Possible topologically non-trivial superconducting order parameter in type-II Weyl semimetal T_d-MoTe_2

TL;DR

This study investigates the superconducting properties of T_d-MoTe_2 under pressure, revealing a two-gap s+s-wave order parameter and suggesting a topologically non-trivial s^{+-} pairing state, potentially making it the first time reversal invariant topological Weyl superconductor.

Contribution

The paper provides experimental evidence for a two-gap s+s-wave superconducting order parameter in T_d-MoTe_2 and proposes a topologically non-trivial s^{+-} pairing symmetry, advancing understanding of topological superconductivity.

Findings

Superfluid density increases with pressure.

Superconducting gap is two-gap s+s-wave symmetric.

No time reversal symmetry breaking detected.

Abstract

MoTe_2, with the orthorhombic T_d phase, is a new type (type-II) of Weyl semimetal, where the Weyl Fermions emerge at the boundary between electron and hole pockets. Non-saturating magnetoresistance (MR), and superconductivity were also observed in T_d-MoTe_2. Understanding the superconductivity in T_d-MoTe_2, which was proposed to be topologically non-trivial, is of eminent interest. Here, we report high-pressure (p_max = 1.3 GPa) muon spin rotation experiments on the temperature-dependent magnetic penetration depth in T_d-MoTe_2. A substantial increase of the superfluid density n_s/m^* and a linear scaling with T_c is observed under pressure. Moreover, the superconducting order parameter in T_d-MoTe_2 is determined to be two gap (s+s)-wave symmetric. We also excluded time reversal symmetry breaking in the SC state with sensitive zero-field ${\mu}$SR experiments. Considering the previous report \cite{Balicas1} on the strong suppression of T_c in T_d-MoTe_2 by disorder, we suggest that s^{+-} (topological order parameter) state is more likely to be realized in MoTe_2 than the s^{++} (trivial) state. Should s^{+-} be the SC gap symmetry, the T_d-MoTe_2 is, to our knowledge, the first known example of a time reversal invariant topological (Weyl) superconductor.

Equations

$$\frac{{\sigma }_{sc}(T)}{{\gamma }_{\mu }}=0.06091\frac{{\Phi }_0}{{\lambda }^2(T)},$$

$$\frac{{\lambda }^{-2}(T)}{{\lambda }^{-2}(0)}={\omega }_1\frac{{\lambda }^{-2}(T,{\Delta }_{0,1})}{{\lambda }^{-2}(0,{\Delta }_{0,1})}+{\omega }_2\frac{{\lambda }^{-2}(T,{\Delta }_{0,2})}{{\lambda }^{-2}(0,{\Delta }_{0,2})},$$

$$\frac{1}{{\lambda }^2}=\frac{4\pi n_s{e}^2}{m^{\ast }{c}^2}\times \frac{1}{1+\xi /l},$$

$$\nu =\frac{1}{2}\sum _{j\in FS}{C}_{\mathrm{j}}sgn({\Delta }_j),$$

$$\nu =sgn({\Delta }_1)-sgn({\Delta }_2).$$

$$\displaystyle P(t)=A_{s}\exp\Big{[}-\frac{(\sigma_{sc}^{2}+\sigma_{nm}^{2})t^{2}}{2}\Big{]}\cos(\gamma_{\mu}B_{int,s}t+\varphi)$$

$$\displaystyle+A_{pc}\exp\Big{[}-\frac{\sigma_{pc}^{2}t^{2}}{2}\Big{]}\cos(\gamma_{\mu}B_{int,pc}t+\varphi),$$

$$\frac{{\lambda }^{-2}(T,{\Delta }_{0,i})}{{\lambda }^{-2}(0,{\Delta }_{0,i})}=1+\frac{1}{\pi }{\int }_0^{2\pi }{\int }_{\Delta {(}_{T,\phi })}^{\infty }(\frac{\partial f}{\partial E})\frac{EdEd\phi }{\sqrt{E^2-{\Delta }_i(T,\phi {)}^2}},$$

$$S(Q)-1=\frac{I_c(Q)-\sum c_i\left|f_i(Q{)}^2\right|}{\sum c_i\left|f_i(Q{)}^2\right|}$$

$$G(r)=\frac{2}{\pi }{\int }_{Q_{\min }}^{Q_{\max }}F(Q)\sin (Qr)\mathrm{d}Q,$$

$$G(r)$$

$$\rho (r)$$

Full text

Possible topologically non-trivial superconducting order parameter in type-II Weyl semimetal $T_{d}$-MoTe2

Z. Guguchia

zg2268@columbia.edu

Department of Physics, Columbia University, New York, NY 10027, USA

  

F.v. Rohr

Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA

  

Z. Shermadini

Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland

  

A.T. Lee

Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA

  

S. Banerjee

Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA

  

A.R. Wieteska

Department of Physics, Columbia University, New York, NY 10027, USA

  

C.A. Marianetti

Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA

  

H. Luetkens

Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland

  

Z. Gong

Department of Physics, Columbia University, New York, NY 10027, USA

  

B.A. Frandsen

Department of Physics, University of California, Berkeley, California 94720, USA

  

S.C. Cheung

Department of Physics, Columbia University, New York, NY 10027, USA

  

C. Baines

Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland

  

A. Shengelaya

Department of Physics, Tbilisi State University, Chavchavadze 3, GE-0128 Tbilisi, Georgia

Andronikashvili Institute of Physics of I.Javakhishvili Tbilisi State University, Tamarashvili str. 6, 0177 Tbilisi, Georgia

  

A.N. Pasupathy

Department of Physics, Columbia University, New York, NY 10027, USA

  

E. Morenzoni

Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland

  

S.J.L. Billinge

Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA

Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, NY 11973, USA

  

A. Amato

Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland

  

R.J. Cava

Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA

  

R. Khasanov

Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland

  

Y.J. Uemura

Department of Physics, Columbia University, New York, NY 10027, USA

††preprint: preprint()

