# Electron-phonon superconductivity in C-doped topological nodal-line
semimetal Zr5Pt3: A muon spin rotation and relaxation ($\mu$SR) study

A Bhattacharyya${}^{1^{*}}$, P P Ferreira${}^{2^{\dagger}}$, K Panda1, F B
Santos2, D T Adroja3,4, K Yokoyama3, T T Dorini5, L T F Eleno2 and A J S
Machado2 1Department of Physics, Ramakrishna Mission Vivekananda Educational
and Research Institute, Howrah 711202, India, Howrah 711202, West Bengal,
India 2Escola de Engenharia de Lorena, Universidade de São Paulo, DEMAR,
Lorena, Brazil 3 ISIS Facility, Rutherford Appleton Laboratory, Chilton,
Didcot, Oxon, OX11 0QX, United Kingdom 4 Highly Correlated Matter Research
Group, Physics Department, University of Johannesburg, Auckland Park 2006,
South Africa 5 Université de Lorraine, CNRS, IJL, Nancy, France
<EMAIL_ADDRESS>
~~~~~~<EMAIL_ADDRESS>
###### Abstract

In the present work we demonstrate that C-doped Zr5Pt3 is an electron-phonon
superconductor (with critical temperature TC = 3.7 K) with a nonsymmorphic
topological Dirac nodal-line semimetal state, which we report here for the
first time. The superconducting properties of Zr5Pt3C0.5 have been
investigated by means of magnetization and muon spin rotation and relaxation
($\mu$SR) measurements. We find that at low temperatures the depolarization
rate is almost constant and can be well described by a single-band $s-$wave
model with a superconducting gap of $2\Delta(0)/k_{\mathrm{B}}T_{\mathrm{C}}$
= 3.84, close to the value of BCS theory. From transverse field $\mu$SR
analysis we estimate the London penetration depth $\lambda_{L}$ = 469 nm,
superconducting carrier density $n_{s}$ = 2$\times$1026 $m^{-3}$, and
effective mass m∗ = 1.584 $m_{e}$. Zero field $\mu$SR confirms the absence of
any spontaneous magnetic moment in the superconducting ground state. To gain
additional insights into the electronic ground state of C-doped Zr5Pt3, we
have also performed first-principles calculations within the framework of
density functional theory (DFT). The observed homogenous electronic character
of the Fermi surface as well as the mutual decrease of $T_{\mathrm{C}}$ and
density of states at the Fermi level are consistent with the experimental
findings. However, the band structure reveals the presence of robust, gapless
fourfold-degenarate nodal lines protected by $6_{3}$ screw rotations and glide
mirror planes. Therefore, Zr5Pt3 represents a novel, unprecedented condensed
matter system to investigate the intricate interplay between superconductivity
and topology.

###### pacs:

71.20.Be, 74.70.Dd, 76.75.+i

Keywords. Nodal-line semi metal; Superconducting gap structure; Muon spin
spectroscopy

##  1 Introduction

The search for new superconductors is a cornerstone in quantum matter physics.
In particular, the study of how the superconducting state evolves with doping
and pressure has been essential to understand the fundamental mechanisms of
Cooper pairs condensation and competion between different coherent states.
However, in the last few years we are seeing a new chapter being written,
where the topological nature of matter is responsible for the emergence of a
wide range of novel quantum states. Topological insulators and symmetry-
protected topological semimetals can exhibit quantizied anomalous Hall effect
[1, 2, 3], ultrahigh electronic mobility [4, 5, 6], negative/giant
magnetoresistence [7, 8], chiral anomaly [9, 10, 11, 12], and gapless, robust
edge states [13, 14, 15], to cite a few examples. When combined with
superconductors, topological materials could also harbor Majorana
quasiparticles with non-Abelian exchange statistics, thus possibly opening the
way for the quantum computation era [16].

In this context, a large number of intermetallic compounds with general
formula M5X3 (M is either a transition or rare earth metal and X is a
metalloid) represent a novel platform to explore a plethora of unique
properties [17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. These compounds can
crystallize in three different prototypes: (i) orthorhombic Yb5Sb3 ($Pnma$,
No. 62), (ii) tetragonal Cr5B3 ($I4/mcm$, No. 140), and (iii) hexagonal Mn5Si3
($P6_{3}/mcm$, No. 193). Among them, the Mn5Si3 structure can interstitially
host a third element (carbon, boron, nitrogen or oxygen) at 2$b$ Wyckoff
position. Physical properties of more than 500 compounds in this hexagonal
structure have been reported so far. Superconductivity is observed only in few
compounds though, such as Zr5Sb3 (TC = 2.3 K) [18], Zr5Pt3 (TC = 6.4 K) [24],
tetragonal (TC = 2.8 K) and hexagonal (TC = 9.4 K) Nb5Ir3 [22], and Nb5Ge3 (TC
= 0.3 K) [26]. In Nb5Ir3, for instance, TC increases to 10.5 K with oxygen
doping [20] and a crossover from multiple to single gap superconductivity with
increasing Pt content was reported in Nb5Ir3-xPtxO [19]. On the other hand,
the addition of oxygen in Zr5Pt3 reduces monotonically the critical
temperature from 6.4 K to 3.2 K (Zr5Pt3O${}_{0.6})$ [24]. Likewise, TC of
Zr5Sb3 decreases with the addition of oxygen until the supresison of the
superconducting phase in Zr5Sb3O [18]. The highest reported TC in this family,
however, belongs to Nb5Ge3C0.3 [23], with T${}_{\mathrm{C}}=15.3$ K.

