# Spectroscopic evidence of multi-gap superconductivity in non-centrosymmetric
AuBe

Soumya Datta1, Aastha Vasdev1, Partha Sarathi Rana2, Kapil Motla3, Anshu
Kataria3, Ravi Prakash Singh3, Tanmoy Das2, and Goutam Sheet1
<EMAIL_ADDRESS>1Department of Physical Sciences, Indian Institute of
Science Education and Research Mohali, Sector 81, S. A. S. Nagar, Manauli, PO
140306, India  2Department of Physics, Indian Institute of Science, Bangalore
560012, India  3Department of Physics, Indian Institute of Science Education
and Research Bhopal, Bhopal 462066, India

###### Abstract

AuBe is a chiral, non-centrosymmetric superconductor with transition
temperature $T_{C}$ $\simeq$ 3.25 K. The broken inversion symmetry in its
crystal structure makes AuBe a possible candidate to host a mixed singlet-
triplet pairing symmetry in its superconducting order parameter ($\Delta$).
This possibility was investigated by transport, thermodynamic, and muon-spin
rotation/relaxation experiments in AuBe. However, this issue was not addressed
using direct spectroscopic probes so far. In addition, certain ambiguities
exist in the description of superconductivity in AuBe based on $\mu$SR
experiments reported earlier. Here we report scanning tunneling spectroscopy
(STS) on AuBe down to 300 mK. We found a signature of two superconducting gaps
(with 2$\Delta_{1}/k_{B}T_{C}$ = 4.37 and 2$\Delta_{2}/k_{B}T_{C}$ = 2.46
respectively) and a clean BCS-like temperature dependence of both the gaps. We
have also performed band structure calculations to identify the different
bands that might give rise to the observed two-gap superconductivity in AuBe.

The nature of superconductivity for a non-centrosymmetric superconductor
(NCSC) varies widely from material to material. Evidence of line nodes in the
superconducting gap is found in numerous NCSCs like Mo3Al2C[1], CeIrSi3[2],
CePt3Si[3, 4], Li2Pt3B[5, 6], and recently Ru7B3[7]. On the other hand, in a
number of NCSCs, only a conventional, isotropic, s-wave, fully-gapped
superconducting phase is reported. Nb0.18Re0.82[8], BiPd[9, 10], T2Ga9 (T =
Rh, Ir)[11, 12] and LaMSi3 (M = Rh[13], Ir[14], Pd, Pt[15]) are a few examples
of that. There are also NCSCs where the superconducting phase is not well
described either by a fully open gap or a nodal gap opened in a single band.
In such a situation, a multi-gap model becomes necessary. LaNiC2[16],
TaRh2B2[17] and Ln2C3 (Ln = La,Y)[18, 19] are examples of such NCSCs. In this
context, AuBe with transition temperature $T_{C}$ $\sim$ 3.25 K[20, 21, 22,
23, 24, 25] is interesting due to multiple reasons. The material contains the
heavy element Au and the crystal structure of the material is non-
centrosymmetric (cubic space-group symmetry of $P2_{1}3$)[21, 22]. This makes
AuBe a potential candidate for a mixed pairing state[4, 5, 7]. Further, the
crystal structure of AuBe is also chiral ($B20$, FeSi type), and such
structure is predicted to host chiral fermions[26, 27, 28]. To note, the $B20$
structure is also the only known crystal structure where bulk magnetic
skyrmions are reported[29, 30, 31]. Further, according to a recent study, the
superconducting pairing of AuBe originates from multiple energy bands[32, 33].
All these collectively make AuBe an attractive system where a search for
unconventional superconducting pairing and exotic quasiparticle excitations
are warranted.

