Emergent $d$-wave altermagnetism in orthogonally twisted bilayer CrPS$_4$

Abstract: Twistronics is a powerful strategy to engineer novel quantum states by controlling the relative orientation between layered materials. Here, we demonstrate that an orthogonally twisted bilayer CrPS$_4$ shows $d$-wave altermagnetism driven purely by structural rotation. Symmetry analysis reveals that the twisted stacking breaks partial translational combined with time-reversal symmetry, leading to a fourfold rotation relation between opposite spin sublattices, enabling altermagnetism. First-principles calculations demonstrate a sizable non-relativistic spin splitting of up to 68 meV around the Fermi level. We further show that the altermagnetic state can be further stabilized through interlayer compression and modulation of the on-site Coulomb interaction. The resulting band structure exhibits pronounced spin-dependent anisotropy, enabling efficient spin to charge conversion reaching $\sim$50% near the Fermi level and sizable giant magnetoresistance. These results establish twisted CrPS$_4$ as a realistic platform for altermagnetism and highlights twistronics as a versatile route for advanced spintronics applications.

• The paper uses symmetry analysis and DFT+U calculations to reveal a d-wave altermagnetic state in 90° twisted CrPS4 layers.
• The study reports non-relativistic spin splitting up to 68 meV with pronounced momentum dependence, key for spin-polarized transport.
• The paper highlights tunable AF coupling via mechanical strain and gating, paving the way for dynamic control in spintronic applications.

Emergent $d$-Wave Altermagnetism in Orthogonally Twisted Bilayer CrPS$_4$

Introduction

The study explores the emergence of $d$-wave altermagnetic (AM) order in orthogonally twisted bilayer CrPS$_4$, leveraging the principles of twistronics to engineer quantum magnetic states in 2D van der Waals (vdW) materials. By performing a combination of symmetry analysis and first-principles DFT+$U$ calculations, the work establishes how a 90° relative twist between antiferromagnetic CrPS$_4$ layers induces a $d$-wave momentum-dependent spin splitting not requiring spin-orbit coupling (SOC) or net magnetization. The findings highlight the potential of twisted CrPS$_4$ as a new experimental platform for AM states, demonstrating strong spin-dependent transport phenomena highly relevant for spintronic applications.

Altermagnetism and the Role of Twistronics

Altermagnetic phases, characterized by spin splitting in momentum space without a net magnetization, combine features of both ferromagnets (FM) and antiferromagnets (AF) without stray fields and with enhanced ultrafast spin dynamics. In crystalline systems, AM arises fundamentally from symmetry rather than relativistic SOC, with spin splittings exhibiting $d$-, $g$-, or $_4$0-wave symmetry-protected patterns in momentum space.

Twistronics enables reconfiguration of interlayer symmetry relations and magnetic couplings in 2D vdW structures via controlled layer rotations. While previous theoretical proposals for 2D AM systems often relied on small twist angles yielding $_4$1-wave patterns with suppressed spin-polarized transport, $_4$2-wave AM—arising at orthogonal (90°) twists—is especially desirable due to reduced nodal surfaces and consequentially strong spin-polarized anisotropic transport.

Theoretical Framework and Symmetry Analysis

Untwisted bilayer CrPS$_4$3 exhibits A-type AF order with significant out-of-plane anisotropy, supported by robust experimental and computational data. For the pristine structure, the [$_4$4] operation (spin inversion + lattice translation) enforces spin degeneracy in the bandstructure. Upon 90° interlayer rotation, this symmetry is broken and replaced by a fourfold rotation operation [$_4$5], which matches spin sublattices between orthogonally rotated layers, forming the essential symmetry condition for $_4$6-wave AM.

The first-principles calculations (DFT+$_4$7) confirm that CrPS$_4$8 remains semiconducting after twisting (bandgap $_4$91.2 eV) and that the new stacking preserves strong AF coupling with out-of-plane magnetic anisotropy, albeit reduced due to diminished orbital overlap.

Electronic Structure and Spin Splitting

The key result is the emergence of substantial, non-relativistic spin splitting near the Fermi energy, with values reaching 68 meV at the valence band maximum and 43 meV at the conduction band minimum. These splittings display a momentum-dependence with $d$0-wave symmetry, reversing sign every 90° in the Brillouin zone—directly reflecting the symmetry properties of the underlying AM state. SOC corrections have negligible impact, further confirming that the AM splitting is a non-relativistic effect.

Efficient tunability of interlayer coupling and electronic correlations is demonstrated. Application of uniaxial pressure (reducing interlayer separation) or increasing Hubbard $d$1 parameter enhances AF coupling and stabilizes the AM phase. Notably, high-dielectric environments or electrostatic gating could be utilized for experimental control, providing a versatile toolbox for tailoring properties of AM states.

Spin-Dependent Anisotropic Transport and Magnetoresistance

The pronounced $d$2-wave spin-momentum coupling leads to highly anisotropic and spin-dependent longitudinal conductivities ($d$3, $d$4) with C$d$5 symmetry, while transverse conductivity $d$6 vanishes due to nodal degeneracy. The spin-to-charge conversion efficiency (SCE) reaches nearly 50% below the Fermi level, and 25% above, comparable to or exceeding other known AM materials such as RuO$d$7 and Mn$d$8Sn. This implies orthogonal charge currents are carried by opposite spin species, establishing a framework for net spin current generation in the absence of global magnetization.

Furthermore, the system exhibits sizable giant magnetoresistance (GMR), with the GMR signal peaking at 25% below the Fermi level. These transport signatures are directly linked to the symmetry-enforced spin anisotropy and provide practical pathways for implementation in next-generation spintronic devices.

Implications and Outlook

This study identifies twisted bilayer CrPS$d$9 as a model system for realization and investigation of $_4$0-wave AM order in 2D magnets. The robust and tunable non-relativistic spin splitting, prominent spin-to-charge conversion, and magnetoresistive responses underscore its technological relevance for spin current generation, information encoding, and non-volatile logic. The highly tunable nature of AF coupling via mechanical/field effects enhances the experimental viability of the proposed platform. The work accentuates twistronics as a generative principle for symmetry-enforced quantum states, potentially extending to other material classes and higher-order AM patterns.

In terms of future directions, the confluence of strong interlayer coupling, out-of-plane AF order, and symmetry control in CrPS$_4$1 provides a blueprint for designing artificial altermagnets with tailored spin physics. The interplay of substrate engineering, electrostatic gating, and twist-angle control could yield further opportunities to dynamically access and manipulate novel altermagnetic phases, paving the way for robust, ultrafast spin logic architecture in 2D heterostructures.

Conclusion

Orthogonally twisted bilayer CrPS$_4$2 is conclusively demonstrated to host a robust $_4$3-wave altermagnetic state driven by structural rotation, manifested as pronounced non-relativistic spin splitting, high spin-to-charge conversion efficiencies, and substantial GMR. These findings position twisted CrPS$_4$4 as an experimentally accessible, symmetry-protected platform for AM-based spintronics, firmly establishing twistronics as a pivotal technique for advanced quantum magnetic engineering in two-dimensional materials.