Nodeless superconductivity and topological nodal states in molybdenum carbide
Tian Shang Yuting Wang Bochen Yu Keqi Xia Darek J. Gawryluk Yang Xu Qingfeng Zhan Jianzhou Zhao Toni Shiroka

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	Preprint: September 9, 2024,  













These authors contributed equally[Corresponding authors:
] tshang@phy.ecnu.edu.cn
Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China


These authors contributed equally
Co-Innovation Center for New Energetic Materials, Southwest University of Science and Technology, Mianyang, 621010, People's Republic of China
School of Science, Southwest University of Science and Technology, Mianyang 621010, P. R. China



Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China


Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China



Center for Neutron and Muon Sciences, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland




Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China



Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China


 [Corresponding authors:
]jzzhao@swust.edu.cn
Co-Innovation Center for New Energetic Materials, Southwest University of Science and Technology, Mianyang, 621010, People's Republic of China 







Center for Neutron and Muon Sciences, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland
Laboratorium für Festkörperphysik, ETH Zürich, CH-8093 Zürich, Switzerland


§ ABSTRACT 
The orthorhombic molybdenum carbide superconductor with T_c = 3.2 K was investigated by muon-spin rotation and relaxation (µSR) measurements and by 
first-principle calculations. The low-temperature superfluid density, determined by transverse-field µSR, suggests a fully-gapped superconducting state in Mo_2C, with a zero-temperature gap Δ_0 = 0.44 meV and a magnetic penetration depth λ_0 = 291 nm. The time-reversal symmetry is preserved in the superconducting state, as confirmed by the absence of an additional muon-spin relaxation in the zero-field µSR spectra. Band-structure calculations indicate that the density of states at the Fermi level is dominated by the
Mo 4d-orbitals, which are marginally
hybridized with the C 2p-orbitals over a wide energy range.
The symmetry analysis confirms that, in the absence of spin-orbit coupling (SOC), Mo_2C hosts twofold-degenerate nodal surfaces and fourfold-degenerate nodal lines. 
When considering SOC, the fourfold-degenerate nodal lines cross the Fermi level and contribute to the electronic properties.
Our results suggest that, similarly to other phases of carbides, also the orthorhombic
transition-metal carbides host topological nodal states and may be potential candidates for future studies of topological superconductivity.





Nodeless superconductivity and topological nodal states in molybdenum carbide
    Toni Shiroka
    
=============================================================================

3pt





§ INTRODUCTION
8pt

The possibilities offered by topological superconductors, ranging from hosting Majorana fermion quasiparticles to potential applications in topological quantum computing <cit.>, have stimulated the researchers to explore different routes to realize them. 
The most obvious approach consists in the introduction of extra carriers into a topological insulator to achieve superconductivity (SC). This route has been frequently attempted
in the copper- or strontium intercalated Bi_2Se_3 topological insulator <cit.>. Another approach utilizes the proximity effect between a conventional s-wave superconductor and a topological insulator or semiconductor <cit.>. 
The surface states of  a topological insulator can lead to a two-dimensional superconducting state with a p+ip pairing at the interfaces, known to support Majorana bound states at the vortices <cit.>. For instance, evidence of topological SC has been reported in
NbSe_2/Bi_2(Se,Te)_3 heterostructures <cit.>, where NbSe_2 represents a typical fully-gapped superconductor.

Despite continued efforts to identify topological SC in accordance with
the aforementioned approaches,
the intricacy of heterostructure fabrication, the rarity of suitable
topological insulators, and the inhomogeneity or disorder
effects induced by carrier doping, have
considerably constrained the investigation and potential applications of topological SC.
A more attractive way to achieve them is to combine
superconductivity and a nontrivial electronic band structure in the same material.
Clearly, it is of fundamental interest to be able to identify such new superconductors with nontrivial band topology, but
with a simple composition. 
For example, topologically protected surface states have been found in superconducting CsV_3Sb_5 <cit.>, β-PdBi_2 <cit.>, and PbTaSe_2 <cit.>, 
all of which are good platforms for studying topological SC. 

