The magnitude of effective intrinsic carrier concentration in silicon varies under the condition of heavy doping concentration. This is due to the band structure of a heavily-doped silicon altered by (a) the formation of impurity band as the spacing between individual impurity atoms becomes smaller and the interaction between adjacent impurity atoms leads to a splitting of impurity levels, (b) the conduction and valence band edges no longer exhibit a parabolic shape which leads to the formation of band tails, and (c) the interaction between the free carriers and impurity atoms leads to a modification of the density of states at band edges, as shown in Fig. 2.17. The electrostatic interaction of the

Fig. 2.16. Resistivities of n-type and p-type silicon at various doping levels at 300 K.

Fig. 2.17. Energy bandgap at high doping level.

minority carriers and the high concentration of the majority carriers leads to a reduction in the thermal energy required to create an electron-hole pair.

The electrostatic potential of the screened minority carrier as a function of distance r from the carrier, can be solved using Poisson’s equation in spherical coordinates (Lanyon and Tuft, 1978)

where Nd is the donor concentration for n-silicon semiconductor. For doping concentration below degenerated level, n(r) is related to the local potential, V(r), by Boltzmann equation so that

For low excess electron concentration, Eq. (2.51) can be simplified using exponential series expansion to

The screened Coulombic potential is

where the screening radius is given by

Equation (2.52) can be solved by recognizing that a Bessel’s differential equation for a real variable r with a solution of the form of Eq. (2.53) is

The electric field distribution for the screened Coulombic field, E(r), is and the unscreened field distribution E_{o}(r) is

The bandgap reduction is the difference in electrostatic energy between the screened and unscreened cases

Using integration by parts, an expression for the bandgap narrowing in the units of eV (electron-Volt) can be found as

At room temperature, bandgap narrowing in silicon in the unit of eV is given by

where Nd is the donor concentration for n-type silicon. In silicon at room temperature, bandgap narrowing becomes significant at doping concentration above 10^{18} cm^{-3}. The mass-action law must be modified for bandgap narrowing as

At room temperature in silicon, the effective intrinsic carrier concentration will increase by a factor of over 70 times when the doping concentration increases from 10^{15}cm^{-3} to 10^{20}cm^{-3}. Figure 2.18 shows the effective intrinsic carrier density and the bandgap reduction as a function of background doping level at 300 K.