# Bandgap Reduction

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 Poissonâ€™s equation in spherical coordinates (Lanyon and Tuft, 1978)

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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

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

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The screened Coulombic potential is

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where the screening radius is given by

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

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

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The bandgap reduction is the difference in electrostatic energy between the screened and unscreened cases **
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Using integration by parts, an expression for the bandgap narrowing in the units of eV (electron-Volt) can be found as

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

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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.