# Superconducting properties of pseudobinary telluride Chevrel Phase Mo4Re2Te8

A. Kataria  Department of Physics, Indian Institute of Science Education and
Research Bhopal, Bhopal, 462066, India  T. Agarwal  Department of Physics,
Indian Institute of Science Education and Research Bhopal, Bhopal, 462066,
India  S. Sharma  Department of Physics, Indian Institute of Science Education
and Research Bhopal, Bhopal, 462066, India  A. Ali  Department of Physics,
Indian Institute of Science Education and Research Bhopal, Bhopal, 462066,
India  R. S. Singh  Department of Physics, Indian Institute of Science
Education and Research Bhopal, Bhopal, 462066, India  R. P. Singh
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Science
Education and Research Bhopal, Bhopal, 462066, India

###### Abstract

Unconventional superconductivity in the Chevrel phase offers a wide structural
aspect to understand the superconducting ground state. A detailed
investigation on the superconducting properties of Re based pseudobinary
telluride Chevrel phase Mo4Re2Te8 is reported. It crystallizes in a trigonal
structure with the space group $R\bar{3}H$ having superconducting transition
temperature at $T_{C}$ = 3.26(3) K. Specific heat measurements suggests a
fully gapped superconducting state; however, the proximity of upper critical
field value from the Pauli limiting field can be attributed to unconventional
nature.

##  I Introduction

Unconventional superconductors are an integral part of a new phase of quantum
matter exhibiting additional broken symmetries (e.g. time reversal or rotation
symmetry) with the global gauge symmetry, U(1). Chevrel phase (CP) is one of
the crucial members of the unconventional superconductor family, providing a
broader aspect of interactions and structural impact on the superconducting
ground state Uemura . This phase has a perplexing array of unconventional
superconducting properties such as extremely high upper critical magnetic
field, multicomponent and multiband superconductivity, magnetic field driven
or magnetically ordered superconductivity, reentrant superconductivity, and
many more hhc ; mbsc ; mfisc ; 2gsc ; nsc ; Hom6x8 ; REmo6X8 . Additionally,
the coupling manner of building blocks (Mo clusters) of classical CPs multi-
folds various crystal structures 6strt ; 7strt2 ; cmo ; scb1  producing
quasi-1D superconductor Q1D  and exhibiting interesting topological properties
Tsc .

The CPs are extensively known for their relatively high value of the upper
critical field, and it is believed that the high spin-orbit splitting by
possibly large Sommerfeld coefficient, $\gamma_{n}$ ascribed to low-lying
phonons responsible for it bC1 ; mc . The observed large superconducting gap
value in CPs is similar to the superconductivity mediated by soft phonons
(low-lying phonons) as recently observed in IrGe IG , BaPd2As2 BP2A2  and
SrPt3P Sp3 . Further, the superconducting properties of CPs depend on the
intercluster Mo-Mo distance scb1 ; scb . The partial substitution at Mo site
can change metallic compound Mo6X8 into the semiconducting sulphides and
selenides rurh ; mre  and in contrast, tellurides of this family become
superconductor mrute ; scb ; MRu . Most studies on the unconventional
superconducting properties of CPs have primarily focused on the rare-earth
intercalated sulfides and selenides. However, to date, the high upper critical
field and pairing mechanism are not understood properly. Along with this, the
superconducting properties of Chevrel phase tellurides are largely unexplored.
It can provide details about the electron-electron interaction, electron-
phonon coupling and spin-orbit coupling effect on the nature of
superconductivity and pairing mechanism, which can be helpful to understand
the unconventionality in the superconducting state of CPs. Detailed studies of
new CPs are essential for deep insight into the superconducting ground state
and gap symmetry.
In this work, we investigate the superconducting properties of a less explored
Re based pseudobinary telluride Chevrel phase Mo4Re2Te8, where Re is partially
substituted at the Mo site. This study comprises two different prospects;
firstly, the presence of Re in the Mo cluster will increase the spin-orbit
coupling strength (SOC $\propto$ Z4) and affect the phononic distribution by
affecting the related interactions of the CP system. Secondly, this compound
can address the ambiguity in Re-based superconductors around the possible
reason for time-reversal symmetry (TRS) breaking. Such as Re6X (X = Zr, Hf,
Ti), a noncentrosymmetric family shows spontaneous field presence, breaking
TRS, regardless of the element X re6zr ; re6hf ; re6ti , while the other non-
centrosymmetric compounds Re3Y (Y = Ta, W) and the Re-B system re3ta ; re3w ;
reb , preserve time-reversal symmetry. Furthermore, the uncertainty in TRS
breaking presence in centrosymmetric Re is also intriguing Re ; Ren  and
raises more questions about the role of associated structure and Re
concentration in time reversal symmetry breaking. In this regard,
investigating more Re based superconductors are essential and Re based
pseudobinary telluride CP Mo4Re2Te8, provides that platform with a new
structural aspect and different Re concentration. We have performed the
temperature-dependent measurements of AC transport, magnetization, and
specific heat in different magnetic fields, which allow deducing the
superconducting characteristics parameters with other electronic parameters of
Mo4Re2Te8. The extracted upper critical field value is close to the Pauli
limiting field, which suggests the possibility of unconventionality in the
superconducting ground state. Specific heat measurement indicates a moderate
electron-phonon coupling with $s$-wave gap symmetry. Moreover, initial band
structure calculations suggests importance of SOC on the electronic states of
Mo4Re2Te8 along with dominance of d state of Re and Mo atoms in density of
states at the Fermi level.

