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Orbital-Zeeman cross correlation in $p$- and $d$-wave altermagnets

Tomonari Mizoguchi, Soshun Ozaki

Abstract

Altermagnets are a novel class of magnets that exhibit a large spin splitting but the total magnetic moment is vanishing. This unconventional spin splitting gives rise to various characteristic phenomena, such as spin current generation. In this paper, we study the orbital-Zeeman (OZ) cross term in altermagnets. Specifically, we consider the Rashba metal and the surface Dirac cones of three-dimensional topological insulators (TIs) in the presence of the altermagnetic order parameters. For the Rashba metals, the $p$-wave order parameter exerts only a limited influence on the OZ term, whereas the $d$-wave one causes the sign change of it when the order parameter becomes sufficiently large. For the TI surface, the $p$-wave order parameter retains the step-function-type dependence of the OZ term as a function of the chemical potential ($μ$) associated with the jump at $μ=0$, observed in the TI surface without magnetism, but its magnitude is reduced. For the $d$-wave case, the magnitude of jump at $μ=0$ is preserved but the OZ term decreases as increasing $|μ|$.

Orbital-Zeeman cross correlation in $p$- and $d$-wave altermagnets

Abstract

Altermagnets are a novel class of magnets that exhibit a large spin splitting but the total magnetic moment is vanishing. This unconventional spin splitting gives rise to various characteristic phenomena, such as spin current generation. In this paper, we study the orbital-Zeeman (OZ) cross term in altermagnets. Specifically, we consider the Rashba metal and the surface Dirac cones of three-dimensional topological insulators (TIs) in the presence of the altermagnetic order parameters. For the Rashba metals, the -wave order parameter exerts only a limited influence on the OZ term, whereas the -wave one causes the sign change of it when the order parameter becomes sufficiently large. For the TI surface, the -wave order parameter retains the step-function-type dependence of the OZ term as a function of the chemical potential () associated with the jump at , observed in the TI surface without magnetism, but its magnitude is reduced. For the -wave case, the magnitude of jump at is preserved but the OZ term decreases as increasing .
Paper Structure (17 sections, 62 equations, 5 figures)

This paper contains 17 sections, 62 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Orbital-Zeeman cross correlation
  4. Results
  5. Rashba metal
  6. Topological insulator surface
  7. Summary
  8. Derivation of Eq. (\ref{['eq:chiOZ_final']})
  9. Analytic results for Rashba metal
  10. Analytic results for topological insulator surface
  11. $J=0$
  12. $s$-wave magnet
  13. $p$-wave magnet
  14. $d$-wave magnet
  15. Detailed calculations of $\chi_{\rm OZ}$ using the orbital magnetization
  16. ...and 2 more sections

Figures (5)

  • Figure 1: Schematic figures of (a) the two-dimensional altermagnet and (b) the altermagnet placed on the three-dimensional TI.
  • Figure 2: Band structures for the $s$-wave case with (a) $J = 0.3$ and (b) $J=0.6$, the $p$-wave case with (c) $J = 0.3$ and (d) $J=0.6$, and the $d$-wave case with (e) $J = 0.3$ and (f) $J=0.6$. We set $\frac{\hbar^2 }{m} = 1$ and $\lambda = 0.3$.
  • Figure 3: $\mu$ dependence of $\chi_{\rm OZ}$ for the Rashba metal for $\lambda = 0.3$ and (a) $J= 0.1$, (b) $J=0.3$, and (c) $J=0.6$. The unit of the vertical axis is $\frac{|e|\mu_{\rm B}}{2\pi\hbar}$.
  • Figure 4: $\mu$ dependence of $\chi_{\rm OZ}$ for the TI surface with $\lambda = 1$ and $T=0$. The unit of the vertical axis is $\frac{|e|\mu_{\rm B}}{2\pi\hbar}$.
  • Figure 5: $\mu$ dependence of $\chi_{\rm OZ}$ for the Rashba metal with $\lambda = 0.3$. The black dots represent the numerical result with $k_{\rm B} T = 0.01$ and the magenta line represent the result for $k_{\rm B} T = 0$ [Eq. (\ref{['eq:chioz_rashba']})]. The unit of the vertical axis is $\frac{|e|\mu_{\rm B}}{2\pi\hbar}$.