In this respect, the binary transition-metal carbides (TMCs)
represent another promising family of materials.
TMCs exhibit essentially four different solid phases, which include the α (Fm3m No. 225)-,
β (Pbcn, No. 60)-, γ (P6m2, No. 187)-, and
η (P6_3/mmc, No. 194)-phase <cit.>. The γ-phase is noncentrosymmetric, while the other three are centrosymmetric. Due to the lack of space inversion, the γ-phase TMCs exhibit exotic topological features. The unconventional three-component fermions with surface Fermi arcs were experimentally observed in γ-phase WC <cit.>.
By applying external pressure, the topological semimetal MoP (isostructural to WC) becomes a superconductor, whose T_c rises up to 4 K (above 90 GPa) <cit.>, thus representing a candidate topological superconductor. Unfortunately, no SC has been observed in WC yet, but the γ-phase MoC (similar to WC and also not superconducting) was predicted to be a topological nodal-line semimetal with drumhead surface states <cit.>. The α-phase TMCs show a relatively high T_c value and some of them
were also predicted to exhibit nontrivial band topologies <cit.>.  
For example, NbC and TaC are fully-gapped superconductors with T_c = 11.5 and 10.3 K, respectively <cit.>. 
At the same time, theoretical calculations suggest that
α-phase TMCs are  nodal line semimetals in the absence of spin-orbit coupling (SOC) <cit.>.

As for the molybdenum carbides, although their SC was already reported in the 1970s <cit.>, their physical properties have been overlooked due to difficulties in synthesizing clean samples. Only recently, the α-phase MoC_x (x < 1) and η-phase Mo_3C_2 (with T_c = 14.3 and 8.5 K) could be synthesized under high-temperature and high-pressure conditions (1700 ^∘C, 6–17 GPa) <cit.> and their superconducting properties studied via different techniques. 
To date, the electronic properties of the other phases
of molybdenum carbides (e.g., the β-phase) remain mostly unexplored.  

In this paper, we report on the superconducting properties of the
β-phase Mo_2C, investigated via magnetization- and muon-spin relaxation and rotation (µSR)
measurements. In addition, we also present numerical density-functional-theory (DFT) band-structure calculations. 
We find that Mo_2C exhibits a fully-gapped superconducting state,
while its electronic band structure suggests that 
it hosts twofold-degenerate nodal surfaces and fourfold-degenerate nodal lines. 
Therefore, the β-phase 
TMCs (of which Mo_2C is a typical example)
may be potential candidates for future studies of topological SC,
similar to the other TMC phases. 




§ EXPERIMENTAL AND NUMERICAL METHODS
8pt

First, we tried to synthesize the β-phase Mo_2C by arc melting
Mo slugs (99.95%, Alfa Aesar) and C rods (99.999+%, ChemPUR).
Similarly to previous studies <cit.>, the obtained
polycrystalline samples showed a mixture of different phases, both before and after the annealing. Akin to the α-phase, the β-phase Mo_2C can be synthesized also under high-temperature and high-pressure conditions (1500–2300 K, 5 GPa) <cit.>. 
However, the resulting Mo_2C samples have a rather low superconducting volume fraction. 
Because of these issues, all our measurements were performed
on high-purity Mo_2C powders (99.5%) produced by
Alfa Aesar.
For the µSR investigation,
the powders were pressed into pellets, while for the magnetization
measurements, performed on a 7-T Quantum Design magnetic property
measurement system, loose powders were used. 
Room-temperature x-ray powder diffraction (XRD) 
was performed on a Bruker D8 diffractometer using Cu Kα radiation. 
The µSR measurements were carried out at the multipurpose surface-muon spectrometer (Dolly) at the πE1 beamline of the Swiss muon source at Paul Scherrer Institut (PSI), Villigen, Switzerland. 
The Mo_2C pellets were mounted on 25-µm thick copper foil to
cover an area 6–8 mm in diameter. The µSR spectra comprised
both transverse-field (TF) and zero-field (ZF) 
measurements, performed upon heating the sample.
The µSR spectra were analyzed by means of the  software package <cit.>.