##  II Experimental Details

Polycrystalline sample of the nominal composition Mo4Re2Te8 was prepared by
solid-state reaction method where the constituent elemental powder of Mo
(99.99%), Re (99.99%), and Te (99.9999%) was mixed together in a
stoichiometric ratio. The palletized form of the mixture was sealed in an
evacuated quartz ampoule and heated between 1150$\degree-$1200$\degree$C for
several days. 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 on a superconducting quantum
interference device of Quantum Design magnetic property measurement system
(MPMS 3, Quantum Design). Transport measurements and specific heat were
carried out on a physical property measurement system (PPMS). The four-probe
technique is used to measure the AC transport, and the two - $\tau$ relaxation
method is used for specific heat measurement.

##  III Results and Discussion

###  a Sample characterization

Figure 1:  Powder x-ray diffraction pattern for Mo4Re2Te8 sample at room
temperature. The solid brown line represents the Rietveld refinement of the
data, with green vertical bars indicating the Bragg reflection peaks. Inset
shows the Mo4Re2Te8 cluster.

The recorded pattern of XRD for Mo4Re2Te8 was refined using Fullprof software
fp . Rietveld refinement confirms the trigonal structure having space group
$R\bar{3}H$, and Fig. 1 shows the refined pattern. The refined lattice
parameters of Mo4Re2Te8 are tabulated together with the parent compound Mo6Te8
in Table I. Mo6-xRexTe8 ($x$ = 2) cluster, the building block of the crystal
structure is displayed in the inset of Fig. 1.

Table 1: Structure parameters of Mo4Re2Te8 obtained from the Rietveld
refinement of x-ray diffraction.

Structure | Trigonal
---|---
Space group | $R\bar{3}H$

Lattice parameters | Mo4Re2Te8 | Mo6Te8 M6T8  |
---|---|---|---
a = b (Å) | 10.233(1) | 10.179(1) |
c (Å) | 11.530(1) | 11.674(2) |
|  $\alpha=\beta=90\degree$, | $\gamma=120\degree$ |

###  b Superconducting and normal state properties

####  1 Electrical Resistivity

Figure 2: (a) Temperature dependence of resistivity of Mo4Re2Te8 is fitted
using BG model shown by the solid purple line. (b) The superconducting drop in
resistivity at the transition temperature, $T_{C}$ = 3.54(2) K. (c) Shows Hall
resistivity, $\rho_{xy}$ measurement at 20 K under $\pm$ 9 T magnetic field.
Figure 3: (a) The magnetic moment versus temperature for Mo4Re2Te8 with inset
showing the magnetization loop at 1.8 K under $\pm$ 4 T magnetic field. (b)
Lower critical field estimated from $M$-$H$ curve (inset) at varying
temperature. (c) Estimated upper critical field by resistivity, specific heat
and magnetization data where dotted lines represent fitting using Eq. (3).
Inset displays resistivity variation with the temperature at different applied
fields.

