of the collision rate of the reactant molecules and the probability that the
reaction occurs after a collision. The collision rates are equal for two
reactions with the same set of reactants $\bm{\lambda}^{\rho}$. Therefore, the
ratio in Eq. (122) reduces to the ratio of the probability that the reaction
occurs. If the probability is independent of the surrounding amounts of
substances $\bm{n}$, the condition (122) holds.

The third condition is a property of the steady state called the complex
balancing [69, 8]. The steady state is said to be complex balanced if, for any
$\bm{y}\in\mathcal{C}$,

$\sum_{\rho\,|\,\bm{\lambda}^{\rho}=\bm{y}}K_{\rho}(\bm{n}^{\mathrm{st}})=\sum_{\rho\,|\,\bm{\lambda}^{\rho}=\bm{y}}K_{-\rho}(\bm{n}^{\mathrm{st}}).$
(123)

In other words, the total flow entering $\bm{y}$ equals the total flow exiting
from $\bm{y}$. This condition is stronger than the steady-state condition
$\sum_{\rho}\mathscr{K}_{\rho}(\bm{n}^{\mathrm{st}}){S{{}_{i}}{{}^{\rho}}}=0$
($i$: closed species) in general. However, every steady state is complex
balanced for a special class of systems called deficiency zero networks [8].

We remark that, for ideal dilute solutions with mass action kinetics [64, 60,
8], the first and the second conditions are always satisfied. Therefore, the
only condition for $d\tilde{\varphi}/dt\leq 0$ is the complex balancing. This
result is known in Refs. [6, 7].

With these conditions, we prove the non-positivity of $d\tilde{\varphi}/dt$ as
follows. First, we rewrite $d\tilde{\varphi}/dt$ in Eq. (100) with the aid of
Eqs. (43) and (64), which are valid also for open systems:

$\displaystyle\frac{d\tilde{\varphi}}{dt}$
$\displaystyle=-\sum_{\rho=1}^{M}\mathscr{K}_{\rho}(\bm{n})\quantity[F^{\rho}(\bm{n})-F^{\rho}(\bm{n}^{\mathrm{st}})]$
$\displaystyle=-\sum_{\rho=\pm 1,\dots,\pm
M}K_{\rho}(\bm{n})\quantity[F^{\rho}(\bm{n})-F^{\rho}(\bm{n}^{\mathrm{st}})].$
(124)

Next, we use the first condition and $\ln x\leq x-1$ for $x>0$ to obtain

$\displaystyle\frac{1}{RT}\frac{d\tilde{\varphi}}{dt}$
$\displaystyle=\sum_{\rho}K_{\rho}(\bm{n})\ln\quantity[\frac{K_{-\rho}(\bm{n})K_{\rho}(\bm{n}^{\mathrm{st}})}{K_{\rho}(\bm{n})K_{-\rho}(\bm{n}^{\mathrm{st}})}]$
$\displaystyle\leq\sum_{\rho}K_{\rho}(\bm{n})\quantity[\frac{K_{-\rho}(\bm{n})K_{\rho}(\bm{n}^{\mathrm{st}})}{K_{\rho}(\bm{n})K_{-\rho}(\bm{n}^{\mathrm{st}})}-1]$
$\displaystyle=\sum_{\rho}K_{\rho}(\bm{n}^{\mathrm{st}})\quantity[\frac{K_{-\rho}(\bm{n})}{K_{-\rho}(\bm{n}^{\mathrm{st}})}-\frac{K_{\rho}(\bm{n})}{K_{\rho}(\bm{n}^{\mathrm{st}})}]$
$\displaystyle=-\sum_{\rho}\quantity[K_{\rho}(\bm{n}^{\mathrm{st}})-K_{-\rho}(\bm{n}^{\mathrm{st}})]\frac{K_{\rho}(\bm{n})}{K_{\rho}(\bm{n}^{\mathrm{st}})}$
$\displaystyle=-\sum_{\bm{y}\in\mathcal{C}}\sum_{\rho\,|\,\bm{\lambda}^{\rho}=\bm{y}}\quantity[K_{\rho}(\bm{n}^{\mathrm{st}})-K_{-\rho}(\bm{n}^{\mathrm{st}})]\frac{K_{\rho}(\bm{n})}{K_{\rho}(\bm{n}^{\mathrm{st}})},$
(125)

where $\sum_{\rho}$ is the sum over $\rho=\pm 1,\dots,\pm M$. It suffices to
show that the last line is zero. To show this, we rearrange the second
condition to obtain
${K_{\rho}(\bm{n})}/{K_{\rho}(\bm{n}^{\mathrm{st}})}={K_{\sigma}(\bm{n})}/{K_{\sigma}(\bm{n}^{\mathrm{st}})}$
if $\bm{\lambda}^{\rho}=\bm{\lambda}^{\sigma}$. Therefore, we can move
$K_{\rho}(\bm{n})/K_{\rho}(\bm{n}^{\mathrm{st}})$ in front of the second
summation in the last line. We now apply the third condition to conclude that
the last line is zero.

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#  Observational properties of a general relativistic instability supernova
from a primordial supermassive star

Takashi J. Moriya,1,2 Ke-Jung Chen,3 Kimihiko Nakajima,1 Nozomu Tominaga,4,5
and Sergei I. Blinnikov6,7,5
1National Astronomical Observatory of Japan, National Institutes of Natural
Sciences, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
2School of Physics and Astronomy, Faculty of Science, Monash University,
Clayton, Victoria 3800, Australia
3 Institute of Astronomy and Astrophysics, Academia Sinica, Taipei 10617,
Taiwan
4 Department of Physics, Faculty of Science and Engineering, Konan University,
8-9-1 Okamoto, Kobe, Hyogo 658-8501, Japan
5 Kavli Institute for the Physics and Mathematics of the Universe (WPI), The
University of Tokyo Institutes for Advanced Study, The University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan
6 National Research Center "Kurchatov institute", Institute for Theoretical
and Experimental Physics (ITEP), 117218 Moscow, Russia
7 Dukhov Automatics Research Institute (VNIIA), Suschevskaya 22, 127055
Moscow, Russia E-mail<EMAIL_ADDRESS>(TJM)

(Accepted 2021 February 26. Received 2021 February 26; in original form 2020
June 22)

###### Abstract

We present the expected observational properties of a general relativistic
instability supernova (GRSN) from the 55,500 $\mathrm{M}_{\odot}$ primordial
(Population III) star. Supermassive stars exceeding
$10^{4}~{}\mathrm{M}_{\odot}$ may exist in the early Universe. They are
generally considered to collapse through the general relativistic instability
to be seed black holes to form supermassive ($\sim
10^{9}~{}\mathrm{M}_{\odot}$) black holes observed as high-redshift quasars.
Some of them, however, may explode as GRSNe if the explosive helium burning
unbinds the supermassive stars following the collapse triggered by the general
relativistic instability. We perform the radiation hydrodynamics simulation of
the GRSN starting shortly before the shock breakout. We find that the GRSN is
characterized by a long-lasting (550 d) luminous ($1.5\times
10^{44}~{}\mathrm{erg~{}s^{-1}}$) plateau phase with the photospheric
temperature of around 5000 K in the rest frame. The plateau phase lasts for
decades when it appears at high redshifts and it will likely be observed as a
persistent source in the future deep near-infrared imaging surveys.
Especially, the near-infrared images reaching 29 AB magnitude that can be
obtained by Galaxy and Reionization EXplorer (G-REX) and James Webb Space
Telescope (JWST) allow us to identify GRSNe up to $z\simeq 15$. Deeper images
enable us to discover GRSNe at even higher redshifts. Having extremely red
color, they can be distinguished from other persistent sources such as high-
redshift galaxies by using color information. We conclude that the deep near-
infrared images are able to constrain the existence of GRSNe from the
primordial supermassive stars in the Universe even without the time domain
information.

###### keywords:

supernovae: general – dark ages, reionization, first stars – early Universe –
quasars: supermassive black holes – stars: Population III

††pubyear: 2021††pagerange:  Observational properties of a general
relativistic instability supernova from a primordial supermassive star –
Observational properties of a general relativistic instability supernova from
a primordial supermassive star

##  1 Introduction

Supermassive black holes (SMBHs) exceeding $\sim 10^{9}~{}\mathrm{M}_{\odot}$
are known to exist at $z>6$ through high-redshift quasar observations (Fan et
al., 2001, 2003; Willott et al., 2007; Mortlock et al., 2011; Morganson et
al., 2012; Kashikawa et al., 2015; Wu et al., 2015; Bañados et al., 2016,
2018; Matsuoka et al., 2019). The age of the Universe at $z>6$ is less than 1
Gyr. Forming SMBHs in 1 Gyr is very challenging and it is one of the frontiers
in the modern astrophysics (see Inayoshi et al. 2020; Haemmerlé et al. 2020
for recent reviews).