The phonon spectrum and the electronic band structure of Mo_2C
were calculated via DFT, within the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) realization <cit.>, as implemented in the Vienna ab initio Simulation Package (VASP) <cit.>. The projector augmented wave (PAW) pseudopotentials were adopted for the calculation <cit.>. Electrons belonging to the outer atomic configuration were treated as valence electrons, here
corresponding to 6 electrons in Mo (4d^55s^1) and 4 electrons in C (2s^22p^2). The kinetic energy cutoff was fixed to 400 eV. 
For the three different crystal structures of Mo_2C,  the atomic
positions and the lattice constants were fully relaxed for the
calculations of the phonon dispersion spectrum. 
The force convergence criterion was set to 1 meV.
For the structure optimization calculations, Monkhorst-Pack grids
of 16 × 16 × 10, 14 × 11 × 13, and
19 × 19 × 21 k-points were used for the space groups
P6_3/mmc, Pbcn, and P3̅1m, respectively.
To obtain the force constants and phonon spectra, we used the
density functional perturbation theory (DFPT)
in combination with the Phonopy package <cit.>.
A supercell of 2× 2× 2 was adopted for the calculation of force constants.
To calculate the phonon spectrum, the Brillouin zone integration
was performed on a Γ-centered mesh of 10 × 10 × 7,
7 × 5 × 6, and 6 × 6 × 7 k-points for
the space groups P6_3/mmc, Pbcn, and P3̅1m, respectively. 
In the P6_3/mmc case, considering that only half of the 2a sites is occupied by C atoms, we simplified the structure such that C atoms are fully occupied only at the corners of the unit cell.
The spin–orbit coupling (SOC) was fully considered in our calculation. 
After optimizing the parameters, the electronic- and phononic band structures, as well as the density of states (DOS)
were calculated.



§ RESULTS AND DISCUSSION
8pt 



 §.§  Crystal structure





The phase purity and the crystal structure of Mo_2C powders were
checked via XRD measurements at room temperature (see Fig. <ref>).
Unlike the arc-melted Mo_2C <cit.>, the purchased Mo_2C
powders show a clean phase. Several phases of molybdenum carbides
have been reported, which exhibit cubic, orthorhombic, hexagonal,
and trigonal structures <cit.>.
In our case, the XRD pattern of Mo_2C, was analyzed by means of
the FullProf Rietveld-analysis suite <cit.> to find that
only the latter three structures, with space groups
Pbcn (No. 60, orthorhombic), P6_3/mmc (No. 194, hexagonal), and
P3̅1m (No. 162, trigonal), reproduce the data reasonably well.
In the insets we depict the corresponding crystal structures,
known as β-, η-, and ζ-phases, respectively.
Among these, the β-phase exhibits the best agreement with
the measured XRD pattern, here reflected in the smallest
goodness-of-fit factor (see Table <ref>). 
Moreover, both the η- and ζ-phases fail to reproduce some
of the low-intensity reflections. For instance, as illustrated
in the inset of Fig. <ref>(a), while neither the η- nor
the ζ-phases admit a reflection at 2θ≈ 30^∘,
the β-phase captures this reflection quite well. 
In conclusion, the Rietveld refinements suggest that the investigated Mo_2C powders
adopt an orthorhombic structure with space group Pbcn, as
further confirmed by the calculated phonon-dispersion spectrum (see below). Furthermore, no impurity phases could be detected, indicating a good sample quality.
The refined crystal-structure information and atomic coordinates for
all the three different phases are listed in
Tables <ref> and <ref>. 