There are two primary outstanding issues within and beyond the previous
reports on superconducting AuBe. The first one is about the type of
superconductivity with respect to its magnetic properties. Amon $et$ $al.$[22]
reported type-II (with Ginzburg-Landau parameter $k_{GL}$ = 0.4)
superconductivity while Singh $et$ $al.$[23] and Beare $et$ $al.$[24] reported
type-I ($k_{GL}$ = 2.3) behavior in AuBe based on their independent dc
magnetization, specific heat, and muon-spin rotation/relaxation ($\mu$SR)
studies. Reber $et$ $al.$[21, 25] attempted to resolve the issue revealing a
crossover from type-II to type-I superconductivity[34, 35] at $\sim$ 1.2 K
based on their resistivity, dc magnetization, ac susceptibility and specific
heat studies. This suggests the possibility that AuBe falls under the class of
type II/I superconductors, the behavior which was for a long time investigated
in intentionally disordered elemental superconductors with $k_{GL}$ $\sim$
1/$\sqrt{2}$[36, 37]. Recently, such crossover was also proposed in
superconductors with non-centrosymmetric crystal structures and/or multiple
bands participating in superconductivity[38, 39]. Coincidentally, both these
properties happen to be relevant in the present context of AuBe. The second
issue lies in the description of the superconducting order parameter. All the
reports up to 2019, including those which have mutually conflicting
conclusions about the type (type I vs. type II) of superconductivity in AuBe,
have agreement on the fact that AuBe is a pure, isotropic s-wave, spin-singlet
superconductor[22, 23, 24, 25]. However, Khasanov $et$ $al.$[32, 33], in their
two successive papers in 2020 proposed an entirely new unconventional multi-
gap mechanism in AuBe. Based on new $\mu$SR experiments[32], the authors
reported that the temperature evolution of the thermodynamic critical field
$B_{C}$ of AuBe could not be explained without considering at least two
different gaps. From a self-consistent two-gap model, it was found that the
corresponding 2$\Delta/k_{B}T_{C}$ were 4.52 and 2.37, respectively. Further,
based on a detailed comparative analysis between the usual single gap model
and a self-consistent two-gap model, the latter’s superiority over the former
was concluded[33]. Hence, in order to acquire a decisive evidence of multi-
band superconductivity or to rule out such a possibility in AuBe, it remains
an important task to measure the superconducting gap(s), spectroscopically in
this system. In addition, detailed studies of the temperature and magnetic
field evolution of the superconducting order parameter are also warranted to
probe the nature of the gaps.

The polycrystalline sample of AuBe was prepared by arc melting method from a
stoichiometric mixture of elemental Au and Be. The preparation and
characterization details are published elsewhere[23]. To directly probe the
superconducting gap(s) in AuBe, we employed low-temperature scanning tunneling
microscopy (STM) and spectroscopy (STS) in a Unisoku system with RHK R9
controller, inside an ultra-high vacuum (UHV) cryostat kept at $\sim$
$10^{-10}$ mbar. The lowest temperature down to which the measurements were
performed was 310 mK. As such experiments are extremely sensitive to the
surface cleanliness, a few layers from the surface were first removed by mild
reverse sputtering in an argon environment in-situ, inside another UHV
preparation chamber that is connected to the main STM/S chamber. Then the
sample with its pristine surface was transferred to the scanning stage at low
temperature for experiments. The Tungsten (W) tip, which was prepared outside
by electrochemical etching, was also cleaned by high-energy electron-beam
bombardment inside the same UHV preparation chamber. This whole process helped
us probe the pristine surface of AuBe. Spectra were further recorded at random
points on the surface, and at each point on the sample surface, we found a
clean spectrum with a fully opened gap. The differential conductance $dI/dV$
was measured using a lock-in based ac modulation technique (amplitude 40
$\mu$V, frequency 3 kHz).

All the spectra we probed show two clear peaks symmetric about $V$ = 0. The
position of these coherence peaks provides a direct measure of the
superconducting energy gap ($\Delta$). However, depending on the different
points on the sample surface probed, the positions of the coherence peaks vary
from $\pm$250 $\mu$V to $\pm$375 $\mu$V, approximately. In Figure 1, we show
eight representative tunneling spectra captured at different points on the
surface of AuBe, all at the lowest temperature. The spectra presented in
Figure 1(a)-(d) visually have a wider spectral gap compared to the spectra
presented in Figure 1(e)-(h). Intrinsic disorder on the surface of the sample
can exhibit such variation, and that was indeed our primary guess.
Nevertheless, to extract the exact value of the order parameter $\Delta$ for
each spectrum, we proceeded further with the analysis. All the spectra were
first normalized $w.r.t$ conductance at 1.5 mV, where they are almost flat.
Then these experimental spectra were compared with numerically generated
spectra using the expression for tunneling current within a single-gap model
given by the following equation.

