# Nodeless time-reversal symmetry breaking in the centrosymmetric
superconductor Sc5Co4Si10 probed by muon-spin spectroscopy

A. Bhattacharyya<EMAIL_ADDRESS>Department of Physics,
Ramakrishna Mission Vivekananda Educational and Research Institute, Belur
Math, Howrah 711202, West Bengal, India  M. R. Lees  Department of Physics,
University of Warwick, Coventry CV4 7AL, United Kingdom  K. Panda  Department
of Physics, Ramakrishna Mission Vivekananda Educational and Research
Institute, Belur Math, Howrah 711202, West Bengal, India  P. P. Ferreira
<EMAIL_ADDRESS>Computational Materials Science Group (ComputEEL/MatSci),
Escola de Engenharia de Lorena, Universidade de São Paulo, DEMAR, Lorena,
Brazil  T. T. Dorini  Université de Lorraine, CNRS, IJL, Nancy, France  Emilie
Gaudry  Université de Lorraine, CNRS, IJL, Nancy, France  L. T. F. Eleno
Computational Materials Science Group (ComputEEL/MatSci), Escola de Engenharia
de Lorena, Universidade de São Paulo, DEMAR, Lorena, Brazil  V. K. Anand
Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Hahn-Meitner Platz
1, D-14109 Berlin, Germany  Department of Physics, University of Petroleum and
Energy Studies, Dehradun, Uttarakhand 248007, India  J. Sannigrahi  School of
Physical Sciences, Indian Institute of Technology Goa, Goa-403401, India  P.
K. Biswas  ISIS Neutron and Muon Source, Rutherford Appleton Laboratory,
Chilton, Didcot Oxon, OX11 0QX, United Kingdom  R. Tripathi  ISIS Neutron and
Muon Source, Rutherford Appleton Laboratory, Chilton, Didcot Oxon, OX11 0QX,
United Kingdom  Jawaharlal Nehru Centre for Advanced Scientific Research,
Jakkur, Bangalore 560064, India  D. T. Adroja<EMAIL_ADDRESS>ISIS Neutron and Muon Source, Rutherford Appleton Laboratory, Chilton, Didcot
Oxon, OX11 0QX, United Kingdom  Highly Correlated Matter Research Group,
Physics Department, University of Johannesburg, PO Box 524, Auckland Park
2006, South Africa

(August 28, 2024)

###### Abstract

We investigate the superconducting properties of Sc5Co4Si10 using low-
temperature resistivity, magnetization, heat capacity, and muon-spin rotation
and relaxation ($\mu$SR) measurements. We find that Sc5Co4Si10 exhibits type-
II superconductivity with a superconducting transition temperature
$T_{\mathrm{C}}=3.5(1)$ K. The temperature dependence of the superfluid
density obtained from transverse-field $\mu$SR spectra is best modeled using
an isotropic Bardeen-Cooper-Schrieffer type $s$-wave gap symmetry with
$2\Delta/k_{\mathrm{B}}T_{\mathrm{C}}=2.84(2)$. However, the zero-field muon-
spin relaxation asymmetry reveals the appearance of a spontaneous magnetic
field below $T_{\mathrm{C}}$, indicating that time-reversal symmetry (TRS) is
broken in the superconducting state. Although this behavior is commonly
associated with non-unitary or mixed singlet-triplet pairing, our group-
theoretical analysis of the Ginzburg-Landau free energy alongside density
functional theory calculations indicates that unconventional mechanisms are
pretty unlikely. Therefore, we have hypothesized that TRS breaking may occur
via a conventional electron-phonon process.

###### pacs:

71.20.Be, 75.10.Lp, 76.75.+i

##  I Introduction

Ternary rare earth-transition metal silicides and germanides have recently
been the subject of several studies because of their wide variety of
structures and phenomenologies [1]. The occurrence of superconductivity is
commonly found in these compounds, with at least 11 superconducting distinct
structural prototypes reported so far [2]. Some of them host $3d$ magnetic
species, such as Fe, Co, and Ni, concomitantly with a superconducting ground
state [3, 4, 5]. For example, two-gap superconductivity in Lu2Fe3Si5 has been
reported from low-temperature specific-heat, penetration depth, and muon
spectroscopy measurements [6, 7, 8]. The coexistence of charge density waves
with magnetic or superconducting order is also reported in $R_{5}T_{4}$Si10
($R=$ rare earth element; $T=$ Co, Ir, Rh, or Os) family [9, 10, 11, 12, 13,
14]. These observations highlight the potential to use this class of materials
to study different and novel quantum states.

The highest superconducting critical temperature $\left(T_{\mathrm{C}}\right)$
reported at ambient pressures is due to Y5Os4Si10 [15], with a
$T_{\mathrm{C}}$ of 9.1 K, followed by Sc5Ir4Si10 ($T_{\mathrm{C}}=8.6$ K),
and Sc5Rh4Si10 ($T_{\mathrm{C}}=8.4$ K). The occurrence of superconductivity
in Sc5Co4Si10 is particularly noteworthy. Sc and Si are not superconducting at
ambient pressure, while Co is a ferromagnetic material with no superconducting
state at all. Superconductivity in materials containing Co is relatively
unusual. Examples include the binary phase CoSi2 with a $T_{\mathrm{C}}=1.2$ K
[16] and Lu3Co4Ge13 single crystals with $T_{\mathrm{C}}$ below 1.4 K [17].
Superconductivity is also reported in layered materials such as
NaCoxO${}_{2}\cdot y$H2O at $\sim 5$ K [18] and, more recently, LaCoSi below 4
K [19]. Additionally, the quasi-one-dimensional carbide Sc3CoC4 [20] becomes
superconducting at approximately 4.5 K.

The relatively high $T_{\mathrm{C}}$ of Sc5Co4Si10 ($T_{\mathrm{C}}=4.9$ K)
indicates that the magnetic character of Co atoms is either suppressed or does
not interfere with the electron pairing mechanism [3]. Sc5Co4Si10 crystallizes
in a tetragonal structure within the centrosymmetric space group $P4/mbm$ (No.
127), with 38 atoms per unit cell [1, 3, 4, 5]. There are three nonequivalent
Sc sites in the structure, one of which forms a chain-like Sc–Si network along
the $c$ axis. One can see from the projection of the Sc5Co4Si10 structure
along the $c$ axis [Fig. 1(a)] that Co and Si atoms give rise to planar
networks of pentagons and hexagons stacked parallel to the basal plane and
connected along the $c$ axis via Co–Si–Co zigzag chains. The pentagon-hexagon
layers are interspersed with layers of Sc [21]. One interesting feature is
that Co–Co distance is of the order of 4 Å, far larger than in pure Co. This
is quite different from what is found in the Mo chalocogenides [22] and
rhodium borides [23], both well studied ternary systems in which the
transition metal atoms form clusters and play an important role in
superconductivity. Nakajima et al. [24] reported the low-temperature heat
capacity of Sc5Ir4Si10, showing that the data is best described by a two-gap
model. Nuclear magnetic resonance measurements on Sc5Co4Si10 suggest that the
density of states at the Fermi level is dominated by the Co-$d$ bands derived
from the pentagon-hexagon layers . In contrast, the $d$ bands of Sc layers,
which separate the pentagon–hexagon sheets, are nearly empty [25, 26].