MoTe2, with the orthorhombic $T_{d}$ phase, is a new type (type-II) of Weyl semimetal Soluyanov ; SunY ; WangZ ; KourtisS ; KDeng ; NXu ; Kaminski ; Tamai1 , where the Weyl Fermions emerge at the boundary between electron and hole pockets. Non-saturating magnetoresistance (MR) Xu ; Ali1 ; Balicas1 ; ZhuZ , and superconductivity PanXC ; KangD ; QiCava were also observed in $T_{d}$-MoTe2. Understanding the superconductivity in $T_{d}$-MoTe2, which was proposed to be topologically non-trivial, is of eminent interest. Here, we report high-pressure ($p_{\rm max}$ = 1.3 GPa) muon spin rotation experiments on the temperature-dependent magnetic penetration depth $\lambda\left(T\right)$ in $T_{d}$-MoTe2. A substantial increase of the superfluid density $n_{s}/m^{*}$ and a linear scaling with $T_{c}$ is observed under pressure. Moreover, the superconducting order parameter in $T_{d}$-MoTe2 is determined to be two gap ($s+s$)-wave symmetric. We also excluded time reversal symmetry breaking in the SC state with sensitive zero-field ${\mu}$SR experiments. Considering the previous report Balicas1 on the strong suppression of $T_{\rm c}$ in MoTe2 by disorder, we suggest that $s^{+-}$ (topological order parameter) state is more likely to be realized in MoTe2 than the $s^{++}$ (trivial) state. Should $s^{+-}$ be the SC gap symmetry, the $T_{d}$-MoTe2 is, to our knowledge, the first known example of a time reversal invariant topological (Weyl) superconductor.

Transition metal dichalcogenides (TMDs) have attracted a lot of attention due to their fascinating physics and promising potential applications Klemm ; Qian ; Xu ; Morosan ; LiY ; Zhang . TMDs share the same formula, MX2, where M is a transition metal (for example, Mo or W) and X is a chalcogenide atom (S, Se and Te). These compounds typically crystallize in a group of related structure types, including 2H-, 1T-, 1T${}^{{}^{\prime}}$- and $T_{d}$-type lattices Clarke ; Puotinen ; Zandt ; Brown . The 2H-MoTe2 is semiconducting, while the 1T${}^{{}^{\prime}}$- and $T_{d}$-MoTe2 are semimetallic and exhibit pseudo-hexagonal layers with zig-zag metal chains. $T_{d}$-MoTe2 and WTe2 materials are considered as a type-II Weyl semimetal Soluyanov ; SunY . The Fermi surfaces in a type-II Weyl semimetal consist of a pair of electron- and hole- pockets touching at the Weyl node, rather than at the point-like Fermi surface in traditional WSM (Type I, e.g. TaAs) systems. The Weyl fermions can also be different from each other in terms of their symmetry-breaking origin. In general, they can arise by breaking either the space-inversion (SIS) or time-reversal symmetry (TRS). At the same time, however, at least one of them must be preserved to realize a bulk superconducting state Hosur ; Ando ; Grushin . These different categories of Weyl semimetals are expected to exhibit distinct topological properties.

MoTe2 exhibits a monoclinic 1T${}^{{}^{\prime}}$-orthorhombic $T_{d}$ structural phase transition at $T_{\rm S}$ ${\sim}$ 250 K QiCava . We note that the 1T${}^{{}^{\prime}}$ structure exhibits the inversion symmetric space group $P$21/$m$, while the $T_{d}$ phase belongs to the non-centrosymmetric space group ${Pmn}$21. Weyl fermions occur in the $T_{d}$ phase where the inversion symmetry is broken Soluyanov . The evidence for the low temperature $T_{d}$ structure in our MoTe2 sample is provided by X-ray pair distribution function (PDF) measurements (see Supplementary section II and Supplementary Figures 7-9). Recently, a surface state connecting bulk hole pockets and bulk electron ones and which also dominates the signal at the Fermi level, was observed by angle-resolved photoemission (ARPES) measurementsLiang . In addition, high field quantum oscillation study of the MR for $T_{d}$-MoTe2, revealed a nontrivial ${\pi}$ Berry’s phase in $T_{d}$-MoTe2, which is a distinguished feature of surface states Luo . Additionally, in the MoxW1-xTe2, experimental signatures of the predicted topological connection between the Weyl bulk states and Fermi arc surface states, which are a unique property of a Weyl semimetal were also reported Zheng .

$T_{d}$-MoTe2 represents a rare example of a material with both a topologically non-trivial band structure and superconductivity. $T_{d}$-MoTe2 is superconducting with $T_{c}$ ${\simeq}$ 0.1 K QiCava . The application of small pressure QiCava or the substitution of Te ion by S Chen can dramatically enhance $T_{c}$. $T_{d}$-MoTe2 is believed to be a promising candidate of topological superconductor (TSC) in the bulk materials. TSCs are special families of those materials with unique electronic states, a full pairing gap in the bulk and gapless surface states consisting of Majorana fermions (MFs) Hosur ; Ando ; Grushin . Due to their scientific importance and potential applications in quantum computing, MFs have attracted lots of attention recently. In general, topological superfluidity, and superconductivity, are well-established phenomena in condensed matter systems. The A-phase of superfluid helium-3 constitutes an example of a charge neutral topological superfluid, whereas Sr2RuO4 LukeTRS , is generally believed to be topological TRS breaking superconductor. However, an example of a TRS invariant topological superconductor Hosur ; Ando is outstanding and $T_{d}$-MoTe2 may be a candidate material for this category. So far, pressure-dependent critical temperatures and fields are the only known properties of the superconducting state of $T_{d}$-MoTe2. Thus, thorough exploration of superconductivity in $T_{d}$-MoTe2 from both experimental and theoretical perspectives are required.

