# Superconducting Properties and $\mu$SR Study of the Noncentrosymmetric
Superconductor Nb0.5Os0.5

D. Singh  Department of Physics, Indian Institute of Science Education and
Research Bhopal, Bhopal, 462066, India  J. A. T. Barker  Physics Department,
University of Warwick, Coventry CV4 7AL, United Kingdom  A. Thamizhavel
Department of Condensed Matter Physics and Materials Science, Tata Institute
of Fundamental Research, Mumbai 400005, India  A. D. Hillier  ISIS Facility,
STFC Rutherford Appleton Laboratory, Harwell Science and Innovation Campus,
Oxfordshire, OX11 0QX, UK  D. McK. Paul  Physics Department, University of
Warwick, Coventry CV4 7AL, United Kingdom  R. P. Singh<EMAIL_ADDRESS>Department of Physics, Indian Institute of Science Education and Research
Bhopal, Bhopal, 462066, India

###### Abstract

The properties of the noncentrosymmetric superconductor ($\alpha$-Mn
structure) Nb0.5Os0.5 is investigated using resistivity, magnetization,
specific heat, and muon spin relaxation and rotation ($\mu$SR) measurements.
These measurements suggest that Nb0.5Os0.5 is a weakly coupled
($\lambda_{e-ph}$ $\sim$ 0.53) type-II superconductor ($\kappa_{GL}$ $\approx$
61) having a bulk superconducting transition temperature $T_{c}$ = 3.07 K. The
specific heat data in the superconductive regime fits well with the single-gap
BCS model indicating nodeless s-wave superconductivity in Nb0.5Os0.5. The
$\mu$SR measurements also confirm s-wave superconductivity with the preserved
time-reversal symmetry.

##  I Introduction

Understanding the mechanism of unconventional superconductivity, where the
structure lacks an inversion symmetry has been a tough challenge ever since
the discovery of the heavy fermion noncentrosymmetric (NCS) superconductor
CePt3Si Bauer2004 ; EBA . The lack of an inversion center in the crystal
structure of the noncentrosymmetric superconductor makes parity an unconserved
quantity. As a result, the superconducting ground state of an NCS
superconductor may exhibit a possible mixing of spin-singlet and spin-triplet
pair states rashba ; sky ; kv ; ia ; pa ; fujimoto1 ; fujimoto2 ; fujimoto3 ;
mdf . The parity mixed superconducting ground state gives rise to several
anomalous superconducting properties, e.g. upper critical field exceeding the
Pauli limit, nodes in the superconducting gap, a helical vortex state, and
time-reversal symmetry breaking.
Several NCS superconducting systems have been investigated to study the
effects of broken inversion symmetry nr1 ; RT1 ; rf ; rhf ; rz3 ; YC ; LC ; rw
; mib ; lip ; mac1 ; rg ; ig ; lrs ; lps ; rb1 , but majority of them appear
to show s-wave superconductivity. Theoretical predictions suggest that NCS
superconductors are prime candidates to exhibit time-reversal symmetry
breaking (TRSB) due to its admixed superconducting ground states. To date only
a few NCS superconductors Re6Zr rz1 , LaNiC2 lnc2 , SrPtAs SPA  and La7Ir3 li1
have been reported to show TRSB. It is a rarely observed phenomena and apart
from NCS superconductors, it has only been observed in a few unconventional
superconductors e.g. Sr2RuO4 sro1 ; sro2 , UPt3 UP1 ; UP2 , PrPt4Ge12 ppg ,
LaNiGa2 LNG , and Lu5Rh6Sn18 LoS . The discrepancy between theory, experiment
and the possibility of realizing an unconventional superconducting state
having TRSB in NCS superconductors are of great interest. To understand the
superconducting mechanism, it is required to study new NCS superconducting
systems by combining bulk measurements such as transport, magnetization, heat
capacity, etc. and local probe techniques like muon spectroscopy. Muon
spectroscopy is one of the most direct methods of detecting the unconventional
superconducting ground state. This technique can accurately determine the
temperature dependence of the magnetic penetration depth and the onset of
time-reversal symmetry breaking in superconductors.
Here we are reporting the superconducting state of a binary NCS compound
($\alpha$ \- $Mn$ structure) Nb0.5Os0.5, having superconducting transition
$T_{c}$ = 3.07 K. Resistivity, magnetization, and specific heat measurements
were carried out to explore the superconducting properties of Nb0.5Os0.5.
$\mu$SR measurements in transverse-field (TF) and longitudinal-field (LF)
configurations are used to probe the flux line lattice (FLL) and time-reversal
symmetry breaking respectively.

