LECTURE 2 CONTENTS

• MAXWELL BOLTZMANN STATISTICS

• FERMI- DIRAC STATISTICS & ITS DISTRIBUTION

• SEMICONDUCTORS AND ITS CLASSIFICATION and

• FERMI ENERGY LEVEL DISTRIBUTION IN INTRINSIC SEMICONDUCTORS.

Maxwell Boltzmann Statistics (Classical law)

This law states that, the total fixed amount of energy is distributed among the various members of an assembly of identical particles in the most proable distribution.

The Maxwell Boltzmann law is )E(i

iie

gn

Where ni ─ number of particles having energy Ei.

gi ─ number of energy states.

Tk

EF

Tk

1and

(Here k ─ Boltzmann constant; T ─ Absolute temperature of the gas, EF ─ Fermi energy)

Therefore, kT/)EE(

ii

Fie

gn

Particles are distinguishable.

Classical particles can have any spin.

Particles do not obey Pauli’s exclusion principle.

Any number of particles may have identical energies.

Fermi-Dirac Statistics (Quantum law)

This statistics applicable to the identical, indistinguishable particles of half spin.

These particles obey Pauli’s exclusion principle and are called fermions (e.g.) Electrons, protons, neutrons …,

In such system of particles, not more than one particle can be in one quantum state.

Fermi Dirac Distribution Law is

1)e(

gn

iEi

i or ni = 1/)( kTEE

i

Fie

g

Example

Let us consider two particles a and a. Let if, these two particles occupy the three energy levels (1,2,3). The number of ways of arranging the particles 31=3 (not more than one particle can be in any one state)

Energy level

Possible distribution in various energy level

1 a A -

2 a - a

3 - A a

Fermi Energy (EF) and Fermi-Dirac Distribution Function f(E) Fermi Energy (EF)

Fermi Energy is the energy of the state at which the probability of electron occupation is ½ at any temperature above 0 K.

It is also the maximum kinetic energy that a free electron can have at 0 K.

The energy of the highest occupied level at absolute zero temperature is called the Fermi Energy or Fermi Level.

The Fermi energy at 0 K for metals is given by

m

hNEF 8

3 23/2

When temperature increases, the Fermi level or Fermi energy also slightly decreases.

The Fermi energy at non–zero temperatures,

22

0

0 121

FFF E

TkEE

Here the subscript ‘0’ refers to the quantities at zero kelvin.

Fermi-Dirac Distribution Function f(E)

The free electron gas in a solid obeys Fermi-Dirac statistics.

Suppose in an assemblage of fermions, there are M(E) allowed quantum states in an energy range between E and E+dE and N(E) is the number of particles in the same range. Then,

The Fermi-Dirac distribution function is defined as,

)/kTE(EEM

EN

F

exp1

1

)(

)(

N(E) / M(E) is the fraction of the possible quantum states which are occupied.

The distribution of electrons among the levels is described by function f (E), probability of an electron occupying an energy level ‘E’.

If the level is certainly empty, then f(E) = 0. Generally the f(E) has a value in between zero and unity.

When E< EF (i.e.,) for energy levels lying below EF, (E –EF) is a negative quantity and hence,

101

1

1

1

e

)E(f

That means all the levels below EF are occupied by the electrons.

Fermi Dirac distribution function at different temperatures

When E > EF (i.e.) for energy levels lying above EF,

(E – EF) is a positive quantity

01

1

1

1

e

)E(f

This equation indicates all the levels above EF are

vacant.

At absolute zero, all levels below EF are completely filled

and all levels above EF are completely empty.This level,

which divides the filled and vacant states, is known as the Fermi energy level.

When E = EF ,2

1

11

1

1

1)(

0

eEf

The probability of finding an electron with energy equal to the Fermi energy in a metal is ½ at any temperature.

At T = 0 K all the energy level upto EF are occupied and all the energy levels above EF are empty .

When T > 0 K, some levels above EF are partially filled while some levels below EF are partially empty.

