Metallic $p$-wave magnet with commensurate spin helix

Abstract

Antiferromagnetic states with spin-split electronic structure give rise to novel spintronic, magnonic, and electronic phenomena despite (near-) zero net magnetization. The simplest odd-parity spin splitting - $p$-wave - was originally proposed to emerge from a collective instability in interacting electron systems. Recent theory identifies a distinct route to realise $p$-wave spin-split electronic bands without strong correlations, termed $p$-wave magnetism. Here we demonstrate an experimental realisation of a metallic $p$-wave magnet. The odd-parity spin splitting of delocalised conduction electrons arises from their coupling to an antiferromagnetic texture of localised magnetic moments: a coplanar spin helix whose magnetic period is an even multiple of the chemical unit cell, as revealed by X-ray scattering experiments. This texture breaks space inversion symmetry but preserves time-reversal ($T$) symmetry up to a half-unit-cell translation - thereby fulfilling the symmetry conditions for $p$-wave magnetism. Consistent with theoretical predictions, our $p$-wave magnet exhibits a characteristic anisotropy in the electronic conductivity. Relativistic spin-orbit coupling and a tiny spontaneous net magnetization further break $T$ symmetry, resulting in a giant anomalous Hall effect (AHE, $σ_{xy}>600\,$S/cm, Hall angle $>3\,\%$), for an antiferromagnet. Our model calculations show that the spin nodal planes found in the electronic structure of $p$-wave magnets are readily gapped by a small perturbation to induce the AHE.

Figures (15)

