CHAPTER 5

FREE ELECTRON

THEORY

Free Electron Theory

Many s

olid

s co

nduct

ele

ctrici

ty.

There

are

ele

ctro

ns

that

are

not

bound t

o a

tom

s but

are

able

to m

ove t

hro

ugh t

he

whole

cry

stal.

Conduct

ing

solid

sfa

llin

totw

om

ain

class

es;

meta

lsand

sem

iconduct

ors

.

and

incr

ease

sby

the

additio

nof

small

am

ounts

of

impurity

. The r

esi

stiv

ity n

orm

ally

decr

ease

s m

onoto

nic

ally

with

decr

easi

ng t

em

pera

ture

.

and

can

be

reduce

dby

the

additio

nof

small

am

ounts

of

impurity

.

Sem

iconduct

ors

tend t

o b

eco

me insu

lato

rs a

t lo

w T

.

68

()

;10

10

metals

RT

mρ

−−

−Ω−

()

()

pure

semiconductor

metal

RT

RT

ρρ

−

Why mobile electrons appear in some

solid

s and others?

When

the

inte

ract

ions

betw

een

ele

ctro

ns

are

consi

dere

dth

isbeco

mes

avery

difficu

ltquest

ion

toansw

er.

The

com

mon

physi

calpro

pert

ies

of

meta

ls;

•G

reat

physi

calst

rength

•H

igh

densi

ty•

Hig

hdensi

ty•

Good

ele

ctri

caland

therm

alco

nduct

ivity,

etc

.

This

chapte

rw

illca

lcula

teth

ese

com

mon

pro

pert

ies

of

meta

lsusi

ng

the

ass

um

ption

that

conduct

ion

ele

ctro

ns

exis

tand

consi

stof

all

vale

nce

ele

ctro

ns

from

all

the

meta

ls;

thus

meta

llic

Na,

Mg

and

Al

will

be

ass

um

ed

tohave

1,

2and

3m

obile

ele

ctro

ns

per

ato

mre

spect

ively

.A

sim

ple

theory

of

‘freeelectronmodel’

whic

hw

ork

sre

mark

ably

well

will

be

desc

ribed

toexpla

inth

ese

pro

pert

ies

of

meta

ls.

Why mobile electrons appear in some

solid

s and not others?

Acc

ord

ing

tofr

ee

ele

ctro

nm

odel

(FEM

),th

evala

nce

ele

ctro

ns

are

resp

onsi

ble

for

the

conduct

ion

of

ele

ctrici

ty,

and

for

this

reaso

nth

ese

ele

ctro

ns

are

term

ed

conduct

ion

ele

ctro

ns.

N

a11￫

1s2

2s2

2p

63s1

This

vala

nce

ele

ctro

n,

whic

hocc

upie

sth

eth

ird

ato

mic

shell,

isth

eele

ctro

nw

hic

his

resp

onsi

ble

chem

ical

pro

pert

ies

of

Na.

Vala

nce

ele

ctro

n (

loose

ly b

ound)

Core

ele

ctro

ns

W

hen

we

bring

Na

ato

ms

togeth

er

tofo

rma

Na

meta

l,

N

ahas

aBCC

stru

cture

and

the

dis

tance

betw

een

neare

stneig

hbours

is3.7

A˚

Na m

eta

l

neare

stneig

hbours

is3.7

A˚

The r

adiu

s of

the t

hird s

hell

in N

a is

1.9

A˚

Solid

state

of

Na

ato

ms

overl

ap

slig

htly.

From

this

obse

rvation

itfo

llow

sth

at

avala

nce

ele

ctro

nis

no

longer

att

ach

ed

toa

part

icula

rio

n,

but

belo

ngs

toboth

neig

hbouring

ions

at

the

sam

etim

e.

The

rem

oval

of

the

vala

nce

ele

ctro

ns

leaves

aposi

tively

charg

ed

ion.

