Carrier Concentration: The fermi level
Introduction:
To obtain equations for the carrier concentrations we must investigate the distribution of carriers over the available energy states. This type of distribution
is not difficult to calculate, but the derivation requires some background in statistical methods.
The fermi level:
Electrons in solids obey Fermi-Dirac statistics.In the development of this type of statistics, one must consider the indistinguishability of the electrons,
their wave nature, and the Pauli exclusion principle. The rather simple result of these statistical arguments is that the distribution of electrons over a range
of allowed energy levels at thermal equilibrium is
where k is Boltzmann's constant (k = 8.62 x 10-5 eV/K = 1.38 X 10-23 J/K).
The function f(E), the Fermi-Dirac distribution function, gives the probability that an available energy state at E will be occupied by an electron at absolute
temperature T. The quantity EF is called the Fermi level, and it represents an important quantity in the analysis of semiconductor behavior.
We notice that, for an energy E equal to the Fermi level energy EF, the occupation probability is
Thus an energy state at the Fermi level has a probability of 1/2 of being occupied by an electron.
At temperatures higher than 0 K, some probability exists for states above the Fermi level to be filled.
For example, at T = T1 in Fig. given below there is some probability f(E) that states above EF are filled, and there is a corresponding probability [1 - f ( E ) ] that states below EF are empty.
The Fermi function is symmetrical about EF for all temperatures; that is, the probability f(EF ΔE) that a state ΔE above EF is filled is the same as the probability [1 - f(EF - ΔE)] that a state ΔE below EF is empty.
The symmetry of the distribution of empty and filled states about EF makes the Fermi level a natural reference point in calculations of electron and hole concentrations in semiconductors.