Relative intensities of indirect transitions: Electron-phonon and hole-phonon interaction matrix elements in Si (TO) and GaP (LA,TA)

Abstract

The electron-phonon ($e$-ph) and hole-phonon ($h$-ph) interaction matrix elements of the indirect transitions in Si and GaP can be evaluated individually by comparing their recently determined ratios together with experimental values of the absorption coefficient to the theoretical expressions for the absorption coefficient. We have obtained values of these matrix elements for the TO phonon of Si and the LA and TA phonons of GaP. For Si, ${S}_{{e\ensuremath{-}\mathrm{p}\mathrm{h}}_{\mathrm{TO}}}=15.3\ifmmode\pm\else\textpm\fi{}2.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ Ry, ${S}_{{h\ensuremath{-}\mathrm{p}\mathrm{h}}_{\mathrm{TO}}}=\ensuremath{-}18.7\ifmmode\pm\else\textpm\fi{}2.8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ Ry, and in GaP ${S}_{{e\ensuremath{-}\mathrm{p}\mathrm{h}}_{\mathrm{TA}}}=19.2\ifmmode\pm\else\textpm\fi{}2.8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ Ry, ${S}_{{h\ensuremath{-}\mathrm{p}\mathrm{h}}_{\mathrm{TA}}}=17.4\ifmmode\pm\else\textpm\fi{}2.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ Ry, and ${S}_{{e\ensuremath{-}\mathrm{p}\mathrm{h}}_{\mathrm{LA}}}=7.6\ifmmode\pm\else\textpm\fi{}1.1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ Ry, ${S}_{{h\ensuremath{-}\mathrm{p}\mathrm{h}}_{\mathrm{LA}}}=13.0\ifmmode\pm\else\textpm\fi{}2.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ Ry. The magnitudes and signs of these matrix elements make it possible to understand more clearly the mechanisms responsible for the relative intensities of indirect transitions in diamond and zinc-blende (DZB) materials (excluding Ge). In Si, constructive interference between the $e$-ph and $h$-ph processes makes the TO phonon dominate, while in the other indirect DZB materials, the small ${\ensuremath{\Gamma}}_{1,c}$ to ${X}_{1,c}$ energy separation makes the longitudinal- (LA- or LO-) phonon-assisted transition with ${\ensuremath{\Gamma}}_{1,c}$ as an intermediate state, the most intense. By considering exact expressions for the coupling coefficient between ${\ensuremath{\Gamma}}_{1,c}$ and ${X}_{1,c}$ we are able to understand the behavior of ternary compounds whose fundamental gap can be adjusted from indirect to direct by alloying. We have also been able to explain why in DZB materials the longitudinal-phonon-assisted transition proceeding via ${\ensuremath{\Gamma}}_{15,c}$ is weak in intensity.