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††thanks: These authors contributed equally††thanks: These authors contributed equally

Nodeless superconductivity and topological nodal states in molybdenum carbide

Tian Shang [email protected] Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Yuting Wang 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    Bochen Yu Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Keqi Xia Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Darek J. Gawryluk Center for Neutron and Muon Sciences, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland    Yang Xu Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Qingfeng Zhan Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Jianzhou Zhao [email protected] Co-Innovation Center for New Energetic Materials, Southwest University of Science and Technology, Mianyang, 621010, People’s Republic of China    Toni Shiroka 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 TcT_{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 Mo2C, with a zero-temperature gap Δ0\Delta_{0} = 0.44 meV and a magnetic penetration depth λ0\lambda_{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 4​d4d-orbitals, which are marginally hybridized with the C 2​p2p-orbitals over a wide energy range. The symmetry analysis confirms that, in the absence of spin-orbit coupling (SOC), Mo2C 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.

††preprint: Preprint: , 15:31

I Introduction

The possibilities offered by topological superconductors, ranging from hosting Majorana fermion quasiparticles to potential applications in topological quantum computing [1, 2, 3, 4], 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 Bi2Se3 topological insulator [5, 6, 7, 8]. Another approach utilizes the proximity effect between a conventional ss-wave superconductor and a topological insulator or semiconductor [8, 9]. The surface states of a topological insulator can lead to a two-dimensional superconducting state with a p+i​pp+ip pairing at the interfaces, known to support Majorana bound states at the vortices [10]. For instance, evidence of topological SC has been reported in NbSe2/Bi2(Se,Te)3 heterostructures [11, 12], where NbSe2 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 CsV3Sb5 [13], β\beta-PdBi2 [14], and PbTaSe2 [15], 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\alpha (F​m​3¯​mFm\overline{3}m No. 225)-, β\beta (P​b​c​nPbcn, No. 60)-, γ\gamma (P​6¯​m​2P\overline{6}m2, No. 187)-, and η\eta (P​63/m​m​cP6_{3}/mmc, No. 194)-phase [16]. The γ\gamma-phase is noncentrosymmetric, while the other three are centrosymmetric. Due to the lack of space inversion, the γ\gamma-phase TMCs exhibit exotic topological features. The unconventional three-component fermions with surface Fermi arcs were experimentally observed in γ\gamma-phase WC [17]. By applying external pressure, the topological semimetal MoP (isostructural to WC) becomes a superconductor, whose TcT_{c} rises up to 4 K (above 90 GPa) [18], thus representing a candidate topological superconductor. Unfortunately, no SC has been observed in WC yet, but the γ\gamma-phase MoC (similar to WC and also not superconducting) was predicted to be a topological nodal-line semimetal with drumhead surface states [19]. The α\alpha-phase TMCs show a relatively high TcT_{c} value and some of them were also predicted to exhibit nontrivial band topologies [19, 20, 21, 22, 23, 24, 25, 26, 27, 16]. For example, NbC and TaC are fully-gapped superconductors with Tc=11.5T_{c}=11.5 and 10.3 K, respectively [25]. At the same time, theoretical calculations suggest that α\alpha-phase TMCs are nodal line semimetals in the absence of spin-orbit coupling (SOC) [25, 19].

As for the molybdenum carbides, although their SC was already reported in the 1970s [26], their physical properties have been overlooked due to difficulties in synthesizing clean samples. Only recently, the α\alpha-phase MoCx (x<1x<1) and η\eta-phase Mo3C2 (with TcT_{c} = 14.3 and 8.5 K) could be synthesized under high-temperature and high-pressure conditions (1700 ∘C, 6–17 GPa) [28, 29, 30] and their superconducting properties studied via different techniques. To date, the electronic properties of the other phases of molybdenum carbides (e.g., the β\beta-phase) remain mostly unexplored.

In this paper, we report on the superconducting properties of the β\beta-phase Mo2C, 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 Mo2C 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\beta-phase TMCs (of which Mo2C is a typical example) may be potential candidates for future studies of topological SC, similar to the other TMC phases.