##  II Experimental Details

The polycrystalline sample of Nb0.5Os0.5 was prepared by arc melting. The
stoichiometric amounts of Nb (99.95$\%$, Alfa Aesar) and Os (99.95$\%$, Alfa
Aesar) were placed on the water cooled copper hearth in an ultrapure argon gas
atmosphere. The sample was inverted and remelted several times to ensure
sample homogeneity and the observed weight loss is negligible. The phase
analysis was done using x-ray diffraction (XRD) at room temperature on a
X’pert PANalytical diffractometer. The magnetization and ac susceptibility
measurements were performed using the magnetic property measurement system
(MPMS 3, Quantum Design Inc.). The electrical resistivity and specific heat
measurements were done using the physical property measurement system (PPMS,
Quantum Design Inc.). The $\mu$SR measurements were carried out using the MuSR
spectrometer at the ISIS facility, Rutherford Appleton Laboratory, Didcot, U.
K. in both longitudinal and transverse geometries.

Figure 1:  Powder XRD pattern for the Nb0.5Os0.5 sample recorded at room
temperature using Cu $K_{\alpha}$ radiation. The solid red line shows the
experimental data. The dotted blue line corresponds to Rietveld refinement to
the pattern.

##  III Results and Discussion

###  III.1 Sample characterization

The powder x-ray diffraction pattern for Nb0.5Os0.5 was collected at room
temperature. Rietveld refinement was performed using the High Score Plus
Software. As observed from Fig. 1, the Nb0.5Os0.5 sample has no impurity
phase. It can be indexed by cubic, noncentrosymmetric $\alpha$ \- $Mn$
structure (space group $I\bar{4}3m$, No. 217) with the lattice cell parameter
a = 9.765(3) Å.

###  III.2 Normal and superconducting state properties

####  III.2.1 Electrical resistivity

Figure 2:  The resistivity measurement $\rho(T)$ for Nb0.5Os0.5 taken in zero
field in a temperature range of 1.85 K $\leq$ T $\leq$ 300 K. The inset shows
$\rho(T)$ measurements as a function of magnetic fields.

The electrical resistivity measurement was done by the ac transport technique
in the temperature range of 1.85 K $\leq$ T $\leq$ 300 K in zero field (see
Fig. 2). The zero resistivity is acquired around $T_{c}^{0}$ $\approx$ 3.1 K.
The normal state resistivity remains almost temperature independent up to the
highest measured temperature, indicating that Nb0.5Os0.5 exhibit poor
metallicity. The low value of the residual resistivity ratio (RRR)
($\frac{\rho(300)}{\rho(10)}$ = 1.05) suggests the dominance of strong
electronic scattering due to the disorder. The resistivity measurements as a
function of temperature were also done under different applied magnetic fields
(up to 3T, see inset of Fig. 2) to calculate the higher critical field.

####  III.2.2 Magnetization

The magnetization measurement was done in zero-field cooled warming (ZFCW) and
field cooled cooling (FCC) mode in an applied field of 5 mT (see Fig. 3(a)).
The superconducting transition temperature was observed around $T_{c}^{onset}$
= 3.07 K, with the transition width of $\Delta T_{c}$ = 0.21 K. Low field M-H
measurements were done at different temperatures to determine the lower
critical field $H_{c1}$(0). It is defined as the first deviation from
linearity in low-field regions in M vs H curves (see Fig. 3(b)). Using the
formula $H_{c1}(T)=H_{c1}(0)(1-(T/T_{c})^{2})$ for the temperature variation
of $H_{c1}(T)$, we estimated $H_{c1}$(0) = 3.06 $\pm$ 0.05 mT.

Figure 3:  (a) The magnetization data for Nb0.5Os0.5 taken in 5 mT field shows
the superconducting transition at $T_{c}$ = 3.07 K. (b) The lower critical
field $H_{c1}$ estimated by the GL formula was 3.06 mT. Inset shows the M vs H
curves taken at various temperatures. (c) The upper critical field $H_{c2}$(T)
obtained from magnetization, ac susceptibility, resistivity, and specific heat
measurements. The dotted lines show the GL fits, yielding $H_{c2}$(0) $\simeq$
5.4 T for Nb0.5Os0.5.

