# Time-reversal-symmetry Breaking in the Superconducting State of ScS

Arushi  Department of Physics, Indian Institute of Science Education and
Research Bhopal, Bhopal, 462066, India  R. K. Kushwaha  Department of Physics,
Indian Institute of Science Education and Research Bhopal, Bhopal, 462066,
India  D. Singh  ISIS Facility, STFC Rutherford Appleton Laboratory, Didcot
OX11 0QX, United Kingdom  A. D. Hillier  ISIS Facility, STFC Rutherford
Appleton Laboratory, Didcot OX11 0QX, United Kingdom  M. S. Scheurer
Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck,
Austria  R. P. Singh<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Science Education and Research Bhopal, Bhopal, 462066, India

###### Abstract

We have studied the electronic properties of ScS, a transition-metal
monochalcogenide with rocksalt crystal structure, using magnetization,
specific heat, transport, and muon spin rotation/relaxation ($\mu$SR)
measurements. All measurements confirm the bulk superconducting in ScS with a
transition temperature of $T_{C}$ = 5.1(5) K. Specific heat together with
transverse-field $\mu$SR measurements indicate a full gap, while our zero-
field $\mu$SR study reveals the presence of spontaneous static or quasi-static
magnetic fields emerging when entering the superconducting state. We discuss
various possible microscopic origins of the observed time-reversal-symmetry
breaking. As none of them can be readily reconciled with a conventional
pairing mechanism, this introduces ScS as a novel candidate material for
unconventional superconductivity.

The study of unconventional superconductors Sigrist , which go beyond the BCS
theory, is a crucial pillar of modern condensed-matter research and involves a
broad range of material classes, ranging from heavy fermion systems HFReview ,
high TC superconductors HT1 ; HT2 , to iron-based systems IB1 ; IB2 , and more
recently moiré superlattices TBG , just to name a few. This research is driven
by the potential of these phases for applications and by fundamental
scientific questions, such as understanding their pairing mechanism,
identifying unifying physical similarities across chemically rather different
sets of materials, and finding ways to probe their microscopic physics.

However, even the identification of unconventional pairing is a challenging
endeavor: while the presence of nodes in the gap function, which can be
protected by symmetry in an unconventional state, is a good indication for
unconventional pairing, there are also more subtle unconventional pairing
states with a full and approximately isotropic gap Sigrist . In this case,
phase sensitive techniques are required, with one example given by studying
the disorder sensitivity of the pairing state Disorder1 ; Disorder2 ;
Disorder3 ; Disorder4 . Another phase-sensitive identification is the
observation of spontaneous internal magnetic fields at the superconducting
transition, indicating that the superconducting order parameter breaks time-
reversal symmetry TRS1 ; TRS2 ; TRS3 ; HF1 . Time-reversal-symmetry-breaking
superconductivity is particularly interesting since it is rare in nature, the
underlying pairing mechanism must involve more than the conventional electron-
phonon coupling SheurerGenRelation , and due to its potential for
technological applications, e.g., for the stabilization of topological edge
modes topo0 , and possibly also for the realization DiodeThSgt ; DiodeTh  of
zero-field superconducting diodes DiodeExp .

Binary transition metal arsenides (TMA) where TM represents a transition metal
and A can be any element from the carbon, pnictogen or chalcogen group, have
been widely studied as they provide an exciting family of candidate materials
where a range of exotic features has been observed over the years. For
instance, NbC, TaC, MoC, VC, and CrC NbC_TaC ; MoC ; VC_CrC  exhibit
superconductivity together with non-trivial topological band structure. These
compounds crystallize in a centrosymmetric cubic structure known as rock salt
structure. Motivated by the interest in and exciting properties of these
materials, we study ScS, which is isostructural to the above-mentioned
compounds.

Figure 1:  a) Powder XRD of ScS obtained at room temperature, shown by open
red circles. The solid blue line is the Rietveld refinement whereas the black
bars show Bragg reflection peaks. Inset: Crystal structure of ScS. b)
Temperature dependence of the magnetic moment shows TC at 5.1 K for ScS. The
data were collected in an applied field of 1 mT via ZFCW and FCC protocols. c)
Normalized specific heat, $C_{\text{el}}/\gamma_{n}T$ is fitted using single
gap $s$-wave model represented by the solid blue line. Inset: Temperature
dependence of the total specific heat in zero-field exhibiting TC at 4.8(1) K.

In this work, we report the macroscopic and microscopic study of
superconducting properties in ScS by means of magnetization, resistivity,
specific heat, and muon spin rotation and relaxation ($\mu$SR) measurements.
Superconductivity in ScS has been known for a long time SCScS1 ; SCScS2 ;
however, a detailed study of the superconducting properties has not yet been
performed. All of our measurements confirm bulk superconductivity with a
transition temperature of 5.1(5) K. Transverse-field (TF) $\mu$SR indicates a
nodeless, approximately isotropic superconducting gap structure together with
a slightly enhanced gap to critical temperature ratio compared to weak-
coupling BCS theory. Zero-field (ZF) $\mu$SR measurements reveal time-reversal
symmetry breaking on entering the superconducting state which makes ScS the
first member of the rocksalt family to exhibit this exotic feature.

