Crystalline spin-orbit interaction and the Zeeman splitting in Pb$_{1-x}$Sn$_x$Te

TL;DR

This paper investigates the ratio of Zeeman splitting to cyclotron energy in Pb$_{1-x}$Sn$_x$Te, revealing its significance in understanding spin-orbit interactions, topological phase transitions, and Dirac electron characteristics in these semiconductors.

Contribution

It provides a combined analytical and numerical analysis of the parameter M, linking it to topological phase transitions and spin-orbit effects in Pb$_{1-x}$Sn$_x$Te.

Findings

M<1 for PbTe, M>1 for SnTe

M=1 at the band inversion point

M can be used to detect topological phase transitions

Abstract

The ratio of the Zeeman splitting to the cyclotron energy ($M=\Delta E_Z / \hbar \omega_c$), which characterizes the relative strength of the spin-orbit interaction in crystals, is examined for the narrow gap IV-VI semiconductors PbTe, SnTe, and their alloy Pb$_{1-x}$Sn$_x$Te on the basis of the multiband $k\cdot p$ theory. The inverse mass $\alpha$, the g-factor $g$, and $M$ are calculated numerically by employing the relativistic empirical tight-binding band calculation. On the other hand, a simple but exact formula of $M$ is obtained for the six-band model based on the group theoretical analysis. It is shown that $M<1$ for PbTe and $M>1$ for SnTe, which are interpreted in terms of the relevance of the interband couplings due to the crystalline spin-orbit interaction. It is clarified both analytically and numerically that $M=1$ just at the band inversion point, where the transition from trivial to nontrivial topological crystalline insulator occurs. By using this property, one can detect the transition point only with the bulk measurements. It is also proposed that $M$ is useful to evaluate quantitatively a degree of the Dirac electrons in solids.