$I(V)\propto\int_{-\infty}^{+\infty}N_{s}(E)N_{n}(E-eV)[f(E)-f(E-eV)]dE$

Here, $N_{s}(E)$ and $N_{n}(E)$ are respectively the normalized density of
states (DOS) of the superconducting sample and the normal metallic tip where
$f(E)$ is the Fermi-Dirac distribution function[40]. Within this single band
model, $N_{s}$ is given by the following expression of the Dynes formula[41].

$N_{s}(E)\propto
Re\left(\frac{(E-i\Gamma)}{\sqrt{(E-i\Gamma)^{2}-\Delta^{2}}}\right)$

In our analysis, $\Gamma$ takes care of all possible reasons for spectral
broadening, including that due to finite quasiparticle lifetime and other
possible interband and intraband scattering effects. Though this model is
routinely used and highly successful to analyze the tunneling spectroscopic
data of a conventional superconductor, it fails to match any experimental
spectrum of AuBe, irrespective of whether the spectrum was visibly wider (like
in Figure 1(a)-(d)) or narrower(like in Figure 1(e)-(h)). To explain the
issue, two such theoretical plots (red and blue curves) are shown on each
experimental spectrum (black circles) presented in Figure 1(d) and (h). The
red line represents the closest match for the upper portions of each spectrum,
especially at the coherence peaks. However, that fails to match the lower
portion near $V$ = 0, underestimating the actual depth of the spectrum. Since
$\Gamma$ is fixed arbitrarily without a complete knowledge about the
microscopic origin of the same, we first tried to fit the spectra by brute
force making $\Gamma$ free. In order to match the lower part, if we tune the
parameter $\Gamma$ down and adjust the $\Delta$ as needed too, we face a
situation (blue line) where the lower part of the theoretical plot matches
perfectly with the experimental spectrum, but now deviates significantly above
and overestimates the actual height of the coherence peaks. Hence, it becomes
clear that though a quasiparticle excitation spectrum of AuBe looks like a
standard one with a pair of clear coherence peaks, it cannot be explained
within the framework of a single-gap $s$-wave model. Replacing the isotropic
$s$-wave $\Delta$ with an anisotropic $\Delta Cos(n\theta)$[42] (where the
integer $n$ can be 1, 2 or higher depending on $p$, $d$ or higher-order
symmetries, respectively) does not give an acceptable description of the data.
Instead, doing that gives rise to a sharp ‘V’-shaped spectrum (not shown in
the figure) in contrast to the experimentally obtained ‘U’-shaped ones. Also,
we did not find a gapless flat spectrum or a spectrum with a zero-bias
conductance peak anywhere on the surface. Hence we considered the simplest
two-gap model for our spectra which was further motivated by the facts about
AuBe reported in the past. Following is a description of such facts.

First, based on the de Haas–van Alphen experiments performed on AuBe, the
presence of multiple bands crossing the Fermi level was reported by Rebar $et$
$al.$[21, 25]. Second, based on density functional theory (DFT) and band-
structure calculations performed independently by Rebar $et$ $al.$[21, 25] and
Amon $et$ $al.$[22] such band crossing was confirmed for at least three
conductive bands. However, though this special type of band structure creates
a possibility of multiband superconductivity in AuBe, it can not be taken as
an evidence of the same. We also noted the reports by Khasanov $et$ $al.$[32,
33], where the authors extracted the thermodynamic critical field $B_{C}$ from
their $\mu$SR experiment on AuBe and explained its temperature dependence
based on a self-consistent two-gap model. In such a scenario, the
quasiparticle excitation spectrum for a two-band superconductor can be
determined simply by adding the two single gap BCS spectra for the two
respective bands[43]. Considering such a picture, the quasiparticle DOS of the
$j$-th band can be written as follows.

$N_{s,j}(E)=N_{j}(E_{F})Re\left(\frac{(E-i\Gamma_{j})}{\sqrt{(E-i\Gamma_{j})^{2}-\Delta_{0j}^{2}}}\right)$,
$j$ = 1, 2

Here $j$ is the band index, $N_{j}(E_{F})$ is the normal state DOS at the
Fermi level corresponding to the $j$th band, and $\Delta_{0j}$ is the
amplitude of the superconducting energy gap formed in the $j$th band. The
tunneling current, which has contributios from both the bands, will now take
the following form.