, at all temperatures

SemiconductorsIntroduction

The materials are classified on the basis of conductivity and resistivity.Semiconductors are the materials which has conductivity, resistivity value inbetween conductor and insulator . The resistivity of semiconductor is in the order of 10−4 to 0.5 Ohm-metre.

It is not that, the resistivity alone decides whether a substance is a semiconductor (or) not , because some alloys have resistivity which are in the range of semiconductor’s resistivity. Hence there are some properties like band gap which distinguishes the materials as conductors, semiconductors and insulators.

semi-conductor is a solid which has the energy band similar to that of an insulator. It acts as an insulator at absolute zero and as a conductor at high temperatures and in the presence of impurities.

Semiconductors are materials whose electronic properties are intermediate between those of metals and insulators.

These intermediate properties are determined by the crystal structure, bonding characteristics and electronic energy bands.

They are a group of materials having conductivities between those of metals and insulators.

Classification of Semiconductors According to their Structure

Amorphous semiconductors-have poor electrical characteristics.

Polycrystalline semiconductors – have better electrical characteristics and lower conductivity.

Single crystal semiconductor – have superior electrical characteristics and higher conductivity. The majority of the semiconductor devices, single-crystal materials are used.

Classification of Semiconductor According to the nature of the current carriers

Ionic semi conductor, in which conduction takes place through the movement of ions and.

Electronic semiconductor, in which conduction takes place through the movement of electrons and no mass transport, is involved

Classification of semiconductors According to the constituent atoms

Elemental semiconductor: All the constituent atoms are of the same kind (i.e) composed of single species of atoms. (eg) germanium and silicon.

Compound semiconductor: They are composed of two or more different elements (eg) GaAS, AlAs etc.,

Crystal structure of silicon and germanium

The structure of Si and Ge, which are having covalent bonding. Covalent bondings are stereo specific; i.e. each bond is between a specific pair of atoms.

The pair of atoms share a pair of electrons (of opposite magnetic spins).

Three dimensional representation of the structures Si, and Ge, with the bonds shown in below figure, the region of high electron probability (shaded).

(a) (b)

Structure of (a) silicon and (b) germanium crystals

All atoms have coordination number 4; each material has an average of 4 valence electrons per atom, and two electrons per bond.

Each atom of a material is coordinated with its neighbours.

(a) (b)

Structure of (a) silicon and (b) germanium crystals

The thermal vibrations on one atom influence the adjacent atoms; the displacement of one atom by mechanical forces, or by an electric field, leads to adjustments of the neighbouring atoms.

The number of coordinating neighbours that each atom has is important. Covalent bonds are very strong.

Structure of (a) silicon and (b) germanium crystals

(a) (b)

Property Silicon (Si) Germanium (Ge)

Atomic number 14 32

Atom/m3 5.02 1028 4.42 1028

Electronic shell configuration 1s22s22p63s23p2 1s22s22p63s23p63d104s24p2

Atomic weight 28.09 72.6

Crystal structure Diamond Diamond

Breakdown field (V/m) ~ 3.0107 ~ 107

Density (gm/m3)2.329106

at 298K5.3234106 at 298 K

Energy gap (eV)1.12 at 300 K1.17 at 77K

0.664 at 291 K0.741 at 4.2K

Dielectric constant 11.7 at 300K 16.2 at 300K

Intrinsic carrier concentration (m3) at 300 K

1.02 1016 2.33 1019

Lattice constant (Å) 5.43107 at 298.3K 5.65791 at 298.15K

Melting point (C) 1412 937.4

Thermal conductivity [Wm1(C1)] 131 at 300K 60 at 300K

Mobility of electrons (m2V1s1) 0.135 at 300K 0.39 at 300K

Mobility of holes (m2V1s1) 0.048 at 300K 0.19 at 300K

Some important properties of elemental semiconductor

Intrinsic Semiconductors

In semiconductors and insulators, when an external electric field is applied the conduction is not possible as there is a forbidden gap, which is absent in metals.

In order to conduct, the electrons from the top of the full valence band have to move into the conduction band, by crossing the forbidden gap.

The field that needs to be applied to do this work will be extremely large.

Eg: Silicon where the forbidden gap is about 1 eV. The distance between these two locations is about 1 Å (1010 m).

