Prediction of low-Z collinear and noncollinear antiferromagnetic compounds having momentum-dependent spin splitting even without spin-orbit coupling

The over 60 years old Rashba-Dresselhaus effect predicts spin-orbit coupling (SOC) induced momentum-dependent spin splitting and spin polarization in materials with noncentrosymmetric structures. Strong SOC induced effects usually require high-atomic number (Z) elements such as rare-earth elements. It has recently been pointed out [Yuan et al., Phys. Rev. B 102, 014422 (2020)] that antiferromagnets could hold SOC-independent spin splitting and spin polarization. In the present work we develop the spatial and magnetic symmetry conditions enabling such antiferromagnet (AFM)-induced spin splitting, dividing the 1651 magnetic space groups into seven different spin splitting prototypes (SST-1 to SST-7). This analysis places the physics of AFM spin splitting (SST-4) within the broader context of symmetry conditions that enable the more familiar forms of spin splitting, such as ferromagnetic Zeeman effect (SST-5), nonmagnetic no spin splitting (SST-6), and the nonmagnetic Rashba and Dresselhaus effects (SST-7). The AFM-induced spin splitting and spin polarization do not necessarily require breaking of inversion symmetry or the presence of SOC, hence can exist even in centrosymmetric, low-Z light element compounds, considerably broadening the material base for spin manipulations. We use the “inverse design” approach of first formulating the target property (here, spin splitting in low-Z compounds not restricted to low symmetry structures), then derive the enabling physical design principles—the magnetic symmetry conditions—to search realizable compounds that satisfy these a priori design principles. This process uncovers 422 magnetic space groups (160 centrosymmetric and 262 noncentrosymmetric) that could hold AFM-induced, SOC-independent spin splitting and spin polarization. We then search for stable compounds following such enabling symmetries. We investigate the electronic and spin structures of some selected prototype compounds by density functional theory (DFT) and find spin textures that are different than the traditional Rashba-Dresselhaus patterns and exist even in the absence of SOC effect. We provide the DFT results for all antiferromagnetic spin splitting prototypes (SST-1, SST-2, SST-3, SST-4), and concentrate on two limits of SST-4 that are particularly unusual: When spin splitting is momentum dependent (just like the Rashba effect) but is enabled in antiferromagnets even in the absence of SOC in the Hamiltonian. This includes examples of (a) centrosymmetric SST-4A compounds (e.g., orthorhombic $\mathrm{LaMn}{\mathrm{O}}_3$ illustrating collinear AFM, as well as cubic $\mathrm{Ni}{\mathrm{S}}_2$ illustrating noncollinear AFM) and (b) noncentrosymmetric SST-4B compounds (e.g., rhombohedral $\mathrm{MnTi}{\mathrm{O}}_3$ illustrating collinear AFM and hexagonal $\mathrm{ScMn}{\mathrm{O}}_3$ illustrating noncollinear AFM). The symmetry design principles outlined here, along with their transformation into an inverse design material search approach and DFT verification, could open the way to their experimental examination.

Figure 1

Classification of spin splitting prototypes (SSTs) in terms of symmetry conditions and the consequences of these symmetry conditions on the spin splitting with or without SOC. Here, $I, \theta ,$ and $T$ are symmetry operations of spatial inversion, time reversion, and translation, respectively. $\theta IT$ are combinations of these operators. The symmetry conditions are phrased in terms of three questions: (1) Is the symmetry $\theta IT$ present in the magnetic space group (MSG)? (DP-I) (2) What is the MSG type? (DP-II) (3) Is the parent space group (PSG) centrosymmetric (CS) or not (NCS)? The consequences of symmetry on the spin splitting patterns at a generic $k$ point are given for both cases with and without SOC.

Figure 2

No spin splitting in AFM CS tetragonal CuMnAs (AFM SST-1). (a) Crystal structure and magnetic moments, where red arrows are used to indicate local magnetic moments; (b) $z$ component of magnetization contour in ($1.53\overline{1}0$) plane which is indicated by green shading in (a); (c) Brillouin zone; (d), (e) spin degenerate energy bands (d) when SOC is off and (e) when SOC is on. The electronic properties are calculated by DFT method using PBE+$U$ functional.

Figure 3

No spin splitting in AFM rocksalt NiO (AFM SST-2). (a) Crystal structure and magnetic moments, where red arrows are used to indicate local magnetic moments; (b) the contour plot of magnetization along [11-2] direction in (110) plane which is indicated by green shading in (a); (c) the first Brillouin zone and high symmetry $k$ paths; (d), (e) spin-degenerate energy bands (d) when SOC is off and (e) when SOC is on. The electronic properties are calculated by DFT method using PBE+$U$ functional.