The temperature dependence of the upper critical field $H_{c2}$(T) was
obtained by measuring the field dependence of superconducting transition
$T_{c}$ in magnetization, ac susceptibility, resistivity, and specific heat
measurements. It is evident from the graph (see Fig. 3(c)) that $H_{c2}$
varies linearly with the temperature and possibly best be fitted by the
Ginzburg-Landau (GL) relation
$H_{c2}(T)=H_{c2}(0)\frac{(1-t^{2})}{(1+t^{2})}$, where t = $T/T_{c}$. By
fitting above equation in the $H_{c2}$-T graph, the specific heat and
magnetization measurements give $H_{c2}$(0) $\simeq$ 5.4 $\pm$ 0.1 T, whereas
resistivity and ac susceptibility measurements give $H_{c2}$(0) $\simeq$ 4.6
$\pm$ 0.1 T. Using the relation $H_{c2}$(0) = $\Phi_{0}/2\pi\xi_{GL}^{2}$
where $\Phi_{0}$ is the quantum flux $(h/2e)$, we obtained $\xi_{GL}(0)$ =
78.12 Å. Other superconducting parameters such as the Ginzburg Landau
parameter $\kappa_{GL}$(0) (= 61), penetration depth $\lambda_{GL}$(0) (= 4774
Å) and the thermodynamic critical field $H_{c}$(0) (= 62.6 mT) were calculated
using the standard relations given in Ref. tin .
For a type-II BCS superconductor in the dirty limit, the orbital limit of the
upper critical field $H_{c2}^{orbital}$(0) is given by the Werthamer-Helfand-
Hohenberg (WHH) EH ; NRW  expression $H_{c2}^{orbital}$(0) = -0.693
$T_{c}\left.\frac{-dH_{c2}(T)}{dT}\right|_{T=T_{c}}$. Using initial slope 2.1
T K-1 from the $H_{c2}$-T phase diagram, $H_{c2}^{orbital}$(0) in the dirty
limit was estimated to be 4.46 T. Within the $\alpha$-model the Pauli limiting
field is given by $H_{c2}^{p}$(0) = 1.86$T_{c}(\alpha/\alpha_{BCS})$ DC .
Using $\alpha$ = 1.81 (from the specific heat measurement), it yields
$H_{c2}^{p}$(0) = 5.85 T. The upper critical field $H_{c2}$(0) calculated
above is close to both the orbital limiting field and Pauli limiting field.
Therefore, it is highly desirable to perform the detailed investigations of
the upper critical field in high quality single crystals of Nb0.5Os0.5.

####  III.2.3 Specific heat

The temperature dependence of the specific heat was collected in zero field.
The normal state low temperature specific heat data above $T_{c}$ can be
fitted with the equation $C/T$ = $\gamma_{n}+\beta_{3}T^{2}+\beta_{5}T^{4}$ to
the limit T $\to$ 0, to extract the electronic contribution ($\gamma_{n}$) and
phononic contribution ($\beta_{3}$, $\beta_{5}$) to the specific heat. The
solid red line in the inset of Fig. 4 shows the best fit to the data which
yields $\gamma_{n}$ = 3.42 $\pm$ 0.01 mJ mol-1 K-2, $\beta_{3}$ = 0.039 $\pm$
0.002 mJ mol-1 K-4, and $\beta_{5}$ = 0.205 $\pm$ 0.004 $\mu$J mol-1 K-6. The
value of $\beta_{3}$ corresponds to a Debye temperature $\theta_{D}$ is 367 K.
The Sommerfeld coefficient is proportional to the density of states
$D_{C}(E_{F})$ at the Fermi level given by $\gamma_{n}$ =
$(\pi^{2}k_{B}^{2}/3)D_{C}(E_{F})$, where using $\gamma_{n}$ = 3.42 $\pm$ 0.01
mJ mol-1 K-2 we obtained $D_{C}(E_{F})$ = 1.45 $\frac{states}{eVf.u}$.

Figure 4:  Single-gap BCS expression given in Eq. (3) fits fairly well for
$\Delta$(0)/$k_{B}T_{c}$ = 1.81 in Nb0.5Os0.5. Inset: The low temperature
specific heat data above $T_{c}$ is fitted to the Debye model shown by solid
red line.

The electron-phonon coupling constant can be calculated using the McMillan
equation WL

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

where $\mu^{*}$ is the Coulomb repulsion parameter, typically given by
$\mu^{*}$ = 0.13 for many intermetallic superconductors. Using $T_{c}$=3.07 K
and $\theta_{D}$ = 367 K for Nb0.5Os0.5, we obtained $\lambda_{e-ph}$ $\simeq$
0.53. This value is comparable to other fully gapped NCS superconductors nr1 ;
RT1 ; sas , suggesting that Nb0.5Os0.5 is a weakly coupled superconductor.
Using the value of $\lambda_{e-ph}$, we have calculated the effective mass for
the quasiparticles $m^{*}$ = 1.53 $m_{e}$ GG . The electronic contribution to
the specific heat can be calculated by subtracting the phononic contribution.
The normalized specific heat jump $\frac{\Delta C_{el}}{\gamma_{n}T_{c}}$ is
1.48 for $\gamma_{n}$ = 3.42 mJ mol-1 K-2, which is close to the value for a
BCS superconductor (= 1.43) in the weak coupling limit. The temperature
dependence of the normalized entropy S in the superconducting state for a
single-gap BCS superconductor is given by