The

charg

edensi

tyass

oci

ate

dth

eposi

tive

++

+

A

vala

nce

ele

ctro

nre

ally

belo

ngs

toth

ew

hole

cryst

al,

since

itca

nm

ove

readily

from

one

ion

toits

neig

hbour,

and

then

the

neig

hbour’s

neig

hbour,

and

soon.

This

mobile

ele

ctro

nbeco

mes

aco

nduct

ion

ele

ctro

nin

aso

lid.

The

charg

edensi

tyass

oci

ate

dth

eposi

tive

ion

core

sis

spre

ad

uniform

lyth

roughout

the

meta

lso

that

the

ele

ctro

ns

move

ina

const

ant

ele

ctro

static

pote

ntial.

All

the

deta

ilsof

the

cryst

al

stru

cture

islo

stw

hen

this

ass

unption

ism

ade.

++

+

Acc

ord

ing

toFE

Mth

ispote

ntial

ista

ken

as

zero

and

the

repuls

ive

forc

ebetw

een

conduct

ion

ele

ctro

ns

are

als

oig

nore

d.

There

fore

,th

ese

conduct

ion

ele

ctro

ns

can

be

consi

dere

das

movin

gin

dependently

ina

square

well

of

finite

depth

and

the

edges

of

well

corr

esp

onds

toth

eedges

of

the

sam

ple

.

Consi

der

am

eta

lw

ith

ash

ape

of

cube

with

edge

length

of

L,

Ψand

Eca

nbe

found

by

solv

ing

Sch

rödin

ger

equation

V

0L/

2L/

2

22

2E

mψ

ψ−

∇=

h0

V=

Sin

ce,

(,

,)

(,,)

xLy

Lz

Lxyz

ψψ

++

+=

•By m

eans

of periodic

boundary

conditio

ns

Ψ’s

are

runnin

g w

aves.

The s

olu

tions

of Sch

rödin

ger

equations

are

pla

ne w

aves,

w

here

V is

the v

olu

me o

f th

e c

ube,

V=

L3

()

11

(,,)

xy

zikxkykz

ikr

xyz

ee

VV

ψ+

+=

=rr

Norm

aliz

ation c

onst

ant

So t

he w

ave v

ect

or

must

satisf

y

where

p,

q,

r ta

kin

g a

ny inte

ger

valu

es;

+ve,

-ve o

r ze

ro.

Na

pλ

=2

,wherek

π λ

=

2Na

pkπ

=2

2k

pp

Na

L

ππ

==

2xk

pLπ

=2

yk

qLπ

=2

zk

rLπ

=;

;

The w

ave funct

ion Ψ

(x,y

,z)

corr

esp

onds

to a

n

energ

y o

f

th

e m

om

entu

m o

f

22

2

kE

m=h

22

22

()

2x

yz

Ek

kk

m=

++

h

(,,)

pk

kk

=h

Energ

y is

com

ple

tely

kin

etic

(,,)

xy

zp

kk

k=h

22

21 2

2

kmv

m=h

22

22

mv

k=h

pk

=h

W

eknow

that

the

num

ber

of

allo

wed

kvalu

es

insi

de

asp

herica

lsh

ell

of

k-s

pace

of

radiu

sk

of

2 2()

,2Vk

gkdk

dk

π=2π

w

here

g(k

)is

the

densi

tyof

state

sper

unit

magnitude

of

k.

The number of allowed states

per unit energy range?

Each

kst

ate

repre

sents

two

poss

ible

ele

ctro

nst

ate

s,one

for

spin

up,

the

oth

er

issp

indow

n.

()

2()

gEdE

gkdk

=()2()dk

gE

gkdE

=dE

22

2

kE

m=h

2dE

k

dk

m=h

2

2mE

k=

h

()

gE

=2()

gk

dk

dE

222V π

kk

2

2mE

h2m

kh

3/2

1/2

23(2

)2

()

Vm

EgE

π=

h

Ground state of the free electron gas

Ele

ctro

ns

are

ferm

ions

(s=

±1/2

)and

obey

Pauli

excl

usi

on

princi

ple

;each

state

can

acc

om

modate

only

one

ele

ctro

n.