Thermodynamics and transport—The polycrystalline sample of ScS was prepared by
arc melting both the constituent elements on a water cooled cooper hearth
under an argon gas atmosphere. To get structural information, powder X-ray
diffraction (XRD) on well grounded powder was performed at 300 K using a
PANalytical diffractometer equipped with CuKα radiation ($\lambda$ = 1.5406
Å). ScS adopts a cubic (NaCl defect) crystal structure, see inset of Fig.
1(a), with space group Fm–3m (No. 225), which was determined by the Rietveld
refinement of room temperature powder XRD data shown in the main panel of Fig.
1(a). The lattice constant is a = 5.172(3) Å. To perform magnetization,
electrical resistivity and specific heat measurements, SQUID (MPMS 3, Quantum
Design) and PPMS were used.

Magnetization $M$ measurements collected in an applied field of
$1\,\textrm{mT}$ via zero-field cooled warming (ZFCW) and field cooled cooling
(FCC) modes confirmed the bulk nature of superconductivity in ScS. As can be
seen in Fig. 1(b), it exhibits a diamagnetic signal at a superconducting
transition temperature of TC,onset = 5.10(5) K, where the electrical
resistivity data also shows a zero drop in resistivity [inset of Fig. 1(b)].
The difference between the diamagnetic signal in FCC and ZFCW indicates the
type II nature of superconductivity in ScS. We extract a Meissner
superconducting volume fraction close to 100 % from magnetization measurement.
Using $M$ $vs$ field $H$ ($M$ $vs$ $T$) curves at different temperatures
(fields) and employing the Ginzburg-Landau relations provided in SI SI  sec.
c, we obtained the lower and upper critical field as HC1(0) = 21.0(2) mT and
HC2(0) = 0.44(1) T. Two important length scales, the penetration depth
$\lambda_{GL}$(0) and coherence length $\xi_{GL}$(0), are found to be 1077(6)
Å and 274(3) Å, respectively. The Ginzburg-Landau parameter $\kappa_{GL}$ =
$\lambda_{GL}$(0)/$\xi_{GL}$(0) = 4(1) indicates type II superconductivity in
ScS. Specific heat measurements at zero-field confirmed bulk superconductivity
by exhibiting a jump at TC = 4.8(1) K, which is shown in the inset of Fig.
1(c). From the total specific heat, the electronic specific heat,
C${}_{\text{el}}$, can be calculated by subtracting the phononic contribution
(see SI SI  sec. d); its temperature dependence is shown in the main panel of
Fig. 1(c). The dimensionless value of electronic specific heat jump at TC,
$\frac{\Delta C_{\text{el}}}{\gamma_{n}T_{C}}$ = 1.13, is lower than the weak-
coupling BCS result ($1.43$). The temperature dependence of the specific heat
below $T_{C}$ follows more closely a nodeless, isotropic superconducting gap
model, see fit in Fig. 1(c) yielding $\Delta$(0)/kBTC = 1.65, than a nodal
$p$-wave or $d$-wave model SI . To estimate the underlying coupling strength
$\lambda_{\text{M}}$ of superconductivity, we employed the McMillan model
McMillanMain  and find $\lambda_{\text{M}}$ = 0.61(5) SI , which indicates
moderately coupled pairing in ScS. Also taking into account the measured
residual resistance, we extract a ratio of BCS coherence length and mean free
path of $\xi_{0}/l_{e}$ = 13.13, signalling dirty limit superconductivity.
Details regarding all the calculated parameters employing magnetization,
electrical resistivity, and specific heat measurements, as well as the fitting
relations, are presented in the SI SI .

Figure 2:  a) Transverse-field time domain spectra collected at 6.5 K and 0.3
K in an applied magnetic field of 40 mT. The solid lines are fits using Eq.
(1). b) Probability distribution of magnetic field for $T<T_{C}$ and
$T>T_{C}$. c) Temperature dependence of $\sigma_{sc}$ (squares), where solid
lines represent fitting using Eq. (2) which employs $s$, $p$, and $d$-wave
models.

