Supercell altermagnets

Altermagnets are compensated magnets with unconventional $d$-, $g$-, and $i$-wave spin order in reciprocal space. So far the search for new altermagnetic candidates has been focused on materials in which the magnetic unit cell is identical to the nonmagnetic one, i.e., magnetic structures with zero propagation vector. Here, we substantially broaden the family of altermagnetic candidates by predicting supercell altermagnets. Their magnetic unit cell is constructed by enlarging the nonmagnetic primitive unit cell, resulting in a nonzero propagation vector for the magnetic structure. This connection of the magnetic configuration to the ordering of sublattices gives an extra degree of freedom to supercell altermagnets, which can allow for the control over the order parameter spatial orientation. We identify realistic candidates ${\mathrm{MnSe}}_2$ with a $d$-wave order, and ${\mathrm{RbCoBr}}_3, {\mathrm{CsCoCr}}_3$, and ${\mathrm{BaMnO}}_3$ with $g$-wave order. We demonstrate the reorientation of the order parameter in ${\mathrm{MnSe}}_2$, which has two different magnetic configurations, whose energy difference is only 5 meV, opening the possibility of controlling the orientation of the altermagnetic order parameter by external perturbations.

Figure 1

Magnetic unit cell of ${\mathrm{MnSe}}_2$-I, shown in the plane normal to $\left(\overline{1}00\right)$. Blue and green spheres represent Mn and Se atoms, respectively. Top: An isosurface of the charge density is plotted in gray. The fractional coordinates of Mn sites in ${\mathbf{a}}_1$ (0 or $1/2$) are written on top of each atom. Bottom: Isosurfaces of the spin density. Cyan and magenta represent positive and negative values of the spin density, respectively. In both figures, the black dashed line represents the nonmagnetic unit cell. A glide mirror plane (red) connects Mn sites with opposite magnetic moments.

Figure 2

Comparison of the magnetic configurations of ${\mathrm{MnSe}}_2$-I (a) and ${\mathrm{MnSe}}_2$-II (d). The magnetic unit cells only differ in the third subcell (from left to right). A glide plane that connects opposite spins is shown in red. (b) and (e) show the band structure along the path $M\mathrm{\Gamma }{m}_y\left(M\right)|R\mathrm{\Gamma }{m}_z\left(R\right)$. The path is divided into two segments, as can be seen in (c) and (f), denoted by dashed red and green lines. Panels (c) and (f) also show the nodal planes of each magnetic configuration. The different orientation of the $d$-wave causes a spin split path $M\mathrm{\Gamma }{m}_y\left(M\right)$ (red) in ${\mathrm{MnSe}}_2$-I, while $R\mathrm{\Gamma }{m}_z\left(R\right)$ (green) is spin degenerate, and vice versa for ${\mathrm{MnSe}}_2$-II.

Figure 3

(a) Comparison of magnetic ($\left\{{\mathbf{a}}_i\right\}$) and nonmagnetic ($\left\{{\mathbf{p}}_i\right\}$) unit cells, for the $g$-wave $AX{B}_3$ supercell altermagnets. (b) BZ (gray) of the magnetic structures $\left\{{\mathbf{b}}_i\right\}$, and BZ (green) of the nonmagnetic phase $\left\{{\mathbf{b}}_i\right\}$. Only the $k_z=0$ plane is shown for simplicity. (c) Four nodal planes in the BZ of the $g$-wave altermagnets $AX{B}_3$; each plane divides domains with opposite spin polarization in the band structure. (d) Crystal structure of the three $g$-wave altermagnets $AX{B}_3$. $X=$ Mn,Co is the magnetic atom, represented by a blue sphere with a red magnetic moment (pointing out/in the $xy$ plane.) $B$ = O,Cl,Br are represented by small green spheres, and $A=$ Ba,Cs,Rb shown as big cyan spheres. (e) $X{B}_3$ octahedron around two magnetic atoms with opposite magnetic moment. A spin density isosurface around the magnetic atoms is shown in cyan/magenta. (f) The DFT band structure of ${\mathrm{BaMnO}}_3$ ($U=2\text{eV}$) shows opposite sign spin splitting in the paths $H\mathrm{\Gamma }$ and $\mathrm{\Gamma }{m}_z\left(H\right)$, since $m_z$ is a transposing symmetry. See Appendix pp3 for the band structure of the other two $AX{B}_3$ materials.

Figure 4

Band structures of the three $g$-wave $AX{B}_3$ supercell altermagnets. Colors red and blue represent opposite spin polarization. Hubbard parameter used was $U=2\mathrm{eV}$.