To further explore superconductivity in $T_{d}$-MoTe2 and its topological nature, it is extremely important to measure the superconducting order parameter of $T_{d}$-MoTe2 on the microscopic level through measurements of the bulk properties. Thus, we concentrate on the high pressure GuguchiaPressure ; Andreica ; MaisuradzePC ; GuguchiaNature muon spin relaxation/rotation (${\mu}$SR) measurements of the magnetic penetration depth $\lambda$ in $T_{d}$-MoTe2, which is one of the fundamental parameters of a superconductor, since it is related to the superfluid density $n_{s}$ via 1/${\lambda}^{2}$ = \mu_{0}$$e^{2}$$n_{s}/m^{*} (where $m^{*}$ is the effective mass). Most importantly, the temperature dependence of ${\lambda}$ is particularly sensitive to the topology of the SC gap: while in a fully gapped superconductor, $\Delta\lambda^{-2}\left(T\right)\equiv\lambda^{-2}\left(0\right)-\lambda^{-2}\left(T\right)$ vanishes exponentially at low $T$, in a nodal SC it vanishes as a power of $T$. The muon-spin rotation (${\mu}$SR) technique provides a powerful tool to measure ${\lambda}$ in the vortex state of type II superconductors in the bulk of the sample, in contrast to many techniques that probe ${\lambda}$ only near the surface Sonier . Details are provided in the Methods section. Zero-field ${\mu}$SR is very powerful for detecting a spontaneous magnetic field due to TRS breaking in exotic superconductors, because internal magnetic fields as small as 0.1 G are detected in measurements without applying external magnetic fields. High-pressure AC susceptibility experiments were also carried out on the same polycrystalline sample of $T_{d}$-MoTe2 using exactly the same cell as the one used for ${\mu}$SR experiments.

Figure 1a shows the temperature dependence of the AC-susceptibility $\chi_{\rm AC}$ of $T_{d}$-MoTe2 in the temperature range between 1.4 K and 4.2 K for selected hydrostatic pressures up to $p$ = 1.9 GPa. Strong diamagnetic response and sharp SC transitions are observed under pressure (Fig. 1), pointing to the high quality of the sample and providing evidence for bulk superconductivity in MoTe2 QiCava . The pressure dependence of $T_{{\rm c}}$ is shown in Fig. 1b. $T_{{\rm c}}$ increases with increasing pressure and reaches a critical temperature $T_{\rm c}$ ${\simeq}$ 4 K at the maximum applied pressure of $p$ = 1.9 GPa in susceptibility experiments. The substantial increase of $T_{\rm c}$ from $T_{\rm c}$ ${\simeq}$ 0.1 K at ambient pressure to $T_{\rm c}$ ${\simeq}$ 4 K at moderate pressures in MoTe2 was considered as a manifestation of its topologically nontrivial electronic structure. Note that a strong pressure-induced enhancement of $T_{\rm c}$ has also been observed in topological superconductors such as Bi2Te3 JLZhang and Bi2Se3 Kirshenbaum . In Fig. 1b, the monoclinic 1T${}^{{}^{\prime}}$-orthorhombic $T_{d}$ structural phase transition temperature $T_{\rm S}$ QiCava is also shown as a function of pressure demonstrating that in the investigated pressure range $p$ = 0 - 1.9 GPa, the system MoTe2 exhibits the orthorhombic $T_{d}$ structure. Moreover, in the investigated pressure region, the system MoTe2 was also confirmed by DFT calculations QiCava to be a Weyl semimetal with a band structure around the Fermi level, which is extremely sensitive to changes in the lattice constants.

Figures 2a and 2b exhibit the transverse-field (TF) ${\mu}$SR-time spectra for MoTe2, measured at $p$ = 0.45 GPa and maximum applied pressure $p$ = 1.3 GPa, respectively. The spectra above (2 K, 3.5 K) and below (0.25 K) the SC transition temperature $T_{{\rm c}}$ are shown. Above $T_{{\rm c}}$ the oscillations show a small relaxation due to the random local fields from the nuclear magnetic moments. Below $T_{{\rm c}}$ the relaxation rate strongly increases with decreasing temperature due to the presence of a nonuniform local magnetic field distribution as a result of the formation of a flux-line lattice (FLL) in the SC state. Magnetism, if present in the samples, may enhance the muon depolarization rate and falsify the interpretation of the TF-${\mu}$SR results. Therefore, we have carried out ZF-${\mu}$SR experiments above and below $T_{{\rm c}}$ to search for magnetism (static or fluctuating) in MoTe2. As shown in 2c no sign of either static or fluctuating magnetism could be detected in ZF time spectra down to 0.25 K. The spectra are well described by a standard Kubo-Toyabe depolarization function Toyabe , reflecting the field distribution at the muon site created by the nuclear moments. Moreover, no change in ZF-${\mu}$SR relaxation rate (see the inset of Fig. 2c) across $T_{c}$ was observed, pointing to the absence of any spontaneous magnetic fields associated with a TRS LukeTRS ; HillierTRS ; BiswasTRS breaking pairing state in MoTe2.

The temperature dependence of the muon spin depolarization rate ${\sigma}_{{\rm sc}}$, which is proportional to the second moment of the field distribution (see Method section), of MoTe2 in the SC state at selected pressures is shown in Fig.  3. Below $T_{{\rm c}}$ the relaxation rate ${\sigma}_{{\rm sc}}$ starts to increase from zero with decreasing temperature due to the formation of the FLL. There is a significant increase of the low-temperature value ${\sigma}_{{\rm sc}}$(0.25 K) as well as $T_{{\rm c}}$ under pressure (see Fig. 3): ${\sigma}_{{\rm sc}}$(5 K) increases by a factor of ${\sim}$ 2 from $p$ = 0 GPa to p = 1.3 GPa. Interestingly, at all pressures the form of the temperature dependence of ${\sigma}_{{\rm sc}}$, which reflects the topology of the SC gap, shows slight upturn at ${\sim}$ 700 mK with indication of the saturation upon further lowering the temperature below ${\sim}$ 700 mK. We show in the following how these behaviours indicate the presence of the two isotropic $s$-wave gaps on the Fermi surface of MoTe2.