$\mu$SR measurements—Muon spin rotation/relaxation experiments were performed
at the ISIS Neutron and Muon facility at the Rutherford Appleton Laboratory,
United Kingdom, using a MuSR spectrometer with 64 detectors in both transverse
and longitudinal directions. A full description of the muon technique is
provided in Ref. Muon, . TF-$\mu$SR measurements were carried out in FCC mode
where the sample was cooled below the transition temperature (to 0.3 K) in the
presence of an external magnetic field. The applied magnetic field was well
above the lower critical field (HC1(0) = 21.0(2) mT) and far below the upper
critical critical field which stabilizes the flux line-lattice in the mixed
superconducting state. Fig. 2(a) shows the asymmetry spectra both above and
below the transition temperature. The spectra at 0.3 K exhibit a faster
relaxation rate than those at 6.5 K, which is due to the inhomogeneous
magnetic field distribution from the flux line lattice. The field distribution
in the vortex state at 0.3 K and 6.5 K using maximum entropy algorithm
(MaxEnt) is shown in Fig. 2(b). At 0.3 K ($T<T_{C}$), there are two peaks
present where one peak corresponds to the applied field sensed by the muons
stopping in the sample holder and the other one represents the field
distribution due to the flux lattice formation. At T = 6.5 K, only one peak is
present at the applied field since ScS is in the normal state. The time domain
spectra were best modelled by a sinusoidal oscillating function with a
Gaussian relaxation plus a sinusoidal oscillation term for muons hitting the
sample holder and a flat background term TF1 ; TF2 :

$\begin{split}A(t)=A_{1}\exp\left(-\frac{1}{2}\sigma^{2}t^{2}\right)\cos(\gamma_{\mu}B_{1}t+\phi)\\\
+A_{bg}\cos(\gamma_{\mu}B_{bg}t+\phi)+A_{0},\end{split}$ (1)

where $A_{1}$, $B_{1}$, and $\sigma$ are the initial asymmetry, internal
field, and the Gaussian muon spin relaxation rate belonging to the sample.
$\gamma_{\mu}/2\pi$ = 135.5 MHz/T is the muon gyromagnetic ratio and A0 is the
flat background. $A_{bg}$ and $B_{bg}$ are the asymmetry and field
contributions coming from the background when muons hit the sample holder. The
relaxation rate corresponding to the superconducting contribution,
$\sigma_{\text{sc}}$, can be calculated after subtracting the nuclear magnetic
dipolar contribution, $\sigma_{\text{ndip}}$, using the following expression:
$\sigma_{\mathrm{sc}}=\sqrt{\sigma^{2}-\sigma_{\mathrm{ndip}}^{2}}$, where
$\sigma_{\text{ndip}}$ is assumed to be constant over the entire temperature
range; the resulting temperature dependence of $\sigma_{\mathrm{sc}}$ is shown
in Fig. 2(c). Since $\sigma_{\mathrm{sc}}$ is related to the magnetic
penetration depth as $\sigma_{sc}\propto\lambda^{-2}$, it contains information
about the form of the superconducting gap structure. Assuming a single,
spherical Fermi surface, it holds TF_sig1 ; TF_sig2 ; TF_sig3

$\frac{\sigma_{sc}(T)}{\sigma_{sc}(0)}=1+2\expectationvalue{\int_{{|\Delta(T,\hat{\boldsymbol{k}})|}}^{\infty}\frac{\partial
f}{\partial
E}\frac{E\mathrm{d}E}{\sqrt{E^{2}-\Delta^{2}(T,\hat{\boldsymbol{k}})}}}.\\\ $
(2)

Here f = [exp(E/$k_{B}T$)+1]-1 is the Fermi-Dirac function and
$\expectationvalue{.}$ represents the average over the Fermi surface.
$\Delta(T,\hat{\boldsymbol{k}})$ = $\Delta_{0}(T)g_{\hat{\boldsymbol{k}}}$ is
the temperature ($T$) and the directional ($\hat{\boldsymbol{k}}$) dependent
superconducting gap. Using spherical coordinates with angles $\phi$ and
$\theta$, we will consider the cases $g_{\hat{\boldsymbol{k}}}=1$,
$|\sin(\theta)|$, and $|\cos(2\phi)|$ for a fully gapped $s$-wave, $p$-wave
with nodal points, and $d$-wave state with nodal lines, respectively. The
temperature dependence of the gap function is approximated by
$\Delta_{0}(T)\simeq\Delta_{0}(0)\tanh\\{1.82(1.018({t^{-1}}-1))^{0.51}\\}$
where $\Delta_{0}(0)$ is the magnitude of the superconducting gap at zero
temperature. As can be seen in Fig. 2(c), the data is best captured by the
fully gapped $s$-wave model, providing the value of the superconducting gap
$\Delta$(0)/kBTC = 2.0 which is slightly greater than the gap value estimated
from the specific heat (1.43) as well as the standard BCS gap value (1.76),
but not inconsistent with the moderately weak coupling constant $\lambda\simeq
0.6$ extracted above GapRatio . A small discrepancy between the gap magnitudes
obtained from the specific heat and $\mu$SR studies may be due to the lack of
specific heat data at low temperatures. To fully understand the gap nature in
ScS, further work will be needed such as a detailed study on the Fermi surface
of ScS.

