Relativistic spin-momentum locking in ferromagnets

Abstract

The relativistic spin-momentum locking has been proven in time-reversal-breaking classes of materials with zero net magnetization in the non-relativistic limit, such as altermagnets and other non-collinear magnets. Using density functional theory calculations, we aim to show relativistic spin-momentum locking in ferromagnets, focusing on a broad class of ferromagnetic materials with magnetic sites connected by rotational symmetry, and compare with fcc Ni. In SrRuO3, the antisymmetric exchange interaction produces a spin canting orthogonal to the easy axis, while in all other cases, spin canting is forbidden. Even when the canted magnetic moment in real space is forbidden, relativistic spin-momentum locking shows sizable contributions in k-space. Using prototypical ferromagnets such as orthorhombic SrRuO3, hexagonal CrTe and CrAs with the NiAs crystal structure, half-Heusler MnPtSb, and fcc Ni, we demonstrate that relativistic spin-momentum locking can generate strong effects in ferromagnets. Subdominant components of centrosymmetric ferro-magnetic materials with magnetic sites connected by rotational symmetry host spin-momentum locking similar to altermagnets, while noncentrosymmetric MnPtSb hosts relativistic p-wave due to the spin-orbit coupling. Fcc Ni shows a more complex behavior with a combination of two spin-momentum locking patterns characteristic of altermagnets. Because ferromagnets typically have larger bandwidths than altermagnets, they provide a promising platform for observing even-wave relativistic spin-momentum locking and associated emergent phenomena. From an application standpoint, relativistic spin-momentum locking governs symmetry-allowed spin Hall currents, spin photocurrents, and other momentum-dependent spin responses in k-space.

Figures (5)

• Figure 1: Fermi surface of SrRuO$_3$ with magnetization $||$$z$-axis for $k_z=0.25$ for (a) the S$_x$ component, (b) the S$_y$ component, (c) the S$_z$ component. Black lines represent the nodal plane for the given spin component. The size of the spin components in the real space is S$_x$ = 0.061 $\mu_B$, S$_y$ = 0.004 $\mu_B$, and S$_z$ = 1.367 $\mu_B$.
• Figure 2: Non-relativistic s-wave spin-momentum locking of the ferromagnetic SrRuO$_3$ in the top part. Relativistic Spin-momentum locking for ferromagnetic SrRuO$_3$ with magnetization vector along the $z$-axis in the bottom part. The S$_z$ component is the main component and inherits the s-wave spin-momentum locking from the non-relativistic case. The S$_x$ component is d$_{xz}$-wave , while the S$_y$ component is a d$_{yz}$-wave. The s-wave is represented with a complete Brillouin zone. Red and blue represent regions of the Brillouin zone with opposite spin-splitting.
• Figure 3: Non-relativistic s-wave spin-momentum locking of the ferromagnetic CrTe in the top part. Relativistic Spin-momentum locking for ferromagnetic CrTe with magnetization vector along the $y$-axis in the bottom part. The S$_y$ component is the main component and inherits the s-wave spin-momentum locking from the non-relativistic case. The S$_x$ component is d$_{xy}$-wave , while the S$_z$ component is a d$_{yz}$-wave. The s-wave is represented with a complete Brillouin zone. Red and blue represent regions of the Brillouin zone with opposite spin-splitting.
• Figure 4: Part of Brillouin zone of the constant 2D energy surface 0.3 eV below the Fermi level of CrTe with magnetization $||$$y$-axis for $k_z=0.25$ for (a) the S$_x$ component, (b) the S$_y$ component, (c) the S$_z$ component. Black lines represent the nodal plane for the given spin component.
• Figure 5: Some of the elements of the relativistic spin-momentum locking for ferromagnetic MnPtSb with magnetization vector along the $z$-axis in the bottom part. The S$_z$ component is the main component and inherits the s-wave spin-momentum locking from the non-relativistic case. The S$_x$ component is p$_x$-p$_y$-wave , while the S$_y$ component is a p$_x$+p$_y$-wave. The s-wave is represented with a complete Brillouin zone. Red and blue represent regions of the Brillouin zone with opposite spin-splitting. For S$_x$ and S$_y$, we also need to add the magnetic quadrupole as reported in the Table \ref{['tab:MnPtSb']}.