Electrical Transport and Quantum Oscillations in the Metallic Spin Supersolid EuCo2Al9

Abstract

The discovery of spin supersolid and its giant magnetocaloric effect has opened a new arena in frustrated quantum magnets and cutting-edge cryogenics. The intermetallic EuCo2Al9 (ECA), for the first time, extends this intriguing phase from Mott insulators to a highly conductive metal [1]. In this work, we systematically study the electrical transport properties of ECA, where itinerant electrons serve as a sensitive probe for the spin supersolid states. We observe anomalies both in the temperature-dependent resistivity and field-dependent magnetoresistance and Hall signals, which are attributed to response of electrons to the Eu2+ spins and their fluctuations. Moreover, Shubnikov-de Haas quantum oscillations at high magnetic field reveal pronounced band splitting in the spin polarized state. Our results reveal an intimate correspondence between electrical transport and magnetic transitions in ECA, deepening the understanding of this metallic spin supersolid.

Figures (4)

• Figure 1: Interplay between conduction electrons and supersolid spins in EuCo$_2$Al$_9$ (ECA). (a) Crystal structure of ECA, where the magnetic moments of Eu$^{2+}$ forms the metallic spin supersolid "Y" (MSY) state at low temperature. (b) Calculated Fermi surfaces of ECA in the paramagnetic (PM) state. (c) Specific heat of ECA at zero field. The two peaks at $T_{N1}\simeq$ 3.5 K and $T_{N2}\simeq$ 1.1 K separate the high-temperature PM state, the fluctuating collinear ordered (CLO$^*$) state, and the MSY state. The specific heat of the nonmagnetic counterpart BaCo$_2$Al$_9$ (BCA) is also shown for comparison, which indicates negligible contributions of phonon and itinerant electrons in the temperature range of interest. (d) Estimation of the electronic specific heat at 0 T using a linear fitting in the $C/T$ versus $T^2$ plot in the PM state. The Sommerfeld coefficient is intercepted to be $\gamma \approx 40$ mJ/mol·K$^2$, suggesting a moderate yet important coupling between electrons and local moments. (e) Longitudinal resistivity $\rho_{xx}$ for electric current $I$ along the $x$ direction (crystallographic $[11\bar{2}0]$ direction). Magnetic field $H$ is applied along the $z$ direction ($c$-axis) from 0 to 3 T. The data show distinct features: a minimum above $T_{N1}$, a peak at $T_{N1}$, and a shoulder-like feature at $T_{N2}$. Sketch illustrates the definition of the $xyz$ coordinates and the measurement configuration. (f) Longitudinal resistivity $\rho_{zz}$ for $I$ along the $z$ direction and $H$ along $x$ direction. There exists only one shoulder-like feature at $T_{N1}$.
• Figure 2: Magneto-transport properties of ECA. (a) Experimentally measured field dependence of $\rho_{xx}$ at representative temperatures, showing a dip in the MSY state and a steep decrease in the metallic spin supersolid "V" (MSV) state. Arrows denote the boundaries between MSY, UUD, MSV and PM phases at 0.35 K. (b) Off-diagonal Hall resistivity $\rho_{yx}$, whose positive sign indicates dominant hole contribution. (c) Simulated magnetoresistance, $\rho_{xx}^\mathrm{cal.}-\rho_0\propto(H-\alpha M)^2$, for $\lvert{\mu_0H}\rvert\leq$4 T, based on the experimental magnetization data in Ref. Nature2026. The parameter $\alpha$ denotes a mean-field coefficient fixed as 0.3 (see main text), which represents the coupling between electrons and local moments. (d) Low-field zoom-in of simulated Hall resistivity, $\rho_{yx}^\mathrm{cal.}\propto(H-\alpha M)/(1+\eta H)$, where a slowly varying hole concentration with respect to field is assumed ($\eta=0.05$).
• Figure 3: Shubnikov-de Haas quantum oscillations (SdH QOs) in ECA. (a) $\rho_{xx}$ for a FIB-milled Hall-bar sample of ECA up to 35 T, showing nonsaturating magnetoresistance and high-field SdH QOs. The sample size is about $8\times1\times0.7$$\mu m^3$ as shown in the inset, with field along the shortest direction ($c$-axis). The curves are shifted vertically by 0.5 $\mu\Omega$·cm for clarity. (b) Oscillatory part $\Delta\rho_{xx}$ of (a) after subtracting smooth backgrounds. The magnetic induction intensity ($B=\mu_0H+\mu_0(1-N)M$) rather than $H$ has been adopted for the precise determination of the oscillation periods. Here $M\sim0.39$ T for the saturation moment of ECA, and the demagnetizing factor $N$ is taken to be 0.5. (c) Fast Fourier transform (FFT) analysis of the QOs at 1.6 K in the 10-35 T window, revealing three major Fermi orbits ($\alpha_1$, $\alpha_2$, and $\gamma$). (d) Temperature variations of the FFT spectra in the 5-14 T window. (e) Temperature variations of the FFT spectra in the 10-35 T window. (f) FFT peak positions of $\alpha_1$, $\alpha_2$, and $\gamma$ orbits with respect to temperature. Solid lines represent linear fittings below and above $T_{N1}$. (g) Fitting of the cyclotron masses of $\alpha_1$, $\alpha_2$, and $\gamma$ orbits from the peak values in (d) and (e), using the standard Lifshitz-Kosevich formula.
• Figure 4: Calculated electronic band structures of ECA. (a) Band dispersions along high-symmetry lines, with all Eu$^{2+}$ spins fixed parallel to the $c$-direction. (b) and (c) Fermi-surface cuts in the paramagnetic and polarized state, respectively, in the first Brillouin zone at $k_z=0$. The largest hole pocket is labeled as the $\Gamma$ orbit. (d) and (e) Fermi-surface cuts showing $\alpha_1$ and $\alpha_2$ orbits around $A$ point in the paramagnetic and polarized state, respectively, in the first Brillouin zone in the $k_z=\pi/c$ plane.