Dominant orbital magnetization in the prototypical altermagnet MnTe

Technical Summary: Dominant Orbital Magnetization in the Prototypical Altermagnet MnTe

Problem Statement
Altermagnetism is a recently identified form of antiferromagnetism characterized by momentum-dependent spin polarization of electronic states and vanishing net magnetization, arising from specific crystalline symmetries. While altermagnets are theoretically predicted to exhibit phenomena typically associated with ferromagnets—such as the anomalous Hall effect (AHE)—when spin-orbit coupling (SOC) is present, the microscopic origin of the resulting weak ferromagnetism remains unclear. Specifically, the relative contributions of spin versus orbital magnetization to this net moment are not well understood. In the prototypical altermagnet $\alpha$-MnTe, which exhibits large nonrelativistic spin splitting and AHE, it is crucial to determine whether the observed weak ferromagnetism is driven primarily by spin canting or by orbital effects, as this distinction is vital for interpreting experiments and designing altermagnetic devices.

Methodology
The authors employed density functional theory (DFT) simulations to quantitatively investigate the intrinsic spin and orbital magnetization of the magnetic ground state of $\alpha$-MnTe.

• Computational Framework: Calculations were performed using the Vienna Ab initio Simulation Package (VASP) with the generalized gradient approximation (PBE) and Hubbard $U$ corrections ($U=4$ eV, $J_H=0.97$ eV) for Mn-3d orbitals. Spin-orbit coupling was included in all calculations.
• Symmetry Analysis: The study analyzed charge density differences and relativistic spin-resolved electronic structures to identify symmetry breaking. The magnetic space group was identified as $Cm'c'm$ (#63.462), resulting from the Néel vector alignment along the $y$-axis, which lowers the symmetry from the parent $P6_3/mmc$ structure.
• Magnetization Calculation: Intrinsic spin and orbital magnetizations were calculated by summing contributions from all Bloch states over the entire Brillouin zone. The orbital magnetization was evaluated using the modern theory of orbital magnetization, involving the Berry curvature and the derivative of Bloch states with respect to the wave vector.
• Doping Simulation: The chemical potential was varied to simulate hole doping, allowing the authors to assess the robustness of spin and orbital magnetization against carrier concentration changes.

Key Results

1. Symmetry and Spin Canting: The introduction of SOC induces a weak ferromagnetism characterized by a slight in-plane rotation of the Néel vector and a small canting angle ($\theta \approx 0.01^\circ$). This canting arises from higher-order SOC-driven interactions rather than first-order Dzyaloshinskii–Moriya interactions.
2. Dominance of Orbital Magnetization: The study reveals a significant net orbital magnetization oriented along the $z$-axis (perpendicular to the Néel vector). The calculated value is approximately 0.176 $\mu_B$ per unit cell. In stark contrast, the net spin magnetization along the same axis is negligible, at only 0.002 $\mu_B$ per unit cell. Thus, the orbital contribution is two orders of magnitude larger than the spin contribution.
3. Robustness vs. Tunability:
   • Orbital Magnetization: The net orbital magnetization remains nearly constant over a wide energy range (up to 0.75 eV below the valence band maximum) and is robust against changes in carrier concentration.
   • Spin Magnetization: The net spin magnetization is highly sensitive to the chemical potential. It can be tuned via hole doping but is strongly suppressed in the insulating phase due to the large band gap, which inhibits the mixing of spin-up and spin-down states necessary for SOC-induced canting.
4. Spin Texture: The spin-resolved density of states and band structures confirm that while the in-plane components ($S_x, S_y$) exhibit characteristic altermagnetic patterns (e.g., $g$-wave polarization), the $S_z$ component is responsible for the weak ferromagnetism and oscillates as a function of energy, consistent with the behavior of SOC-driven antisymmetric interactions.

Significance and Claims
The paper establishes that in $\alpha$-MnTe, the net magnetization arising from SOC is dominated by the orbital contribution rather than the spin contribution. This finding challenges the conventional view that weak ferromagnetism in such systems is primarily a spin-canting phenomenon. The authors argue that this dominance of orbital magnetization is crucial for understanding the microscopic origin of the anomalous Hall effect and other transport phenomena in altermagnets.

The results suggest that future studies and device designs involving altermagnets must explicitly account for orbital degrees of freedom. The robustness of the orbital magnetization against doping implies that orbital-based phenomena may be more stable in practical applications than spin-based ones in these materials. The authors note a discrepancy between their theoretical total magnetization ($\approx -0.178 \mu_B$) and experimental values ($10^{-4}$ to $10^{-3} \mu_B$), attributing this to the compensation effects of altermagnetic domains, which tend to cancel out the net moment in bulk measurements.

In conclusion, this work highlights the importance of incorporating orbital magnetization into the theoretical description of altermagnetic materials, proposing that the field of altermagnets should extend beyond spintronics to include "orbitronics."