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

where f($\xi$) = [exp(E($\xi$)/$k_{B}T$)+1]-1 is the Fermi function, E($\xi$)
= $\sqrt{\xi^{2}+\Delta^{2}(t)}$, where $\xi$ is the energy of normal
electrons measured relative to the Fermi energy, y = $\xi/\Delta(0)$,
$\mathit{t=T/T_{c}}$, and $\Delta(t)$ =
tanh[1.82(1.018(($\mathit{1/t}$)-1))0.51] is the BCS approximation for the
temperature dependence of the energy gap. The normalized electronic specific
heat is then calculated from the normalized entropy by

$\frac{C_{el}}{\gamma_{n}T_{c}}=t\frac{d(S/\gamma_{n}T_{c})}{dt}.$ (3)

The $C_{el}$ below $T_{c}$ is described by Eq. (3) whereas above $T_{c}$ its
equal to $\gamma_{n}T_{c}$. Figure 4 shows the fitting of the specific heat
data using Eq. (3), which yields $\alpha$ = $\Delta(0)/k_{B}T_{c}$ = 1.81
$\pm$ 0.02. The obtained value is close to the BCS value $\alpha_{BCS}$ =
1.764 in the weak coupling limit, suggesting single-gap BCS like
superconductivity in Nb0.5Os0.5.
In the $\alpha$ model, BCS parameter $\alpha_{BCS}$ is replaced by $\alpha$
which can be determined using the formula $\Delta
C_{el}/\gamma_{n}T_{c}=1.426(\alpha/\alpha_{BCS})^{2}$ DC . Substituting the
value of normalized specific heat jump $\Delta C_{el}/\gamma_{n}T_{c}$ = 1.48
for our sample, we get $\alpha$ = 1.8, which is in good agreement with the
fitted value.

Table 1: Normal and superconducting properties of Nb0.5Os0.5 Parameter | unit | value
---|---|---
$T_{c}$ | K | 3.07
$H_{c1}(0)$ | mT | 3.06
$H_{c2}(0)$ | T | 5.4
$H_{c}(0)$ | mT | 62.6
$H_{c2}^{orbital}(0)$ | T | 4.46
$H_{c2}^{P}(0)$ | T | 5.85
$\xi_{GL}$ | Å | 78.12
$\lambda_{GL}$ | Å | 4774
$\kappa_{GL}$ |  | 61
$\gamma$ | mJmol-1K-2 | 3.42
$\beta$ | mJmol-1K-4 | 0.039
$\theta_{D}$ | K | 367
$\lambda_{e-ph}$ |  | 0.53
DC(Ef) | states/ev f.u | 1.45
$\Delta C_{el}/\gamma_{n}T_{c}$ |  | 1.48
$\Delta(0)/k_{B}T_{c}$ |  | 1.81

####  III.2.4 Muon spin relaxation and rotation

The superconducting ground state of Nb0.5Os0.5 was further analyzed by $\mu$SR
relaxation and rotation measurements. The zero-field muon spin relaxation
(ZF-$\mu$SR) spectra was collected below ($T$ = 40 mK) and above ($T$ = 3.5 K)
the transition temperature ($T_{c}$ = 3.07 K) as displayed in Fig. 5. The
absence of any oscillatory component in the spectra confirms that there are no
atomic moments, generally associated with the ordered magnetic structure. In
the absence of atomic moments, muon-spin relaxation in zero field is given by
the Gaussian Kubo-Toyabe (KT) function RSH

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

where $\sigma_{\mathrm{ZF}}$ accounts for the relaxation due to static,
randomly oriented local fields associated with the nuclear moments at the muon
site. The spectra well described by the function

$A(t)=A_{1}G_{\mathrm{KT}}(t)\mathrm{exp}(-\Lambda t)+A_{\mathrm{BG}},$ (5)

where $A_{1}$ is the initial asymmetry, $\Lambda$ is the electronic relaxation
rate, and $A_{\mathrm{BG}}$ is the time-independent background contribution
from the muons stopped in the sample holder. By fitting both the ZF-$\mu$SR
spectra (Fig. 5 ) with the Eq. (5), yields the similar set of parameters
within the sensitivity of the instrument. In the superconducting state, if the
spin-triplet component is present, an additional relaxation should be observed
lnc2 ; rz1 ; SPA ; li1 ; sro1 . It is clearly absent in Fig. 5, where
identical relaxation signals can be observed on the either side of the
superconducting transition temperature. This leads to the conclusion that the
time-reversal symmetry is preserved in Nb0.5Os0.5 within the detection limit
of $\mu$SR.

