# Superconducting ground state study of valence skip compound AgSnSe2

A. Kataria  Department of Physics, Indian Institute of Science Education and
Research Bhopal, Bhopal, 462066, India  Arushi  Department of Physics, Indian
Institute of Science Education and Research Bhopal, Bhopal, 462066, India  S.
Sharma  Department of Physics and Astronomy, McMaster University, Hamilton,
Ontario L8S 4M1, Canada  T. Agarwal  Department of Physics, Indian Institute
of Science Education and Research Bhopal, Bhopal, 462066, India  M. Pula
Department of Physics and Astronomy, McMaster University, Hamilton, Ontario
L8S 4M1, Canada  J. Beare  Department of Physics and Astronomy, McMaster
University, Hamilton, Ontario L8S 4M1, Canada  S. Yoon  TRIUMF, Vancouver,
British Columbia V6T 2A3, Canada  Department of Physics, Sungkyunkwan
University, Suwon 16419, Korea  Y. Cai  TRIUMF, Vancouver, British Columbia
V6T 2A3, Canada  Quantum Matter Institute, The University of British Columbia,
Vancouver, BC V6T 1Z4, Canada  K. M. Kojima  TRIUMF, Vancouver, British
Columbia V6T 2A3, Canada  G. M. Luke  Department of Physics and Astronomy,
McMaster University, Hamilton, Ontario L8S 4M1, Canada  TRIUMF, Vancouver,
British Columbia V6T 2A3, Canada  R. P. Singh<EMAIL_ADDRESS>Department
of Physics, Indian Institute of Science Education and Research Bhopal, Bhopal,
462066, India

###### Abstract

The valence-skipped superconductors are natural candidates for unconventional
superconductivity, as they can exhibit a negative effective, attractive
interaction for electron-pairing. This work reports comprehensive XRD,
magnetization, specific heat and muon spin rotation and relaxation
measurements ($\mu$SR) on a valence-skipped compound: AgSnSe2. The temperature
dependence of the electronic specific heat ($C_{el}(T)$) and of the upper
critical field ($H_{c2}(T)$) provide evidence of two-gap superconductivity,
which is also confirmed by our transverse-field $\mu$SR measurements. Our
zero-field $\mu$SR measurements suggest preserved time-reversal symmetry in
the superconducting ground state of AgSnSe2.

##  I Introduction

Understanding the microscopic pairing mechanism of unconventional
superconductors is one of the most challenging problems in condensed matter
physics. It has recently attracted much interest due to its possible
applications in quantum technologies. In unconventional superconductors, the
electron-pair formation is not mediated by phonons but by different
mechanisms, which may include charge, spin or magnetic fluctuations sf ; mf ;
cf .

Valence-skipped materials are an exciting class of materials which may become
unconventional superconductors. In these materials, due to valence-
skipping/valence fluctuations, negative-$U$ centers are formed, which causes
an attractive on-site interaction between the electrons hf ; nsc ; cm ; as ,
and promotes superconductivity with relatively high $T_{c}$ compared to
conventional superconductors htc ; htc2 ; vs ; vs2 ; vs3 ; vs4 . This valence-
skipped induced negative-$U$ superconductivity has previously been reported in
Tl-doped PbTe, K- and Na-doped BaBiO3 tlpt ; BBO3 ; kbbo3 , with a high
$T_{c}$ value. Valence-skipped materials also exhibit intriguing
unconventional properties and other quantum phenomena, including charge
density wave (CDW), pseudogap, and charge Kondo effect ke ; hf ; cf . However,
the exact superconducting pairing mechanism in these valence-skipped compounds
is still elusive. With the possibility of achieving a high value of $T_{c}$ in
relatively low carrier density materials, a valance-skipped superconducting
pairing mechanism could be a novel root to realize high $T_{c}$ materials,
motivating detailed microscopic studies which are not available for these
materials to date.

