EuAuSb: An odd-parity helical variation on altermagnetism

TL;DR

This work investigates EuAuSb, a triangular-lattice Dirac semimetal exhibiting a topological Hall effect linked to a magnetically ordered phase. Using single-crystal neutron diffraction, the authors identify an incommensurate helical order in which Eu$^{2+}$ layers rotate by about $120^ olinebreak^ oindent$ between adjacent layers, with a propagation vector $ abla$ along $c$ of $\ \approx 0.63$–$0.67$, and they observe a first-order incommensurate-to-commensurate transition under an in-plane field near $\,H_{c1} \sim 0.9$ T. Density functional theory yields exchange constants $J_1 \approx 1.49$ meV and $J_2 \approx 0.57$ meV compatible with the helix and reveals spin-splitting near the Fermi level that is odd under $\mathcal{P}$ and $\mathcal{T}$, i.e., an odd-parity form of altermagnetism. The study links a depression in the static in-plane moment sum at fields where harmonics disappear to quantum spin fluctuations and shows that the spiral order drives nondegenerate electronic states despite fully compensated magnetism, signaling EuAuSb as a novel member of odd-wave spin-split materials with potential transport relevance in spintronics and topological physics.

Abstract

EuAuSb is a triangular-lattice Dirac semimetal in which a topological Hall effect has been observed to develop in association with a magnetically-ordered phase. Our single-crystal neutron diffraction measurements have identified an incommensurate helical order in which individual ferromagnetic Eu$^{2+}$ layers rotate in-plane by $\sim$120$^{\circ}$ from one layer to the next. An in-plane magnetic field distorts the incommensurate order, eventually leading to a first order transition to a state that is approximately commensurate and that is continuously polarized as the bulk magnetization approaches saturation. From an analysis of the magnetic diffraction intensities versus field, we find evidence for a dip in the ordered in-plane moment at the same field where the topological Hall effect is a maximum, and we propose that this is due to field-induced quantum spin fluctuations. Our electronic structure calculations yield exchange constants compatible with the helical order and show that the bands near the Fermi level lose their spin degeneracy via a mechanism similar to that in the collinear altermagnets. We find that, unlike the even symmetry seen in the altermagnets, the spin-splitting in EuAuSb has odd-wave symmetry similar to that recently found in a number of coplanar magnetic materials.

Figures (5)

• Figure 1: (a) Crystal stucture of EuAuSb with space group $P6_3/mmc$, and magnetic moment directions for the Eu planes. The drawing was produced using VESTA Momma2011. (b) Inverse of magnetic susceptibility for the $c$-axis and in-plane directions, showing linear Curie-Weiss behavior. (c) Zoom in of low temperature region of the inverse magnetic susceptibility, showing the change in slope below the magnetic ordering transition of 3.9 K. The Curie-Weiss fit to the high temperature data is shown by the black dashed lines. (d) Temperature dependence of $(0,0,L)$ scans, showing appearance of additional magnetic peaks below the magnetic ordering temperature. Bragg peaks at $(0,0,2)$, $(0,0,4)$ and $(0,0,6)$ positions are present at all temperatures. The dashed black line shows the Eu$^{2+}$ form factor, corrected for absorption, footprint, and Lorentz factor. The weak peaks at $L=3.65$ and 4.21 rlu are from Al powder rings.
• Figure 2: (a) Magnetic field dependence of the intensities for the Bragg ($I_2$) and magnetic first and second harmonic peaks ($I_{2-\zeta}$ and $I_{2\zeta}$ respectively). The black dashed lines are fits, showing that in the low field range, the first and second harmonic peak intensities have the expected quadratic field dependence. (b) Field dependence of the magnetic signal in the vicinity of the $(0,0,2-\zeta)$ magnetic peak, showing an incommensurate to commensurate transition at $\sim$1 T. The weaker feature at lower $L$ is the second harmonic. Data was collected at 1.6 K. (c) Magnetization divided by magnetic field as a function of field, showing a transition at $\sim0.9$ T. The 4 K data is offset by a constant of 0.2 $\mu_B$/f.u./T to simplify comparison with the 2.4 K data. (d), and (e) Fitted peak positions and FWHM from the fits of the 1.6 K data shown in panel (a).
• Figure 3: (a) Color plot showing intensity of the magnetic $(0,0,2-\zeta)$ peak ($\sqrt{I}$ is plotted to enhance weak features), collected by fixing the magnetic field and sweeping the temperature. $T_c$ data points were extracted from this data set by fitting critical type behavior. The incommensurate to commensurate ($H_{c1}$) and high field transition ($H_{c2}$) values were extracted from the field-dependent data shown in Fig. \ref{['fig2']}. The position of the minimum in the M/H data plotted in Fig. \ref{['fig2']}(c) is also shown. (b) Color plot showing $\zeta$ extracted from the position of the magnetic $(0,0,2-\zeta)$ peak.
• Figure 4: Plot of $|F|^2_{\rm tot}$, $|F(2-\zeta)|^2+|F(2+\zeta)|^2$, and $\sqrt{\Delta|F(2)|^2}$ (designated $\Delta|F(2)|$ in the legend), normalized at $H=0$, as a function of magnetic field at $T=1.6$ K.
• Figure 5: Calculated Fermi surface intersections with the $k_x$-$k_y$ plane for fixed values of $k_z$, evaluated at a chemical potential ($-0.3$ eV) consistent with photoemission results. Left panels show results for the nonmagnetic state at $k_z=0$ and $k_z=0.33(2\pi/c)$. Right panel shows results at $k_z=0$ for commensurate helical order, where the larger real-space unit cell results in a reduced volume in reciprocal space, with bands at $k_z=\pm0.33(2\pi/c)$ folded back to $k_z=0$. Red and blue colors indicate opposite spin polarizations for states along the same radial k vector. (Possible spin splitting of other Fermi pockets is too small to resolve in this figure.)