Relativistic Spin-momentum locking in altermagnets

Abstract

Spin-momentum locking in altermagnets has been deeply explored in the non-relativistic limit. Including spin-orbit coupling, altermagnets exhibit antisymmetric exchange interactions, leading to spin cantings. Therefore, the spin-momentum locking differs among the three spin components Sx, Sy, and Sz, forming the relativistic spin-momentum locking. We consider orthorhombic YVO3 and hexagonal MnTe. For YVO3, the relativistic locking comprises s-, dxy -, and dxz-wave. In MnTe, the dominant component Sy of MnTe inherits the polarized charge distribution and the non-relativistic spin-momentum locking bulk g-wave, but the breaking of the C6z rotational symmetry by the Neel vector lowers the symmetry from g-wave to d-wave. The relativistic spin-momentum locking for MnTe is composed of dxz-, dyz- and s-wave. Despite small magnitudes in real space, the canted spin components contribute significant spectral weight in k-space, impacting k-space properties such as the spin-Hall conductivity.

Figures (4)

• Figure 1: Non-relativistic d-wave spin-momentum locking of G-type YVO$_3$ in the top part. Relativistic Spin-momentum locking for G-type YVO$_3$ with Néel vector along the $z$-axis in the bottom part. The S$_z$ component is the dominant component and inherits the d$_{xz}$-wave spin-momentum locking from the non-relativistic case. The S$_x$ component is s-wave due to the weak ferromagnetism, while the S$_y$ component is a d$_{xy}$-wave. Red and blue represent regions of the Brillouin zone with opposite spin-splitting. The weak ferromagnetism is represented by the s-wave with a complete Brillouin zone.
• Figure 2: Spin-resolved band structure of MnTe with n$||$$y$-axis for the S$_y$ components along the inequivalent directions (a) L$_1$-$\Gamma$-L$_2$, (b) L'$_1$-$\Gamma$-L'$_2$, (c) H$_1$-$\Gamma$-H$_2$ and (d) H'$_1$-$\Gamma$-H'$_2$. The position of these k-points in the Brillouin zone is reported in Figure \ref{['RSML_MnTe']} of the main text. The plots focus on the top of the valence band from -0.3 eV up to the Fermi level. The Fermi level is set to zero.
• Figure 3: Spin-resolved band structure of MnTe with n$||$$y$-axis for the S$_x$ components along the inequivalent directions (a) L$_1$-$\Gamma$-L$_2$, (b) L'$_1$-$\Gamma$-L'$_2$, (c) H$_1$-$\Gamma$-H$_2$ and (d) H'$_1$-$\Gamma$-H'$_2$. The position of these k-points in the Brillouin zone is reported in Figure \ref{['RSML_MnTe']} of the main text. The plots focus on the top of the valence band from -0.3 eV up to the Fermi level. The Fermi level is set to zero.
• Figure 4: Non-relativistic g-wave spin-momentum locking of MnTe in the top part. Relativistic spin-momentum locking for MnTe with Néel vector along the $y$-axis in the bottom part. The S$_y$ is the dominant component and inherits the non-relativistic spin-momentum locking, but the Néel vector lowers the symmetry from g-wave to d$_{yz}$-wave. The S$_z$ component is s-wave due to the weak ferromagnetism, while the spin-momentum locking of the S$_x$ component is d$_{xz}$-wave, which is rotated by $\frac{\pi}{2}$ with respect to the S$_y$ component. The solid lines represent the nodal planes, while the dashed lines represent the nodal lines broken by the Néel vector. Red and blue represent regions of the Brillouin zone with opposite spin-splitting. The weak ferromagnetism is represented by the s-wave with a complete Brillouin zone. The arrows in the middle of the hexagonal Brillouin zone for the relativistic case represent the Néel vector along the $y$-axis.