The metal chalcogenide superconductor, AgSnSe2, where Ag partially substitutes
Sn in SnSe, is a novel valence-skipped system ass2 ; ass2sc . The
$ab$-$initio$ band structure calculations indicate SnSe is a nontrivial
crystalline topological material ss1 ; ss . X-ray absorption (XAS) and x-ray
photoemission spectroscopy (XPS) studies of AgSnSe2 suggested the presence of
valency states Sn2+ and Sn4+, which might create dynamic negative-$U$ centers
and contribute to superconductivity xps . Further, 119Sn̈ Mössbauer
spectroscopy suggests valence skip behaviour with excess Sn moss . The
coexisting superconductivity and nontrivial band topology in AgSnSe2 make it a
unique valence-skipped system and motivate further studies of this compound. A
thorough microscopic investigation requires understanding the correlation
between unconventional superconductivity, valence fluctuations and the effect
of nontrivial band topology on the superconducting ground state of AgSnSe2 and
valence-skipped compounds in general.

In this paper, we report the bulk and microscopic superconducting properties
of AgSnSe2 via magnetization, specific heat and muon spin rotation and
relaxation ($\mu$SR) measurements. Our measurements confirm the
superconducting transition temperature $T_{c}$ = 4.91(2) K. Transverse-field
(TF) $\mu$SR measurements, along with the temperature dependence of the
specific heat and the upper critical field suggest a two isotropic $(s+s)$
superconducting gap structure. Zero-field (ZF) muon spin relaxation
measurements indicate that within our resolution, time-reversal symmetry (TRS)
is preserved in the superconducting state of AgSnSe2.

##  II Experimental Details

The polycrystalline sample was synthesized using the solid-state reaction
method. The required elemental powders Ag (99.999%), Sn (99.99%), and Se
(99.999%) were mixed in a stoichiometric ratio. The mixture was then
palletized and heated at 800$\degree$C for 48 hours in a vacuum-sealed tube,
followed by water quenching. At room temperature, the powder x-ray diffraction
(XRD) pattern was collected using a PANalytical diffractometer equipped with
Cu$K_{\alpha}$ radiation ($\lambda$ = 1.5406 Å). Magnetic measurements were
performed using a Quantum Design MPMS XL SQUID magnetometer with RSO
(reciprocating sample option) insert, and for measurements down to 0.5 K, an
IQuantum He3 insert was used. A physical property measurement system (PPMS,
Quantum Design) was used to measure the specific heat of the sample. Zero-
field and transverse-field $\mu$SR measurements were performed at TRIUMF,
Center for Molecular and Materials Science, Vancouver, Canada msr . In the TF
geometry, the applied magnetic field was perpendicular to the initial muon
spin direction. The $\mu$SR data were analyzed using the musrfit software
musrfit .

##  III Results and Discussion

Figure 1:  Refined room temperature powder XRD pattern of AgSnSe2 with inset
showing the NaCl crystal structure with equal probability of occupying edge
state by Ag and Sn atoms.

The crystal structure of AgSnSe2 is shown in the inset of Fig. 1 depicting the
partial substitution of Ag at the Sn site in SnSe. Rietveld refined powder XRD
pattern of AgSnSe2 is shown in Fig. 1, confirming the crystallization in a
cubic NaCl structure with $Fm$-$3m$ space group. The obtained lattice
parameters from the refinement, a = b = c = 5.6865(2) Å, are in good agreement
with previously reported values str .

###  a Magnetization

Figure 2:  (a) The magnetic measurements under ZFCW and FCC mode with the
magnetic hysteresis loop at 1.8 K in the inset. (b) The temperature variation
of the lower critical field and the inset shows the $M$ dependence on the
internal magnetic field. (c) The upper critical field temperature variation
fitted by the GL equation and the two-gap model are represented by dashed and
solid lines, respectively. Inset shows the temperature-dependent magnetization
at the different applied fields.

The temperature-dependent magnetic measurements are performed in zero-field-
cooled warming (ZFCW) and field-cooled cooling (FCC) mode under 1 mT magnetic
field, as shown in Fig. 2(a). The observed superconducting transition
temperature of AgSnSe2, $T_{c}$ = 4.91(2) K, is close to previously reported
values ass2 ; ass2sc . The separation between ZFCW and FCC curves indicates
the presence of strong flux pinning. The magnetization variation under a high
magnetic field at temperature 1.8 K (inset of Fig. 2(a)) also suggests strong
pinning with complex vortex nature as a small area under the magnetization
loop with the fish-tail effect is observed fte . An irreversible magnetic
field is observed at $H_{irr}$ = 0.41(1) T, above which vortices start to
unpin. The field-dependent magnetization measurements at different
temperatures below $T_{c}$ are used to estimate the lower critical field
value.

