CHAPTER 5FREE ELECTRON

THEORY

Free Electron TheoryMany solids conduct electricity.

There are electrons that are not bound to atoms but are able to move through the whole crystal.

Conducting solids fall into two main classes; metals and semiconductors.

and increases by the addition of small

amounts of impurity. The resistivity normally decreases monotonically with decreasing temperature.

and can be reduced by the addition of

small amounts of impurity.

Semiconductors tend to become insulators at low T.

6 8( ) ;10 10metalsRT m

( ) ( )pure semiconductor metalRT RT

Why mobile electrons appear in some solids and others?

When the interactions between electrons are considered this becomes a very difficult question to answer.

The common physical properties of metals;• Great physical strength• High density• Good electrical and thermal conductivity, etc.

This chapter will calculate these common properties of metals using the assumption that conduction electrons exist and consist of all valence electrons from all the metals; thus metallic Na, Mg and Al will be assumed to have 1, 2 and 3 mobile electrons per atom respectively. A simple theory of ‘ free electron model’ which works remarkably well will be described to explain these properties of metals.

Why mobile electrons appear in some solids and not others? According to free electron model (FEM), 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.

Valance electron (loosely bound)

Core electrons

When we bring Na atoms together to form a Na metal,

Na has a BCC structure and the distance between nearest neighbours is 3.7 A˚ The radius of the third shell in Na is 1.9 A˚

Solid state of Na atoms overlap slightly. From this observation it follows that a valance electron is no longer attached to a particular ion, but belongs to both neighbouring ions at the same time.

Na metal

The removal of the valance electrons leaves a positively charged ion.

The charge density associated the positive ion cores is spread uniformly throughout the metal so that the electrons move in a constant electrostatic potential. All the details of the crystal structure is lost when this assunption is made.

+

+ + +

+ +

A valance electron really 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.

According to FEM this potential is taken as zero and the repulsive force between conduction electrons are also ignored.

Therefore, these conduction electrons can be considered as moving independently in a square well of finite depth and the edges of well corresponds to the edges of the sample.

Consider a metal with a shape of cube with edge length of L, Ψ and E can be found by solving Schrödinger equation

0 L/2

V

L/2

22

2E

m

0V Since,

( , , ) ( , , )x L y L z L x y z

• By means of periodic boundary conditions Ψ’s are running waves.

The solutions of Schrödinger equations are plane waves,

where V is the volume of the cube, V=L3

So the wave vector must satisfy

where p, q, r taking any integer values; +ve, -ve or zero.

( )1 1( , , ) x y zi k x k y k zik rx y z e e

V V

Normalization constant

Na p 2,where k

2Na p

k

2 2k p p

Na L

2xk p

L

2yk q

L

2

zk rL

; ;

The wave function Ψ(x,y,z) corresponds to an energy of

the momentum of

Energy is completely kinetic

2 2

2

kE

m

22 2 2( )

2 x y zE k k km

( , , )x y zp k k k

2 221

2 2

kmv

m 2 2 2 2m v k p k

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

2

2( ) ,

2

Vkg k dk dk

where g(k) is the density of states per unit magnitude of k.

The number of allowed states per unit energy range?

Each k state represents two possible electron states, one for spin up, the other is spin down.

( ) 2 ( )g E dE g k dk ( ) 2 ( )dk

g E g kdE

2 2

2

kE

m 2dE k

dk m

2

2mEk

( )g E 2 ( )g kdk

dE2

22

V

kk

2

2mE

2

m

k

3/ 2 1/ 22 3(2 )

2( )

Vm Eg E

Ground state of the free electron gas

Electrons are fermions (s=±1/2) and obey Pauli exclusion principle; each state can accommodate only one electron.

The lowest-energy state of N free electrons is therefore obtained by filling the N states of lowest energy.

