3D bulk-resolved $g$-wave magnetic order parameter symmetry in the metallic altermagnet CrSb

Abstract

Electronic phases of matter, such as magnetism and superconductivity, are defined and distinguished by their order parameters that quantify the spontaneous symmetry breaking underlying each phase. The simplest cases are the uniform magnetisation of ferromagnets and isotropic gap function of conventional superconductors. Unconventional superconductors often have a nodal gap function, where the gap changes sign at nodes on the Fermi surface. This concept of unconventional or nodal order parameter symmetry has recently been extended to numerous magnetic systems, including altermagnets, in which up- and down-spin species are non-degenerate around the Fermi surface. Here we demonstrate that magnetic quantum oscillation measurements can provide a direct, bulk-sensitive, 3D mapping of the order parameter in an unconventional magnet. By rotating a magnetic field through high- and low-symmetry directions of the CrSb Brillouin zone, we show that this material's altermagetic band structure leads to the loss of mirror symmetry for each spin-split Fermi sheet away from highly symmetric nodal orientations. In momentum space, the difference between up and down spins follows the profile of the $\mathcal{Y}_{4}^{-3}=zy(3x^2-y^2)$ spherical harmonic - analogous to a $g$-orbital of the hydrogen atom. While notoriously difficult to resolve in unconventional superconductors, our work demonstrates that the order parameter symmetry of unconventional magnets can be conclusively determined through quantum-oscillatory quasiparticle spectroscopy. Our results empirically establish CrSb as a prototypical $g$-wave metallic altermagnet, which in pristine form possesses low residual resistivities down to $\sim$1 $μΩ$cm, opening numerous avenues for next-generation spintronic device applications

Figures (14)