Figure 1: (a) Unit cell of Sc5Co4Si10 viewed along the $c$ axis. The Co (blue)
and Si (green) atoms form pentagons and hexagons stacked parallel to the basal
plane interspersed with Sc (red) atoms. (b) Temperature dependence of the
magnetic susceptibility $\chi\left(T\right)$ for Sc5Co4Si10, collected in
zero-field-cooled (ZFC) and field-cooled (FC) modes in an applied field of 1
mT. The inset shows the resistivity versus temperature for Sc5Co4Si10 in zero
applied field. (c) Temperature dependence of the normalized resistivity for
Sc5Co4Si10 in different applied magnetic fields. (d) Temperature dependence of
the upper critical field $H_{\mathrm{C2}}$ of Sc5Co4Si10 determined using
resistivity measurements made as a function of either temperature or applied
magnetic field. $H_{\mathrm{C2}}(0)$ is estimated from the fit shown by a red
line using the GL expression
$H_{\mathrm{C2}}(T)=H_{\mathrm{C2}}(0)\left(1-t^{2}\right)/\left(1+t^{2}\right)$.
The inset shows the lower critical field $H_{\mathrm{C1}}$ for Sc5Co4Si10 as a
function of the reduced temperature $t=T/T{{}_{\mathrm{C}}}$ fitted using the
GL expression $H_{\mathrm{C1}}(T)=H_{\mathrm{C1}}(0)(1-t^{2}$).

Motivated by these results, we have conducted a systematic study of the
superconducting properties of Sc5Co4Si10 employing electrical resistivity,
magnetization, heat capacity, and muon-spin rotation and relaxation ($\mu$SR)
measurements. The thermal, transport, and magnetic measurements indicate a
bulk type-II superconducting ground state with $T_{\mathrm{C}}=3.5(1)$ K,
$\mu_{0}H_{\mathrm{C1}}=1.4(1)$ mT, and $\mu_{0}H_{\mathrm{C2}}=1.33(2)$ T.
The temperature dependence of the superfluid density estimated from
transverse-field (TF) $\mu$SR measurements shows the presence of a fully-
gapped superconducting order parameter. This finding is also supported by heat
capacity measurements at low temperatures. Surprisingly, zero-field (ZF)
$\mu$SR measurements reveal a finite, spontaneous internal magnetic field
within the superconducting state, providing clear evidence for time-reversal
symmetry breaking. A first analysis based on Ginzburg-Landau’s theory
alongside density functional theory suggests that both non-unitary triplet
pairing state and mixed singlet-triplet state are quite unlikely due to the
complex, closed topography of the Fermi surface. Therefore, we argue that
conventional mechanisms could carry TRS breaking due to the highly symmetric
multiband low-energy states. Sc5Co4Si10, therefore, joins the shortlist of
unconventional TRS breaking superconducting materials with fully-gapped
$s$-wave symmetry.

##  II Methods

###  II.1 Experimental Details

Polycrystalline Sc5Co4Si10 samples were prepared by arc melting stoichiometric
amounts of high purity elemental Sc (99.9%), Co (99.99%), and Si (99.9999%) in
an argon atmosphere with a Zr getter. The ingot was flipped and remelted
several times to improve the phase homogeneity. The sample was then annealed
in a dynamic vacuum of $10^{-6}$ torr for 18 days at a temperature of
$1050\,^{\circ}$C, followed by a quench to room temperature by switching off
the heater power. The homogeneous single phase nature of the sample with a
5:4:10 composition for Sc:Co:Si was confirmed using a ZEISS GeminiSEM 500,
which was used to perform energy-dispersive X-ray spectroscopy (EDX).
Electrical resistivity measurements were carried out in different applied
fields using a Quantum Design Physical Property Measurement System (QD-PPMS).
The $dc$-magnetization measurements were performed using a Quantum Design
Magnetic Property Measurement System SQUID magnetometer. Heat capacity down to
0.5 K was measured using a QD-PPMS with a 3He insert.

Muon spin rotation and relaxation ($\mu$SR) measurements were carried out
using the MuSR spectrometer at the ISIS Pulsed Neutron and Muon Source, United
Kingdom [27, 28]. The powdered sample was mounted on a high purity silver
plate (99.995%) using GE varnish, and then covered with silver foil. The
sample was cooled to temperatures as low as 400 mK using a He-3 system. 100%
spin-polarized positive muons ($\mu^{+}$) were implanted into the sample. Each
$\mu^{+}$ decays with a mean lifetime of $2.2\,\mu$s, releasing a positron.
The time dependence of the polarization of the implanted muons is given by
$P_{\mathrm{\mu}}(t)=G(t)P_{\mathrm{\mu}}(0)$, where $G(t)$ corresponds to the
$\mu^{+}$ spin auto-correlation function. The time-dependent asymmetry $A(t)$,
which is proportional to $P_{\mu}(t)$, is given by
$A(t)=\left[{N_{\mathrm{F}}(t)-\alpha
N_{\mathrm{B}}(t)}\right]/\left[{N_{\mathrm{F}}(t)+\alpha
N_{\mathrm{B}}(t)}\right]$, where $N_{\mathrm{B}}(t)$ and $N_{\mathrm{F}}(t)$
are the number of positrons counted in the backward and forward detectors,
respectively, and $\alpha$ is an instrumental calibration constant determined
in the normal state in a small (2 mT) transverse magnetic field. All the
$\mu$SR data were analyzed using the WiMDA data analysis program [29]. The
TF-$\mu$SR data were collected at different temperatures between 0.4 and 5 K
in a transverse-field of 40 mT ($>H_{\mathrm{C1}}(0)$, which is $\approx
1.4(1)$ mT). The ZF data were collected between 1.3 and 6.2 K, and active
compensation coils were used to reduce any stray magnetic field at the sample
position to below $\sim 0.1\,\mu$T.

###  II.2 Computational methods

Calculations were performed using density functional theory (DFT) as
implemented in the Vienna ab initio simulation package (VASP) [30, 31, 32].
Spin-polarized and spin-orbit coupling calculations were performed with the
projector-augmented wave (PAW) method using a plane-wave basis set of 500 eV
[33, 34] using the Perdew-Burke-Ernzerhof (PBE) functional [35]. The following
electrons were treated explicitly: 3$p^{6}$ 4$s^{2}$ 3$d^{3}$ (Sc), 3$d^{8}$
4$s^{1}$ (Co), and 3$s^{2}$ 3$p^{2}$ (Si). Electronic relaxations were carried
out until the self-consistent calculation loops reached an energy convergence
of less than $10^{-5}$ eV and structural relaxations were performed until
forces on all atoms were below 0.01 eV/Å. A k-point mesh of $8\times 8\times
24$ was used to sample the Brillouin zone (BZ).

Figure 2: (a) Temperature dependence of heat capacity $C_{\mathrm{P}}(T)$ for
$0.4\leqslant T\leqslant 5$ K measured in different applied magnetic fields.
The inset shows the fit of $C_{\mathrm{p}}$ in the normal state. (b)
Electronic contribution to the zero-field heat capacity, $C_{\mathrm{e}}$, as
a function of temperature. The solid line indicates the fit to
$C_{\mathrm{e}}$ with the isotropic Bardeen-Cooper-Schrieffer expression and
the normal state contribution ($\gamma_{ns}T$). The inset shows the field
dependence of $\Delta\gamma$. The linear trend follows the behavior expected
for conventional BCS-type superconductors. The blue dashed line represents the
extrapolation of normal state electronic heat capacity $\gamma_{n}(T)$.

##  III Results and Discussion

###  III.1 Physical characterization

The temperature dependence of the resistivity $\rho\left(T\right)$ of
Sc5Co4Si10 in zero applied field and in different applied fields are shown in
Fig. 1(c). There is a sharp superconducting transition at
$T_{\mathrm{C}}=3.5(1)$ K. The high temperature resistivity data exhibits
metallic behavior. The estimated residual resistivity ratio
$\rho(220K)/\rho(4K)$ = 2.04, this value is close to that seen in the
isostructural superconductor Lu5Ir4Si10 [36] The temperature dependence of the
magnetic susceptibility $\chi\left(T\right)$ of Sc5Co4Si10 in an applied
magnetic field of 1 mT is shown in Fig. 1(b). $\chi(T)$ reveals a clear
signature of superconductivity below the superconducting transition
temperature $T_{\mathrm{C}}=3.5(1)$ K as well, with a superconducting volume
close to 90% at 2.0 K, thus confirming the bulk nature of the
superconductivity.