In order to investigate the symmetry of the SC gap, we note that ${\lambda}(T)$ is related to the relaxation rate ${\sigma}_{{\rm sc}}(T)$ by the equation Brandt :

[TABLE]

where ${\gamma_{\mu}}$ is the gyromagnetic ratio of the muon, and ${\Phi}_{{\rm 0}}$ is the magnetic-flux quantum. Thus, the flat $T$-dependence of ${\sigma}_{{\rm sc}}$ observed at various pressures for low temperatures (see Fig. 3) is consistent with a nodeless superconductor, in which $\lambda^{-2}\left(T\right)$ reaches its zero-temperature value exponentially.

To proceed with a quantitative analysis, we consider the local (London) approximation (${\lambda}$ ${\gg}$ ${\xi}$, where ${\xi}$ is the coherence length) and employ the empirical ${\alpha}$-model. The model, widely used in previous investigations of the penetration depth of multi-band superconductors Bastian ; Tinkham ; carrington ; padamsee ; GuguchiaPRB , assumes that the gaps occuring in different bands, besides a common $T_{{\rm c}}$, are independent of each other. The superfluid density is calculated for each component separately GuguchiaPRB and added together with a weighting factor. For our purposes, a two-band model suffices, yielding:

[TABLE]

where ${\lambda}(0)$ is the penetration depth at zero temperature, ${\Delta_{0,i}}$ is the value of the $i$-th SC gap ($i=1$, 2) at $T=0$ K, and ${\omega}_{i}$ is the weighting factor which measures their relative contributions to ${\lambda^{-2}}$ (i.e. ${\omega}_{1}+{\omega}_{2}=1$).

The results of this analysis are presented in Fig. 4a-c, where the temperature dependence of ${\lambda^{-2}}$ for MoTe2 is plotted at various pressures. We consider two different possibilities for the gap functions: either a constant gap, $\Delta_{0,i}=\Delta_{i}$, or an angle-dependent gap of the form $\Delta_{0,i}=\Delta_{i}\cos 2\varphi$, where $\varphi$ is the polar angle around the Fermi surface. The dashed and the solid lines represent a fit to the data using a $s$-wave and a $s$ + $s$-wave models, respectively. The analysis appears to rule out the simple $s$-wave model as an adequate description of ${\lambda^{-2}}$($T$) for MoTe2. The two gap $s$ + $s$-wave scenario with a small gap ${\Delta}_{1}$ ${\simeq}$ 0.12(3) meV and a large gap ${\Delta}_{2}$ (with the pressure independent weighting factor of ${\omega}_{2}$ = 0.87), describes the experimental data remarkably well. A $d$-wave gap symmetry was also tested, shown with black dotted line in Fig. 4a, but was found to be inconsistent with the data. This conclusion is supported by a ${\chi}^{2}$ test, revealing a ${\chi}^{2}$ for the $d$-wave model by ${\sim}$ 30 ${\%}$ higher than the one for ($s+s$)-wave model for $p$ = 0.45 GPa. The pressure dependence of all the parameters extracted from the data analysis within the ${\alpha}$ model are plotted in Figs. 5(a-b). From Fig. 5a a substantial decrease of ${\lambda}\left(0\right)$ (increase of ${\sigma}_{{\rm sc}}$) with pressure is evident. At the maximum applied pressure of $p$ = 1.3 GPa the reduction of ${\lambda}\left(0\right)$ is approximately 25 ${\%}$ compared to the value at $p$ = 0.45 GPa. The small gap ${\Delta}_{1}$ ${\simeq}$ 0.12(3) meV stays nearly unchanged by pressure, while the large gap ${\Delta}_{2}$ increases from ${\Delta}_{2}$ ${\simeq}$ 0.29(1) meV at $p$ = 0.45 GPa to ${\Delta}_{2}$ ${\simeq}$ 0.49(1) meV at $p$ = 1.3 GPa, i.e., by approximately 70 ${\%}$.

In general, the penetration depth ${\lambda}$ is given as a function of $n_{\rm s}$, $m^{*}$, ${\xi}$ and the mean free path $l$ as

[TABLE]

For systems close to the clean limit, ${\xi}$/$l$ ${\rightarrow}$ 0, the second term essentially becomes unity, and the simple relation 1/${\lambda}^{2}$ $\propto$ $n_{s}/m^{*}$ holds. Considering the $H_{c2}$ values of MoTe2, reported in Ref.QiCava we estimated ${\xi}$ ${\simeq}$ 26 nm and 14 nm for $p$ = 0.45 GPa and 1 GPa, respectively. At ambient pressure, the in-plane mean free path $l$ was estimated to be $l$ ${\simeq}$ 100 - 200 nm Luo . No estimates are currently available for $l$ under pressure. However, in-plane $l$ is most probably independent on pressure, considering the fact that the effect of compression is mostly interlayered (i.e., intralayer Mo-Te bond length is almost not changed, especially in our investigated pressure region), thanks to the unique anisotropy of the van der Waals structure. Thus, in view of the short coherence length and relatively large $l$, we can assume that MoTe2 lies close to the clean limit Frandsen . With this assumption, we obtain the ground-state value $n_{s}/(m^{*}/m_{e}$) ${\simeq}$ 0.9 ${\times}$ 1026 m*-3*, 1.36 ${\times}$ 1026 m*-3*, and 1.67 ${\times}$ 1026 m*-3* for $p$ = 0.45 GPa, 1 GPa, and 1.3 GPa respectively. Interestingly, $n_{s}/(m^{*}/m_{e}$) increases substantially under pressure and this will be discussed below.