Figure 1: (a) Hexagonal Zr5Pt3C unit cell with space group $P6_{3}/mcm$ (no.
193). Purple, orange, and yellow spheres represent Zr, Pt, and C,
respectively. (b) First Brillouin zone of Zr5Pt3 system with the path along
high-symmetry points. The $a$–$b$, $a$–$c$, and $c$–$b$ planes are shown in
(c), (d) and (e), respectively. (f) Temperature variation of the ac magnetic
susceptibility in presence of an applied field of 1 mT, collected in FC and
ZFC modes. The inset shows the field dependence of the isothermal
magnetization at 2 K.

Recently, Renosto et al. reported a comprehensive study of the effects of
carbon doping on the superconducting properties of Zr5Pt3Cx [25]. It was found
that TC first increases from 6 K ($x=0$) to 7 K ($x=0.3$) and further
decreases monotonously for $x>0.3$ [25]. However, by measuring the lower and
upper critical fields and the temperature dependence of penetration lenght and
specific heat, the authors argued that Zr5Pt3C0.3 strongly deviates from the
conventional s-wave pairing behavior, suggesting thus an unconventional
superconductivity driven by carbon doping.

Motivated by these results, we show, using muon spin relaxation and rotation
($\mu$SR) measurements combined with first-principles electronic-structure
calculations, that a conventional, single-gap s-wave superconducting order
parameter within the weak-coupling limit is sufficient to describe the
superconducting ground state of Zr5Pt3C0.5, in contrast to the recent
proposals. This is the first systematic $\mu$SR study on the superconducting
properties of doped M5X3-type compounds. Furthermore, we demonstrate that
Zr5Pt3 is a topological nodal-line semimetal protected by nonsymmorphic
symmetries. In contrast to Dirac/Weyl semimetals, where the conduction and
valence bands touch at discrete points in the Brillouin zone and disperse
linearly in all momentum directions [27], in nodal-line semimetals the linear
band-crossing points form closed loops in momentum space, opening the way to
novel quantum phases, such as nearly flat drumhead-like surface states [28,
29, 30]. In this vein, Zr5Pt3-type compounds represent an unprecedented, large
family of superconducting nodal-line semimetals, still to be explored in more
details.

##  2 Experimental Details

###  2.1 Sample Preparation and Physical Properties Measurements

A Zr5Pt3C0.5 sample was prepared using typical arc melting process in argon
gas atmosphere on a water-cooled copper crucible using highly pure Zr, Pt, and
C in a stoichiometric ratio. The arc melted ingot was remelted several times
to confirm its homogeneity. After that, the sample was annealed at 1200 oC for
a week in a sealed vacuum quartz tube. X-ray powder diffraction was carried
out using Cu-$K_{\alpha}$ radiation. The magnetization measurements were
performed using VSM-SQUID, 9T-PPMS (Quantum Design Inc.) using both field-
cooled (FC) and zero-field-cooled (ZFC) protocols.

###  2.2 $\mu$SR Measurements

$\mu$SR measurements were carried out at the ISIS Neutron and Muon Source of
STFC Rutherford Appleton Laboratory, UK, on the MUSR spectrometer. 64
detectors are placed in transverse and longitudinal directions to detect the
asymmetry of the positrons [31]. In the sample, 100% spin-polarized positive
muons are implanted, which decay with a half-life of 2.2$\mu$s, decaying into
a positron and a pair of neutrinos. As the positrons are preferentially
emitted along the muon spin vectors, information on the local magnetic field
distribution at the muon stopping site can be obtained by detecting the
asymmetry of the emitted positrons. The time-dependent symmetry $A(t)$ of the
$\mu$SR spectra is given by $A(t)=\frac{N_{\mathrm{F}}(t)-\alpha
N_{\mathrm{B}}(t)}{N_{\mathrm{F}}(t)+\alpha N_{\mathrm{B}}(t)}$, where
$N_{\mathrm{F}}(t)$ and $N_{\mathrm{B}}(t)$ are the number of positrons
counted in the forward and backward detectors respectively, and $\alpha$ is an
instrumental calibration factor. ZF-$\mu$SR is carried out in the longitudinal
set up of the detectors. A correction coil is applied to neutralize any stray
magnetic fields up to 10-3 G. The transverse field measurements were carried
with detectors in a transverse arrangement, with a field of 300 G ( well below
the upper critical field $H_{\mathrm{c2}}$=6.3 T) applied perpendicular to the
initial muon polarization direction [32].

The Zr5Pt3C0.5 sample was powdered and placed in a high purity (99.995%)
silver sample holder using diluted GE-varnish and then wrapped with thin
silver foil, since the signal from muons stopping in silver depolarizes at a
negligible rate. All data analysis were done using WiMDA [33] software.

