Warping- and inversion-asymmetry-induced cyclotron-harmonic transitions in InSb

Selection rules are obtained for harmonics of the cyclotron resonance transition in InSb, such as $2{\ensuremath{\omega}}_{c}(\ensuremath{\Delta}n=2,\ensuremath{\Delta}{m}_{s}=0)$, $3{\ensuremath{\omega}}_{c}(\ensuremath{\Delta}n=3,\ensuremath{\Delta}{m}_{s}=0)$, etc., and spin-shifted harmonics such as $2{\ensuremath{\omega}}_{c}+{\ensuremath{\omega}}_{s}(\ensuremath{\Delta}n=2,\ensuremath{\Delta}{m}_{s}=\ensuremath{-}1)$, etc., where $\ensuremath{\Delta}n$ and $\ensuremath{\Delta}{m}_{s}$ are the changes in the Landau quantum number and the $z$ component of the spin angular momentum. These transitions are induced by warping and inversion-asymmetry effects. The complete $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}$ Hamiltonian is obtained to second order in $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ and to first order in the applied magnetic field $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ for the coupled conduction band (${\ensuremath{\Gamma}}_{6}$), light- and heavy-hole valence bands (${\ensuremath{\Gamma}}_{8}$) and split-off valence band (${\ensuremath{\Gamma}}_{7}$). This Hamiltonian treats the interactions with higher bands as second-order perturbations, and includes terms proportional to three new parameters which arise from the spin-orbit splitting of these higher bands. A group-theoretical analysis is carried out for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ in the (1\ifmmode\bar\else\textasciimacron\fi{}10) plane including ${k}_{H}\ensuremath{\ne}0$, ${k}_{H}$ being the momentum component along the direction of the applied magnetic field. The selection rules for the intra-conduction-band transition $2{\ensuremath{\omega}}_{c}$, $2{\ensuremath{\omega}}_{c}+{\ensuremath{\omega}}_{s}$, and $3{\ensuremath{\omega}}_{c}$ are in agreement with experiment but with one important exception: that a strong $2{\ensuremath{\omega}}_{c}$ transition observed for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}\ensuremath{\parallel}[001]$ in the polarization $\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}\ensuremath{\perp}\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ is not predicted by the above group-theoretical analysis.