We performed the magnetization measurements on a spherical sample piece,
allowing accurate demagnetising field determination. We calculate the internal
field of the sample by using $H_{internal}$ = $H_{applied}$ \- $NM$, where $N$
= 1/3 is the demagnetising factor of a sphere, and $M$ is the magnetization
Coh_Leng . The inset of Fig. 2(b) shows the variation of $M$ versus
$H_{internal}$, and the point of linear deviation in the low-field region is
considered as $H_{c1}$ for the respective isotherm. The temperature dependence
of the lower critical field, $H_{c1}$, is well described using the Ginzburg-
Landau (GL) equation,

$H_{c1}(T)=H_{c1}(0)\left[1-\left(\frac{T}{T_{c}}\right)^{2}\right]$ (1)

where the lower critical field is estimated to be $H_{c1}(0)$ = 5.76(6) mT
(Fig. 2(b)).

From the magnetization measurement, the shift in $T_{c}$ to a lower value with
an increasing magnetic field is noted to extract the upper critical field
(inset of Fig. 2(c)). We attempted to fit the $H_{c2}$ versus $T$ curve by the
GL equation,

$H_{c2}(T)=H_{c2}(0)\left[\frac{(1-t^{2})}{(1+t^{2})}\right]$ (2)

where $t=T/T_{c}$ is the reduced temperature. Fig. 2(c) shows the relatively
poor fitting of the $H_{c2}$ curve from the single-band GL model providing
$H_{c2}^{GL}$(0) = 2.13(3) T. Therefore, a two-band model was used to analyze
the $H_{c2}$ variation. The two-band equation for the upper critical field is
written in the parametric form as mb2 ; pts2 ; txs ; lfo ; ns2 ,

$\ln t=-\frac{1}{2}\left[U(s)+U(\eta s)+\frac{\lambda_{0}}{w}\right]+\\\
\left(\frac{1}{4}\left[U(s)-U(\eta
s)-\frac{\lambda_{-}}{w}\right]^{2}+\frac{\lambda_{eh}\lambda_{he}}{w^{2}}\right)^{1/2},$
(3)

where

$U(s)=\psi\left(s+\frac{1}{2}\right)-\psi\left(\frac{1}{2}\right),$ (4)

$H_{c2}=\frac{2\Phi_{0}T_{c}th}{D_{e}}$, $\eta=\frac{D_{h}}{D_{e}}$ and the
parameter $h$ changes from 0 to 1 as $T$ varies from $T_{c}$ to 0 with $t$
being the reduced temperature. The variables
$\lambda_{ee},\lambda_{hh},\lambda_{he},\lambda_{eh}$ are the matrix elements
of the BCS coupling constants and $\lambda_{-}=\lambda_{ee}-\lambda_{eh}$,
$\lambda_{0}=\lambda_{-}^{2}+\lambda_{eh}\lambda_{he}$,
$w=\lambda_{ee}\lambda_{hh}-\lambda_{he}\lambda_{eh}$. $D_{e}$ and $D_{h}$ are
the electron and hole diffusivity, respectively while $\Phi_{0}$ is the
magnetic flux quantum and $\psi$ is the digamma function. The fit of the
$H_{c2}$ curve using the two-band theory yields $H_{c2}^{2G}$(0) = 2.18(7) T.
The obtained values of the upper critical field from the two theories are much
smaller than the Pauli paramagnetic limit, expressed as
$H_{c2}^{P}(0)=1.86T_{c}=9.13(3)$T Pauli_1 ; Pauli_2 . Further, Ren et al.
ass2sc  reported the anomalous broadening of the magnetic-field-induced
resistive transition and an increase in the upper critical field value in the
low-temperature region in AgSnSe2 which might be associated with multiple
superconducting gap presence, as suggested by our upper critical field results
and also observed in MgB2 mgb22 .

