### Carrier concentration of doped semiconductors

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The carrier concentration in doped semiconductors changes with temperature, and experiences weak ionization region, intermediate ionization region, strong ionization region, transition region and intrinsic excitation region from low temperature to high temperature. The carrier concentration of doped semiconductors can be calculated and analyzed by quantum statistics theory.

There are a large number of energy states in the conduction band and valence band. After adding donor impurities and acceptor impurities, energy states are introduced into the forbidden band. For a certain energy state E, the probability P(E) that an electron occupies it is given by the Fermi-Dirac function:

In the formula, E_{F} is called the Fermi level. The definition of the Fermi level is the chemical potential of electrons in a solid, and the probability that an electron occupies the Fermi level E_{F} is exactly 1/2, that is, the probability that an electron with energy E_{F} appears is 1/2.

Figure 1 shows the distribution of electrons in the semiconductor energy band according to the Fermi-Dirac function. The Fermi-Dirac function is symmetric for the Fermi level E_{F}, so if the number of electron energy states in the conduction band and the valence band is the same, the number of electrons in the conduction band and the number of holes in the valence band are also the same, the Fermi level is located at the midline of the forbidden band, as shown in Fig. 1(a). This case is an intrinsic semiconductor, and the Fermi level of an intrinsic semiconductor is denoted by E_{i}.

1) N-type semiconductor carrier concentration The relationship between electron concentration and temperature in N-type silicon is shown in Figure 2. At low temperatures, the electron concentration increases with increasing temperature. At 100K, the impurities are all ionized, and the intrinsic excitation starts to play a major role after the temperature is higher than 500K, entering the intrinsic region. In the range of 100~500K, the impurities are all ionized, and the carrier concentration is basically equal to the impurity concentration.

(1) Low temperature weak ionization region (the case where the donor energy level is partially ionized): when the temperature is low, most of the donor impurity energy levels are still occupied by electrons, and only a small amount of donor impurities are ionized, forming a small amount of electrons into the conduction band, and the number of electrons from the valence band to transition to the conduction band by intrinsic excitation is negligible. In this weak ionization case, it can be considered that the electrons in the conduction band are all provided by the ionized donor impurity.

At this time, the Fermi level E_{F} and the electron concentration n_{0} is

where E_{C} is the energy at the bottom of the conduction band; N_{C} is the effective density of states of the conduction band; N_{D} is the concentration of the donor impurity; k is the Boltzmann constant; T is the absolute temperature; △E_{D} is the ionization energy of the donor impurity, △E_{D}= E_{C}-E_{D}.

(2) Strongly ionized region (the case where the donor maximum is ionized): the region corresponding to room temperature is a strongly ionized region, and the ionized donor concentration in this region is almost equal to the donor impurity concentration N_{D}.

Equations (1-5) show that the Fermi level EF is dependent on temperature and donor impurity concentration. Under the general doping concentration, N_{C}>N_{D} is negative, and the Fermi level E_{F} is located in the forbidden band. At a certain temperature, the larger the N_{D}, the closer the E_{F} is to the guide band.

When the donor impurity concentration N_{D} is constant, the higher the temperature, the closer the E_{F} is to the intrinsic Fermi level E_{i}. When the donor impurities are all ionized, the electron concentration n_{0} is

n_{0}=N_{D} (1-6)

At this time, the carrier concentration has nothing to do with temperature, and this temperature region is called the saturation region.

The higher the impurity concentration, the higher the temperature at which full ionization is achieved. It is generally believed that the shallow energy levels are all ionized at room temperature, which ignores the influence of impurity concentration. Taking phosphorus-doped N-type silicon as an example, at room temperature, N_{C}=2.8×10^{19}cm^{–}^{3}, △E_{D}=0.044eV, kT=0.026eV, it can be calculated that the upper limit of concentration ND when phosphorus is fully ionized is about 3×10^{17}cm^{–}^{3}. At room temperature, the intrinsic carrier concentration of silicon is about 1.5 × 10^{10}cm^{–}^{3}. Therefore, the concentration of phosphorus in silicon should be in the range of 10^{11}~3×10^{1}^{7}cm^{–}^{3}, which can be considered as the main impurity ionization, and it is in the saturation region where all the impurities are ionized.

2) P-type semiconductor carrier concentration

(1) Low temperature weak ionization region (the case where the acceptor energy level is partially unionized): the Fermi level E_{F} and hole concentration P_{0} is

where E_{V} is the top energy of the valence band; N_{V} is the effective density of states of the valence band; N_{A} is the acceptor impurity concentration; ΔE_{A} is the ionization energy of the acceptor impurity, ΔE_{A}=E_{A}-E_{V}.

(2) Strong ionization region (the case where most of the acceptor has been ionized, that is, the saturation region): In this case, the Fermi level E_{F} is

When the acceptor impurities are all ionized, the hole concentration is

P_{0}=N_{A} （1-10）

Equation (1-10) shows that in the saturation region, the hole concentration increases proportionally with the acceptor concentration and is independent of temperature.

In summary, the carrier concentration and Fermi level of doped semiconductors are determined by temperature and impurity concentration. For N-type semiconductors, the larger the N_{D}, the higher the E_{F} position; for P-type semiconductors, the larger the N_{A}, the lower the E_{F} position.