For an electron which behaves as a free particle with effective mass, m*, the density of states was derived in section , yielding: (2.6.3) Ec is the bottom of the conduction band below which the density of states is zero. The density of states for holes in the valence band is given by: (2.6.4) 2.6.2. Calculation of the Fermi integral The carrier density in a semiconductor, is obtained by integrating the product of the density of states and the probability density function over all possible states. For electrons in the conduction band the integral is taken from the bottom of the conduction band, labeled, Ec, to the top of the conduction band: (2.6.5) Where gc(E) is the density of states in the conduction band and f(E) is the Fermi function. This general expression is illustrated with Figure for a parabolic density of states function with Ec = 0. The figure shows the density of states function, gc(E), the Fermi function, f(E), as well as the product of both, which is the density of electrons per unit volume and per unit energy, n(E). The integral corresponds to the crosshatched area. Figure 2.6.1 : The carrier density integral. Shown are the density of states, gc(E), the density per unit energy, n(E), and the probability of occupancy, f(E). The carrier density, no, equals the crosshatched area. The actual location of the top of the conduction band does not need to be known as the Fermi function goes to zero at higher energies. The upper limit can therefore be replaced by infinity. We also relabeled the carrier density as no to indicate that the carrier density is the carrier density in thermal equilibrium. (2.6.6) Using equations () and () this integral becomes:
The temperature dependence at low temperatures is somewhat more complex as it depends on whether or not the material is compensated. Figure was calculated for silicon containing both donors and acceptors. At 70 K the electron density is the donor density but still larger than the acceptor density. Under such conditions the activation energy, EA, equals half of the ionization energy of the donors or (Ec - Ed)/2. At lower temperatures where the electron density is lower than the acceptor density, the activation energy equals the ionization energy or Ec - Ed. This behavior is explained by the fact that the Fermi energy in compensated material is fixed at the donor energy. The donors levels are always partially empty as electrons are removed from the donor atoms to fill the acceptor energy levels. If the acceptor density is smaller than the electron density – as is true for uncompensated material - the Fermi energy does change with temperature and the activation energy approaches half of the ionization energy. Lightly semiconductors suffer from freeze-out at relatively high temperature. Higher-doped semiconductors freeze-out at lower temperatures. Highly-doped semiconductors do not contain a single donor energy, but rather an impurity band which overlaps with the conduction or valence band. The overlap of the two bands results in free carriers even at zero Kelvin. Degenerately doping a semiconductor therefore eliminates freeze-out effects. 2.6.4.5 Temperature dependence of the carrier density in non-degenerate material The analysis of non-degenerate semiconductors can be further expanded to include carrier freeze-out in the presence of donors as well as acceptors. Equation () can be written as: (2.6.40) For n-type material (Nd > Na), this equation can be simplified to: (2.6.41) the hole density, po, has been ignored and the acceptors are assumed to be fully ionized since the Fermi energy is expected to be above the energy of even the deepest acceptors. The density N* is given by: (2.6.42) The solution to the quadratic equation is: (2.6.43) Example 2.6c 4H-SiC is with 2 x 1017 cm-3 nitrogen donor atoms (Ec – Ed = 90 meV). Use Nc = 4 x 1020 cm-3.