One possible path to form SMBHs in such a short timescale is through
Population III supermassive stars (SMSs) having
$10^{4}-10^{6}~{}\mathrm{M}_{\odot}$. Although the typical mass of Population
III stars is predicted to be much less than
$10^{4}-10^{6}~{}\mathrm{M}_{\odot}$ (e.g., McKee & Tan, 2008; Hosokawa et
al., 2011; Susa et al., 2014; Hirano et al., 2015; Sugimura et al., 2020),
SMSs can be formed by preventing $\mathrm{H_{2}}$ cooling through, e.g.,
intense the Lyman-Werner ultraviolet background radiation photodissociating
$\mathrm{H_{2}}$ (e.g., Omukai, 2001; Oh & Haiman, 2002; Bromm & Loeb, 2003;
Sugimura et al., 2014; Chon et al., 2016; Chon et al., 2018; Regan et al.,
2017). SMSs are predicted to collapse through general relativistic instability
(e.g., Iben, 1963; Chandrasekhar, 1964; Fowler, 1966; Osaki, 1966; Shibata &
Shapiro, 2002; Shibata et al., 2016; Umeda et al., 2016; Uchida et al., 2017;
Nagele et al., 2020), forming seed massive black holes (BHs). SMSs can be
formed at $z\simeq 15-20$ (e.g., Agarwal et al., 2012; Dijkstra et al., 2014;
Habouzit et al., 2016) to leave the seed BHs and they can grow to SMBHs in 1
Gyr to explain the existence of SMBHs at $z>6$.

The general relativistic instability of SMSs does not necessarily lead to
their collapse to BHs (Fuller et al., 1986; Montero et al., 2012; Chen et al.,
2014; Nagele et al., 2020). In particular, Chen et al. (2014) showed that a
non-rotating Population III SMS with 55,500 $\mathrm{M}_{\odot}$ explodes as a
supernova (SN) through the explosive helium burning following the collapse.
Although SMSs causing such a general relativistic instability SNe (GRSNe) may
be limited in a small mass range, they could be bright enough to be observed
in the future near-infrared (NIR) transient surveys (Whalen et al., 2013).
They also have distinctive chemical signatures that can be traced by the
stellar archaeology of unusual extreme metal poor stars (Johnson et al.,
2013).

In this paper, we investigate the observational properties of the GRSN from
the 55,500 $\mathrm{M}_{\odot}$ SMS presented by Chen et al. (2014). The
observational properties of the GRSN were previously investigated by Whalen et
al. (2013). They showed that future NIR transient surveys can discover the
GRSN. In this work, we present our new radiation hydrodynamics simulation of
the GRSN and suggest that it is not required to conduct transient surveys to
discover the GRSN given its extremely long timescale keeping its brightness.
Having deep multi-band images in NIR would be enough to identify them.

The rest of the paper is organized as follows. We first show our method of the
GRSN investigation in Section 2. Then we discuss its properties in the rest
frame in Section 3. Then we show its observational properties when it appears
in the early Universe and discuss how to find it in Section 4. We conclude
this paper in Section 5. We adopt the standard $\Lambda$CDM cosmology with
$H_{0}=70~{}\mathrm{km~{}s^{-1}}~{}\mathrm{Mpc^{-1}}$, $\Omega_{M}=0.3$, and
$\Omega_{\Lambda}=0.7$ throughout this paper.

##  2 Method

###  2.1 GRSN model

We take the GRSN explosion model of the 55,500 $\mathrm{M}_{\odot}$ Population
III star presented by Chen et al. (2014). We adopt their two-dimensional
explosion model. The evolution of the 55,500 $\mathrm{M}_{\odot}$ star is
followed by using the KEPLER one-dimensional stellar evolution code (Weaver et
al., 1978). The star is found to explode with the one-dimensional calculation
with KEPLER (Chen et al., 2014). The radius of the star at the time of the
explosion is 256 $\mathrm{R}_{\odot}$.

The one-dimensional stellar structure at 600 s before the maximum compression
at the center in the one-dimensional calculation, which is sufficiently long
before the explosive burning leading to the explosion, is transferred to the
multi-dimensional hydrodynamics code CASTRO (Almgren et al., 2010; Zhang et
al., 2011). The subsequent hydrodynamic evolution with nucleosynthesis is
followed until shortly before the shock breakout by two-dimensional
hydrodynamics. We adopt the result from the multi-dimensional simulation
because of the strong mixing found in the GRSN which affect the abundance
profile and, therefore, synthetic LCs (Chen et al., 2014). The explosion
energy is $9\times 10^{54}~{}\mathrm{erg}$. Because the core temperature
($<10^{9}$ K) and density ($<10^{3}~{}\mathrm{g~{}cm^{-3}}$) is low at the
explosive helium burning, little ${}^{56}\mathrm{Ni}$ is produced during the
explosion and the total ${}^{56}\mathrm{Ni}$ mass in the ejecta is less than
0.1 $\mathrm{M}_{\odot}$.

We obtain the angle-averaged hydrodynamic and abundance profiles from the two-
dimensional simulation and use it as the initial condition for our one-
dimensional radiation hydrodynamics simulation described in the next section.
For each physical quantity at a given radius, we take ten directions from
polar angles of $0.1\pi$, $0.2\pi$, $0.3\pi$, …, $\pi$ in the two-dimensional
result and then take the average value of the ten directions for the one-
dimensional calculation. In this way we approximately take the effect of
multi-dimensional mixing found in the two-dimensional calculation into
account. Detailed chemical mixing patterns in the GRSN model are presented in
Fig. 4 of Chen et al. (2014). The angle-averaged density and abundance
profiles are shown in Fig. 1. A hydrogen+helium layer exists from the surface
to around 31,000 $\mathrm{M}_{\odot}$. The layers below are mostly composed of
helium, oxygen, neon, and magnesium.

Figure 1:  Top: Initial density profile for the GRSN LC calculation by STELLA.
Bottom: Abundance profile for the GRSN LC calculation. The top axis shows the
velocity of the mass coordinate during the homologous expansion.

###  2.2 Light-curve calculation

The numerical LC calculation is performed by using the one-dimensional
radiation hydrodynamics code STELLA (Blinnikov et al., 1998, 2000, 2006). In
addition to the hydrodynamics equations, STELLA implicitly treats time-
dependent equations of the angular moments of intensity averaged over a
frequency bin with the variable Eddington method. STELLA calculates the
spectral energy distributions (SEDs) at each time step and obtains multicolour
LCs by convolving filter functions with the SEDs. We adopt the standard 100
frequency bins from 1 Å to $5\times 10^{4}$ Å on a log scale. STELLA has been
widely used for the LC modeling of hydrogen-rich SNe (e.g., Blinnikov et al.,
2000; Baklanov et al., 2005; Tominaga et al., 2011; Moriya et al., 2011, 2016,
2018b; Goldberg et al., 2019, 2020), including hydrogen-rich Population III
SNe (Tolstov et al., 2016; Moriya et al., 2019). STELLA is also shown to
provide very similar SN II LCs to those obtained by the Monte Carlo radiation
transfer approach (Tsang et al., 2020).

We put the initial condition that is shortly before the shock breakout
provided by the CASTRO simulation and no energy is artificially injected. We
do not set mass cut because no compact remnant remains in the GRSN explosion.

Because we have the rest-frame SED information at each time step, we shift the
SEDs at given redshifts and apply filter functions at the observer frame to
estimate the observational properties of the GRSN at high redshifts. We adopt
the following filter functions in NIR when we discuss the observational
properties of the GRSN. The mean wavelength of each filter is also mentioned.
The H band filter ($1.77~{}\mathrm{\mu m}$) from Euclid, which is the reddest
filter in Euclid, is adopted (Maciaszek et al., 2016). We take the J129
($1.29~{}\mathrm{\mu m}$), H158 ($1.58~{}\mathrm{\mu m}$), and F184
($1.84~{}\mathrm{\mu m}$) filters from the Nancy Grace Roman Space Telescope
(RST, previously known as
WFIRST)111https://roman.gsfc.nasa.gov/science/Roman_Reference_Information.html.
The K band filter ($2.15~{}\mathrm{\mu m}$) is adopted from ULTIMATE-
Subaru222https://ultimate.naoj.org/english/, which is planned to have a
similar K band filter to the $K_{S}$ band filter of
Subaru/MOIRCS333https://subarutelescope.org/Observing/Instruments/MOIRCS/imag_sensitivity.html.
We take three filters from Galaxy and Reionization EXplorer (G-REX), which is
a proposed wide-field surveyor at $2-5~{}\mathrm{\mu m}$ (A.K. Inoue, private
communication), i.e., the F232 ($2.32~{}\mathrm{\mu m}$), F303
($3.03~{}\mathrm{\mu m}$), and F397 ($3.97~{}\mathrm{\mu m}$). These filters
were previously proposed for the Wide-field Imaging Surveyor for High-redshift
(WISH) satellite444https://wishmission.org and are currently considered for
G-REX. Finally, we take three broad filters from James Webb Space Telescope
(JWST)/NIRCam555https://jwst-docs.stsci.edu/near-infrared-camera/nircam-
instrumentation/nircam-filters: F277W ($2.77~{}\mathrm{\mu m}$), F356W
($3.56~{}\mathrm{\mu m}$) and F444W ($4.44~{}\mathrm{\mu m}$).

