Strain- and Field-Tunable Nonrelativistic Spin Splitting and Wave-Symmetry-Dependent Spin Transport in Twisted Bilayer Altermagnets

Abstract

Magnetism-driven nonrelativistic spin splitting (NRSS) provides a pathway toward efficient, spin-orbit-free spintronics. In centrosymmetric two-dimensional antiferromagnets, spin-polarized transport is symmetry-forbidden due to the combined space-time inversion ($PT$) symmetry. Here, by employing first-principles density functional theory and spin-group symmetry analysis, we demonstrate that twisting two antiferromagnetic or ferromagnetic monolayers of CoCl$_2$, AX$_2$ (A = Mn, V; X = Cl, Br, I), NiF$_2$, NiBr$_2$, FeS, CoS, MnTe$_2$, MnSe$_2$, and RuSe induces finite NRSS even in the absence of spin-orbit coupling. The relative twist breaks $[C_2||P]$ and $[E||C_{nz}]$ symmetries, giving rise to momentum-dependent spin polarization with distinct $d$-, $g$-, and $i$-wave altermagnetic patterns across the Brillouin zone. Using symmetry-invariant $k\cdot p$ modeling, we extract linear spin-splitting coefficients $α^{(1)}$ ranging from 800-1100 meVÅ, comparable to SOC-induced Rashba-Dresselhaus strengths observed in noncentrosymmetric semiconductors. An out-of-plane electric field ($\mathcal{E}_z$) introduces Zeeman-type band splitting up to 110 meV at 10 MV/cm, while biaxial strain tunes the NRSS magnitude nearly linearly without altering symmetry. Crucially, the strain $u_{xx-yy}$ reduces the spin point group symmetry and drives reversible $g/i \rightarrow d$ wave-type transitions, resulting in finite spin conductivity and an enhanced spin-splitter angle (up to 18$^\circ$). These results extend the concept of altermagnetism to twisted bilayer geometries and establish a general route for realizing exchange-driven, nonrelativistic spin currents through symmetry engineering without requiring heavy elements or spin-orbit coupling.

Figures (6)

• Figure 1: Definition of the local momentum coordinates used in the symmetry-based $k\cdot p$ analysis of twisted bilayers. (a) Square (tetragonal) moiré Brillouin zone, showing the generic point $M_c$ along the $X$–$M$–$Y$ direction. The local coordinate system $(q_x',q_y')$ is defined with respect to $M_c$. (b) Hexagonal moiré Brillouin zone, highlighting the generic point $K_c$ along the $K_1$–$K_2$ direction and the generic point $M_c$ along the $M_1$–$M_2$ direction, with $q_x'$ and $q_y'$ defined relative to $K_c$ and $M_c$. The shaded regions indicate the irreducible momentum sectors used to construct the symmetry-allowed nonrelativistic $k\cdot p$ Hamiltonian.
• Figure 2: Crystal structures, moiré Brillouin zones (BZs), and spin-resolved band structures of representative twisted bilayer altermagnets. (a) Relaxed moiré superlattice of hexagonal tb-MnBr$_2$ and (b) tetragonal tb-CoCl$_2$, where red and blue spheres denote opposite spin-polarized transition-metal atoms and halogen atoms are shown in brown (Br) and yellow (Cl). Dashed lines indicate the nonrelativistic spin-group symmetry operation $[C_2||C_{2[010]}]$ connecting opposite spin sublattices. (c,f) Construction of the moiré BZ from the individual layer BZs (red/blue hexagons or squares for hexagonal/tetragonal lattices) to the reduced moiré BZ (black), with high-symmetry paths indicated. (d,e) Spin-resolved band structures of tb-MnBr$_2$ along $\Gamma$–M–K–$\Gamma$ and tb-CoCl$_2$ along $\Gamma$–X–M–Y–$\Gamma$, calculated within PBE. Although spin degeneracy persists along high-symmetry lines, the reduced symmetry of twisted bilayers enables nonrelativistic spin splitting at generic $\mathbf{k}$ points.
• Figure 3: Nonrelativistic spin splitting (NRSS) and symmetry-resolved $k\cdot p$ analysis in representative twisted bilayer altermagnets. (a) Momentum-resolved spin-splitting energy $\Delta E(\mathbf{k})$ at the VBM of tetragonal tb-FeS, exhibiting a $d$-wave altermagnetic pattern. (b) $\Delta E(\mathbf{k})$ at the CBm of tetragonal tb-CoCl$_2$, showing a $g$-wave symmetry. (c) $\Delta E(\mathbf{k})$ at the CBm of hexagonal tb-MnBr$_2$, characteristic of an $i$-wave altermagnetic texture. (d) Spin-resolved band dispersion near $M_c$ for tb-FeS (VBM), comparing DFT results (solid lines) with the symmetry-invariant $k\cdot p$ model (dashed lines). (e,f) Same comparison for the CBm of tetragonal tb-CoCl$_2$ and hexagonal tb-MnBr$_2$, respectively. The excellent agreement validates the symmetry-derived effective Hamiltonian and confirms the nonrelativistic origin of the observed spin splitting.
• Figure 4: Tuning of nonrelativistic spin splitting by biaxial strain and electric field. (a--c) Linear spin-splitting coefficient $\alpha^{(1)}$ as a function of biaxial strain for tetragonal tb-FeS, tetragonal tb-CoCl$_2$, and hexagonal tb-MnBr$_2$, respectively. Both compressive and tensile strain preserve crystal symmetry but continuously tune the NRSS magnitude. (d) Spin-resolved band structure of tetragonal tb-FeS under an out-of-plane electric field $\mathcal{E}_z = 10$ MV/cm, showing Zeeman-type spin splitting at the $\Gamma$ point. (e,f) Electric-field-induced spin splitting $\Delta$ at the VBM and CBm for representative hexagonal and tetragonal twisted bilayers, demonstrating a nearly linear dependence on $\mathcal{E}_z$, repectively.
• Figure 5: Uniaxial strain-induced wave-type transition and emergence of spin-polarized transport. (a) $g$-wave momentum-space spin-splitting texture at the VBM of tetragonal tb-CoCl$_2$. (b) Corresponding texture under $u_{xx-yy}=1\%$ strain, showing a symmetry-driven $g \rightarrow d$ transition. (c,d) Spin conductivity $\sigma^z_{xy}$ and spin-splitter angle (SSA) as a function of energy for unstrained and strained tb-CoCl$_2$. (e) $i$-wave spin-splitting texture at the CBm of hexagonal tb-MnTe$_2$. (f) Conversion to a $d$-wave pattern under $u_{xx-yy}=1\%$ strain. (g,h) Corresponding spin conductivity and SSA for tb-MnTe$_2$. Uniaxial strain activates finite spin currents and enhances charge-to-spin conversion without invoking spin--orbit coupling.