Interplay between Relativistic Spin-Momentum Locking and Breaking of Inversion Symmetry: conditions for p-wave magnetism

Abstract

We investigate the interplay between relativistic spin-momentum locking arising from altermagnetism and various forms of inversion symmetry breaking. Depending on the symmetry breaking, this can give rise to Rashba-type spin-orbit coupling (SOC), Weyl-type SOC, or the coexistence of two distinct spin-momentum lockings. We focus on the altermagnetic Ca2RuO4 as a testbed material. Our results reproduce the experimentally observed ground state, which is an A-centered magnetic order with the Neel vector along the b-axis, hosting spin cantings along the a- and c-axes but without weak ferromagnetism. Ca2RuO4 exhibits relativistic spin-momentum locking, with different even-parity wave orders for the three spin components. We interpret the experimental results on doped samples as evidence for a transition from a pure altermagnetic phase to a weak ferromagnetic phase. Under ferroelectric- and antiferroelectric-like distortions, there are no qualitative changes in the non-relativistic spin-momentum locking and in the weak ferromagnetism. However, we observe the rise of the Rashba or Weyl-type SOC. Using numerical and analytical models, we investigate which nodal planes persist when inversion symmetry is broken in the relativistic case. The spin-momentum locking of the other components adopt a p-wave character in the case of Rashba; in contrast, Weyl-type SOC disrupts all nodal planes, leaving only nodal lines. Finally, to simulate a stripe phase with structural distortions along the z-axis, we studied a modulated electric field inducing atomic displacements within one Ca2RuO4 layer. This produces a magnetic phase transition to an exotic altermagnetic state with two non-relativistic spin-momentum lockings hosting weak ferromagnetism. Our research presents a comprehensive analysis of various possible scenarios in altermagnets with breaking of inversion symmetries under relativistic effects

Figures (17)

• Figure 1: Crystal structure of the orthorhombic Ca$_2$RuO$_4$ shown from the (a) top view and (b) side view. Grey, red and blue spheres indicate the Ru, O and Ca atoms, respectively. The arrows represent the spins in the experimental ground state (A-centered) with the Néel vector aligned along the b direction. Red and blue arrows represent the spin-up and spin-down in the non-relativistic limit. For clarity, Ca and O atoms are omitted in panel (a). Spin canting is illustrated with the components along the $a$ and $c$ directions scaled to 35% of the principal $b$-axis component, to visualize the spin canting better. $a$-, $b$- and $c$-axis, are equivalent to $x$-, $y$- and $z$-axis, respectively.
• Figure 2: a) Spin-momentum locking in the non-relativistic case. Red and blue sectors represent regions of the Brillouin zone with opposite non-relativistic spin-splitting. b) Brillouin zone of Ca$_2$RuO$_4$ (space group Pbca n. 61) for the A-centered magnetism. In our notation, the high-symmetry points with subscripts 1 and 2 show altermagnetism along the path towards the $\Gamma$ point. We project the bulk Brillouin zone on the principal surfaces (100), (010) and (001). The projected high-symmetry points have an overline. Given the geometrical position of the k-points with opposite non-relativistic spin-splitting, the altermagnetic surface states survived on the (010) surface (colored in green), while the other two surfaces are blind to AM (colored in red).
• Figure 3: Band structure of Ca$_2$RuO$_4$ in the A-centered magnetic phase without SOC along the path R$_1$-$\Gamma$-R$_2$ between -1.4 and -0.9 eV. Blue and red bands represent the spin-up and spin-down channels, respectively. The Fermi level is set to zero energy. The band gap is 0.84 eV.
• Figure 4: In the top part, relativistic spin-momentum locking of Ca$_2$RuO$_4$ with Néel vector along the $y$-axis composed of d$_{yz}$, d$_{xz}$, and d$_{xy}$ for the S$_x$, S$_y$, and S$_z$ components, respectively. Red and blue sectors represent regions of the Brillouin zone with opposite non-relativistic spin-splitting. The S$_y$ component is the dominant component and inherits the non-relativistic spin-momentum locking. Once we introduce the Rashba Hamiltonian H$_R$ with an electric field along the z-direction, one symmetry-protected nodal plane is destroyed for the S$_x$ and S$_y$ components, but the spin-momentum locking of the component parallel to the electric field survives. The relativistic spin-momentum locking is composed of p$_y$ for S$_x$, p$_x$ for S$_y$, while the d$_{xy}$ for S$_z$ remains intact.
• Figure 5: Shift of the Ru-atoms in the (a) ferroelectric-like distortions, (b) antiferroelectric-like distortions, and (c) stripe cases. Red and gray balls represent the oxygen and ruthenium atoms, respectively. The black arrows indicate the displacements of the Ru atoms. Only the case of shifts along the $z$-axis is shown here. The black lines represent the unit cell of the system.