3. electrons in solids 08 - vgtu in solids.pdf · electrons, protons and some other microparticles...

$: 3. Electrons in solids 08 - VGTU in solids.pdf · Electrons, protons and some other microparticles have only spin quantum numbers that are fractional numbers (-1/2, 1/2, 3/2,…)$
ELEKTRONIKOS PAGRINDAI 2008

VGTU EF ESK [email protected]

1

3. ELECTRONS IN METALS AND SEMICONDUCTORS

Large numbers of free electrons can exist in a metal or semiconductor.

Statistical methods are the most useful for the investigation of systems that

consist of a large number of particles.

The type of statistics developed to describe such distributions depends on the

type of particles and also on the possible interactions between them.

Classical statistics is applied to the systems of classical particles. Classical

physics assumes that the particles are distinguishable (easily recognised), can

be named numbering them and that each state can hold an unlimited number

of particles. These assumptions yield the Maxwell-Boltzmann distribution

function that will be denoted by fB(W).

Systems of microparticles are investigated by quantum statistics.

There are two groups of microparticles.

Photons and phonons are indistinguishable and have spin quantum numbers

that are integers: 1, 2, 3,… An unlimited number of photons and phonons can

be accommodated in an allowed state. Such particles have been investigated

by Bose and Einstein. Now they are called bosons. The Bose-Einstein

distribution function is used to describe distribution of bosons.



2

Electrons, protons and some other microparticles have only spin quantum

numbers that are fractional numbers (-1/2, 1/2, 3/2,…). These microparticles are

also indistinguishable, as bosons, but they obey Pauli’s exclusion principle. This

results in the Fermi-Dirac distribution function denoted by fF(W). Such

particles are called fermions.

We will be interested in the properties of electrons in solids. So first of all we will

examine fermions. Fermions reveal their characteristic features meeting each

other and occupying allowed states.

… Characteristic features of fermions reveal when number of particles NΣ is large

and near to the number of allowed states G.

If NΣ<<G, fermions do not meet occupying allowed states. Then the system of

fermions is non-degenerate. We will see that classical statistics may be

applied to the non-degenerate system of fermions.

Statistical methods. Classical and quantum statistics

Statistical

physics

Quantum

statistics

Bose-

Einstein

Fermi-

DiracMaxwell-

Boltzmann

NΣ/G<<1



3

Distribution functions

Distribution functions are used to characterise the systems of particles.

There are two types of distribution functions.

Let us consider a system containing NΣ particles.

The number of particles dN having energy in the range from W to W+dW is

proportional to NΣ and dW:

The proportionality coefficient f(W) is distribution function.

Ratio dN/NΣ=f(W)dW means probability dP that particle energy is in the range

from W to W+dW .

WWfNN d)(d Σ=

… Ratio dP/dW is probability density.

So distribution function means probability density.

W

P

W

NNWf

d

d

d

/d)( == Σ



4

At dW = 1, we have that dP = f(W). So distribution function means the

probability of a particle to have energy in a unit range given at energy value

W.

Distribution functions

W

P

W

NNWf

d

d

d

/d)( == Σ

Because N(W)dW = dN, the product N(W)dW means the number of particles dN

with energy from W to W+dW .

At dW = 1, we have that dN = N(W). So distribution function N(W) describes the

number of particles per unit of energy centred at energy value W.

At given distribution function, we can find parameters characterizing the system

(mean energy, most probable momentum, mean-square-root velocity, etc.)

Besides f(W), the other distribution function is used: N(W)=NΣf(W).

∫∞

=

0

1d)( WWf ∫∞

Σ=

0

d)( NWWN



5

Energy distribution of electrons in metals

Objectives:

Analysis of properties of fermions and distribution of electrons in metals

(derivation and analysis of the energy distribution function for electrons in

metals)

Steps:

1. Analysis of density of states of free electrons.

2. Analysis of probabilities of allowed states ocupation.

3. Derivation and analysis of the energy distribution function for electrons in

metals.



6

Density of states

The state of a classical particle is defined by the set of three co-ordinates and

three components of momentum.

… A point of the six-dimensional space with coordinates x, y, z , px, py, pzcorresponds to the classical particle.

According to the uncertainty principle and Heisenberg’s inequalities:

3h≥zyx pppzyx ∆∆∆∆∆∆

... A cell of the six-dimensional space corresponds to a microparticle. Its

volume is h3.

… The density of the allowed states of microparticles in the six-dimensional space

is limited.

Let us consider that a microparticle is free and can move in the volume V. Then

Vppp zyx /h3≥∆∆∆ … The cell of volume h3/V of the three-dimensional

space with co-ordinates px, py, pz corresponds to the

free microparticle.