• Figure 1: Nodal planes bisect $g$-wave-symmetric spin-split Fermi surface sheets in CrSb. a, The crystal structure of CrSb, with alternating magnetic moments on the Cr sites oriented along the $c$-axis. Red indicates spin-up, with blue corresponding to spin-down. b, The trigonal arrangement of the antimony ions means that the mapping of a red chromium site to a blue one cannot be performed by translation and time-reversal operations alone, but necessarily requires a screw rotation, which is a point-group operation. c-e, The primary Fermi surface sheet of CrSb is a closed 3D ellipsoidal pocket, shaped like a dogbone. Here we indicate its spin-split character with red and blue colouring, with the distinctive $g$-wave flowerlike symmetry of the order parameter visible looking along the $c$-axis. Nodal planes -- where spin degeneracy is imposed by symmetry -- are given by thin shaded slabs, bisecting the Fermi surface where the up- and down-spin sheets cross. Here we introduce the azimuthal angle $\varphi$ defined as the inclination from $a$ to $ab$, i.e. from [100] towards [110], and the polar angle $\theta$ from $c$ towards the $ab$ plane, i.e. from [001] to [100] or equivalently [001] to [110]. f, Visual depiction of the $\mathcal{Y}_{4}^{-3}=zy(3x^2-y^2)$ real spherical harmonic, which characterises the $g$-wave symmetry profile of
• Figure 2: (cont.) altermagnetic spin-splitting in CrSb. g, Quantum oscillations in the background-subtracted magnetic torque $\Delta \tau$, rescaled to the same maximal amplitude for ease of presentation (see Methods for experimental and analytical details) collected at magnetic field H orientations as indicated, and h, their fast Fourier transform (FFT) frequency spectra. A singular FFT peak is observed in the three nodal orientations, which splits into two peaks in the antinodal plane (red data). All measurements were performed at 0.4 K. i,j$\Delta \tau$ and corresponding FFT spectra for small rotations of $\theta$ in the antinodal plane away from the $ab$ direction towards $c$. For H aligned along $ab$, only one frequency is observed, which elsewhere splits into two distinct peaks. Once H has rotated by just 4$\degree$ away from $ab$ towards $c$, these two frequency components are separated by a considerable 0.6 kT. The waveforms themselves exhibit a pronounced beating structure, which is acutely sensitive to small modulations of $\theta$.
• Figure 3: Mapping the altermagnetically spin-split Fermi surface of CrSb. a, Heatmap of the spectral intensity of quantum oscillatory frequencies versus rotation angle in the nodal $c-a$ plane and b, simulated frequency profile from DFT calculations (see Methods). $\mathcal{A_\perp}$ denotes the enclosed reciprocal space area of a given frequency component. c,d Selected $\Delta \tau$ curves and normalized FFTs that contributed to the heatmap in a (see Extended Data Fig. 4 for the full $\Delta \tau$ dataset). All data were collected at 0.4 K. e, Data and f, Simulation for the antinodal $c-ab$ rotation plane, with waveforms and normalised FFTs in g,h (and full datasets in Extended Data Fig. 5). Here spin-splitting between non-degenerate Fermi sheets is predicted by the simulation, which is clearly resolved experimentally close to $\theta = 90\degree$. In i, we present a zoomed-in view having high-pass filtered the raw data, to clearly show the non-degenerate spin-up and -down frequency branches crossing the nodal plane at $\theta = 90\degree$, which are separated by $\sim$1 kT. j,k Fermi surface rendering of the spin-down sheets of the dogbone and web pockets and l,m their spin-up counterparts. n, The full Fermi surface of CrSb, including all spin-up and spin-down sheets.
• Figure 4: Confirmation of $g$-wave altermagnetic spin texture in CrSb. a, Heatmap of quantum oscillation frequencies versus rotation angle and b, simulated spin-up (red) and spin-down (blue) angular frequency profile computed by DFT. Here we rotate by an angle $\alpha$ through a plane of low symmetry at a tilt of 14$\degree$ from the $ab$ plane, which is depicted in Extended Data Fig. 2. $\alpha$ is defined such that $\cos(\theta) = \sin(14\degree)\sin(\alpha)$ and $\tan(\varphi) = \cos(14\degree)\tan(\alpha)$. Symmetry-enforced nodal planes are crossed at $\varphi = 0\degree$ and at $\varphi = \pm 60\degree$. At all other orientations in this low-symmetry plane, significant spin-splitting is observed. c,d, Selected $\Delta \tau$ curves and corresponding FFTs that contributed to the heatplot in a (the full dataset is presented in Extended Data Fig. 6). e, The tilted rotation plane through which H was swept, whereby the arc that is traced out crosses a nodal plane every 60$\degree$ increment of $\varphi$, due to the $g$-wave symmetry profile.
• Figure 5: Spin-split effective mass study. a, Quantum oscillations at incremental temperatures for $\theta = 84.8\degree$, $\varphi = 21.4\degree$ and b, their corresponding frequency spectra, with temperatures indicated. $\Delta \tau$ was high-pass filtered to focus on two frequency components, at $f_1 =$ 3.41 kT and $f_2 =$ 3.82 kT (see Methods), which come from spin-split up and down sheets of the primary dogbone Fermi surface. c, Temperature dependence of QO amplitude fitted to the Lifshitz-Kosevich formula LK, yielding effective cyclotron masses of $m_1^* =$ 1.97(5) $m_e$ and $m_2^* =$ 2.09(5) $m_e$. The ratio of $\sqrt{f_1/f_2}$ is equal to that of $m_1^*/m_2^*$, as expected for these two frequency components being two spin-split daughters of the same mother Fermi sheet. d, Visualisation of the spin-up and spin-down orbits around the dogbone cross-section, which can enclose markedly different areas for the same field orientation. The effective cyclotron mass grows with increasing orbit area. e, Raw torque $\tau$ without background-subtraction at two orientations as indicated. Close to high-symmetry, the overall cantilever deflection is small, hence $\tau$ is dominated by the oscillatory component of magnetisation.