We can evaluate various superconducting parameters using our $H_{c2}$ and
$H_{c1}$ values. The superconducting coherence length, $\xi_{GL}$(0) is
calculated using GL theory as Coh_Leng ,
$H_{c2}(0)=\left(\frac{\Phi_{0}}{2\pi\xi_{GL}(0)^{2}}\right)$, providing
$\xi_{GL}$(0) = 12.2(4) nm. The London penetration depth, $\lambda_{GL}$ is
obtained from the lower critical field, $H_{c1}$(0) using the expression lam ,

$H_{c1}(0)=\frac{\Phi_{0}}{4\pi\lambda_{GL}^{2}(0)}\left(\mathrm{ln}\frac{\lambda_{GL}(0)}{\xi_{GL}(0)}+0.12\right)$
(5)

which gives $\lambda_{GL}$(0) = 309(13) nm for $H_{c1}$(0) = 2.07(1) mT and
$\xi_{GL}$(0) = 12.2(4) nm. The GL parameter,
$\kappa_{GL}=\lambda_{GL}(0)/\xi_{GL}(0)$ = 25(2), is indicating the strong
type-II superconducting nature in AgSnSe2. Using these parameters and the
relation $H_{c1}H_{c2}=H_{c}^{2}\mathrm{ln}\kappa_{GL}$ lam , the
thermodynamic critical field value, $H_{c}(0)$ = 62(7) mT is also estimated.

###  b Specific heat

The temperature-dependent specific heat of AgSnSe2 under zero magnetic field
is shown in the Fig. 3(a). The significant jump in the specific heat around
temperature $T_{c,midpoint}$ = 4.74(8) K further confirms the bulk nature of
superconductivity in our sample. The normal-state specific heat above $T_{c}$
was fit by the Debye relation,

$C=\gamma_{n}T+\beta_{3}T^{3}+\beta_{5}T^{5}$ (6)

where $\gamma_{n}T$ is the electronic contribution, $\beta_{3}T^{3}$ is the
phononic contribution and $\beta_{5}T^{5}$ is the anharmonic contribution in
the specific heat. The best fit to the data yields Sommerfeld coefficient,
$\gamma_{n}$ = 4.6(1) mJ$\cdot$mol-1 K-2, Debye constant, $\beta_{3}$ =
0.68(1) mJ$\cdot$mol-1 K-4 and $\beta_{5}$ = 7.5(1) $\mu$J$\cdot$mol-1 K-5.

Further, $\gamma_{n}$ is used to evaluate the density of states at the Fermi
level, $D_{C}(E_{\mathrm{F}})$ via the relation,
$\gamma_{n}=\left(\frac{\pi^{2}k_{B}^{2}}{3}\right)D_{C}(E_{\mathrm{F}})$,
where $k_{B}$ = 1.38$\times$10-23 J K-1. $D_{C}(E_{\mathrm{F}})$ is calculated
to be 1.97(6) states eV-1f.u-1. The Debye temperature is obtained from
$\beta_{3}$ using the expression,
$\theta_{D}=\left(\frac{12\pi^{4}RN}{5\beta_{3}}\right)^{\frac{1}{3}}$ =
255(3) K, where $R$ = 8.314 J mol-1K-1 is the gas constant. From $\theta_{D}$
and $T_{c}$, the electron-phonon coupling constant, $\lambda_{e-ph}$ can also
be calculated from the inverted McMillan’s equation McMillan ,

$\lambda_{e-ph}=\frac{1.04+\mu^{*}\mathrm{ln}(\theta_{D}/1.45T_{c})}{(1-0.62\mu^{*})\mathrm{ln}(\theta_{D}/1.45T_{c})-1.04}$
(7)

here $\mu^{*}$ is the screened Coulomb repulsion and is taken as 0.13. For
$\theta_{D}$ = 255 K and $T_{c,midpoint}$ = 4.74 K, resulting $\lambda_{e-ph}$
= 0.69(1). The obtained value of $\lambda_{e-ph}$ for AgSnSe2 is in the range
of moderately coupled superconductors.

Figure 3:  (a) Specific heat fitting in the normal region. (b) Normalized
electronic specific heat data, $C_{el}/\gamma_{n}T$, fitted using two
different models $s$-wave shown by the dotted yellow line and two-gap
$s+s$-wave shown by the solid green line.