To search for the possible magnetism (static or fluctuating) in ScS, we have
performed ZF $\mu$SR measurements as this technique is extremely sensitive to
tiny magnetic fields associated with TRS-breaking phases; these measurements
were performed in the presence of an active compensation system in order to
cancel the stray magnetic field within the range of $0.01\,\textrm{G}$. The
time domain spectra were taken above (8.0 K) and below (0.3 K) the transition
temperature TC as shown in Fig. 3(a). There is a significant difference in the
relaxation rate observed across TC, hinting towards the spontaneous emergence
of magnetic fields in the superconducting state. For non-magnetic samples, the
depolarization can be best described with the function given below:

$A(t)=A_{0}G_{\mathrm{KT}}(t)\mathrm{exp}(-\Lambda t)+A_{1},$ (3)

where $\Lambda$ is the Lorentzian relaxation component, A1 is the flat
background, A0 is asymmetry signal coming from the sample; furthermore, GKT is
the static Kubo-Toyabe function provided as ZF_KT

$G_{\mathrm{KT}}(t)=\frac{1}{3}+\frac{2}{3}(1-\Delta^{2}t^{2})\mathrm{exp}\left(\frac{-\Delta^{2}t^{2}}{2}\right),$
(4)

with $\Delta$ representing the Gaussian muon spin depolarization rate which
accounts for the randomly oriented, static nuclear moments experienced at the
muon site. The rise in the extracted $\Lambda(T)$, see Fig. 3(b), in the
superconducting state confirms the presence of spontaneous magnetic fields. To
exclude the possibility of an impurity induced relaxation, we have performed
an additional longitudinal measurement at 0.3 K. A magnetic field of 30 mT was
sufficient to decouple the muon spins from the internal magnetic field [Fig.
3(a)]. It suggests the presence of a static or quasi-static magnetic field.

Figure 3:  a) Asymmetry spectra recorded at two different temperatures, 0.3 K
and 8.0 K, in zero magnetic field (small open circles); data at a small
longitudinal field (30 mT) are shown by large filled circles. The solid lines
are the corresponding fits to the data using Eq. (3). b) Shows the temperature
dependence of the relaxation rate, $\Lambda$, featuring an increase below the
superconducting transition temperature $T_{C}$.

Discussion—Taken together, our measurements of specific heat and penetration
depth point towards a fully developed nodeless superconducting gap, while our
zero-field $\mu$SR data indicate the emergence of weak magnetic moments below
a temperature very close to the superconducting $T_{C}$. To explore the
implications for the superconducting state in ScS, we will next discuss and
critically evaluating four possible microscopic origins, labeled as scenario
(i-iv) below, of this phenomenology.

In scenario (i), we assume that the superconducting phase is reached by a
single phase transition, which is natural as there are no indications of
multiple superconducting transitions. The order parameter must then transform
under one of the (10) irreducible representations (IRs) of the normal-state
point group $O_{h}$. We further assume that the observed magnetic moments
result from the time-reversal-symmetry breaking of the superconducting state
itself. As is well-known, this is only possible if it transforms under a two-
or higher-dimensional IR—in our case, one of the six IRs $E_{g}$, $T_{1g}$,
$T_{2g}$, $E_{u}$, $T_{1u}$, $T_{2u}$, leading to a total of $10$ possible
time-reversal-symmetry-breaking candidate states Sigrist . All of them are
required by symmetry to have at least one nodal direction (in some cases even
nodal planes) around the $\Gamma$ point. However, first-principle calculations
FirstPrinciple  predict two Fermi surfaces enclosing the $\Gamma$ point
completely, such that any of these superconductors is expected to have nodes.
While this seems to be at odds with our specific heat and penetration depth
data, it is possible that these nodes are not resolved in our measurements. As
such, we believe that specific heat measurements at lower temperature or more
accurate penetration depth studies, e.g., using the tunnel diode resonator
technique TunnelDiodeRes , might help decide whether this scenario is realized
in ScS or not.

Another possibility, scenario (ii), to reconcile broken time-reversal symmetry
and a full gap is based on having two consecutive superconducting transitions
that are so close that they cannot be resolved. When the dominant repulsive
Cooper-channel interactions are between the three symmetry-unrelated Fermi
sheets of the system, a natural compromise in this ‘frustrated’ situation
might be to have non-trivial complex phases of the order parameter between
these sheets and a fully gapped $s+i\,s$ state might be realized (see, e.g.,
the toy model discussion in LaNi  or the mechanism for time-reversal-symmetry
breaking superconductivity in the iron-based superconductors Andrey ; BaKFeAs
); such a state will give rise to magnetic moments spisFields  that can be
detected by the muons. Note, however, that by virtue of transforming under the
trivial (and thus one-dimensional) IR of $O_{h}$, such a state can only be
reached by two, possibly very close, phase transitions. We hope that future
high-quality crystals will exhibit a sharper phase transition signature in the
specific heat, possibly allowing to confirm or rule out this scenario.