The

low

est

-energ

yst

ate

of

Nfr

ee

ele

ctro

ns

isth

ere

fore

obta

ined

by

filli

ng

the

Nst

ate

sof

low

est

energ

y.

Thus

all

state

sare

fille

dup

toan

energ

yE

F,kn

ow

nas

Ferm

ien

erg

y,

obta

ined

by

inte

gra

ting

densi

tyof

state

sbetw

een

0and

EF,

should

equal

N.

Hence

Rem

em

ber

3/2

1/2

23(2

)2

()

Vm

EgE

π=

h

FF

EE

VV

==

=∫

∫

Solv

e f

or

EF

(Ferm

i energ

y);

2/3

22

3

2F

NE

mVπ

=

h

3/2

1/2

3/2

23

23

00

()

(2)

(2)

23

FF

F

VV

NgEdE

mE

dE

mE

ππ

==

=∫

∫h

h

The o

ccupie

d s

tate

s are

insi

de t

he F

erm

i sp

here

in k

-space

show

n b

elo

w;

radiu

s is

Ferm

i w

ave n

um

ber

kF. 2

2

2

FF

kE

m=h

kz

Ferm

i su

rface

E=

E

2/3

22

3

2F

NE

mVπ

=

h

2F

e

Em

=

ky

kx

E=

EF

kF

From

th

ese

two e

quation k

F

can b

e found a

s,1/3

23

F

Nk

Vπ

=

Th

e s

urfa

ce o

f th

e F

erm

i sp

here r

ep

resen

t th

e

bo

un

dary b

etw

een

occu

pie

d a

nd

un

occu

pie

d

k

sta

tes a

t ab

so

lute

zero

fo

r t

he f

ree e

lectr

on

gas.

Typic

al

valu

es

may

be

obta

ined

by

usi

ng

monovale

nt

pota

ssiu

mm

eta

las

an

exam

ple

;fo

rpota

ssiu

mth

eato

mic

densi

tyand

hence

the

vala

nce

ele

ctro

ndensi

tyN

/Vis

1.4

02x10

28

m-3

soth

at

19

3.4010

2.12

FE

JeV

−=

×=

10.746

kA

−=

°

Fe

rmi (d

egenera

cy)

Tem

pera

ture

TF by

10.746

Fk

A−

=°

FB

FE

kT

=

42.4610

FF

B

ET

Kk

==

×

It

isonly

at

ate

mpera

ture

of

this

ord

er

that

the

part

icle

sin

acl

ass

ical

gas

can

att

ain

(gain

)kin

etic

energ

ies

as

hig

has

EF

.

Only

at

tem

pera

ture

sabove

TF

will

the

free

ele

ctro

ngas

behave

like

acl

ass

icalgas.

Fe

rmim

om

entu

mF

FP

k=h

Fe

FP

mV

=

These

are

the

mom

entu

mand

the

velo

city

valu

es

of

the

ele

ctro

ns

at

the

state

son

the

Ferm

isu

rface

of

the

Ferm

isp

here

.

So,

Ferm

iSphere

pla

ys

import

ant

role

on

the

behavio

ur

of

meta

ls.6

10.8610

FF

e

PV

ms

m

−=

=×

2/3

22

32.12

2F

NE

eVm

Vπ

=

=

h

1/3

21

30.746

F

Nk

AVπ

−

=

=°

Typ

ical

valu

es o

f m

on

ovale

nt

po

tassiu

m m

eta

l;

V

61

0.8610

FF

e

PV

ms

m

−=

=×

42.4610

FF

B

ET

Kk

==

×

The free electron gas at finite temperature

At

ate

mpera

ture

Tth

epro

babili

tyof

occ

upation

of

an

ele

ctro

nst

ate

of

energ

yE

isgiv

en

by

the

Ferm

idis

trib

ution

funct

ion 1

Fe

rmi

dis

trib

ution

funct

ion

dete

rmin

es

the

pro

babili

tyof

findin

gan

ele

ctro

nat

the

energ

yE.

