$d$-Wave Polarization-Spin Locking in Two-Dimensional Altermagnets

Abstract

We report the emergence of an uncharted phenomenon, termed $d$-wave polarization-spin locking (PSL), in two-dimensional (2D) altermagnets. This phenomenon arises from nontrivial Berry connections, resulting in perpendicular electronic polarizations in the spin-up and spin-down channels. Symmetry-protected $d$-wave PSL occurs exclusively in $d$-wave altermagnets with tetragonal layer groups. To identify 2D altermagnets capable of exhibiting this phenomenon, we propose a symmetry-eigenvalue-based criterion, and a rapid method by observing the spin-momentum locking. Using first-principles calculations, monolayer Cr$_2$X$_2$O (X = Se, Te) characterizes promising candidates for $d$-wave PSL, driven by the unusual charge order in these monolayers. This unique polarization-spin interplay leads to spin-up and spin-down electrons accumulating at orthogonal edges, enabling potential applications as spin filters or splitters in spintronics. Furthermore, $d$-wave PSL introduces an unexpected spin-driven ferroelectricity in conventional antiferromagnets. Such magnetoelectric coupling positions $d$-wave PSL as an ideal platform for fast antiferromagnetic memory devices. Our findings not only expand the landscape of altermagnets, complementing conventional collinear ferromagnets and antiferromagnets, but also highlight tantalizing functionalities in altermagnetic materials, potentially revolutionizing information technology.

Figures (4)

• Figure 1: (a) A schematic of 2D $d$-wave altermagnet in the $xy$ plane. The red and blue arrows represent the two spin sublattices connected by transformation $[C_2||C^+_{4z}]$. (b) First Brillouin zone with four TRIM $\Gamma = (0, 0)$, X = $(\frac{1}{2},0)$, Y=$(0, \frac{1}{2})$ and M = $(\frac{1}{2},\frac{1}{2})$. The green/orange colored regions represent opposite spin splitting. (c) Two configurations of $d$-wave PSL. Top/bottom: spin-up and down electronic polarizations are along $y$/$x$ and $x$/$y$ direction, with vanishing ionic polarization, spin-up and down electrons (red and blue balls) accumulate at surfaces perpendicular to $y$/$x$ and $x$/$y$ direction, respectively.
• Figure 2: (a) Atomic configuration of monolayer Cr$_2$Se$_2$O. The two spin sublattices are connected by $[C_2||C^+_{4z}]$. (b) Spin-resolved band structure along high symmetric $\vec{k}$-path. Under $[C_2||C^+_{4z}]$, the spin-up (down) bands along M-X-$\Gamma$ (yellow colored region) are transformed to spin-down (up) bands along M-Y-$\Gamma$ (green colored region). Spin-resolved band structure for a ribbon with length $L_x$ = 20.5 a$_0$ along (c) $x$ and (e) $y$ direction, insert is the 1D FBZ. The charge density distribution for the in-gap Bloch states at (d) $\bar{\Gamma}$ and $\bar{Y}$ and (e) $\bar{\Gamma}$ and $\bar{X}$ point.
• Figure 3: (a) Orbital-resolved band structure of Hamiltonian $\hat{H}_{Cr_d}+\hat{H}_{Se_p}+\hat{H}_{O_p}$ in the spin-up channel. (b) Ligand field splitting for the CrSe$_4$O$_2$ distorted octahedra. The $x$, $y$ and $z$ axes shown are local coordinate, to be distinguished from the global coordinate shown in (c). (c) Actual ionic charge in the spin-up channel (here the core ionic charge and electrons are ignored). (d)-(e) The evolution of the Wannier charge centers along $k_x$ and $k_y$ directions as indicated by the insert.
• Figure 4: (a) Schematic 180$^{\circ}$ switching of Néel vector. (b) Energy evolution per unit cell with respect to magnetic moment rotation angle $\Theta$ in the $yz$ plane, as shown by the insert.