To evaluate the superconducting gap parameter, we analyzed the electronic-
specific heat below $T_{c}$. The electronic contribution is extracted by
subtracting the lattice contribution from the total specific heat. The
normalized superconducting gap, $\Delta(0)/k_{B}T_{c}$ is estimated by fitting
the data to the BCS expectation of electronic-specific heat, which is
expressed as cel ,

$\frac{C_{el}}{\gamma_{n}T_{c}}=t\frac{d}{dt}\left[\frac{-6\Delta(0)}{\pi^{2}k_{B}T_{c}}\int_{0}^{\infty}[\textit{f}\ln(f)+(1-f)\ln(1-f)]\\\
dy\right]\\\ $ (8)

here $t=T/T_{c}$, f($\xi$) = [exp(E($\xi$)/$k_{B}T$)+1]-1 is the Fermi
function, E($\xi$) = $\sqrt{\xi^{2}+\Delta^{2}(t)}$, with $E(\xi$) being the
energy of the normal electrons measured relative to the Fermi energy. y =
$\xi/\Delta(0)$ and $\Delta(t)$ =
tanh$\\{1.82(1.018[(\mathit{1/t})-1])\\}^{0.51}$ is the BCS approximation for
the temperature dependence of energy gap. The normalized $C_{el}(T)$ fitting
deviates from a single $s$-wave model, as shown in the Fig. 3(b), whereas a
two-gap isotropic $s+s$ model fits the data better. The same two-gap model has
also been used in other multi-gap superconductors such as MgB2 mgb2 ,
Lu2Fe3Si5 l2f3s5 , and La7Ni3 l7n3 . In this model, the two parameters
$\Delta_{1}(0)/k_{B}T_{c}$ and $\Delta_{2}(0)/k_{B}T_{c}$ correspond to the
two different superconducting gaps, and the weighted contribution, $x$ of the
partial Sommerfeld coefficient are the three fitting parameters involved. Best
fitting of the data yields $x$ = 0.17(6), $\Delta_{1}(0)/k_{B}T_{c}$ = 0.80(8)
and $\Delta_{2}(0)/k_{B}T_{c}$ = 2.16(3) (Fig. 3(b)). A significant difference
in the two gap values and gap ratio $\frac{\Delta_{2}(0)}{\Delta_{1}(0)}$
$\simeq$ 2.7(3) is observed with the smaller gap, $\Delta_{1}$ having
relatively weak contribution (0.17) in the superconducting state. The larger
gap value, $\Delta_{2}(0)/k_{B}T_{c}$ = 2.16, is higher than the BCS value in
the weak coupling limit, suggesting strong electron-phonon coupling in
AgSnSe2. A previous report on polycrystalline AgSnSe2 reported a single
isotropic gap (also somewhat larger than that expected in the weak coupling
limit of BCS theory); it is possible that the apparent absence of a two-gap
nature in that study might be due to sample differences or could indicate that
a multi-gap analysis was not performed ass2sc .

###  c Muon spin rotation and relaxation

To further understand the superconducting gap structure of the valence-skipped
superconductor AgSnSe2 at the microscopic level, we carried out transverse-
field $\mu$SR measurements. The measurements were performed in the field-
cooled protocol in a field of 0.1 T. Representative TF asymmetry spectra above
and below $T_{c}$ are plotted in a rotating reference frame of 0.095 T for
clarity as shown in Fig. 4(a), where a clear difference between the two is
observable. Below $T_{c}$, the precession signal shows significant damping
with time due to the inhomogeneous field distribution from flux-line-lattice
(FLL) in the mixed state. The time domain spectra were fit to analyze the
temperature dependence of superconducting parameters. The TF asymmetry spectra
are described by two sinusoidally oscillating functions, with the Gaussian
decaying component characterizing the sample signal and a temperature-
independent weakly decaying exponential component characterizing the
background signal,

$A_{TF}(T)=A[F\mathrm{exp}\left(\frac{-\sigma^{2}t^{2}}{2}\right)\mathrm{cos}(wt+\phi)+\\\
(1-F)\mathrm{exp}(-\sigma_{bg})\mathrm{cos}(w_{bg}t+\phi)]$ (9)

where $\phi$ is the initial phase of the muons entering the sample and $A$ is
the total TF asymmetry. $F$ is the fraction of the signal coming from the
sample, and $w,w_{bg}$ are the frequencies of the oscillatory muon signal in
the sample and background, respectively. $\sigma$ is the total Gaussian muon
relaxation rate. Above the transition temperature $T_{c}$, $\sigma$ becomes
temperature independent, reflecting the nuclear contribution in the normal
state called $\sigma_{n}$ (represented by a horizontal dashed line in Fig.
4(b)). The total relaxation rate, $\sigma$ consists contribution of both
superconducting and nuclear parts and can be written as,

$\sigma^{2}=\sigma_{sc}^{2}+\sigma_{n}^{2},$ (10)

where $\sigma_{sc}$ is the relaxation in the vortex state.

