A unified approach to modelling the charge state of monatomic hydrogen and other defects in crystalline silicon

Abstract

There are a number of existing models for estimating the charge states of defects in silicon. In order of increasing complexity, these are (a) the Fermi-Dirac distribution, (b) the Shockley-Last model, (c) the Shockley-Read-Hall model, and (d) the Sah-Shockley model. In this work, we demonstrate their consistency with the general occupancy ratio α, and show that this parameter can be universally applied to predict the charge states of both monovalent and multivalent deep levels, under either thermal equilibrium or steady-state conditions with carrier injection. The capture cross section ratio is shown to play an important role in determining the charge state under non-equilibrium conditions. The application of the general occupancy ratio is compared with the quasi-Fermi levels, which are sometimes used to predict the charge states in the literature, and the conditions where the latter can be a good approximation are identified. The general approach is then applied to the prediction of the temperature- and injection level-dependent charge states for the technologically important case of multivalent monatomic hydrogen, and several other key monovalent deep levels including Fe, Cr, and the boron-oxygen complex in silicon solar cells. For the case of hydrogen, we adapt the model of Herring et al., which describes the charge states of hydrogen in thermal equilibrium, and generalize it for non-equilibrium conditions via the inclusion of the general occupancy ratio, while retaining the pre-factors which make the model more complete. Based on these results, the impact of temperature and injection on the hydrogenation of the key monovalent defects, and other pairing reactions, are discussed, demonstrating that the presented model provides a rigorous methodology for understanding the impact of charge states.