$I(V)\propto\sum_{j=1,2}\alpha_{j}\int_{-\infty}^{+\infty}N_{sj}(E)N_{n}(E-eV)[f(E)-f(E-eV)]dE$

Here $\alpha_{j}$ is the relative contribution of the $j$-th band to the
tunneling current. When we tried to fit the spectra using this model, it
became extremely successful over the entire energy range. We have presented
such theoretical plots with green lines in Figure 1(d) and (h). It is clear
that such a plot matches both the upper and lower part of each spectrum this
time very well. The extracted values of two superconducting gaps ($\Delta_{1}$
and $\Delta_{2}$) and the corresponding two broadening parameters
($\Gamma_{1}$ and $\Gamma_{2}$) are also mentioned for each spectrum. To note,
the pair of the superconducting gap values did not vary noticeably from
$\Delta_{1}$ $\simeq$ 320$\pm$10 $\mu$eV, and $\Delta_{2}$ $\simeq$ 180$\pm$10
$\mu$eV across the spectra. The parameters which actually do vary are the
relative contributions of each band to the total tunneling current, i.e
$\alpha_{1}$ and $\alpha_{2}$. For the spectra represented in Figure 1(a)-(d),
we found $\alpha_{1}>\alpha_{2}$, and for the spectra represented in Figure
1(e)-(h), the opposite.

In MgB2, the two gaps are distinctly visible from experiments like Andreev
reflection spectroscopy, tunneling spectroscopy, etc.[44, 45, 46, 47, 48, 49].
In such experiments, two pairs of coherence peaks corresponding to the two
gaps appear in the quasiparticle DOS. In contrast, the spectra that we
recorded on AuBe do not have distinct multi-gap features, and visually they
look like a single-gap BCS spectrum. However, from our detailed analysis, we
found that a usual single band model cannot explain such spectra while a
simple two-gap model can. Visually the spectra of AuBe appear different from
that of MgB2, primarily, because the amplitudes of the two gaps in AuBe are
close. Schopohl $et$ $al.$[50] and Noat $et$ $al.$[51] explained such
situations with interband scattering and tunneling of quasiparticles. They
described multiple characteristic features of the multi-band spectra like
damped quasiparticle peaks, kinks near the peaks, dips beyond the peaks, etc.
that may appear as a consequence of such interband physics. From a close
visual inspection of our spectra, when we compare the experimental data (black
circles) with the best single gap fits (blue lines) in Figure 1(d) and (h), we
actually can notice the first two features mentioned above. However, it is
important to note that a simple multi-gap model proposed by Suhl $et$
$al.$[43] is successful to fit (green lines) our experimental data with very
high fidelity, and in this model, more complicated factors like interband
scattering, k-selective tunneling etc. were not taken into account. At the
same time, when Iavarone et al.[47] reported a distinct two-gap
superconductivity in MgB2 by tunneling spectroscopy, Eskildsen et al.[52] also
reported tunneling spectroscopy on the same material. Interestingly, when the
latter group attempted to explain their spectra using the usual single gap
Dynes equation, their theoretical fit overestimated the coherence peak. This
can be compared with our fittings (blue lines) in Figure 1(d) and (h) using
the single gap Dynes model. Now, we also note that the existence of two gaps
cannot be proved simply based on the analysis of certain spectra. If two
different bands participate in the superconductivity of AuBe, they are
expected to evolve with temperature and external magnetic field independently
and differently, unless the inter-band scattering is too strong.

