k→·p→perturbation theory in III-V compounds and alloys: a reexamination

Abstract

From the recent optical conduction-electron spin-resonance (CESR) measurements of the $g$ factors ${g}^{*}$ in III-V compounds and the known effective masses ${m}^{*}$, in the framework of the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}$ perturbation theory, we determine an experimental value of the interband matrix element ${P}^{2}=(\frac{2}{{m}_{0}}){|〈S|{p}_{x}|X〉|}^{2}$ coupling the conduction band and the upper valence bands. ${P}^{2}$ ranges from 21 \ifmmode\pm\else\textpm\fi{} 1.5 eV in InP to 29 \ifmmode\pm\else\textpm\fi{} 1 eV in GaAs. This unexpected strong variation can be justified by a crude tight-binding calculation, evidencing the combined influence of ionicity and cell dimension. We show that ${g}^{*}$ can be calculated with a precision of 10% in a three-band calculation, whereas a multiband approximation is required for ${m}^{*}$. The good agreement between our CESR measurements of the $g$ factors in ${\mathrm{Ga}}_{1\ensuremath{-}x}{\mathrm{In}}_{x}\mathrm{As}$ and ${\mathrm{Ga}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}\mathrm{As}$ and the calculated values by $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}$ theory shows the correctness of this theory in alloys. Moreover, it is possible to obtain a satisfactory fit of the effective-mass data previously unexplained within simple $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}$ theory by using a multiband model and correct values of ${P}^{2}$. The modifications to $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}$ theory involving random potentials and strains are then not necessary at the precision of the experimental data available up to now.