 §.§ Magnetization measurements






We first characterized the SC of Mo_2C powders by magnetic susceptibility, carried out in  
a 5-mT field, using both field-cooled (FC) and zero-field-cooled (ZFC) protocols. As indicated by the arrow in Fig. <ref>(a), a clear diamagnetic signal appears
below the superconducting transition at T_c = 3.2 K. The reduced
T_c value compared to the previously reported T_c ∼ 6 K
(whose SC fraction was less than 1%)
is most likely attributed to the varying C-content <cit.>.
Such a variable T_c against C-content has been previously reported
in the α- and η-phase of molybdenum carbides <cit.>,
where the light C atom is expected to modify significantly the electron-phonon
coupling and the phonon frequencies and, ultimately, also T_c.

 
A large diamagnetic response (i.e., χ_V∼ -0.8 at 2 K)
indicates a bulk SC in Mo_2C, as further confirmed by our TF-µSR measurements. 
The field-dependent magnetization curves M(H), collected at a few temperatures below T_c, are plotted in the inset of Fig. <ref>(b). The estimated lower
critical fields H_c1 as a function of temperature are summarized in Fig. <ref>(b). This yields a lower critical field μ_0H_c1 = 6.4(2) mT for Mo_2C at zero temperature (see solid line).  




 §.§ Transverse-field MuSR










To study the gap symmetry and superconducting pairing of Mo_2C,
we performed systematic TF-µSR measurements in an applied
field of 30 mT (i.e., much higher than H_c1) at various temperatures. 
In a TF-µSR measurement, the magnetic field is applied perpendicular to the muon-spin direction, leading to the precession of the muon spin. 
By performing TF-µSR, one can quantify the additional field-distribution broadening due to the flux-line lattice (FLL) and, thus, determine the superfluid density in type-II superconductors. 
Figure <ref>(a) plots two representative superconducting- and  normal-state TF-µSR spectra for Mo_2C. 
The enhanced muon-spin relaxation in the superconducting state is clearly visible
and it is due to the formation of a FLL during the field-cooling process, which generates an inhomogeneous field distribution <cit.>.
The broadening of field distribution in the superconducting state is clearly reflected in the fast Fourier transform (FFT) 
of the TF-µSR spectra [see Figs. <ref>(b)-(c)]. 
To describe the field distribution, the TF-µSR spectra can be modelled using <cit.>: 


    A_TF(t) = ∑_i=1^n A_i cos(γ_μ B_i t + ϕ) e^- σ_i^2 t^2/2 +
    	A_bgcos(γ_μ B_bg t + ϕ).


Here A_i, A_bg and B_i, B_bg 
are the initial asymmetries and local fields sensed by implanted muons in the 
sample and sample holder,
γ_μ/2π = 135.53 MHz/T 
is the muon gyromagnetic ratio, ϕ is a shared initial phase, and σ_i 
is the Gaussian relaxation rate of the ith component. 
Here, we find that Eq. (<ref>) with n = 2 [solid line in Fig. <ref>(b)] shows a better agreement with the experimental data than with n = 1 [dashed line in Fig. <ref>(b)]. 
In the normal state, the derived muon-spin relaxation rates σ_i(T) are small and independent of temperature while, below T_c, they start to increase due to the onset of the FLL
and the increased superfluid density (see inset in Fig. <ref>).
The effective Gaussian relaxation rate σ_eff can be calculated from σ_eff^2/γ_μ^2 = ∑_i=1^2 A_i [σ_i^2/γ_μ^2 - (B_i - ⟨ B ⟩)^2]/A_tot <cit.>, where ⟨ B ⟩ = (A_1 B_1 + A_2 B_2)/A_tot and A_tot = A_1 + A_2. By considering the constant nuclear
relaxation rate σ_n in the narrow temperature
range (∼0.3–5 K) investigated here, confirmed also 
by ZF-µSR measurements (see Fig. <ref>), the
superconducting Gaussian relaxation rate can be extracted from
σ_sc = √(σ_eff^2 - σ^2_n). 