II Experimental and numerical methods

First, we tried to synthesize the β\beta-phase Mo2C by arc melting Mo slugs (99.95%, Alfa Aesar) and C rods (99.999+%, ChemPUR). Similarly to previous studies [31], the obtained polycrystalline samples showed a mixture of different phases, both before and after the annealing. Akin to the α\alpha-phase, the β\beta-phase Mo2C can be synthesized also under high-temperature and high-pressure conditions (1500–2300 K, 5 GPa) [32]. However, the resulting Mo2C samples have a rather low superconducting volume fraction. Because of these issues, all our measurements were performed on high-purity Mo2C 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\alpha radiation. The µSR measurements were carried out at the multipurpose surface-muon spectrometer (Dolly) at the π\piE1 beamline of the Swiss muon source at Paul Scherrer Institut (PSI), Villigen, Switzerland. The Mo2C 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 musrfit software package [33].

The phonon spectrum and the electronic band structure of Mo2C were calculated via DFT, within the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) realization [34], as implemented in the Vienna ab initio Simulation Package (VASP) [35, 36]. The projector augmented wave (PAW) pseudopotentials were adopted for the calculation [37, 38]. Electrons belonging to the outer atomic configuration were treated as valence electrons, here corresponding to 6 electrons in Mo (4​d5​5​s14d^{5}5s^{1}) and 4 electrons in C (2​s2​2​p22s^{2}2p^{2}). The kinetic energy cutoff was fixed to 400 eV. For the three different crystal structures of Mo2C, 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 16×16×1016\times 16\times 10, 14×11×1314\times 11\times 13, and 19×19×2119\times 19\times 21 kk-points were used for the space groups P​63/m​m​cP6_{3}/mmc, P​b​c​nPbcn, and P​3¯​1​mP\bar{3}1m, respectively. To obtain the force constants and phonon spectra, we used the density functional perturbation theory (DFPT) in combination with the Phonopy package [39, 40, 41]. A supercell of 2×2×22\times 2\times 2 was adopted for the calculation of force constants. To calculate the phonon spectrum, the Brillouin zone integration was performed on a Γ\Gamma-centered mesh of 10×10×710\times 10\times 7, 7×5×67\times 5\times 6, and 6×6×76\times 6\times 7 kk-points for the space groups P​63/m​m​cP6_{3}/mmc, P​b​c​nPbcn, and P​3¯​1​mP\bar{3}1m, respectively. In the P​63/m​m​cP6_{3}/mmc case, considering that only half of the 2aa 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.

III Results and discussion

III.1 Crystal structure

Refer to caption
Figure 1. : Room-temperature x-ray powder diffraction patterns and Rietveld refinements for Mo2C powders using different space groups: P​b​c​nPbcn (a), P​63/m​m​cP6_{3}/mmc (b), and P​3¯​mP\bar{3}m (c). The left inset in (a) shows an enlarged plot of intensity for 2θ\theta between 26 and 30∘. The open red circles and the solid black lines represent the experimental patterns and the refinement profiles, respectively. The blue lines at the bottom show the residuals, i.e., the difference between the calculated and the experimental data. The vertical bars mark the calculated Bragg-peak positions. The unit-cell crystal structures are shown in the right insets of each panel. Here, the blue and black spheres represent the Mo and C atoms, respectively.

The phase purity and the crystal structure of Mo2C powders were checked via XRD measurements at room temperature (see Fig. III.1). Unlike the arc-melted Mo2C [31], the purchased Mo2C powders show a clean phase. Several phases of molybdenum carbides have been reported, which exhibit cubic, orthorhombic, hexagonal, and trigonal structures [29, 30, 32, 42, 43]. In our case, the XRD pattern of Mo2C, was analyzed by means of the FullProf Rietveld-analysis suite [44] to find that only the latter three structures, with space groups P​b​c​nPbcn (No. 60, orthorhombic), P​63/m​m​cP6_{3}/mmc (No. 194, hexagonal), and P​3¯​1​mP\bar{3}1m (No. 162, trigonal), reproduce the data reasonably well. In the insets we depict the corresponding crystal structures, known as β\beta-, η\eta-, and ζ\zeta-phases, respectively. Among these, the β\beta-phase exhibits the best agreement with the measured XRD pattern, here reflected in the smallest goodness-of-fit factor (see Table III.1). Moreover, both the η\eta- and ζ\zeta-phases fail to reproduce some of the low-intensity reflections. For instance, as illustrated in the inset of Fig. III.1(a), while neither the η\eta- nor the ζ\zeta-phases admit a reflection at 2​θ≈30∘2\theta\approx 30^{\circ}, the β\beta-phase captures this reflection quite well. In conclusion, the Rietveld refinements suggest that the investigated Mo2C powders adopt an orthorhombic structure with space group P​b​c​nPbcn, 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 III.1 and III.1.

