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

Sunil Ghimire  Ames National Laboratory, Ames, Iowa 50011, USA  Department of
Physics & Astronomy, Iowa State University, Ames, Iowa 50011, USA  Kamal R.
Joshi  Ames National Laboratory, Ames, Iowa 50011, USA  Department of Physics
& Astronomy, Iowa State University, Ames, Iowa 50011, USA  Elizabeth H.
Krenkel  Ames National Laboratory, Ames, Iowa 50011, USA  Department of
Physics & Astronomy, Iowa State University, Ames, Iowa 50011, USA  Makariy A.
Tanatar  Ames National Laboratory, Ames, Iowa 50011, USA  Department of
Physics & Astronomy, Iowa State University, Ames, Iowa 50011, USA  Marcin
Kończykowski  Laboratoire des Solides Irradiés, CEA/DRF/lRAMIS, École
Polytechnique, CNRS, Institut Polytechnique de Paris, F-91128 Palaiseau,
France  Romain Grasset  Laboratoire des Solides Irradiés, CEA/DRF/lRAMIS,
École Polytechnique, CNRS, Institut Polytechnique de Paris, F-91128 Palaiseau,
France  Paul C. Canfield  Ames National Laboratory, Ames, Iowa 50011, USA
Department of Physics & Astronomy, Iowa State University, Ames, Iowa 50011,
USA  Ruslan Prozorov<EMAIL_ADDRESS>Ames National Laboratory, Ames,
Iowa 50011, USA  Department of Physics & Astronomy, Iowa State University,
Ames, Iowa 50011, USA

(3 July 2024)

###### Abstract

London, $\lambda_{L}(T)$, and Campbell, $\lambda_{C}(T)$, penetration depths
were measured in single crystals of a topological superconductor candidate
$\text{AuSn}_{4}$. At low temperatures, $\lambda_{L}(T)$ is exponentially
attenuated and, if fitted with the power law, $\lambda(T)\sim T^{n}$, gives
exponents $n>4$, indistinguishable from the isotropic single $s-$wave gap
Bardeen-Cooper-Schrieffer (BCS) asymptotic. The superfluid density fits
perfectly in the entire temperature range to the BCS theory. The
superconducting transition temperature, $T_{c}=2.40\pm 0.05\>\text{K}$, does
not change after 2.5 MeV electron irradiation, indicating the validity of the
Anderson theorem for isotropic $s-$wave superconductors. Campbell penetration
depth before and after electron irradiation shows no hysteresis between the
zero-field cooling (ZFC) and field cooling (FC) protocols, consistent with the
parabolic pinning potential. Interestingly, the critical current density
estimated from the original Campbell theory decreases after irradiation,
implying that a more sophisticated theory involving collective effects is
needed to describe vortex pinning in this system. In general, our
thermodynamic measurements strongly suggest that the bulk response of the
$\text{AuSn}_{4}$ crystals is fully consistent with the isotropic $s-$wave
weak-coupling BCS superconductivity.

##  I Introduction

In recent years, superconductors with topological features in their electronic
bandstructure have attracted significant interest for various novel features
predicted by a well-developed theory. For example, emerging zero-energy
excitations called Majorana fermions [1]. On the material side, the search for
topological superconductors (TSCs) is very active but so far has yielded only
a few “candidates” whose topological properties have not yet been fully
confirmed experimentally, including UTe2 [2], Sr2RuO4 [3, 4, 5], UPt3 [6],
2M-WS2 [7], and MxBi2Se3 with M=Cu [8, 9]. The subject of this study,
$\text{AuSn}_{4}$, is another promising TSC candidate with theoretically
predicted non-trivial topological characteristics [10, 11, 12, 13].