Figure 4:  (a) Transverse-field spectra recorded in an applied magnetic field
of 0.1 T at temperature 5.37 K ( > $T_{c}$) and 0.02 K (< $T_{c}$) plotted in
a rotating reference frame of 0.095 T, where solid lines are the fits using
Eq. (9). (b) Temperature dependence of total relaxation rate, $\sigma$. The
dashed line represents the $s$-wave fitting, the solid line represents the
two-isotropic $s+s$-wave model, and $\sigma_{n}$ is the fitting parameter.

The temperature-dependence of the superconducting muon depolarization rate,
$\sigma_{sc}$, can be expressed in the semiclassical approximation as,

$\frac{\sigma_{sc}(T,\Delta_{0,i})}{\sigma_{sc}(0,\Delta_{0,i})}=\frac{\lambda_{sc}^{-2}(T,\Delta_{0,i})}{\lambda_{sc}^{-2}(0,\Delta_{0,i})}=\\\
1+\frac{1}{\pi}\int_{0}^{2\pi}\int_{\Delta(T,\phi)}^{\infty}\left(\frac{\delta
f}{\delta E}\right)\frac{EdEd\phi}{\sqrt{E^{2}-\Delta_{i}(T,\phi)^{2}}}$ (11)

where $f(E)=[1+\exp(E/k_{B}T)]^{-1}$ is the Fermi function and
$\Delta_{i}(T,\phi)=\Delta_{0,i}\delta(T/T_{c})g(\phi)$. The temperature
variation $\delta(T/T_{c})=\tanh[1.82[1.018(T_{c}/T-1)]^{0.51}]$ and $g(\phi)$
refers to the angular dependence of the superconducting gap function having
azimuthal angle $\phi$.

For an isotropic $s$-wave gap, $g(\phi)$ becomes equal to one. Combining Eqs.
(10) and (11), the $\sigma$ variation with temperature can be fitted with
$\sigma_{n}$ as a fitting parameter. The $\sigma$ versus T curve fitted with
various models is shown in the Fig. 4(b). The isotropic $s$-wave model with a
gap value $\Delta_{0}$ = 0.68 (5) meV and $\chi^{2}$ = 1.43 does not properly
fit the temperature variation of $\sigma$. Hence a function consisting
weighted linear combination of two distinct superconducting gaps is
considered, which corresponds to the model used to fit the specific heat data
l2f3s5 ; l7n3 ,

$\frac{\sigma_{sc}(T)}{\sigma_{sc}(0)}=x\frac{\sigma_{sc}(T,\Delta_{0,1})}{\sigma_{sc}(0,\Delta_{0,1})}+(1-x)\frac{\sigma_{sc}(T,\Delta_{0,2})}{\sigma_{sc}(0,\Delta_{0,2})}\\\
$ (12)

where $\Delta_{0,1}$ and $\Delta_{0,2}$ are the two gap values, respectively,
and $x$ is the weighting factor, measuring the relative contributions to
superconducting parameters. The two-gap $s+s$ wave model fits well than the
single gap $s$-wave, for $x$ = 0.20(4), with the respective gap values of
$\Delta_{0,1}(0)$ = 0.29(6) meV ($\Delta_{0,1}(0)/k_{B}T_{c}$ = 0.77(9) and
$\Delta_{0,2}(0)$ = 0.80(5) meV ($\Delta_{0,2}(0)/k_{B}T_{c}$ = 2.12(5)) and
$\chi^{2}$ = 1.25 (Fig. 4(b)). The ratio of superconducting gaps
$\frac{\Delta_{0,2}(0)}{\Delta_{0,1}(0)}$ $\simeq$ 2.7(5) is consistent with
the value estimated from the specific heat measurement.