We present the temperature ($T$) dependence of a typical spectrum in Figure
2(a), where the colored circles represent the experimentally obtained spectra.
With increasing $T$, the coherence peaks gradually decrease, and all the gap
features disappear at 1.77 K. The corresponding theoretical fits within the
two-band model[43] are shown on top of each experimental spectrum as black
lines where the values of $\alpha_{1}$(0.8) and $\alpha_{2}$(0.2) were kept
unchanged over the entire $T$ range. The two gaps extracted from the fits are
plotted with $T$ in Figure 2(b) with red ($\Delta_{1}$) and blue
($\Delta_{2}$) triangles. Each gap independently follows a BCS-like
dependence[40] up to 1.77 K, where they merge and disappear. The smaller gap
($\Delta_{2}$) slightly deviates from the BCS-line at higher $T$, which is
typical in the presence of a non-zero inter-band scattering[43]. Such a
situation, where two gaps independently evolve with temperature until they
disappear at the same $T_{C}$, is valid for a multi-gap superconductor where
interband scattering is negligible. This further validates the model we have
used for our analysis. The extracted values of the two broadening parameters
($\Gamma_{1}$ and $\Gamma_{2}$) are presented in the inset of Figure 2(b)
throughout the $T$ range. To note, the $\Gamma$s do not increase with
temperature as expected for a strong coupling superconductor. Rather they
decrease slightly with increasing $T$ or remain almost constant considering
the same energy scale of $\Delta$s. In the pioneering paper by Dynes et
al.[41], the authors pointed out that such behavior is a describing feature of
a weakly coupled superconductor (like Al) vis-a-vis a strongly coupled one
(like PbBi). Our observation is consistent with the previous reports[22, 25,
23] that AuBe is a weakly coupled superconductor with a specific heat jump
near $T_{C}$, $\Delta C/\gamma_{n}T_{C}\approx$ 1.26, and coupling strength
$\lambda_{e-p}\approx$ 0.5. From our two-gap analysis of the tunneling
spectra, we found that 2$\Delta_{1}/k_{B}T_{C}$ = 4.37 and
2$\Delta_{2}/k_{B}T_{C}$ = 2.46, respectively. The former one is slightly
underestimated, while the latter one is slightly overestimated compared to the
values 4.52 and 2.37 respectively, as was reported by Khasanov et al. based on
$\mu$SR experiments[32]. In the presence of a very small yet finite interband
scattering, these differences are in accordance with the two-gap model[43].

To gain further information about the pairing mechanism, we now focus on the
magnetic field dependence of the spectra. The colored circles in Figure 3(a)
show the experimentally obtained spectra, all measured at 310 mK, and the
black lines represent the corresponding two-band fits. All the superconducting
features, including the coherence peaks, disappear at 17 kG. The evolution of
the extracted two gaps (larger $\Delta_{1}$ and smaller $\Delta_{2}$) with
magnetic field are shown in Figure 3(b). Up to 10 kG, both the gaps tend to
decrease slowly in a linear fashion, and beyond that, they decrease faster
until becoming zero at 17 kG. Two important conclusions can be drawn from this
observation. First, the gradual transition supports the type-II behavior in
AuBe as reported by Amon $et$ $al.$[22] but contradicts the type-I behavior
reported by Singh $et$ $al.$[23] and Beare $et$ $al.$[24]. However, this
contradiction can be easily resolved considering the proposed type II/I
superconductivity by Reber $et$ $al.$[25]. As our magnetic measurements are
performed at $\sim$ 310 mK, which is far below the crossover point 1.2 K[21,
25], the type-II behavior is normal and visible. On the second note, the
reported critical field $H_{C}$ of AuBe from various bulk measurements varies
between 259 G to 335 G[22, 25, 23, 24, 32], but from our magnetic field
dependence, we found the local critical field $H_{C(l)}$ as high as 17 kG. It
is interesting to note that Reber $et$ $al.$[25] also reported a considerably
higher resistive upper critical field $H_{C(\rho)}$ compared to the
thermodynamic upper critical field $H_{C}$ found from heat capacity and
magnetization measurements. To explain this enhancement, the authors have
eliminated the possibility of defects or impurity phases at the surface and
concluded it to be an intrinsic surface behavior with a possible topological
protection. However, based on our data, such a possibility can be neither
confirmed nor ruled out. The field dependence of $\Gamma_{1}$ and $\Gamma_{2}$
are presented in the inset of Figure 3(b). With increasing magnetic field,
$\Gamma_{1}$ increases almost linearly where, $\Gamma_{2}$ increases slowly up
to $\sim$10 kG and then increases faster.

In order to understand the origin of multiband superconductivity in AuBe, we
investigated the band structure, density of states, and the Fermi surface of
the system through first-principles electronic structure calculations. Ab-
initio electronic structure of AuBe was calculated using the Density Fuctional
Theory (DFT)[53] implemented in Quantum Espresso[54]. The calculated band
structure and the topology of the Fermi surfaces are consistent with the
previous reports[22, 25]. In our calculations, the experimental crystal
structure was used with a relaxed cell parameter of 4.709$\AA$. A full-
relativistic pseudo-potential with Perdew-Burke-Ernzerof (PBE)[55] exchange-
correlation potential was used in the projected augmented wave(PAW)[56]
method, both with and without spin-orbit coupling (SOC). Self-consistent
charge-density convergence was achieved on a $10\times 10\times 10$ Monkhorst-
Pack[57] $k$-grid. The energy cut-off for the calculation is $60$ $Ry$ and the
Fermi-surfaces and Femi-velocities reported here were obtained on a $k$-grid
of $20\times 20\times 20$ without the SOC.