Figure 5:  Zero field $\mu$SR spectra collected below (40 mK) and above (3.5
K) the superconducting transition temperature. The solid lines are the fits to
Gaussian Kubo-Toyabe (KT) function given in Eq. (5).  Figure 6:
Representative TF $\mu$SR signals collected at (a) 3.5 K and (b) 0.1 K in an
applied magnetic field of 30 mT. The solid lines are fits using Eq. (6).

Transverse-field muon spin rotation (TF-$\mu$SR) measurements were done to
gain information on the superconducting gap structure of Nb0.5Os0.5. Asymmetry
spectra was recorded above (3.5 K) and below (0.1 K) the transition
temperature $T_{c}$ in a transverse field of 30 mT as shown in Fig. 6. The
TF-$\mu$SR precession signal were fitted using an oscillatory decaying
Gaussian function

$G_{\mathrm{TF}}(t)=A_{1}\mathrm{exp}\left(\frac{-\sigma^{2}t^{2}}{2}\right)\mathrm{cos}(w_{1}t+\phi)+A_{2}\mathrm{cos}(w_{2}t+\phi),$
(6)

where $w_{1}$ and $w_{2}$ are the frequencies of the muon precession signal
and background signal respectively, $\phi$ is the initial phase offset and
$\sigma$ is the Gaussian muon-spin relaxation rate. Figure 6(a) shows the
signal in the normal state where depolarization rate is small, attributed to
homogeneous field distribution throughout the sample. The significant
depolarization rate in the superconducting state shown in the Fig. 6(b) is due
to the flux line lattice (FLL) in the mixed state of the superconductor, which
gives rise to the inhomogeneous field distribution. The depolarization arising
due to the static fields from the nuclear moments $\sigma_{\mathrm{N}}$ is
assumed to be temperature independent and adds in quadrature to the
contribution from the field variation across the flux line lattice
$\sigma_{\mathrm{FLL}}$:

$\sigma^{2}=\sigma_{\mathrm{N}}^{2}+\sigma_{\mathrm{FLL}}^{2}.$ (7)

The muon-spin relaxation rate in the superconducting state
$\sigma_{\mathrm{FLL}}$ is related to the London magnetic penetration depth
$\lambda$ and thus to the superfluid density $n_{s}$ by the equation

$\frac{\sigma_{\mathrm{FLL}}(T)}{\sigma_{\mathrm{FLL}}(0)}=\frac{\lambda^{-2}(T)}{\lambda^{-2}(0)}.$
(8)

For an s-wave BCS superconductor in the dirty limit, the temperature
dependence of the London magnetic penetration depth is given by

$\frac{\lambda^{-2}(T)}{\lambda^{-2}(0)}=\frac{\Delta(T)}{\Delta(0)}\mathrm{tanh}\left[\frac{\Delta(T)}{2k_{B}T}\right],$
(9)

where $\Delta$(T) = $\Delta_{0}$$\delta(T/T_{c})$. The temperature dependence
of the gap in the BCS approximation is given by the expression
$\delta(T/T_{c})$ = tanh[1.82(1.018($\mathit{(T_{c}/T})$-1))0.51]. Taking the
dirty limit expression for Nb0.5Os0.5 and combining Eq. (7), (8) and (9), a
model was obtained for a dirty limit single-gap s-wave superconductor, where
$\sigma$(T) above ${T}_{c}$ is equal to $\sigma_{\mathrm{N}}$ and below
${T}_{c}$ is given by Eq. (10) which contain contributions from both
$\sigma_{\mathrm{N}}$ and $\sigma_{\mathrm{FLL}}$.

$\sigma(T)=\sqrt{\sigma_{\mathrm{FLL}}^{2}(0)\frac{\Delta^{2}(T)}{\Delta^{2}(0)}\mathrm{tanh}^{2}\left[\frac{\Delta(T)}{2k_{B}T}\right]+\sigma_{\mathrm{N}}^{2}}.$
(10)

Figure 7:  The temperature dependence of the muon-spin relaxation rate
$\sigma$(T) collected at an applied field of 30 mT. The solid blue line shows
the s-wave fit for a dirty limit superconductor using Eq. (10).