###  2.3 Computational Methods

First-principes electronic-structure calculations were performed within the
Kohn-Sham scheme [34] of the Density Functional Theory (DFT) [35] with full-
relativistic projector augmented plane wave pseudopotentials [36] as
implemented in Quantum Espresso [37, 38]. Exchange and correlation (XC)
effects were treated with the local density approximation (LDA) as described
by the Perdew-Zunger (PZ) parametrization [39]. We have used a wave function
energy cut-off of 80 Ry (1 Ry $\approx$ 13.6 eV), and 800 Ry for the charge
density and potential kinetic energy cut-off. The Monkhorst-Pack scheme [40]
was used for a $8\times 8\times 12$ $k$-point sampling in the first Brillouin
zone. A denser $16\times 16\times 24$ $k$-point sampling was further used to
obtain the band structure, density of states and Fermi surface. Self-
consistent-field (SCF) calculations were carried out using a Marzari-
Vanderbilt smearing [41] of 0.005 Ry. All latice parameters and internal
degrees of freedom were relaxed in order to guarantee a ground-state
convergence of 10-5 Ry in total energy and 0.5 mRy/$a_{0}$ ($a_{0}\approx
0.529\,$Å) for forces acting on the nuclei.

##  3 Results and discussion

###  3.1 Crystal structure and Magnetization

X-ray powder diffraction shows that Zr5Pt3 crystallizes in hexagonal
Mn5Si3-type structure ($P6_{3}/mcm$, No. 193), as shown in Fig. 1(a). The
temperature dependence of the magnetic susceptibility, $\chi(T)$, in the
presence of an applied field of 10 G is presented in Fig. 1(b). The low field
dc magnetic susceptibility clearly shows a superconducting phase transition at
T${}_{\mathrm{C}}=3.7$ K. Moreover, the isothermal field dependence of
magnetization at 2 K [see the inset in Fig. 1(b)] confirms the presence of a
type-II superconductivity.

Figure 2: (a) Time dependence of transverse field $\mu$SR asymmetry spectra
collected at (a) $T$ = 0.3 K ($<$TC) and (b) $T$ = 5 K ($>$TC) in the presence
of an applied field $H$ = 300 G. The red solid line shows the fit to the data
using Eq. (1). (c) and (d) display the corresponding maximum entropy spectra
(below and above TC).

###  3.2 TF-$\mu$SR analysis

TF-$\mu$SR measurements were performed from 0.3 K to 5 K. The observed
asymmetries, at 0.3 K and 5 K, below and above $T_{\mathrm{C}}$ respectively,
can be found in Fig 2(a)-(b). The respective maximum entropies are shown in
Fig. 2(c) and (d). In the superconducting state at 0.3 K, it is clear that the
spectra strongly depolarize due to the inhomogeneous distribution of the
internal field derived from the formation of the vortex state. Meanwhile, at 5
K the spectra depolarization is quite negligible, which can be attributed to
the nuclear moments of the silver background. The time spectra can be fitted
using two Gaussian oscillatory functions [42, 43, 44],

$G_{\mathrm{TF}}(t)=A_{\mathrm{sc}}\cos(\omega_{\mathrm{sc}}t+\phi)\exp(-\frac{\sigma_{\mathrm{T}}^{2}t^{2}}{2})+A_{\mathrm{bg}}\cos(\omega_{\mathrm{bg}}t+\phi),$
(1)

where the initial asymmetries associated with both sample and background are
represented by $A_{\mathrm{sc}}$ and $A_{\mathrm{bg}}$, respectively,
$\omega_{\mathrm{sc}}$ and $\omega_{\mathrm{bg}}$ are the frequencies of the
muon spin precession frequency from the sample and background, and $\phi$ is
the phase offset. The Gaussian depolarization rate $\sigma_{T}$ has two
contribution below $T_{\mathrm{C}}$, where $\sigma_{\mathrm{sc}}$ is derived
from a superconducting component and $\sigma_{\mathrm{nm}}$(=0.0852 $\mu$s-1)
is derived from a nuclear magnetic dipolar moment that is constant over the
whole temperature range, later assisted by the ZF-$\mu$SR. Thus, the
contribution from the vortex lattice, $\sigma_{sc}$, was calculated by
quadratically subtracting $\sigma_{\mathrm{nm}}$ obtained from the fitting of
the spectra measured above $T_{\mathrm{C}}$. The field shift is $\Delta B$ =
BSC-Bapp, where BSC indicates the superconducting field induced by the vortex
lattice and Bapp is the applied field, as shown in Fig. 3(b). As the sample
goes through the transition into the superconducting state, there is a strong
negative shift in the peak field, which is a unique characteristic of the
vortex lattice [45, 46].

