Fermi surface, possible unconventional fermions, and unusually robust resistive critical fields in the chiral-structured superconductor AuBe

The noncentrosymmetric superconductor (NCS) AuBe is investigated using a variety of thermodynamic and resistive probes in magnetic fields of up to 65 T and temperatures down to 0.3 K. Despite the polycrystalline nature of the samples, the observation of a complex series of de Haas–van Alphen (dHvA) oscillations has allowed the calculated band structure for AuBe to be validated. This permits a variety of BCS parameters describing the superconductivity to be estimated, despite the complexity of the measured Fermi surface. In addition, AuBe displays a nonstandard field dependence of the phase of dHvA oscillations associated with a band thought to host unconventional fermions in this chiral lattice. This result demonstrates the power of the dHvA effect to establish the properties of a single band despite the presence of other electronic bands with a larger density of states, even in polycrystalline samples. In common with several other NCSs, we find that the resistive upper critical field exceeds that measured by heat capacity and magnetization by a considerable factor. We suggest that our data exclude mechanisms for such an effect associated with disorder, implying that topologically protected superconducting surface states may be involved.

Figure 1

(a) The differential susceptibility $\mathrm{d}M/\mathrm{d}H$ plotted versus magnetic field ${\mu }_{0}H$ for a polycrystalline AuBe rod at $T=0.68$ K, recorded during a 40-T pulsed-magnet shot. Several different series of dHvA oscillations are visible. Data for both rising and falling fields are shown. The “fur” on the data is not noise, but comprises the “belly” orbit dHvA oscillations from the copper of the susceptometer coil. (b) The red trace shows dHvA oscillations recorded using a 10-T pulsed magnet shot ($T=0.66$ K) where the sample is the same as that used in (a). The green curve represents the application of a low-pass filter to the red trace; this removes the higher-frequency de Haas–van Alphen oscillations so that the lower-frequency series can be seen more readily.

Figure 2

(a) Fourier spectrum ($20–38$ T window) of the data in Fig. 1 plotted with a logarithmic amplitude scale. A band of several series of dHvA oscillations (labeled $F$) and their higher harmonics (labeled $2F...6F$) are clearly visible, along with a peak at 59.5 kT due to the belly orbits in the Cu coil of the susceptometer. (b) Inset: Fourier transforms (linear amplitude scale) of data from 40-T pulses recorded at a series of higher temperatures $T=1.5–26$ K. The same field window as in (a) has been applied. The frequency range has been chosen to show just the fundamental frequencies in the band of oscillations labeled $F$ in (a). (c) Logarithmic amplitudes (points) of the harmonics of the $F=3620$ T series of dHvA oscillations versus harmonic index $p$; the red line is a guide to the eye, showing an approximate linear decrease with $p$. The green curve is a fit of Eq. (2) for $g{m}^{*}=2$. (d) Fourier transforms of dHvA oscillations from a series of 10-T pulsed-magnet shots at temperatures $T=0.66–8.0$ K. (e) Field-axis expansion of Fourier transforms (linear amplitude scale) of data from 40-T pulses for three example temperatures, showing frequencies in the range $490–1200$ T. The same field window as in (a) has been applied. (f) (Fourier $\mathrm{amplitude})/T$ versus temperature for the $F=3510$ T series of dHvA oscillations. The data window was centered on $B_{\mathrm{m}}=12$ T. The red line is a fit of Eq. (4) to the data, yielding $m^{*}=0.58\pm 0.04m_{\mathrm{e}}$.

Figure 3

(a) Calculated electronic structure of AuBe including spin-orbit coupling and (b)–(e) the predicted Fermi-surface sections shown within the simple-cubic Brillouin zone. For clarity, the predicted Fermi pockets are shown here without the slight doubling of surfaces due to band splitting caused by the spin-orbit coupling. (b) Hole ellipsoids centered on the zone-edge $M$ points. (c) An electron approximate superellipsoid centered on the zone-corner $R$ point, plus a small, approximately spherical, electronlike pocket at the zone-center $\mathrm{\Gamma }$ point; (d) a “monster” [31] spanning most of the Brillouin zone; and (e) a second, electron approximate superellipsoid centered on the zone-corner $R$ point. As is conventional, the terms “electron” and “hole” are used for Fermi-surface sections for which the effective masses are, respectively, positive and negative [31].