The temperature dependence of muon depolarization rate $\sigma$ was collected
in an applied field of 30 mT as shown in Fig. 7. The depolarization rate
$\sigma$ remains temperature independent up to $T_{c}$ attributing to random
nuclear magnetic moments, then after $T_{c}$, $\sigma$ increases due to the
formation of well-ordered FLL. The best fit to the $\sigma(T)$ data were
obtained with the single-gap BCS model (Eq. (10)) shown by the solid blue line
in Fig. 7, where we have obtained $\sigma_{\mathrm{N}}$ = 0.366 $\pm$ 0.002
$\mu$s-1, $\sigma$(0) = 0.444 $\pm$ 0.001 $\mu$s-1, and $\Delta$(0) = 0.50
$\pm$ 0.02 meV. The value of $\alpha$ = $\Delta(0)/k_{B}T_{c}$ = 1.89 is close
to the value ($\alpha$ = 1.81) obtained from the low temperature specific heat
measurement. Thus, the TF- $\mu$SR measurements together with the specific
heat measurement confirm that Nb0.5Os0.5 is a s-wave superconductor.
The penetration depth $\lambda$(0) at T = 0 K can be directly calculated
($\sigma_{\mathrm{FLL}}$(0) = 0.251 $\pm$ 0.001 $\mu$s-1) from the relation
JES ; EHB

$\frac{\sigma_{\mathrm{FLL}}^{2}(0)}{\gamma_{\mu}^{2}}=0.00371\frac{\Phi_{0}^{2}}{\lambda^{4}(0)},$
(11)

where $\gamma_{\mu}$/2$\pi$ = 135.53 MHz/T is the muon gyromagnetic ratio and
$\Phi_{0}$ is the magnetic flux quantum. The value of penetration depth
$\lambda$(0) is 6538$\pm$13 Å. The estimated value is little higher than the
$\lambda_{GL}$(0), which could be due to the dirty limit superconductivity in
Nb0.5Os0.5.
Uemura et al. showed in 1991 that the superconductors can be classified into a
conventional/unconventional superconductor KK ; YJU  based on the ratio of the
transition temperature ($T_{c}$) to the Fermi temperature ($T_{F}$). It was
shown that the unconventional, exotic superconductors fall in the range of
0.01 $\leq$ $\frac{T_{c}}{T_{F}}$ $\leq$ 0.1.

Figure 8:  The Uemura plot showing the superconducting transition temperature
$T_{c}$ vs the effective Fermi temperature $T_{F}$, where Nb0.5Os0.5 is shown
as a solid red square. Other data points plotted between the blue solid lines
is the different families of unconventional superconductors.

The Fermi temperature can be calculated using the relation

$k_{B}T_{F}=\frac{\hbar^{2}}{2}(3\pi^{2})^{2/3}\frac{n_{s}^{2/3}}{m_{e}[1+\lambda_{e-ph}]},$
(12)

where ns is the density of paired electrons and $\lambda_{e-ph}$ is the
electron-phonon coupling constant. Using the Sommerfeld coefficient for
Nb0.5Os0.5 ck , we have calculated the number density of electrons ne = 2.94
$\times$ 1030 $m^{-3}$. The estimated value of l (0.56 Å) $\ll$ $\xi_{0}$
(14091 Å), means that in Nb0.5Os0.5 the density of paired electrons will be
given by ns $\simeq$ ne $\frac{\textit{l}}{\xi_{0}}$ = 1.17 $\times$ 1026
$m^{-3}$. The above result is verified from the magnetic penetration depth
$\lambda$ calculated from the muon analysis, where the density of paired
electrons is given by ns =
$\frac{m_{e}(1+\lambda_{e-ph})}{\mu_{0}e^{2}\lambda^{2}}$ $\simeq$ 1.01
$\times$ 1026 $m^{-3}$.
Using the value of ns in Eq. (12), it yields $T_{F}$ = 662 K, giving the ratio
$\frac{T_{c}}{T_{F}}$ = 0.0046, just outside the range of unconventional
superconductors as shown by a solid red square in Fig. 8, where blue solid
lines represent the band of unconventional superconductors. A similar result
is obtained if we express the superfluid density in term of the muon spin-
relaxation rate $\sigma(0)$ $\propto$ $\lambda(0)^{-2}$ $\propto$
$\rho_{s}$(0) as in the original Uemura plot.

##  IV Conclusion

The transport, magnetization, and heat capacity measurements confirm type-II,
s-wave superconductivity in Nb0.5Os0.5 having transition temperature $T_{c}$ =
3.07 K. The upper and lower critical fields estimated to be $H_{c1}$$\simeq$
3.06 mT and $H_{c2}$$\simeq$ 5.4 T respectively. The TF-$\mu$SR measurements
further confirm s-wave superconductivity. The ZF-$\mu$SR measurements show no
evidence of long-range magnetic ordering and any additional relaxation channel
in the superconducting state. It confirms that time-reversal symmetry is
preserved in Nb0.5Os0.5. This result contradicts the possibility of time-
reversal symmetry breaking in NCS superconductors due to the admixed pairing
states (spin-singlet/spin-triplet). Several other NCS superconductors
(weakly/strongly correlated) reported to show the similar result. It suggest
some other mechanism may be involved, which control the TRSB in NCS
superconductors. In order to understand the presence and absence of time-
reversal symmetry breaking in NCS superconducting compounds, it is clearly
important to search the new NCS superconductor.