The effective magnetic penetration depth λ_eff can 
then be calculated using σ_sc^2(T)/γ^2_μ = 0.00371Φ_0^2/λ_eff^4(T) <cit.>.
Figure <ref> summarizes the temperature-dependent inverse square of magnetic penetration depth λ_eff^-2(T), which is proportional to the superfluid density ρ_sc(T). 
The various models used to analyze the ρ_sc(T) data, are generally described by the relation:
 



    ρ_sc(T) = 1 + 2 ⟨∫^∞_Δ_kE/√(E^2-Δ_k^2)∂ f/∂ EdE ⟩_FS.


Here, f = (1+e^E/k_BT)^-1 is the Fermi function and ⟨⟩_FS represents an average over the Fermi surface <cit.>; Δ_k(T) = Δ(T) δ_k is an angle-dependent
gap function, where Δ is the maximum gap value and δ_k is the 
angular dependence of the gap, equal to 1, cos2ϕ, and sinθ 
for an s-, d-, and p-wave model, respectively, where ϕ 
and θ are the azimuthal angles. The temperature-dependent gap is assumed to follow Δ(T) = Δ_0 tanh{1.82[1.018(T_c/T-1)]^0.51} <cit.>, where Δ_0 is the zero-temperature gap value.
The s- and p-wave models (see black solid and red dashed lines in Fig. <ref>) yield the same zero-temperature magnetic penetration depth λ_0 =291(3) nm, but different zero-temperature energy gaps Δ_0 =  0.44(1) and 0.60(1) meV, respectively. The magnetic penetration depth of the β-phase Mo_2C is much higher than that of the α-phase MoC_x (λ_0 ∼ 132 nm) and the η-phase Mo_3C_2 (λ_0 ∼ 197 nm) <cit.>.
A possible d-wave model (green dash-dotted line in Fig. <ref>) provides a gap size Δ_0 = 0.55(1) meV. This is comparable to the p-wave model, 
but λ_0 = 255(3) nm is much shorter than that of both the s- and p-wave models. 
As can be clearly seen in Fig. <ref>, below ∼1.2 K, the d-wave model deviates significantly from the experimental data. 
At the same time, also the p-wave model shows a poor agreement with data in the 0.7–1.6 K range. The s-wave model, on the other hand,
reproduces the experimental data quite well over the entire temperature
range studied. In addition, the temperature-independent
λ_eff^-2(T) at T < 1 K (i.e., 1/3T_c) definitely
excludes possible gap nodes and suggests that a fully-gapped
superconducting state occurs in Mo_2C.









 §.§ Zero-field MuSR


ZF-µSR is one of the few techniques sensitive enough to detect the tiny 
spontaneous magnetic field occurring below the superconducting transition temperature. 
Similarly, it is suitable also for detecting a possible short-range magnetic order or magnetic fluctuations. In view of this, 
we performed also ZF-µSR measurements on Mo_2C. The ZF-µSR spectra collected in the normal- and superconducting states of Mo_2C are presented in Fig. <ref>.
The lack of a fast decay and of coherent oscillations in the ZF-µSR data confirms the absence of magnetic order and/or fluctuation in Mo_2C. As a consequence, owing to the absence of magnetic 
fields of electronic origin, the muon-spin relaxation is 
mainly due to the randomly oriented nuclear magnetic moments.
Considering that both Mo and C atoms have relatively small nuclear moments (<1 µ_n), Mo_2C exhibits a very
weak muon-spin relaxation. Therefore, the ZF-µSR spectra can be modeled by a Lorentzian-type Kubo-Toyabe relaxation function G_KT = [1/3 + 2/3(1 -Λ_ZFt) e^-Λ_ZFt] <cit.>, where Λ_ZF represents the zero-field Lorentzian relaxation rate. 
As shown by solid lines in Fig. <ref>, the ZF-µSR spectra of Mo_2C were fitted to A_ZF(t) = A_s G_KT + A_bg, where A_s is the same as A_i in Eq. (<ref>). The obtained muon-spin relaxation rates are Λ_ZF = 0.021(1) µs^-1 at 0.3 K and 0.019(1) µs^-1 at 10 K.
Obviously, the relaxation rates are almost identical in the superconducting- and the normal state of Mo_2C, differing less than 
their standard deviations. The absence of an additional muon-spin relaxation below T_c definitely excludes a possible time-reversal symmetry (TRS) breaking in the superconducting state of Mo_2C.
Hence, combined with TF-µSR data, our ZF-µSR 
results suggest a conventional fully-gapped bulk SC with a preserved TRS in the β-phase Mo_2C superconductor.  









