Single-gap Isotropic $s-$wave Superconductivity
in Single Crystals $\text{AuSn}_{4}$

Figure 1 shows the low-temperature dependence of the change in the London penetration depth, $\Delta\lambda(T)=\lambda(T)-\lambda(T_{min}=0.4\>\text{K})$ before (blue circles) and after 2.5 C/cm² electron irradiation (red circles). The upper left inset shows the exponent $n$ determined from the power-law fitting, $\Delta\lambda(T)\sim At^{n}$, as a function of the upper fitting limit, $t_{max}=T_{max}/T_{c}$. The solid lines in the main frame show an example of such a fitting with $t_{max}=0.4$. The results show a robust and consistent behavior with $n\geq 4$, indicating experimentally indistinguishable from the exponential temperature dependence. The exponent, $n$, decreased after irradiation as it should be in an $s-$wave superconductor [31, 32].

The upper right inset of Fig.1 shows $\Delta\lambda(T)$ of the same sample in its pristine state and after 2.15 C/cm² electron irradiation as a function of absolute temperature $T$. One might think that for some reason (e.g., defect annealing and recombination), there was no increase in disorder after irradiation. This is not the case, as the saturation value above $T_{c}$ increased substantially. The saturation is determined by the skin depth in the normal state, $\delta_{\text{skin}}=\sqrt{\rho/\mu_{0}\pi f}$, where $\mu_{0}=4\pi\times 10^{-7},\text{H/m}$ is the vacuum permeability, and $\rho$ is the resistance. We did not measure resistivity in this $\text{AuSn}_{4}$ sample, but we directly compared resistivity from transport measurements and extracted from the skin depth on the same samples in other compounds and always found good quantitative agreement [33, 34]. The upper critical fields are small, $H^{\parallel ab}_{c2}=130\>\text{Oe}$ and $H^{\parallel c}_{c2}=90\>\text{Oe}$ [11]. Combined with the trend of measured magnetoresistance [13], the expected variation above $T_{c}$ is negligible. An increase in $\delta_{\text{skin}}$ at a fixed frequency, $f$, is due to an increase in resistivity, which is indicative of the increased scattering. Therefore, the fact that the superconducting transition $T_{c}$ remains unchanged is consistent with the Anderson theorem for isotropic $s-$wave superconductors [35, 36]. We observe similar robust superconductivity in another low$-T_{c}$ superconductor with non-trivial topology, LaNiGa₂ [37].

The exponential temperature dependence of $\lambda(T)$ can be fitted with the well-known low-temperature asymptotic BCS, $\Delta\lambda(T)=\lambda(0)\sqrt{\frac{\pi\delta}{2t}}e^{-\frac{\delta}{t}}$ [19], where the ratio $\delta=\Delta(0)/T_{c}$ was fixed at $\delta\approx 1.76$, leaving only one free parameter $\lambda(0)$. The fitting is shown in the top panel of Fig.2. It produces $\lambda(0)=150\>\text{nm}$ in the pristine state (blue fitting curve and blue data symbols) and $\lambda(0)=258\>\text{nm}$ after 2.15 C/cm² electron irradiation (red curve and symbols). With these numbers, we can calculate the superfluid density in the full temperature range using $\rho_{s}(T)\equiv\left(\lambda(0)/\lambda(T)\right)^{2}=\left(1+\Delta\lambda(T)/\lambda(0)\right)^{-2}$. The bottom panel of Fig.2 shows $\rho_{s}(T)$ by blue and red circles for the pristine and irradiated states of the same sample, respectively. The theoretical lines of the clean (blue) and dirty (red) limits were calculated self-consistently using the Eilenberger formalism [39]. We note that the analytical dirty limit formula, $\rho_{s}=\left(\Delta(T)/\Delta(0)\right)\tanh{\left(\Delta(T)/2T\right)}$ reproduces the numerical calculation precisely [38]. Examining Fig.2 we conclude that the classical BCS theory describes the experimental data well.

To summarize our findings from measurements of the London penetration depth, $\lambda(T)$, several independent parameters: (1) low-temperature behavior of $\lambda(T)$; (2) full temperature range behavior of $\rho_{s}$; (3) disorder-independent $T_{c}$ before and after electron irradiation, fully agree with the BCS theory for the isotropic $s-wave$ gap with the ratio $\delta=\Delta(0)/T_{c}\approx 1.76$. This is the nature of superconductivity in the bulk of $\text{AuSn}_{4}$ crystals. However, our measurements would not pick up a tiny signal coming from the surface atomic layers, so unconventional topological features are still possible.

The temperature variation of the magnetic penetration depth before (top panel) and after (bottom panel) electron irradiation, measured in various dc magnetic fields applied along the $c-$ axis, is shown in Fig.3. The field values are shown next to each curve. Solid lines correspond to zero-field cooling (ZFC) in all curves, and dotted lines correspond to field cooling (FC) protocols. For one curve, this is shown by arrows. The ZFC and FC curves are indistinguishable, implying that the process is totally reversible, which indicates a parabolic shape of the pinning potential.