The temperature dependence of resistivity, $\rho(T)$ of the compound Mo4Re2Te8
in zero magnetic field is shown in Fig. 2(a). The onset of sharp drop-in
resistivity is recorded at temperature $T_{C,onset}$ = 3.54(2) K with the
transition width of $\Delta T$ = 0.3 K (Fig. 2(b)). The observed $T_{C}$
matches well with the reported value mrute ; MRu . The low temperature (in
normal state) variation of $\rho(T)$ demonstrates the metallic nature of
Mo4Re2Te8 with the increasing behaviour. Hence, the $\rho(T)$ temperature
dependence is analysed by using the equation,

$\rho(T)=\rho_{0}+C\left(\frac{T}{\theta_{R}}\right)^{n}\int_{0}^{\theta_{R}/T}\frac{x^{n}}{(e^{x}-1)(1-e^{-x})}dx\\\
$ (1)

where the second term is the Bloch-Grüneisen expression accounting for the
electron-phonon scattering bg . $\rho_{0}$ is the residual resistivity due to
defect scattering, $\theta_{R}$ is debye temperature from resistivity
measurements, and $C$ is a material dependent property. $n$ depends on the
nature of the interaction. Best fitting of $\rho(T)$ below 100 K, is for $n$ =
3 providing $\theta_{R}$ = 143(2)K, $C$ = 0.51(1) m$\ohm$-cm and $\rho_{0}$ =
2.95(1) m$\ohm$-cm. Moreover, the resistivity trend in high temperature region
for Mo4Re2Te4 consists of negative curvature ($d\rho/dT$) with the nearly
saturated resistivity, indicating an additional contribution in the system
with the metallic nature. This behaviour is reminiscent of the other Chevrel
phase, skutterudites, and clathrates compounds consisting of similar type of
cluster structures 2gsc ; cpr ; cla ; sku ; cla1 . Moreover, the same trend is
also observed in La7X3 (X = Rh, Ir) family and YCo2, where the electronic
scattering is related to the excitations with a broad spectrum of energy is
stated the possible reason for the resistivity trend l7x3 ; co2 .

Further, the carrier density, $n$, is estimated from the Hall measurement. The
longitudinal resistivity, $\rho_{xy}$ measured at 20 K in field dependence,
shown in the Fig. 2(c). The observed positive slope indicates the dominant
hole carriers and provides the Hall resistivity, $R_{H}$ = $5.2(1)\times
10^{-10}$ m-3C. Considering only the single band contribution, the estimated
number density $n$ is 1.21(3) $\times$ 1028 m-3, which is again in good
agreement with the known CPs and their selenides substitutes n ; ncp .

####  2 Magnetization

Figure 4: (a) The low-temperature normal region fitting of $C/T$ vs $T^{2}$
curve for Mo4Re2Te8 is represented by the solid line. (b) Shows the variation
of specific heat jump under magnetic field. (c) Electronic specific heat data,
$C_{el}$, is well described by BCS single gap $s$-wave model shown by solid
black line.

The magnetic moment under temperature variation of Mo4Re2Te8 was measured at 1
mT using both zero-field-cooled warming (ZFCW) and field-cooled cooling (FCC)
mode, as shown in Fig. 3(a). The onset of the diamagnetic signal of the
compound is noted at $T_{C,onset}$ = 3.26(3) K, where the deviation of FCC
curve from the ZFCW depicts the flux pinning in the sample. The inset of Fig.
3(a) shows the magnetization loop under magnetic field variation $\pm$ 4 T at
1.8 K of Mo4Re2Te8. The full magnetization loop confirms the type-II
superconductivity with bulk pinning in the sample representing an irreversible
nature of magnetization, $H_{irr}$ = 1.54 T, and a further increase in the
applied magnetic field de-pins the vortices.