Figure 3(a) presents $\sigma_{\mathrm{sc}}(T)$ as a function of temperature,
which is proportional to the superfluid density, thus providing details about
the gap structure. It is clear that at low temperature the
$\sigma_{\mathrm{sc}}$ is almost constant, which indicates a fully gapped
superconducting state. The normalized superfluid density was modelled using
[47, 48, 49, 50, 51]

$\displaystyle\frac{\sigma_{sc}(T)}{\sigma_{sc}(0)}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{-2}(T)}{\lambda^{-2}(0)}$ $\displaystyle=$
$\displaystyle
1+\frac{1}{\pi}\int_{0}^{2\pi}\int_{\Delta(T)}^{\infty}(\frac{\delta f}{\delta
E})\times\frac{EdEd\phi}{\sqrt{E^{2}-\Delta(T})^{2}}.$

Here, $f$ is the Fermi function, that can be represented by
$f=[1+\exp(-E/k_{\mathrm{B}}T)]^{-1}$, and
$\Delta(T,0)=\Delta_{0}\delta(T/T_{\mathrm{C}})g(\phi)$, whereas $g(\phi)$ is
the angular dependence of the gap function. The azimuthal angle in the
direction of the Fermi surface is denoted by $\phi$. The temperature variation
of the superconducting gap is approximated by the relation
$\delta(T/T_{\mathrm{C}})=\tanh\\{{1.82[1.018(T_{\mathrm{C}}/T-1)]^{0.51}}\\}$.
$g(\phi$), the spatial dependence, is substituted by 1 for $s-$wave symmetry.
Using this, we find that the data is best modelled considering a single
isotropic $s$-wave gap of 0.59 meV, which yields
$2\Delta(0)/k_{\mathrm{B}}T_{\mathrm{C}}=3.84$, close to the value of 3.53
predicted for weak-coupling BCS superconductors. Below T$\mathrm{C}$ the
electronic specific heat capacity (Ce) is well described by BCS gap model
[25].

Figure 3: (a) The temperature variation of superconducting depolarization rate
$\sigma_{\mathrm{sc}}(T)$. The solid red line shows the fit using $s-$wave
model. Inset shows the total muon spin depolarization rate
$\sigma_{\mathrm{T}}$ as a function of temperature. (b) The relative change of
the internal field normalized to external applied field as a function of
temperature, where $\Delta B$ = BSC \- Bapp.

The depolarization rate ($\sigma_{\mathrm{sc}}$) recorded below
$T_{\mathrm{C}}$ is correlated with the superfluid density or the penetration
depth. For a triangular lattice,
$\frac{\sigma_{\mathrm{sc}}^{2}}{\gamma_{\mathrm{\mu}}^{2}}=\frac{0.00371\times\phi_{\mathrm{0}}^{2}}{\lambda^{4}}$
[52, 53, 54], where $\phi_{\mathrm{0}}$ is the flux quantum number (2.07
$\times$10-15T m2) and $\gamma_{\mathrm{\mu}}$ is the muon gyromagnetic ratio
$\gamma_{\mathrm{{\mu}}}/2\pi$ = 135.5 MHz T-1. Superfluid density can also be
related to quantities at the atomic order, as with other phenomenological
parameters that characterize the superconducting state. Using London’s theory,
[52] $\lambda_{\mathrm{L}}^{2}=\frac{m^{*}c^{2}}{4\pi n_{\mathrm{s}}e^{2}}$,
where $m^{*}=(1+\lambda_{\mathrm{e-ph}})m_{\mathrm{e}}$ is the effective mass
and $n_{\mathrm{s}}$ is the density of superconducting carriers. Within this
simple picture, $\lambda_{\mathrm{L}}$ is independent of magnetic field.
$\lambda_{\mathrm{e-ph}}$ is the electron-phonon coupling parameter that can
be estimated from the Debye temperature ($\Theta_{\mathrm{D}}$) and
$T_{\mathrm{C}}$ using the semi-empirical McMillan equation [55],

$\lambda_{\mathrm{e-ph}}=\frac{1.04+\mu^{*}\ln(\Theta_{\mathrm{D}}/1.45T_{\mathrm{C}})}{(1-0.62\mu^{*})\ln(\Theta_{\mathrm{D}}/1.45T_{\mathrm{C}})-1.04},$
(3)

where $\mu^{*}$ is a repulsive electron-electron pseudopotential with typical
values at the order of $\mu^{*}$ = 0.1, which gives
$\lambda_{\mathrm{e-ph}}=0.584$. Since Zr5Pt3C0.5 is a type-II superconductor,
it implies that the density of the normal state carriers is approximately
equal to the density of superconducting carriers ($n_{\mathrm{s}}\approx
n_{\mathrm{e}}$). Therefore, the density of the superconducting carriers
($n_{s}$), their effective-mass ($m^{*}$), and London penetration depth
($\lambda_{L}$) can be estimated, respectively, as
$m^{*}=1.584m_{\mathrm{e}}$, $n_{\mathrm{s}}=2.026\times 10^{26}$ carriers/m3,
$\lambda_{\mathrm{L}}(0)$ = 469 nm, for Zr5Pt3C0.5.

Figure 4: Time evolution of ZF-$\mu$SR asymmetry spectra at 0.3 K (black
squares) and 5 K (green circles), presented together. The red line is the
least squares fit to the data using Eq. (4).

