The Intrinsic Fermi-Level Position

We have qualitatively argued that the Fermi energy level is located near the center of the forbidden bandgap for the intrinsic semiconductor. We can specifically calculate the intrinsic Femi-level position. Since the electron and hole concentrations are equal, we have

(1)

If we take the natural log of both sides of this equation and solve for E_Fi, we obtain

(2)

From the definitions for N_c and N_v, respectively Equation (2) may be written as

(3a)

The first term, 1/2 (E_c + E_v), is the energy exactly midway between E_c and E_v, or the midgap energy. We can define

so that

(3b)

If the electron and hole effective masses are equal so that m^*_P = m^*_n , then the intrinsic Fermi level is exactly in the center of the bandgap. If m^*_P > m^*_n , the intrinsic Fermi level is slightly above the center, and if m^*_P < m^*_n: it is slightly below the center of the bandgap. The density of states function is directly related to the carrier effective mass; thus a larger effective mass means a larger density of states function. The intrinsic Fermi level must shift away from the band with the larger density of states in order to maintain equal numbers of electrons and holes.