chapter 5 (electron theory)

8/18/2019 Chapter 5 (Electron Theory)

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CHAPTER 5

FREE ELECTRON THEORY

Free Electron Theory Many solids conduct electricity.

There are electrons that are not bound to atoms but are able to move through thewhole 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 withdecreasing temperature.

and can be reduced by the addition of

small amounts of impurity.

Semiconductors tend to become insulators at low T.

6 8

( ) ;10 10metals RT m

( ) ( ) pure semiconductor metal

RT RT


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Why mobile electrons appear in some

solids and others?

When the interactions between electrons are considered thisbecomes 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 metalsusing the assumption that conduction electrons exist and consistof all valence electrons from all the metals; thus metallic Na, Mg

and Al will be assumed to have 1, 2 and 3 mobile electrons peratom respectively.

A simple theory of ‘ free electron model’ which worksremarkably well will be described to explain these properties of metals.

Why mobile electrons appear in somesolids and not others?

According to free electron model (FEM), thevalance electrons are responsible for theconduction of electricity, and for this reasonthese 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


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When we bring Na atoms together to form a Nametal,

Na has a BCC structure and the distance betweennearest 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 thisobservation it follows that a valance electron is nolonger attached to a particular ion, but belongs toboth neighbouring ions at the same time.

Na metal

The removal of the valance electrons leavesa positively charged ion.

The charge density associated the positiveion cores is spread uniformly throughout the

metal so that the electrons move in aconstant electrostatic potential. All thedetails of the crystal structure is lost whenthis assunption is made.

+

+ + +

+ +

A valance electron really belongs to the wholecrystal, since it can move readily from one ion toits neighbour, and then the neighbour’sneighbour, and so on.

This mobile electron becomes a conductionelectron in a solid.

According to FEM this potential is taken as zeroand the repulsive force between conductionelectrons are also ignored.


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Therefore, these conduction electrons can beconsidered as moving independently in a squarewell 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

2 E

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 z

i k x k y k z ik r x y z e e

V V

Normalization constant

Na p 2,where k 2 Na pk 2 2k p p Na L

2 x

k p L

2 y

k q L

2

z k r

L

; ;


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

the momentum of

Energy is completely kinetic

2 2

2

k E

m

2 2 2 2( )2

x y z E k k k

m

( , , ) x y z

p k k k

2 221

2 2

k

mv m

2 2 2 2

m v k p k

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

2

2( ) ,

2

Vk g k dk dk

where g(k) is thedensity of states perunit magnitude of k.


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The number of allowed states

per unit energy range?

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

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

g E g k dE

2 2

2

k E

m

2

dE k

dk m

2

2mE k

( ) g E 2 ( ) g k dk

dE 2

22

V

k k

2

2mE

2

m

k

3/ 2 1/ 2

2 3 (2 )

2( )

V m E g E

Ground state of the free electron gas

Electrons are fermions (s=±1/2) and obeyPauli exclusion principle; each state canaccommodate only one electron.

The lowest-energy state of N freeelectrons is therefore obtained by fillingthe N states of lowest energy.


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Thus all states are filled up to an energy EF,known as Fermi energy, obtained byintegrating density of states between 0 and EF,should equal N. Hence

Remember

Solve for EF (Fermi energy);

2/ 32 2

3

2 F

N E

m V

3/ 2 1/ 2

2 3 (2 )

2( )

V m E g E

3/ 2 1/ 2 3/ 2

2 3 2 3

0 0

( ) (2 ) (2 )2 3

F F E E

F

V V N 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

2

F

F

e

k E

m

kz

ky

kx

Fermi surfaceE=EF

kF

2/ 32 2

3

2 F

N E

m V

From these two equation kFcan be found as,

1/ 323

F

N k

V

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

states at absolute zero for the free electron gas.


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Typical values may be obtained by usingmonovalent potassium metal as an example; forpotassium the atomic density and hence the

valance electron density N/V is 1.402x1028 m-3 sothat

Fermi (degeneracy) Temperature TF by

193.40 10 2.12 F

E J eV

10.746

F k A

F B F E k T

42.46 10 F F B

E T K k

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

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

Fermi momentum

These are the momentum and the velocity valuesof the electrons at the states on the Fermisurface of the Fermi sphere.

So, Fermi Sphere plays important role on thebehaviour of metals.

F F P k F e F P m V

6 1

0.86 10 F

F

e

P

V msm


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2/ 32 2

3 2.122

F N E eV m V

1/ 32

13 0.746 F N

k AV

6 10.86 10 F

F

e

P V ms

m

42.46 10 F

F

B

E T K

k

Typical values of monovalent potassium metal;


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The free electron gas at finite temperature

At a temperature T the probability of occupationof an electron state of energy E is given by theFermi distribution function

Fermi distribution function determines theprobability of finding an electron at the energyE.