To determine the lower critical field $H_{\mathrm{C1}}$, field-dependent
magnetization $M\left(H\right)$ curves for Sc5Co4Si10 were collected at
various temperatures up to $T_{\mathrm{C}}$ using a ZFC-protocol. For each
temperature, $H_{\mathrm{C1}}\left(T\right)$ was determined as the field where
$M\left(H\right)$ first deviates from linearity. $H_{\mathrm{C1}}$ as a
function of the reduced temperature $T/T_{\mathrm{C}}$ is shown in Fig. 1(d).
The temperature variation of $H_{\mathrm{C1}}$ is reasonably well described by
the Ginzburg-Landau (GL) expression
$H_{\mathrm{C1}}(T)=H_{\mathrm{C1}}(0)(1-t^{2}$), where $t=T/T_{\mathrm{C}}$,
yielding $\mu_{0}H{{}_{\mathrm{C1}}}(0)=1.4(1)$ mT.

The upper critical field $H_{\mathrm{C2}}$ for Sc5Co4Si10 was estimated using
the resistivity data collected at different temperatures and applied magnetic
fields. The temperature of the midpoint of the resistivity transition is taken
as the transition temperature $T_{\mathrm{C}}\left(H\right)$ in the field $H$.
Figure 1(d) shows $H_{\mathrm{C2}}(T)$, which follows a nearly linear behavior
in the temperature range close to $T_{\mathrm{C}}$ that can be fitted with the
phenomenological Ginzburg-Landau expression
$H_{\mathrm{C2}}(T)=H_{\mathrm{C2}}(0)\frac{1-t^{2}}{1+t^{2}}$. The fit gives
$\mu_{0}H_{C2}=1.33(2)$ T, well below the Pauli paramagnetic limit
$1.84T_{\mathrm{C}}\left[\mathrm{K}\right]=6.44$ T [37].

Heat capacity $C_{\mathrm{P}}$ as a function of temperature for $0.45\leqslant
T\leqslant 5$ K is shown in Fig. 2(a) in different applied magnetic fields.
The inset in Fig. 2(a) shows the temperature variation of the heat capacity in
zero applied magnetic fields. A clear signature of the superconducting
transition is observed below 3.5 K in $C_{\mathrm{P}}(T)$. Above
$T_{\mathrm{C}}$, in the normal state, $C_{\mathrm{P}}\left(T\right)$ was
found to be independent of the external magnetic field and can be described
using $C_{\mathrm{P}}(T)=\gamma T+\beta T^{3}$, where $\gamma$ is the
electronic Sommerfeld coefficient and $\beta T^{3}$ is the lattice (phonon)
contribution to the specific heat, yielding $\gamma=35.6(5)$ mJ mol-1K-2 and
$\beta=0.276(1)$ mJ mol-1K-4. Using the Debye model, the Debye temperature is
given by $\Theta_{\mathrm{D}}=(\frac{12\pi^{4}}{5\beta}nR)^{1/3}$, where
$R=8.314$ Jmol-1K-1 is the gas constant and $n=19$ is the number of atoms per
formula unit in Sc5Co4Si10. Using this relationship, $\Theta_{\mathrm{D}}$ is
estimated to be 511(3) K. Figure 2(b) shows the temperature dependence of the
electronic specific heat, $C_{\mathrm{e}}\left(T\right)$, obtained by
subtracting the phonon contribution from $C_{\mathrm{P}}\left(T\right)$.
$C_{\mathrm{e}}\left(T\right)$ can be used to investigate the superconducting
gap symmetry. Using the maximum entropy construction shown in Fig. 2(b) the
heat capacity jump at TC is $\Delta C_{e}$ = 96 mJ/mol K2, which yield $\Delta
C_{e}/\gamma T_{\mathrm{C}}\approx$ 0.90, which is smaller than the 1.43
expected for weak-coupling BCS superconductors [38]. We analyzed the
superconducting state electronic heat capacity within the framework of BCS
model [38]. The solid red curve in Fig. 2(b) corresponds to the fit according
to $C_{\rm e}=C_{\rm BCS}+\gamma_{\rm ns}T$, where $C_{\rm BCS}$ represents
the theoretical weak-coupling fully gapped isotropic s-wave BCS
superconductivity corresponding to $\gamma_{\rm s}=21$ mJ/mol K2, and
$\gamma_{\rm ns}T$ represents the contribution due to the nonsuperconduting
fraction of the sample. The nonsuperconducting contribution might have its
origin in defects in materials as well as due to the presence of small
inhomogeneous/impurity phase(s).

The inset of Fig. 2(b) presents the magnetic field dependence of the
Sommerfeld coefficient $\Delta\gamma=\gamma(H)-\gamma(0)$ from values of
$\gamma$ obtained by extrapolating $C_{\mathrm{e}}\left(T\right)$ to $T\sim$ 0
K. Interestingly, the linear trend follows the behavior expected for a
conventional BCS-type superconductor.

Figure 3: Time evolution of TF-$\mu$SR asymmetry spectra for Sc5Co4Si10
recorded at (a) 0.4 K (below $T_{\mathrm{C}}$) and (b) 4.5 K (above
$T_{\mathrm{C}}$). The solid red lines represent the fit to the data using Eq.
1. (c) and (d) display the probability of internal field distributions
obtained using the maximum entropy method, below and above $T_{\mathrm{C}}$,
respectively. Figure 4: (a) Temperature dependence of the superconducting
depolarization rate $\sigma_{\mathrm{sc}}(T)$ in the presence of an applied
magnetic field of 40 mT. The solid red and dotted blue lines were determined
using Eq. III.2 considering both $s$-wave and $d$-wave symmetries,
respectively. The inset shows the total muon spin depolarization rate
$\sigma_{\mathrm{TF}}$ as a function of temperature. (b) Temperature
dependence of the internal field. Figure 5: (a) Time evolution of ZF-$\mu$SR
asymmetry spectra at 1.3 K (black squares) and 6.2 K (green circles). The red
and blue lines are the least square that fit the data using Eq. 3. (b) Time
dependence of LF-$\mu$SR asymmetry spectra in different applied fields. The
solid lines are the least square that fit the data using Eq. 3. The line to H
= 0 is same as in Fig.(a) and fit lines to other spectra is with
$G_{\mathrm{KT}}$(t)=1 in Eq. 3. (c) Temperature dependence of the ZF-
relaxation rate, $\sigma_{\mathrm{ZF}}$. The solid red line is a guide to the
eye. (d) Temperature dependence of $\sigma_{\mathrm{KT}}$, which is almost
constant over the temperature range analyzed.

###  III.2 Superconducting gap structure

To investigate the superconducting gap structure in Sc5Co4Si10, we have
performed TF-$\mu$SR measurements. Figures 3(a) and 3(b) display the
TF-$\mu$SR asymmetry spectra recorded at temperatures below and above
$T_{\mathrm{C}}$, respectively, with an applied magnetic field of 40 mT, well
above $H_{\mathrm{C1}}(0)$. The corresponding field distributions determined
using the maximum entropy method are shown in Figs. 3(c) and 3(d). The
presence of a flux-line lattice in the superconducting state results in an
inhomogeneous field distribution within the sample, which in turn induces a
faster decay in the asymmetry spectra below $T_{\mathrm{C}}$. The time
evolution of the TF-$\mu$SR data at all temperatures above and below
$T_{\mathrm{C}}$ is best described by a sinusoidal oscillatory function damped
with a Gaussian relaxation arising from muons implanted in the sample and an
oscillatory background term from muons in the silver sample holder that do not
depolarize [39, 40, 41, 42]:

$G_{\mathrm{TF}}(t)=A_{1}\cos\left(\omega_{\mathrm{1}}t+\varphi\right)\exp\left(-\frac{\sigma_{\mathrm{TF}}^{2}t^{2}}{2}\right)+A_{\mathrm{2}}\cos\left(\omega_{2}t+\varphi\right).$
(1)

Here, $A_{1}$ is the initial asymmetry of muons in the sample, and $A_{2}$ is
the asymmetry of the background. $\omega_{\mathrm{1}}$ and
$\omega_{\mathrm{2}}$ are the muon precession frequencies within the sample,
and the sample holder, respectively, and $\varphi$ is an initial phase offset.
Finally, $\sigma_{\mathrm{TF}}$ is the total muon spin relaxation rate and
consists of two contributions. One is due to the inhomogeneous field variation
across the superconducting vortex lattice, $\sigma_{\mathrm{sc}}$. The other
is a regular state contribution, $\sigma_{\mathrm{n}}$ which is taken to be
temperature-independent over the entire temperature range studied and was
obtained from spectra measured above $T_{\mathrm{C}}$. Fitting reveals $A_{1}$
is 0.910(1) and $A_{2}$ is 0.090(1) of the total initial asymmetry, while
$\sigma_{\mathrm{n}}=0.293\leavevmode\nobreak\ \mu s^{-1}$. Using
$\sigma_{\mathrm{TF}}^{2}=\left(\sigma_{\mathrm{sc}}^{2}+\sigma_{\mathrm{n}}^{2}\right)$
we obtain the superconducting contribution $\sigma_{\mathrm{sc}}$.

