Impact of spin--orbit coupling on orbital diamagnetism in a narrow-gap semiconductor $\mathrm{Pb}_{1-x}\mathrm{Sn}_x\mathrm{Te}$

Abstract

We study the influence of spin--orbit coupling (SOC) on orbital magnetism in $\mathrm{Pb}_{1-x}\mathrm{Sn}_x\mathrm{Te}$, a narrow-gap semiconductor. Using the $π$-matrix method, we calculate material-specific Landau levels and evaluate the magnetization, fully including interband effects. The system exhibits diamagnetism for both $x = 0$ and $x = 0.35$, with the latter showing a stronger response due to its smaller gap. The magnitude of diamagnetism increases monotonically with SOC strength, particularly in strong magnetic fields. To clarify the underlying mechanism, we introduce the free--Zeeman--Dirac (fZD) model and fit its parameters to the calculated Landau levels. The analysis reveals that SOC enhances the Dirac-type interband contribution relative to the Zeeman term, leading to increased diamagnetism. These results demonstrate that SOC can play a key role in orbital magnetism through interband effects.

Figures (8)

• Figure 1: Contributions of the fZD model coefficients to the magnetization. (a) Magnetization with only the free-electron term $C_{\mathrm{f}}$ finite. (b) Additional contribution due to the Zeeman term $C_{\mathrm{Z}}$, with $C_{\mathrm{f}} = 5\times10^{-4}\ \mathrm{eV\,T^{-1}}$ fixed. The magnetization for $C_{\mathrm{Z}} = 0$ has been subtracted. (c) Magnetization with only the Dirac term $C_{\mathrm{D}}$ finite. A cutoff function is used in (c) to ensure convergence.
• Figure 2: Magnetization as a function of the Zeeman ($C_{\mathrm Z}$) and Dirac ($C_{\mathrm D}$) coefficients within the fZD model at $B=2\,\mathrm{T}$.
• Figure 3: Magnetic-field dependence of Landau levels at the L point for (a) $x=0$ (PbTe) and (b) $x=0.35$ with $\kappa_{\mathrm{soc}}=1$. Blue thick lines show the $\pi$-matrix results, and orange thin lines indicate the fitted fZD model.
• Figure 4: Magnetic field dependence of magnetization in $\mathrm{Pb}_{1-x}\mathrm{Sn}_x\mathrm{Te}$ with varying SOC strength for (a) $x=0$ and (b) $x=0.35$.
• Figure 5: Magnetization as a function of SOC scaling parameter $\kappa_{\mathrm{soc}}$ in $\mathrm{Pb}_{1-x}\mathrm{Sn}_x\mathrm{Te}$ for (a) $x=0$ and (b) $x=0.35$.