###  3.3 ZF-$\mu$SR analysis

ZF-$\mu$SR is used to investigate the existence of any spontaneous magnetic
moment in the superconducting ground state [56, 57, 58]. The evolution of the
ZF-asymmetry spectra over time is displayed in Fig. 4 for $T$ = 0.3 K
($<T_{\mathrm{C}}$) and $T$ = 5 K ($>T_{\mathrm{C}}$). The spectra below and
above $T_{\mathrm{C}}$ are found to be similar, ruling out the presence of any
magnetic ordering, which implies that time-reversal symmetry (TRS) is
preserved in the superconducting state of Zr5Pt3C0.5. The ZF-$\mu$SR data was
fitted using a Lorentzian function with a constant background [59]:

$G_{\mathrm{ZF}}(t)=A_{\mathrm{0}}\exp{(-\lambda t)}+A_{\mathrm{bg}}.$ (4)

Here, $A_{\mathrm{0}}$ is the sample asymmetry and $A_{\mathrm{bg}}$ is the
background asymmetry, which are almost independent of the temperature. The
parameter $\lambda$ is the rate of relaxation resulted from the nuclear
moments. In Fig. 4, the red line indicates the obtained fit for ZF-$\mu$SR
data using Eq. 4. The parameters obtained from the ZF-$\mu$SR asymmetry data
are as follows: $\lambda$ = 0.291 $\mu\mathrm{s}^{-1}$ at 0.3 K and $\lambda$
= 0.242 $\mu\mathrm{s}^{-1}$ at 5 K. The shift in the relaxation rate is
within the error bar, indicating that TRS is preserved in Zr5Pt3C0.5.

##  4 Theoretical Calculations

The electronic band structure, partial density of states (DOS), and Fermi
surface of Zr5Pt3, Zr5Pt3C0.5, and Zr5Pt3C including spin-orbit coupling (SOC)
effects are shown in Figures 5(a)-(c), with the corresponding DOS at the Fermi
level, $N(E_{F})$, and optimized lattice parameters. With the increasing of C
content $x$, the $a$ ($c$) lattice parameter decreases (increases)
monotonically, reflecting a stronger in-plane hybridization between C-p and
Zr-d states. There are six bands crossing the Fermi energy in Zr5Pt3 [see
Figure 5(a)], which results in a high density of states at the Fermi level of
18.3 eV-1, with approximately 68 % of these carriers derived from the Zr-d
manifold. The change in $N(E_{F})$ with increasing $x$ from 0 to 0.5 in
Zr5Pt3Cx is almost $-5.5\,$%, which can be partily associated with the quasi-
rigid band tuning of the chemical potential and the respective suppression of
the $\epsilon$ sheet in the Fermi surface up to $x=1$.

(a) Zr5Pt3 (b) Zr5Pt3C0.5 (c) Zr5Pt3C

Figure 5: Electronic structure, partial density of states, and Fermi surface
of (a) Zr5Pt3, (b) Zr5Pt3C0.5, and (c) Zr5Pt3C. Also shown are the total DOS
at the Fermi level, $N(E_{F})$, and the relaxed lattice parameters. The
gapless nodal lines protected by the nonsymmorphic 63 screw rotation symmetry
of space group $P6_{3}/mcm$ are highlighted by solid red lines in the band
structure diagram. The color map in the Fermi surfaces indicates the
contribution of Zr-d orbitals in the low-energy states’ wave functions.

Although its complex, disconnected multiband nature, the homogeneous
distribution of the electronic character of the Fermi surface shows an evident
contrast from the usual signature presented by the Fermi surface of multiband
superconductors, which generally presents very distinct orbital characters and
an anisotropic hybridization on distinct sheets, and, as consequence, the
average of the electron-phonon scattering connecting different points at Fermi
surface are disjointed relative to the band index [60, 61, 62, 63, 64, 65, 66,
67, 68]. The mutual monotonic decrease of the DOS at the Fermi level and the
observed superconducting critical temperature with the carbon doping [25] is
compatible with the BCS theory as well. Therefore, in the light of our
theoretical and experimental findings, the single-band electron-phonon s-wave
pairing within weak coupling limit may be considered a coherent mechanism for
superconductivity in Zr5Pt3 system.

Interestingly, our calculations reveal that Zr5Pt3 is a nonsymmorphic
topological Dirac nodal-line semimetal [69, 70, 71, 72], as explained below.
In the presence of SOC all bands are twofold-degenerate due to the presence of
both time-reversal and inversion symmetries. Along $\Gamma$–$A$ we can observe
the presence of several linear band crossings in the vicinity of the Fermi
level. Such fourfold-degenerate gapless Dirac nodes along $\Gamma$–$A$ are
protected against SOC by $C_{3}$ rotational symmetry of the double point group
$C_{6v}$, since both twofold-degenerate electronic dispersions that originate
the band crossings, with distinct irreducible representations, possess
different rotation eingevalues on the out-of-plane axis [73, 74, 75, 76].
