Equilibrium Distribution of Electrons and Holes

The distribution (with respect to energy) of electrons in the conduction band is given by the density of allowed quantum states times the probability that a state is occupied by an electron. This statement is written in equation form as

(1)

where f_F(E) is the Fermi-Dirac probability function and g_c (E) is the density of quantum states in the conduction band. The total electron concentration per unit volume in the conduction band is then found by integrating Equation (1) over the entire conduction-band energy.

Similarly, the distribution (with respect to energy) of holes in the valence bend is the density of allowed quantum states in the valence hand multiplied by the probability that a state is not occupied by an electron. We may express this as

(2)

The total hole concentration per unit volume is found by integrating this function over the entire valence-band energy.

To find the thermal-equilibrium electron and hole concentrations, we need to determine the position of the Fermi energy E­_F with respect to the bottom of the conduction-band energy E_c and the top of the valence-band energy E_v. To address this question, we will initially consider an intrinsic semiconductor. An ideal intrinsic semiconductor is a pure semiconductor with no impurity atoms and no lattice defects in the crystal (e.g., pure silicon). We have argued in the previous chapter that, for an intrinsic semiconductor at T = 0 K, all energy states in the valence band are filled with electrons and all energy states in the conduction band are empty of electrons. The Fermi energy must, therefore, be somewhere between E_c and E_v. (The Fermi energy does not need to correspond to an allowed energy.)

As the temperature begins to increase above 0 K, the valence electrons will gain thermal energy. A few electrons in the valence band may gain sufficient energy to jump to the conduction band. As an electron jumps from the valence band to the conduction band, an empty state, or hole, is created in the valence band. In an intrinsic semiconductor, then, electrons and holes are created in pairs by the thermal energy so

Figure 1.1 (a) Density of states functions, Fermi-Dirac probability function, and areas representing electron and hole concentrations for the case when E_F is near the midgap energy; (b) expanded view near the conduction band energy; and (c) expanded view near the valence band energy.

that the number of electrons in the conduction band is equal to the number of holes in the valence band.

Figure 1.la shows a plot of the density of states function in the conduction band g_c(E), the density of states function in the valence band g_v(E), and the Fermi-Dirac probability function for T > 0 K when E_F is approximately halfway between E_c and E_v. If we assume, for the moment, that the electron and hole effective masses are equal, then g_c(E) and g_v (E) are symmetrical functions about the midgap energy (the energy midway between E_c and E_v). We noted previously that the function , f_F (E) for E > E_F is symmetrical to the function 1 – f_F (E) for E < E_F about the energy E = E_F. This also means that the function f_F (E) for E = E_F + dE is equal to the function 1 – f_F(E) for E = E_F - dE.

Figure 1.1b is an expanded view of the plot in Figure 1.la showing f_F(E) and g_c (E) above the conduction band energy E_c. The product of g_c(E) and f_F(E) is the distribution of electrons n(E) in the conduction band given by Equation (1). This product is plotted in Figure 1.la. Figure 1.lc is an expanded view of the plot in Figure 1.la showing [ 1 – f_F ( E )] and g_v(E) below the valence band energy E_v. The product of g_v (E) and [ 1 – f_F (E)] is the distribution of holes p(E) in the vale d band given by Equation (2). This product is also plotted in Figure 1. la. The areas under these curves are then the total density of electrons in the conduction band and the total density of holes in the valence band. From this we see that if g_c(E) and g_v(E) are symmetrical, the Femi energy must be at the midgap energy in order to obtain equal electron and hole concentrations. If the effective masses of the electron and hole are not exactly equal, then the effective density of states functions g_c(E) and g_v(E) will not be exactly symmetrical about the midgap energy. The Fermi level for the intrinsic semiconductor will then shift slightly from the midgap energy in order to obtain equal electron and hole concentrations.