Thus all states are filled up to an energy EF, known as Fermi energy, obtained by integrating density of states between 0 and EF,

should equal N. Hence

Remember

Solve for EF (Fermi energy);

2/32 23

2F

NE

m V

3/ 2 1/ 22 3(2 )

2( )

Vm Eg E

3/ 2 1/ 2 3/ 22 3 2 3

0 0

( ) (2 ) (2 )2 3

F FE E

F

V VN g E dE m E dE mE

The occupied states are inside the Fermi sphere in k-space

shown below; radius is Fermi wave number kF.

2 2

2F

Fe

kE

m

kz

ky

kx

Fermi surfaceE=EF

kF

2/32 23

2F

NE

m V

From these two equation kF

can be found as,1/323

F

Nk

V

The surface of the Fermi sphere represent the boundary between occupied and unoccupied k

states at absolute zero for the free electron gas.

Typical values may be obtained by using monovalent potassium metal as an example; for potassium the atomic density and hence the valance electron density N/V is 1.402x1028 m-3 so that

Fermi (degeneracy) Temperature TF by

193.40 10 2.12FE J eV 10.746Fk A

F B FE k T

42.46 10FF

B

ET K

k

It is only at a temperature of this order that the particles in a classical gas can attain (gain) kinetic energies as high as EF .

Only at temperatures above TF will the free electron gas behave like a classical gas.

Fermi momentum

These are the momentum and the velocity values of the electrons at the states on the Fermi surface of the Fermi sphere.

So, Fermi Sphere plays important role on the behaviour of metals.

F FP k F e FP mV

6 10.86 10FF

e

PV ms

m

2/32 232.12

2F

NE eV

m V

1/3213

0.746F

Nk A

V

6 10.86 10FF

e

PV ms

m

42.46 10FF

B

ET K

k

Typical values of monovalent potassium metal;

The free electron gas at finite temperature At a temperature T the probability of occupation

of an electron state of energy E is given by the Fermi distribution function

Fermi distribution function determines the probability of finding an electron at the energy E.

( ) /

1

1 F BFD E E k Tf

e

EFE<EF E>EF

0.5

fFD(E,T)

E

( ) /

1

1 F BFD E E k Tf

e

Fermi Function at T=0 and at a finite temperature

fFD=? At 0°K

i. E<EF

ii. E>EF

( ) /

11

1 F BFD E E k Tf

e

( ) /

10

1 F BFD E E k Tf

e

Fermi-Dirac distribution function at various

temperatures,

T>0

T=0

n(E,T)

E

g(E)

EF

n(E,T) number of free electrons per unit energy range is just the area under n(E,T) graph.

( , ) ( ) ( , )FDn E T g E f E T

Number of electrons per unit energy range according to the free electron model?

The shaded area shows the change in distribution between absolute zero and a finite temperature.

Fermi-Dirac distribution function is a symmetric function; at finite temperatures, the same number of levels below EF is emptied and same number of levels above EF are filled by electrons.

T>0

T=0

n(E,T)

E

g(E)

EF

Heat capacity of the free electron gas From the diagram of n(E,T) the change in the

distribution of electrons can be resembled into triangles of height 1/2g(EF) and a base of 2kBT so 1/2g(EF)kBT electrons increased their energy by kBT.

T>0

T=0

n(E,T)

E

g(E)

EF

The difference in thermal energy from the value at T=0°K

21( ) (0) ( )( )

2 F BE T E g E k T

Differentiating with respect to T gives the heat capacity at constant volume,

2( )v F B

EC g E k T

T

2( )

33 3

( )2 2

F F

FF B F

N E g E

N Ng E

E k T

2 23( )

2v F B BB F

NC g E k T k T

k T

3

2v BF

TC Nk

T

Heat capacity ofFree electron gas

Transport Properties of Conduction Electrons

Fermi-Dirac distribution function describes the behaviour of electrons only at equilibrium.

If there is an applied field (E or B) or a temperature gradient the transport coefficient of thermal and electrical conductivities must be considered.