One additional complication for both of these scenarios is that we have
estimated the coherence length $\xi_{0}$ to be larger than the mean-free path
$l_{e}$, indicating a significant amount of disorder in the superconductor.
Therefore, common wisdom BW ; GolubovMazin  would imply that any of the
aforementioned unconventional superconducting states should be completely
suppressed by impurity scattering. However, more recent theory DisorderSOCFu ;
OurDisorderSOC ; BrydonScattering ; PdTeScattering ; Jonathan  has reveal that
spin-orbit coupling can protect unconventional pairing states, which is also
supported by experiment Ando2012 ; Ando2014 ; Welp . Moreover, disorder can
even be the driving force inducing a time-reversal-symmetry-breaking
superconductor, which leads us to scenario (iii): as demonstrated in recent
theoretical works on $d$-wave superconductors Brian ; DisorderCuprates , when
two superconducting pairing channels are in close competition, strong disorder
can locally induce complex admixtures of these orders leading to local
currents, even if the clean sample was in a time-reversal-symmetric state.
Finally, as pointed out in SrPtAs , in a granular sample like the one studied
here, a single sign-changing superconducting order parameter can give rise to
flux trapping in voids formed by three or more crystallites, defining scenario
(iv).

Importantly, all of these scenarios, (i-iv), require effectively repulsive
Cooper-channel interactions, at least for parts of the Fermi surface, and thus
necessitate BrydonRelation ; SheurerGenRelation  an unconventional pairing
mechanism in the sense that the pairing state cannot be understood in terms of
electron-phonon coupling alone. The only conceivable picture to explain our
observations with a conventional pairing mechanism requires assuming that
there are local magnetic moments that are strongly screened in the normal
state. At the onset of superconductivity, this screening might be reduced and
could lead to the enhancement of the relaxation rate in Fig. 3(b) below
$T_{C}$; this was recently proposed TaS2  for $4Hb$-TaS2 where signs of Kondo
screening have been observed KondoScreening . However, for ScS, we did not
find any signs of screening of magnetic moments in the metallic state,
rendering this scenario improbable.

Conclusion and outlook—We have presented transport, magnetization, specific
heat, and $\mu$SR experiments in the superconducting and normal state of ScS,
which crystallizes in rocksalt (NaCl) structure. All measurements confirmed
the bulk nature of superconductivity, with a transition temperature TC = 4.6
K. We extracted various superconducting and normal-state parameters of ScS.
Specific heat data and the temperature-dependence of the penetration depth
following from TF $\mu$SR measurements are most naturally explained by a fully
established superconducting gap. Surprisingly, our zero-field $\mu$SR data
reveal time-reversal-symmetry-breaking moments at the onset of
superconductivity. We have discussed several possible microscopic origins of
these moments, which suggest that the underlying pairing glue cannot arise
solely from the electron-phonon coupling, but rather requires repulsive
components. As such, our results establish ScS as a system exhibiting complex
superconducting properties that deserve further investigation. In particular,
$\mu$SR measurements in high-quality single crystals, specific heat
measurements at low temperatures, as well as complementary penetration depth
measurements TunnelDiodeRes  and controlled disorder studies should be able to
elucidate the microscopics of superconductivity in ScS.

R. P. S. acknowledge Science and Engineering Research Board, Government of
India for the Core Research Grant CRG/2019/001028. Department of Science and
Technology, India (Grant No. SR/NM/Z-07/2015) for the financial support and
Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR) for managing
the project. Arushi acknowledges the funding agency, University Grant
Commission (UGC) of Government of India for providing SRF fellowship. We thank
ISIS, STFC, UK for the beamtime to conduct the $\mu$SR experiments
[RB2068032].

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Supplementary Information to "Time-reversal-symmetry Breaking in the
Superconducting State of ScS"

##  a Synthesis and Structural Characterization

The starting materials for the preparation of ScS were scandium ingot, and
sulphur powder which were weighed in a ratio of 1:1.2 and melted together on a
water-cooled copper hearth under an argon gas atmosphere. The as-cast ingots
were flipped several times and remelted in order to achieve homogeneity.
Rietveld refinement of powder XRD confirmed the crystal structure as cubic and
single phase nature within the detection limit of the technique. Other
parameters obtained from the refinement, such as atomic positions, cell
volume, are summarized in Table S1.

Table S1: Structure parameters of ScS obtained from the Rietveld refinement of
XRD

Structure | Cubic
---|---
Space group | Fm-3m
Lattice parameters |
a (Å) | 5.172(3)
V${}_{\text{Cell}}$ (Å3) | 138.34(3)

Atom | Wyckoff position | x | y | z
---|---|---|---|---
Sc | 4b | 0.5 | 0.5 | 0.5
S | 4a | 0 | 0 | 0

##  b Electrical Resistivity

The temperature dependence of electrical resistivity for 1.9 K $\leq$ T $\leq$
300 K in zero applied field is shown in Fig. S1. The residual resistivity
ratio (RRR = $\rho$(300K)/$\rho$(10K)) is found to be 3.2 which indicates the
poor metallic nature and presence of disorder in the system. The normal-state
resistivity was analyzed in the framework of the Bloch-Grüneisen (BG) model;
more specifically, we write