However, pairs of twofold-degenerate bands along $A$–$L$ merge into single,
degenerate bands along $L$–$H$–$A$, that is, along $k_{z}=\pi/c$ plane, which
creates symmetry-enforced fourfold-degenerate nodal lines (solid red lines in
Figure 5). These nodal lines are protected by 63 screw rotations with centre
of symmetry in the 2$b$ Wyckoff position and axial glide mirror planes of the
$P6_{3}/mcm$ space group, which means that they are robust against any
pertubations as long as the nonsymmorphic symmetries hold. In fact, the
presence of nonsymmorphic symmetries, such as screw rotations and glide mirror
symmetries, support a nontrivial band connection at the Brillouin zone
boundary [75].

The nonsymmorphic protection of the nodal lines are demonstrated in Figure
5(b), where we show the band structure of Zr5Pt3C0.5. In this structure only
half of the 2$b$ Wyckoff positions are filled, thus explicitly breaking the
center of symmetry of the $6_{3}$ screw axis and the fractional translation of
the glide mirror planes. Consequently, the nodal lines are gapped due to the
spin-orbit coupling effect, as represented by the dashed red lines along
$L$–$H$–$A$, and the Fermi surface acquire a distorsive topography. However,
when the nonsymmorphic symmetries are preserved, by completely filling out the
$2b$ positions in Zr5Pt3C [Figure 5(c)], the nodal-line states remain
protected. At this point it is important to note that, on average, effective
point-group symmetries can be preserved in a homogeneously, perfectly randomly
disordered alloy, thus we expect that the topological nodal-line phase could
be observed even upon a high level of alloying [77, 78, 79, 80, 81], as in the
case of Zr5Pt3C0.5.

##  5 Conclusions

In summary, we presented the superconducting properties of the interstitial
carbon-doped compound Zr5Pt3C0.5 using magnetization and muon spin rotation
and relaxation measurements. Magnetization data confirms the bulk
superconductivity at TC = 3.7 K. The depolarization rate of muon spin
($\sigma_{\mathrm{sc}}$) in the FC mode is almost constant at low temperatures
and can be well modeled considering a fully gapped isotropic $s$-wave
superconducting order parameter, with 2$\Delta$/kBT${}_{\mathrm{C}}=3.84$,
close to 3.53, the value expected for BCS superconductors. The London
penetration depth, superconducting carrier density and its effective mass are
also estimated from TF-$\mu$SR analysis. Furthermore, zero field $\mu$SR
confirms that there is no spontaneous magnetic moment, thus demonstrating that
time-reversal symmetry is preserved in the superconducting ground state.
Additionally, we showed through DFT calculations that Zr5Pt3 is a topological
Dirac nodal-line semimetal protected by 63 screw rotations and glide mirror
planes of the $P6_{3}/mcm$ space group. Therefore, this work puts forward a
large, unprecedented class of superconducting topological nodal-line
semimetals to realize novel quantum states of matter.

####

Acknowledgments

PPF, LTFE and AJSM gratefully acknowledge the financial support of the São
Paulo Research Foundation (FAPESP) under Grants 2019/05005-7, 2019/14359-7,
and 2020/08258-0. KP acknowledge the financial support from DST India, for
Inspire Fellowship (IF170620). AB would like to acknowledge DST India, for
Inspire Faculty Research Grant (DST/INSPIRE/04/2015/000169), the SERB, India
for core research grant support. and UK-India Newton funding for funding
support. DTA would like to thank the Royal Society of London for Newton
Advanced Fellowship funding and International Exchange funding between UK and
Japan. The research was partially carried out using high-performance computing
resources made available by the Superintendência de Tecnologia da Informação
(STI), Universidade de São Paulo. The authors also acknowledge the National
Laboratory for Scientific Computing (LNCC/MCTI, Brazil) for providing HPC
resources of the SDumont supercomputer, which have contributed to the research
results reported within this paper. This study was financed in part by the
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) –
Finance Code 001.