One of the essential findings of this paper is the observation of two-gap superconductivity in $T_{d}$-MoTe2. Recent ARPES Liang experiments on MoTe2 revealed the presence of three bulk hole pockets (a circular hole pocket around the Brillouin zone center and the two butterfly-like hole pockets) and two bulk electron pockets, that are symmetrically distributed along the ${\Gamma}$-X direction with respect to the Brillouin zone center ${\Gamma}$. Since several bands cross the Fermi surface in MoTe2, two-gap superconductivity can be understood by assuming that the SC gaps open at two distinct types of bands. Now the interesting question arises: How consistent is the observed two-gap superconductivity with the possible topological nature of superconductivity in $T_{d}$-MoTe2? Note that the superconductor $T_{d}$-MoTe2 represents a case of time-reversal-invariant Weyl semimetals, which has broken inversion symmetry. Recently, the microscopic interactions and the SC gap symmetry in a time-reversal-invariant topological superconductivity in Weyl semimetals was studied based on the fluctuation-exchange approach Hosur . In a Weyl semimetal, the FSs carry nonzero Chern numbers of the Berry‘s phase gauge field. A simple formula Qi , relates the FS Chern number to the topological invariant ${\nu}$ for a time-invariant TSC, considering a set of FSs with Chern numbers $C_{\rm j}$:

[TABLE]

where $\Delta_{\rm j}$ is the pairing gap function on the $j$-th FS. A TSC is implied by ${\nu}{\neq}$ 0. It was shown that for isotropic Weyl nodes (Hosur ), the superconducting gap function projected onto the FSs in any time-symmetric gapped phase must have the form $\Delta_{\rm j}$(k) = $\Delta_{\rm j}$, i.e., it must be independent of k. Using further theoretical considerations Hosur , the topological invariant defined in Eq. 4 is expressed as

[TABLE]

Therefore the necessary condition for the TSC is that ${\Delta}_{\rm 1}$ and ${\Delta}_{\rm 2}$ have opposite signs. So, for TSC the gaps can be momentum independent on each FS but must alternate in sign between different FSs. As suggested in Ref. Hosur , for the TSC to be favoured over the trivial state, in which ${\Delta}_{\rm 1}$ and ${\Delta}_{\rm 2}$ have the same sign, the useful conditions are Coulomb repulsion as well as attractive interactions, such as those mediated by phonons, and a fine-tuning of their relative strengths. In this way, the net interaction can be made to change sign over large momenta, comparable to the separation of the Weyl nodes. It was also suggested Hosur that strong disorder may be more detrimental to topological superconductivity than to ordinary $s$-wave superconductivity due to inter-FS scattering processes. ${\mu}$SR experiments alone can not distinguish between $s^{+-}$ (topological) and $s^{++}$ (trivial) pairing states. However, considering the strong suppression of $T_{\rm c}$ in MoTe2 by disorder Balicas1 ; Daniel , we suggest that $s^{+-}$ state is more likely to be realized than the $s^{++}$ state. Further phase sensitive experiments are desirable to distinguish between $s^{+-}$ and $s^{++}$ states in MoTe2.

Besides the two-gap superconductivity, another interesting observation is the strong enhancement of the superfluid density ${\lambda}^{-2}(0)$ ${\propto}$ $n_{s}/(m^{*}/m_{e}$) and its linear scaling with $T_{c}$ (see Fig. 5c). $n_{s}/(m^{*}/m_{e}$) increases by factor of ${\sim}$ 1.8 between $p$ = 0.45 GPa and 1.3 GPa. Note that our DFT calculations also show the strong changes in the electronic band structure (see Supplementary Note I and Supplementary Figure 6). The nearly linear relationship between $T_{\rm c}$ and the superfluid density was first noticed in hole-doped cuprates in 1988-89 Uemura1 ; Uemura3 , and possible relevance to the crossover from Bose Einstein Condensation to BCS condensation have been discussed in several subsequent papers Uemura4 ; Uemura5 ; Uemura6 . The linear relationship was noticed mainly with systems following the line of the ratio between $T_{\rm c}$ and their effective Fermi temperature $T_{\rm F}$ being about $T_{\rm c}$/ $T_{\rm F}$ ${\sim}$ 0.05, which means about 4-5 times reduction of $T_{\rm c}$ from the ideal Bose Condensation temperature for a non-interacting Bose gas composed of the same number of Fermions pairing without changing their effective masses. The present results on MoTe2 and NbSe2 Le in Figure 5c demonstrate that a linear relation holds for these systems but with the ratio $T_{\rm c}$/$T_{\rm F}$ being reduced further by a factor of 16-20. It was also noticed Shengelaya that electron-doped cuprates follow another line with their $T_{\rm c}$/$T_{\rm F}$ reduced by a factor of ${\sim}$ 4 from the line of hole doped cuprates. Since the present system MoTe2 and NbSe2 falls into the clean limit, the linear relation is unrelated to pair breaking, and can be regarded to hold between $T_{\rm c}$ and $n_{s}/m^{*}$.

In a naive picture of BEC to BCS crossover, systems with small $T_{\rm c}$/$T_{\rm F}$ (large $T_{\rm F}$) are considered to be in the ”BCS” side, while the linear relationship between $T_{\rm c}$ and $T_{\rm F}$ is expected only in the BEC side. Figure 5c indicates that the BEC-like linear relationship may exist in systems with $T_{\rm c}$/$T_{\rm F}$ reduced by a factor 4 to 20 from the ratio in hole doped cuprates, presenting a new challenge for theoretical explanations.