 §.§  Band-structure calculations
  

According to XRD refinements (see Fig. <ref>), in Mo_2C, the orthorhombic crystal structure shows the best agreement with the 
XRD pattern. To confirm the crystal structure of Mo_2C, we performed comparative first-principle calculations of the phonon dispersion spectra of Mo_2C by using the space groups Pbcn (β-phase), P6_3/mmc (η-phase), and P3̅1m (ζ-phase), respectively. 
As shown in Figs. <ref>(a)-(c), no soft phonon modes could be
identified in the spectra of these structures, implying that all of them are dynamically stable and can be synthesized experimentally. This is consistent with the mixture of different phases
we find in the samples obtained by arc melting. The calculated total energies versus the unit-cell volumes are summarized in Fig. <ref>(d) 
for the three crystal structures. Among them, the β-phase Mo_2C has the lowest energy at the equilibrium volume, while this is highest for the η-phase. Therefore, the β-phase molybdenum carbides can be stabilized at a relatively low pressure and temperature compared to the other phases <cit.>.
Since both the experiment and the theory confirm that Mo_2C adopts the β-phase, we calculated the electronic-band structures solely for this phase. The theoretical results for the other phases can be found elsewhere <cit.>.






The calculated electronic band structures for the β-phase 
Mo_2C, are summarized in Fig. <ref>.
Close to the Fermi level, the electronic bands are dominated by the 4d-orbitals of Mo atoms, while the contribution from the C 2p-orbitals is almost negligible. Indeed, over a wide range of energies, 
the contribution from the C-2p orbitals is less than 4.4%. This situation is also reflected in the DOS shown in the right panels.
The estimated DOS at the Fermi level is about 1.93 states/(eV f.u.) [= 7.72 states/(eV cell)/Z, with Z = 4, the number of Mo_2C formula units per unit cell]. Such a relatively high DOS suggests a 
good metallicity for Mo_2C, consistent with previous electrical resistivity data <cit.>. After including the SOC the bands separate, since SOC breaks the band degeneracy and brings one of the bands closer to the Fermi level [see Fig. <ref>(b)]. The band splitting due to the SOC is rather weak, here visible only along the Z–U line near the Fermi level. Although the band splitting along the S–Y line is quite significant, these bands are too far away from the Fermi level to have any meaningful influence on the electronic properties of Mo_2C. In the β-phase Mo_2C, the band splitting E_SOC is up to 100 meV. This is comparable to the α-phase NbC, but much smaller than in TaC <cit.>. 





	
The Pbcn space group of Mo_2C is nonsymmorphic and it has an inversion symmetry. 
After inspecting the band structure without SOC across the whole Brillouin zone [see Fig. <ref>(a)], the bands along the X–S–Y, 
Z–U, and R–T–Z directions turn out to be 
twofold degenerate, while the bands along U–R are fourfold 
degenerate (without considering the spin degree of freedom).
By using symmetry arguments <cit.>, the X–U–S–R and U–R–Z–T planes are twofold degenerate nodal surfaces due to the combined 
presence of a screw rotation and time-reversal symmetries. 
The fourfold degenerate nodal lines along U–R are protected 
by the combination of glide-mirror- and PT symmetries.
In the presence of SOC, the fourfold-degenerate bands are broken into two twofold-degenerate bands [see Fig. <ref>(b)], 
except for the fourfold-degenerate nodal lines along the 
R–T–Z direction, which are protected by a combination of glide-mirror- and PT symmetries.
Therefore, similar to other phases of carbides <cit.>, β-Mo_2C with nodal lines crossing the Fermi level could also be material candidates for future studies of topological superconductivity. 