The superconductivity in orthorhombic $\text{AuSn}_{4}$ with a transition
temperature to the superconducting state, $T_{c}=2.4\>\text{K}$, was
discovered in 1962 [14]. This compound is isostructural to $\text{PtSn}_{4}$
[15] and $\text{PdSn}_{4}$ [16], which are not superconductors. The first
principal study suggests semimetallic behavior with type I nodes [12]. The
magneto-trasnport measurements show two-dimensional (2D) superconductivity in
$\text{AuSn}_{4}$ [11, 17]. Recently, ARPES measurements supported by DFT
calculations [13] revealed nearly degenerate polytypes in $\text{AuSn}_{4}$
crystals, making it a unique case of a three-dimensional (3D) electronic band
structure with properties of a low-dimensional layered material. Thermodynamic
magnetization and specific heat measurement in $\text{AuSn}_{4}$ single
crystals are consistent with conventional nodeless $s-$wave Bardeen-Cooper-
Schrieffer (BCS) [18, 19] superconductivity [11]. Scanning tunneling
microscopy (STM) measurements determined the superconducting gap to $T_{c}$
ratio close to the $s-$wave BCS value of $\Delta/T_{c}=1.76$ [13]. However,
other STM measurements suggest unconventional 2D superconductivity with a
mixture of $p-$wave surface states and $s-$wave bulk [10]. Clearly, more
measurements are required for an objective and conclusive determination of the
nature of superconductivity in $\text{AuSn}_{4}$.

Here, we probe the bulk nature of superconductivity in $\text{AuSn}_{4}$
single crystals by measuring London and Campbell penetration depths using a
highly sensitive tunnel-diode resonator (TDR). Furthermore, we examine the
response to a controlled non-magnetic point-like disorder induced by 2.5 MeV
electron irradiation. We conclude that $\text{AuSn}_{4}$ is a robust isotropic
$s-$wave superconductor in the bulk. However, we cannot exclude the
possibility that it could have a different type of superconductivity in the
surface atomic layers, where the STM is most sensitive.

##  II Samples and Methods

Single crystals of $\text{AuSn}_{4}$ were grown with excess Sn flux [20, 21,
13]. High-purity Au and Sn were mixed in a 12:88 ratio in a fritted crucible
and sealed in a quartz ampoule under an Ar gas atmosphere. The ampoule was
heated to $1100\>^{\circ}$C over 12 hours, then cooled to $250\>^{\circ}$C in
12 hours, and significantly slower to $230\>^{\circ}$C over 90 hours. The
ampoule was held at this temperature for 48 hours prior to removal from the
furnace.

The London penetration depth, $\lambda(T)$, was measured using a sensitive
frequency-domain self-oscillating tunnel-diode resonator (TDR) operating at a
frequency of around 14 MHz. The measurements were performed in a cryostat 3 He
with a base temperature of $\approx$ 400 mK, which is $0.17T_{c}$, allowing us
to examine the low-temperature limit, which starts below approximately
$T_{c}/3$, where the superconducting gap is approximately constant. The
experimental setup, measurement protocols, and calibration are described in
detail elsewhere [22, 23, 24, 25, 26]. In the experiment, the variation,
$\Delta\lambda(T)=\lambda(T)-\lambda(0.4\>\text{K})$, is extracted from the
resonant frequency shift. The small excitation magnetic field of
$20\>\text{Oe}$ ensures a true Campbell regime when vortices are gently
perturbed. For TDR measurements, the samples were cut into cuboids that are
typically of size $0.6\times 0.4\times 0.1\>\textrm{mm}^{3}$.

Point-like disorder was introduced at the SIRIUS facility in the Laboratoire
des Solides Irradiés at École Polytechnique in Palaiseau, France. Electrons,
accelerated in a pelletron-type linear accelerator to 2.5 MeV, knock out ions,
creating vacancy-interstitial Frenkel pairs [27, 28]. During irradiation, the
sample is immersed in liquid hydrogen at around 20 K. This ensures efficient
heat removal upon impact and prevents immediate recombination and migration of
the produced atomic defects. The acquired irradiation dose is determined by
measuring the total charge collected by a Faraday cage located behind the
sample. As such, the acquired dose is measured in the “natural” units of
$\text{C/cm}^{2}$, which is equal to $1\>\text{C/cm}^{2}\equiv 1/e\approx
6.24\times 10^{18}$ electrons per cm2. Upon warming to room temperature, some
defects recombine, and some migrate to various sinks (dislocations, surfaces,
etc.). This leaves a metastable population, about 70%, of point-like defects
[29, 30]. Importantly, the same sample has been measured before and after
electron irradiation.

