Giant momentum-dependent spin splitting in centrosymmetric low-$Z$ antiferromagnets

The energy vs crystal momentum $E$($k$) diagram for a solid (band structure) constitutes the road map for navigating its optical, magnetic, and transport properties. By selecting crystals with specific atom types, composition, and symmetries, one could design a target band structure and thus desired properties. A particularly attractive outcome would be to design energy bands that are split into spin components with a momentum-dependent splitting, as envisioned by Pekar and Rashba [Zh. Eksp. Teor. Fiz. 47, 1927 (1964)], enabling spintronic application. The current paper provides “design principles” for wave-vector dependent spin splitting (SS) of energy bands that parallel the traditional Dresselhaus and Rashba spin-orbit coupling (SOC)–induced splitting, but originates from a fundamentally different source—antiferromagnetism. We identify a few generic antiferromagnetic (AFM) prototypes with distinct SS patterns using magnetic symmetry design principles. These tools allow also the identification of specific AFM compounds with SS belonging to different prototypes. A specific compound—centrosymmetric tetragonal $\mathrm{Mn}{\mathrm{F}}_2$—is used via density functional band-structure calculations to quantitatively illustrate one type of AFM SS. Unlike the traditional SOC-induced effects restricted to noncentrosymmetric crystals, we show that antiferromagnetic-induced spin splitting broadens the playing field to include even centrosymmetric compounds, and gives SS comparable in magnitude to the best known (“giant’” SOC effects, even without traditional SOC, and consequently does not rely on the often-unstable high atomic number elements required for high SOC. We envision that use of the current design principles to identify an optimal antiferromagnet with spin-split energy bands would be beneficial for efficient spin-charge conversion and spin-orbit torque applications without the burden of requiring compounds containing heavy elements.

Figure 1

Crystal structure, band structure, and spin splitting of the centrosymmetric AFM tetragonal $\mathrm{Mn}{\mathrm{F}}_2$. (a) Magnetic unit cell where red arrows indicate local magnetic moment; (b) contour plot of magnetization along $z$ in $z=c/2$ plane; (c) DFT calculated band structure with our calculated magnetic symmetry representations (see Appendix pp3), using the notations of Ref. [38] with numbers in parentheses indicating the dimension of the representation (i.e., degeneracies). The top four valence bands are denoted by $V1$, $V2$, $V3$, $V4$ and the yellow screen highlights the gaps between valence and conduction bands. Inset of (c) shows the BZ and the blow-up bands around $R$ point. The blue to red color scale denotes the calculated out-of-plane spin polarization. Panels (d) and (e) show DFT calculated wave-vector dependence of the spin splitting between pairs of valence bands $V1$ and $V2$ [in (d)] and between $V3$ and $V4$ [in (e)] for different scaling of SOC ${\lambda }_{\mathrm{SOC}}$ (numerical coefficient $0<{\lambda }_{\mathrm{SOC}}<1$). Inset of (d) shows the spin splitting vs the amplitude of the spin-orbit coupling ${\lambda }_{\mathrm{SOC}}$ at Γ (0, 0, 0), $R$ (0, 0.5, 0.5), and the middle point of $\mathrm{\Gamma }\text{−}M$ (0.25, 0.25, 0). All DFT calculations use PBE exchange correlation functional [39] with on-site Coulomb interaction on Mn-$3d$ orbitals of $U=5\mathrm{eV}$, $J=0\mathrm{eV}$, and the experimental crystal structure [37].

Figure 2

DFT band structures of centrosymmetric (CS) $\mathrm{Mn}{\mathrm{F}}_2$ in NM and AFM without SOC. In all cases we use the experimentally observed centrosymmetric tetragonal structure [37]: (a) NM with SOC set to zero; (b) AFM with SOC set to zero. Out-of-plane spin polarizations are mapped to color scale from blue to red. The integer number attached to each band and $\mathit{k}$ point is the degeneracies.

Figure 3

Spin textures in AFM $\mathrm{Mn}{\mathrm{F}}_2$ with SOC on $k_z=\pi /2c$ plane. (a) Out-of-plane spin texture of $V1$ band, (b) out-of-plane spin texture of $V2$ band, (c) in-plane spin texture of $V1$ band, and (d) in-plane spin texture of $V2$ band.

Figure 4

Spin textures of the top two valence bands ($V1$ and $V2$) in AFM $\mathrm{Mn}{\mathrm{F}}_2$ on two $k_Z$ planes: (a) (b) $k_z=0$ plane, and (d) and (e) $k_z=\pi /c$ plane. For each $k_z$ plane, the labels of the high-symmetry $\mathit{k}$ points are shown by a diagram on the left side of each horizonal panel. The in-plane spin polarizations are indicated by black arrows, while black dot means the in-plane polarization at this $\mathit{k}$ is zero; the out-of-plane spin polarizations are mapped by colors from blue to red.

Figure 5

DFT band structures of centrosymmetric $\mathrm{Mn}{\mathrm{F}}_2$ in NM and AFM with SOC. In all cases we use the experimentally observed centrosymmetric tetragonal structure [37]: (a) NM with SOC; (b) AFM with SOC. Out-of-plane spin polarizations are mapped to color scales from blue to red. The integer numbers attached to bands are degeneracy factors.