Chapter 6: Free Electron Fermi Gas

Free Electron Theory• Conductors fall into 2 main classes; metals &

semiconductors. Here, we focus on metals.• A metal is loosely defined as a solid with valence

electrons that are not tightly bound to the atoms but are relatively easily able to move through the whole crystal.

• The room temperature conductivity of metals is in the range

and it increases on the addition of small amounts of impurities. The resistivity normally decreases monotonically with decreasing temperature & can be reduced by the addition of small amounts of impurity.

6 8( ) ;10 10metalsRT m

Question! Why do mobile (conducting) electrons occur in some solids & not in others?•When the interactions between electrons areconsidered,this becomes a

Very Difficult Question to Answer.D

Some Common Physical Properties of Metals:Great Physical Strength, High Density, Good

Electrical and Thermal Conductivity, etc.•Here, we outline a surprisingly simple theorywhich does a very good job of explaining many ofthese common properties of metals.

“The Free Electron Model” •We begin with the assumption that

conductionelectrons exist & that they consist of thevalence electrons from the metal atoms.•Thus, metallic Na, Mg & Al are assumed to

have1, 2 & 3 mobile electrons per atom, respectively.•This model seems at first as if it is

Way Too Simple!•But, this simple model works remarkablywell for many metals & it will be used to helpto explain many properties of metals.

• According to this

Free Electron Model (FEM)the valence electrons are responsible for the conduction of electricity, & for this reason these electrons are called

“Conduction Electrons”.• As an example, consider Sodium (Na). • The electron configuration of the Free Na Atom is:

1s2 2s2 2p6 3s1

• The outer electron in the third atomic shell(n = 3, ℓ = 0)

is the electron which is responsible for the chemical properties of Na.

Valence Electron(loosely bound)Core Electrons

• When Na atoms are put together to form a Na metal,

• Na has a BCC structure & the distance between nearest neighbors is 3.7 A˚• The radius of the third shell in Na iis 1.9 A˚

• In the solid the electron wavefunctions of the Na atoms overlap slightly. From this observation it follows that a valence electron is no longer attached to a particular ion, but belongs to both neighboring ions at the same time.

Na metal

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

• Consider a metal with a cubic shape with edge length L: Ψ and E can be found by solving the Schrödinger equation:

0 L/2

V

-L/2

22

2E

m

0V

Since

Use periodic boundary conditions& get the Ψ’s as travelling plane waves.

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

• The solutions to the Schrödinger equation 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 2

k p pNa L

2xk p

L

2yk q

L

2zk r

L

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

• The corresponding momentum is:

• The 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 is:

2

2( ) ,

2

Vkg k dk dk

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

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

k3/ 2 1/ 2

2 3(2 )

2( )

Vm Eg E

Ground State of the Free Electron Gas

• Electrons are Fermions (s = ± ½) and obey the 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 The Fermi energy, obtained by integrating the density of states between 0 and EF, The result should equal N. Remember that

• Solving for EF (Fermi energy);2/32 23

2F

NE

m V

3/ 2 1/ 22 3

(2 )2

( )V

m 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 as shown below; the radius is Fermi wave number kF. 2 2

2F

Fe

kE

m

kz

ky

kx

Fermi SurfaceE = EF

kF

From these two equations, kF can be found as,

1/323F

Nk

V

The surface of the Fermi sphere represents the boundary between occupied & unoccupied k states at absolute zero for the free electron gas.

2/32 23

2F

NE

m V

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

• The Fermi (degeneracy) Temperature TF is given 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 & velocity values of the electrons at the states on the Fermi surface of the Fermi sphere.

• So, the Fermi Sphere plays an important role in the behavior 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 for monovalent potassium metal;

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

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

( ) /

1

1 F BFD E E k Tf

e

20

Fermi-Dirac Distribution & The Fermi-Level

Main Application: Electrons in a Conductor •The Density of States g(E) specifies

how many states exist at a given energy E. •The Fermi Function f(E) specifies how many of theexisting states at energy E will be filled with electrons.

