The Intrinsic Carrier Concentration

For an intrinsic semiconductor, the concentration of electrons in the conduction band is equal to the concentration of holes in the valence band. We may denote n_i and p_i as the electron and hole concentrations, respectively, in the intrinsic semiconductor. These parameters are usually referred to as the intrinsic electron concentration and intrinsic hole concentration. However, n_i = p_i, so normally we simply use the parameter n_i as the intrinsic carrier concentration, which refers to either the intrinsic electron or hole concentration.

The Fermi energy level for the intrinsic semiconductor is called the intrinsic Fermi energy, or E_F = E_Fi.  Then we can write

(1)

and

(2)

If we take the product of Equations (1) and (2), we obtain

(3)

or

(4)

where E_g is the bandgap energy. For a given semiconductor material at a constant temperature, the value of n_i is a constant, and independent of the Fermi energy.

The intrinsic carrier concentration for silicon at T = 300 K may be calculated by using the effective density of states function values from Table 1.1. The value of n_i calculated from Equation (4) for E_g = 1.12 eV is n_i = 6.95 × 10⁹ cm^-3. The commonly accepted value of n_i for silicon at T = 300 K is approximately 1.5 × 10¹⁰ cm^-3. This discrepancy may arise from several sources. First, the values of the effective masses are determined at a low temperature where the cyclotron resonance experiments are performed. Since the effective mass is an experimentally determined parameter, and since the effective mass is a measure of how well a particle moves in a crystal, this parameter may be a slight function of temperature. Next, the density of states function for a semiconductor was obtained by generalizing the model of an electron in a three-dimensional infinite potential well. This theoretical function may also not agree exactly with experiment. However, the difference between the theoretical value and the experimental value of n_i is approximately a factor

Table 1.1 Effective density of states function and effective mass values

Table 1.2 Commonly accepted value of n_i at T = 300K.

of 2, which, in many cases, is not significant. Table 1.2 lists the commonly accepted values of n_i for silicon, gallium arsenide, and germanium at T = 300 K.

The intrinsic carrier concentration is a very strong function of temperature. Figure 1.1 is a plot of n_i from Equation (4) for silicon, gallium arsenide, and germanium as a function of temperature. As seen in the figure, the value of n_i for these semiconductors may easily vary over several orders of magnitude as the temperature changes over a reasonable range.

Figure 1.1 The intrinsic carrier concentration of Ge, Si, and GaAs as a function of temperature.