The Fermi Level

• The electrons in solids obey Fermi-Dirac statistics.
• This statistics accounts for indistinguishability of electrons, their wave nature, and Pauli exclusion principle.
• The Fermi-Dirac distribution function f(E) of the electrons over a range of permitted energy levels at thermal equilibrium can be given by

                                        (2.7)

here k is the Boltzmann's constant (= 8.62 x  10^-5eV/K = 1.38 x 10^-23 J/K).

• Gives the probability that an available energy state at E will be occupied by the electron at absolute temperature T.
• E_F is called as Fermi level and is the measure of the average energy of electrons in lattice    an extremely important quantity for analysis of device behavior.
• Note that for (E - E_F) > 3kT (called as Boltzmann approximation), f(E)≈ exp[- (E- E_F )/kT]   this is referred to  as Maxwell-Boltzmann (MB) distribution (followed  by the gas atoms).
• The probability which an energy state at   E_F will be occupied by electron is half at all temperatures.

• At 0 K, distribution takes a simple rectangular form, with all states below E_F occupied, and all states above E_F are empty.
• At temperature T > 0 K, there is a finite probability of the states above  E_F to be occupied and states below  E_F to be empty.
• The F-D distribution function is quite symmetric, that is the probability f(  E_F +  ΔE ) that a state E above  E_F is filled same as the probability [1- f(  E_F - ΔE)] that a state E below the  E_F is empty.
• The symmetry about EF makes Fermi level a natural reference point for calculation of electron and hole concentrations in semiconductor.
• Note that the f(E) is the probability of occupancy of an available state at energy E, thus, if there is no available state at E (for example within band gap of the semiconductor), there is no possibility of finding the electron there.

• For the intrinsic materials, the Fermi level is close to middle of the band gap (the difference between effective masses of the electrons and holes accounts for this small deviation from mid gap).
• In the n-type material, electrons in conduction band outnumber the holes in valence band, therefore, the Fermi level lies closer to conduction band.
• Likewise, in p-type material, the holes in valence band outnumber the electrons in conduction band, threfore, the Fermi level lies closer to valence band.
• The probability of the occupation f(E) in conduction band and the probability of the vacancy [1- f(E)] in valence band are quite small, although, the densities of available states in these bands are large, thus a small change in the f(E) can cause large changes in carrier concentrations.

Figure The density of states N(E), the Fermi-Dirac distribution function f(E), and the carrier concentration as functions of energy for (a) intrinsic, (b) n-type, and (c) p-type semiconductors at the thermal equilibrium.

• Note that since the function f(E) is symmetrical about the E_F, a large electron concentration implies the small hole concentration, and vice versa of it.
• In the n-type material, the electron concentration in conduction band increases as the  E_F moves closer to E_C; therefore, (  E_C -  E_F ) gives the measure of n.
• Likewise, in p-type material, the hole concentration in the valence band increases as  E_F moves closer to E_v; therefore, (  E_v -  E_F ) gives a measure of p.

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