•I

•The Fermi Function f(E) specifies, underequilibrium conditions, the probability that anavailable state at an energy E will be occupied by anelectron. It is a probability distribution function.

EF = Fermi Energy or Fermi Level

k = Boltzmann Constant

T = Absolute Temperature in K

Fermi-Dirac Statistics

EF is called The Fermi Energy.Note the following:

• When E = EF, the exponential term = 1 and FFD = (½).

• In the limit as T → 0:L

L

• 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.

The Fermi Energy EF is essentially the same as theChemical Potential μ.

β (1/kT)

Fermi-Dirac DistributionConsider T 0 K

For E > EF : 0)(exp1

1)( F

EEf

1)(exp1

1)( F

EEf

E

EF

0 1 f(E)

For E < EF :

A step function!

EFE<EF E>EF

0.5

fFD(E,T)

E

( ) /

1

1 F BFD E E k Tf

e

Fermi Function at T=0 & at a Finite Temperature

fFD=? At 0°K

a. E < EF

b. E > EF

( ) /

11

1 F BFD E E k Tf

e

( ) /

10

1 F BFD E E k Tf

e

Fermi-Dirac DistributionTemperature Dependence

25

If E = EF then f(EF) = ½ . If then

So, the following approximation is valid:That is, most states at energies 3kT above EF are empty.

If then lSo, the following approximation is valid:So, 1 f(E) = probability that a state is empty, decays to zero.

So, most states will be filled.

kT (at 300 K) = 0.025eV, Eg(Si) = 1.1eV, so 3kT is very small in comparison to other

energies relelevant to the electrons.

kTEE 3F

1exp F

kT

EE

kT

EEEf

)(exp)( F

kTEE 3F

1exp F

kT

EE

kT

EEEf Fexp1)(

Fermi-Dirac Distribution: Consider T > 0 K

Fermi-Dirac DistributionTemperature Dependence

T = 0

T > 0

As the temperature increases from T = 0,The Fermi-Dirac Distribution “smears out”.

T = 0

The Fermi “Temperature” is defined asTF ≡ (EF)/(kB).

T > 0

T = TF

• As the temperature increases from T = 0,The Fermi-Dirac Distribution “smears out”.

• When T >> TF, FFD approaches a decaying exponential.

T >> TF

At T = 0 the Fermi Energy EF is the energyof the highest occupied energy level.•If there are a total of N electrons, then is easy to show that EF has the form:

3/23

23/2

1

2/3

1

)3

(8

)3

(

)(3

1

L

N

m

hNEE

E

EN

F

F

Fermi-Dirac Distribution FunctionFermi-Dirac Distribution Function

0.00 0.25 0.50 0.75 1.00 1.25 1.500.0

0.2

0.4

0.6

0.8

1.0

1.2n(k)

(k)/F

F/kBT

T = 0 50 20 10

( ) /

1( )

1Bk k Tn k

e

0 1 2 3

-4

-2

0

T lnT

Temperature

TT-Dependence of the Chemical Potential-Dependence of the Chemical Potential

Classical

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.

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

T>0

T=0

n(E,T)

E

g(E)

EF

kkyy

kkxx

kkzz

Fermi SurfaceFermi Surface

1/ 323Fk n

2 / 32 3

2 8F

h n

m

/F F BT k

3

53

5

F

F

NU

u

Fermi-Dirac Distribution FunctionFermi-Dirac Distribution Function

0.00 0.25 0.50 0.75 1.00 1.25 1.500.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4n(k)

(k)/F

F/kBT = 50

( ) /

1( )

1Bk k Tn k

e dn

d

FF

T

T

Examples of Fermi SystemsExamples of Fermi Systems

Electrons in metals (dense system)

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) ( )( )

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 Flectron Gas

Low-Temperature Heat CapacityLow-Temperature Heat Capacity

2

2el BF

TC Nk

T

3

el lat

lat

C C C

C T

Copper

• Total metallic heat capacity at low temperatures

where γ & β are constants found plotting Cv/T as a function of T2

3C T T ElectronicHeat capacity

Lattice HeatCapacity

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