A field gradient of approximately 1V/ (1010 m) = 1010Vm1 is necessary to move an electron from the top of the valence band to the bottom of the conduction band.

The other possibility by which this transition can be brought about is by thermal excitation.

At room temperature, the thermal energy that is available can excite a limited number of electrons across the energy gap. This limited number accounts for semi-conduction. When the energy gap is large as in diamond, the number of electrons that can be excited across the gap is extremely small.

In intrinsic semiconductors, the conduction is due to the intrinsic processes (without the influence of impurities).

A pure crystal of silicon or germanium is an intrinsic semiconductor. The electrons that are excited from the top of the valence band to the bottom of the conduction band by thermal energy are responsible for conduction.

The number of electrons excited across the gap can be calculated from the Fermi-Dirac probability distribution.

f(E) = ]}Tk/)EE{exp[ BF1

1

# The Fermi level EF for an intrinsic semiconductor lies midway in the forbidden gap.

# The probability of finding an electron here is 50%, even though energy levels at this point are forbidden.

# Then (EEF) is equal to Eg /2, where Eg is the magnitude of the energy gap.

EF

E

O O.5F(E)

Conduction band

Valence band

Eg

1.0

The Fermi level in an intrinsic semiconductor lies in the middle of the energy gap.

# For a typical semiconductor like silicon, Eg = 1.1 eV, so that (EEF) is 0.55 eV, which is more than twenty times

larger than the thermal energy kBT at room temperature

(=0.026 eV).

# The probability f(E)of an electron occupying energy level E becomes f(E) = exp(Eg / 2kBT ).

# The fraction of electrons at energy E is equal to the probability f(E). The number n of electrons promoted across the gap,

n = N exp(Eg / 2kBT)

where N is the number of electrons available for excitation from the top of the valence band.

The promotion of some of the electrons across the gap leaves some vacant electron sites in the valence band. These are called holes.

An intrinsic semiconductor contains an equal number of holes in the valence band and electrons in the conduction band, that is ne = nh.

Under an externally applied field, the electrons, which are excited into the conduction band by thermal means, can accelerate using the vacant states available in the conduction band.

At the same time, the holes in the valence band also move, but in a direction opposite to that of electrons.

The conductivity of the intrinsic semiconductor depends on the concentration of these charge carriers, ne and nh.

In the case of metals, the drift velocity acquired by the free electrons in an applied field.

The mobility of conduction electrons and holes, e and h, as the drift velocity acquired by them under unit field gradient.

The conductivity of an intrinsic semiconductor as

i = ne e e + nh e h

where e is the electronic charge, ne and nh are concentrations of electrons and holes per unit volume.

Fermi levelThe number of free electrons per unit volume in an intrinsic semiconductor is

kT

EEp

kTmn cF

/*e ex

h

22

23

2

The number of holes per unit volume in an intrinsic semiconductor is

KT

EE

h

Tkm FVh exp.2

22

3

2

p =

Since n = p in intrinsic semiconductors.

kT

EEv

h

Tkm

kT

EEp

h

Tkm FhcF*e exp

22ex

22

23

2

23

2

KT

EEvm

kT

EEm F

hCF

e expexp2

3

23

or

kT

EE

m

me cv

e

hkTEF

exp2

3

*

2

Taking log on both sides,

kT

EEexplog

m

mlog

kT

E cve*

e

he

F

2

32

kT

EE

m

mog

kT

E cv*

e

he

F l2

32

2log

4

3e

cv*

e

h EE

m

mkTor Ef =

If we assume that, *h

*e mm

2

cvF

EEE

Thus, the Fermi level is located half way between the valence and conduction band and its position is independent of temperature. Since mh* is greater than me*, EF is just

above the middle, and rises slightly with increase in temperature

[ since loge1 = 0]

Valen ce band

C o ndu ctio n band

E v

E g

(E 2g / )

E c

E F (a )

(b )

Position of Fermi level in an intrinsic semiconductor at various temperatures (a) at T = 0 K, the Fermi level in the middle of the forbidden gap (b) as temperature increases, EF shifts upwards