Table 1.: Refined lattice parameters and goodness of fits (including RR factors and χr2\chi_{\mathrm{r}}^{2}) for Mo2C powders utilizing different space groups. The orthorhombic β\beta-phase shows clearly the best fit parameters.
Space group P​b​c​nPbcn (No. 60) P​63/m​m​cP6_{3}/mmc (No. 194) P​3¯​1​mP\bar{3}1m (No. 162)
Structure orthorhombic hexagonal trigonal
(β\beta-phase) (η\eta-phase) (ζ\zeta-phase)
aa (Å) 4.7332(2) 3.0070(2) 5.2085(2)
bb (Å) 6.0292(2) 3.0070(2) 5.2085(2)
cc (Å) 5.2055(2) 4.7279(2) 4.7282(2)
VcellV_{\mathrm{cell}} (Å3) 148.553(9) 37.021(3) 111.084(8)
RPR_{\mathrm{P}} 3.31 4.23 4.24
RWPR_{\mathrm{WP}} 4.79 6.37 6.38
RexpR_{\mathrm{exp}} 1.82 1.83 1.82
χr2\chi_{\mathrm{r}}^{2} 6.91 12.1 12.2
Table 2.: Refined atomic coordinates and occupations of the three different crystal structures of Mo2C powders. Only the orthorhombic structure, with P​b​c​nPbcn space group, is relevant in our case.

P​b​c​nPbcn
Atom Wyckoff xx yy zz Occ. Mo1 8dd 0.2470(4) 0.1250 0.0798(2) 1 C1 4cc 0.00000 0.3750(2) 0.2500 1
P​63/m​m​cP6_{3}/mmc
Atom Wyckoff xx yy zz Occ. Mo1 2cc 0.3333 0.6667 0.2500 1 C1 2aa 0.0000 0.0000 0.0000 0.5
P​3¯​1​mP\bar{3}1m
Atom Wyckoff xx yy zz Occ. Mo1 6kk 0.3333 0.0000 0.2500 1.00 C1 1aa 0.0000 0.0000 0.0000 0.97 C2 1bb 0.0000 0.0000 0.5000 0.03 C3 2cc 0.3333 0.6667 0.0000 0.26 C4 2dd 0.3333 0.6667 0.5000 0.74

III.2 Magnetization measurements

Refer to caption
Figure 2. : (a) Magnetic susceptibility of Mo2C vs. temperature, measured in an applied field of 5 mT using both ZFC and FC protocols. (b) Temperature-dependent lower critical field Hc​1​(T)H_{c1}(T) for Mo2C. The solid line is a fit to Hc​1​(T)=Hc​1​(0)​[1−(T/Tc)2]H_{c1}(T)=H_{c1}(0)[1-(T/T_{c})^{2}]. Inset: field-dependent magnetization curves recorded at different temperatures below TcT_{c}. The lower critical field Hc​1H_{c1} was determined as the field value where M​(H)M(H) starts deviating from linearity (see dashed line).

