Non-collinear Altermagnetic Phases in the Mott Insulator NiS$_2$

Abstract

Altermagnets (A$\ell$Ms) constitute a novel family of magnetic materials characterized by the absence of net magnetization and the presence of spin-polarized band structures. Whereas A$\ell$M phases were initially proposed in collinear structures, the recently discovered noncollinear chiral A$\ell$Ms stand out for their distinct hedgehog spin texture and multifunctionality in spintronics. In this work, we deepen the characterization of these systems by constructing a Landau theory for noncollinear achiral A$\ell$Ms. Furthermore, we demonstrate that the achiral symmetry of the crystal is reflected in the spin texture in reciprocal space, which presents only spatial-even multipoles. These multipoles, distinguished from those in collinear A$\ell$Ms via the high-order secondary order parameters, can couple to many phenomena such as the spin Hall effect and piezomagnetic effect. To exemplify our theory, we study the noncollinear achiral magnet NiS$_2$ within the framework of altermagnetism, showcasing both spin Hall and piezomagnetic effects in a prototypical correlated Mott insulator that provides an ideal platform to explore the interplay between strong electronic correlations, crystal symmetry, and altermagnetic spin textures. Interestingly, altermagnetism emerges in two magnetic ordered phases of NiS$_2$ upon lowering the temperature. The non-collinearity strengthens the robustness of A$\ell$M order, as the anti-ferromagnetism induced by the strong correlations will not impose effective time-reversal symmetry as in the collinear case. Our results suggest non-collinear achiral A$\ell$Ms as a promising platform for spintronics applications due to the potential to achieve various spin textures with different magnetic orders.

Figures (8)

• Figure 1: Classification diagram of altermagnetic systems. The horizontal and vertical axes denote (non-)collinearity and (a)chirality, respectively. For each class, the spin texture in reciprocal space is shown schematically with colors and arrows.
• Figure 2: NiS$_2$ crystal structure (a) and magnetic structure with only Ni atoms shown from high- (b) and low- (c,d) temperature (high-T, low-T) phases. The magnetic phase transition happens twice, upon lowering the temperature, from paramagnetic to high-T and high-T to low-T phase. Ni atoms lying at light grey planes ($\alpha$, $\beta$, and $\gamma$), perpendicular to the cubic (111) direction, are equivalent in high-T phase and inequivalent between $\alpha$ and $\beta$/$\gamma$ planes. The standard unit cell of low-T phase is shown in (d) with top view at the upper panel. The lower panel in (d) shows the relation between different unit cells: the black solid lines and grey dashed lines denote the primitive cells of high-T and low-T phases, respectively.
• Figure 3: Fermi surfaces and spin textures of NiS$_2$ for the high-T phase (a) and the low-T phase (b). Both phases have finite band gaps. The Fermi surfaces are shown at $E=E_f-0.55\ \mathrm{eV}$ and $E=E_f-0.12\ \mathrm{eV}$ for the high-T and low-T phases, respectively. To illustrate the quadrupolar spin texture, slices at $k_z=\pm0.2\,\pi/c$ and at $k_z=0$ are shown next to each Fermi surface plot. The $s_z$ component in the high-T phase is indicated by the arrow colors in (a). For the low-T phase, all spins are rotated counterclockwise by $\pi/6$, with colors denoting the coplanar spin angles for clarity.
• Figure 4: Electronic properties of NiS$_2$ and calculated spin Hall conductivity at high-T phase and piezomagnetism at low-T phase. (a) The spin texture in the $k_z = 0/\pi$ plane taken at $E= E_f - 0.30/-0.50$ (eV) for valence band, and $E= E_f + 0.35/+0.55$ (eV) for conductance band, respectively. (b) The band structure along high symmetry lines without and with SOC in solid black and dashed red lines, respectively. The mBJ approximation is adapted with the mBJ potential CMBJ = 1.15 mbj_1mbj_2. (c) The SHE effect respective to the energy region in (b). The two independent spin conductivity elements $\sigma^{I/II,x}_{yz},\sigma^{I/II,x}_{zy}$ are shown in blue and red in the left panel, and the contributions from $\chi^{I/II}$ are in dashed and solid lines, respectively. The charge conductivity element is shown in the right panel. (d) Band structure and piezomagnetism under uniaxial strain along $x$ and $y$ directions. The strained band structures and corresponding two-dimensional Fermi surfaces evolutions are shown with the same background colors. The right-upper panel shows the relation between strain strength and $M^{y}$.
• Figure S1: Surface spectrum of NiS$_2$ in (a) the high-T phase and (b) the low-T phase. With the terminations denoted in (a) and (b), the surface states are shown along high symmetry lines without SOC in (c,d) and with SOC in (e,f).