abeer alshammari,2016 1

abeer alshammari,2016 2

Many 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

Free Electron Model

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

Schematic model of metallic crystal, such as Na, Li, K, etc.

The equilibrium positions of the atomic cores are positioned on

the crystal lattice and surrounded by a sea of conduction

electrons.

For Na, the conduction electrons are from the 3s valence

electrons of the free atoms. The atomic cores contain 10

electrons in the configuration: 1s22s2p6.

In this model, electrons are completely free to move about, free from collisions, except for a surface potential that keeps the electrons inside the metal.

In the alkali metals, with a bcc structure, the cores take up about 15% of the volume of the crystal, but in the noble metals (Cu, Ag, Au), with an fcc structure, the atomic cores are relatively larger and maybe close to contacting or in contact with each other.

PAUL Drude

(1863-1906)

• resistivity ranges from 108 m (Ag) to 1020 m (polystyrene)

• Drude (circa 1900) was asking why? He was working prior to the

development of quantum mechanics, so he began with a classical

model:

• positive ion cores within an electron gas that follows Maxwell-

Boltzmann statistics

• following the kinetic theory of gases- the electrons in the gas move

in straight lines and make collisions only with the ion cores – no

electron-electron interactions.

• He envisioned instantaneous collisions in which electrons lose any energy gained from the electric

field.

• The mean free path was approximately the inter-ionic core spacing.

• Model successfully determined the form of Ohm’s law in terms of free electrons and a relation

between electrical and thermal conduction σ / κ , but failed to explain electron heat capacity and

the magnetic susceptibility of conduction electrons, and mean free path

abeer alshammari,2016 5

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

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

According to FEM this potential is taken as zero and the repulsive force betweenconduction 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.

Fermi Gas model

abeer alshammari,2016

6

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 Since 0V

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

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

Energy Levels In One Dimension

𝐻𝛹 = 𝐸𝛹

Energy Levels In One Dimension2 2

22n n n

dH

m dx

Orbital: solution of a 1-e Schrodinger equation

0 0n n L

sinn

nA x

L

2n L

n

Particle in a box

2sin

n

A x

1,2,n

22

2n

n

m L

Since the n x) is a continuous function and is equal to zero beyond the length L, the boundary conditions for the wave function are

These solutions correspond to standing waves with a different number of nodes within the potential well as is shown in Fig.

Pauli-exclusion principle: No two electrons can occupy the same quantum state.

Quantum numbers for free electrons: (n, ms ) ,sm

Degeneracy: number of orbitals having the same energy.

Fermi energy εF = energy of topmost filled orbital when system is in ground state (at T=0K).

22

2

FF

n

m L

2F

Nn

N number of valence electrons

where n describes the orbital n (x) , and ms describes the projection of the spin momentum on a quantization axis

For the onedimensional system of N electrons we find

In metals the value of the Fermi energy is of the order of 5 eV.

(a) Occupation of energy levels according to the Pauli

exclusion principle, (b) The distribution function f(E), at T = 0°K and T> 0°K.

Effect Of Temperature On The Fermi-DiracDistribution

Fermi-Dirac distribution :

1

1f

e

1

Bk T

Chemical potential μ = μ(T) is determined by N d g f g = density of states

At T = 0: 1

for0

f

→ 0 F

1

2f For all T :

For ε >> μ : f e

(Boltzmann distribution)

3D e-gas

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

At T = 0, Fermions occupy the lowest energy levels.Near T = 0, there is little chance that thermal agitation will kick a Fermion to an energy greater than EF.

Free Electron Gas In Three Dimensions

2 2 2 2

2 2 22

d d dH

m dx dy dz

r r

Particle in a box (fixed) boundary conditions:

0, , , , ,0, , , , ,0 , , 0y z L y z x z x L z x y x y L n n n n n n

sin sin sinyx z

nn nA x y z

L L L

n

Periodic boundary conditions:

Standing

waves

, , , , , , , ,x y z x L y z x y L z x y z L k k k k

→

→iA e k r

k

2 ii

nk

L

0, 1, 2,in

2 2

2

k

m

k

Traveling

waves

1, 2,in

If the electrons are confined to a cube of edge L, the solution is the standing wave

our wavefunction is periodic in x, y, and z directions with period L,

i k kp

kk → ψk is a momentum eigenstate with eigenvalue ℏk.

p km

k

v

3

3

42

8 3F

VN k

1/323

F

Nk

V

2 2

2

FF

k

m

2/32 23

2

N

m V

FF

kv

m

1/323 N

m V

In the ground state a system of N electrons occupies states with lowest possible

energies. Therefore all the occupied states lie inside the sphere of radius kF

• The Fermi energy and the Fermi wavevector (momentum) are

determined by the number of valence electrons in the system.

• In order to find the relationship between N and kF, we need to

count the total number of orbitals in a sphere of radius kF

which should be equal to N. There are two available spin

states for a given set of kx, ky, and kz.