We first characterized the SC of Mo2C 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. III.2(a), a clear diamagnetic signal appears below the superconducting transition at Tc=3.2T_{c}=3.2 K. The reduced TcT_{c} value compared to the previously reported Tc∼6T_{c}\sim 6 K (whose SC fraction was less than 1%) is most likely attributed to the varying C-content [32]. Such a variable TcT_{c} against C-content has been previously reported in the α\alpha- and η\eta-phase of molybdenum carbides [28, 29, 30], where the light C atom is expected to modify significantly the electron-phonon coupling and the phonon frequencies and, ultimately, also TcT_{c}. A large diamagnetic response (i.e., χV∼−0.8\chi_{\mathrm{V}}\sim-0.8 at 2 K) indicates a bulk SC in Mo2C, as further confirmed by our TF-µSR measurements. The field-dependent magnetization curves M​(H)M(H), collected at a few temperatures below TcT_{c}, are plotted in the inset of Fig. III.2(b). The estimated lower critical fields Hc​1H_{c1} as a function of temperature are summarized in Fig. III.2(b). This yields a lower critical field μ0\mu_{0}Hc​1H_{c1} = 6.4(2) mT for Mo2C at zero temperature (see solid line).

III.3 Transverse-field µSR

Refer to caption
Figure 3. : (a) Representative TF-µSR spectra of Mo2C collected in its superconducting- (0.3 K) and normal state (5 K) in an applied magnetic field of 30 mT. Fast Fourier transforms of the relevant time spectra at (b) 0.3 K and (c) 5 K. The dashed- and solid lines are fits to Eq. (1) using one and two oscillations, respectively. Note the clear broadening of field distribution in the superconducting state. Fits with two oscillations show a goodness-of-fit χr2\chi_{\mathrm{r}}^{2} ∼\sim 1.1, which is relatively smaller than χr2\chi_{\mathrm{r}}^{2} ∼\sim 1.7 of the one-oscillation fits.

To study the gap symmetry and superconducting pairing of Mo2C, we performed systematic TF-µSR measurements in an applied field of 30 mT (i.e., much higher than Hc1H_{\mathrm{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 III.3(a) plots two representative superconducting- and normal-state TF-µSR spectra for Mo2C. 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 [45, 46, 47]. 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. III.3(b)-(c)]. To describe the field distribution, the TF-µSR spectra can be modelled using [48]:

ATF​(t)=∑i=1nAi​cos⁡(γμ​Bi​t+ϕ)​e−σi2​t2/2+Abg​cos⁡(γμ​Bbg​t+ϕ).A_{\mathrm{TF}}(t)=\sum\limits_{i=1}^{n}A_{i}\cos(\gamma_{\mu}B_{i}t+\phi)e^{-\sigma_{i}^{2}t^{2}/2}+A_{\mathrm{bg}}\cos(\gamma_{\mu}B_{\mathrm{bg}}t+\phi). (1)

Here AiA_{i}, AbgA_{\mathrm{bg}} and BiB_{i}, BbgB_{\mathrm{bg}} are the initial asymmetries and local fields sensed by implanted muons in the sample and sample holder, γμ\gamma_{\mu}/2π\pi = 135.53 MHz/T is the muon gyromagnetic ratio, ϕ\phi is a shared initial phase, and σi\sigma_{i} is the Gaussian relaxation rate of the iith component. Here, we find that Eq. (1) with n=2n=2 [solid line in Fig. III.3(b)] shows a better agreement with the experimental data than with nn = 1 [dashed line in Fig. III.3(b)]. In the normal state, the derived muon-spin relaxation rates σi​(T)\sigma_{i}(T) are small and independent of temperature while, below TcT_{c}, they start to increase due to the onset of the FLL and the increased superfluid density (see inset in Fig. III.3). The effective Gaussian relaxation rate σeff\sigma_{\mathrm{eff}} can be calculated from σeff2/γμ2=∑i=12Ai​[σi2/γμ2−(Bi−⟨B⟩)2]/Atot\sigma_{\mathrm{eff}}^{2}/\gamma_{\mu}^{2}=\sum_{i=1}^{2}A_{i}[\sigma_{i}^{2}/\gamma_{\mu}^{2}-\left(B_{i}-\langle B\rangle\right)^{2}]/A_{\mathrm{tot}} [48], where ⟨B⟩=(A1​B1+A2​B2)/Atot\langle B\rangle=(A_{1}\,B_{1}+A_{2}\,B_{2})/A_{\mathrm{tot}} and Atot=A1+A2A_{\mathrm{tot}}=A_{1}+A_{2}. By considering the constant nuclear relaxation rate σn\sigma_{\mathrm{n}} in the narrow temperature range (∼\sim0.3–5 K) investigated here, confirmed also by ZF-µSR measurements (see Fig. III.3), the superconducting Gaussian relaxation rate can be extracted from σsc=σeff2−σn2\sigma_{\mathrm{sc}}=\sqrt{\sigma_{\mathrm{eff}}^{2}-\sigma^{2}_{\mathrm{n}}}.

