Uniaxial strain tuned magnetism of the altermagnet candidate h-FeS

Abstract

Altermagnets are collinear magnetic materials with 'alter'nating local crystalline environments, characterized by joint spin and crystalline symmetries that enable ferromagnetic-like transport properties but with vanishing net magnetization. Hexagonal FeS (h-FeS) is a recently identified altermagnet candidate that shows a spontaneous anomalous Hall effect (AHE) accompanied by a tiny net magnetization. Here, we show that both the spontaneous AHE and magnetization can be effectively suppressed by an in-plane compressive strain. Since neutron diffraction measurements show that the applied uniaxial strain only modifies the in-plane domain population but does not affect the in-plane magnetic structure, the major effect of the applied strain is to tune the small $c$-axis ferromagnetic moment. Our results demonstrate a strong correlation between the tiny net magnetization and the spontaneous AHE in h-FeS, and show that uniaxial strain provides an effective knob to tune both properties in this altermagnet candidate for spintronic applications.

Figures (20)

• Figure 1: (a) Crystal structure of h-FeS above $T_{\rm{M}}$. Two S octahedra surrounding Fe ions are explicitly shown. The upper left inset shows the Fe spin configurations inside the two octahedra, where the spins initially along the [$1\bar{1}0$] direction exhibit a slight canting toward the Fe-S bonds (dashed lines). (b) Evolution of the orbital states for 3$d$ electrons under octahedral, trigonal, and lower-symmetry crystal fields. (c) Top view of the preferred spin directions in the FeS$_{6}$ octahedron in the free state and under compressive strain along the [$1\bar{1}0$] direction (magenta arrows). (d) Three magnetic domains (I, II, and III) in one Fe layer, connected by the $C_3$ symmetry. (e) Two remaining magnetic domains (II and III) under compressive strain along the [$1\bar{1}0$] direction. (f) Temperature dependence of the magnetic susceptibility $M/H$ (with and without strain, left axis) and the resistivity (right axis). A magnetic field of 1 T was applied along the $c$-axis for the $M/H$ data. The inset schematically shows a side-view of the spin configurations. Dashed vertical line marks $T_{\rm{M}}$. The canting angles are intentionally enlarged for better visualization.
• Figure 2: (a)-(d) Magnetic-field dependence of the Hall resistivity measured at selected temperatures under compressive strain applied along [$1\bar{1}0$]. The solid line in (b) represents a linear fit to the data, as described in the text. For clarity, the data are vertically offset by 0.2 $\mu \Omega\cdot\rm{cm}$. (e) and (f) Temperature dependence of the fitted anomalous Hall resistivity and the $B$-linear Hall coefficient at different strain magnitudes. (g) Magnetic-field dependence of the Hall resistivity measured at selected temperatures under compressive strain applied along [110]. Insets in (a) and (g) illustrate the corresponding strain directions.
• Figure 3: (a) Isothermal magnetization along the $c$-axis with and without compressive strain applied along [$1\bar{1}0$] at 100 K ($< T_{\rm{M}}$) and 160 K ($> T_{\rm{M}}$). Insets show the zoom-in view of data within $\pm$ 1 T. (b) Isothermal magnetization along the $c$-axis at selected temperatures after subtracting the high-field linear part. Light and dark colors denote data with and without strain, respectively. Data are evenly offset 0.002 $\mu_{\rm{B}}$/f.u. for clarity.
• Figure 4: (a)-(c) Neutron diffraction patterns in the ($H$, $K$, 1) plane measured at 300 K without strain, 300 K with strain, and 120 K with strain, respectively. White hexagons indicate the Brillouin zone boundaries. Dashed circles in (a) mark the Bragg peaks whose integrated intensities are shown in (d). Compressive strain is applied along the horizontal axis in (b) and (c). (d) Integrated intensities (square symbols with error bars) of six Bragg peaks under the three conditions. The bar chart represents the corresponding calculated intensities SM. (e) and (f) INS spectra measured along the [0, 0, $L$] direction at 8 K and 250 K, respectively. (g) Energy dependence of the excitation intensity at (0, 0, 1) measured at 8 K and 250 K. The dashed curve shows a fit to the spin-wave gap using an error function SM. The inset displays the temperature dependence of the diffraction intensity at the (0, 0, 1) Bragg peak. (h) Left: low-energy excitation spectrum around (0, 0, 1) at 250 K. Right: the corresponding energy dependence of the intensity at (0, 0, 1).
• Figure S5: (a) One piece of representative h-FeS single crystal against a millimeter grid. (b) X-ray Laue diffraction pattern taken with the X-ray beam along the $c$-axis. The pattern corresponds to the sample orientation shown in (a).