In Figure 4(a), we have plotted the Fermi surface of AuBe, where the
corresponding Fermi velocity is shown with color gradients. The band
dispersion along the high symmetry directions and the orbital projected DOS
are plotted in Figure 4(c) and 4(d), respectively. The corresponding high
symmetry points within the first Brillouin zone are represented in the inset
of Figure 4(d). From the DOS, it is evident that there are significant
contributions from Be $p$-orbitals, followed by Au $p$-orbital and so on.
Because of the dominant role of Be $p$-orbital on the Fermi surface, the SOC,
as well as the interaction effects are relatively weak, and the
superconducting state is expected to be of phonon-mediated BCS type. The Fermi
velocity plot in Figure 4(a) indicates that the large electron pocket at the
$\Gamma$-point has significantly less Fermi velocity and large DOS. On the
other hand, the DOS of the hole pocket at the $M$-point is significantly less,
while the Fermi velocity is relatively larger. Therefore, it is rational to
conclude that the pocket at the $\Gamma$-point leads to a larger
superconducting gap ($\Delta_{1}$ $\simeq$ 320$\pm$10 $\mu$eV) while that at
the $M$-point causes a smaller superconducting gap ($\Delta_{2}$ $\simeq$
180$\pm$10 $\mu$eV). Hence, we have identified the bands that are responsible
for multiband superconductivity in AuBe, and we have also found the distinct
bands that give rise to the experimentally measured larger and smaller gaps,
respectively.

In order to explain a detailed tunneling spectroscopic study on the vortex
lattice of MgB2 reported by Eskildsen et al.[52], Koshelev and Golubov[58]
developed a general theory for a two-band superconductor under weak interband
scattering. In such a superconductor, the two bands develop two different
field scales, which can be revealed by the distributions of the order
parameters and the local DOS. Consequently, two bands attain their normal
state DOS at two different rates with respect to increasing magnetic field
strength. Considering a negligible difference in the coupling constants, the
ratio of the diffusion constants of the two bands is the only parameter to
determine this relative rate. For MgB2 as an example, the ratio 0.2 best
explains the experimental results of Eskildsen et al.[52]. To understand this
effect in our present context for AuBe, we calculated the magnetic field
dependence of the DOS for the respective bands using the formula for
$N_{sj}(E)$ described earlier and the fitting parameters used in Figure 3(b).
As it can be seen in Figure 4(b), the larger DOS $N_{s1}$ (corresponding to
the electron pocket at the $\Gamma$-point and the larger gap $\Delta_{1}$)
initially rises faster with increasing field to attain the normal state DOS
(i.e. $N_{s}$=1) compared to the smaller DOS $N_{s2}$ (corresponding to the
hole pocket at the M-point and the smaller gap $\Delta_{2}$). However, though
$N_{s2}$ initially starts with a slower rate, latter it increases faster and
attains the normal state DOS almost at the same field value $N_{s1}$ does.
From visual interpretation, this result can be compared with the situation
where the ratio of the diffusion constants is 1 in the proposed model. This
suggests that, unlike in MgB2, the two bands responsible for the
superconductivity in AuBe have comparable transport characteristics. Possibly
for the same reason, unlike in MgB2, the two gaps are not distinctly resolved
in the tunneling spectra of AuBe. To note, the model developed by Koshelev and
Golubov[58] had its own limitation as it assumed a large Ginzburg Landau
parameter ($\kappa$). It serves the purpose for MgB2 as $\kappa\gtrsim$ 10 for
that material[58]. However, in our present case for AuBe, reported $\kappa$
mostly varies between 0.4[23] to 0.75[25] depending on the clean to dirty
limit (except a widely different value 2.34 initially reported by [22]). Some
recent theoretical works[59, 60] based on an intertype domain between type I
and II can be interesting in this context and also relevant for multiband
superconductors with low $\kappa$ like AuBe.

