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

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 $\overrightarrow{\mathrm{k}}·\overrightarrow{\mathrm{p}}$ perturbation theory, we determine an experimental value of the interband matrix element $P^2=\left(\frac{2}{m_0}\right){\left|〈S\right|p_x\left|X〉\right|}^2$ coupling the conduction band and the upper valence bands. $P^2$ ranges from 21 ± 1.5 eV in InP to 29 ± 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 ${\mathrm{Ga}}_{1-x}{\mathrm{In}}_x\mathrm{As}$ and ${\mathrm{Ga}}_{1-x}{\mathrm{Al}}_x\mathrm{As}$ and the calculated values by $\overrightarrow{\mathrm{k}}·\overrightarrow{\mathrm{p}}$ 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 $\overrightarrow{\mathrm{k}}·\overrightarrow{\mathrm{p}}$ theory by using a multiband model and correct values of $P^2$. The modifications to $\overrightarrow{\mathrm{k}}·\overrightarrow{\mathrm{p}}$ theory involving random potentials and strains are then not necessary at the precision of the experimental data available up to now.