The temperature variation of the penetration depth/superfluid density was
modeled using [43, 44, 45]

$\displaystyle\frac{\sigma_{\mathrm{sc}}(T)}{\sigma_{\mathrm{sc}}(0)}$
$\displaystyle=\frac{\lambda^{-2}\left(T,\Delta_{\mathrm{0,i}}\right)}{\lambda^{-2}\left(0,\Delta_{\mathrm{0,i}}\right)},$
$\displaystyle=1+\frac{1}{\pi}\int_{0}^{2\pi}\int_{\Delta\left(T\right)}^{\infty}\left(\frac{\delta
f}{\delta E}\right)\frac{EdEd\phi}{\sqrt{E^{2}-\Delta^{2}(T,\phi})},$ (2)

where $f=\left[1+\exp\left(E/k_{\mathrm{B}}T\right)\right]^{-1}$ is the Fermi
function and
$\Delta_{\mathrm{i}}(T,0)=\Delta_{\mathrm{0,i}}\delta(T/T_{\mathrm{C}})\mathrm{g}(\phi)$
is the value of the superconducting gap at 0 K. The temperature dependence of
the superconducting gap is approximated by the relation
$\delta(T/T_{\mathrm{C}})=\tanh\left[1.82[1.018\left(T_{\mathrm{C}}/T-1\right)\right]^{0.51}]$,
where $\mathrm{g}(\phi$) is the angular dependence of the superconducting gap
function. $\mathrm{g}(\phi$) is replaced by 1 for a $s$-wave gap or
$|\cos(2\phi)|$ for a $d$-wave gap with line nodes [46, 47]. We find that the
data are best modeled [see Fig. 4(a)] using a single isotropic $s$-wave gap of
0.43(1) meV, which yields a gap to $T_{\mathrm{C}}$ ratio
2$\Delta/k_{\mathrm{B}}T_{\mathrm{C}}=2.84(2)$, indicating a weak-coupling
regime. This value of $2\Delta/k_{B}T_{C}$ is significantly smaller than the
BCS expected value of 3.53 in weak coupling limit. Usually, a lower value of
$2\Delta/k_{B}T_{C}$ is caused by the presence of an anisotropic
superconducting gap structure in momentum space or due to the presence of a
nonsuperconducting fraction in the sample. As our $\mu$SR data reveal an
isotropic $s$-wave superconducting gap structure, we attribute this reduction
in value of $2\Delta/k_{B}T_{C}$ to the presence of a small nonsuperconducting
fraction in the sample. This is further supported by the fact that the
superconducting state electronic heat capacity could be described very well by
fully gapped ($2\Delta/k_{B}T_{C}$ = 3.53) isotropic $s$-wave BCS
superconductivity by adding a contribution from the nonsuperconducting
fraction as discussed above [see Fig. 2(b)]. The muon spin depolarization rate
associated with the superconducting state, $\sigma_{\mathrm{sc}}$, is related
with the penetration depth via
$\sigma_{\mathrm{sc}}(T)=0.06091\frac{\gamma_{\mu}\Phi_{0}}{\lambda_{L}^{2}(T)}$
[48], where $\Phi_{0}=2.609\times 10^{-15}$ Wb is the magnetic flux quantum,
$\gamma_{\mu}=2\pi\times 135.5$ MHz/T is the muon gyromagnetic ratio. This
gives $\lambda_{\mathrm{L}}(0)=259(4)$ nm from the $s$-wave model fit. The
London model provides a direct relation between $\lambda_{L}(T)$ and
($m^{*}/n_{\mathrm{s}})(\lambda_{\mathrm{L}}^{2}=m^{*}c^{2}/4\pi
n_{\mathrm{s}}e^{2})$ where
m${}^{*}=(1+\lambda_{\mathrm{e-ph}})m_{\mathrm{e}}$ is the effective mass in
units of the bare electron mass $m_{\mathrm{e}}$, and $n_{\mathrm{s}}$ is the
carrier density. $\lambda_{\mathrm{e-ph}}$ is calculated from $\Theta_{D}$ and
$T_{\mathrm{C}}$ using McMillan equation
$\lambda_{\mathrm{e-ph}}=\frac{1.04+\mu^{*}\ln(\Theta_{\mathrm{D}}/1.45T_{\mathrm{C}})}{(1-0.62\mu^{*})\ln(\Theta_{\mathrm{D}}/1.45T_{\mathrm{C}})-1.04}$.
The superconducting carrier density is then estimated to be
$n_{\mathrm{s}}=2.7(3)\times 10^{26}$ carriers m-3 and the effective-mass
enhancement $m^{*}=1.51(2)\,m_{\mathrm{e}}$. Details of similar calculations
can be found in Refs. [49, 50, 51, 52]. In Fig. 4(b), we have plotted the
internal magnetic field as a function of temperature. As the sample goes
through the superconducting state ($T<T_{\mathrm{C}}$), the internal magnetic
field distribution in the mixed state is less than the applied fields, which
clearly demonstrates the Meissner field expulsion in the superconducting
state.

###  III.3 Time-reversal symmetry breaking

Figure 5(a) shows the time evolution of the zero-field muon-spin relaxation
asymmetry in Sc5Co4Si10 at temperatures above (6.2 K) and below (1.3 K)
$T_{\mathrm{C}}$. Below $T_{\mathrm{C}}$, the muon-spin relaxation becomes
faster. It should be noted that there is no signature of muon spin precession,
_i.e._ oscillations in $A(t)$, that would follow any internal magnetic field
produced by an ordering of the electronic moments.

The ZF-$\mu$SR spectra can be well described by a damped Gaussian Kubo-Toyabe
(KT) function

$G_{\mathrm{ZF}}(t)=A_{\mathrm{3}}G_{\mathrm{KT}}(t)e^{-\lambda_{\mathrm{ZF}}t}+A_{\mathrm{bg}},$
(3)

where

$G_{\mathrm{KT}}(t)=\left[\frac{1}{3}+\frac{2}{3}\left(1-\sigma_{\mathrm{KT}}^{2}t^{2}\right)\exp\left({-\frac{\sigma_{\mathrm{KT}}^{2}t^{2}}{2}}\right)\right].$
(4)

$A_{\mathrm{3}}$, $A_{\mathrm{bg}}$, $\lambda_{\mathrm{ZF}}$, and
$\sigma_{\mathrm{KT}}$ are the initial asymmetry from muons implanted in the
sample, the asymmetry arising from muons landing in the silver sample holder,
the electronic relaxation rate originating from any electronic moment present
in the sample, and the Kubo-Toyabe depolarization rate, respectively.
$A_{\mathrm{bg}}$ is found to be temperature independent. The contribution to
$\sigma_{\mathrm{KT}}$ comes from 45Sc with nuclear spin $I$=7/2 (natural
abundance 100%) and 59Co, $I$=7/2 (100%). 28Si does not have a nuclear spin,
$I$=0 and hence it does not contribute to $\sigma_{\mathrm{KT}}$. A similar
value of $\sigma_{\mathrm{KT}}$ is also obtained for Sc5Rh6Sn18 [53].