Transport coefficientsTransport coefficients

σ,Electricalconductivityσ,Electricalconductivity

K,ThermalconductivityK,Thermal

conductivity

Total heat capacity at low temperatures

where γ and β are constants and they can be found drawing Cv/T as a function of T2

3C T T

ElectronicHeat capacity

Lattice HeatCapacity

Equation of motion of an electron with an applied electric and magnetic field.

This is just Newton’s law for particles of mass me and charge (-e).

The use of the classical equation of motion of a particle to describe the behaviour of electrons in plane wave states, which extend throughout the crystal. A particle-like entity can be obtained by superposing the plane wave states to form a wavepacket.

e

dvm eE ev B

dt

SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS

The velocity of the wavepacket is the group velocity of the waves. Thus

So one can use equation of mdv/dt

1

e e

d dE k pv

m mdk dk

SSSSSSSSSSSSS S

2 2

2 e

kE

m

p k

e

dv vm eE ev B

dt

SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS

= mean free time between collisions. An electron loses all its energy in time

(*)

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

The additional term cause the velocity v

to decay exponentially with a time constant

when the applied E is removed.

e

vm

The Electrical Conductivty In the presence of DC field only, eq.(*) has the

steady state solution

Mobility determines how fast the charge carriers move with an E.

e

ev E

m

SSSSSSSSSSSSS S

a constant ofproportionality

(mobility)

ee

e

m

Mobility forelectron

Electrical current density, J

Where n is the electron density and v is drift velocity. Hence

( )J n e v N

nV

2

e

neJ E

m

SSSSSSSSSSSSSSSSSSSSSSSSSSSS

J ESSSSSSSSSSSSSSSSSSSSSSSSSSSS

2

e

ne

m

e

ev E

m

SSSSSSSSSSSSS S

Electrical conductivity

Ohm’s law1

L

RA

Electrical Resistivity and Resistance

Collisions In a perfect crystal; the collisions of electrons are

with thermally excited lattice vibrations (scattering of an electron by a phonon).

This electron-phonon scattering gives a temperature dependent collision time which tends to infinity as T 0.

In real metal, the electrons also collide with impurity atoms, vacancies and other imperfections, this result in a finite scattering time even at T=0.

( )ph T

0

The total scattering rate for a slightly imperfect crystal at finite temperature;

So the total resistivity ρ,

This is known as Mattheisen’s rule and illustrated in following figure for sodium specimen of different purity.

0

1 1 1

( )ph T

Due to phonon Due to imperfections

02 2 20

( )( )

e e eI

ph

m m mT

ne ne T ne

Ideal resistivity Residual resistivity

Residual resistance ratioResidual resistance ratio = room temp. resistivity/ residual resistivity

and it can be as high as for highly purified single crystals.610

Temperature

pure

impure

Rela

tive

resi

stance

Collision time

10 15.3 10 ( )pureNaresidual x m 7 1( ) 2.0 10 ( )sodiumRT x m

28 32.7 10n x m

em m 1422.6 10

mx s

ne

117.0 10x s

61.1 10 /Fv x m s( ) 29l RT nm( 0) 77l T m

can be found by taking

at RT

at T=0

Fl v Taking ; and

These mean free paths are much longer than the interatomic distances, confirming that the free electrons do not collide with the atoms themselves.

Thermal conductivity, K

metals non metalsK K

1

3 V FK C v l VC

Due to the heat tranport by the conduction electrons

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

l

Fv

Bk T F

Fl v 21

2F e Fm v

where is the specific heat per unit volume

is the mean free path; and Fermi energy

is the mean speed of electrons responsible for thermal conductivity since only electron states within about of change their occupation as the temperature varies.

2 2 221 1 2

( )3 3 2 3

BV F B F

F e e

N T nk TK C v k

V T m m

2

2v BF

TC Nk

T

where

Wiedemann-Franz law

2

e

ne

m

2 2

3B

e

nk TK

m

228 22.45 10

3

K kx W K

T e

B

The ratio of the electrical and thermal conductivities is independent of the electron gas parameters;

8 22.23 10K

L x W KT

Lorentz number

For copper at 0 C