7

Density of states

... Lets find the number of states u in the momentum

range from p to p+dp.… A spherical shell corresponds to the momentum

range from p to p+dp. Its volume is

( ) pppppVp dπ43

π4d

3

π4d 233 ≅−+=

ppV

V

Vu

pd

h

π4

/h

d2

33==

k2mWp =k

k

d2

d WW

mp = kk

2/3

3d

h

π24WWm

Vu =

WWgWWWmV

u d)(dh

π24c

2/3

3=−=

c2/3

3h

π24)( WWm

VWg −=

Density of states:

Then



8

The Fermi level is the highest energy level that cabn be occupied by an electron in

a solid at absolute zero temperature.

At higher temperatures the Fermi level corresponds to the value of energy at

which the Fermi-Dirac distribution function has value 0.5.

Fermi-Dirac function

Any allowed energy level may be occupied by two electrons having different spin

quantum numbers. On the other hand, not all available energy states are filled.

Usually electrons cannot gain sufficient energy to occupy very high levels.

( )[ ] 1k/exp

1)(

FF +−

=TWW

Wf WF – is the constant called the Fermi level,

…

The probability that allowed energy level is occupied by electrons is

given by the Fermi-Dirac function.

At 0 K all available states up to the energy

level WF are filled, whereas all levels above

are empty.

Probability of occupancy of the Fermi level

is always 0.5, independent of temperature.



9

Energy distribution of electrons in a metal

The number of electrons in a given energy range depends on the number of

available states in the range and also on the probability of occupation of the

states. Because two electrons with different spin quantum numbers can occupy

every energy level,

WWfWgufN d)()(22d FF ==

WWNWWfNN d)(d)(d == Σ

)()(2)()( FΣ WfWgWfNWN ==

( )[ ] ;1k/exph

π28)(

F

c

3

2/3

+−

−=

TWW

WW

n

mWf

3/22

FccFπ8

3

2

h

==−

n

mWWW

The Fermi energy in a metal depends upon the density of free electrons. The

difference WF –Wc is positive.

… The Fermi level lies in the partially filled allowed energy band.



10

Statistics of electrons in metals. Problems

1. Using distribution function, derive expression for the Fermi energy in a

metal (find how the Fermi energy depends on the density of conduction

electrons).

2. Find the mean kinetic energy of an electron in a metal at absolute zero

temperature.

3. Supposing that WFc= 7 eV , find the mean kinetic energy of an electron in a

metal at absolute zero temperature.

4. Calculate the Fermi energy and the mean kinetic energy of an electron at

0 K in copper given that there is one conduction electron per atom, that the

density of copper is 8920 kg/m3 and its atomic mass is 63.54.

Fck5

3WW =

3/22

Fcπ8

3

2

h

=

n

mW eV2.4...

5

3Fck ≅== WW



11

Distributions of particles in non-degenerate systems.

Equilibrium condition

Objectives:

1. Analysis and properties of electrons in the non-degenerate systems

(general properties of electrons in semiconductors).

2. Derivation of equilibrium condition.

Steps of distribution analysis:

1. Derivation and analysis of the energy distribution function for electrons in

the non-degenerate systems (electrons in semiconductors).

2. Analysis of the non-degenaration condition.

3. Distribution functions for particles in the non-degenerate systems.

4. Use of the distribution functions for calculation of averaqge values

(parameters characterizing systems).



11

Distributions of particles in non-degenerate systems

WWfWgufN d)()(22d FF ==

c2/3

3h

π24)( WWm

VWg −=

( )[ ] 1k/exp

1)(

FF +−

=TWW

Wf

)()(2)()( FΣ WfWgWfNWN ==

( )[ ] 1k/exph

π28)(

F

c

3

2/3

+−

−=

TWW

WW

n

mWf

According to the Fermi-Dirac statistics:



12

Distribution functions for a non-degenerate system

The density of free electrons in semiconductors is small. So, free electrons in a

semiconductor form a non-degenerate collection.