$\rho(T)=\rho_{0}+\rho_{\text{BG}}(T)$ (S1a) where $\rho_{0}$ is the
temperature-independent residual resistivity, resulting from scattering due to
the presence of defects, and $\rho_{\text{BG}}$ is the BG resistivity, given
by BG1
$\rho_{\text{BG}}(T)=C\left(\frac{T}{\Theta_{D}}\right)^{n}\int_{0}^{\Theta_{D}/T}\frac{x^{n}}{(e^{x}-1)(1-e^{-x})}\mathrm{d}x,$
(S1b)

which results from the scattering of electrons and phonons. In Eq. (S1b),
$\Theta_{D}$ is the Debye temperature, $C$ is a material-dependent prefactor
and $n$ takes values from 2 to 5 depending upon the nature of electron
scattering BG2 . The orange solid line in Fig. S1 represents the fit to the
data and yields $\rho_{0}$ = 19.3(1) $\mu\Omega$-cm, $\Theta_{D}$ = 278(5) K,
$C$ = 80.7(1) $\mu\Omega$-cm, and $n$ = 3. In order to extract the carrier
density for ScS, we have measured the Hall resistivity ($\rho_{xy}$) as a
function of magnetic field, which is shown in the inset of Fig. S1.
$\rho_{xy}$ data was fit with a linear line that is shown by blue curve and
yields the Hall coefficient $R_{H}=1.35(9)\times 10^{-10}\ohm$ $m$ T-1. The
positive value of $R_{H}$ indicates that holes are the dominant carriers in
ScS. Using $n$ = $\frac{1}{R_{H}e}$, we find a carrier density of $n$ =
4.6(3)$\times 10^{28}m^{-3}$.

Figure S1:  Temperature dependence of resistivity in the applied zero field
over the temperature range 1.9 K $\leq$ T $\leq$ 300 K. The solid line shows
the fit to the BG model in Eq. (S1b). The lower right inset shows the enlarged
view of the $\rho$(T) data, with superconducting transition at TC,onset =
5.1(2) K, and the inset in the top left corner displays the magnetic field
dependence of the Hall resistivity.

##  c Critical Fields

Figure S2:  a) Estimated lower critical field, HC1, as a function of
temperature, where the solid line is a fit according to Eq. (S2). The inset
shows low-field magnetization curves at various temperatures (1.8 $\leq$ T
$\leq$ 5 K). b) Temperature dependence of upper critical field obtained from
specific heat, AC transport, and magnetization data. The solid lines represent
the fits using Eq. (S3). The inset shows the temperature dependence of the
resistivity at different applied fields.

To determine the lower critical field, HC1, and its temperature dependence, we
use the magnetization versus field curves at different temperatures, see inset
of Fig. S2(a), and define HC1 via the field strength at which $M$ deviates
from linearity ($M\propto H$). These data points were fitted with the
Ginzburg-Landau equation

$H_{C1}(T)=H_{C1}(0)\left[1-\left(\frac{T}{T_{C}}\right)^{2}\right],$ (S2)

which yields HC1(0) = 21.0(2) mT.

The upper critical field, $H_{C2}(T)$, was estimated from the field dependence
of three different physical quantities: i) the specific heat $C(T,H)$, ii) the
magnetization $M(T,H)$, and iii) $\rho(T,H)$; to this end, the transition
temperature for magnetization is defined as the temperature that corresponds
to the onset of superconductivity and the midpoint of transition temperature
in the case of specific heat and resistivity measurements. As shown in Fig.
S2(b), the resulting $H_{C2}(T)$ are roughly the same in the three approaches
and, most importantly, all show approximately linear behavior close to $T_{C}$
and can be fit well with

$H_{C2}(T)=H_{C2}(0)\left[\frac{(1-t^{2})}{(1+t^{2})}\right],\quad
t=\frac{T}{T_{C}}.$ (S3)

This provides HC2(0) = 0.37(1) T, 0.44(1) T, and 0.49(1) T for specific heat,
magnetization, and resistivity, respectively.

There are two different mechanisms by which a magnetic field interacts with
the conduction electrons in a nonmagnetic superconductor and destroys
superconductivity. One is orbital pair breaking where the magnetic field
couples to the orbital motion of the electron, and the other is the Pauli
paramagnetic limiting effect, in which the interaction of the magnetic field
is with the electronic spin. The orbital limit for an upper critical field is
given by the Werthamer-Helfand-Hohenberg (WHH) expression WHH1 ; WHH2

$H_{C2}^{orbital}(0)=-\alpha
T_{C}\left.\frac{dH_{C2}(T)}{dT}\right|_{T=T_{C}}$ (S4)

The initial slope $\frac{-dH_{C2}(T)}{dT}$ at $T=T_{C}$ is estimated to be
0.07(1) T/K. Considering $\alpha$ = 0.693 for dirty limit superconductors,
gives the orbital limiting upper critical field $H_{C2}^{orbital}(0)$ = 0.27
T. The value obtained is less than the value of the critical field at
$T/T_{C}$ = 0.55 which suggests that single gap model cannot accurately model
this system and multigap effects should be considered.