In the presence of an external DC magnetic field, Abrikosov vortices penetrate the sample and form a vortex lattice. Then the measured penetration depth, $\lambda_{m}$, has two contributions, the usual London penetration depth that in this section we explicitly denote as $\lambda_{L}$, and the Campbell penetration depth $\lambda_{C}$, which is a characteristic length scale over which a small ac perturbation is transmitted elastically by a vortex lattice into the sample [40, 41, 42, 43]. More specifically, the amplitude of the ac perturbation must be small enough so that the vortices remain in their potential well, and their motion is described by the reversible linear elastic response. In this case, $\lambda_{m}^{2}={\lambda_{L}^{2}+\lambda_{C}^{2}}$ [44, 45]. This requirement of a very small amplitude makes most conventional ac susceptibility techniques inapplicable for the measurements of the Campbell length. Specialized frequency domain resonators with sufficient sensitivity to a small excitation ac magnetic field are needed [46, 47]. Until now, only a few experimental studies have been published [46, 48, 49, 47, 50].

Figure 3 shows the temperature-dependent variation of the magnetic penetration depth, $\lambda_{m}(T)=\lambda_{L}(0)+\Delta\lambda_{m}(T)$, for different values of the dc magnetic field applied parallel to the sample $c-$axis. For $\lambda_{L}(0)$, we have used the values obtained from the BCS fit; see the upper panel of Fig. 2. Then, we assumed that, above $T_{c}$, the resistivity is field independent, so we adjusted other curves to match that value. The top panel shows a pristine state, and the bottom panel shows the same sample after electron irradiation.

Generally speaking, the Campbell penetration depth can exhibit a hysteresis upon warming and cooling, indicating an anharmonic (non-parabolic) pinning potential and/or strong pinning [46, 51, 43, 50]. Therefore, there are two types of measurement protocols: zero-field cooling (ZFC) and field cooling (FC). In the ZFC protocol, the Campbell length is measured on warming after the sample was cooled in a zero magnetic field and the target field was applied at the base temperature (solid lines in Fig. 3). In the FC protocol, measurements are performed on cooling in a target magnetic field applied above $T_{c}$ (dotted lines in Fig. 3). For both pristine and irradiated states, $\lambda_{m}(T)$ shows a monotonic increase with temperature, and there is no hysteresis between the ZFC and FC protocols. To aid in visualizing the effect of irradiation, the scales of the axes in Fig. 3 are the same in the top and bottom panels. It is clear that the measured penetration depth has increased after electron irradiation.

Figure 4 shows the Campbell penetration depth as a function of an applied magnetic field, $H$, evaluated from the data shown in Fig. 3 at a fixed temperature of $T=0.5\>\text{K}$ for a FC protocol comparing pristine (blue symbols) and irradiated (red symbols) states of the same sample. The Campbell length $\lambda_{C}$ increases after irradiation. In the simple Campbell model [40, 41], $\lambda_{C}^{2}=\phi_{0}H/\alpha$, where $\phi_{0}$ is the magnetic flux quantum and $\alpha$ is the curvature of the pinning potential, $\alpha=d^{2}U/dr^{2}$. The critical current density $j_{c}=\alpha r_{p}/\phi_{0}=Hr_{p}/\lambda_{C}^{2}$, where $r_{p}$ is the radius of the pinning potential, usually assumed to be the coherence length, $\xi$. We note that this critical current is not the same as the persistent current in other magnetization measurements because of magnetic relaxation. With an operational frequency of 14 MHz, unattainable in conventional ac susceptometry, our estimate of the critical current density is much closer to the true critical value. The dc magnetization gives an even lower current density as a result of a long time window of data acquisition.

In a more general picture, $\alpha$ is determined by the elementary pinning forces [52, 42, 43]. In the original model with a fixed $r_{p}$, the Campbell length is expected to scale as $\lambda_{C}\sim\sqrt{H}$, but Fig.4 shows a practically linear temperature dependence, especially after irradiation. This indicates that vortex pinning in $\text{AuSn}_{4}$ is more complicated with a field-dependent radius of the pinning potential, which is possible, for example, in a collective pinning theory when the vortex lattice evolves from the single-vortex pining regime to the vortex bundle regime [53]. In addition, it is known that the coherence length increases with the magnetic field [54]. Therefore, if $\xi\sim H$, then $\lambda_{C}$ will be a linear function of the applied field. As for the difference between pristine and irradiated states, it is possible that the collective pinning in the pristine state is replaced by the disordered vortex phase after electron irradiation, and one cannot directly compare the critical current densities using the same formula. In any case, the nature of pinning in $\text{AuSn}_{4}$ requires further investigation.