Figure 2:  Velocity evolution at the surface of the progenitor shortly after
the beginning of the STELLA calculation.

Figure 3:  Bolometric LC of the GRSN. The horizontal axes in the top panel are
in the log scale and those in the bottom panel are in the linear scale. The
top panel shows the analytically estimated bolometric LC decay rate after the
shock breakout ($\propto t^{-0.35}$) and the bottom panel shows the nuclear
decay rate of ${}^{56}\mathrm{Co}$.

Figure 4:  Properties at the photosphere.

##  3 GRSN properties in the rest frame

Fig. 2 shows the velocity evolution at the outermost layers at the beginning
of the LC calculation. The initial condition ($t=0$) is shortly before the
shock breakout and the shock front continues to travel towards the progenitor
surface. The shock breakout occurs at $t\simeq 500~{}\mathrm{s}$. After the
shock breakout, the homologous expansion of the ejecta is set at $t\simeq
5000~{}\mathrm{s}$ and the velocity of each mass shell is fixed from the
moment. The velocity of the mass coordinate during the homologous expansion is
shown in Fig. 1.

The bolometric LC of the GRSN is shown in Fig. 3. Fig. 4 presents the
velocity, temperature, mass coordinate, and radius of the photosphere which is
where the Rosseland mean optical depth becomes $2/3$. We first see the shock
breakout signal peaking at 14 s with $10^{47}~{}\mathrm{erg~{}s^{-1}}$. The
shock breakout signal lasts for several hundred seconds which roughly
correspond to the light-crossing time of the progenitor (600 s).

The bolometric LC evolution after the shock breakout roughly follows $\propto
t^{-0.35}$ as expected from the analytical models (e.g., Chevalier & Fransson,
2008; Rabinak & Waxman, 2011; Kozyreva et al., 2020). The bolometric LC keeps
declining until 17 d when the outermost layers get close to the hydrogen
recombination temperature at $\simeq 6000-5000~{}\mathrm{K}$. The subsequent
bolometric LC evolution can be interpreted simply with $L\propto R^{2}T^{4}$,
where $L$ is the bolometric luminosity, $R$ is the photospheric radius, and
$T$ is the photospheric temperature. The bolometric luminosity increases from
17 d to 250 d when the recession velocity of the photosphere is slower than
the hydrodynamic expansion velocity of the ejecta. In other words, the
photospheric temperature is set at the recombination temperature but the
photospheric radius increases (Fig. 4), making the bolometric luminosity to
increase.

The bolometric luminosity increases until 250 d when the recession velocity of
the photosphere and the hydrodynamic expansion velocity matches. From this
time, the photosphere is kept at the same radius and it also keeps the same
recombination temperature. Thus, the bolometric luminosity becomes constant
and the plateau phase appears. This physical condition is the same as that
found in Type IIP SNe during their plateau phase (Grassberg et al., 1971; Falk
& Arnett, 1977). The plateau luminosity ($\simeq 1.5\times
10^{44}~{}\mathrm{erg~{}s^{-1}}$) is much larger than those observed in Type
IIP SNe (Bersten & Hamuy, 2009) because of the extremely larger explosion
energy. Exceeding $10^{44}~{}\mathrm{erg~{}s^{-1}}$, the GRSN can be regarded
as a member of superluminous SNe (Moriya et al., 2018a; Gal-Yam, 2019, for
recent reviews). The plateau continues up to around 800 d when the photosphere
reaches the bottom of the layers containing hydrogen (see the mass coordinate
of the photosphere in Fig. 4). The hydrogen recombination ends at this point
and the photospheric temperature drops suddenly. The plateau duration is about
550 d in the rest frame which is also much longer than those found in Type IIP
SNe (about 100 d, Bersten & Hamuy 2009; Anderson et al. 2014). The total
radiated energy during the plateau phase amounts to $\simeq 7\times
10^{51}~{}\mathrm{erg}$.

The plateau phase is followed by the sudden drop in luminosity and then the
“tail” phase starts. The bolometric luminosity keeps decreasing during the
tail phase. The tail phase luminosity is not powered by the radioactive decay
of ${}^{56}\mathrm{Co}$ unlike the case of Type IIP SNe because of the small
amount of ${}^{56}\mathrm{Ni}$ (less than 0.1 $\mathrm{M}_{\odot}$) in the
ejecta. The LC decline rate is different from that of ${}^{56}\mathrm{Co}$ as
shown in Fig. 3. The luminosity source in this tail phase is still thermal
energy from the initial explosion as in the previous hydrogen recombination
phase. About a half of magnesium, which is the most abundant element in the
hydrogen-free core (Fig. 1), remain ionized even after the plateau phase
because of its low ionization energy. Therefore, the opacity at the hydrogen-
free core remains high (of the order of $0.01~{}\mathrm{cm^{2}~{}g^{-1}}$)
even after the hydrogen recombination phase and the photosphere is kept at the
surface of the hydrogen-free layers during the epochs presented in Fig. 3.
This is why the photospheric radius increase linearly after the hydrogen
recombination phase, i.e., the photosphere is kept at the surface of the
hydrogen-free core. The bolometric luminosity decreases as the temperature
decreases due to the adiabatic cooling. In the photospheric temperature
evolution, we find a sharp increase at around 1150 days. The exact reason for
the jump is not clear and it could be a numerical artifact.

Our bolometric LC behavior is different from that presented in the previous
study in Whalen et al. (2013) in which the same initial explosion model is
used but the numerical LC calculation is performed in a different code. The
bolometric LC from Whalen et al. (2013) shows a slow luminosity decline
starting from the shock breakout peak. Their LC model does not show the
plateau phase caused by the hydrogen recombination as we found in our model.
We believe that the existence of the plateau phase is expected from the
presence of the massive hydrogen-rich envelope as in the case of Type IIP SNe
(Grassberg et al., 1971; Falk & Arnett, 1977). We have compared our results
with predictions of the analytic model by Nagy & Vinkó (2016) (see also Szalai
et al. 2019 for the update of that model). We find a good agreement of
synthetic light curves produced by STELLA with this simple model on the
plateau stage. The earlier bolometric LC behaivor in our model is also
consistent with the analytic models as discussed earlier in this section. The
reasons for the difference between our LC model and the model in Whalen et al.
(2013) remain unclear.

Figure 5:  GRSN LCs at $z=15$ with the G-REX and JWST filters in the observer
frame.

##  4 Discovering GRSNe in the future NIR surveys

We discuss the observational properties of GRSNe when they appear in the early
Universe based on the GRSN model presented in the previous section.

###  4.1 GRSNe as persistent sources

Fig. 5 shows the GRSN LCs at $z=15$ with the G-REX and JWST filters. The
observed wavelengths correspond to $0.1-0.3~{}\mathrm{\mu m}$ in the rest
frame. The SEDs at $0.1-0.3~{}\mathrm{\mu m}$ are strongly affected by metal
absorption and have steep gradients. This causes large differences in the
observed magnitudes even if the differences in the filter central wavelengths
are small. This effect of absorption becomes larger with higher redshifts (cf.
Fig. 6).

The most prominent feature is the long plateau phase lasting for three
decades. It originate from the luminous plateau phase lasting for 550 d in the
rest frame (Section 3). Because of the time dilution, the long plateau phase
becomes even longer and lasts for decades when GRSNe appear at high redshifts.

The long plateau lasting for more than a decade would be observed as a
persistent source rather than a transient during the NIR survey, because the
major NIR survey missions are conducted by satellite telescopes and their
operation period is usually set to 5 years. Fig. 6 shows the predicted
magnitudes of the GRSN plateau as a function of redshift. We find that GRSNe
up to $z\simeq 15$ can be observed with G-REX and JWST if a deep imaging
survey reaching at least $\simeq 29~{}\mathrm{AB~{}mag}$ is conducted. The
deeper images enable us to reach higher redshifts (Fig. 6).