( )[ ] 1k/exp F >>− TWW )(eee)( Bk/k/k/

FF WfAWf TWTWTW ==≅ −−

WWWmV

N TWTWdee

h

)2(π4d k/k/

c3

2/3F −−= ∫ −−=

max

c

F deeh

)2(π4 k/c

/

3

2/3

Σ

W

W

TWkTWWWW

mVN

∫∞

−−=0

kk/

kk/k/

3

2/3

Σ deeeh

)2(π4kcF WW

mVN

TWTWTW

TWTWTm

V

Nn

k/k/

3

2/3Σ cF ee

h

)kπ2(2 −== TWTW

Tm

n k/

2/3

3k/ cF e

)kπ2(2

he =

( ) kkΣkk/

k2/3Σ d)(dek

π

2d k WWfNWWT

NN

TW == −− TWeWTWf

k/k

2/3k

k)k(π

2)(

−−=

2/3

02

ea

dxx ax π∫∞

− =

The collection of fermions is non-degenerate if



13


The density of free electrons in semiconductors is small. So, free electrons in a

semiconductor form a non-degenerate collection.

The collection of fermions is non-degenerate if

( )[ ] 1k/exp F >>− TWW )(eee)( Bk/k/k/

FF WfAWf TWTWTW ==≅ −−

WWWmV

N TWTWdee

h

)2(π4d k/k/

c3

2/3F −−=

∫ −−=max

c

F deeh

)2(π4 k/c

/

3

2/3

Σ

W

W

TWkTWWWW

mVN

)()(2)()( FΣ WfWgWfNWN == c2/3

3h

π24)( WWm

VWg −=

∫ −−=max

c

F deeh

)2(π4 k/c

/

3

2/3

Σ

W

W

TWkTWWWW

mVN



14


∫∞

−−=0

kk/

kk/k/

3

2/3

Σ deeeh

)2(π4kcF WW

mVN

TWTWTW

TWTWTm

V

Nn

k/k/

3

2/3Σ cF ee

h

)kπ2(2 −== TWTW

Tm

n k/

2/3

3k/ cF e

)kπ2(2

he =

( ) kkΣkk/

k2/3Σ d)(dek

π

2d k WWfNWWT

NN

TW == −−

TWeWTWf

k/k

2/3k

k)k(π

2)(

−−=

2/3

02

ea

dxx ax π∫∞

− =



15

TWeWTWf

k/k

2/3k

k)k(π

2)(

−−=

... electrons form a non-degenerate collection, if their density is small.

This condition is usually satisfied for semiconductors.

The difference WF-Wc is negative in this case and the Fermi level lies below the

bottom of the conduction band.

The distribution function for non-

degenerate collection:


1eeek/)(k/)(k/)( cFcF <<= −−−−− TWWTWWTWW

( )1

kπ22

he

2/3

3k/)( cF <<=−

Tm

nTWW( )[ ] 1k/exp cF <<− TWW

TWTW

Tm

n k/

2/3

3k/ cF e

)kπ2(2

he =

WF<Wc– kT



15

k

k/

k

2/3 d)k(π

2d k WeWT

NN

TW−−Σ=

...)(...,)( == pfpN

m

dpW

m

pW

pd,

2k

2

k ==

peTm

pNN Tmp d

)k(π

2d k2/

2/3

22−Σ=

vmpmvp dd, == ...)(...,)( == pfpN

vevT

mNN Tmv d

kπ

2d k2/2

2/32−Σ

= ...)(...,)( == vfvN

Distributions of classical particles and electrons in the non-degenerate

systems



15

The mean and the most probable values of classical particles and electrons

in the non-degenerate systems

∫∞

=0

d)( xxxfx TW k2

3k =kWx =

vx =

xxfxx d)(0

22 ∫∞

=

m

Tv

π

k22=

0=(x)f'

2

rms~ vvv ==

m

Tv

k3~ =

kWx =2

kkt

TW =

px = Tmp k2t =

m

Tv

k2t =



16

Equilibrium condition

Besides the statistical methods, thermodynamical methods are used for

investigation of particle systems. The state of a system is characterized

using macroscopic variables such as pressure, temperature, and volume.

According to thermodynamics at constant temperature and constant

preasure, the free Gibbs energy variation is given by

dG = µ dN,

where µ is the chemical potential of the system, dN is variation in the

number of oparticles in the system.

dN

dG = -µ1 dN + µ2 dN = 0;

µ1 = µ2 WF1= WF2

… The parts of a system have the

same Fermi energy level in

equilibrium.



17

Distribution in a non-degenerate system. Problems

1. Find the mean energy and the root-mean-square speed of an electron in

silicon at T = 0 K and T = 300 K.

2. How many times does the mean energy of electrons in germanium increase

with temperature increasing from 10 to 100 0C?

3. Find the most probable kinetic energy of a particle obeying Maxwell-

Boltzmann statistics at T = 0.

4. Find the maximal density of electrons in a non-degenerate system at 300 K.

3. electrons in solids 08 - vgtu in solids.pdf · electrons, protons and some other microparticles...

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