Refer to caption
Figure 4. : Temperature-dependent inverse square of the effective magnetic penetration depth of Mo2C, as determined from TF-µSR measurements in an applied magnetic field of 30 mT. The solid, dashed, and dash-dotted lines represent fits to the ss-, pp-, and dd-wave model, with χr2∼1.9\chi_{\mathrm{r}}^{2}\sim 1.9, 5.1, and 14, respectively. The inset shows the two muon-spin relaxation rates σi​(T)\sigma_{i}(T) required to fit the TF-µSR data, where the dashed line indicates the TcT_{c} = 2.9 K.

The effective magnetic penetration depth λeff\lambda_{\mathrm{eff}} can then be calculated using σsc2​(T)/γμ2=0.00371​Φ02/λeff4​(T)\sigma_{\mathrm{sc}}^{2}(T)/\gamma^{2}_{\mu}=0.00371\Phi_{0}^{2}/\lambda_{\mathrm{eff}}^{4}(T) [49, 50]. Figure III.3 summarizes the temperature-dependent inverse square of magnetic penetration depth λeff−2​(T)\lambda_{\mathrm{eff}}^{-2}(T), which is proportional to the superfluid density ρsc​(T)\rho_{\mathrm{sc}}(T). The various models used to analyze the ρsc​(T)\rho_{\mathrm{sc}}(T) data, are generally described by the relation:

ρsc​(T)=1+2​⟨∫Δk∞EE2−Δk2​∂f∂E​dE⟩FS.\rho_{\mathrm{sc}}(T)=1+2\,\Bigg{\langle}\int^{\infty}_{\Delta_{\mathrm{k}}}\frac{E}{\sqrt{E^{2}-\Delta_{\mathrm{k}}^{2}}}\frac{\partial f}{\partial E}\mathrm{d}E\Bigg{\rangle}_{\mathrm{FS}}. (2)

Here, f=(1+eE/kB​T)−1f=(1+e^{E/k_{\mathrm{B}}T})^{-1} is the Fermi function and ⟨⟩FS\langle\rangle_{\mathrm{FS}} represents an average over the Fermi surface [51]; Δk​(T)=Δ​(T)​δk\Delta_{\mathrm{k}}(T)=\Delta(T)\delta_{\mathrm{k}} is an angle-dependent gap function, where Δ\Delta is the maximum gap value and δk\delta_{\mathrm{k}} is the angular dependence of the gap, equal to 1, cos⁡2​ϕ\cos 2\phi, and sin⁡θ\sin\theta for an ss-, dd-, and pp-wave model, respectively, where ϕ\phi and θ\theta are the azimuthal angles. The temperature-dependent gap is assumed to follow Δ​(T)=Δ0​tanh​{1.82​[1.018​(Tc/T−1)]0.51}\Delta(T)=\Delta_{0}\mathrm{tanh}\{1.82[1.018(T_{\mathrm{c}}/T-1)]^{0.51}\} [51, 52], where Δ0\Delta_{0} is the zero-temperature gap value. The ss- and pp-wave models (see black solid and red dashed lines in Fig. III.3) yield the same zero-temperature magnetic penetration depth λ0=291​(3)\lambda_{\mathrm{0}}=291(3) nm, but different zero-temperature energy gaps Δ0\Delta_{0} = 0.44(1) and 0.60(1) meV, respectively. The magnetic penetration depth of the β\beta-phase Mo2C is much higher than that of the α\alpha-phase MoCx (λ0\lambda_{0} ∼\sim 132 nm) and the η\eta-phase Mo3C2 (λ0\lambda_{0} ∼\sim 197 nm) [29, 28]. A possible dd-wave model (green dash-dotted line in Fig. III.3) provides a gap size Δ0\Delta_{0} = 0.55(1) meV. This is comparable to the pp-wave model, but λ0\lambda_{\mathrm{0}} = 255(3) nm is much shorter than that of both the ss- and pp-wave models. As can be clearly seen in Fig. III.3, below ∼\sim1.2 K, the dd-wave model deviates significantly from the experimental data. At the same time, also the pp-wave model shows a poor agreement with data in the 0.7–1.6 K range. The ss-wave model, on the other hand, reproduces the experimental data quite well over the entire temperature range studied. In addition, the temperature-independent λeff−2​(T)\lambda_{\mathrm{eff}}^{-2}(T) at T<T< 1 K (i.e., 1/3​Tc1/3T_{c}) definitely excludes possible gap nodes and suggests that a fully-gapped superconducting state occurs in Mo2C.