## References

## References

  * [1] Haldane F 2004 Physical Review Letters 93 206602
  * [2] Xu G, Weng H, Wang Z, Dai X and Fang Z 2011 Physical Review Letters 107 186806
  * [3] Weng H, Yu R, Hu X, Dai X and Fang Z 2015 Advances in Physics 64 227–282
  * [4] Shekhar C, Nayak A K, Sun Y, Schmidt M, Nicklas M, Leermakers I, Zeitler U, Skourski Y, Wosnitza J, Liu Z et al. 2015 Nature Physics 11 645–649
  * [5] Liang T, Gibson Q, Ali M N, Liu M, Cava R and Ong N 2015 Nature Materials 14 280–284
  * [6] Zhao Y, Liu H, Zhang C, Wang H, Wang J, Lin Z, Xing Y, Lu H, Liu J, Wang Y et al. 2015 Physical Review X 5 031037
  * [7] Li H, He H, Lu H Z, Zhang H, Liu H, Ma R, Fan Z, Shen S Q and Wang J 2016 Nature Communications 7 1–7
  * [8] Gao W, Hao N, Zheng F W, Ning W, Wu M, Zhu X, Zheng G, Zhang J, Lu J, Zhang H et al. 2017 Physical Review Letters 118 256601
  * [9] Zyuzin A and Burkov A 2012 Physical Review B 86 115133
  * [10] Parameswaran S, Grover T, Abanin D, Pesin D and Vishwanath A 2014 Physical Review X 4 031035
  * [11] Xiong J, Kushwaha S K, Liang T, Krizan J W, Hirschberger M, Wang W, Cava R J and Ong N P 2015 Science 350 413–416
  * [12] Huang X, Zhao L, Long Y, Wang P, Chen D, Yang Z, Liang H, Xue M, Weng H, Fang Z et al. 2015 Physical Review X 5 031023
  * [13] Hsieh D, Qian D, Wray L, Xia Y, Hor Y S, Cava R J and Hasan M Z 2008 Nature 452 970–974
  * [14] Hsieh D, Xia Y, Wray L, Qian D, Pal A, Dil J, Osterwalder J, Meier F, Bihlmayer G, Kane C et al. 2009 Science 323 919–922
  * [15] Zhang H, Liu C X, Qi X L, Dai X, Fang Z and Zhang S C 2009 Nature physics 5 438–442
  * [16] Beenakker C 2013 Annu. Rev. Condens. Matter Phys. 4 113–136
  * [17] Sürgers C, Gajdzik M, Fischer G v, Löhneysen H v, Welter E and Attenkofer K 2003 Physical Review B 68 174423
  * [18] Lv B, Zhu X, Lorenz B, Wei F, Xue Y, Yin Z, Kotliar G and Chu C 2013 Physical Review B 88 134520
  * [19] Xu Y, Jöhr S, Das L, Kitagawa J, Medarde M, Shiroka T, Chang J and Shang T 2020 Physical Review B 101 134513
  * [20] Zhang Y, Wang B, Xiao Z, Lu Y, Kamiya T, Uwatoko Y, Kageyama H and Hosono H 2017 npj Quantum Materials 2 1–7
  * [21] Zheng C, Mattausch H and Simon A 2002 Journal of alloys and compounds 347 79–85
  * [22] Cort B, Giorgi A and Stewart G 1982 Journal of Low Temperature Physics 47 179–185
  * [23] Bortolozo A D, Dos Santos C, Jardim R d F, Ritter C, Devishvili A, Rotter M, Gandra F and Machado A J d S 2012 Journal of Applied Physics 111 123912
  * [24] Hamamoto S and Kitagawa J 2018 Materials Research Express 5 106001
  * [25] Renosto S, Lang R, Manesco A, Rodrigues Jr D, Santos F, Machado A, Baldan M and Diez E 2018 arXiv preprint arXiv:1809.05008
  * [26] Claeson T, Ivarsson J and Rasmussen S 1977 Journal of Applied Physics 48 3998–3999
  * [27] Bernevig A, Weng H, Fang Z and Dai X 2018 Journal of the Physical Society of Japan 87 041001
  * [28] Weng H, Liang Y, Xu Q, Yu R, Fang Z, Dai X and Kawazoe Y 2015 Physical Review B 92 045108
  * [29] Yu R, Weng H, Fang Z, Dai X and Hu X 2015 Physical Review Letters 115 036807
  * [30] Chan Y H, Chiu C K, Chou M and Schnyder A P 2016 Physical Review B 93 205132
  * [31] Lee S L, Cywinski R and Kilcoyne S 1999 Muon science: Muons in physics, chemistry and materials vol 51 (CRC press)
  * [32] Bhattacharyya etal A STFC ISIS Neutron and Muon Source
  * [33] Pratt F 2000 Physica B: Condensed Matter 289 710–714
  * [34] Kohn W and Sham L J 1965 Physical Review 140 A1133
  * [35] Hohenberg P and Kohn W 1964 Physical Review 136 B864
  * [36] Dal Corso A 2014 Computational Materials Science 95 337–350
  * [37] Giannozzi P, Baroni S, Bonini N, Calandra M, Car R, Cavazzoni C, Ceresoli D, Chiarotti G L, Cococcioni M, Dabo I et al. 2009 Journal of physics: Condensed matter 21 395502
  * [38] Giannozzi P, Andreussi O, Brumme T, Bunau O, Nardelli M B, Calandra M, Car R, Cavazzoni C, Ceresoli D, Cococcioni M et al. 2017 Journal of Physics: Condensed Matter 29 465901
  * [39] Perdew J P and Zunger A 1981 Physical Review B 23 5048
  * [40] Monkhorst H J and Pack J D 1976 Physical Review B 13 5188
  * [41] Marzari N, Vanderbilt D, De Vita A and Payne M C 1999 Physical Review Letters 82 3296