In conclusion, we provide the first microscopic investigation of the superconductivity in $T_{d}$-MoTe2. Specifically, the zero-temperature magnetic penetration depth ${\lambda}\left(0\right)$ and the temperature dependence of ${\lambda^{-2}}$ were studied in the Type-II Weyl-semimetal $T_{d}$-MoTe2 by means of ${\mu}$SR experiments as a function of pressure up to p ${\simeq}$ 1.3 GPa. Remarkably, the temperature dependence of $1/\lambda^{2}\left(T\right)$ is inconsistent with the presence of nodes in the gap as well as with a simple isotropic $s$-wave type of the order parameter. However, it is well described by a two-gap ($s+s$) - wave scenario, indicating multigap superconductivity in MoTe2. We also excluded time reversal symmetry breaking in the high-pressure SC state with sensitive zero-field ${\mu}$SR experiments, clasifying MoTe2 as time-reversal-invariant superconductor with broken inversion symmetry. In this type of superconductor, a two-gap ($s+s$)-wave model is consistent with a topologically non-trivial superconducting state if the gaps ${\Delta}_{\rm 1}$ and ${\Delta}_{\rm 2}$, at different Fermi surfaces, have opposite signs. ${\mu}$SR experiments alone can not distinguish between sign changing $s^{+-}$ (topological) and $s^{++}$ (trivial) pairing states. However, considering the previous report on the strong suppression of $T_{\rm c}$ in MoTe2 by disorder, we suggest that $s^{+-}$ state is more likely to be realized in MoTe2 than the $s^{++}$ state. Should $s^{+-}$ be the SC gap symmetry, the high pressure state of MoTe2 is, to our knowledge, the first known example of a Weyl superconductor, as well as the first example of a time reversal invariant topological superconductor. Finally, we observed a linear correlation between $T_{c}$ and the zero-temperature superfluid density ${\lambda^{-2}}(0)$ in MoTe2, which together with the observed two-gap behaviour points to the unconventional nature of superconductivity in $T_{d}$-MoTe2. We hope, the present results will stimulate theoretical investigations to obtain a microscopic understanding of the relation between superconductivity and the topologically non-trivial electronic structure of $T_{d}$-MoTe2.

I METHODS

Sample preparation: High quality single crystals and polycrystalline samples were obtained by mixing of molybdenum foil (99.95 ${\%}$) and tellurium lumps (99.999+${\%}$) in a ratio of 1:20 in a quartz tube and sealed under vacuum. The reagents were heated to 1000oC within 10 h. They dwelled at this temperature for 24 h, before they were cooled to 900oC within 30 h (polycrystalline sample) or 100 h (single crystals). At 900oC the tellurium flux was spined-off and the samples were quenched in air. The obtained MoTe2 samples were annealed at 400oC for 12 h to remove any residual tellurium.

Pressure cell: Pressures up to 1.3 GPa were generated in a single wall piston-cylinder type of cell made of CuBe material, especially designed to perform ${\mu}$SR experiments under pressure GuguchiaPressure ; Andreica . As a pressure transmitting medium Daphne oil was used. The pressure was measured by tracking the SC transition of a very small indium plate by AC susceptibility. The filling factor of the pressure cell was maximized. The fraction of the muons stopping in the sample was approximately 40 ${\%}$.

${\mu}$SR experiment:

In a ${\mu}$SR experiment nearly 100 ${\%}$ spin-polarized muons ${\mu}$+ are implanted into the sample one at a time. The positively charged ${\mu}$+ thermalize at interstitial lattice sites, where they act as magnetic microprobes. In a magnetic material the muon spin precesses in the local field $B_{\rm\mu}$ at the muon site with the Larmor frequency ${\nu}_{\rm\mu}$ = $\gamma_{\rm\mu}$/(2{\pi})$$B_{\rm\mu} (muon gyromagnetic ratio $\gamma_{\rm\mu}$/(2${\pi}$) = 135.5 MHz T*-1*). Using the $\mu$SR technique important length scales of superconductors can be measured, namely the magnetic penetration depth $\lambda$ and the coherence length $\xi$. If a type II superconductor is cooled below $T_{\rm c}$ in an applied magnetic field ranged between the lower ($H_{c1}$) and the upper ($H_{c2}$) critical fields, a vortex lattice is formed which in general is incommensurate with the crystal lattice with vortex cores separated by much larger distances than those of the unit cell. Because the implanted muons stop at given crystallographic sites, they will randomly probe the field distribution of the vortex lattice. Such measurements need to be performed in a field applied perpendicular to the initial muon spin polarization (so called TF configuration).

${\mu}$SR experiments under pressure were performed at the ${\mu}$E1 beamline of the Paul Scherrer Institute (Villigen, Switzerland, where an intense high-energy ($p_{\mu}$ = 100 MeV/c) beam of muons is implanted in the sample through the pressure cell. The low background GPS (${\pi}$M3 beamline) and low-temperature LTF instruments were used to study the single crystalline as well as the polycrystalline samples of MoTe2 at ambient pressure.

Analysis of TF-${\mu}$SR data:

The TF ${\mu}$SR data were analyzed by using the following functional form:Bastian

[TABLE]

Here $A_{\rm s}$ and $A_{\rm pc}$ denote the initial assymmetries of the sample and the pressure cell, respectively. $\gamma/(2{\pi})\simeq 135.5$ MHz/T is the muon gyromagnetic ratio, ${\varphi}$ is the initial phase of the muon-spin ensemble and $B_{\rm int}$ represents the internal magnetic field at the muon site. The relaxation rates ${\sigma}_{\rm sc}$ and ${\sigma}_{\rm nm}$ characterize the damping due to the formation of the FLL in the SC state and of the nuclear magnetic dipolar contribution, respectively. In the analysis ${\sigma}_{\rm nm}$ was assumed to be constant over the entire temperature range and was fixed to the value obtained above $T_{\rm c}$ where only nuclear magnetic moments contribute to the muon depolarization rate ${\sigma}$. The Gaussian relaxation rate, ${\sigma}_{\rm pc}$, reflects the depolarization due to the nuclear moments of the pressure cell. The width of the pressure cell signal increases below $T_{c}$. As shown previously MaisuradzePC , this is due to the influence of the diamagnetic moment of the SC sample on the pressure cell, leading to the temperature dependent ${\sigma}_{\rm pc}$ below $T_{c}$. In order to consider this influence we assume the linear coupling between ${\sigma}_{\rm pc}$ and the field shift of the internal magnetic field in the SC state: ${\sigma}_{\rm pc}$($T$) = ${\sigma}_{\rm pc}$($T$ ${\textgreater}$ $T_{\rm c}$) + $C(T)$({\mu}_{\rm 0}$$H_{\rm int,NS} - {\mu}_{\rm 0}$$H_{\rm int,SC}), where ${\sigma}_{\rm pc}$($T$ ${\textgreater}$ $T_{\rm c}$) = 0.25 {\mu}$$s^{-1} is the temperature independent Gaussian relaxation rate. {\mu}_{\rm 0}$$H_{\rm int,NS} and {\mu}_{\rm 0}$$H_{\rm int,SC} are the internal magnetic fields measured in the normal and in the SC state, respectively. As indicated by the solid lines in Figs. 2b,c the ${\mu}$SR data are well described by Eq. (1). The good agreement between the fits and the data demonstrates that the model used describes the data rather well.

