Carrier Concentrations in an Intrinisc Semiconductor and Fermi Level

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     ρE  = Density of electrons in this energy interval
The density distribution function ρE  is expressed as,
where     N(E) = Density of states in the conduction band (number of states per eV per m3)
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 EC. 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 /oK
T = Temperature in K
E= 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 EC to ∞,
dnE = ρE dE = N(E) f(E) dE
This integral gives the value of n as,

and       mn = 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
In a semiconductor, the highest energy level in the valence band is. Then N(E) is given by,
The Fermi-Dirac probability function for holes {1 - f(E)} and given by,
This integral gives the value of p as,

and          mp = Effective mass of a hole.