Figure 4

Solid points represent effective masses derived using Eq. (4) versus corresponding experimental dHvA frequencies. The error bars are the uncertainties given by the simplex fitting routine and the different colored points and guidelines are explained in the text. Masses and frequencies corresponding to a selection of predicted extremal orbits from the band structure calculations (including spin-orbit interactions) are shown as hollow points. The hollow light-blue points (calculated for $\mathbf{H}\parallel \left[100\right]$; dotted line is a guide to the eye) correspond to the hole ellipsoids centered on the zone-edge $M$ points [Fig. 3]. The hollow green, red-filled points (calculated for $\mathbf{H}\parallel \left[100\right]$) are from the two electron approximate superellipsoids centered on the zone-corner $R$ points [Figs. 3 and 3]. The black, hollow point corresponds to a possible “monster” extremal orbit [Fig. 3].

Figure 5

Magnetic field dependence of parameters describing the $F\approx 170$ T series of dHvA oscillations: (a) effective mass $m^{*}$, (b) frequency $F$, and (c) phase $\aleph$ [see Eq. (7)] of the oscillations. The field dependencies of these parameters are derived from Fourier analysis of $\mathrm{d}M/\mathrm{d}H$ data using inverse field windows of $\frac{1}{24}{\mathrm{T}}^{-1}$ width symmetrically disposed (in inverse field) about the center field ($B_{\mathrm{m}})$ values given on the lower axis.

Figure 6

(a) Heat capacity $C$ of AuBe, divided by temperature $T$, plotted as a function of $T^2$ at fields ${\mu }_{0}H=0$, 50, and 300 mT. The transition to the superconducting state is marked by the near-vertical jump close to $T^2=10{\mathrm{K}}^2$ in the $H=0$ data set. (b) The low-temperature electronic component of the heat capacity versus $T$ at fields of ${\mu }_{0}H=0$, 50, and 300 mT. The curve is a fit to the standard BCS model for an isotropic, fully gapped superconductor in the weak-coupling limit [44].

Figure 7

(a) The dc magnetization $M$ (red) versus field $H$ at a temperature of $T=1.8$ K. Sharp transitions and a small supercooling, evident in the hysteresis of the critical field, are apparent in $M\left(H\right)$. (b) The real component ${\chi }^{\prime }$ (purple) and the imaginary component ${\chi }^{\prime \prime }$ (green) of the ac magnetic susceptibility versus $H$, again at $T=1.8$ K.

Figure 8

Real component ${\chi }^{\prime }$ (a) and imaginary component ${\chi }^{\prime \prime }$ (b) of the magnetic susceptibility of a polycrystalline bar of AuBe plotted against $H$ for a series of constant temperatures in the range $3.1\mathrm{K}\ge T\ge 0.3\mathrm{K}$. Field sweeps in the temperature range $0.3–1.6$ K were done in the ${}^3\mathrm{He}$ cryostat, while those in the range $1.8–3.1$ K employed the MPMS. The jump in peak magnitudes between 1.6 and 1.8 K is caused by a difference in drive amplitude. (c) Isothermal resistivity data, shown as resistivity $\rho$ divided by the low-temperature normal-state resistivity ${\rho }_{\mathrm{n}}$ versus magnetic field $H$. The data were recorded at a series of approximately equally spaced temperatures covering the range $3.0\mathrm{K}\ge T\ge 0.3$ K.

Figure 9

(a) Inset: critical fields versus temperature for AuBe. The solid symbols are critical fields measured using resistivity (circles = onset of resistive transition, diamonds = highest field at which zero resistance occurs), while the hollow symbols represent critical fields derived from thermodynamic probes (magnetization, susceptibility and heat capacity). Data from the magnetization of a small single crystal are shown as hollow squares. (b) Analogous phase diagram for a representative collection of NCSs, presented in reduced units $H_{\mathrm{c}}/H_{\mathrm{c}}(T=0), T/T_{\mathrm{c}}$. Here, $H_{\mathrm{c}}(T=0)$ is derived from standard BCS fits to critical fields derived from thermodynamic probes. As in (a), open symbols represent thermodynamic critical fields and filled points are resistive critical fields. Data for the various NCSs are from the following sources: AuBe (this work), BiPd [60], ${\mathrm{LaRhSi}}_3$ [61], ${\mathrm{BaPtSi}}_3$ [62], ${\mathrm{LaPdSi}}_3$ [56], ${\mathrm{LaPtSi}}_3$ [56].