin of hall ef-fect

a Lorentz force

Bve

The Lorentz force tends to deflect the electrons downwards and this results in the rapid build up of a negative charge den-sity in –y side of the bar

In steady state, the Lorentz force on the electrons is just balanced by

the force due to the Hall field

Bve

HEe

HE

jBRE HH


Transport properties of the conduction electronsThe Hall Effect

In steady state, the Lorentz force on the electrons is just balanced by

the force due to the Hall field

Bve

HEe

HE

BveEev

me

z

x

B

v

kji

e

00

00

xxe eEvm /

)(0 BvEe xy

a

neRH /1 HRand

BjRneBjBvE xHxxy )/(


Transport properties of the conduction electronsThe Hall Effect

Metal Group )/(1 NeRH

Na I +0.9Ka +1.1

Cu IB +1.3Au +1.5

Be II -0.2Mg +1.5

Cd IIB -2.2

Al III +3.5

Hall coefficient of various met-als

Predicted as the number of conduction electrons per atom

HR is positive a charge carriers are holes


Summary

The Free Electron Model: By ignoring all interaction, we describe the electrons as a free particles confined in size L of specimen.

How we describe these electrons ? Using quantum mechanism, electron has wave and particle duality

)(

2/12/1

11),,( zkykxkirki zyxe

Ve

Vzyx

)(22

222222

zyx kkkmm

k

kdkdkkg s

3)()( dkk

Vkd

L 23

33

482

,2 2

2

dkVk

2/12/33222222

2

)2(2

2

22)(2)(

m

VmVm

k

mVk

d

dkkgg

m

k

V

N

mF

F 2

3

2

223/222

2/3320

2/12/3320

)2(3

)2(2

)( FmV

dmV

dgNFF

3/222 3

V

NkF


Fermi energy:

Fermi wavenumber:

Fermi temperature:

Fermi momentum:

Fermi velocity:

eV12.2J1040.33

219

3/222

V

N

mF

746.03

3/22

V

NkF

Å-1

16 ms1086.0 e

F

e

FF m

k

m

Pv

FF kP

Kk

TB

FF

41046.2

SummaryThe Free Electron Model: For potassium metal

of N/V = 1.402x1028 m-3

Atomic scale

Much higher than Debye T

At finite T the probability of the occupation of an electron state of energy E

1

1),( /)( TkBe

Tf

EFE<EF E>EF

0.5

fFD(E,T)

E


Summary

The Free Electron Model:

The # of electrons per unit energy = ),()(),( TfgTn

T > 0

n(e,T)

e

g(e)

eF

T = 0

TkB~

The amount of energy absorbed by electrons at finite T = shaded area ⅹ kBT

Thermal energy =

,))((2

1~)0()( 2TkgETE BF )(

3

2FF gN

FBv T

TNkC

2

3

ElectronicHeat capacity

Lattice HeatCapacity

3TTC


SummaryTransport properties of the conduction electrons

BveEedt

vdmF e

a impose collision by adding v

me

BveEev

dt

vdme

Force applied to an electron in the presence of E&M fields

Em

ev

e

a

Steady state solution when B=0

The Ohm’s law for the electric current density J

EEm

nevnqj

e

2

ee

nem

ne 2

a

0

1

)(

11

Tph

term for perfect crystal(phonon scattering)

term for scattering with impurity

00

222)(

1

)(

111

Tne

m

Tne

m

ne

mI

e

ph

eea

Ideal resistivity Residual resistivity

From elementary kinetic theorylvCK FV3

1 a

e

B

m

TnkK

3

22



Wiedemann-Franz law

e

B

m

TnkK

3

22 e

e

nem

ne 2

2822

KW1045.23

e

k

T

K B

Lorentz number

Scattering associated with electrical and thermal conductivity by electrons:

a Absorption and emission of phonon fk

ik

q

fk

ik

q

xk

yk

0 Tx

HotCold

A typical relaxation processes associated with heat-current carrying state

E

xk

yk

k

A typical relaxation processes associated with electric-current carrying state


D

0 5TI

TI

T

K

D T

2TK0TK



TK





Wiedemann-Franz law fails


At very low T ( ), scattering dominated by impurity0T const.phAt low T ( ), the average phonon energy is and momentum of T TkB

a mean free path and scattering time are inversely proportional to the phonon

#

3th

TFD pT )/(

At high T ( ), the typical phonon energy is .T Bk

a mean free path and scattering time are inversely proportional to the phonon

#

1Tph

e

B

m

TnkK

3

22 e

e

nem

ne 2


The Hall Effect

In steady state, the Lorentz force on the electrons is just balanced by the force due to the Hall field

Bve

HEe

HE

jBRE HH

BveEev

me

z

x

B

v

kji

e

00

00

xxe eEvm /

)(0 BvEe xy

a

neRH /1 HRand

BjRneBjBvE xHxxy )/(

Summary

Science

Solid state physics 04-free electrons in metals