As the temperature of the metal increases, the atoms of crystal starts
vibrating in an arbitrary manner. Due to the additional gain of
energy from thermal vibrations, many valence electrons become free and
start moving randomly in the crystal. These free electrons collide
with other vibrating atoms, losing and receiving energy. The direction
of the resultant motion after the collision is unpredictable. Hence
any analytical method can not give exact positions or energies as a
function of time. Every electron transfers average or mean energy in
each collision, travels mean free path between the collisions and
takes average or mean energy in each collision, travels mean free path
between the collision and takes average or mean free time between any
two collisions. Hence each electron has a statistical variation
avound these average or mean values.

The statistical methods use distribution function and set of laws
which are used to predict the behaviour of large group of items and
can not say anything about the behaviour of a particular item.
Accurate determination of such density distribution function is very
difficult, because of nonuniform potential energies possessed by the
electrons.

The distribution function which explains satisfactory the distribution
of energies among different electrons within a crystal at a given
temperature is developed by Fermi and Dirac. This is called Fermi-Dirac
energy distribution function. This function enables to find the number
of free electrons dn per unit volume which may possess a value of
energy in the range E to E + dE at a given temperature T.

The number of free electrons dn per unit volume which may occupy energy level between E to E + dE is given by,

where ρ

where ρ

_{E}= Density of electrons in this energy interval
The

where N(E) = Density of states in the conduction band (number of states per eV per m

**density distribution function**ρ_{E}is expressed as,where N(E) = Density of states in the conduction band (number of states per eV per m

^{3})
and f(E) = Probability that the quantum state with energy E is occupied by an electron

The probability function f(E) is called Fermi-Dirac probability function.

__1.1 Concentration of Electrons (n) in Conduction Band__
In a semiconductor, the lowest energy level in the conduction band is E

The γ is a proportionality constant and given by,

where m = Mass of electron = 9.107 x 10

_{C}. Then N(E) is given by,The γ is a proportionality constant and given by,

where m = Mass of electron = 9.107 x 10

^{-31}kg
q = Charge on electron = 1.6 x 10

^{-19}C
h = Planck's constant in J-s = 6.625 x 10cm

^{-34}J-s
From the principles of quantum mechanics, the Fermi-Dirac probability function is given by,

where k = Boltzmann's constant in eV /

where k = Boltzmann's constant in eV /

^{o}K
T = Temperature in K

E

_{F }= Fermilevel or characteristics energy for the crystal in eV
E = Energy level occupied by an electron eV

The concentration of electrons (n) in the conduction band can be obtained by integrating equation of from E

dn_{C }to ∞,_{E}= ρ

_{E}dE = N(E) f(E) dE

This integral gives the value of n as,

and m

_{n }= Effective mass of electron

**ќ**= Expressed in joules per degree kelvin

h = Planck's constant in joule-seconds

__1.2 Concentration of Holes (p) in valence Band__

The Fermi-Dirac probability function for holes {1 - f(E)} and given by,

This integral gives the value of p as,

and m

_{p }= Effective mass of a hole.

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