The lower critical field value, $H_{C1}$(0) is extracted from the
magnetization dependence on the magnetic field measured at different
temperatures up to $T_{C}$, as shown in the inset of Fig. 3(b). The linear
deviation from Meissner effect (entering in the vortices region) is considered
as the $H_{C1}$ for the respective isotherm. The temperature dependence of
$H_{C1}$ for Mo4Re2Te8 is shown in Fig. 3(b) and fitted by using the Ginzburg-
Landau equation given as,

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

The estimated value of $H_{C1}(0)$, is 0.49(1) mT. The upper critical field,
$H_{C2}(0)$, value is computed via variation of transition temperature in
various applied magnetic fields. The applied, increasing magnetic field shifts
the superconducting transition temperature to the lower value as observed in
resistivity, magnetization and specific heat measurements. The inset of Fig.
3(c) indicates the same via resistivity measurement, where $T_{C}$ is recorded
at 90% drop in normal state resistivity value. The upper critical field value,
$H_{C2}$ variation with reduced temperature, $t=T/T_{C}$ was fitted with the
Ginzburg-Landau equation,

$H_{C2}(T)=H_{C2}(0)\left[\frac{(1-t^{2})}{(1+t^{2})}\right].$ (3)

The well-fitted data is shown in Fig. 3(c), yielding upper critical field,
$H_{C2}(0)$ = 5.08(7) T, 5.62(6) T and 5.78(4) T from magnetization, specific
heat and resistivity data, respectively. The effect of an applied external
magnetic field is described by two mechanisms involved (1) orbital decoupling
and (2) Pauli limiting effect. The orbital limiting field is where the
increased kinetic energy of one electron breaks the Cooper pair, given by the
Wartherm-Helfand-Hohenberg (WHH) expression WHH_1 ; WHH_2 ,

$H_{C2}^{orbital}(0)=-\alpha
T_{C}\left.\frac{dH_{C2}(T)}{dT}\right|_{T=T_{C}}$ (4)

here $\alpha$ is considered as 0.69 for a dirty limit superconductor. The
initial slope $\frac{-dH_{C2}(T)}{dT}$, in the vicinity of $T_{C}$ is 1.58(1)
T for Mo4Re2Te8, which evaluates the orbital limiting field,
$H_{C2}^{orbital}(0)$ = 3.55(4) T. From BCS theory of superconductors, another
effect Pauli limiting field is represented as $H_{C2}^{P}$(0) = 1.84 $T_{C}$
Pauli_1 ; Pauli_2 , with $T_{C}$ = 3.26(3) K, $H_{C2}^{P}$(0) becomes 5.99(5)
T. Moreover, the relative strength of two magnetic effects, orbital and Pauli
limiting field, is measured by the Maki parameter maki ,
$\alpha_{M}=\sqrt{2}H_{C2}^{orb}(0)/H_{C2}^{P}(0)$ = 0.84(2). The close value
of $\alpha_{M}$ to one indicates the non-negligible effect of Pauli limiting
field in the Cooper pair breaking. Along with this, the proximity of upper
critical field value ($H_{C2,\rho}(0)=5.78(4)$ T) with Pauli limiting field is
also compelling and suggests the possible unconventional nature associated
with the system similar to other Re based superconductors re6zr ; re5ta ; HPG
.

