Bandgap narrowing in moderately to heavily doped silicon

Abstract

A model of bandgap reduction in silicon through the stored electrostatic energy of majority-minority carrier pairs is developed and compared with experimental results in the doping range from 3 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">17</sup> to 1.5 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">20</sup> /cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> at room temperature. An analytic expression for the bandgap reduction in nondegenerate material is obtained <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\delta\epsilon _{g} = 3q^{2}/(16\pi \epsilon) \cdot (q^{2}n/\epsilon</tex> kT) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> having a square-root dependence on the majority carrier concentration. At room temperature this becomes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\delta\epsilon _{g} = 22.5 (n/10^{18})^{1/2}</tex> meV. In degenerate material, the bandgap reduction is independent of temperature, following the relationship <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\delta \epsilon _{g} = 162 (n/10^{20})^{1/6}</tex> meV. The experimental data at room temperature are in excellent agreement with this theory. Plots of bandgap narrowing as a function of doping level are presented for a number of temperatures.