Figure 5(c) shows that the temperature dependence of the electronic relaxation
rate, $\lambda_{\mathrm{ZF}}\left(T\right)$, is constant within error above
$T_{\mathrm{C}}$, and then increases below $T_{\mathrm{C}}$ down to the lowest
temperature measured. Over the temperature range studied, the Kubo-Toyabe
relaxation rate $\sigma_{\mathrm{KT}}(T)$ remains almost constant, as shown in
Fig. 5(d) while $\lambda_{ZF}$ increases below $T_{\mathrm{C}}$, most probably
due to a Lorentzian contribution to the distribution of static fields. Later
we have shown that a weak longitudinal field is sufficient to decouple the
relaxation (Figure 5 b), suggesting that the increase in
$\lambda_{\mathrm{ZF}}$ is due to static rather than dynamic fields.

The increase in $\lambda_{\mathrm{ZF}}(T)$ indicates the appearance of a
finite, spontaneous internal magnetic field with a Lorentzian distribution
correlated with the superconductivity. The increase in
$\Delta\lambda_{\mathrm{ZF}}$ below $T_{\mathrm{C}}$ is 0.017 $\mu$s-1, which
corresponds to a characteristic field strength
$\Delta\lambda_{\mathrm{ZF}}/\gamma_{\mu}=0.020$ mT, where
$\gamma_{\mu}=2\pi\times 135.5$ MHz/T is the muon gyromagnetic ratio. Assuming
the relaxation is due to the static field we can then use the increase in the
exponential relaxation rate, to the field strength. Similar estimates have
been made elsewhere, for example, for Sr2RuO4 the estimated characteristics
field strength is 0.05 mT [54], similar calculations have also been performed
for UPt3 [55]. This observation provides strong evidence that time-reversal
symmetry is broken in the superconducting state of Sc5Co4Si10. Similar small
changes in $\lambda_{ZF}\left(T\right)$ have provided evidence of TRS breaking
in superconducting LaNiC2 [56], $A_{5}$Rh6Sn18 [$A=$ Y, $R$, or Sc] [53, 57,
58], SrPtAs [59], the binary lanthanides ${\text{La}_{7}}{\text{(Ir, Rh,
Pd)}}_{3}$ [60, 61, 62], the recently discovered ${\text{Zr}}_{3}{\text{Ir}}$
[63, 64], and, but more controversially, in the layered perovskite
${\text{Sr}}_{2}{\text{RuO}}_{4}$ [54, 65, 66, 67, 68]. Theoretical analysis
has argued that TRS broken state occurs below $T_{\mathrm{C}}$ within the BCS
formalism in multiband superconductors [69]. On the other hand, the TRS
breaking has also been proposed theoretically for fully-gap BCS
superconductors based on the loop supercurrent state, as proposed for multi-
orbital systems with complex crystal structures [70, 71].

A longitudinal magnetic field of 25 mT [Fig. 5(b)] removes any relaxation due
to the spontaneous fields and is sufficient to fully decouple the muons from
this relaxation channel. This indicates that the associated magnetic fields
are static or quasistatic on the time scale of the muon precession. These
observations further support the broken TRS in the superconducting ground
state of Sc5Co4Si10.

###  III.4 Theoretical calculations

To bridge the gap between our experimental results suggesting a nodeless,
$s$-wave gap structure with broken time-reversal symmetry and a
superconducting order parameter coherently underlying such phenomenology, we
have used the group theoretical formulation of the Ginzburg-Landau theory
alongside first-principles electronic-structure calculations and isotropic
single-particle Bogoliubov-de Gennes model to constrain the possible
symmetries allowed for the Sc5Co4Si10 superconducting order parameter, as
explained below.

Considering a continuous gauge-symmetry breaking phase transition, it is
possible to fully determine the simplest allowed symmetries of the
superconducting momentum-dependent pairing potential
$\hat{\Delta}(\mathbf{k})$ close to the critical temperature by the
$\mathcal{D}$-dimensional irreducible representations (IRs) of the crystalline
space group [71]. For the specific case of the tetragonal space group $P4/mbm$
with point group symmetry $D_{4h}$, there are 8 one-dimensional IRs and 2 two-
dimensional IRs available. Based on this, one examine all the possible singlet
and triplet order parameters which might occur in Sc5Co4Si10 assuming a non-
negligible spin-orbit coupling. Below, we assume a TRS breaking phase, as
suggested by muon-spin spectroscopy measurements.

Figure 6: Spin-polarized Fermi surface (FS) projected onto the effective
quasiparticle velocity of Sc5Co4Si10. (a) FS representation at the first
Brillouin zone. (b–e) Fermi surface sections in the (1,0,0) plane, (b) without
and (c) with SOC effects, and in the (1,1,0) plane, (d) without and (e) with
SOC effects.

In the case of singlet states there are five representations of $SO(3)\times
D_{4h}$, namely ${}^{1}A_{1g}$, ${}^{1}A_{2g}$, ${}^{1}B_{1g}$, ${}^{1}B_{2g}$
and ${}^{1}E_{g}$, four of which are one-dimensional with a single complex
component, and just one, ${}^{1}E_{g}$, is two-dimensional with two complex
components and three types of solution [47]. However, it turns out that TRS
breaking superconducting ground states require that two or more components of
their pairing potential acquire different complex phases with $\mathcal{D}>1$.
These conditions stem from the fact that TRS breaking states under the TRS
operation should give rise to a new state with symmetrically non-related order
parameter [71]. As such, among the seven possible singlet states, just
${}^{1}E_{g}(c)$ satisfies the conditions above. In this case, by minimizing
the Ginzburg-Landau free energy up to quartic terms, the broken-time reversal
singlet has a $(1,i)$-type ground state with the simplest gap function
compatible with its $SO(3)\times D_{4}(E)\times i$ residual symmetry group
being [47]

$\displaystyle\hat{\Delta}(\mathbf{k})=i\left[X(\mathbf{k})+iY(\mathbf{k})\right]Z(\mathbf{k})\hat{\sigma}_{y},$
(5)

where $X(\mathbf{k})$, $Y(\mathbf{k})$ and $Z(\mathbf{k})$ are the basis
functions which transform as $k_{x}$, $k_{y}$, and $k_{z}$, respectively,
under point group operations, and $\hat{\sigma}_{y}$ is the second Pauli
matrix. The gap function of the quasiparticle spectrum is then obtained by
diagonalising the Bogoliubov-de Gennes Hamiltonian [58]

$\displaystyle\mathcal{H}_{BdG}=\begin{pmatrix}\hat{h}(\mathbf{k})&\hat{\Delta}(\mathbf{k})\\\
\hat{\Delta}^{\dagger}(\mathbf{k})&-\hat{h}^{T}(-\mathbf{k})\\\
\end{pmatrix},$ (6)

where $\hat{h}(\mathbf{k})$ is the single-particle Hamiltonian. Thus, for the
sake of simplicity, considering the isotropic single-electron dispersion
relation, it follows [58]

$\displaystyle\Delta(\mathbf{k})\propto\left|k_{z}\right|\sqrt{k_{x}^{2}+k_{y}^{2}},$
(7)

for the singlet state, yielding a gap structure with line nodes at $k_{z}=0$
and two-point nodes at the north ($\theta=0$) and south ($\theta=\pi$) poles
of the Fermi surface.