Superconducting characteristic length parameters have been calculated from
upper and lower critical field values. The relation between $H_{C2}(0)$ and
Ginzburg-Landau coherence length, $\xi_{GL}(0)$ Coh_Leng ,
$H_{C2}(0)=\frac{\Phi_{0}}{2\pi\xi_{GL}(0)^{2}}$, where $\Phi_{0}$ = 2.07
$\times 10^{-15}$ Tm2 is the quantum flux, gives $\xi_{GL}(0)$ = 80(1) Å. The
lower critical field, $H_{C1}(0)$ and coherence length, $\xi_{GL}(0)$ is
related to the penetration depth as follows MIB ,

$H_{C1}(0)=\frac{\Phi_{0}}{4\pi\lambda_{GL}^{2}(0)}\left(\mathrm{ln}\frac{\lambda_{GL}(0)}{\xi_{GL}(0)}+0.12\right)$
(5)

providing $\lambda_{GL}(0)$ = 1325(40) nm. The value of the Ginzburg-Landau
parameter, $\kappa_{GL}=\frac{\lambda_{GL}(0)}{\xi_{GL}(0)}$, is calculated to
be 165(7), indicating a strong type-II superconductivity in Mo4Re2Te8
consistent with the previously studied CPs. The thermodynamic critical field
is also evaluated using the relation,
$H_{C}=\sqrt{\frac{H_{C1}(0)H_{C2}(0)}{{\ln}\kappa_{GL}}}$ MIB  and obtains
the value, $H_{C}$ = 22(1) mT for Mo4Re2Te8.

####  3 Specific heat

The bulk superconductivity is further probed by specific heat versus
temperature measurement at zero applied magnetic field for Mo4Re2Te8. Fig.
4(a) shows the jump in specific heat data at $T_{C,mid}$ = 3.23(8) K
considering the mid-point. Above $T_{C}$, the normal state region of specific
heat is fitted using the Debye relation, $C/T=\gamma_{n}+\beta_{3}T^{2}$,
where $\gamma_{n}$ is the Sommerfeld coefficient, and $\beta_{3}$ is the
lattice constant. Fitting of the data provides the parameters as $\gamma_{n}$
= 24.7(8) mJmol-1K-2 and $\beta_{3}$ = 5.95(4) mJmol-1 K-4. Further, the
change in superconducting transition temperature with applied magnetic field
is shown in Fig. 4(b).

Specific heat measurement also give access to calculate several other
parameters. Lattice constant parameter, $\beta_{3}$ is used to extract the
information about the phonons and provides the Debye temperature $\theta_{D}$.
Using the relation, $\theta_{D}=(\frac{12\pi^{4}RN}{5\beta_{3}})^{1/3}$, where
$R$ is the universal gas constant, and $N$ = 14 the number of atoms per
formula unit, the estimated value is 166(1) K. The obtained value of
$\theta_{D}$ from specific heat matches with the value from resistivity
measurement. The Sommerfeld coefficient, $\gamma_{n}$ is related to single
particle density of states at the Fermi level by expression,
$\gamma_{n}=(\frac{\pi^{2}k_{B}^{2}}{3})D_{c}(E_{\mathrm{F}})$ and the
operation yields $D_{c}(E_{\mathrm{F}})$ = 10.5(3) states eV-1 f.u.-1. The
information on electron-phonon coupling strength can also be extracted from
McMillian theory, where the dimensionless quantity, $\lambda_{e-ph}$ is stated
by the relation mcm ,

$\lambda_{e-ph}=\frac{1.04+\mu^{*}\mathrm{ln}(\theta_{D}/1.45T_{C})}{(1-0.62\mu^{*})\mathrm{ln}(\theta_{D}/1.45T_{C})-1.04}$
(6)

here $\mu^{*}$ is typically assumed to be 0.13 for intermetallic
superconductors. The values $\theta_{D}$ = 166(1) K and $T_{C}$ = 3.23(8) K,
assort the value $\lambda_{e-ph}$ = 0.67(2), suggest that the telluride
chevrel phase Mo4Re2Te8 is a moderately-coupled superconductor. Even though,
the high DOS at the Fermi level, large contribution to the lattice specific
heat, $\beta_{3}$ and parameter $\gamma_{n}/T_{C}$ = 7.6 mJmol-1 K-3 of
Mo4Re2Te8 are comparable to SnMo6S8, PbMo6S8 and Ag1-xMo6S8, possessing strong
electron-phonon coupling, Mo4Re2Te8 has moderate coupling. The ratio
$\gamma_{n}/T_{C}$ indicates the scale of transition temperature with the
density of states at $E_{\mathrm{F}}$ 2gsc ; mbsc .

