Approximate dependence of the spontaneous emission rate on electron and hole concentrations

The total spontaneous emission rate (R_sp) is an important quantity, which is widely used in the particle rate equation of light-emitting diodes and semiconductor lasers. It is important to find the explicit dependence of R_sp on electron and hole concentrations. Such an approximate analytical expression is proposed both for two- and three-dimensional systems. Accuracy of the approximation is within 15% for undoped and 25% for doped semiconductors. Dependence of the relative error on the electron-hole concentrations and a quantitative comparison of the approximate, and exact expressions are illustrated