Why is the $d$-Wave spin splitting in CuF$_2$ bulk-like?

TL;DR

This work addresses why CuF$_2$ exhibits bulk-like $d$-wave nonrelativistic spin splitting ($NRSS$) while other MF$_2$ compounds show planar NRSS. It develops a magnetic multipole framework and analyzes structural distortions via phonon calculations, comparing the monoclinic CuF$_2$ ground state to a hypothetical tetragonal $P4_2/mnm$ phase. The key finding is that antipolar F displacements generate an additional totally symmetric magnetic octupole component, yielding an extra NRSS channel and transforming the pattern from planar to bulk-like; spin–orbit coupling can further induce a canting moment and a new octupole, adding subtle splittings. This demonstrates that NRSS patterns can be engineered through structural control (e.g., pressure or strain), with potential implications for spintronic devices that operate without strong relativistic effects.

Abstract

With the advent of nonrelativistic spin splitting in collinear compensated antiferromagnets, several candidate materials have also been proposed, among which the family of transition-metal difluorides stands out as a prominent example. Within this family, most members exhibit planar $d$-wave spin splitting, whereas CuF$_2$ shows bulk $d$-wave splitting with an explicit $k_z$ dependence. In this work, we show that this transition from planar to bulk $d$-wave splitting in CuF$_2$ is primarily driven by the antipolar displacements of the F ions, which are absent in the tetragonal rutile structure of the other family members. Our calculations reveal that these additional structural distortions introduce an extra plane of anisotropic magnetization density, giving rise to an additional totally symmetric component of the magnetic octupole tensor. The $k$-space representation of this octupole component, consequently, dictates an additional direction of spin splitting, thereby transforming the $d$-wave spin splitting pattern from planar to bulk-like. We further analyze the effect of spin-orbit coupling on the magnetic octupoles and the resulting spin splitting in the band structure. Our work highlights the possibility of controlling the pattern of nonrelativistic spin splitting through structural modifications, for example, via the application of external pressure.

Figures (10)

• Figure 1: Schematic illustrations of planar vs. bulk $d$-wave spin splitting. The (a) side and (b) top views of the planar $d$-wave spin splitting across different parts of the BZ. The same (c) side and (d) top views of the bulk-$d$ wave spin splitting. The red and blue planes indicate the nodal planes of degenerate up ($E_{\uparrow}$)- and down-spin ($E_{\downarrow}$) polarized bands. The $\pm$ sign shows the sign of the energy splitting, $\Delta E = E_{\uparrow} - E_{\downarrow}$. The reversal of the NRSS energy, as shown in (d), for $k_z > 0$ ( left) and $k_z < 0$ ( right) depicts the $k_z$ dependence of the bulk $d$-wave splitting.
• Figure 2: (a) Crystal Structure of $\mathrm{CuF}_2$. The Cu sublattices, viz., Cu1 and Cu2, are indicated in different colors. The CuF$_6$ octahedra around the Cu1 and Cu2 are rotated by 90° with respect to each other. (b) The AFM configuration of CuF$_2$.
• Figure 3: Band structures of $\mathrm{CuF_2}$, computed for the (a) non-magnetic configuration and (b-c) the AFM configuration (see Fig. \ref{['fig2']}b). NRSS along (b) $k_x$-$k_z$, and (c) $k_y$-$k_z$ directions are apparent from the band structures in the presence of antiferromagnetism.
• Figure 4: (a) Phonon band structure of CuF$_2$ in its hypothetical $P4_2/mnm$ structure. Unstable phonon modes, marked with a black circle at the $\Gamma$-point, give rise to the antipolar displacements of the F atoms ($\Gamma_5^+$). Stable phonon modes, marked with blue and brown circles, are linked to the $\Gamma_2^+$ and $\Gamma_1^+$ distortions, respectively. (b) Structural change from tetragonal $P4_2/mnm$ (structure: 1) to the ground state monoclinic $P2_1/c$ (structure: 4). Structure: 2 is the intermediate $P2_1/c$ structure, which includes only the atomic distortion of the unstable $\Gamma$-phonon modes, i.e., $\Gamma_5^+$ atomic distortion. Upon optimizing the ionic positions of structure:2, we obtain structure:3, which includes additional atomic distortions $\Gamma_2^+$ and $\Gamma_1^+$. Atomic displacements of the $\Gamma_5^+$, $\Gamma_2^+$, and $\Gamma_1^+$ are represented by the black, blue, and brown arrows. Upon optimizing the lattice parameters, we obtain the final ground state $P2_1/c$ structure (structure: 4), which includes all the atomic distortions and modification of the lattice parameters. The change of the lattice angle $\gamma$ to 97.84$^\circ$ gives rise to a shearing strain in the $ab$ plane, producing two inequivalent Cu-Cu bonds with bond lengths of 3.53 and 3.82 Å.
• Figure 5: (a) The variation in Cu1–Cu2 distance and the electronic band gap as a function of amplitude of distortion (see text for details). (b) The changes in the Cu–F bond lengths as a function of distortion amplitude. The inset shows the CuF$_6$ octahedron, indicating different Cu-F bonds, color-coded with their corresponding variations shown in (b).