To identify non-transient GRSNe, we need to distinguish them from other
persistent sources such as high-redshift galaxies. Fig. 7 shows the color-
color diagrams in which GRSNe at plateau, high-redshift galaxies, high-
redshift quasars, and dwarf stars in our Galaxy. The high-redshift galaxy
templates are produced with the stellar radiation provided by BPASS (v2.1;
Eldridge et al. 2017) in addition to nebular emission (both lines and
continua) as calculated in a self consistent manner with Cloudy (see Nakajima
et al. 2018 for more details). Here we adopt “young” and “evolved” galaxies,
with a continuous star formation history at the ages of $10$ and $500$ Myr,
the metallicity of $0.1$ and $0.5$ solar metallicity, and the dust reddening
of $E(B-V)=0.0$ and $0.2$ assuming the Calzetti et al. (2000)’s extinction
law, respectively. The young population shows intense emission lines with a
rest-frame equivalent width of [OIII]$\lambda 5007$+H$\beta$ as large as $\sim
1000$ Å, which corresponds to extreme emission line galaxies such as green pea
galaxies (e.g., Cardamone et al., 2009). The evolved population represents
continuum-selected galaxies like Lyman-break galaxies (e.g., Steidel et al.,
2014). Both galaxy populations are then placed at redshifts of $z=6-12$ with
an IGM attenuation prescribed by Inoue et al. (2014). At these redshifts,
strong optical emission lines such as H$\alpha$ and [O iii] $\lambda 5007$
fall in the NIR filters of G-REX and JWST/NIRCam. The nebular component in our
galaxy-SEDs is thus essential to obtain realistic predictions of NIR colors of
high-redshift galaxies, which are inferred to present intense H$\alpha$ and [O
iii]$+$H$\beta$ emission lines (e.g., Smit et al. 2014; Roberts-Borsani et al.
2016). The quasar models are based on a composite of spectra of quasars at
$z=1-2$ observed in the rest-frame ultraviolet to near-infrared wavelength
(Selsing et al., 2016). The composite is redshifted and IGM-attenuated in the
same way as the galaxy models to mock up the high-redshift quasars. The dwarf
stars are plotted by using the NIR spectra obtained by SpeX (Rayner et al.,
2003) and presented in (Cushing et al., 2005; Rayner et al., 2009)666We
obtained the spectra at
http://irtfweb.ifa.hawaii.edu/~spex/IRTF_Spectral_Library/.. We took all the
dwarf spectra with the spectral coverage up to $4~{}\mathrm{\mu m}$ for G-REX
and $5~{}\mathrm{\mu m}$ for JWST and convolved the filter functions. When
there is a gap in the dwarf spectra caused by the telluric absorption, we
simply interpolated the gap.

The color-color diagram (Fig. 7) shows that GRSNe are much redder than high-
redshift galaxies, high-redshift quasars, and dwarf stars and they can be
easily distinguished from GRSNe at the plateau phase. This is because the
temperature of the plateau is only $\simeq 5000-6000~{}\mathrm{K}$ in the rest
frame and they become extremely red if they are at high redshifts. This
temperature at the plateau phase is determined by the hydrogen recombination
temperature. Thus, the color during the plateau phase is a solid prediction
that does not depend much on the actual mass and energy of the GRSN
explosions.

Figure 6:  Magnitudes of the GRSN at the plateau phase as a function of
redshift. The SED at 500 d after shock breakout presented in Section 3 is used
to estimate the plateau magnitudes. The horizontal dashed line shows a typical
limiting magnitude of planned imaging surveys (29 mag).

Figure 7:  Color-color diagrams of the GRSN at the plateau phase (500 d after
the shock breakout). We also put high-redshiift galaxies and dwarf stars in
our Galaxy for comparison.  Figure 8: GRSN LCs at $z=15$ and 20 during the
cooling phase after the shock breakout with the selected NIR filters.

Figure 9:  Peak magnitudes and rise times of GRSNe during the cooling phase
shortly after the shock breakout as a function of redshift.

###  4.2 Discovering GRSNe with NIR transient surveys

As discussed previously by Whalen et al. (2013), it is also possible to find
GRSNe with NIR transient surveys. As presented in Fig. 3, GRSNe are brightest
shortly after the shock breakout and their luminosity keeps declining until 17
d. GRSNe at this phase can be observed at $z\gtrsim 15$ thanks to their large
luminosity and high temperature. Fig. 8 shows examples of the GRSN LCs during
the declining phase. The LCs rise in about a month and declines. Fig. 9 shows
the peak luminosity of the first peak and the rise time to the peak in the NIR
filters as a function of redshift. NIR transient surveys reaching 28.5 AB mag
are required to find high-redshift GRSNe. The timescale is about a month.
Thus, deep NIR transient surveys reaching down to 28.5 AB mag with a cadence
of around 10 d are required to identify high-redshift GRSNe shortly after the
explosion.

##  5 Conclusions

We have presented the observational properties of the GRSN from the 55,500
$\mathrm{M}_{\odot}$ Population III star. The progenitor has the radius of 256
$\mathrm{R}_{\odot}$ and the explosion energy is $9\times
10^{54}~{}\mathrm{erg}$. We take the results of the GRSN explosion simulation
up to shortly before the shock breakout conducted previously by Chen et al.
(2014). We put it as the initial condition to the radiation hydrodynamics code
STELLA to investigate its observational properties. The overall observational
properties of the GRSN is similar to those of Type IIP SNe, but the GRSN has
the much longer (550 d in the rest frame) and more luminous ($1.5\times
10^{44}~{}\mathrm{erg~{}s^{-1}}$) plateau phase. The photospheric temperature
during the plateau is characterized by the hydrogen recombination temperature
($\simeq 5000-6000~{}\mathrm{K}$) as is the case for Type IIP SNe. The plateau
phase was not predicted to exist in the previous study by Whalen et al. (2013)
in which the same initial explosion model is used to investigate the GRSN
properties, but the existence of the plateau phase is expected from the
presence of the massive hydrogen-rich envelope as in the case of Type IIP SNe.
The plateau phase lasts for many decades when the GRSN appears at high
redshifts and the GRSN is not recognized as a transient in the future NIR
imaging surveys. However, it can be distinguished from other persistent
sources such as high-redshift galaxies and local dwarf stars by using color
information. The deep NIR images reaching 29 AB mag obtained by G-REX and JWST
will allow us to search for the GRSN up to $z\simeq 15$. The deeper images
allow us to reach higher redshifts. The NIR transient surveys reaching
$28-28.5~{}\mathrm{AB~{}mag}$ with a cadence of 10 d may identify the GRSN at
$z\gtrsim 15$ during the adiabatic cooling phase after the shock breakout as a
transient.

Because the exact mass range of the GRSN explosions is unclear, it is
difficult to predict the expected number of the GRSN detection in the deep NIR
images obtained by the future NIR surveys. Still, the extremely red color of
high-redshift GRSNe makes them easy to identify. The existence or lack of the
GRSNe can be constrained in the future deep NIR images without conducting
transient surveys. Such a constraint provides valuable information on the fate
of SMSs possibly leading to the formation of SMBHs in the early Universe.

## Acknowledgements

We thank the anonymous referee for constructive comments that improved this
paper. T.J.M. thanks the organisers of the First Star VI conference at
Universidad de Concepción, Chile, where this work was initiated. T.J.M. is
supported by the Grants-in-Aid for Scientific Research of the Japan Society
for the Promotion of Science (JP18K13585, JP20H00174). K.C. acknowledges
support from the Ministry of Science and Technology (Taiwan, R.O.C.) grant
number MOST 107-2112-M-001-044-MY3. S.B. is supported by grant RSF 19-12-00229
for the development of STELLA code. This research has been supported in part
by the RFBR (19-52-50014)-JSPS bilateral program. Numerical computations were
in part carried out on PC cluster at Center for Computational Astrophysics
(CfCA), National Astronomical Observatory of Japan.