In the case of triplet states, there are also five different IRs of the
corresponding double group: the four one-dimensional triplets $A_{1u}$,
$A_{2u}$, $B_{1u}$, $B_{2u}$, each with a single complex order parameter, and
a two-dimensional representation $E_{u}$, with a complex two-component order
parameter and three types of solution [47]. Using the same argument as above,
just the $E_{u}(c)$ representation with a ground state $(1,i)$ and a
$D_{4}(E)\times i(E)$ residual symmetry group breaks TRS, in which the
simplest gap function assumes the general form [47]

$\displaystyle\hat{\Delta}(\mathbf{k})=i\left[\left(AZ(\mathbf{k}),iAZ(\mathbf{k}),BX(\mathbf{k})+iBY(\mathbf{k})\right)\cdot\hat{\mathbf{\sigma}}\right]\hat{\sigma}_{y},$
(8)

where the undetermined coefficients $A$ and $B$ depend on the Fermi surface
topography and electronic correlations.

Following the same steps as above, the simplest gap function of the isotropic
single-particle Bogoliubov-de Gennes quasiparticle spectrum for the non-
unitary triplet pairing state is given by [58]

$\displaystyle\Delta(\mathbf{k})\propto\left|A\left|k_{z}\right|-\sqrt{A^{2}k_{z}^{2}+B^{2}(k_{x}^{2}+k_{y}^{2})}\right|,$
(9)

resulting in gap structures with two point nodes at both the north and south
Fermi surface poles.

In both cases, if the Fermi surface is absent in the regions where the
superfluid density is dominated by the low-energy nodal excitations, the
superconducting gap would remain fully open over the entire Fermi surface,
resembling a nodeless $s$-wave gap structure. This argument is widely used
elsewhere [72, 58, 57, 64] to overcome the apparent incompatibility of TRS
breaking in conventional systems . However, even in those cases, we believe
that this scenario is quite unlikely since the topography of the Fermi surface
in real materials is generally very complex and unlikely to meet such
conditions.

For instance, 12 distinct low-energy bands cross the Sc5Co4Si10 Fermi level,
resulting in highly complex topography with a three-dimensional character (see
Figure 6). Despite the small influence of SOC in low-energy states, several
degenerate points are gapped when including the relativistic effects
explicitly. The two ball-shaped $\Gamma$-centered pockets have their
topography heavily modified by spin-orbit interactions, giving rise to a
single lobulated molten spheroid. However, the Dirac-type degeneracies at
high-symmetry points are protected against SOC by the double point-group
symmetries [73, 74, 75, 76]. There are two closed pockets crossing the
geometrical poles and several closed electrons- and hole-pockets at the
equator along $k_{z}$ direction, as shown in Fig. 6. Therefore, if any, the
detection of point nodes would be favored since low-energy states are present
in those regions where they are symmetrically allowed. Recently, Huddart et
al. Huddart  _et al._ [77] conducted a systematic investigation of muon-
stopping sites in TRS breaking superconductors. They have demonstrated that
muons lead to self-limited and small high-energy localized perturbations in
the electronic structure. Therefore, the observation of time-reversal symmetry
breaking superconducting states is unlikely to originate from muon-induced
perturbations, including in the present case.

The conventional electron-phonon mechanism, however, is believed to be enough
to give rise to exotic low-symmetry order parameters in particular conditions
[78]. Since the interband and intraband electron scattering channel are
determined by symmetry, it is argued that there is no special need for the
resulting superconducting order parameter to preserve the
$\mathcal{G}\otimes\mathcal{T}$ symmetry, where $\mathcal{G}$ is the crystal
point group and $\mathcal{T}$ is the time-reversal operation. This effect can
appear due to the competition between phonon and Coulomb interactions of
highly symmetric systems containing multiple electron- and hole-pockets
centered at high-symmetry points [78]. Several exotic superconducting
compounds have been analyzed recently within this theoretical picture [79, 80,
81, 82, 83, 84]. In those cases, it is expected multidimensional gaps of equal
magnitude (that is, hard to differentiate) on each of the Fermi surface sheets
within the quasiparticle spectrum. Our calculations show that the Fermi
surface topography of Sc5Co4Si10 may allow, therefore, a fully-gapped
superconducting state with TRS breaking due to conventional mechanisms.

In some particular cases, an unconventional singlet-triplet pairing can look
like a nodeless $s$-wave pairing if the magnitude of the two superconducting
gaps is very similar [62]. Therefore, additional hypotheses, including the
loop supercurrent state [70], cannot be ruled out entirely in the present
case. We hope that the work presented here will motivate further theoretical
and experimental studies addressing the essential and still open questions on
the nature of the pairing mechanisms in superconductors that exhibit TRS
breaking in what appear to be fully gapped systems.

##  IV Summary

In summary, we have performed a systematic study of the superconducting
properties of Sc5Co4Si10 using low-temperature resistivity, magnetization,
heat capacity, and muon spin rotation and relaxation measurements. Our
measurements show that Sc5Co4Si10 is a type-II superconductor with
$T_{\mathrm{C}}=3.5(1)$ K, $\mu_{0}H_{\mathrm{C1}}=1.4(1)$ mT, and
$\mu_{0}H_{\mathrm{C2}}=1.33(2)$ T. The temperature dependence of the
superfluid density obtained from transverse-field $\mu$SR measurements is best
modeled assuming an isotropic, single $s$-wave gap structure with
2$\Delta/k_{\mathrm{B}}T_{\mathrm{C}}=2.84(2)$. These findings are supported
by the heat capacity measurements. In addition, the appearance of spontaneous
magnetic fields in the ZF-$\mu$SR spectra provides strong evidence for TRS
breaking, suggesting Sc5Co4Si10 may exhibit unconventional pairing, despite
the observed full-gapped state. Our theoretical assessment of the symmetry
allowed pairing states and the particular features of the Sc5Co4Si10 Fermi
surface demonstrate that non-unitary or mixed singlet-triplet order parameters
are unlikely, due to the presence of low-energy states in regions where
gapless, nodal excitations would be expected. This analysis suggests that
conventional mechanisms may be responsible for TRS breaking in Sc5Co4Si10. Our
work paves the way for further studies on this large class of superconductors.
Furthermore, it highlights the importance of Sc5Co4Si10 in building a complete
understanding of why some materials with solid evidence for TRS breaking
simultaneously exhibit behavior expected from more conventional
superconductors.

###### Acknowledgements.

AB would like to thank Science & Engineering Research Board for the CRG
Research Grant (CRG/2020/000698). DTA is grateful to the EPSRC UK for funding
(Grant Ref: EP/W00562X/1), the JSPS for an invitation fellowship, and the
Royal Society of London for the International Exchange funding and Newton
Advanced Fellowship between UK-China. KP would like to acknowledge the DST
India for an Inspire Fellowship (IF170620). This study was financed in part by
the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil
(CAPES) – Finance Code 001. We gratefully acknowledge the São Paulo Research
Foundation (FAPESP) under Grants 2019/05005-7 and 2020/08258-0. The research
was partially carried out using high-performance computing resources made
available by the Superintendência de Tecnologia da Informação (STI),
Universidade de São Paulo. The authors also acknowledge the National
Laboratory for Scientific Computing (LNCC/MCTI, Brazil) for providing HPC
resources of the SDumont supercomputer, which have contributed to the research
results reported within this paper. Finally, we would like to thank the ISIS
Facility (DOI: 10.5286/ISIS.E.RB1810871) for $\mu$SR beamtime.