Analysis of ${\lambda}(T)$:

${\lambda}$($T$) was calculated within the local (London) approximation (${\lambda}$ ${\gg}$ ${\xi}$) by the following expression Bastian ; Tinkham :

[TABLE]

where $f=[1+\exp(E/k_{\rm B}T)]^{-1}$ is the Fermi function, ${\varphi}$ is the angle along the Fermi surface, and ${\Delta}_{i}(T,{\varphi})={\Delta}_{0,i}{\Gamma}(T/T_{\rm c})g({\varphi}$) (${\Delta}_{0,i}$ is the maximum gap value at $T=0$). The temperature dependence of the gap is approximated by the expression ${\Gamma}(T/T_{\rm c})=\tanh{\{}1.82[1.018(T_{\rm c}/T-1)]^{0.51}{\}}$,carrington while $g({\varphi}$) describes the angular dependence of the gap and it is replaced by 1 for both an $s$-wave and an $s$+$s$-wave gap, and ${\mid}\cos(2{\varphi}){\mid}$ for a $d$-wave gap.

II Acknowledgments

The ${\mu}$SR experiments were carried out at the Swiss Muon Source (S${\mu}$S) Paul Scherrer Insitute, Villigen, Switzerland. X-ray PDF measurements were conducted on beamline 28-ID-2 of the National Synchrotron Light Source II, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-SC0012704. Z. Guguchia gratefully acknowledges the financial support by the Swiss National Science Foundation (SNFfellowship P2ZHP2-161980). The material preparation at Princeton was supported by the Gordon and Betty Moore Foundation EPiQS initiative, grant GBMF-4412. Z.G. and Y.J.U. thank Prof. Andrew Millis for useful discussions. Work at Department of Physics of Columbia University is supported by US NSF DMR-1436095 (DMREF) and NSF DMR-1610633. Work in the Billinge group was supported by U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences (DOE-BES) under contract No. DE-SC00112704. S. Banerjee acknowledges support from the National Defense Science and Engineering Graduate Fellowship program. A.S. acknowledges support from the SCOPES grant No. SCOPES IZ74Z0-160484.

III SUPPLEMENTAL MATERIAL

III.1 I. DFT calculations of the electronic band structure

We used van der Waals density (vdW) functional and the projector augmented wave (PAW) method PAW , as implemented in the VASP code VASP . We adopted the generalized gradient approximation (GGA) proposed by Perdew et al (PBE) PBE and DFT-D2 vdW functional proposed by Grimme et al Grimme ; Harl as a nonlocal correlation. Spin-orbit coupling (SOC) is included in all cases. A plane wave basis with a kinetic energy cutoff of 500 eV was employed. We used a $\Gamma$-centered k-point mesh of 15$\times$9$\times$5. Optimized lattice parameters of $T_{d}$ phase are $a$=3.507, $b$=6.371, and $c$=13.743Å, close to the previous experimental values; $(a,b,c)$=(3.468, 6.310, 13.861) Tamai and (3.458, 6.304,13.859) Wang .

Figure 6 shows the calculated band structure of $T_{d}$-MoTe2 at ambient $p$ and for $p$ = 1.3 GPa, showing the strong changes in the electronic band structure, induced by pressure. This is consistent with the strong pressure dependence of $n_{s}/m^{*}$, observed experimentally.

III.2 II. Confirmation of the orthorhombic ($T_{d}$) to monoclinic ($T^{{}^{\prime}}$) phase transition in MoTe2

Total scattering measurements were carried out on the XPD (28-ID-2) beamline at the National Synchrotron Light Source II (NSLS-II), Brookhaven National Laboratory. A finely ground powder of MoTe2 was prepared in an inert Argon chamber, and sealed in 1.02mm (OD) polyimide capillaries. Diffraction patterns were collected in a Debye-Scherrer geometry with an X-ray energy of 67.127 keV ($\lambda=0.18470$ Å) using a large-area 2D PerkinElmer detector (20482 pixels with 200 $\mu m^{2}$ pixel size). The detector was mounted with a sample-to-detector distance of 344.79 mm, to achieve a balance between $q$-resolution and $q$-range. The sample was measured at 100K and 300K using an Oxford CS-700 cryostream, allowing ample time for the material to thermalize. The experimental geometry, $2\theta$ range, and detector misorientations were calibrated by measuring a crystalline nickel powder directly prior to data collection at each temperature point, with the experimental geometry parameters refined using the PyFAI program kieffer_pyfai_2013 .