()/

1

1F

BFD

EE

kT

fe

−=

+

f FD(E,T

)

()/

1

1F

BFD

EE

kT

fe

−=

+Fermi Function at T=0

and at a finite temperature

f F

D=

? A

t 0°K

i.E<

EF

11

f=

=

EF

E<

EF

E>

EF

0.5

E

ii.E>

EF

()/

11

FB

FD

EE

kT

fe

−=

=+

()/

10

1F

BFD

EE

kT

fe

−=

=+

Fermi-Dirac distribution function at

various temperatures,

n(E

,T)

g(E

)

n(E

,T)

num

ber

of

free

N

um

ber

of

ele

ctro

ns

per

unit

energ

yra

nge

acc

ord

ing

toth

efr

ee

ele

ctro

nm

odel?

The

shaded

are

ash

ow

sth

ech

ange

indis

trib

ution

betw

een

abso

lute

zero

and

afinite

tem

pera

ture

.

T>

0

T=

0

EE

F

n(E

,T)

num

ber

of

free

ele

ctro

ns

per

unit

energ

yra

nge

isju

stth

eare

aunder

n(E

,T)

gra

ph.

(,)

()

(,)

FD

nET

gEf

ET

=

Fe

rmi-

Dir

ac

dis

trib

ution

funct

ion

isa

sym

metr

icfu

nct

ion;

at

finite

tem

pera

ture

s,th

esa

me

num

ber

of

levels

belo

wE

Fis

em

ptied

and

sam

enum

ber

of

levels

above

EF

are

fille

dby

ele

ctro

ns.

n(E

,T)

g(E

)

T>

0

T=

0

EE

F

Heat capacity of the free electron gas

Fr

om

the

dia

gra

mof

n(E

,T)

the

change

inth

edis

trib

ution

of

ele

ctro

ns

can

be

rese

mble

din

totr

iangle

sof

heig

ht

1/2

g(E

F)and

abase

of

2k

BT

so1/2

g(E

F)k

BT

ele

ctro

ns

incr

ease

dth

eir

energ

yby

kBT.

kBT.

T>

0

T=

0

n(E

,T)

E

g(E

)

EF

The

diffe

rence

inth

erm

al

energ

yfr

om

the

valu

eat

T=

0°K

21

()

(0)

()(

)2

FB

ET

EgE

kT

−

D

iffe

rentiating w

ith r

esp

ect

to T

giv

es

the

heat

capaci

ty a

t co

nst

ant

volu

me,

2()

vF

B

EC

gE

kT

T∂=

=∂

2(

)3

FF

NEgE

=3

33

()2

2F

FB

F

NN

gE

EkT

== 2

23

()

2v

FB

B

BF

NC

gE

kT

kT

kT

==

3 2v

B

FTC

Nk

T

=

H

eat

cap

acit

y o

fFree e

lectr

on

gas

Transport Properties of Conduction Electrons

Fe

rmi-

Dirac

dis

trib

ution

funct

ion

desc

ribes

the

behavio

ur

of

ele

ctro

ns

only

at

equili

brium

.

Ifth

ere

isan

applie

dfield

(Eor

B)

or

ate

mpera

ture

gra

die

nt

the

transp

ort

coeff

icie

nt

of

therm

al

and

ele

ctrica

lco

nduct

ivitie

sm

ust

be

therm

al

and

ele

ctrica

lco

nduct

ivitie

sm

ust

be

consi

dere

d.

Tran

sp

ort

co

eff

icie

nts

σ,E

lectr

ical

co

nd

ucti

vit

yK

,Th

erm

al

co

nd

ucti

vit

y

Tota

l heat

capaci

ty a

t lo

w t

em

pera

ture

s

w

here

γ a

nd

βare

const

ants

and t

hey c

an

3C

TT

γβ

=+

Ele

ctro

nic

Heat

capaci

tyLa

ttic

e H

eat

Capaci

ty

w

here

γ a

nd

βare

const

ants

and t

hey c

an

be f

ound d

raw

ing C

v/T

as

a f

unct

ion o

f T

2

Equation

of

motion

of

an

ele

ctro

nw

ith

an

applie

dele

ctric

and

magnetic

field

.

