Improved optical transitions theory for superlattices and periodic systems; new selection rules

Abstract

Using the superlattice (SL) eigenvalues $E_{\mu ,\nu }^{c,v}$ and eigenfunctions $\varphi_{\mu\nu}^{c,v}(z) $, obtained within the theory of finite periodic systems, where $\mu$ indicates the subbands and $\nu$ the intra-subband levels, we calculate optical transitions in SLs and periodic systems. Based on the eigenfunction parity symmetries, studied in the previous paper, new symmetry selection rules (SSR) are derived and photoluminescence and infrared spectra for different types of SLs are calculated. The narrow peaks clustered in groups that couldn't be explained before in Nakamura's et al. for blue emitting devices, are now fully understood. Among the various properties and differences that we discuss in the paper, we notice that many transitions that were experimentally observed but theoretically forbidden, are now perfectly possible. Since the number of matrix-elements allowed by the SSR is generally too large, we devote the second part of this paper to show that one can obtain the same spectra when, besides the SSR, other leading order selection rules closely related to intra-subband symmetries are introduced. These rules reduce the number of matrix evaluations from $\sim n^2$$n_cn_v/2$ to $\sim n$$n_cn_v/2$, i.e., depending on the SL, from about 1000 to 100. We comment also on a third rule, that picks up the contributions of the surface and edge states, and show that it reduces further the number of transitions to $N_s\leq n_cn_v$. With these rules, the main peaks are conserved and their number practically matches with that of the actual spectrum. Excellent agreements with experimental results are found.