Raw 2D diffraction patterns are azimuthally integrated to obtain the 1D scattering intensity $I(Q)$ where $Q=4\pi\sin\theta/\lambda$. Standardized corrections are then made to the data and $I(Q)$ is normalized by dividing by the total scattering cross-section of the sample, resulting in the structure function $S(Q)$

[TABLE]

where $I_{c}$ is the coherent scattering intensity as a function of the momentum transfer $Q$ and $c_{i}$ and $f_{i}$ are the concentration and X-ray scattering factor, per atom type $i$. Because the scattering cross section becomes very small at high-$Q$, an important result of this normalization is that the high angle data are significantly amplified egami_underneath_2003 . This is illustrated in Figure 1(a,b) where we show the standard integrated XRD patterns $I(Q)$ versus the reduced total scattering structure function $F(Q)$ (insets), where $F(Q)=Q[S(Q)-1]$. While the raw diffraction pattern appears to have almost no distinguishable peaks near the upper limit of the shortened $Q$-range, the form factor normalized $F(Q)$ contains well resolved, high amplitude peaks extending to a $Q$-range twice the maximum value shown for $I(Q)$. The high degree of structural coherence observed for MoTe2 is in contrast to diffraction studies of similar layered TMDs, where diffuse scattering from turbostratic disorder and other defects often dominates the high angle signalpetkov_application_2000 .

The full $Q$-range where structure can be resolved, including both Bragg and diffuse components, is shown in the insets of Figure 7. The structure function is Fourier transformed to obtain the pair distribution function (PDF), using PDFgetX3 juhas_pdfgetx3:_2013 within xPDFsuite yang_xpdfsuite:_2014 . The experimental PDF, $G(r)$, is the truncated Fourier transform of $F(Q)$

[TABLE]

For both MoTe2 measurements (100K and 300K), the $Q_{max}$ was chosen to be 22.5 Å*-1* to give the best tradeoff between statistical noise and real-space resolution.

The PDF is a histogram of interatomic distances that gives the probability of finding pairs of atoms in a material, separated by a distance $r$. $G(r)$ can be calculated from a structure model according to

[TABLE]

Here, $\rho_{0}$ is the atomic number density of the material and $\rho(r)$ is the atomic pair density, which is the mean weighted density of neighbor atoms at distance $r$ from an atom at the origin. The sums in $\rho(r)$ run over all atoms ($N$) in the model, with periodic boundary conditions. $f_{i}$ is the scattering factor of atom $i$, $\langle f\rangle$ is the average scattering factor and $r_{ij}$ is the distance between atoms $i$ and $j$. Eqs. 10 are used to fit the PDF calculated from a model to the experimentally measured PDFs. The PDFgui program was used to construct unit cells from reference structures, carry out the refinements, and determine the agreement between simulated PDFs and data, quantified by the residual function $R_{w}$ farrow_pdffit2_2007 .

We included three candidate structure models for local structure refinements of MoTe2 PDFs measured at 100K and 300K. The monoclinic 1T*′-MoTe2 (SG: $P2_{1}/m$) structure reported by B.E. Brown brown_crystal_1966 , the orthorhombic 1Td*-MoTe2 (SG: $Pmn2_{1}$) structure, also referred to as the $\gamma$ form, given in Wang et al. Wang , and a hexagonal 2H polytype (SG:$P6_{3}/mmc$) reported by D. Puotinen and R.E. Newnham puotinen_crystal_1961 . The 2H structure was ruled out as both 1T structures gave significantly better fits for both low-temperature and high-temperature datasets. The T*′* and Td PDF refinements were performed conservatively by adjusting just a few parameters in the structure models: lattice ($a$,$b$,$c$), scale, delta-2 (a parameter for correlated motion effects), one isotropic atomic displacement parameter (ADP) for the 4 Mo atoms, and one ADP for the 8 Te atoms, with both structures containing 12 atoms in the unit cell. For the monoclinic phase, an additional symmetry allowed $\beta$ angle was refined. Atomic positions were not refined. The instrumental resolution parameters $Q_{damp}=0.0217$ and $Q_{broad}=0.009$ were determined through refinements of the nickel calibrant, and kept fixed for all other refinements.

PDFs for MoTe2 are refined over an $r$-range from $1.5<r<15$ Å, slightly larger than the $c$-axis length for both T*′* and Td models. The PDF at low-$r$ contains structural information from the full $F(Q)$ range and as a result, refinements are very sensitive to deviations from the average structure. This is advantageous when characterizing the local atomic environment for MoTe2, with both candidate models containing layers of edge-sharing, distorted MoTe6 octahedra (Figure 7) within the refinement range selected. The approach is different from traditional powder XRD Rietveld refinements of similar materials, where a substantial amount of diffuse scattering is neglected, and phase identification is restricted to a small number of low angle Bragg reflections, thus increasing the number of possible structures that may fit equivalently to the same data.

The results of the PDF analysis are summarized in Figure 8. For MoTe2 measured at 100K there is a clear improvement in the agreement factor ($R_{w}$) when the data is refined to the orthorhombic Td model, with smaller residuals indicating better fits. The difference in refinement quality is more significant above the orthorhombic to monoclinic phase transition ($T_{s}~{}\sim~{}250K$) and the 300K PDF is in good agreement with the T*′* model. By varying a small number of structural parameters we ensure that these differences in $R_{w}$ accurately represent a structural transformation. Additionally, the refined lattice constants differ minimally from the starting values provided in the reference T*′* and Td models. The best-fit PDF at 100K using the Td structure, yields refined lattice parameters of $a=3.474$Å, $b=6.328$Å, $c=13.884$Å, and small ADPs, $U_{iso}$(Mo)$=0.0043$; $U_{iso}$(Te)$=0.0056$. For MoTe2 at 300K, the monoclinic structure refines $a=6.335$Å, $b=3.471$Å, $c=13.853$Å; $\beta=93.886$\mathrm{\SIUnitSymbolDegree}$$ and $U_{iso}$(Mo)$=0.0058$; $U_{iso}$(Te)$=0.0064$.

In Figure 9, we index the Bragg profile to support the results from the more quantitative PDF refinements. Here we simply simply compare the raw diffraction patterns measured at 100K and 300K to powder diffraction patterns calculated from the candidate models using the program VESTA momma_vesta:_2008 . We plot the data over a $2\theta$ range where the T*′* and Td models have Bragg peaks that are well resolved from one another to highlight distinguishable features that differ between the phases in the measured XRD pattern.