A tunneling theory of exciton photoluminescence for neutral acceptors in silicon: A study of the systems Si: (B, In), Si: (Al, In), Si: (Ga, In), Si: (B, Al), Si: (B, Ga), and Si: (Al, Ga)

Abstract

Rate equations for the densities of free excitons and excitons bound to two different neutral acceptors in silicon are solved for steady state in the absence of saturation. These rate equations explicitly include terms for forward and reverse tunneling of bound excitons from one type of neutral impurity to another. Both tunneling rates are calculated using a simple model of an exciton in a one-dimensional semi-infinite double potential well. The steady-state solutions of the rate equations yield an expression for the ratio of the bound exciton luminescence intensity as a function of the impurity concentrations. The relative photoluminescence intensities for the systems Si: (B, In), Si: (Al, In), Si: (Ga, In), Si: (B, Al), Si: (B, Ga), and Si: (Al, Ga) are calculated for the relevant free-exciton capture cross-section ratios. This model predicts no exciton tunneling for any of the above systems for the low-impurity concentration range of 1012–1013 cm−3. For those systems with large differences in the bound exciton energy levels such as Si: (B, In), Si: (Al, In), and Si: (Ga, In), and having indium concentrations exceeding 1015 cm−3, it predicts quenching of shallow impurity bound exciton luminescence because the forward exciton tunneling rate from the shallow level to the deep level of indium dominates and the reverse exciton tunneling rate from indium to the shallow impurities is negligible. For the systems with small differences in the bound exciton energy levels such as Si: (B, Al) and Si: (B, Ga), the theory predicts enhancement of shallow impurity bound exciton luminescence beyond certain concentrations depending upon the free-exciton capture cross-section ratios because in these cases the reverse exciton tunneling rate dominates. For the system Si: (Al, Ga) in which the difference in the bound exciton energy levels is very small, gallium bound exciton luminescence dominates when the gallium concentration exceeds 1016 cm−3.