The symmetry associated with the superconducting gap can be understood by
analyzing the electronic specific heat temperature dependence in the
superconducting region. The electronic contribution to the specific heat is
calculated by subtracting the phononic contribution from the total specific
heat at zero field. While theoretically can be computed from
$C_{el}=t\frac{dS}{dt}$, with $S$ being the entropy in the superconducting
region and within BCS approximation has the form,

$S=-\frac{6\gamma_{n}}{\pi^{2}}\left(\frac{\Delta(0)}{k_{B}}\right)\int_{0}^{\infty}[\textit{f}\ln(f)+(1-f)\ln(1-f)]dy$
(7)

where $f(\xi)=[\exp(E(\xi))/k_{B}T)+1]^{-1}$ is the Fermi function.
$E(\xi)=\sqrt{\xi^{2}+\Delta(t)^{2}}$ is the excitation energy of the
quasiparticle measured relative to the Fermi level, with $y=\xi/\Delta(0)$ and
$\Delta(t)$ is the temperature-dependent gap function. In the isotropic
$s$-wave BCS approximation, gap function can be written as, $\Delta(t)$ =
tanh$\\{1.82(1.018[(\mathit{1/t})-1])\\}^{0.51}$ with $t=T/T_{C}$. Fig. 4(c)
displays the electronic specific heat, $C_{el}$ variation with temperature,
fitted with the BCS $s$-wave model. The well fitted data in the
superconducting region yields $\Delta(0)/k_{B}T_{C}$ = 1.81(4) while the
specific heat jump value $\Delta C_{el}/\gamma_{n}T_{C}$ = 1.44(8). Both these
values, $\Delta(0)/k_{B}T_{C}$ and $\Delta C_{el}/\gamma_{n}T_{C}$ are close
to the BCS predicted value (1.76 and 1.43) in a weak coupling limit. Hence,
indicates Mo4Re2Te8 as a weakly coupled superconductor. However, to know the
exact structure of the superconducting gap and the nature of the pairing
state, low-temperature specific heat below 1.9 K is required.

####  4 Electronic properties and the Uemura plot

Figure 5:  Band dispersion and density of states of Mo4Re2Te8 calculated with
GGA+SO.

To get an insight about electronic properties, we have performed band
structure calculations using the full potential linear augmented plane wave
(FP-LAPW) method as implemented in Wien2k wien2k1 ; wien2k2 . We used $k$
point mesh of 10 $\times$ 10 $\times$ 10 within the first Brillouin zone, and
the generalized gradient approximation (GGA) exchange-correlation functional
of Perdew, Burke and Ernzerhof (PBE) PBE . SOC was included for all the
elements in the calculations. The energy and charge convergence criteria was
set to 0.01 meV and 10-4 electronic charge per f.u., respectively. The
Mo4Re2Te8 structure was simulated by substituting Re at the place of two Mo
atoms in Mo6Te8. The electronic band structure between high symmetry points bz
in the Brillouin zone and DOS of Mo4Re2Te8 are shown in Fig. 5. The inclusion
of SOC for Re significantly modifies the band dispersion while SOC in Mo and
Te do not have much influence, indicating high SOC of Re plays an important
role in Mo4Re2Te8. Two doubly degenerate bands, B1 and B2 shown in red and
blue colour respectively, are dispersing in large energy window appear in the
vicinity of Fermi level ($E_{\mathrm{F}}$) and crosses $E_{\mathrm{F}}$ at
different $k$ points of the Brillouin zone. Fermi level lies almost at the
middle of band B1 while band B2 remains almost empty. The system is close to
cubic (Rhombohedral angle = 92.12°) and the high symmetry points P, Z, Q, P1
and Q1 are almost equivalent where a small electron pocket (band B2) is seen
in the band structure. At the same time, a large hole pocket is centred around
$\Gamma$ point (band B1) as also observed from the Hall measurement suggesting
dominant hole carriers.

