Fermi Level

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

F (E) = 1/ (1+e^(E-E_F^)/KT)   (7)

where k is Boltzmann's constant (= 8.62 x   eV/K = 1.38 x 10^-3 J/K)

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

• The probability that an energy state at E_Fwill be occupied by an electron is 1/2 at all temperatures.
• At 0 K, the distribution takes a simple rectangular form, with all states below E_F occupied, and all states above E_F empty.
• At T > 0 K, there is a finite probability of states above E_F to be occupied and states below E_F to be empty.
• The F-D distribution function is highly symmetric, i.e., the probability f (E_F+ΔE) that a state E above E_Fis filled is the same as the probability [1- f (E_F-ΔE)] that a state E below E_Fis empty.
• This symmetry about EF makes the Fermi level a natural reference point for the calculation of electron and hole concentrations in the semiconductor.
• Note: f (E) is the probability of occupancy of an available state at energy E, thus, if there is no available state at E (e.g., within the band gap of a semiconductor), there is no possibility of finding an electron there.
• For intrinsic materials, the Fermi level lies close to the middle of the band gap (the difference between the effective masses of electrons and holes accounts for this small deviation from the mid gap).
• In n-type material, the electrons in the conduction band outnumber the holes in the valence band, thus, the Fermi level lies closer to the conduction band.
• Similarly, in p-type material, the holes in the valence band outnumber the electrons in the conduction band, thus, the Fermi level lies closer to the valence band.
• The probability of occupation f(E) in the conduction band and the probability of vacancy [1- f(E)] in the valence band are quite small, however, the densities of available states in these bands are very large, thus a small change in f(E) can cause large changes in the carrier concentrations.