Within BCS theory, the value of the Pauli paramagnetic limiting field is given
by $H_{C2}^{P}(0)$ = $c_{P}\,T_{C}$ where $c_{P}$ = 1.86 T/K Pauli1 ; Pauli2 .
It yields $H_{C2}^{P}(0)$ = 9.8 T, which is much higher than the obtained
value of HC2(0). The Maki parameter Maki , defined as
$\alpha_{M}=\sqrt{2}H_{C2}^{orb}(0)/H_{C2}^{P}(0)$, is a measure of the
relative strength of Pauli and orbital limits for the upper critical field; we
get $\alpha_{M}$ = 0.04, which implies that orbital effects crucially
determine HC2(0) while the Pauli limiting field is less relevant for ScS.

To determine the Ginzburg-Landau coherence length $\xi_{\text{GL}}$(0), the
vale of HC2(0) as extracted above from our measurements has been used in the
relation $H_{C2}(0)=\frac{\Phi_{0}}{2\pi\xi_{\text{GL}}^{2}}$ Coh_Leng , where
$\Phi_{0}$ = 2.07 $\times$ 10-15 Tm2 is the magnetic flux quantum. It yields
$\xi_{\text{GL}}$(0) = 274(3) Å. In turn, the respective values of
$\xi_{\text{GL}}$(0) = 274(3) Å and HC1(0) = 21.0(2) mT are used to calculate
the Ginzburg-Landau penetration depth, $\lambda_{\text{GL}}$, using pene

$H_{C1}(0)=\frac{\Phi_{0}}{4\pi\lambda_{GL}^{2}(0)}\left(\mathrm{ln}\frac{\lambda_{GL}(0)}{\xi_{GL}(0)}+0.12\right).$
(S5)

This yields $\lambda_{\text{GL}}$(0) = 1077(6) Å. The Ginzburg-Landau
parameter
$\kappa_{\text{GL}}=\frac{\lambda_{\text{GL}}(0)}{\xi_{\text{GL}}(0)}$, which
distinguishes between type I and type II superconductivity, is evaluated to be
4.0(1). Being significantly larger than $1/\sqrt{2}$, this value demonstrates
the type II nature of the superconductivity in ScS. The thermodynamic critical
field HC has been estimated by using the relation pene
$H_{C1}(0)H_{C2}(0)=H_{C}^{2}\mathrm{ln}\kappa_{\text{GL}}$, which yields HC =
82(2) mT.

##  d Specific Heat

The temperature dependence of the specific heat $C(T)$ in the temperature
range 1.9 K to 9.0 K in the zero applied field is shown in the inset of Fig.
S3. Above TC, the specific heat can be well described with the expression

$\frac{C}{T}=\gamma_{n}+\beta_{3}T^{2},$ (S6)

where $\gamma_{n}$ is the Sommerfeld coefficient, which represents the
electronic contribution, and $\beta_{3}$ describes the phononic contribution.
The fit to the data yields: $\gamma_{n}$ = 5.17(5) mJmol-1K-2, and $\beta_{3}$
= 98.6(9) $\mu$Jmol-1K-4. The Sommerfeld coefficient, $\gamma_{n}$, is used to
estimate the density of states at the Fermi level, $D_{C}(E_{\mathrm{F}})$,
via the free-electron relation,
$\gamma_{n}=\left(\frac{\pi^{2}k_{B}^{2}}{3}\right)D_{C}(E_{\mathrm{F}})$,
where kB = 1.38$\times$10-23 J K-1. This gives $D_{C}(E_{\mathrm{F}})$ =
2.19(2) states eV-1f.u-1. Debye temperature, $\Theta_{D}$ is estimated via
$\Theta_{D}=\left(\frac{12\pi^{4}RN}{5\beta_{3}}\right)^{\frac{1}{3}}$, where
R = 8.314 J mol-1K-1 is a gas constant, $N$ is the number of atoms per formula
unit, and $\beta_{3}$ = 98.6(9) $\mu$Jmol-1K-4 is the Debye constant extracted
from Eq. (S6). After employing the values, we obtained $\Theta_{D}$ = 340(3)
K. The electron-phonon coupling parameter, $\lambda_{\text{M}}$ can be
estimated using McMillan’s model McMillan  which relates $\Theta_{D}$ and TC
as given below:

$\lambda_{\text{M}}=\frac{1.04+\mu^{*}\mathrm{ln}(\Theta_{D}/1.45T_{C})}{(1-0.62\mu^{*})\mathrm{ln}(\Theta_{D}/1.45T_{C})-1.04}$
(S7)

where $\mu^{*}$ is the screened Coulomb repulsion and the typical value of
0.13 will be used here. From the values of $\Theta_{D}$ = 340(3) K and TC =
4.8(1) K, we obtain $\lambda_{\text{M}}$ = 0.61(5), which classifies ScS as a
moderately coupled superconductor.

Figure S3:  Solid lines show fits to normalized specific heat,
$C_{\text{el}}/\gamma_{n}T$ data using $p$ wave and $d$ wave models.