Refer to caption
Figure 5. : ZF-µSR spectra of Mo2C, recorded in its superconducting (0.3 K)- and its normal (10 K) states. Solid lines through the data are fits using the Kubo-Toyabe relaxation function.

III.4 Zero-field µSR

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 Mo2C. The ZF-µSR spectra collected in the normal- and superconducting states of Mo2C are presented in Fig. III.3. 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 Mo2C. 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 µn), Mo2C exhibits a very weak muon-spin relaxation. Therefore, the ZF-µSR spectra can be modeled by a Lorentzian-type Kubo-Toyabe relaxation function GKT=[13+23​(1−ΛZF​t)​e−ΛZF​t]G_{\mathrm{KT}}=[\frac{1}{3}+\frac{2}{3}(1-\Lambda_{\mathrm{ZF}}t)\,\mathrm{e}^{-\Lambda_{\mathrm{ZF}}t}] [53, 45], where ΛZF\Lambda_{\mathrm{ZF}} represents the zero-field Lorentzian relaxation rate. As shown by solid lines in Fig. III.3, the ZF-µSR spectra of Mo2C were fitted to AZF​(t)=As​GKT+AbgA_{\mathrm{ZF}}(t)=A_{\mathrm{s}}G_{\mathrm{KT}}+A_{\mathrm{bg}}, where AsA_{\mathrm{s}} is the same as AiA_{i} in Eq. (1). The obtained muon-spin relaxation rates are ΛZF\Lambda_{\mathrm{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 Mo2C, differing less than their standard deviations. The absence of an additional muon-spin relaxation below TcT_{c} definitely excludes a possible time-reversal symmetry (TRS) breaking in the superconducting state of Mo2C. Hence, combined with TF-µSR data, our ZF-µSR results suggest a conventional fully-gapped bulk SC with a preserved TRS in the β\beta-phase Mo2C superconductor.

Refer to caption
Figure 6. : The calculated phonon dispersion spectra for the β\beta- (a), η\eta- (b), and ζ\zeta- phase (c) of Mo2C. The calculated total energies vs. unit-cell volume for the three crystal structures are shown in (d). Here, V0V_{0} stands for the experimental volume (see details in Table III.1).

III.5 Band-structure calculations

According to XRD refinements (see Fig. III.1), in Mo2C, the orthorhombic crystal structure shows the best agreement with the XRD pattern. To confirm the crystal structure of Mo2C, we performed comparative first-principle calculations of the phonon dispersion spectra of Mo2C by using the space groups P​b​c​nPbcn (β\beta-phase), P​63/m​m​cP6_{3}/mmc (η\eta-phase), and P​3¯​1​mP\bar{3}1m (ζ\zeta-phase), respectively. As shown in Figs. III.4(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. III.4(d) for the three crystal structures. Among them, the β\beta-phase Mo2C has the lowest energy at the equilibrium volume, while this is highest for the η\eta-phase. Therefore, the β\beta-phase molybdenum carbides can be stabilized at a relatively low pressure and temperature compared to the other phases [28, 29, 30, 32]. Since both the experiment and the theory confirm that Mo2C adopts the β\beta-phase, we calculated the electronic-band structures solely for this phase. The theoretical results for the other phases can be found elsewhere [19, 23, 54].