##  V Acknowledgments

R. P. S. acknowledges Science and Engineering Research Board, Government of
India for the Ramanujan Fellowship through Grant No. SR/S2/RJN-83/2012 and
Newton Bhabha funding. We thank ISIS, STFC, UK for the muon beamtime to
conduct the $\mu$SR experiments.

## References

  * (1) E. Bauer, G. Hilscher, H. Michor, Ch. Paul, E. W. Scheidt, A. Gribanov, Yu. Seropegin, H. No$\ddot{e}$l, M. Sigrist, and P. Rogl, Phys. Rev. Lett. 92, 027003 (2004).
  * (2) E. Bauer and M. Sigrist,Non-centrosymmetric Superconductor:Introduction and Overview (Heidelberg, Springer-Verlag 2012).
  * (3) L. P. Gor’kov, E. I. Rashba, Phys. Rev. Lett. 87, 037004 (2001).
  * (4) S. K. Yip, Phys. Rev. B 65, 144508 (2002).
  * (5) K. V. Samokhin, E. S. Zijlstra, and S. K. Bose, Phys. Rev. B 69, 094514 (2004).
  * (6) I. A. Sergienko and S. H. Curnoe, Phys. Rev. B 70, 214510 (2004).
  * (7) P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, Phys. Rev. Lett. 92, 097001 (2004).
  * (8) S. Fujimoto, Phys. Rev. B 72, 024515 (2005).
  * (9) S. Fujimoto, J. Phys. Soc. Jpn. 75, 083704 (2006).
  * (10) S. Fujimoto, J. Phys. Soc. Jpn. 76, 051008 (2007).
  * (11) M. Sigrist, D. F. Agterberg, P. A. Frigeri, N. Hayashi, R. P. Kaur, A. Koga, I. Milat, K. Wakabayashi, and Y. Yanase, J. Magn. Magn. Mater. 310, 536 (2007).
  * (12) A. B. Karki, Y. M. Xiong, N. Haldolaarachchige, S. Stadler, I.Vekhter, P. W. Adams, D. P. Young, W. A. Phelan, and J. Y. Chan, Phys. Rev. B 83, 144525 (2011).
  * (13) C. S. Lue, H. F. Liu, C. N. Kuo, P. S. Shih, J-Y Lin, Y. K. Kuo, M. W. Chu, T-L Hung, and Y. Y. Chen, Superconduct. Sci. and Tech. 26, 055011 (2013).
  * (14) Bin Chen, Yang Guo, Hangdong Wang, Qiping Su, Qianhui Mao, Jianhua Du, Yuxing Zhou, Jinhu Yang, and Minghu Fang, Phys. Rev. B 94, 024518 (2016).
  * (15) D. Singh, A. D. Hillier, A. Thamizhavel, and R. P. Singh, Phys. Rev. B 94, 054515 (2016).
  * (16) J. Chen, M. B. Salamon, S. Akutagawa, J. Akimitsu, J. Singleton, J. L. Zhang, L. Jiao, and H. Q. Yuan, Phys. Rev. B 83, 144529 (2011).
  * (17) S. Kuroiwa, Y. Saura, J. Akimitsu, M. Hiraishi, M. Miyazaki, K. H. Satoh, S. Takeshita, and R. Kadono, Phys. Rev. Lett. 100, 097002 (2008).
  * (18) P. K. Biswas, M. R. Lees, A. D. Hillier, R. I. Smith, W. G. Marshall, and D. McK. Paul, Phys. Rev. B 84, 184529 (2011).
  * (19) T. Klimczuk, F. Ronning, V. Sidorov, R. J. Cava, and J. D. Thompson, Phys. Rev. Lett. 99, 257004 (2007).
  * (20) Y. Qi, J. Guo, H. Lei, Z. Xiao, T. Kamiya, and H. Hosono, Phys. Rev. B 89, 024517 (2014).
  * (21) A. B. Karki, Y. M. Xiong, I. Vekhter, D. Browne, P. W. Adams,D. P.Young, K. R. Thomas, J.Y. Chan,H.Kim, and R. Prozorov, Phys. Rev. B 82, 064512 (2010).
  * (22) T. Shibayama, M. Nohara, H. Aruga, Katori, Y. Okamoto, Z. Hiroi, and H. Takagi, J. Phys. Soc. Jpn. 76, 073708 (2007).
  * (23) K.Wakui, S. Akutagawa, N. Kase, K. Kawashima, T. Muranaka, Y. Iwahori, J. Abe, and J.Akimitsu, J. Phys. Soc. Jpn. 78, 034710 (2009).
  * (24) V. K. Anand, A. D. Hillier, D. T. Adroja, A. M. Strydom, H. Michor, K. A. McEwen, and B. D. Rainford, Phys. Rev. B 83, 064522 (2011).