In summary, we have presented a detailed scanning tunneling spectroscopic
investigation on non-centrosymmetric AuBe. From our analysis, we find that a
single band model fails to explain the quasiparticle excitation data, but a
simple two-band model provides a clean description of the same. From our two-
gap analysis of tunneling spectra, we found that 2$\Delta_{1}/k_{B}T_{C}$ =
4.37 and 2$\Delta_{2}/k_{B}T_{C}$ = 2.46, respectively for the two bands.
These values are consistent with the previous reports[32, 33]. Our magnetic
field-dependent data support type-II behavior at sub-Kelvin temperature and
show the survival of a finite local spectral gap far above the bulk critical
field of AuBe. Our electronic band structure calculations, which are
consistent with the previous reports[22, 25], support phonon-mediated BCS type
superconductivity in this compound. Based on our Fermi surface analysis, we
propose that the electron pocket at the $\Gamma$-point leads to the larger
superconducting gap while the hole pocket at the $M$-point causes the smaller
one.

R.P.S. would like to acknowledge financial support from the Science and
Engineering Research Board (SERB)-Core Research Grant (grant No.
CRG/2019/001028). G.S. acknowledges financial support from the Swarnajayanti
fellowship awarded by the Department of Science and Technology (DST), Govt. of
India (grant No. DST/SJF/PSA-01/2015-16).

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Figure 1:  Eight representative tunneling spectra probed at different points
on the sample surface, all recorded at $T$ $\sim$ 310 mK. The spectra (red
circles) represented in (a)-(d) have the dominant contribution from the larger
gap ($\Delta_{1}$ $\simeq$ 320$\pm$10 $\mu$eV), and those represented in
(e)-(h) have the same from the smaller gap ($\Delta_{2}$ $\simeq$ 180$\pm$10
$\mu$eV). The verticle lines are guide to the eye and correspond to the
biasing of $\pm$320 $\mu$V (red) and $\pm$180 $\mu$V (blue), respectively. For
visual clarity, best fittings within a single-gap $s$-wave model (red lines
matching the upper part only, and the blue lines matching the lower part only)
and the two-gap $s$-wave model (green lines matching the whole spectrum) are
presented in (d) and (h) only. Figure 2: (a) The temperature dependence of a
typical tunneling conductance spectra with the theoretical fits in the absence
of any magnetic field. At the lowest $T$, $\Delta_{1}$=310 $\mu$eV,
$\Delta_{2}$=180 $\mu$eV, $\Gamma_{1}$=50 $\mu$eV, $\Gamma_{2}$=15 $\mu$eV for
the spectrum. The relative contributions $\alpha_{1}$=0.8 and $\alpha_{2}$=0.2
remain constant throughout the $T$ range. (b) Evolution of the two gaps
($\Delta_{1}$ and $\Delta_{2}$) with $T$, extracted from the plot (a) along
with two-gaps BCS fits. In the inset: the $T$ evolution of the two broadening
parameters ($\Gamma_{1}$ and $\Gamma_{2}$) associated with the two gaps
extracted similarly. The errors of the fitting parameters are shown in the
respective plots. Figure 3: (a) Magnetic field dependence of tunneling
conductance spectra recorded and kept throughout at $T$ = 310 mK along with
corresponding theoretical fits. (b) Evolution of $\Delta_{1}$ and $\Delta_{2}$
with the magnetic field extracted from the plot (a). $Inset$: the field
evolution of the two broadening parameters ($\Gamma_{1}$ and $\Gamma_{2}$)
associated with the two gaps. The fitting errors of the parameters are also
shown in the respective plots. The same spectrum is used for $T$ and $H$
dependence keeping the position of the tip unchanged on the sample surface.
Figure 4: (a) The Fermi surface of AuBe with the Fermi velocities at different
pockets represented with color gradients. (b) Normalized zero-bias DOS
corresponding to two different bands as a function of the reduced magnetic
field $h$ = $H$/$H_{c(l)}$. $H_{c(l)}$ = 17 kG is the spectroscopically
measured local critical field. The normalization is done by dividing the
$[dI/dV]_{V=0}$ meV with $[dI/dV]_{V=1.5}$ meV. $Inset$: An STM topographic
image of the sample surface. (c) The band dispersion along the high symmetry
directions, where band split is visible due to moderate spin-orbit coupling.
For visual clarity, a slight doubling of the surfaces due to this band
splitting is not presented in (a). (d) total and orbital-resolved DOS
considering the spin-orbit coupling. $Inset$: the high symmetry paths joining
the high symmetry points within the first Brillouin zone of simple-cubic
shape.