## References

  * Parthé and Chabot [1984] E. Parthé and B. Chabot, in  _Handbook on the Physics and Chemistry of Rare Earths_ , Vol. 6, edited by K. A. Gschneidner, Jr. and L. Eyring (Elsevier Science, North-Holland, Netherlands, 1984) pp. 113–334.
  * Braun [1984] H. F. Braun, J. Less Common. Met. 100, 105 (1984).
  * Braun and Segre [1980] H. F. Braun and C. U. Segre, Solid State Commun. 35, 735 (1980).
  * Braun [1980] H. F. Braun, Phys. Lett. A 75, 386 (1980).
  * Segre and Braun [1981] C. U. Segre and H. F. Braun, Phys. Lett. A 85, 372 (1981).
  * Nakajima  _et al._ [2008] Y. Nakajima, T. Nakagawa, T. Tamegai, and H. Harima, Phys. Rev. Lett. 100, 157001 (2008).
  * Gordon  _et al._ [2008] R. T. Gordon, M. D. Vannette, C. Martin, Y. Nakajima, T. Tamegai, and R. Prozorov, Phys. Rev. B 78, 024514 (2008).
  * Biswas  _et al._ [2011] P. K. Biswas, G. Balakrishnan, D. M. Paul, M. R. Lees, and A. D. Hillier, Phys. Rev. B 83, 054517 (2011).
  * Yang  _et al._ [1991] H. D. Yang, P. Klavins, and R. N. Shelton, Phys. Rev. B 43, 7688 (1991).
  * Ghosh  _et al._ [1993] K. Ghosh, S. Ramakrishnan, and G. Chandra, Phys. Rev. B 48, 4152 (1993).
  * Becker  _et al._ [1999] B. Becker, N. G. Patil, S. Ramakrishnan, A. A. Menovsky, G. J. Nieuwenhuys, J. A. Mydosh, M. Kohgi, and K. Iwasa, Phys. Rev. B 59, 7266 (1999).
  * Galli  _et al._ [2000] F. Galli, S. Ramakrishnan, T. Taniguchi, G. J. Nieuwenhuys, J. Mydosh, S. Geupel, J. Lüdecke, and S. van Smaalen, Phys. Rev. Lett. 85, 158 (2000).
  * Kuo  _et al._ [2003] Y.-K. Kuo, F. Hsu, H. Li, H. Huang, C. Huang, C. Lue, and H. Yang, Phys. Rev. B 67, 195101 (2003).
  * van Smaalen  _et al._ [2004] S. van Smaalen, M. Shaz, L. Palatinus, P. Daniels, F. Galli, G. J. Nieuwenhuys, and J. A. Mydosh, Phys. Rev. B 69, 014103 (2004).
  * Hausermann-Berg and Shelton [1987] L. S. Hausermann-Berg and R. N. Shelton, Phys. Rev. B 35, 6659 (1987).
  * Matthias and Hulm [1953] B. T. Matthias and J. K. Hulm, Phys. Rev. 89, 439 (1953).
  * Rai  _et al._ [2015] B. K. Rai, I. W. H. Oswald, J. K. Wang, G. T. McCandless, J. Y. Chan, and E. Morosan, Chem. Mater. 27, 2488 (2015).
  * Takada  _et al._ [2003] K. Takada, H. Sakurai, E. Takayama-Muromachi, F. Izumi, R. A. Dilanian, and T. Sasaki, Nature 422, 53 (2003).
  * He  _et al._ [2021] Z. He, R. Huang, K. Zhou, Y. Liu, S. Guo, Y. Song, Z. Guo, S. Hu, L. He, Q. Huang, L. Li, J. Zhang, S. Wang, J. Guo, X. Xing, and J. Chen, Inorg. Chem. 60, 6157 (2021).
  * Scherer  _et al._ [2010] W. Scherer, C. Hauf, M. Presnitz, E.-W. Scheidt, G. Eickerling, V. Eyert, R.-D. Hoffmann, U. Rodewald, A. Hammerschmidt, C. Vogt, and R. Pöttgen, Angew. Chem. Int. Edit. 49, 1578 (2010).
  * Braun  _et al._ [1980] H. F. Braun, K. Yvon, and R. M. Braun, Acta. Crystall. B-Stru. 36, 2397 (1980).
  * Fischer [1978] Ø. Fischer, Appl. Phys. 16, 1 (1978).
  * Vandenberg and Matthias [1977] J. M. Vandenberg and B. T. Matthias, Science 198, 194 (1977).
  * Nakajima  _et al._ [2009] Y. Nakajima, G. J. Li, and T. Tamegai, J. Phys. Conf. Ser. 150, 052180 (2009).
  * Koyama  _et al._ [1999] T. Koyama, H. Sugita, S. Wada, and K. Tsutsumi, J. Phys. Soc. Jpn. 68, 2326 (1999).
  * Lue  _et al._ [2008] C. S. Lue, R. F. Liu, Y. F. Fu, C. Cheng, and H. D. Yang, Phys. Rev. B 77, 115130 (2008).
  * Lee  _et al._ [1999] S. L. Lee, R. Cywinski, and S. H. Kilcoyne,  _Muon Science: Muons in Physics, Chemistry and Materials_ , Vol. 51 (SUSSP Publications and IOP Publishing, Bristol, 1999).
  * Adroja  _et al._ [2018] D. T. Adroja, P. K. Biswas, A. Bhattacharyya, A. Hillier, and V. K. Anand, https://doi.org/10.5286/ISIS.E.92918257 10.5286/ISIS.E.RB1810871 (2018).
  * Pratt [2000] F. L. Pratt, Physica B 289–290, 710 (2000).
  * Kresse and Hafner [1993] G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).
  * Kresse and Furthmüller [1996] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
  * Kresse and Furthmüller [1996] G. Kresse and J. Furthmüller, Comp. Mater. Sci. 6, 15 (1996).
  * Blöchl [1994] P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
  * Kresse and Joubert [1999] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Singh  _et al._ [2005] Y. Singh, R. Nirmala, S. Ramakrishnan, and S. Malik, Physical Review B 72, 045106 (2005).
  * Werthamer  _et al._ [1966] N. R. Werthamer, E. Helfand, and P. C. Hohenberg, Phys. Rev. 147, 295 (1966).
  * Bardeen  _et al._ [1957] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).
  * Bhattacharyya  _et al._ [2018a] A. Bhattacharyya, D. T. Adroja, M. Smidman, and V. K. Anand, Sci. China Phys. Mech. Astron. 61, 1 (2018a).
  * Bhattacharyya  _et al._ [2019a] A. Bhattacharyya, D. T. Adroja, K. Panda, S. Saha, T. Das, A. J. S. Machado, O. V. Cigarroa, T. W. Grant, Z. Fisk, A. D. Hillier, and P. Manfrinetti, Phys. Rev. Lett. 122, 147001 (2019a).
  * Adroja  _et al._ [2021] D. T. Adroja, A. Bhattacharyya, Y. J. Sato, M. R. Lees, P. K. Biswas, K. Panda, V. K. Anand, G. B. G. Stenning, A. D. Hillier, and D. Aoki, Phys. Rev. B 103, 104514 (2021).
  * Bhattacharyya  _et al._ [2021] A. Bhattacharyya, P. P. Ferreira, K. Panda, S. H. Masunaga, L. R. de Faria, L. E. Correa, F. B. Santos, D. T. Adroja, K. Yokoyama, T. T. Dorini, R. F. Jardim, L. T. F. Eleno, and A. J. S. Machado, J. Phys.: Condens. Matter 34, 035602 (2021).
  * Prozorov and Giannetta [2006] R. Prozorov and R. W. Giannetta, Supercond. Sci. Tech. 19, R41 (2006).
  * Adroja  _et al._ [2015] D. T. Adroja, A. Bhattacharyya, M. Telling, Y. Feng, M. Smidman, B. Pan, J. Zhao, A. D. Hillier, F. L. Pratt, and A. M. Strydom, Phys. Rev. B 92, 134505 (2015).
  * Panda  _et al._ [2019] K. Panda, A. Bhattacharyya, D. T. Adroja, N. Kase, P. K. Biswas, S. Saha, T. Das, M. R. Lees, and A. D. Hillier, Phys. Rev. B 99, 174513 (2019).