For an ideal vortex lattice, the relation between the magnetic penetration
depth, $\lambda$ and the superconducting depolarization rate $\sigma_{sc}$ for
GL parameter $\kappa$ $\geq$ 5 is given by SigSC_2 ; sigmsc_3 ,

$\frac{\sigma_{sc}^{2}(T)}{\gamma_{\mu}^{2}}=\frac{0.00371\Phi_{0}^{2}}{\lambda^{4}(T)}$
(13)

where $\gamma_{\mu}/2\pi$ = 135.5 MHz/T is the muon gyromagnetic ratio and
$\Phi_{0}$ = 2.068 $\times$ 10-15 Wb is the magnetic flux quantum. The
estimated magnetic penetration depth at $T$ = 0 K is $\lambda(0)$ = 901(35)
nm. The obtained value is significantly different from the value estimated
from magnetization, probably due to the different approximation considerations
while calculating. Such a large difference in $\lambda$ has also been observed
in some other compounds RRP ; Z2I .

To further investigate the superconducting ground state of AgSnSe2, ZF-$\mu$SR
measurements were also performed. The ZF-$\mu$SR time domain spectra were
measured at various temperatures on either side of the superconducting
transition temperature. Fig. 5 shows representative asymmetry spectra above
and below $T_{c}$. The ZF-$\mu$SR spectra were analyzed by fitting the time-
domain asymmetry variation using the damped Gaussian Kubo-Toyabe function
($G_{\mathrm{KT}}$) KT ,

$A(t)=A_{0}[fG_{\mathrm{KT}}(t)\mathrm{exp}(-\Lambda t)+\\\
(1-f)\mathrm{exp}(-\lambda_{BG}t)]+A_{BG},$ (14)

where

$G_{\mathrm{KT}}(t)=\frac{1}{3}+\frac{2}{3}(1-\Delta^{2}t^{2})\mathrm{exp}\left(\frac{-\Delta^{2}t^{2}}{2}\right)$
(15)

and $A_{0}$ and $A_{BG}$ are the initial asymmetries corresponding to the
sample and the background, respectively. $\Delta$ represents the muon spin
relaxation due to the randomly oriented, static nuclear moments experienced at
the muon site and $\Lambda$ accounts for the electronic relaxation rate. The
temperature independence of $\Lambda$ is shown in the inset of Fig. 5,
confirming the absence of any spontaneous internal magnetic field in the
superconducting state within our detection limit. The RMS deviation in value
of $\Lambda$ provides an upper limit on a spontaneous magnetic field of 0.01
mT. The obtained value is smaller than the value reported for many TRS-
breaking superconductors s2ro4 . However, there is still the possibility of
time-reversal symmetry breaking with a spontaneous field less than 0.01 mT;
such a small TRS breaking internal field has been previously observed in some
superconductors such as UPt3, Re6Hf, La7Ni3 upt3 ; re6hf ; l7n3 .

Figure 5:  Zero-field asymmetry spectra collected below ($T$ = 0.02 K) and
above ($T$ = 5.37 K) the transition temperature, $T_{c}$ where the solid lines
represent the fit to the data using Eqs. (14) and (15). The inset shows the
temperature variation of the electronic relaxation rate, $\Lambda$.

We can use Uemura’s classification Umera  scheme to place AgSnSe2 in the
context of other superconductors based on the ratio of
$T_{c}$/$T_{\mathrm{F}}$. For unconventional superconductors, the ratio
generally falls in 0.01 $\leq$ $T_{c}$/$T_{\mathrm{F}}$ $\leq$ 0.1, however
for conventional superconductors, $T_{c}$/$T_{\mathrm{F}}$ $\leq$ 0.0003. The
effective Fermi temperature, $T_{\mathrm{F}}$ of AgSnSe2 is extracted by
considering the 3D Fermi surface expression Tf ,

$k_{B}T_{\mathrm{F}}=\frac{\hbar^{2}}{2}(3\pi^{2})^{2/3}\frac{n^{2/3}}{m^{*}},$
(16)

where $n$ is the carrier density, and $m^{*}$ is the effective mass. $m^{*}$
is calculated from Sommerfeld coefficient via relation, $m^{*}={(\hbar
k_{\mathrm{F}})^{2}\gamma_{n}/\pi^{2}nk_{B}^{2}}$, with $\gamma_{n}$ = 84.51
Jm${}^{-3}K^{-2}$, $k_{\mathrm{F}}$ is the Fermi vector, and $n$ is taken from
ass2sc . The estimated value $T_{\mathrm{F}}$ = 16200(610) K places AgSnSe2
well outside the unconventional superconductors band, lying close to
conventional superconductors, as shown in Fig. 6. The Uemura classification
and no evidence of time-reversal symmetry breaking in AgSnSe2 suggest the
conventional superconducting pairing mechanism.

