Uniaxial strain tuned magnetism of the altermagnet candidate h-FeS

1. Problem Statement

Altermagnetism is a recently identified third class of magnetic order (distinct from ferromagnetism and antiferromagnetism) characterized by collinear magnetic order that breaks time-reversal symmetry while maintaining zero net magnetization. A key signature of altermagnets is the spontaneous Anomalous Hall Effect (AHE), which arises from momentum-dependent spin splitting and alternating spin polarization rather than net magnetization.

However, the study of altermagnetism faces two major challenges:

1. Scarcity of Observations: Despite many proposed candidates, the spontaneous AHE has been observed in very few materials, limiting comprehensive studies of its response to external tuning.
2. Unclear Mechanisms: The relationship between the spontaneous AHE and specific magnetic structural details (such as spin canting and domain populations) remains elusive.
3. Tunability: There is a need to identify effective "knobs" to control altermagnetic properties for spintronic applications.

The specific material under investigation is hexagonal FeS (h-FeS), a recently identified altermagnet candidate that exhibits a spontaneous AHE accompanied by a tiny net magnetization due to spin canting. The authors aim to determine how uniaxial strain affects the magnetic structure, the spontaneous AHE, and the net magnetization in h-FeS.

2. Methodology

The researchers employed a multi-modal experimental approach combining transport, magnetization, and neutron scattering techniques on high-quality single crystals of h-FeS (composition $\approx$ Fe$_{0.93}$S).

• Sample Synthesis: Single crystals were grown via chemical vapor transport using Fe powder, S powder, and iodine as a transport agent.
• Strain Application: A custom-made mechanical strain device (fabricated from brass or aluminum) was used to apply uniaxial compressive strain along specific in-plane directions ([1$\bar{1}$0] and [110]) with an estimated magnitude of $\sim$100 MPa.
• Electrical Transport: Hall resistivity ($\rho_{yx}$) was measured under magnetic fields to extract the spontaneous AHE component ($\Delta\rho_{yx}$) by fitting high-field data to separate ordinary and anomalous contributions.
• Magnetization: Isothermal magnetization ($M$) and magnetic susceptibility ($M/H$) were measured using a Vibrating Sample Magnetometer (VSM) to quantify the net magnetization and spin canting.
• Neutron Diffraction: Performed at the Spallation Neutron Source (CORELLI) to probe the magnetic structure and domain populations under strain. This allowed the authors to distinguish between changes in the magnetic structure itself versus changes in domain populations.
• Inelastic Neutron Scattering (INS): Conducted at ARCS to measure spin-wave excitations and determine the spin-wave gap, providing insight into the energy barriers between magnetic domains.
• Symmetry Analysis: Theoretical symmetry analysis of the conductivity tensor was performed to understand the constraints imposed by strain on the AHE.

3. Key Contributions

1. Strain Tuning of Altermagnetism: The study demonstrates that in-plane compressive strain is an effective control parameter for tuning both the spontaneous AHE and the tiny net magnetization in h-FeS.
2. Decoupling Structure from Domains: Using neutron diffraction, the authors proved that the applied strain does not alter the fundamental in-plane magnetic structure (A-type antiferromagnetic order) but instead selectively suppresses the population of specific magnetic domains.
3. Correlation Mechanism: The work establishes a direct correlation between the spontaneous AHE and the weak ferromagnetic moment (spin canting) in h-FeS, showing that both are suppressed simultaneously by strain.
4. Microscopic Origin: The suppression is attributed to strain-induced breaking of local $C_3$ symmetry in the FeS$_6$ octahedra, which further quenches orbital angular momentum, thereby weakening spin-orbit coupling (SOC) and reducing spin canting.

4. Key Results

• Suppression of Spontaneous AHE:

  • In the absence of strain, h-FeS exhibits a clear spontaneous AHE above the Morin transition temperature ($T_M \approx 156$ K).
  • Applying compressive strain along [1$\bar{1}$0] or [110] significantly suppresses the magnitude of the spontaneous AHE.
  • The ordinary Hall effect and conventional AHE (field-dependent) remain largely unchanged, indicating the electronic band structure is not drastically altered by the strain magnitude used.

• Suppression of Net Magnetization:

  • The tiny net magnetization (order of $10^{-3} \mu_B$/Fe) arising from spin canting along the $c$-axis is also suppressed by the strain.
  • The differential AFM susceptibility (slope of the linear background) remains unaffected, confirming that the strain specifically targets the ferromagnetic component (canting) rather than the antiferromagnetic order.

• Neutron Diffraction Findings (Domain Repopulation):

  • Without strain, three equivalent in-plane magnetic domains (I, II, III) are equally populated.
  • Under compressive strain, the domain aligned with the strain direction (Domain I) is strongly suppressed, while the other two domains (II and III) become dominant.
  • The ratio of domain populations shifts from $\sim$1:1:1 (unstrained) to $\sim$0.02:0.49:0.49 (strained).
  • Crucially, the intensities of magnetic Bragg peaks indicate that the magnetic structure itself remains unchanged; the observed changes in AHE and magnetization are driven by this domain redistribution.

• Spin-Wave Gap and Dynamics:

  • INS measurements reveal a very small spin-wave gap ($< 0.5$ meV) above $T_M$ in the easy-plane configuration.
  • This negligible gap explains why even moderate strain ($\sim$100 MPa) is sufficient to lift the energy degeneracy between domains and repopulate them.

• Symmetry Analysis:

  • Theoretical analysis confirms that while strain reduces the magnetic point group symmetry, it does not forbid the spontaneous AHE. Instead, the reduction in AHE is physically linked to the reduction in spin canting (which is driven by SOC) and the specific alignment of the remaining domains.

5. Significance

• Validation of Altermagnetism: The results provide strong evidence that h-FeS is a genuine altermagnet, where the spontaneous AHE is intrinsically linked to the underlying magnetic symmetry and spin canting.
• Control Mechanism for Spintronics: The study identifies uniaxial strain as a highly effective "knob" for tuning altermagnetic properties. This is crucial for the development of strain-engineered spintronic devices where magnetic states need to be switched or modulated without external magnetic fields.
• Insight into Spin Canting: The work clarifies the role of spin canting in altermagnets. It suggests that in h-FeS, the spontaneous AHE behaves similarly to that in ferromagnets (proportional to net magnetization) but with an anomalously large anomalous Hall coefficient ($|R_S| \sim 100 \mu\Omega\cdot\text{cm}\cdot\text{T}^{-1}$), likely due to the unique band structure of the altermagnet.
• General Applicability: The findings suggest that similar strain-tuning mechanisms could be applicable to other altermagnet candidates (like $\alpha$-MnTe and Mn$_3$Sn) that exhibit small net magnetizations and spontaneous AHEs, opening new avenues for material design and optimization.

In conclusion, this paper successfully demonstrates that the magnetic properties of the altermagnet candidate h-FeS, specifically its spontaneous AHE and net magnetization, can be effectively tuned by uniaxial strain through the mechanism of magnetic domain repopulation and the suppression of spin canting.