Strong hybridization between Mo/Re $d$ states and Te $p$ states is observed
and seen in the DOS plot. The calculated value of total DOS at
$E_{\mathrm{F}}$, is 9.87 states eV-1f.u.-1 in qualitative agreement with
values obtained from the specific heat experiment.

Figure 6:  A plot between the $T_{C}$ and the $T_{\mathrm{F}}$ representing
the unconventional superconductors family. Mo4Re2Te8 is shown, which is close
to conventional superconductors band region regardless of the other Chevrel
phase which lies in the unconventional band Unconv_1 ; Unconv_2 .

Further, in order to quantify the London penetration depth, $\lambda_{L}$,
electronic mean free path, $l_{e}$ and to verify the dirty limit
superconductivity for the Mo4Re2Te8 a set of equations are implemented. The
Fermi wave vector, $k_{\mathrm{F}}$ is estimated from quasiparticle number
density by the relation, $k_{\mathrm{F}}=(3\pi^{2}n)^{1/3}$, here
$n=1.21(3)\times 10^{28}$ m-3 from normal Hall measurement, hence
$k_{\mathrm{F}}$ = 0.71(2) $\text{\AA}^{-1}$. Sommerfeld coefficient,
$\gamma_{n}$ and $k_{\mathrm{F}}$ is used for effective mass value estimation
by the relation, $m^{*}=(\hbar k_{\mathrm{F}})^{2}\gamma_{n}/\pi^{2}nk_{B}$,
with $k_{B}$ being the Boltzmann constant, $m^{*}$ becomes 3.2(3) $m_{e}$.
Moreover, in consideration of Drude’s model, the mean free path is defined as,
$l_{e}$ = $v_{\mathrm{F}}\tau$, where $\tau$ is the scattering time and given
by $\tau^{-1}=ne^{2}\rho_{0}/m^{*}$. $v_{\mathrm{F}}$ is the Fermi velocity
stated as $v_{\mathrm{F}}=\hbar k_{\mathrm{F}}/m^{*}$. Including the
respective values of $m^{*}$, $k_{\mathrm{F}}$, $n$, and residual resistivity,
$\rho_{0}$ = 2.95(1) m$\ohm$-cm, the scattering time, $\tau$ and the Fermi
velocity $v_{\mathrm{F}}$ are evaluated, which provides mean free path,
$l_{e}$ = 0.81(9) Å. From BCS theory Coh_Leng , the coherence length $\xi_{0}$
is approximated as $0.18\hbar v_{\mathrm{F}}/k_{B}T_{C}$, hence becoming
1074(202) Å for $T_{C}$ = 3.23 K. The huge difference between the coherence
length and mean free path, $\xi_{0}\gg$ $l_{e}$ puts Mo4Re2Te8 in the dirty
limit superconductor. Further, London penetration depth $\lambda_{L}$, is
expressed as, $\lambda_{L}=\left(\frac{m^{*}}{\mu_{0}ne^{2}}\right)^{1/2}$ and
calculated to be 86(1) nm. The characteristics nature of poor metal with the
short mean free path of the known CPs is also observed in our telluride CP
compound Mo4Re2Te8.

Chevrel phase has been classified as unconventional superconductor based
according to their $\frac{T_{C}}{T_{\mathrm{F}}}$ ratio provided by Uemura et
al. Uemura . The ratio for Chevrel phase lies in the range 0.01 $\leq$
$\frac{T_{C}}{T_{\mathrm{F}}}$ $\leq$ 0.1 with high-$T_{C}$, organic, heavy-
fermion, and other unconventional superconductors. For 3D system, assuming
spherical Fermi surface, the Fermi temperature, $T_{\mathrm{F}}$ is given by
the relation Tf

$k_{B}T_{{\mathrm{F}}}=\frac{\hbar^{2}}{2}(3\pi^{2})^{2/3}\frac{n^{2/3}}{m^{*}},$
(8)

where $n$ is the quasiparticle number density per unit volume, and $m^{*}$ is
the effective mass of quasiparticles. Considering the values of $n$ and
$m^{*}$ listed in Table II, from equation 8, we get $T_{{\mathrm{F}}}$ =
6980(845) K for Mo4Re2Te8. The ratio $\frac{T_{C}}{T_{{\mathrm{F}}}}$ = 0.0005
placing Mo4Re2Te8 in between the region of Re based compounds having Re24Ti5
r24t5  on its left and other compounds on its right side of same composition
re6hf ; re6zr ; Re ; re5ta , with the Re element Re , as shown in Fig. 6 .