• The volume in the k space which is occupies by this state is

equal to (2 / L) . Thus in the sphere of (4 / 3) k the total number of states is

Fermi velocity

which depends only of the electron concentration

The surface of the Fermi sphere represent the boundary between

occupied and unoccupied k states at absolute zero for the free

electron gas.

abeer alshammari,2016 12

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

Fermi (degeneracy) Temperature TF by

F B FE k T

Density of states:

3/2

2 2

2

2

V m

3

23F

VN k

3/2

2 2

2

3

FmV

→

3

2F

F

ND

3

2 F F

ND

•The Density of States D(E) specifies how many states exist at a given

energy E.

•The Fermi Function f(E) specifies how many of the existing states at

energy E will be filled with electrons.

Fermi-Dirac distribution functionis a symmetric function; at finitetemperatures, the same numberof levels below EF is emptied andsame number of levels above EF

are filled by electrons.

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 (½)g(Ef) and a base of 2kBT so (½)g(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) ( )( )

2F BE T E g E k T

•The electron energy levels are mostly filled up to the Fermi energy.

•So, only a small fraction of electrons, approximately T/TF, can be excited to higher levels –

because there is only about kBT of thermal energy available.

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

N(E,T)=g(E) f(E,T)

• Differentiating With Respect To T Gives The Heat

Capacity At Constant Volume,

2( )v F B

EC g E k T

T

2( )

3

3 3( )

2 2

F F

F

F B F

N E g E

N Ng E

E k T

2 23( )

2v F B B

B F

NC g E k T k T

k T

3

2v B

F

TC Nk

T

Heat Capacity of

Free Flectron Gas

Low-Temperature Heat Capacity

2

2el B

F

TC Nk

T

3

el lat

lat

C C C

C T

Copper

3

• Total metallic heat capacity at low

temperatures

Where γ & β are constants found plotting

cv/T as a function of T2

3C T T Electronic

Heat capacity

Lattice Heat

Capacity

abeer alshammari,2016 18

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 coefficients

σ,Electricalconductivity

K,Thermalconductivity

• Equation of motion of an electron with an appliedelectric 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 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

• 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

2 2

2 e

kE

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 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 Conductivity

• 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

a constant ofproportionality

(mobility)

e

e

e

m

Mobility for

electron

• 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

J E

2

e

ne

m

e

ev E

m

Electrical conductivity

Ohm’s law

1

LR

A

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 2

0

( )( )

e e e

I

ph

m m mT

ne ne T ne

Ideal resistivity Residual resistivity

Residual Resistance Ratio

Residual resistance ratio = room temp. Resistivity/ residual resistivity

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

Temperature

pure

impure

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 14

22.6 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 interatomicdistances, confirming that the free electrons do not collide with theatoms themselves.

Thermal Conductivity, K

metals non metalsK K

1

3V FK C v l VC

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

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 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 TK C v k

V T m m

2

2v B

F

TC Nk

T

where

>

Wiedemann-Franz Law

2

e

ne

m

2 2

3

B

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

abeer alshammari,2016 31

• A uniform magnetic field B is

applied in the y-direction.

• If the charge carriers are electrons

moving in the negative x-

direction with a velocity vd, they

will experience an upward

magnetic force fb.

• The electrons will be deflected

upward, making the upper edge

negatively charged and the lower

edge positively charged.

Hall Effect

Hall Effect

z B

x E

zyx vInitially,

z

x

zyx

B

E

vvv

)BvE(v1

F

e

dt

dm

)(1

)(1

xy

yxx

Bvevdt

dmF

BvEevdt

dmF

y

x

net force in

x direction

net force in

y direction

Hall Effect

As a result, electrons

move in the y direction

and an electric field

component appears in the

y direction, Ey. This will

continue until the Lorentz

force is equal and

opposite to the electric

force due to the buildup of

electrons – that is, a

steady condition arises.B

)(

)(

xyy

yxx

BvEemv

BvEemv

Hall Effect

m

eBC

xCy

y

yCx

x

vm

Eev

vm

Eev

Hall Effect

xx Em

eBEE Cy

e

vmE

vm

Eev

xCy

xCy

y

0

e

vmE

Em

ev

xx

xx

Hall Effect

The Hall coefficient is defined as:

neBE

m

ne

Em

eB

Bj

ER

1

x

2

x

x

yH

For copper:

n = 8.47 × 1028 electrons/m3.

The sign and magnitude of RH

gives the sign of the charge carriers and their density.In most metals, the charge carriers are electrons and the charge density determined from the Hall effect measurements agrees with calculated values for metals which release a single valence electron and charge density is approximately equal to the number of valence electrons per unit volume.

abeer alshammari,2016 37

•The Hall Effect experiment suggests that a carrier can have a positive charge.

•These carriers are “holes” in the electron sea - the absence of an electron acts as a net positive

charge. These were first explained by Heisenberg.

•We can’t explain why this would happen with our free electron theory.

Failure of Fermi gas model

Some successes:

1. electrical conductivity

2. heat capacity

3. thermal conductivity

Some failures:

1. physical differences between conductors, insulators,

semiconductors, semi-metals

2. positive Hall coefficients – positive charge carriers ??