The foundations of Shockley's equation for the average electron–hole-pair creation energy in semiconductors

Abstract

Energetic carriers in semiconductors thermalize by impact-ionization, which generates electron–hole pairs (EHPs), and by energy losses to phonons. The average EHP creation energy is typically about three times the energy gap. In 1960, Shockley derived a simple equation for the average EHP creation energy with a single free parameter that fits experimental values for a wide range of materials, but the underlying assumptions, as stated, have been widely criticized as lacking justification. Modified expressions derived by improved approximations have been proposed but do not fare better. Here, we revisit the foundations of Shockley's equation and provide a robust justification for the kinetic-energy component as a model averaging procedure and then apply a similar procedure to the phonon component of the equation. The phonon result retains Shockley's form, but the interpretation and justification are now on par with those of the kinetic-energy term. The single-parameter fit to the data remains unchanged, i.e., the present analysis accounts for the exceptional applicability of Shockley's equation.