## Data availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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# Superconducting and normal state properties of high entropy alloy Nb-Re-Hf-
Zr-Ti investigated by muon spin relaxation and rotation

Kapil Motla  Department of Physics, Indian Institute of Science Education and
Research Bhopal, Bhopal, 462066, India  Pavan Meena  Department of Physics,
Indian Institute of Science Education and Research Bhopal, Bhopal, 462066,
India  Arushi  ISIS Facility, STFC Rutherford Appleton Laboratory, Harwell
Science and Innovation Campus, Oxfordshire, OX11 0QX, UK  D. Singh  ISIS
Facility, STFC Rutherford Appleton Laboratory, Harwell Science and Innovation
Campus, Oxfordshire, OX11 0QX, UK  P. K. Biswas  ISIS Facility, STFC
Rutherford Appleton Laboratory, Harwell Science and Innovation Campus,
Oxfordshire, OX11 0QX, UK  A. D. Hillier  ISIS Facility, STFC Rutherford
Appleton Laboratory, Harwell Science and Innovation Campus, Oxfordshire, OX11
0QX, UK  R. P. Singh<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Science Education and Research Bhopal, Bhopal, 462066, India

(30-12-2020)

###### Abstract

Superconducting high entropy alloy (HEA) are emerging as a new class of
superconducting materials. It provides a unique opportunity to understand the
complex interplay of disorder and superconductivity. We report the synthesis
and detail bulk and microscopic characterization of Nb60Re10Zr10Hf10Ti10 HEA
alloy using transport, magnetization, specific heat, and muon spin
rotation/relaxation ($\mu$SR) measurements. Bulk superconductivity with
transition temperature $T_{C}$ = 5.7 K confirmed by magnetization,
resistivity, and heat capacity measurements. Zero-field $\mu$SR measurement
shows that the superconducting state preserves time-reversal symmetry, and
transverse-field measurements of the superfluid density are well described by
an isotropic s-wave model.

##  I Introduction

High entropy alloys (HEA) are a new class of multi-component material with
outstanding mechanical, thermal, physical, and chemical properties metals ;
alloy . High entropy alloys are typically a solid solution containing more
than five principal elements 2004Yeh ; 2004 ; cantor . Configurations entropy
(S) is also used as a parameter to define high entropy alloys. It should be
comparable to $\Delta S_{mix}$ $>$ 1.5 $R$ for five-element alloy 1.5 . The
high entropy plays a vital role in the stabilization of the disordered solid
solution phase prl . It forms single-phase solid solutions with simple lattice
despite having multiple components and complex chemistry. These disorder
alloys have a broad spectrum of applications, such as anti-oxidation, anti-
corrosion, thermoelectric, soft magnet, and radiation tolerance materials
anti-oxy ; anti-corr ; radi-tol  over conventional alloys. In 2014, another
fascinating electronic property, superconductivity, was observed first time in
Ta34Nb33Hf8Zr14Ti11 HEA with a transition temperature of 7.3 K prl . It
extends the possible applications of HEAs beyond their structural and
mechanical properties-related applications. Since then, few other HEA or
medium entropy alloy superconductors have been reported, which crystallized in
BCC, FCC, CsCl, hexagonal, and tetragonal crystal structure alpha Mn ; FCC ;
Hexagonal ; Hexa ; tetragonal . To date, the maximum Tc was achieved in
(ScZrNbTa)0.65(RhPd)0.35 HEA with CsCl lattice-type crystal structure hightc .
Most of the effort on superconducting HEA is focused on discovering new
superconducting HEAs families and enhancing superconducting transition
temperature using the different combinations of 3$d$, 4$d$ and 5$d$ elements.
Its multi-component nature offers the tunability of superconducting parameters
and crystal structure. Some of these alloys show exotic superconducting
properties such as retention of superconducting properties in very high-
pressure pressure hea , Debye temperature in elemental range alpha Mn ; Hexa ;
muon hea  and phonon mediated superconducting similar to binary and ternary
superconducting alloys despite its highly disordered nature, where the
occurrence of regular phonon mode is unlikely. Understanding of
superconductivity in high-entropy alloys remains a challenge, and the
unavailability of microscopic measurements on superconducting HEA makes it
difficult to understand the superconducting pairing mechanism of these
disordered alloys muon hea . In this paper, we report the synthesis and
detailed investigation of superconducting HEA Nb0.6Re0.1Zr0.1Hf0.1Ti0.1 using
bulk measurements of magnetization, heat capacity, and resistivity along with
microscopic probe of the vortex lattice and superconducting ground state with
muon spin spectroscopy. These measurements confirmed bulk type-II
superconductivity in Nb0.6Re0.1Zr0.1Hf0.1Ti0.1 having superconducting
transition temperature TC $\simeq$ 5.7 K. Superconducting ground state can be
described as a nodeless isotropic gap with a moderate coupling strength and
preserved time-reversal symmetry.

##  II Experimental Details

The standard arc melting technique was used to synthesize a polycrystalline
sample of Nb0.60Re0.10Zr0.10Hf0.10Ti0.10. Elements Re (99.99$\%$), Nb
(99.99$\%$), Hf (99.99$\%$), Zr (99.99$\%$), and Ti (99.99$\%$) with high
purity were taken in the stoichiometric ratio and loaded in the cavity of the
copper hearth. The chamber was filled with argon, and the cooling of the
copper hearth was provided with continuous water flow. Firstly, we melted the
titanium getter to absorb the oxygen, if any, present in the chamber and then
melted the sample several times to ensure homogeneity. After several melting,
the sample was shiny, and the weight loss was negligible. Sample composition
was confirmed by energy dispersive X-ray spectrometer, and phase purity was
confirmed by X-ray diffractometer (XRD) by using X’pert PANalytical
diffractometer where Cu-Kα ($\lambda$ = 1.5405 Å) radiation was used. Bulk
superconductivity was confirmed by magnetization measurement using a
superconducting quantum interference device (SQUID) from Quantum Design. Four
probe and two tau methods were adopted to perform resistivity and specific
heat, respectively using Quantum Design Physical Property Measurement System
(PPMS). In order to investigate microscopic superconducting ground state and
vortex lattice, the MuSR spectrometer was used at ISIS Neutron and Muon
Source, STFC Rutherford Appleton Laboratory, United Kingdom, in zero field and
transverse fields configurations.

Figure 1:  Powder X-ray diffraction pattern collected at room temperature by
using CuKα radiation and EDAX spectra shown in inset.

##  III Results and Discussion

###  a Structural characterization

The X-ray data from the sample, at ambient conditions, was collected using the
PANalytical X’Pert Pro instrument. The observed diffracted pattern was indexed
with Full Prof software, and the refined pattern is displayed in Fig. 1. It
crystallized in centrosymmetric a BCC, space group Im-3m, structure with unit
cell parameter a = b = c = 3.320184(7) Å. The broadness of the peaks may be
due to the disorder created by different radii of elements within the sample
prl . The EDAX was performed at five different positions (one spectra was
shown in the inset of Fig. 1) on HEA sample, which confirms the homogeneity
and the averaged composition is found out to be Nb60Re10Zr10Hf10Ti10.

Figure 2:  The temperature dependent resistivity in the range of 1.9 $K$-300
$K$ and the left inset shows the sharp transition at 5.2(1) $K$. the right
inset shows field dependent resistivity at 10 $K$ in the range $\pm$ 9 $T$ and
slope of resistivity vs magnetic field is negative. Figure 3: (a) Temperature
dependent of the DC magnetic moment in both ZFCW and FCC mode under the
applied field of 1 $mT$ (left).(b) Temperature dependent of lower critical
field curved fitted with Gingburg-Landau equation (eqn. 1) and inset of (b)
shows the field dependent magnetization within temperature range from 1.8 $K$
to 5.2 $K$ (middle). (c) Temperature dependence data of the upper critical
field fitted with the Gingburg-Landau equation (eqn.2) and the inset of (c)
magnetization versus temperature within range of field 50 $mT$ to 5 $T$ is
shown(right).

###  b Electrical resistivity

The electrical transport study of Nb0.60Re0.10Zr0.10Hf0.10Ti0.10 and nature
was revealed by temperature dependent resistivity measurement between
temperature range 1.9 $K$ to 300 $K$ without applying any external field. The
inset of Fig. 2 is displayed the sharp transition $T_{C}^{onset}$ at 5.30(5)
$K$ and true zero resistive value $T_{C}^{base}$ = 5.03(5) $K$. We take the
mid-value of the normal state to zero resistive states as the transition
temperature $T_{C}^{mid}$ = 5.16(5) $K$. Above the transition temperature, the
normal state resistivity was found to be almost temperature-independent and is
shown in Fig 2. The residual resistivity ratio (RRR)
($\rho_{300}$/$\rho_{10}$) is found out to be 1.2 which is comparable to other
HEA superconductors muon hea ; TNZHT ; Cava . We have also measured Hall
resistivity to study the nature of charge carrier and density in a normal
state of the sample. The field-dependent hall resistivity measurement was done
at 10 $K$, which is shown in the inset of Fig. 2. The observed $\rho$ (H) data
was linear fit within $\pm$ 9 $T$ magnetic field, and the slope provides the
hall coefficient RH = - 1.08(2)$\times$10-10 $\ohm mT^{-1}$. The negative sign
of RH indicates that the electrons are the charge carriers. Using the
expression RH = - 1/$ne$, where $n$ and $e$ are the carrier density and
electronic charge, respectively. This expression yields the charge carrier
density $n$ = 6.1(1) $\times$ 1028 $m^{-3}$, which is comparable to the non-
centrosymmetric (HfNb)0.10(MoReRu)0.90 and (ZrNb)0.10(MoReRu)0.90 high entropy
alloy superconductors muon hea .