This

isju

stN

ew

ton’s

law

for

part

icle

sof

mass

me

and

charg

e(-

e).

e

dv

meE

evB

dt=−

−×

rur

rur

This

isju

stN

ew

ton’s

law

for

part

icle

sof

mass

me

and

charg

e(-

e).

The

use

of

the

class

ical

equation

of

motion

of

apart

icle

todesc

ribe

the

behavio

ur

of

ele

ctro

ns

inpla

ne

wave

state

s,w

hic

hexte

nd

thro

ughout

the

cryst

al.

Apart

icle

-lik

eentity

can

be

obta

ined

by

superp

osi

ng

the

pla

ne

wave

state

sto

form

aw

avepack

et.

The v

elo

city

of

the w

avepack

et

is t

he g

roup

velo

city

of

the w

aves.

Thus

So o

ne c

an u

se e

quation o

f m

dv/d

t

1

ee

ddE

kp

vm

mdk

dk

ω=

==

=

rur

rh

rr

h

22

2e

kE

m

pkω

==

=

hh h

So o

ne c

an u

se e

quation o

f m

dv/d

t

e

dv

vm

eE

evB

dt

τ

+=−

−×

rr

urr

ur

τ=

mean fre

e t

ime b

etw

een c

olli

sions.

An e

lect

ron

lose

s all

its

energ

y in t

imeτ

(*)

In

the a

bse

nce

of

a m

agnetic

field

, th

e a

pplie

d E

re

sults

a c

onst

ant

acc

ele

ration b

ut

this

will

not

cause

a c

ontinuous

incr

ease

in c

urr

ent.

Sin

ce

ele

ctro

ns

suffer

colli

sions

with

phonons

ele

ctro

ns

The a

dditio

nal te

rm

cause

the v

elo

city

v t

o

deca

y e

xponentially

with a

tim

e c

onst

ant

when

the a

pplie

d E

is

rem

oved.

e

vm

τ

r

τ

The Electrical Conductivty

In

the p

rese

nce

of

DC fie

ld o

nly

,eq.(

*)

has

the

steady s

tate

solu

tion

ev

Emτ

=−

rur

e

e mτµ=

Mobili

ty f

or

ele

ctro

n

M

obili

ty d

ete

rmin

es

how

fast

the c

harg

e c

arr

iers

m

ove w

ith a

n E

.

em

a c

onst

ant

of

pro

port

ionalit

y(m

obili

ty)

e

em

ele

ctro

n

Ele

ctrica

l cu

rrent

densi

ty,

J

W

here

n is

the e

lect

ron d

ensi

ty a

nd v

is

drift

velo

city

.H

ence

()

Jn

ev

=−

Nn

V=

2 τur

ur2

neτ

e

ev

Emτ

=−

rur

2 e

ne

JE

m

τ=

urur

JE

σ=

urur

2 e

ne m

τσ=

Ele

ctr

ical co

nd

ucti

vit

y

Ohm

’s law

1ρ

σ=

LR

Aρ=

Ele

ctri

cal Resi

stiv

ity a

nd R

esi

stance

Collisions

In

aperf

ect

cryst

al;

the

colli

sions

of

ele

ctro

ns

are

with

therm

ally

exci

ted

latt

ice

vib

rations

(sca

ttering

of

an

ele

ctro

nby

aphonon).

This

ele

ctro

n-p

honon

scatt

ering

giv

es

ate

mpera

ture

dependent

colli

sion

tim

e()

phT

τte

mpera

ture

dependent

colli

sion

tim

ew

hic

hte

nds

toin

finity

as

T0.