Among the 8 bands crossing the Fermi level, only two of them 
contribute significantly to the DOS and have the largest Fermi surfaces (FSs). These two bands are highlighted in purple and cyan 
in Fig. <ref>(a), and their corresponding FSs are depicted in Figs. <ref>(b) and (c), respectively. Clearly, these two bands form distinct FSs, 
even though both are due to Mo 4d-orbitals. 
The purple band exhibits two small hole pockets near the Brillouin center, which are much smaller than the analogous electron 
pocket of the cyan band. Near the Brillouin boundary of the purple band two cylinder-like FSs extend along the Γ–Z direction. By contrast, in the cyan band, such FSs extend along the Γ–Y direction. 
Clearly, the FSs of the orthorhombic Mo_2C are more three dimensional and more complex than those of the α-phase TMCs.
In the latter case, the largest FSs consist of three cylinders along the k_i (i = x, y, z) directions.
Such cylinder-like FSs originate from the strong hybridization between the transition metal d-orbitals and C p-orbitals. 
By contrast, the p–d hybridization is rather weak in the orthorhombic Mo_2C.  
The cylinderlike FSs are known to play an important role in the SC of high-T_c iron-based materials <cit.>. This may also be the case for α-phase TMCs, which have relatively
high T_c values in comparison to other carbide phases.




§ DISCUSSION


Now, we briefly discuss the different phases of molybdenum carbides. 
To date, there are mainly two phases of molybdenum carbides that have been reported to become superconducting at low temperature, namely α-MoC_x and η-Mo_3C_2. The γ-MoC and ζ-phase Mo_2C adopt a noncentrosymmetric hexagonal and a centrosymmetric trigonal structure, respectively, but no SC has been reported in these phases yet. Recent theoretical work predicts that by introducing hole carriers, the γ-phase MoC could show SC with a T_c up to 9 K <cit.>. Here, by using the µSR technique, we reveal
that it is the β-phase Mo_2C, instead, to represent the third
member of molybdenum carbides to show bulk SC. 
Among the latter, the α- and β-phases show the highest and the lowest T_c, i.e., ∼ 15 K <cit.> and ∼ 3.2 K, respectively. While the η-Mo_3C_2-phase shows an intermediate T_c of 7.4 K <cit.>.
The highest T_c in the α-phase TMCs is most likely due to their
strong p–d hybridization and, thus, to an enhanced electron-phonon coupling.
We recall that, the strong p–d hybridization produces large
cylinder-like FSs <cit.>, which play an important
role also in the SC of high-T_c iron-based materials <cit.>. 
As for the β-phase Mo_2C, band-structure calculations indicate
a rather weak p–d hybridization (see Fig. <ref>), which
may justify their comparatively low T_c value. 

The low-temperature superfluid density, determined by TF-µSR
in our study, suggests a fully-gapped superconducting state in the β-phase Mo_2C. A µSR study has not yet been performed in the α-phase MoC_x and η-phase Mo_3C_2. 
This is related to the difficulties in synthesizing sufficient amounts of material under the demanding conditions (1700 ^∘C, 6–17 GPa) required in these cases <cit.>. 
According to our previous TF-µSR studies, the α-phase NbC and TaC also exhibit a fully-gapped superconducting state <cit.>. We expect also the α-phase MoC_x to show similar SC properties to NbC and TaC.
In fact, the electronic specific heat of α-phase MoC_x
(and η-phase Mo_3C_2) shows an exponential temperature dependence
in the superconducting state, consistent with a nodeless SC <cit.>. Further, the small
zero-temperature energy gap (Δ_0 < 1.76 k_BT_c)
and a reduced specific-heat jump at T_c (ΔC/γT_c < 1.43) suggest a weakly coupled SC in the various phases of superconducting molybdenum carbides. Taking into account the preserved TRS in the superconducting state, as well as an upper critical field H_c2 well below the Pauli limit <cit.>, we conclude that the molybdenum
carbides exhibit a spin-singlet pairing, independent of
their crystal structure (phase). 