Having extracted the values of the Sommerfeld coefficient $\gamma_{n}$,
carrier density $n$ and residual resistance $\rho_{0}$, we can estimate the
ratio of mean free path and coherence length $\xi_{0}$/$l_{e}$. Based on the
Drude model, the mean free path is given by $l_{e}=v_{F}\tau$ where $\tau$ is
the scattering time and is defined as $\tau^{-1}$ = $ne^{2}\rho_{0}/m^{*}$.
The effective mass $m^{*}$ is given by the expression: $m^{*}$ = ($\hbar
k_{F})^{2}\gamma_{n}$/$\pi^{2}nk_{B}^{2}$ where $k_{F}$ = (3$\pi^{2}n$)1/3
when considering a spherical Fermi surface. We find $k_{F}$ = 1.11(2)
$\text{\AA}^{-1}$ and $m^{*}$ = 5.2(5) $m_{e}$ employing the values of $n$ and
$\gamma_{n}$ obtained above. The Fermi velocity $v_{F}=\hbar k_{F}/m^{*}$ is
found to be 2.5(2) $\times$ 105 m/s and the mean free path $l_{e}$ is 51(4) Å.
Within the BCS framework, the coherence length is given via the expression:
$\xi_{0}$ = 0.18$\hbar v_{F}$/$k_{B}T_{C}$ and by substituting the values of
$v_{F}$ and $T_{C}$, we obtained $\xi_{0}$ = 667(59) Å. All parameters are
listed in Table S2 together with all other parameters extracted from our
measurements. The ratio $\xi_{0}$/$l_{e}$ indicates the presence of
significant disorder in the superconducting phase (consistent with the rather
low RRR value discussed above).

Table S2: Superconducting and normal-state parameters of ScS Parameters | Units | ScS
---|---|---
$T_{C}$ | K | 5.1
$H_{C1}(0)$ | mT | 21.03
$H_{C2}(0)$ | T | 0.44
$H_{C2}^{P}(0)$ | T | 9.8
$H_{C2}^{Orb}(0)$ | T | 0.26
$\xi_{GL}$ | Å | 274
$\lambda_{GL}$ | Å | 1077
$k_{GL}$ |  | 4
$\gamma_{n}$ | mJ mol-1 K-2 | 5.16
$\Theta_{D}$ | K | 340
$\Delta C_{el}/\gamma_{n}T_{C}$ |  | 1.13
$\xi_{0}/l_{e}$ |  | 13.13
$\xi_{0}$ | Å | 667
$l_{e}$ | Å | 51
$v_{F}$ | 105m s-1 | 2.46
$n$ | 1028m-3 | 4.59
$m^{*}$/me |  | 5.21

From total specific heat, electronic specific heat, C${}_{\text{el}}$, can be
calculated subtracting the phononic contribution,
C${}_{\text{ph}}=\beta_{3}T^{3}$; its temperature dependence is shown in the
main panel of Fig. S3. The value of the electronic specific heat jump at the
superconducting transition is found to be $\frac{\Delta
C_{\text{el}}}{\gamma_{n}T_{C}}$ = 1.13, which is smaller than the BCS weak
coupling result ($1.43$). A reduced magnitude of the jump has been observed in
specific heat studies of several superconductors. These studies point toward
the contribution of impurities, inhomogeneity, or the existence of regions
which do not take part in superconductivity lowjump . In our sample, XRD
confirms the presence of a single phase, which disfavors this explanation.
Another possible origin of the reduced jump is a more complex momentum and
Fermi-surface dependent, gap structure distinct from the BCS model.

The electronic specific heat in the superconducting state is calculated from
the normalized entropy expression, which can be written as SH1 ; SH2 ; SH3

$\frac{S(t)}{\gamma_{n}T_{C}}=-\frac{6}{\pi^{2}}\int_{0}^{\infty}\mathrm{d}\epsilon\,\,[f\ln
f+(1-f)\ln(1-f)],$ (S8)

which depends only on the reduced temperature $t=T/T_{C}$. In Eq. (S8),
$f=[\exp(\textit{E}/k_{B}T)+1]^{-1}$ is the Fermi function and E($\epsilon$) =
$\sqrt{\epsilon^{2}+\Delta_{k}^{2}}$ where $\epsilon$ is the relative
energy(measured in units of $T_{C}$). $\Delta_{k}$ represents the temperature
and angle dependence similar to as described in the main text. The normalized
entropy S is related to the normalized electronic specific heat by

$\frac{C_{\text{el}}}{\gamma_{n}T_{C}}=t\frac{d(S/\gamma_{n}T_{C})}{dt}\\\ $
(S9)

We have analyzed the temperature dependence of the normalized specific heat
with the aforementioned models where the fitting is represented by the solid
lines shown in Fig. S3. As can be seen, the nodal $p$\- and $d$-wave ansätze
can not provide a fit in the measured temperature range. While specific heat
can in principle be used to distinguish between single- and multi-gap $s$-wave
behavior, the available temperature range does not allow to distinguish
reliably between different $s$-wave pairing scenarios.

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