###  c Magnetization

We have performed temperature-dependent DC magnetization in zero-field cooled
warming (ZFCW) and field-cooled cooling (FCC) mode under 1 $mT$ external
magnetic field. A clear indication of diamagnetic signal was observed
corresponding to the onset of superconducting transition temperature at
$T_{C}$ = 5.60 (4) $K$ and is shown in Fig. 3(a). The FCC data indicates the
strong pinning nature in Nb0.60Re0.10Zr0.10Hf0.10Ti0.10 alloy. Lower critical
field $H_{C1}$ was determined using field-dependent magnetization data within
a range of temperature from 1.8 K up to $T_{C}$ as shown inset of Fig. 3(b).
Up to a certain value of field, magnetization varies linearly and the point
from which it starts deviating from the linearity or the Meissner line is
taken as critical field for a particular temperature. The lower critical field
value $H_{C1}$ at 0 $K$ was estimated after fitting the data points using
Ginzburg-Landau expression

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

The extrapolation of $H_{C1}(T)$ to 0 $K$ yields $H_{C1}$(0) = 7.95(7) $mT$.
In order to estimate the upper critical field, temperature-dependent
magnetization was carried out with varying magnetic fields. The onset of
superconducting transition was considered as the criteria for the upper
critical field, and transition temperature decreased with an increase of an
applied external magnetic field, as shown in Fig. 3(c). $H_{C2}$ versus
temperature shows a linear behaviour near $T_{C}$ and can be described with
the Ginzburg-Landau 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, the fitting reveals the upper
critical value at 0 $K$ is 6.31(4) $T$. The Gingburg-Landau coherence length
parameter can be evaluated using the value of $H_{C2}$(0) according to the
expression

$H_{C2}(0)=\frac{\Phi_{0}}{2\pi\xi_{GL}^{2}}$ (3)

where $\Phi_{0}$ is the flux quantum ($\Phi_{0}$ = 2.07$\times$10-15T m2) and
substituting the $H_{C2}$(0) = 6.31(4) $T$, we get coherence length
$\xi_{GL}$(0) = 7.2(1) $nm$. The magnetic penetration depth $\lambda_{GL}$(0)
depends on the lower critical field $H_{C1}$(0) and $\xi_{GL}$(0) according to
the expression

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

After substituting the calculated parameters $H_{C1}$(0) = 7.95(7) $mT$ and
$\xi_{GL}$(0) = 7.2(1) $nm$, we obtained $\lambda_{GL}$(0) = 279(2) $nm$. The
Gingburg-Landau parameter $\kappa_{GL}$ differentiate the type of
superconductivity and expressed as
$\kappa_{GL}$=$\frac{\lambda_{GL}(0)}{\xi_{GL}(0)}$. The $\xi_{GL}$ = 7.2(1)
$nm$ and $\lambda_{GL}$(0) = 279(2) $nm$, we obtained $\kappa_{GL}$ = 39(1) >>
${\frac{1}{\sqrt{2}}}$ indicate that the Nb60Re10Zr10Hf10Ti10 HEA is a strong
type II superconductor. In type II superconductors, the applied magnetic field
is responsible for breaking the Cooper pair by two types of mechanism; orbital
and Pauli limiting effects (briefly described in ref. Dsingh ).

Figure 4: (a) $C/T$ vs $T^{2}$ curve above $T_{C}$ fitted by equation 8. (b)
The observed data shown by red square circle was fitted with BCS isotropic gap
(blue line) equation. (c) Normalised $\gamma$ of superconducting state is
plotted with normalised field. Dashed red curve represents the linear behavior
at lower values and indicate the $s$-wave superconductivity.

For BCS superconductor, the orbital limiting field $H_{C2}^{orb}$(0) is
calculated via Werthamar-Helfand-Hohenberg (WHH) model

$H_{C2}^{orbital}(0)=-\alpha
T_{C}\left.\frac{dH_{C2}(T)}{dT}\right|_{T=T_{C}}$ (5)

where $\alpha$ can take 0.693 value in dirty limit. The ratio of BCS coherence
length and electronic mean free path value $\xi_{0}$/l >1 [detailed in
section(f)] and suggest the dirty limit superconductivity in
Nb0.60Re0.10Zr0.10Hf0.10Ti0.10. The slope of temperature dependent upper
critical field ${\frac{dH_{C2}}{dT}}$ at $T=T_{C}$ is estimated to be -1.08(3)
$T$, which yields $H_{C2}^{orb}$ = 4.2(1) $T$, and another effect, Pauli
limiting field $H_{P}$ can be addressed within the BCS theory using the
expression H${}_{c2}^{P}$ = 1.84 $T_{C}$ and the estimate value of
$H_{C2}^{P}$ = 10.3 (1) $T$.
In order to measure the relative strength of orbital and Pauli limiting field,
Maki parameter is calculated and expressed as
$\alpha_{M}=\sqrt{2}H_{C2}^{orb}(0)/H_{C2}^{p}(0)$. The value obtained for
$\alpha_{M}$= 0.57, suggesting the small influence of paramagnetic effect and
mainly orbital limiting field is responsible for pair breaking.

###  d Specific heat

Specific heat measurement was performed on Nb0.60Re0.10Zr0.10Hf0.10Ti0.10
alloy with applying external field and in zero fields. The sample was mounted
on the platform with Apiezon N grease to establish good thermal contact. A
pronounced jump in heat capacity was observed at transition temperature
$T_{C}$ at 5.4(1) $K$ as shown in Fig. 4(a). The $T_{C}$ value estimated by
specific heat measurement is comparable with resistivity and magnetization.
The specific heat data in the normal state (above $T_{C}$) was well fitted
with the following equation:

$\frac{C}{T}=\gamma_{n}+\beta_{3}T^{2}$ (6)

This fitting provides the electronic parameter (Sommerfeld coefficient)
$\gamma_{n}$ = 7.1(1) $mJmol^{-1}$K-2 and the lattice contribution parameter
to the specific heat, $\beta_{3}$ = 0.140(2) $mJmol^{-1}$K-4. Both the
parameters, $\gamma_{n}$ and $\beta_{3}$ have paramount importance in
calculating the density of states ($D_{c}(E_{F})$), Debye temperature
($\theta_{D}$), and electron-phonon coupling constant ($\lambda_{e-p}$). The
following quantities are calculated from the expressions provided in Dsingh
and the estimated values are: $D_{c}(E_{F})$ = 3.01(4) $\frac{states}{eVf.u}$,
$\theta_{D}$ = 240(1) $K$, and $\lambda_{e-p}$ = 0.70. The value of electron
phonon coupling strength $\lambda_{e-p}$ is comparable with other High entropy
alloy superconductors tetragonal ; poly , which suggests moderately coupled
superconductivity in Nb0.60Re0.10Zr0.10Hf0.10Ti0.10.
To further reveal the superconducting ground state behavior, electronic
specific heat $C_{el}$ in the superconducting state can be estimated by
excluding the phonon contribution from the total specific heat $C(T)$ by the
expression $C_{el}$ = $C(T)$ \- $\beta_{3}T^{3}$. For a single gap BCS
superconductor, the normalized entropy in the superconducting state 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\\\
$ (7)

where f($\xi$) = [exp(E($\xi$)/$k_{B}T$)+1]-1 is the Fermi function, E($\xi$)
= $\sqrt{\xi^{2}+\Delta^{2}(t)}$, where E($\xi$) is the energy of the normal
electrons 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
energy gap. The normalized electronic specific heat below $T_{C}$ is related
to the first derivative of normalized entropy as

$\frac{C_{el}}{\gamma_{n}T_{C}}=t\frac{d(S/\gamma_{n}T_{C})}{dt}\\\ $ (8)

The fitting provides the superconducting gap value
$\frac{\Delta(0)}{k_{B}T_{C}}$ = 2.57, which is very high in magnitude with
respect to the BCS predicted value and slightly more than the equimolar
Nb20Re20Zr20Hf20Ti20 HEA maneesha . The electronic heat capacity data together
with electron phonon coupling strength provide the evidence of isotropic $s$
\- wave moderately coupled superconductivity in Nb60Re10Zr10Hf10Ti10 . To
further unveil the superconducting gap structure, we carried out field-
dependent specific heat measurements. We first calculated the electronic
specific heat with respective fields by subtracting the corresponding phononic
part ($\beta_{3}$T3) from total specific heat (C). The field-dependent
electronic specific heat in the superconducting state was fitted with the
expression low BCS

$\frac{C_{el}}{T}=\gamma+\frac{A}{T}\exp\left(\frac{-bT_{c}}{T}\right)$ (9)

here A is material dependent quantity, and b = $\frac{\Delta(0)}{k_{B}T_{c}}$
superconducting gap. Fig. 4(c) shows the normalized $\gamma$ rises linearly
with applied external magnetic field H and both the parameters $\gamma$ and H
are normalized by $\gamma_{n}$ and the upper critical field $H_{C2}$(0),
respectively. The linear increase in $\gamma$ is because the quasiparticle
excitation is confined within the vortex cores and a number of vortices vary
with field $H$ for an isotropic superconducting gap structure low BCS ; gama H
. Thus, the linear behavior of $\gamma(H)$ proportional to field $H$ at low
fields indicates the fully isotropic $s$-wave superconducting gap. In the
presence of nodes, the $\gamma(H)$ is expected to vary as square root to the
applied field.