Doping with valency-skipped elements can enhance or induce superconductivity
in low-carrier-density systems such as topological semimetals or
semiconductors. This could be a prototype for realizing novel quantum phases
called topological superconductivity (TSC), where the topological phase and
superconductivity coexist tsc . An example is In-doped SnTe, a possible
topological superconductor, where the valence-skipped state of In induces
superconductivity in a topological crystalline insulator SnTe ist ; ist2 .
Moreover, Ag-doped SnSe and other systems, including In-doped GeTe and K-doped
BaBiO3, exhibit nontrivial topological band structures igt ; bbo3sc . Thus,
the aforementioned material corresponds to the category of possible
topological superconductors and presents AgSnSe2 as a candidate for TSC.
However, detailed studies and band structure calculations are required to
investigate the origin of multigap superconductivity and to address the effect
of the nontrivial band topology with the possible negative-$U$ induced
superconductivity in AgSnSe2.

Figure 6:  A plot between the superconducting transition temperature $T_{c}$ and the effective Fermi temperature $T_{\mathrm{F}}$ with AgSnSe2. The data points between the shaded region represent the band of unconventionality Unconv_1 ; Unconv_2 . Table 1: Parameters in the superconducting and normal state of AgSnSe2 Parameters | Unit | AgSnSe2
---|---|---
$T_{C}$ | K | 4.91(2)
$H_{c1}(0)$ | mT | 5.83(6)
$H_{c2}^{GL}(0)$ | T | 2.14(3)
$H_{c2}^{2G}(0)$ | T | 2.18(7)
$H_{c2}^{P}(0)$ | T | 9.13(3)
$\xi_{GL}(0)$ | nm | 12.2(4)
$\lambda_{GL}(0)$ (Mag.) | nm | 309(13)
$\lambda(0)$ ($\mu$SR) | nm | 907(35)
$k_{GL}(0)$ |  | 25(2)
$\gamma_{n}$ | mJ mol-1 K-2 | 4.6(1)
$\theta_{D}$ | K | 255(3)
$\lambda_{e-ph}$ |  | 0.69(1)
$\Delta_{2}/\Delta_{1}^{SH}$ |  | 2.7(3)
$\Delta_{2}/\Delta_{1}^{\mu SR}$ |  | 2.7(5)
$T_{\mathrm{F}}$ | K | 16200(610)
$T_{c}/T_{\mathrm{F}}$ |  | 0.0003(1)
$m^{*}$/$m_{e}$ |  | 1.88(2)

##  IV CONCLUSION

Detailed measurements of magnetization and specific heat have confirmed the
bulk superconductivity of AgSnSe2 at a transition temperature of 4.91(3) K.
Microscopic investigations were also carried out using muon spin rotation and
relaxation measurements on this valence-skipped superconductor. The results of
our TF-$\mu$SR measurements suggest the presence of two isotropic $s+s$
superconducting gaps, which is supported by the temperature variation of
specific heat and upper critical field measurements. Furthermore, our
ZF-$\mu$SR study confirms the preserved time-reversal symmetry in the
superconducting state, suggesting a conventional superconducting state in
valence-skipping mediated superconductors, despite the fact that in these
compounds, the attractive interaction between electrons is mediated via
negative -U due to valence-skipping/valence fluctuations, which is a non-BCS
pairing mechanism. However, the role of nontrivial topological states in the
superconducting ground state requires further investigation.

##  V Acknowledgments

A. Kataria acknowledges the funding agency Council of Scientific and
Industrial Research (CSIR), Government of India, for providing SRF fellowship
(Award No: 09/1020(0172)/2019-EMR-I). R. P. S. acknowledges Science and
Engineering Research Board, Government of India, for the CRG/2019/001028 Core
Research Grant. G.M. Luke (McMaster) acknowledges the support of the Natural
Sciences and Engineering Research Council (Canada).

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