##  III Results

###  III.1 London penetration depth

Figure 1: Main Panel: Low-temperature temperature variation of the London
penetration depth $\Delta\lambda(T)=\lambda(T)-\lambda(0.4\>\text{K})$ as a
function of normalized temperature, $t=T/T_{c}$, for pristine (blue circles)
and irradiated at 2.5 C/cm2 (red circles) single crystal of $\text{AuSn}_{4}$.
Lines show fits to the power law, $\Delta\lambda(T)\sim At^{n}$, with the
upper range of $t_{max}=0.4$. The top right inset shows the $\Delta\lambda(T)$
in the whole temperature range, showing sharp superconducting transition with
onset $T_{c}=2.4\>\text{K}$ for both pristine and electron irradiated state.
The top left inset shows the exponent $n$ versus the upper limit of the power-
law fitting, $t_{max}=T_{max}/T_{c}$, indicating robustness of the power law,
experimentally indistinguishable from exponential.

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].

Figure 2: Top panel. Fit to the BCS low-temperature asymptotic,
$\Delta\lambda(T)=\lambda(0)\sqrt{\frac{\pi\delta}{2t}}e^{-\frac{\delta}{t}}$
with a fixed ratio $\delta=\Delta(0)/T_{c}\approx 1.76$ leaving only one free
parameter, $\lambda(0)=150\>\text{nm}$ in the pristine sample (blue fitting
curve and blue data symbols) and $\lambda(0)=258\>\text{nm}$ after 2.15 C/cm2
electron irradiation (red curve and symbols). Bottom panel: Superfluid density
calculated from the data, $\rho_{s}(T)=(1+\Delta\lambda(T)/\lambda(0))^{-2}$.
Solid lines show self-consistent full temperature range calculations using
Eilenberger formalism for pristine (blue line) and irradiated (red line)
states. The known analytical expression for the $s-$wave dirty limit is shown
in [38].

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.

###  III.2 Campbell penetration depth

Figure 3: Temperature variation of the measured magnetic penetration depth,
$\lambda_{m}$, before (top panel) and after (bottom panel) electron
irradiation, measured with the various dc magnetic fields applied along the
$c-$ axis. The field values are shown. Solid lines correspond to zero-field
cooling (ZFC), 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 completely reversible,
indicating the pinning potential’s parabolic shape. Note that the axes scales
are the same in the top and bottom panels, aiding in a visual comparison of
the effect of irradiation.

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: Campbell penetration depth,
$\lambda_{C}^{2}=\sqrt{\lambda_{m}^{2}-\lambda_{L}^{2}}$ 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.

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.

##  IV Conclusions

We report measurements of London, $\lambda_{L}(T)$, and Campbell,
$\lambda_{C}(T)$, penetration depths in single crystals of the topological
superconductor candidate $\text{AuSn}_{4}$ to elucidate the nature of
superconductivity in the bulk. Several independent parameters studied before
and after 2.5 MeV electron irradiation unambiguously point to isotropic single
$s-$wave gap weak coupling BCS superconductor. Specifically, the superfluid
density before and after electron irradiation overlaps almost perfectly with
the parameter-free theoretical BCS curves in the full temperature range for
clean and dirty limits, respectively. The Campbell penetration depth before
and after electron irradiation does not show hysteresis between the ZFC and FC
data, indicating a parabolic shape of the pinning potential. However, the
$H-$linear behavior of $\lambda_{C}$ implies either the field-dependent
Labusch parameter, $\alpha$, or the radius of the pinning potential, $r_{p}$,
or both. Considering the low pinning in AuSn4 single crystals and the point-
like nature of the induced defects, such a field dependence may be expected in
the vortex bundle regimes within the collective pinning theory [53].

###### Acknowledgements.

We thank Hermann Suderow for the fruitful discussion. This work was supported
by the US DOE, Office of Science, BES Materials Science and Engineering
Division under the contract $\\#$ DE-AC02-07CH11358. The authors acknowledge
support from the EMIRA French network (FR CNRS 3618) on the SIRIUS platform.

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