2.18.1 For a Pure or Intrinsic Compound (semiconductor or insulator)

Concentrations of electrons (n_e) and holes ((n_h) are equal at any temperature and are given as

(2.71)

(2.72)

where N_c and N_v are defined as effective conduction and valence band density of states and are expressed as

and

(2.73)

Here m_e* and m_h* are the effective mass of electron and holes respectively, k is the Boltzmann’s constant, h is the Planck’s constant and T is the temperature. The value of N_c and N_v is ~10¹⁹ cm^-3 at 300 K.

Electron and hole concentrations in the conduction and valence bands are equal i.e.

(2.74)

Fermi energy is defined as

(2.75)

where .

In oxides, m_e* and m_h* are generally 2-10 times larger than the mass of free electron, m_e. Since, the atomic density of solids is about 10²³ cm^-3, the density of states is about four order of magnitude lower than that in semiconductors.

Moreover, if the second term in equation (2.75) is small and is actually zero if m_e* = m_h* , then Fermi level is quite close to the center of the band gap.

Hence for an intrinsic compound where n_e = n_h.

(2.76)

Fermi level usually remains in the middle of the band gap but can shift up or down when materials is doped. Typically E_F moves up, from the center of the bandgap towards E_C , for n-type doping and moves down for p-type doping.

The above expressions have striking similarity to the concentration of lattice defects where

(2.77)

The density of electronic states may be thought of as equivalent to the density of vacancies in the lattice sites.

The excitation of electrons across the band gap can be depicted by a chemical defect reaction as follows

The equilibrium constant is

(2.78)

At 300K

(2.79)

Here K_i is not unit-less unlike the reaction constant in the defect reactions because n_e and n_h have the units of cm^-3.