Table 2: Parameters in the superconducting and normal state of Mo4Re2Te8 Parameters | Unit | Mo4Re2Te8
---|---|---
$T_{C}$ | K | 3.26(3)
$H_{C1}(0)$ | mT | 0.49(1)
$H_{C2}^{\rho}(0)$ | T | 5.78(4)
$H_{C2}^{P}(0)$ | T | 5.99(5)
$H_{C2}^{Orb}(0)$ | T | 3.55(4)
$\xi_{GL}(0)$ | Å | 80(1)
$\lambda_{GL}(0)$ | nm | 1325(40)
$k_{GL}$ |  | 165(7)
$\gamma_{n}$ | mJ mol-1 K-2 | 24.7(8)
$\theta_{D}$ | K | 166(1)
$\Delta C_{el}/\gamma_{n}T_{C}$ |  | 1.44(8)
$\Delta(0)/k_{B}T_{C}$ |  | 1.81(4)
$v_{{\mathrm{F}}}$ | 105m s-1 | 2.6(2)
$n$ | 1028m-3 | 1.21(3)
$T_{{\mathrm{F}}}$ | K | 6980(845)
$T_{C}/T_{{\mathrm{F}}}$ |  | 0.0005(1)
$m^{*}$/$m_{e}$ |  | 3.2(3)

Irrespective of short mean free path, high DOS at Fermi level with moderate
electron-phonon coupling in pseudobinary compound Mo4Re2Te8, the exotic
character of the extremely high upper critical field value of CPs is absent.
As these parameters but with the strong electron-phonon coupling were
accounted to be the possible reason for high upper critical field value in
well-known CPs bC1 ; mc . Hence, strong electron-phonon coupling might be a
certain possible requirement to enhance the upper critical field beyond a
limit in CPs. However, the comparable value of the upper critical field to the
Pauli paramagnetic limit also suggests an unconventional nature in its
superconducting ground state. Notably, such a relatively large value of the
upper critical field has not been observed in any other Te superconductors,
excluding the iron-based chalcogenide superconductor fst . Further, the Uemura
classification of Mo4Re2Te8 places it in an interesting position between the
two different Re concentration systems. Hence, more experimental details are
required to elucidate the associated pairing mechanism with the
superconducting ground state in this Chevrel phase, and muon spectroscopy is
required to study the time-reversal symmetry.

##  IV CONCLUSION

To summarize, pseudobinary telluride Chevrel phase Mo4Re2Te8 is investigated,
and the superconducting transition at $T_{C}$ = 3.26(3) K is recorded. XRD
analysis confirms the phase purity and trigonal structure crystallization. AC
resistivity shows the poor metallic character of the sample with a short mean
free path. The magnetization measurements suggest the type-II
superconductivity with an upper critical field, $H_{C2}(0)$ = 5.78(4) T, close
to the Pauli limiting field. The specific heat data measured in zero applied
magnetic field presents the jump value $\Delta C_{el}/\gamma_{n}T_{C}$ =
1.44(8) with fully gapped superconductivity in Mo4Re2Te8. Initial band
structure calculations also depict significant contribution of Re in DOS at
the Fermi level with notable spin-orbit coupling impact of Re in band
dispersion. A Detailed microscopic study (e.g. muon spectroscopy) combined
with detailed theoretical calculations of electronic structure is required to
understand the superconducting ground state and the role of the Re on the
superconducting properties of this compound.

##  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 Core Research Grant
CRG/2019/001028.

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