• Figure 1: Magnetic structure of a $p$-wave magnet in direct space and spin-split electronic bands in momentum space.a, Ferromagnets have spins along the $z$-axis in direct space and a uniform spin polarisation $S_z(\bm{k})$ in $\bm{k}$-space (right). b, $p$-wave magnetism (conceptual): Non-collinear spin modulation in direct space and alternating spin polarisation in momentum ($\bm{k}$-)space. Red and blue bars: momentum-resolved expectation value of $S_x(\bm{k})$ at the chemical potential. For commensurate order with wavelength $\lambda_\mathrm{mag} = N$ (even integer $N$), this state is unchanged under $180^\circ$ spin rotation followed by translation of half a lattice constant, termed $[C_{2\perp}\parallel\bm{t}_{1/2}]$ symmetry. The spin polarisation vector is $\bm{\alpha}$Ezawa2024_electric_neel. Violet disk and circular arrow: spin rotation plane in direct space. c, Minimal electronic structure model for a $p$-wave magnet depicted in the magnetic Brillouin zone according to Eq. (\ref{['main:TwoBandPerturb']}). The spin-splitting of bands causes two-fold electronic anisotropy. Red (blue) indicate spin parallel (antiparallel) to $k_x$. d, Spin polarisation $\left<S_x\right>$ when $m_z=\lambda= 0$ in Eq. \ref{['main:TwoBandPerturb']}, which represents a sum of all occupied states -- i.e., of the two bands in (c). e, Electronic structure in presence of a tiny net magnetization $m_z$. f, Anomalous Hall conductivity $\sigma_{xy}$ induced by Berry curvature $\Omega_{xy}$ appears in the $k_x$-$k_y$ plane when $m_z \neq 0$ and $\lambda \neq 0$, due to lifting the band degeneracy of the nodal plane.
• Figure 2: Commensurate spin helix in direct space, consistent with $p$-wave magnetism, revealed by resonant elastic X-ray scattering (REXS).a, Crystal structure of hexagonal Gd$_3$(Ru$_{1-\delta}$Rh$_{\delta}$)$_4$Al$_{12}$. b, Tuning the magnetic propagation vector $\bm{k}_\mathrm{mag}$ by chemical substitution of Rh for Ru. $T_{\mathrm{N}1}$ (dotted line) is the magnetic transition temperature from the paramagnetic state, while $T_{\mathrm{N}2}$ (solid line) corresponds to the transition into the $p$-wave state. Inset: Evolution of the magnetic ordering vector $\bm{k}_\mathrm{mag}$ with a shift of the Fermi energy $E_\mathrm{F}$ due to Rh-doping $\delta$: $\bm{k}_\mathrm{mag} = (1/6, 1/6, 0)$ appears for $\delta = 0.04-0.07$, c.f. shaded region in the main panel. For $\delta=0$ and $\delta=0.2$ the magnetic wavevectors in the ground state are $(0.272,0,0)$ and $(0.5,0,0)$, respectively. Momenta are expressed in reciprocal lattice units (r.l.u.) and only half of the first hexagonal Brillouin zone is depicted, sliced at $k_z = 0$. c, Single crystal REXS experiments in reflection geometry, with incoming and outgoing beams $\bm{k}_\mathrm{i}$, $\bm{k}_\mathrm{f}$ spanning the scattering plane (light gray). In polarisation analysis, $I_{\pi- \pi^\prime}$ and $I_{\pi- \sigma^\prime}$ components of $\bm{k}_\mathrm{f}$ are separated by rotating the detector around an analyzer plate by the angle $\gamma$ (Methods). d, Crystallographic unit cells (hexagonal, black grid) and reconstructed magnetic unit cell (orthorhombic, black box). The requirements for $p$-wave magnetism, $[C_{2\perp}\parallel \bm{t}_{1/2}]$ and $[\mathcal{T}\parallel\bm{t}_{1/2}]$ symmetry, are satisfied by the black arrow. e, Due to the formation of three $p$-wave magnetic domains, six magnetic reflections appear in a given Brillouin zone. f-h, Polarisation analysis using magnetic satellites around the fundamental $(2, 5, 0)$ reflection. Insets: Geometry of $\bm{k}_\mathrm{i}$ relative to the modulated magnetization projected into the scattering plane, $\bm{m}_{\mathrm{ip}}$. i, Illustration: Commensurate spin helix with short wavelength $\lambda_\mathrm{mag}$ in direct space.
• Figure 3: Anisotropic electronic transport properties.a, Scanning electron micrograph of the rotationally symmetric focused ion beam device. The meander-shaped arms release the strain on the device, which arises due to thermal expansion mismatch of device and substrate. The false colour regions indicate the target material (blue) and the electrical contacts (orange), which are composed of evaporated gold (Au) wires and ion beam-deposited platinum (Pt), respectively. The contacts utilised in the directional transport measurements are labelled and colour coded. The scale bar is $5\,$µ m. b, Device geometry with crystal axes $a$, $c$ and three pairs of current contacts $I_a$, $I_b$, $I_c$; the corresponding longitudinal voltage drops $V_a$, $V_b$, $V_c$ are measured simultaneously. The magnetic field $\bm{B}$ is rotated by an angle $\phi$ to the $a$-axis. c, Temperature-dependent traces of longitudinal resistance (left) and magnetization $M$ (right), with two antiferromagnetic transitions $T_{\mathrm{N}1}$ and $T_{\mathrm{N}2}$. d, Change in longitudinal resistance $R_{xx}^a$ when switching between different $p$-wave domains by a small magnetic field of $0.9\,$T. Insets: deduced direction of the spin polarisation vector $\bm{\alpha}$ in momentum space, and the periodic spin texture in direct space. The gray wedge indicates the sextant of the magnetic field angle over which the depicted $p$-wave domain is stable. Two-fold anisotropy emerges, with higher resistance for current $I_a$ (center arrow) parallel to $\bm{\alpha}$. Open circles: Data at $30\,\mathrm{K}>T_{\mathrm{N}1}$ plotted for comparison. e,f Simultaneous measurements for other contact pairs, demonstrating the anisotropic resistance is due to the relative alignment between current and $\bm{\alpha}$.
• Figure 4: Giant anomalous Hall effect (AHE) induced by a tiny net magnetization $m_z$ and spin-orbit coupling (SOC).a, Comparison of net magnetization $M(B)$ measured at $T= 2\,$K and $20\,$K. The magnetic field is applied along the $c$-axis. Black triangles correspond to a transition field ($B_{\mathrm{c}1}$) from the $p$-wave magnetic phase (commensurate, C) to the high-temperature Phase-II (incommensurate, IC). Inset: small spontaneous magnetization $m_z$ appears around zero field below $B_\mathrm{coerc}$ (red triangles) at low temperature. b, Hall conductivity with large zero-field (anomalous) value $\sigma_{xy}^\mathrm{A}$ reaching up to $600 \, \mathrm{\Omega}^{-1}\mathrm{cm}^{-1}$, measured on a bulk single crystal. c,d, Hall conductivity $\Delta \sigma_{xy}$ after subtraction of components proportional to the magnetic field and to the magnetization (Supplementary Fig. \ref{['main:Efig_AHC_subtraction']}). $\Delta \sigma_{xy}$ is enhanced in the $p$-wave magnetic Phase-I. Open circles marking the phase boundaries in panel c are obtained from magnetization measurements in Extended Data Fig. \ref{['main:Efig_Magnetization']}. Red arrows in panel d indicate the direction of the field sweep. e, As compared to other bulk antiferromagnets with AHE, our slightly distorted $p$-wave state in Gd$_3$Ru$_4$Al$_{12}$ has large $\sigma_{xy}^\mathrm{A}$ induced by $m_z$, large Hall angle $\sigma_{xy}^\mathrm{A}/ \sigma_{xx}$, and a low saturation magnetic field $B_\mathrm{sat}$. Raw data for the bar plot are provided in Extended Table \ref{['main:Table_AHC']}. f, Calculated anomalous Hall conductivity from the low-energy model Eq. \ref{['main:TwoBandPerturb']} plotted against $m_z$/$p$. We consider the breaking of $p$-wave state around $m_z / p \sim 1.5$, where anomalous Hall conductivity is suppressed to zero (see Supplementary Note \ref{['main:conversion_exchange_splitting']} for details).
• Figure E1: Fermi surfaces of a $p$-wave band.a, Band dispersion with $p$-wave splitting, spin-orbit coupling (SOC), and time-reversal breaking $m_z = 0.01$. Here, we set $p = 1.2$ and $\lambda = 0.4$, where $p$-wave exchange splitting is larger than the size of SOC and the net magnetization is small but non-zero. b,c, Fermi surfaces of a $p$-wave split band with ($m_z = 0$, $\lambda = 0$) and without ($m_z = 0.01$, $\lambda = 0.4$) time-reversal symmetry, respectively. The two bands are degenerate on the $k_x=0$ plane for $m_z = 0$. However, the degeneracy is lifted due to band hybridization when $m_z$ is finite in Eq. (\ref{['main:TwoBandPerturb']}). The equal energy surface is calculated at $E = 10$ in panels (b, c). Contrary to the low-energy model for a $p$-wave Pomeranchuk instability, $\left|\left<S_x\right>\right|<\hbar/2$ for the $p$-wave magnet of conduction electrons coupled to a spin helix (Methods, Supplementary Note \ref{['main:SI_Pomeranchuk']}).