Refer to caption
Figure 7. : Electronic band structures of Mo2C calculated by ignoring (a) and by considering (b) the spin-orbit coupling. Here, the Mo dd-orbitals and the C pp-orbitals are shown in red and blue, respectively. The total and partial (Mo and C atoms) density of states without SOC and with SOC are shown on the right side of each panel. The primitive-cell Brillouin zone, including the high-symmetry points, is shown in the inset of panel (a).

The calculated electronic band structures for the β\beta-phase Mo2C, are summarized in Fig. III.5. Close to the Fermi level, the electronic bands are dominated by the 4​d4d-orbitals of Mo atoms, while the contribution from the C 2​p2p-orbitals is almost negligible. Indeed, over a wide range of energies, the contribution from the C-2​p2p 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)/ZZ, with Z=4Z=4, the number of Mo2C formula units per unit cell]. Such a relatively high DOS suggests a good metallicity for Mo2C, consistent with previous electrical resistivity data [32]. 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. III.5(b)]. The band splitting due to the SOC is rather weak, here visible only along the ZZ–UU line near the Fermi level. Although the band splitting along the SS–YY line is quite significant, these bands are too far away from the Fermi level to have any meaningful influence on the electronic properties of Mo2C. In the β\beta-phase Mo2C, the band splitting ESOCE_{\mathrm{SOC}} is up to 100 meV. This is comparable to the α\alpha-phase NbC, but much smaller than in TaC [25].

Refer to caption
Figure 8. : (a) Close-up view of the electronic band structure without SOC for Mo2C. The bands that cross the Fermi level are highlighted in purple and cyan. (b)-(c) Representative Fermi surfaces of Mo2C using the same color code for the bands as shown in panel (a).

The P​b​c​nPbcn space group of Mo2C is nonsymmorphic and it has an inversion symmetry. After inspecting the band structure without SOC across the whole Brillouin zone [see Fig. III.5(a)], the bands along the XX–SS–YY, ZZ–UU, and RR–TT–ZZ directions turn out to be twofold degenerate, while the bands along UU–RR are fourfold degenerate (without considering the spin degree of freedom). By using symmetry arguments [55], the XX–UU–SS–RR and UU–RR–ZZ–TT 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 UU–RR 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. III.5(b)], except for the fourfold-degenerate nodal lines along the RR–TT–ZZ direction, which are protected by a combination of glide-mirror- and PT symmetries. Therefore, similar to other phases of carbides [25, 19], β\beta-Mo2C 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. III.5(a), and their corresponding FSs are depicted in Figs. III.5(b) and (c), respectively. Clearly, these two bands form distinct FSs, even though both are due to Mo 4dd-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\Gamma–ZZ direction. By contrast, in the cyan band, such FSs extend along the Γ\Gamma–YY direction. Clearly, the FSs of the orthorhombic Mo2C are more three dimensional and more complex than those of the α\alpha-phase TMCs. In the latter case, the largest FSs consist of three cylinders along the kik_{i} (i=x,y,zi=x,y,z) directions. Such cylinder-like FSs originate from the strong hybridization between the transition metal dd-orbitals and C pp-orbitals. By contrast, the pp–dd hybridization is rather weak in the orthorhombic Mo2C. The cylinderlike FSs are known to play an important role in the SC of high-TcT_{c} iron-based materials [56, 57, 58]. This may also be the case for α\alpha-phase TMCs, which have relatively high TcT_{c} values in comparison to other carbide phases.