( ) /

1

1 F B FD E E k T

f e

EFEEF

0.5

f FD(E,T)

E

( ) /

1

1 F B FD E E k T

f e

Fermi Function at T=0and at a finite temperature

f FD=? At 0°K

i. EEF

( ) /

11

1 F B FD E E k T

f e

( ) /

10

1 F B FD E E k T

f e


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Fermi-Dirac distribution function at

various temperatures,

T>0

T=0

n(E,T)

E

g(E)

EF

n(E,T) number of freeelectrons per unit energyrange is just the areaunder n(E,T) graph.

( , ) ( ) ( , ) FD

n E T g E f E T

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

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


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Fermi-Dirac distribution function is asymmetric function; at finitetemperatures, the same number of levelsbelow 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 thedistribution of electrons can be resembled intotriangles of height 1/2g(EF) and a base of 2kBT so1/2g(EF)kBT electrons increased their energy bykBT.

T>0

T=0

n(E,T)

E

g(E)

EF

The difference in thermalenergy from the value atT=0°K

21( ) (0) ( )( )

2 F B E T E g E k T


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Differentiating with respect to T gives theheat capacity at constant volume,

2( )v F B

E C g E k T

T

2( )

3

3 3( )

2 2

F F

F

F B F

N E g E

N N g E

E k T

2 23( ) 2

v F B B

B F

N C g E k T k T k T

3

2v B

F

T C Nk

T

Heat capacity of Free electron gas

Transport Properties of Conduction Electrons

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

If there is an applied field (E or B) or atemperature gradient the transport coefficient of thermal and electrical conductivities must beconsidered.

Transport coefficients

σ,Electricalconductivity

K,Thermalconductivity


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Total heat capacity at low temperatures

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

2

3

C T T Electronic

Heat capacityLattice Heat

Capacity

Equation of motion of an electron with an appliedelectric and magnetic field.

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

The use of the classical equation of motion of aparticle to describe the behaviour of electrons inplane wave states, which extend throughout thecrystal. A particle-like entity can be obtained bysuperposing the plane wave states to form awavepacket.

e

dvm eE ev B

dt


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

So one can use equation of mdv/dt

1

e e

d dE k pv

m md k d k

2 2

2e

k E

m

p k

e

dv vm eE ev B

dt

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

(*)

In the absence of a magnetic field, the applied Eresults a constant acceleration but this will notcause a continuous increase in current. Sinceelectrons 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

v

m


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The Electrical Conductivty

In the presence of DC field only, eq.(*) has thesteady state solution

Mobility determines how fast the charge carriersmove with an E.

e

ev E

m

a constant of proportionality

(mobility)

e

e

e

m

Mobility for

electron

Electrical current density, J

Where n is the electron density and v is driftvelocity. Hence

( ) J n e v N

nV

2

e

ne J E m

J E

2

e

ne

m

e

ev E

m

Electrical conductivity

Ohm’s law1

L R

A

Electrical Resistivity and Resistance


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Collisions

In a perfect crystal; the collisions of electrons arewith thermally excited lattice vibrations(scattering of an electron by a phonon).

This electron-phonon scattering gives atemperature dependent collision timewhich tends to infinity as T 0.

In real metal, the electrons also collide with

impurity atoms, vacancies and otherimperfections, this result in a finite scatteringtime even at T=0.

( ) ph T

0

The total scattering rate for a slightly imperfectcrystal at finite temperature;

So the total resistivity ρ,

This is known as Mattheisen’s rule and illustrated infollowing figure for sodium specimen of differentpurity.

0

1 1 1

( ) ph T

Due to phonon Due to imperfections

02 2 2

0( )( )

e e e

I

ph

m m m

T ne ne T ne

Ideal resistivity Residual resistivity


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Residual resistance ratio

Residual resistance ratio = room temp. resistivity/ residual resistivity

and it can be as high as for highly purified single crystals.6

10

Temperature

pure

impure

Collision time

10 15.3 10 ( )

pureNaresidual x m

7 1( ) 2.0 10 ( )

sodium RT x m

28 32.7 10n x m

em m 14

2 2.6 10

m x s

ne

117.0 10 x s

61.1 10 /

F v x m s

( ) 29l RT nm

( 0) 77l T m

can be found by taking

at RT

at T=0

F l v Taking ; and

These mean free paths are much longer than the interatomicdistances, confirming that the free electrons do not collide with theatoms themselves.


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Thermal conductivity, K

metals non metals K K

1

3 V F K C v l V C

Due to the heat tranport by the conduction electrons

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

l

F v

Bk T

F

F l v 21

2 F e F m 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 conductivitysince only electron states within about of change theiroccupation as the temperature varies.

2 2 221 1 2( )

3 3 2 3

BV F B F

F e e

N T nk T K C v k

V T m m

2

2v B

F

T C Nk

T

where

Wiedemann-Franz law

2

e

ne

m

2 2

3

B

e

nk T K

m

228 2

2.45 103

K k x W K

T e

B

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

8 22.23 10

K L x W K

T

Lorentznumber

For copper at 0 C

chapter 5 (electron theory)

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