  * (25) M. Smidman, A. D. Hillier, D. T. Adroja, M. R. Lees, V. K. Anand, R. P. Singh, R. I. Smith, D. M. Paul, and G. Balakrishnan, Phys. Rev. B 89, 094509 (2014).
  * (26) N. Kase and J. Akimitsu, J. Phys. Soc. Jpn. 78, 044710 (2009).
  * (27) Mojammel A. Khan, A. B. Karki, T. Samanta, D. Browne, S. Stadler, I. Vekhter, Abhishek Pandey, P. W. Adams, D. P. Young, S. Teknowijoyo, K. Cho, R. Prozorov, and D. E. Graf Phys. Rev. B 94, 144515 (2016).
  * (28) R.P. Singh, A.D. Hillier, B. Mazidian, J. Quintanilla, J.F. Annett, D. McK. Paul, G. Balakrishnan, and M.R. Lees, Phys. Rev. Lett. 112, 107002 (2014).
  * (29) A. D. Hillier, J. Quintanilla, and R. Cywinski, Phys. Rev. Lett. 102, 117007 (2009).
  * (30) 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).
  * (31) J. A. T. Barker, D. Singh, A. Thamizhavel, A. D. Hillier, M. R. Lees, G. Balakrishnan, D McK. Paul, and R. P. Singh, Phys. Rev. Lett. 115, 267001 (2015).
  * (32) 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 et al., Nature (London) 394, 558 (1998).
  * (33) J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, Phys. Rev. Lett. 97, 167002 (2006).
  * (34) G. M. Luke, A. Keren, L. P. Le, W. D. Wu, Y. J. Uemura, D. A. Bonn, L. Taillefer, and J. D. Garrett, Phys. Rev. Lett. 71, 1466 (1993).
  * (35) P. D. de Reotier, A. Huxley, A. Yaouanc, J. Flouquet, P. Bonville, P. Impert, P. Pari, P. C. M. Gubbens, and A. M. Mulders, Phys. Lett. A 205, 239 (1995).
  * (36) A. Maisuradze, W. Schnelle, R. Khasanov, R. Gumeniuk, M. Nicklas, H. Rosner, A. Leithe-Jasper, Y. Grin, A. Amato, and P. Thalmeier, Phys. Rev. B 82, 024524 (2010).
  * (37) A. D. Hillier, J. Quintanilla, B. Mazidian, J. F. Annett, and R. Cywinski, Phys. Rev. Lett. 109, 097001 (2012).
  * (38) A. Bhattacharyya, D. T. Adroja, J. Quintanilla, A. D. Hillier, N. Kase, A. M. Strydom, and J. Akimitsu, Phys. Rev. B 91, 060503 (2015).
  * (39) M. Tinkham, Introduction to Superconductivity, 2nd ed. (McGraw-Hill, New York, 1996).
  * (40) E. Helfand and N. R. Werthamer, Phys. Rev. 147, 288 (1966).
  * (41) N. R. Werthamer, E. Helfand, and P. C. Hohenberg, Phys. Rev. 147, 295 (1966).
  * (42) D. C. Johnston, Supercond. Sci. Technol. 26, 115011 (2013).
  * (43) W. L. McMillan, Phys. Rev. 167, 331 (1968).
  * (44) M. Isobe, A.Masao, and N. Shirakawa, Phys. Rev. B 93, 054519 (2016).
  * (45) G.Grimvall, Phys. Scr. 14(1-2), 63 (1976).
  * (46) R. S. Hayano, Y. J. Uemura, J. Imazato, N. Nishida, T.Yamazaki, and R. Kubo, Phys. Rev. B 20, 850 (1979).
  * (47) J. E. Sonier, J. H. Brewer, and R. F. Kiefl, Rev. Mod. Phys. 72, 769 (2000).
  * (48) E. H. Brandt, Phys. Rev. B 37, 2349 (1988).
  * (49) K. Hashimoto, K. Cho, T. Shibauchi, S. Kasahara, Y. Mizukami, R. Katsumata, Y. Tsuruhara, T. Terashima, H. Ikeda, M. A. Tanatar, H. Kitano, N. Salovich, R. W. Giannetta, P. Walmsley, A. Carrington, R. Prozorov, and Y. Matsuda, Science 336, 1554 (2012).
  * (50) Y. J. Uemura et al., Phys. Rev. Lett. 62, 2317 (1989).
  * (51) C.Kittel, Introduction to Solid State Physics 8th edn. (Wiley, New York, 2005).