IV 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\alpha-MoCx and η\eta-Mo3C2. The γ\gamma-MoC and ζ\zeta-phase Mo2C 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\gamma-phase MoC could show SC with a TcT_{c} up to 9 K [19]. Here, by using the µSR technique, we reveal that it is the β\beta-phase Mo2C, instead, to represent the third member of molybdenum carbides to show bulk SC. Among the latter, the α\alpha- and β\beta-phases show the highest and the lowest TcT_{c}, i.e., ∼15\sim 15 K [29, 30] and ∼3.2\sim 3.2 K, respectively. While the η\eta-Mo3C2-phase shows an intermediate TcT_{c} of 7.4 K [28]. The highest TcT_{c} in the α\alpha-phase TMCs is most likely due to their strong pp–dd hybridization and, thus, to an enhanced electron-phonon coupling. We recall that, the strong pp–dd hybridization produces large cylinder-like FSs [25, 19], which play an important role also in the SC of high-TcT_{c} iron-based materials [56, 57, 58]. As for the β\beta-phase Mo2C, band-structure calculations indicate a rather weak pp–dd hybridization (see Fig. III.5), which may justify their comparatively low TcT_{c} value.

The low-temperature superfluid density, determined by TF-µSR in our study, suggests a fully-gapped superconducting state in the β\beta-phase Mo2C. A µSR study has not yet been performed in the α\alpha-phase MoCx and η\eta-phase Mo3C2. 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 [29, 30, 28]. According to our previous TF-µSR studies, the α\alpha-phase NbC and TaC also exhibit a fully-gapped superconducting state [25]. We expect also the α\alpha-phase MoCx to show similar SC properties to NbC and TaC. In fact, the electronic specific heat of α\alpha-phase MoCx (and η\eta-phase Mo3C2) shows an exponential temperature dependence in the superconducting state, consistent with a nodeless SC [29, 30, 28]. Further, the small zero-temperature energy gap (Δ0\Delta_{0} ¡ 1.76 kB\mathrm{k}_{\mathrm{B}}TcT_{c}) and a reduced specific-heat jump at TcT_{c} (Δ\DeltaCC/γ\gammaTcT_{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 Hc2H_{\mathrm{c2}} well below the Pauli limit [25, 29, 28], 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\alpha-phase MoC, possesses a nonzero ℤ2\mathbb{Z}_{2} topological invariant and Dirac surface states [19]. 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 [25]. In case of a large SOC, such nodal loops become gapped. Since the 4​d4d Nb and Mo atoms exhibit a weaker intrinsic SOC than the 5​d5d 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\alpha-phase MoC and NbC might be good candidates for observing the exotic two-dimensional surface states. Further, although the γ\gamma-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 TcT_{c} of up to 9 K [19]. Since the γ\gamma-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\gamma-phase MoP and WC [59, 17]. By applying external pressure, the topological semimetal MoP becomes a superconductor, whose TcT_{c} reaches 4 K (above 90 GPa) [18], thus representing a possible candidate topological superconductor. Here, we also find that the β\beta-phase Mo2C 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.

V Conclusion

To summarize, we studied the superconducting properties of Mo2C mostly by means of the µSR technique, as well as via numerical band-structure calculations. The latter show that the phonon dispersion spectrum of Mo2C provides the lowest total energy in case of the orthorhombic β\beta-phase (with P​b​c​nPbcn space group), a result consistent with the experiment. Magnetization measurements confirm the bulk superconductivity of Mo2C, with a TcT_{c} of 3.2 K. The temperature dependence of the superfluid density reveals a nodeless superconducting state, which is well described by an isotropic ss-wave model. The lack of spontaneous magnetic fields below TcT_{c} indicates that time-reversal symmetry is preserved in the superconducting state of Mo2C. Electronic band-structure calculations suggest that the density of states at the Fermi level is dominated by the Mo-4​d4d electrons, while the contribution of the C-2​p2p electrons is negligible over a broad energy range. As a consequence, the pp–dd hybridization is rather weak in the β\beta-phase Mo2C, resulting in a relatively low TcT_{c} value. Topological nodal states including nodal surfaces and nodal lines could be identified in the Mo2C electronic band structure near the Fermi level. This finding, together with the intrinsic superconductivity, suggests that the β\beta-phase Mo2C, too, is a potential candidate for studies of topological SC, similar to the other phases of molybdenum carbides.

Acknowledgements.
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).

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