Microscopic Origin of Piezomagnetism in Mn$_3$Sn: A Dual Real- and $k$-Space Picture

Technical Summary: Microscopic Origin of Piezomagnetism in Mn3Sn

Problem Statement
The antiferromagnet Mn3Sn, known for its non-collinear magnetic structure and large anomalous Hall effect, exhibits a piezomagnetic effect where applied strain induces a net magnetization. While previous experimental and theoretical studies have established that this effect macroscopically arises from the rotation of magnetic moments on specific Mn atoms under uniaxial pressure, the microscopic mechanism driving this induction remains unclear. Specifically, the relationship between real-space spin reorientation and the underlying electronic structure changes in momentum space (k-space) has not been fully elucidated.

Methodology
The authors performed comprehensive first-principles calculations using the Vienna Ab initio Simulation Package (VASP) based on Density Functional Theory (DFT). Key methodological features include:

• Exchange-Correlation: Treated within the Generalized Gradient Approximation (GGA) using the PBE functional.
• Relativistic Effects: Spin-orbit coupling (SOC) was included self-consistently, which is essential for accurately describing the non-collinear antiferromagnetic states of Mn3Sn.
• Computational Parameters: A plane-wave energy cutoff of 600 eV and a $\Gamma$-centered $16\times16\times16$ Monkhorst-Pack k-point mesh were employed.
• Strain Simulation: The study compared the zero-pressure optimized structure ($a=b=5.567$ Å, $c=4.432$ Å) with a uniaxially compressed structure along the x-axis (approx. 0.4% strain, resulting in $a=5.545$ Å, $b=5.566$ Å, $c=4.435$ Å). Atomic positions were fully relaxed in the strained configuration.
• Analysis Framework: The study utilizes a dual perspective, analyzing spin moments in real space and electronic structures (Fermi surfaces, band structures, and orbital decompositions) in k-space.

Key Contributions and Results

1. Real-Space Picture: Spin Reorientation
   The analysis confirms that under uniaxial compression along the x-axis, the magnetic symmetry (point group $mm'm'$) remains unchanged. However, the balance of magnetic moments is disrupted.

   • In the unstressed state, Mn1 and Mn2 spins are aligned at $\phi = 180^\circ$, while Mn3–Mn6 are at $\phi = \pm 60.02^\circ$, resulting in a negligible net magnetization ($M_x \approx 0.002 \mu_B$).
   • Under compression, the magnitude of local spin moments decreases slightly, but the azimuthal angles of Mn3–Mn6 rotate by approximately $0.4^\circ$ within the xy plane (to $\pm 60.39^\circ$).
   • This small reorientation breaks the delicate cancellation balance, increasing the net magnetization to $M_x = 0.06 \mu_B$.

2. k-Space Picture: Fermi Surface and Band Evolution
   The study identifies specific Fermi surface features as crucial to the piezomagnetic response.

   • Fermi Surface Splitting: On the $k_xk_z$ plane containing $\Gamma$ and $Z$ points, compression lifts the pseudo-degeneracy of bands. Two pseudo-degenerate Fermi surfaces, which are nearly degenerate in the unstrained state due to weak SOC on Mn atoms, split under strain.
   • Spin Polarization: This splitting enhances the spin component oriented along the negative $k_x$ direction. Regions near the Fermi surface contribute approximately 10% of the total induced spin magnetization, reflecting the spin response of itinerant carriers—a mechanism distinct from piezomagnetism in insulators.
   • Fermi Surface Shrinkage: On planes containing $Y$ and $T$ points, compression causes a shrinkage of the Fermi surface, reducing the occupied region and enhancing spin polarization in the negative $k_x$ direction within the reduced area.
   • Band Shifts: The Mn3–Mn6 dominated bands exhibit a positive shift in the Fermi wave-number ($\Delta k_F = +0.0078$ Å$^{-1}$) along $k_z$, while Mn1–Mn2 dominated bands shift negatively ($\Delta k_F = -0.0027$ Å$^{-1}$). This directly accounts for the observed band splitting and Fermi surface separation.

3. Orbital Decomposition and Real-K Space Correlation
   The paper establishes a direct correspondence between the real-space spin rotation and k-space electronic modifications through orbital analysis:

   • The bands dominated by Mn3–Mn6 (which rotate under strain) are characterized by significant contributions from $d_{xz}$ and $d_{xy}$ orbitals.
   • Under compression, these bands shift toward higher energy and exhibit a positive $k_z$ shift.
   • Conversely, bands dominated by Mn1 and Mn2 (which do not rotate) show minimal shifts.
   • The phase of the single-electron wave function with finite $k_y$ creates differences in orbital character between Mn3/Mn5 and Mn4/Mn6, further refining the understanding of the electronic response.

Significance and Claims
The authors claim that their dual real- and k-space analysis provides deep insight into the microscopic origins of strain-driven magnetization in Mn3Sn. The primary significance of this work lies in:

• Mechanism Elucidation: Demonstrating that the piezomagnetic effect is not merely a macroscopic symmetry breaking but is driven by specific electronic mechanisms: the lifting of pseudo-degeneracy at the Fermi surface and the subsequent splitting of bands dominated by specific Mn sites and orbitals.
• Itinerant Carrier Role: Highlighting that the Fermi-surface-induced component is a key microscopic source of the response, distinguishing metallic antiferromagnets like Mn3Sn from insulating piezomagnets.
• Unified Framework: Providing a clear link between real-space spin dynamics (rotation of Mn3–Mn6 moments) and k-space electronic structure (band shifts and orbital weight redistribution).

The paper concludes that this framework offers new insight into the mechanism of piezomagnetism and may serve as a foundation for future studies on pressure-induced magnetic phenomena and their potential applications in spintronics, without proposing specific new experimental protocols or applications beyond these general directions.