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 $2{\omega }_c(\Delta n=2,\Delta m_s=0)$, $3{\omega }_c(\Delta n=3,\Delta m_s=0)$, etc., and spin-shifted harmonics such as $2{\omega }_c+{\omega }_s(\Delta n=2,\Delta m_s=-1)$, etc., where $\Delta n$ and $\Delta m_s$ 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 $\overrightarrow{\mathrm{k}}·\overrightarrow{\mathrm{p}}$ Hamiltonian is obtained to second order in $\overrightarrow{\mathrm{k}}$ and to first order in the applied magnetic field $\overrightarrow{\mathrm{H}}$ for the coupled conduction band (${\Gamma }_6$), light- and heavy-hole valence bands (${\Gamma }_8$) and split-off valence band (${\Gamma }_7$). 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 $\overrightarrow{\mathrm{H}}$ in the (1¯10) plane including $k_H\ne 0$, $k_H$ being the momentum component along the direction of the applied magnetic field. The selection rules for the intra-conduction-band transition $2{\omega }_c$, $2{\omega }_c+{\omega }_s$, and $3{\omega }_c$ are in agreement with experiment but with one important exception: that a strong $2{\omega }_c$ transition observed for $\overrightarrow{\mathrm{H}}\parallel \mathrm{}\left[001\right]\mathrm{}$ in the polarization $\overrightarrow{\mathrm{E}}\perp \overrightarrow{\mathrm{H}}$ is not predicted by the above group-theoretical analysis.