Figure 4

SOC induced spin splitting in AFM orthorhombic $\mathrm{Mn}{\mathrm{S}}_2$ (AFM SST-3A). (a) Crystal structure and magnetic moments, where red arrows are used to indicate local magnetic moment; (b) magnetization contour plot in the middle (001) plane; (c) Brillouin zone and high symmetry $k$ paths. (d) Spin degenerate energy bands when SOC is off and (e) spin split energy bands when SOC is on. (f), (g) Isosurface of spin splitting between top two valence bands (VB1, VB2) at 20 meV (red surface) and −20 meV (blue surface) in the Brillouin zone of top view and side perspective view when SOC is off (f) and when SOC is on (g); the electronic properties are calculated by DFT method using PBE+$U$ functional.

Figure 5

Spin polarization and spin splitting in centrosymmetric collinear AFM orthorhombic $\mathrm{LaMn}{\mathrm{O}}_3$ (AFM SST-4A). (a) Crystal structure and magnetic moments, where red arrows are used to indicate local magnetic moments; (b) the contour plot of $y$ component of magnetization contour in (001) plane which is indicated by green shading in (a); (c) the first Brillouin zone; (d) energy bands when SOC is off and (e) energy bands when SOC is on. (f), (g) Isosurfaces of spin splitting in the first Brillouin zone between the top two valence bands (VB1, VB2) at 20 meV (red) and −20 meV (blue) when SOC is off (f) and when SOC is on (g). (i), (h) Spin textures of the top two valence bands VB1 and VB2 when SOC is off (h) and when SOC is on (i). The electronic properties are calculated by DFT method using PBE+$U$ functional.

Figure 6

Spin polarization and spin splitting in centrosymmetric noncollinear AFM cubic $\mathrm{Ni}{\mathrm{S}}_2$ (AFM SST-4A). (a) Crystal structure and magnetic moments, where red arrows are used to indicate local magnetic moments; (b) the contour plot of $z$ component of magnetization in (001) plane which is indicated by blue rectangular in (a); (c) the first Brillouin zone; (d) energy bands when SOC is off and (e) energy bands when SOC is on. (f), (g) Isosurfaces of spin splitting between the top two valence bands (VB1, VB2) at 300 meV (absolute value) in the Brillouin zone when SOC is off (f) and when SOC is on (g). (h), (i) Spin textures of the top two valence bands VB1 and VB2 when SOC is off (h) and when SOC is on (i). The electronic properties are calculated by DFT method using PBE+$U$ functional.

Figure 7

Spin polarization and spin splitting in noncentrosymmetric collinear AFM rhombohedral $\mathrm{MnTi}{\mathrm{O}}_3$ (AFM SST-4B). (a) Crystal structure and magnetic moments; red arrows are used to indicate local magnetic moments; (b) the $y$ component of magnetization contour in (001) plane which is indicated by green shading in (a); (c) the first Brillouin zone; (d) energy bands when SOC is off and (e) energy bands when SOC is on. (f), (g) Isosurfaces of spin splitting between the top two valence bands (VB1, VB2) at 45 meV (red) and −45 meV (blue) in the Brillouin zone when SOC is off (f) and when SOC is on (g). (i), (h) Spin textures of the top two valences bands VB1 and VB2 when SOC is off (h) and when SOC is on (i). The electronic properties are calculated by DFT method using PBE+$U$ functional.

Figure 8

Spin polarization and spin splitting in noncentrosymmetric noncollinear AFM hexagonal $\mathrm{ScMn}{\mathrm{O}}_3$ (AFM SST-4B). (a) Crystal structure and magnetic moments; red arrows are used to indicate local magnetic moments; (b) $x$ component of magnetization contour plot in (001) plane which is indicated by green shading in (a); (c) Brillouin zone; (d) energy bands when SOC is off and (e) energy bands when SOC is on. (f), (g) Isosurfaces of spin splitting between the top two valence bands (VB1, VB2) at 25 meV (absolute value) in the BZ of top view and side perspective when SOC is off (f) and when SOC is on (g). (h), (i) Spin textures of the top two valence bands VB1 and VB2 when SOC is off (h) and when SOC is on (i). The electronic properties are calculated by DFT method using PBE+$U$ functional.