Finally, we discuss the topological aspects of molybdenum carbides. The α-phase MoC, possesses a nonzero ℤ_2 topological invariant and Dirac surface states <cit.>. 
The isostructural NbC, contains three closed node lines in the bulk band structure (without considering SOC) of its first Brillouin zone. These are protected by time-reversal and space-inversion symmetry <cit.>.
In case of a large SOC, such nodal loops become gapped. Since the 4d Nb and
Mo atoms exhibit a weaker intrinsic SOC than the 5d Ta atoms,
the SOC effects should be modest in both NbC and MoC. Consequently, the node lines — predicted by calculations neglecting SOC effects — are most likely preserved in  both the above carbides. As such, the α-phase MoC and NbC might 
be good candidates for observing the exotic two-dimensional surface states. Further, although the γ-phase MoC is not superconducting in its pristine form, it is predicted to be a topological nodal-line material, exhibiting drumhead surface states. 
After introducing hole carriers, its SC can be tuned to reach a T_c of up to 9 K <cit.>.  
Since the γ-phase MoC adopts a noncentrosymmetric hexagonal structure, it can be classified as a topological Weyl semimetal. Indeed, three-component fermions were experimentally observed in the γ-phase MoP and WC <cit.>. By applying external pressure, the topological semimetal MoP becomes a superconductor, whose T_c reaches 4 K (above 90 GPa) <cit.>, thus representing 
a possible candidate topological superconductor. Here, we also find that the β-phase Mo_2C hosts twofold-degenerate nodal surfaces and fourfold-degenerate nodal lines near the Fermi level. In the case of SOC, the fourfold degenerate
nodal lines cross the Fermi level and, hence, could contribute to the
superconducting pairing. In general, all the various phases of molybdenum carbides are promising for studying topological superconductivity.






§ CONCLUSION


To summarize, we studied the superconducting properties of 
Mo_2C mostly by means of the µSR technique, 
as well as via numerical band-structure calculations.
The latter show that the phonon dispersion spectrum of Mo_2C
provides the lowest total energy in case of the orthorhombic β-phase
(with Pbcn space group), a result consistent with the experiment.
Magnetization measurements confirm the bulk superconductivity
of Mo_2C, with a T_c of 3.2 K. The temperature dependence of the superfluid density reveals a nodeless 
superconducting state, which is well described by an isotropic s-wave
model. The lack of spontaneous magnetic fields below T_c indicates 
that time-reversal symmetry is preserved in the superconducting state of Mo_2C. 
Electronic band-structure calculations suggest that the density of states
at the Fermi level is dominated by the Mo-4d electrons, while the
contribution of the C-2p electrons is negligible over a broad
energy range. As a consequence, the p–d hybridization is rather weak
in the β-phase Mo_2C, resulting in a relatively low
T_c value. Topological nodal states including nodal surfaces and nodal lines could be identified
in the Mo_2C electronic band structure near the Fermi level.
This finding, together with the intrinsic superconductivity,
suggests that the β-phase Mo_2C, too, is a potential candidate for 
studies of topological SC, similar to the other phases of molybdenum carbides. 




 
The authors thank Weikang Wu for fruitful discussions.
This work was supported by the Natural Science Foundation of Shanghai
(Grant Nos. 21ZR1420500 and 21JC­140­2300), Natural Science
Foundation of Chongqing (Grant No. CSTB-2022NSCQ-MSX1678), National
Natural Science Foundation of China (Grant No. 12374105), Fundamental
Research Funds for the Central Universities, and the Schwei­ze­rische
Na­ti­o­nal­fonds zur För­der­ung der Wis­sen­schaft­lichen
For­schung (SNF) (Grant Nos. 200021_169455 and No. 200021_188706). 
We also acknowledge the allocation of beam time at the Swiss muon source (Dolly µSR spectrometer).







