Solid-state analogue of the relativistic gases.

Abstract

An exact analogy is developed here between particles in a nonparabolic band of a semiconductor (nonparabolicity parameter ${E}_{0}$\ensuremath{\sim}(1/2)${E}_{G}$, energy gap ${E}_{G}$) and a gas of particles of rest mass ${m}_{0}$ moving with relativistic velocities. The connection is made by replacing the velocity of light in the relativistic theory by the velocity v=(${E}_{0}$/${m}_{0}^{\mathrm{*}}$${)}^{1/2}$, where ${m}_{0}^{\mathrm{*}}$ is the effective mass at the band extremum. For InSb (conduction band), v\ensuremath{\sim}1.1\ifmmode\times\else\texttimes\fi{}${10}^{8}$ cm/s. The rest energy ${\ensuremath{\varepsilon}}_{0}$\ensuremath{\equiv}${m}_{0}$${c}^{2}$ of the relativistic theory is then replaced by ${E}_{0}$\ensuremath{\equiv}${m}_{0}^{\mathrm{*}}$${v}^{2}$. A compressive stress, for example, which increases nonparabolicity corresponds to a gas which is rendered more relativistic because ${E}_{0}$ has been decreased relative to kT. The density of states corresponding to the dispersion relation ${\ensuremath{\Elzxh}}^{2}$${k}^{2}$/2${m}_{0}^{\mathrm{*}}$ =E(1+E/2${E}_{0}$) is shown to be of the same form as that for an ideal special relativistic gas. The well-known relativistic change of mass with velocity is also reproduced exactly in the solid-state case, and not only approximately as hitherto supposed. The thermodynamic quantities such as the mean number of particles, the mean energy, and the pressure of the gas are all given in terms of modified Bessel functions ${K}_{\ensuremath{\nu}}$ of the argument kT/${E}_{0}$.