Twist-induced altermagnetism in a metallic van der Waals antiferromagnet

1. Problem Statement

Altermagnetism (AM) is a magnetic state characterized by spin-polarized electronic bands coexisting with zero net magnetization. This state is highly desirable for spintronics due to the absence of stray fields and ultrafast spin dynamics. However, realizing AM in two-dimensional (2D) materials is challenging because:

• Symmetry Constraints: Conventional 2D van der Waals (vdW) antiferromagnets (e.g., CrI3, CrSBr) often possess inversion ($\mathcal{P}$) or translational ($t$) symmetries connecting opposite spin sublattices, which enforces spin degeneracy and forbids AM.
• Material Limitations: The metallic vdW material $\text{Fe}_3\text{GaTe}_2$ is a promising candidate due to its high Curie temperature and metallic nature, but it naturally exhibits a ferromagnetic (FM) ground state.
• Doping Limitations: While Cobalt (Co) doping in $(\text{Fe}_{1-x}\text{Co}_x)_3\text{GaTe}_2$ can induce an antiferromagnetic (AF) ground state, the resulting bilayer structure retains inversion symmetry ($\mathcal{P}$), preventing the necessary breaking of combined $\mathcal{PT}$ symmetry required for AM.

The Core Challenge: Identifying a strategy to break $\mathcal{PT}$ symmetry in a metallic vdW antiferromagnet while preserving the specific rotational/mirror symmetries required for altermagnetic order, without destroying the AF interlayer coupling.

2. Methodology

The authors employed a combination of first-principles calculations and symmetry analysis:

• Computational Framework: Density Functional Theory (DFT) calculations were performed using the VASP package.
  • Exchange-correlation was treated using the Generalized Gradient Approximation (GGA) and validated with the Local Density Approximation (LDA).
  • Van der Waals interactions were modeled using the DFT-D2 approximation.
  • Spin-orbit coupling (SOC) was included to assess relativistic effects.
• System Modeling:
  • Bilayer $\text{Fe}_3\text{GaTe}_2$: Established as a baseline FM system.
  • Doped Bilayer $\text{Fe}_2\text{CoGaTe}_2$: Modeled with Co substitution at the central Fe sites ($x=1/3$) to induce AF coupling.
  • Twisted Bilayer: A specific twist angle of 21.79° was applied to the doped bilayer. This angle was chosen because it breaks global inversion symmetry while inducing zero strain in the sublattices.
• Analysis Tools:
  • Spin-Space Group (SSG) Analysis: To classify magnetic symmetries and determine allowed spin splittings.
  • TB2J Code: Used to calculate isotropic exchange interactions ($J_{ij}$).
  • VAMPIRE Package: Used for atomistic simulations to estimate magnetic ordering temperatures ($T_N$).
  • Wannier90: Used to construct maximally localized Wannier functions for analyzing orbital contributions.

3. Key Contributions

1. Demonstration of Twist-Induced AM in a Metal: The paper provides the first theoretical evidence that twisting a metallic vdW antiferromagnet ($\text{Fe}_2\text{CoGaTe}_2$) can induce a robust altermagnetic state.
2. Symmetry Breaking Mechanism: The authors elucidate how a 21.79° twist breaks the global $\mathcal{PT}$ symmetry (which connects opposite spin sublattices) while preserving specific rotational symmetries ($\mathcal{C}_2$) that enforce the characteristic momentum-dependent spin splitting of AM.
3. Microscopic Exchange Mechanism: A detailed orbital-resolved analysis reveals how Co doping alters the balance between ferromagnetic (inter-plane) and antiferromagnetic (in-plane) exchange interactions, driving the FM-to-AF transition.
4. Alternative Pathways: The study proposes Janus configurations (Se substitution) and Fe-vacancy-induced displacements as alternative routes to achieve AM in bulk $\text{Fe}_3\text{GaTe}_2$-based systems without twisting.

4. Key Results

Electronic and Magnetic Structure

• Ground State Transition: Co doping transforms $\text{Fe}_3\text{GaTe}_2$ from FM to AF. The interlayer exchange coupling ($J_{int}$) becomes strongly antiferromagnetic ($-2.44$ meV/atom in the untwisted state).
• Twist Effect: Upon twisting (21.79°), the AF interlayer coupling is preserved ($J_{int} \approx -0.54$ meV/atom), though reduced due to increased interlayer spacing (2.94 Å $\to$ 3.25 Å).
• Altermagnetic Order: The twisted structure belongs to a non-centrosymmetric magnetic space group ($P\bar{3}m1$ reduced to specific SSG operations). The system is identified as an $i$-wave altermagnet.
• Spin Splitting:
  • Non-relativistic limit: The system exhibits massive spin splitting up to 138 meV (approx. 0.2 eV below the Fermi level) and 70 meV at the Fermi energy.
  • Symmetry Pattern: Spin polarization alternates every 30° in momentum space, forming a star-like pattern with sixfold symmetry.
  • SOC Effects: When SOC is included, spin degeneracy is lifted along directions perpendicular to the twofold rotation axes but remains protected along the axes themselves.

Magnetic Exchange Interactions

• Inter-plane vs. In-plane:
  • Inter-plane ($J_{Fe1-Fe2}$, $J_{Fe1-Co}$): Strongly Ferromagnetic (FM). Dominated by $d_{z^2}$-$d_{z^2}$ and $d_{xz}$-$d_{xz}$ orbital overlaps.
  • In-plane ($J_{Fe1-Fe1}$, $J_{Co-Co}$): Antiferromagnetic (AF). Mediated by superexchange pathways ($d_{x^2-y^2}$-$d_{x^2-y^2}$ and $d_{z^2}$-$d_{z^2}$).
• Role of Electrons: The introduction of extra electrons via Co doping stabilizes the AF in-plane interactions, reducing the overall FM order and lowering the magnetic ordering temperature ($T_N$).
• Ordering Temperature: The calculated $T_N$ for twisted $\text{Fe}_2\text{CoGaTe}_2$ is 280 K (overestimated compared to experimental ~130 K, but correctly predicts a reduction by a factor of ~2.1 compared to pristine $\text{Fe}_3\text{GaTe}_2$).

Fermi Surface

• The spin-resolved Fermi surface exhibits "star-like" sheets with sixfold symmetry.
• While regions are spin-polarized, the total magnetization is zero due to symmetry-enforced compensation.

5. Significance

• Platform for 2D Altermagnetism: This work establishes twistronics as a versatile tool to induce altermagnetism in metallic systems, overcoming the limitations of naturally occurring 2D magnets that possess prohibitive symmetries.
• Spintronic Applications: The combination of metallic character, zero net magnetization, and large spin splitting (138 meV) makes twisted $\text{Fe}_2\text{CoGaTe}_2$ a prime candidate for next-generation spintronic devices, such as spin transistors and spin-orbit torque devices, which require efficient electrical control of spin currents without stray fields.
• Fundamental Physics: The study deepens the understanding of how lattice symmetry (twisting) and chemical doping (Co substitution) can be synergistically used to engineer complex magnetic ground states and break specific symmetry operations ($\mathcal{PT}$) while preserving others.

In summary, the paper successfully demonstrates that twisting a Co-doped metallic bilayer $\text{Fe}_3\text{GaTe}_2$ creates a robust, high-temperature altermagnetic state with significant spin splitting, offering a new design paradigm for ultrathin, high-efficiency spintronic nanodevices.