###  e $\mu$SR Spectroscopy

Muon spin relaxation in zero fields and muon spin rotation in transverse field
measurements were done on a MuSR spectrometer ISIS facility at Rutherford
Appleton Laboratory, U. K. Forty pulses of spin-polarized muons enter MuSR in
each second, which are implanted into the sample and these muons sits at the
interstitial positions in the powder sample. The muon spin precess with Larmor
frequency, decays into a positron and two neutrinos. The positron decay into
the initial muon spin direction, is recorded through 64 scintillator
detectors, and these detectors are positioned in a circular design around the
sample chamber. In zero field muon spin relaxation measurement, two sets of
the detectors were positioned around the sample, while in transverse field
muon spin rotation, the detectors were positioned in four configurations, and
each configuration consist of 16 detectors which are named as forward,
backward, top and bottom. The time domain asymmetry spectra were calculated
using the expression A(t) = $\left(\frac{F(t)-\alpha B(t)}{F(t)+\alpha
B(t)}\right)$ and $\alpha$ is the relative efficiency of counts between any
two of the four detectors aidy . This asymmetry function gives information
regarding muon spin polarization at a time scale. The powder sample was mixed
with GE varnish and mounted on a silver holder. The benefits of using a silver
holder are getting time-independent signal in ZF-$\mu$SR and a non-decaying
oscillation in TF-$\mu$SR configurations. The low temperature was achieved
using 3He sorption cryostat. In the ZF configuration, the sample was first
cooled to the lowest temperature in the zero field, and then data was taken
while warming without any applied field. The effect of the magnetic field at
sample position due to the earth’s magnetic field or from the neighboring
instruments are cancelled by a set of correction coil. In TF configuration, an
external magnetic field was applied perpendicular to the direction of muon
spin. The sample was field cooled to create a flux line lattice and data were
taken from base temperature to above the transition temperature.

Figure 5:  (a) Zero field $\mu$SR asymmetry spectra is shown from above and
below TC. The black line represents the fit to the Kubo-Toyabe equation. (b)
Transverse field $\mu$SR asymmetry spectra collected at 1.39 $K$ and 7.5 $K$
under applied field of 30 $mT$. These red and blue lines are the fit to the
data using equation 12.

####  1 ZF-$\mu$SR Relaxation

$\mu$SR measurements in zero field can be used to detect spontaneous
magnetization associated with broken time-reversal symmetry in the
superconducting state.

Figure 6: (a) Temperature dependent variation of $\lambda^{-2}$ fitted well
with isotropic $s$-wave BCS superconducting model. (b) Internal magnetic field
distribution inside the sample with respect to temperature. Red data points
are the observed field from the sample and blue data points are the background
field contribution from the sample holder. (c) Field distribution in vortex
state and normal state (inset of Fig. 6(c).

The ZF-$\mu$SR is the best tool to investigate the minimal magnetic field
without applying an external magnetic field. We collected the relaxation
spectra above and far below the superconducting transition temperature, which
is shown in Fig. 5(a). The decaying relaxation behavior indicates that no
oscillation present above and below $T_{C}$, and the relaxation comes from the
randomly oriented nuclear moments. ZF-$\mu$SR relaxation data above and below
$T_{C}$ were fitted very well with the Kubo-Toyabe equation, which is given as
KT

$G_{\mathrm{KT}}(t)=\frac{1}{3}+\frac{2}{3}(1-\sigma^{2}_{\mathrm{ZF}}t^{2})\mathrm{exp}\left(\frac{-\sigma^{2}_{\mathrm{ZF}}t^{2}}{2}\right),$
(10)

where $\sigma_{ZF}$ is the muon spin relaxation rate. The asymmetry spectra
can be explained with expression as

$A(t)=A_{1}G_{\mathrm{KT}}(t)\mathrm{exp}(-\Lambda t)+A_{\mathrm{BG}}$ (11)

here A1 is the sample asymmetry, ABG is the background asymmetry signal coming
due to the stopping of muons in the sample holder, and $\Lambda$ is electronic
relaxation rate. The fitting of Eq. (11) above and below the TC, yield no
noticeable change of relaxation rate in the detection limit (10 $\mu$T) of
instruments. The change corresponding to both the relaxation are found as
$\sqrt{\Delta}/\gamma_{\mu}$ = 0.01 Oe and $\Lambda/\gamma_{\mu}$ = 0.04 Oe,
which are outside the scope of detection for $\mu$SR instrument. Hence, the
time reversal symmetry is preserve in Nb60Re10Zr10Hf10Ti10.

####  2 TF-$\mu$SR

To detect the superconducting gap symmetry and the field distribution in the
mixed state, TF-$\mu$SR measurements have been performed on
Nb0.60Re0.10Zr0.10Hf0.10Ti0.10 alloy. In this experiment, we first cool the
sample in an applied magnetic field of 30mT, which is greater than the lower
critical field but less than the $H_{c2}$ ($H_{c1}$ < $H^{ext}$ << $H_{C2}$)
to create a well-ordered flux line lattice. Asymmetry spectra in TF-$\mu$SR
were recorded in temperature range from 1.39 $K$ to 7.5 $K$ and only two
spectra at 1.39 $K$ and above $T_{C}$ are displayed in Fig. 5(b). The
distribution of field in the entire temperature scale was extracted from the
TF-$\mu$SR time spectra by the Maximum Entropy method (ME) ME . Below the TC,
Fig. 6(c) is showing the evidence of depolarization, and inset of Fig. 6(c))
is clearly showing the homogeneous field above $T_{C}$. The variation of muon
precession in asymmetry spectra was observed because of the interaction of
muon spin with internal field distribution in a sample. Above $T_{C}$, muon
spin relax due to the presence of randomly oriented nuclear dipolar moments,
and below $T_{C}$, the relaxation arises from the inhomogeneous magnetic field
due to the formation of flux line lattice in the mixed state plus the nuclear
contribution. The asymmetry spectra of TF-$\mu$SR is well fitted in all
temperature range by the Gaussian oscillatory decaying function, and it is
described below as KT1 ; KT2

$\begin{split}A(t)=\sum_{i=1}^{N}A_{i}\exp\left(-\frac{1}{2}\sigma_{i}^{2}t^{2}\right)\cos(\gamma_{\mu}B_{i}t+\phi)\\\
+A_{bg}\cos(\gamma_{\mu}B_{bg}t+\phi),\end{split}$ (12)

where $A_{i}$ and $A_{bg}$ are the sample and sample holder related asymmetry
signal respectively, $\phi$ is the initial phase offset and $\sigma$ is rate
of Gaussian depolarization of muon spin. $B_{i}$ is mean field contribution of
$i^{th}$ component of the Gaussian distribution, and $B_{bg}$ is the
background field contribution coming from the sample holder.
Above $T_{C}$, there is a constant relaxation rate that occurs due to the
random orientation of magnetic moment of nuclear spins and is temperature
independent. The internal field dependence on temperature is shown in Fig.
6(b), which shows a background magnetic field contribution that is constant
for the temperature range measured. The internal magnetic field distribution
in the mixed state denoted by $<$B$>$ is less than the applied fields for $T$
< $T_{C}$, which clearly shows the Meissner field expulsion in the
superconducting state. At temperature ($T$ > $T_{C}$), it again recovers the
applied field value and overlaps with the background signal. The total
depolarization rate is not only from the flux lattice ($\sigma_{FLL}$) but
also arises due to the nuclear magnetic moment ($\sigma_{n}$), which is
temperature independent. $\sigma_{n}$ contribution was calculated from the
above $T_{C}$ data points and $\sigma_{FLL}$ is estimated from subtracting the
contribution of $\sigma_{n}$ from the total depolarization rate as
$\sigma_{FLL}^{2}=\sigma^{2}-\sigma_{n}^{2}$, where $\sigma$ is the total
depolarization rate. $\sigma_{FLL}$ is related to the London penetration depth
via ($\sigma_{FLL}$ $\propto$ 1/$\lambda^{-2}$). For small applied fields
where H << $H_{C2}$, the inverse square of the London penetration depth can be
calculated via the formula:

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

where $\Phi_{0}$ is the magnetic flux quantum and $\gamma_{\mu}$/2$\pi$ =
135.5 $MHz/T$ is muon gyromagnetic ratio. Within London approximation, the
calculated London penetration depth at 0 $K$ is $\lambda^{\mu}$ = 391(18)
$nm$, which is bit higher than the calculated value of $\lambda$ = 279(2) $nm$
from magnetization. The temperature dependence of penetration depth for
$s$-wave BCS superconducting gap is given by the expression BCS L1 ; BCS L2 ;
BCS L3

$\begin{split}\frac{\lambda^{-2}(T)}{\lambda^{-2}(0)}=1+2\int_{\Delta(T)}^{\infty}(\frac{\partial
f}{\partial E})\frac{EdE}{\sqrt{E^{2}-\Delta(T)^{2}}},\end{split}$ (14)

Where $\lambda$(0) is the London penetration depth at 0 $K$, and $f$ =
[1+$\exp$(E/kBT)]-1 is the Fermi function and $\Delta$ (T) is the BCS
superconducting gap function which is defined by $\Delta$ (T) =
$\Delta_{0}$tanh[1.82(1.018($\mathit{(T_{C}/T})$-1))0.51]. The experimental
data of $\lambda^{-2}$ (T) was best fitted with the BCS s-wave modal given by
Eq. (14) and the obtained TC = 5.2(2) $K$ is also in good agreement with the
values measured from magnetization and specific heat measurements for
Nb60Re10Zr10Hf10Ti10. The ratio of superconducting gap at 0 $K$,
$\Delta_{0}$/kBTc = 2.65(5), which shows great deviation from the BCS value
(1.73) and comparable to estimated value by specific heat measurement
(2.57(8).

Figure 7: Uemura plot representing the superconducting transition temperature
versus Fermi temperature for different superconducting families. Two solid
blue lines represents the unconventional band of superconductors and the solid
red circle for Nb60Re10Zr10Hf10Ti10 alloy away from the unconventional band.

###  f Electronic property

To supplement the observed result of experiment measurements, we have
determined the electronic parameter such as the BCS coherence length,
electronic mean free path and the Fermi velocity. We have used estimated
parameter charge carrier density (measured from hall measurement) shown inset
of Fig. 2, Sommerfeld coefficient by normal heat capacity, and resistivity at
transition temperature. The Sommerfeld coefficient is related with effective
mass and carrier density of quasiparticle by the equation Thinkam

$\gamma_{n}=\left(\frac{\pi}{3}\right)^{2/3}\frac{k_{B}^{2}m^{*}n^{1/3}}{\hbar^{2}}$
(15)

where $k_{B}$, $n$ and $m^{*}$ are Boltzmann constant, carrier density and
effective mass of quasiparticle, respectively. Using the values of Sommerfeld
coefficient $\gamma_{n}$ = 7.1(1) $mJ-mol^{-1}$K-2 and carrier density $n$ =
6.1(1)$\times{10}^{28}m^{-3}$, we estimated the effective mass of quasi-
particle $m^{*}$ = 9.9(2) $m_{e}$.
The Fermi velocity is directly related to the carrier density and effective
mass, which can be expressed as

$n=\frac{1}{3\pi^{2}}\left(\frac{m^{*}v_{\mathrm{f}}}{\hbar}\right)^{3}.$ (16)

Using the estimated value of effective mass $m^{*}$ = 9.9 $m_{e}$ and carrier
density $n$ = 6.1(1)$\times{10}^{28}m^{-3}$ of quasiparticle, we get the Fermi
velocity $v_{F}$ = 1.5$\times$ 105 $ms^{-1}$.
The mean free path is dependent on Fermi velocity $v_{f}$, effective mass
$m^{*}$ and the residual resistivity by the expression

$\textit{l}=\frac{3\pi^{2}{\hbar}^{3}}{e^{2}\rho_{0}m^{*2}v_{\mathrm{F}}^{2}}$
(17)

The previous estimated value of Fermi velocity and effective mass with the
residual resistivity $\rho_{0}$ = 76(1) $\mu\ohm-cm$, which yield mean free
path $l$ = 9.6 Å.
BCS coherence length can be estimated using Fermi velocity and transition
temperature by the expression as

$\xi_{0}=\frac{0.18{\hbar}{v_{F}}}{k_{B}T_{C}}$ (18)

Using the Fermi velocity parameter value and transition temperature $T_{C}$ =
5.6 $K$, we get the ratio of BCS coherence length to electronic mean free path
$\xi(0)$/$l$ = 38 > 1, which strongly suggested that the sample is in the
dirty limit of superconductivity. Other estimated parameters are listed in
Table I.
The Fermi temperature can be expressed as; $k_{B}T_{F}$ =
${\frac{{\hbar}^{2}}{2m^{*}}(3{\pi}^{2}n)^{2/3}}$ where $k_{B}$, $m^{*}$ =
9.9(2) $m_{e}$, and $n$ = 6.1(1)$\times{10}^{28}m^{-3}$ are the Boltzmann
constant, effective mass, and carrier density of quasi-particle, respectively.
The expression gives Fermi temperature, TF 6480(236) $K$. Uemura $et$ $al$. U1
; U2 ; U3  give a classification to distinguish the conventional and
unconventional nature of superconductor by the ratio of TC/TF. The range of
unconventional superconductor is 0.01 $\leq$ TC/TF $\leq$ 0.1, and rest of the
range represents the conventional superconductor. The ratio of TC/TF = 0.00086
as shown in Fig. 7, which places Nb60Re10Zr10Hf10Ti10 away from the
unconventional superconductor line, and categories the Nb60Re10Zr10Hf10Ti10 as
a conventional superconductor.

Table 1: Superconducting and normal state parameters of Nb60Re10Zr10Hf10Ti10 PARAMETERS | UNIT | Nb0.6Re0.1Zr0.1Hf0.1Ti0.1
---|---|---
$T_{C}$ | K | 5.60(4)
$H_{C1}(0)$ | mT | 7.95(7)
H${}_{C2}^{mag}$(0) | T | 6.31(1)
$H_{C2}^{P}(0)$ | T | 10.3(1)
$\xi_{GL}$ | $nm$ | 7.2(1)
$\lambda_{GL}$ | $nm$ | 279(2)
$\lambda(0)^{\mu}$ | $nm$ | 391(18)
$k_{GL}$ |  | 39(1)
$\Delta(0)/k_{B}T_{C}$(specific heat) |  | 2.57(8)
$\Delta(0)/k_{B}T_{C}$(muon) |  | 2.65(5)
$m^{*}/m_{e}$ |  | 9.9(2)
$\xi_{0}/l_{e}$ |  | 38(5)
$v_{F}$ | ${10}^{5}ms^{-1}$ | 1.50(4)
$n_{s}$ | 1028m-3 | 6.1(1)
$T_{F}$ | K | 6480(236)

##  IV Conclusion

A detailed investigation of the superconducting and normal state properties of
the superconducting HEA Nb60Re10Zr10Hf10Ti10 has been conducted, using
detailed magnetization, electrical resistivity, heat capacity, and $\mu$SR
measurements. It is a type-II superconductor with a nodeless, isotropic
superconducting gap and is well described by the BCS theory in the dirty
limit. In the superconducting ground state, time-reversal symmetry has been
preserved in the detection limit of $\mu$SR experiments. Combined results from
all of the experimental techniques used to calculate the electronic properties
and Umera plot. It is shown that superconductivity in Nb60Re10Zr10Hf10Ti10 is
in the dirty limit, and the Uemura plot placed it in the region of
conventional superconductor. To understand the superconducting properties of
HEAs, it is essential to search and perform a microscopic study of new
superconducting HEAs.

##  V Acknowledgments

Kapil Motla acknowledges the CSIR funding agency (Award no;
09/1020(0123)/2017-EMR-I), Council of Scientific and Industrial Research
(CSIR) Government of India for providing SRF fellowship. R. P. S. acknowledge
Science and Engineering Research Board, Government of India for the Core
Research Grant CRG/2019/001028. We thank ISIS, STFC, UK for the beamtime to
perform the $\mu$SR experiments.

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