  * [42] Bhattacharyya A, Adroja D T, Smidman M and Anand V 2018 Science China Physics, Mechanics & Astronomy 61 1–22
  * [43] Adroja D T, Bhattacharyya A, Biswas P K, Smidman M, Hillier A D, Mao H, Luo H, Cao G H, Wang Z and Wang C 2017 Physical Review B 96 144502
  * [44] Bhattacharyya A, Adroja D T, Panda K, Saha S, Das T, Machado A, Cigarroa O, Grant T, Fisk Z, Hillier A et al. 2019 Physical Review Letters 122 147001
  * [45] Brandt E 1988 Journal of low temperature physics 73 355–390
  * [46] Brandt E H 2003 Physical Review B 68 054506
  * [47] Prozorov R and Giannetta R W 2006 Superconductor Science and Technology 19 R41
  * [48] Adroja D T, Bhattacharyya A, Telling M, Feng Y, Smidman M, Pan B, Zhao J, Hillier A, Pratt F and Strydom A 2015 Physical Review B 92 134505
  * [49] Adroja D T, Bhattacharyya A, Smidman M, Hillier A, Feng Y, Pan B, Zhao J, Lees M R, Strydom A and Biswas P K 2017 Journal of the Physical Society of Japan 86 044710
  * [50] Bhattacharyya A, Panda K, Adroja D T, Kase N, Biswas P, Saha S, Das T, Lees M and Hillier A D 2020 Journal of Physics: Condensed Matter 32 085601
  * [51] Adroja D T, Bhattacharyya A, Sato Y, Lees M, Biswas P, Panda K, Stenning G B, Hillier A and Aoki D 2021 Physical Review B 103 104514
  * [52] Sonier J E, Brewer J H and Kiefl R F 2000 Reviews of Modern Physics 72 769
  * [53] Chia E E, Salamon M, Sugawara H and Sato H 2004 Physical Review B 69 180509
  * [54] Amato A 1997 Reviews of Modern Physics 69 1119
  * [55] McMillan W 1968 Physical Review 167 331
  * [56] Bhattacharyya A, Adroja D T, Quintanilla J, Hillier A, Kase N, Strydom A and Akimitsu J 2015 Physical Review B 91 060503
  * [57] Bhattacharyya A, Adroja D T T, Kase N, Hillier A, Akimitsu J and Strydom A 2015 Scientific reports 5 1–8
  * [58] Bhattacharyya A, Adroja D T, Kase N, Hillier A, Strydom A and Akimitsu J 2018 Physical Review B 98 024511
  * [59] Panda K, Bhattacharyya A, Adroja D T, Kase N, Biswas P, Saha S, Das T, Lees M and Hillier A 2019 Physical Review B 99 174513
  * [60] Floris A, Sanna A, Massidda S and Gross E 2007 Physical Review B 75 054508
  * [61] Bersier C, Floris A, Sanna A, Profeta G, Continenza A, Gross E and Massidda S 2009 Physical Review B 79 104503
  * [62] Flores-Livas J A and Sanna A 2015 Physical Review B 91 054508
  * [63] Kawamura M, Akashi R and Tsuneyuki S 2017 Physical Review B 95 054506
  * [64] Ferreira P, Santos F, Machado A, Petrilli H and Eleno L 2018 Physical Review B 98 045126
  * [65] Pascut G L, Widom M, Haule K and Quader K F 2019 Physical Review B 100 125114
  * [66] Bhattacharyya A, Ferreira P, Santos F, Adroja D T, Lord J, Correa L, Machado A, Manesco A and Eleno L T 2020 Physical Review Research 2 022001
  * [67] Zhao Y, Lian C, Zeng S, Dai Z, Meng S and Ni J 2020 Physical Review B 101 104507
  * [68] Correa L E, Ferreira P P, de Faria L R, Dorini T T, Fisk Z, Torikachvili M S, Eleno L T and Machado A J 2021 arXiv preprint arXiv:2102.04812
  * [69] Burkov A, Hook M and Balents L 2011 Physical Review B 84 235126
  * [70] Fang C, Chen Y, Kee H Y and Fu L 2015 Physical Review B 92 081201
  * [71] Fang C, Weng H, Dai X and Fang Z 2016 Chinese Physics B 25 117106
  * [72] Yu R, Fang Z, Dai X and Weng H 2017 Frontiers of Physics 12 1–14
  * [73] Young S M, Zaheer S, Teo J C, Kane C L, Mele E J and Rappe A M 2012 Physical Review Letters 108 140405
  * [74] Yang B J and Nagaosa N 2014 Nature communications 5 1–10
  * [75] Yang B J, Morimoto T and Furusaki A 2015 Physical Review B 92 165120
  * [76] Ferreira P P, Manesco A L, Dorini T T, Correa L E, Weber G, Machado A J and Eleno L T 2021 Physical Review B 103 125134
  * [77] Dziawa P, Kowalski B, Dybko K, Buczko R, Szczerbakow A, Szot M, Łusakowska E, Balasubramanian T, Wojek B M, Berntsen M et al. 2012 Nature materials 11 1023–1027
  * [78] Narayan A, Di Sante D, Picozzi S and Sanvito S 2014 Physical Review Letters 113 256403
  * [79] Yan C, Liu J, Zang Y, Wang J, Wang Z, Wang P, Zhang Z D, Wang L, Ma X, Ji S et al. 2014 Physical Review Letters 112 186801
  * [80] Lu H, Zhang X, Bian Y and Jia S 2017 Scientific Reports 7 1–10
  * [81] Thirupathaiah S, Morozov I, Kushnirenko Y, Fedorov A, Haubold E, Kim T, Shipunov G, Maksutova A, Kataeva O, Aswartham S et al. 2018 Physical Review B 98 085145