In

real

meta

l,th

eele

ctro

ns

als

oco

llide

with

impurity

ato

ms,

vaca

nci

es

and

oth

er

imperf

ect

ions,

this

resu

ltin

afinite

scatt

eri

ng

tim

eeven

at

T=

0.

()

phT

τ

0τ

The t

ota

l sc

att

ering r

ate

for

a s

lightly im

perf

ect

cr

yst

alat

finite t

em

pera

ture

;

So t

he t

ota

l re

sist

ivity ρ

,

0

11

1

()

phT

ττ

τ=

+

Due t

o p

honon

Due t

o im

perf

ect

ions

So t

he t

ota

l re

sist

ivity ρ

,

This

is

know

n a

s M

att

heis

en’s

rule

and illu

stra

ted in

follo

win

g f

igure

for

sodiu

m s

peci

men o

f diffe

rent

purity

.

02

22

0

()

()

ee

e

I

ph

mm

mT

ne

ne

Tne

ρρ

ρτ

ττ

==

+=

+

Ideal re

sist

ivity

Resi

dual re

sist

ivity

Residual resistance ratio

Resi

dual re

sist

ance

ratio =

room

tem

p.

resi

stiv

ity/

resi

dual re

sist

ivity

and it

can b

e a

s hig

h a

s

for

hig

hly

puri

fied s

ingle

cry

stals

.6

10

Tem

pera

ture

pure

impure

Collision tim

e τ

σ

10

15.310(

)pureNa

residual

xm

σ−

=Ω−

71

()

2.010(

)sodium

RT

xm

σ−

=Ω−

em

m=

14

22.610

mx

sneσ

τ−

=

can b

e f

ound b

y

takin

g

at

RT

28

32.710

nx

m−

=ne

11

7.010

xs

−

61.110

/Fv

xm

s=

()29

lRT

nm

=

(0)

77lT

mµ=

=

at

T=

0

Fl

vτ

=Ta

kin

g

;

a

nd

These

mean

free

path

sare

much

longer

than

the

inte

rato

mic

dis

tance

s,co

nfirm

ing

that

the

free

ele

ctro

ns

do

not

colli

de

with

the

ato

ms

them

selv

es.

Thermal conductivity, K

metals

nonmetals

KK

−

Due t

o t

he h

eat

tranport

by t

he c

onduct

ion e

lect

rons

Ele

ctro

ns

com

ing

from

ahott

er

regio

nof

the

meta

lca

rry

more

therm

al

energ

yth

an

those

from

aco

ole

rre

gio

n,

resu

ltin

gin

anet

flow

of

heat.

The

therm

alco

nduct

ivity

1 3V

FK

Cvl

=V

C

net

flow

of

heat.

The

therm

alco

nduct

ivity

l

Fv

BkT

Fε

Fl

vτ

=2

1 2F

eF

mv

ε=

where

is

the s

peci

fic

heat

per

unit v

olu

me

is t

he m

ean f

ree p

ath

;

a

nd

Ferm

i ener

gy

isth

em

ean

speed

of

ele

ctro

ns

resp

onsi

ble

for

therm

al

conduct

ivity

since

only

ele

ctro

nst

ate

sw

ithin

about

of

change

their

occ

upation

as

the

tem

pera

ture

vari

es.

22

22

11

2(

)3

32

3

BV

FB

F

Fe

e

NT

nkT

KCv

kV

Tm

m

ππ

ττ

ετ

==

=2 2

vB

FTC

Nk

T

π

=

where

Wiedem

ann-Franz law

2 e

ne m

τσ=

22

3

B e

nkT

Km

πτ

=

The r

atio o

f th

e e

lect

rica

l and t

herm

al co

nduct

ivitie

s is

independent

of

the

ele

ctro

n g

as

para

mete

rs;

22

82

2.4510

3

Kk

xW

KT

e

πσ

−−

==

Ω

B

ele

ctro

n g

as

para

mete

rs;

82

2.2310

KL

xW

KT

σ−

−=

=Ω

Lore

ntz

num

ber

For

copper

at

0 C