  * Pang  _et al._ [2015] G. M. Pang, M. Smidman, W. B. Jiang, J. K. Bao, Z. F. Weng, Y. F. Wang, L. Jiao, J. L. Zhang, G. H. Cao, and H. Q. Yuan, Phys. Rev. B 91, 220502 (2015).
  * Annett [1990] J. F. Annett, Adv. Phys. 39, 83 (1990).
  * Brandt [2003] E. H. Brandt, Physical Review B 68, 054506 (2003).
  * Chia  _et al._ [2004] E. E. M. Chia, M. B. Salamon, H. Sugawara, and H. Sato, Phys. Rev. B 69, 180509 (2004).
  * Amato [1997] A. Amato, Rev. Mod. Phys. 69, 1119 (1997).
  * McMillan [1968] W. L. McMillan, Phys. Rev. 167, 331 (1968).
  * Bhattacharyya  _et al._ [2019b] A. Bhattacharyya, K. Panda, D. T. Adroja, N. Kase, P. K. Biswas, S. Saha, T. Das, M. R. Lees, and A. D. Hillier, J. Phys.: Condens. Matter 32, 085601 (2019b).
  * Bhattacharyya  _et al._ [2018b] A. Bhattacharyya, D. T. Adroja, N. Kase, A. D. Hillier, A. M. Strydom, and J. Akimitsu, Phys. Rev. B 98, 024511 (2018b).
  * Luke  _et al._ [1998] G. M. Luke, Y. Fudamoto, K. M. Kojima, M. I. Larkin, J. Merrin, B. Nachumi, Y. J. Uemura, Y. Maeno, Z. Q. Mao, Y. Mori, H. Nakamura, and M. Sigrist, Nature 394, 558 (1998).
  * Luke  _et al._ [1993] G. Luke, A. Keren, L. Le, W. Wu, Y. Uemura, D. Bonn, L. Taillefer, and J. Garrett, Physical review letters 71, 1466 (1993).
  * Hillier  _et al._ [2009] A. D. Hillier, J. Quintanilla, and R. Cywinski, Phys. Rev. Lett. 102, 117007 (2009).
  * Bhattacharyya  _et al._ [2015a] A. Bhattacharyya, D. Adroja, N. Kase, A. Hillier, J. Akimitsu, and A. Strydom, Sci. Rep. 5, 12926 (2015a).
  * Bhattacharyya  _et al._ [2015b] A. Bhattacharyya, D. T. Adroja, J. Quintanilla, A. D. Hillier, N. Kase, A. M. Strydom, and J. Akimitsu, Phys. Rev. B 91, 060503 (2015b).
  * Biswas  _et al._ [2013] P. K. Biswas, H. Luetkens, T. Neupert, T. Stürzer, C. Baines, G. Pascua, A. P. Schnyder, M. H. Fischer, J. Goryo, M. R. Lees, H. Maeter, F. Brückner, H.-H. Klauss, M. Nicklas, P. J. Baker, A. D. Hillier, M. Sigrist, A. Amato, and D. Johrendt, Phys. Rev. B 87, 180503 (2013).
  * Barker  _et al._ [2015] J. A. T. Barker, D. Singh, A. Thamizhavel, A. D. Hillier, M. R. Lees, G. Balakrishnan, D. M. Paul, and R. P. Singh, Phys. Rev. Lett. 115, 267001 (2015).
  * Singh  _et al._ [2020] D. Singh, M. S. Scheurer, A. D. Hillier, D. T. Adroja, and R. P. Singh, Phys. Rev. B 102, 134511 (2020).
  * Mayoh  _et al._ [2021] D. A. Mayoh, A. D. Hillier, G. Balakrishnan, and M. R. Lees, Phys. Rev. B 103, 024507 (2021).
  * K. P.  _et al._ [2019] S. K. P., D. Singh, P. K. Biswas, G. B. G. Stenning, A. D. Hillier, and R. P. Singh, Phys. Rev. Materials 3, 104802 (2019).
  * Shang  _et al._ [2020] T. Shang, S. K. Ghosh, J. Z. Zhao, L.-J. Chang, C. Baines, M. K. Lee, D. J. Gawryluk, M. Shi, M. Medarde, J. Quintanilla, and T. Shiroka, Phys. Rev. B 102, 020503 (2020).
  * Luke  _et al._ [2000] G. M. Luke, Y. Fudamoto, K. M. Kojima, M. I. Larkin, B. Nachumi, Y. J. Uemura, J. E. Sonier, Y. Maeno, Z. Q. Mao, Y. Mori, and D. F. Agterberg, Physica B 289–290, 373 (2000).
  * Xia  _et al._ [2006] J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, Phys. Rev. Lett. 97, 167002 (2006).
  * Kashiwaya  _et al._ [2019] S. Kashiwaya, K. Saitoh, H. Kashiwaya, M. Koyanagi, M. Sato, K. Yada, Y. Tanaka, and Y. Maeno, Phys. Rev. B 100, 094530 (2019).
  * Grinenko  _et al._ [2021] V. Grinenko, S. Ghosh, R. Sarkar, J.-C. Orain, A. Nikitin, M. Elender, D. Das, Z. Guguchia, F. Brückner, M. E. Barber, J. Park, N. Kikugawa, D. A. Sokolov, J. S. Bobowski, T. Miyoshi, Y. Maeno, A. P. Mackenzie, H. Luetkens, C. W. Hicks, and H.-H. Klauss, Nat. Phys. 17, 748 (2021).
  * Wilson and Das [2013] B. J. Wilson and M. P. Das, J. Phys.: Condens. Matter 25, 425702 (2013).
  * Ghosh  _et al._ [2021] S. K. Ghosh, J. F. Annett, and J. Quintanilla, New J. Phys. 23, 083018 (2021).
  * Ghosh  _et al._ [2020] S. K. Ghosh, M. Smidman, T. Shang, J. F. Annett, A. D. Hillier, J. Quintanilla, and H. Yuan, J. Phys.: Condens. Matter 33, 033001 (2020).
  * Mackenzie and Maeno [2003] A. P. Mackenzie and Y. Maeno, Rev. Mod. Phys. 75, 657 (2003).
  * Young  _et al._ [2012] S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, and A. M. Rappe, Phys. Rev. Lett. 108, 140405 (2012).
  * Yang and Nagaosa [2014] B.-J. Yang and N. Nagaosa, Nat. Commun. 5, 4898 (2014).
  * Yang  _et al._ [2015] B.-J. Yang, T. Morimoto, and A. Furusaki, Phys. Rev. B 92, 165120 (2015).
  * Ferreira  _et al._ [2021] P. P. Ferreira, A. L. R. Manesco, T. T. Dorini, L. E. Correa, G. Weber, A. J. S. Machado, and L. T. F. Eleno, Phys. Rev. B 103, 125134 (2021).
  * Huddart  _et al._ [2021] B. Huddart, I. Onuorah, M. Isah, P. Bonfa, S. Blundell, S. Clark, R. De Renzi, and T. Lancaster, Physical review letters 127, 237002 (2021).
  * Agterberg  _et al._ [1999] D. F. Agterberg, V. Barzykin, and L. P. Gor’kov, Phys. Rev. B 60, 14868 (1999).
  * Mao  _et al._ [2003] Z. Mao, M. Rosario, K. Nelson, K. Wu, I. Deac, P. Schiffer, Y. Liu, T. He, K. Regan, and R. J. Cava, Phys. Rev. B 67, 094502 (2003).
  * Kreisel  _et al._ [2013] A. Kreisel, Y. Wang, T. A. Maier, P. J. Hirschfeld, and D. J. Scalapino, Phys. Rev. B 88, 094522 (2013).
  * Singh  _et al._ [2017] D. Singh, J. A. T. Barker, A. Thamizhavel, D. M. Paul, A. D. Hillier, and R. P. Singh, Phys. Rev. B 96, 180501 (2017).
  * Huang  _et al._ [2018] Q. Huang, T. Le, L. Che, L. Yin, J. Li, J. Yang, M. Fang, and X. Lu, Mater. Res. Express 6, 016001 (2018).
  * Singh  _et al._ [2018] D. Singh, S. K. P., J. A. T. Barker, D. M. Paul, A. D. Hillier, and R. P. Singh, Phys. Rev. B 97, 100505 (2018).
  * Shang  _et al._ [2018] T. Shang, G. M. Pang, C. Baines, W. B. Jiang, W. Xie, A. Wang, M. Medarde, E. Pomjakushina, M. Shi, J. Mesot, H. Q. Yuan, and T. Shiroka, Phys. Rev. B 97, 020502 (2018).