Multi-Band Superconductivity in Strongly Hybridized 1T’-WTe₂/NbSe₂ Heterostructures

Inducing superconductivity by proximity in materials with non-trivial band topology [1, 2, 3] has become a method of choice in the search for unconventional forms of superconducting pairing [4, 5]. Prominent examples are demonstrations of unconventional superconductivity in semiconductor nanowires with strong spin-orbit coupling [4], atomic chains and islands [5, 6], as well as at the surfaces and edges of three-dimensional topological insulators [7] and semimetals [8]. At the one-dimensional (1D) edges of 2D topological insulators [9, 10], such as the quantum spin Hall (QSH) state [11], the presence of non-Abelian parafermions have been predicted [12].

In any material system, topologically trivial or non-trivial, the proximity effect can be understood in a microscopic picture to result from Andreev reflection of quasiparticles at the superconductor / normal metal interface. The strength of the induced superconducting pairing is directly linked to the transparency of the interface, i.e. the tunneling coupling at the contact between the proximity-coupled materials. While strong induced pairing with a large order parameter may often be desired, e.g. to mitigate disorder or the effects of an applied magnetic field [4], it is often assumed that the tunnel-coupling does not induce any fundamental changes to the bandstructure. In stark contrast, exchange of charge, especially across atomically abrupt interfaces, can be expected to give rise to atomic-bonding or at least perturbations of the electronic structure, which can lead to profound changes in the electronic band dispersion [13].

Here, we investigate the hybrid electronic structure of such strongly-coupled heterostructures of the quantum spin Hall (QSH) candidate 1T’-WTe₂, grown by in-situ van-der-Waals epitaxy on the type-II superconductor 2H-NbSe₂. Scanning tunneling spectroscopy down to 500 mK allows us to resolve the superconducting local density of states (LDOS) which we analyze within a self-consistent multi-band framework based on the McMillan equations [14]. We are able to show that superconductivity is induced in a semi-metallic WTe₂ monolayer bulk, which we understand as a result of a strong inter-layer tunneling across the heterointerface, leading hybridization of electronic states. A material-specific mean-field tight-binding model is able to capture the hybrid electronic structure in the normal state, allowing direct quantitative comparison with the measured LDOS. As a result, our model is able to predict the magnitude of the induced superconducting order parameter, quantitatively, robust to magnetic fields beyond 2 T. Owing to strong hybridization, we observe a significant weakening of the topological edge state signature, measurable by an enhancement of the LDOS in the normal state at the edge, concomitant with a slight enhancement of the induced superconducting order parameter.

Tungsten ditelluride (WTe₂) and related members of the transition metal dichalcogenide family with 1T’ crystal structure have attracted much attention as candidates material system to realize type-II Weyl [15] fermions and higher-order topology [16] in its 3D bulk. Evidence of high-order semi-metallic properties have recently been confirmed in superconducting Josephson junction measurements [17, 18] showing that induced supercurrents are well-localized to the materials 1D edges (“hinges”). In atomic monolayers, a time-reversal symmetry protected 2D topologically insulating state [19] has been predicted, arising from inversion of the transition metal $d$- and the chalcogen $p$-orbitals, highly tunable by electric fields and strain [19]. While free-standing 1T’-WTe₂ monolayers had initially been predicted to be semi-metallic with a negative fundamental band gap [19], a positive gap was later confirmed by density functional theory and layer-dependent optical spectroscopy [20]. Quantum spin Hall insulating behaviour and the presence of highly-confined 1D metallic states at the edge have since been confirmed in electron transport [21, 22], angle-resolved photoemission spectroscopy (ARPES), and scanning probe measurements [23, 24] including microwave impedance microscopy [25]. However, the sensitivity of the WTe₂ band gap to strain [26] and electric fields [27] remains a matter of intense debate, as significant variations in the magnitude and definition of the measured QSH bulk are being observed [20, 21, 23, 22, 24, 28, 27]. Recent reports indicate that the bulk gap observed may indeed arise from 2D electronic interactions [28] challenging the band-insulating picture [29, 30].

The study of superconductivity remains of fundamental interest in both the QSH insulating [19, 31, 12] and semi-metallic [32, 33] states of 1T’-WTe₂, given theoretical predictions of non-trivial pairing. Intrinsic low-density superconductivity has recently been reported in the 2D bulk of electrostatically doped semi-metallic WTe₂ monolayers [34, 35], and is believed to be of non-trivial pairing [32, 33]. Similarly, the 1D helical edge of any quantum spin Hall insulating state has been predicted as a potential host for parafermions in the superconducting state [19, 10, 9, 31, 12].

Superconductivity up to relatively high critical temperatures and magnetic fields can be achieved by proximity-coupling to a stable intrinsic superconductor, such as the layered type-II superconductor 2H-NbSe₂ ($T_{C}\simeq 7.2$ K, $B_{C2}\simeq 5$ T) [36, 37, 38, 39]. Further to these studies, here we show that strong coupling to the superconductor gives rise to a hybrid multi-band electronic structure, which needs to be taken into account when interpreting the detailed functional form of the induced superconducting local density of states.

In summary, we have reported signatures of multi-band superconductivity in strongly-coupled 1T’-WTe₂/NbSe₂ heterostructures, grown by van-der-Waals epitaxy. Analyzing the superconducting density of states down to 500 mK in scanning probe spectroscopy, we have shown that strong hybridization of electronic states gives rise to a semimetallic density of states in the 2D bulk even in nominally band-insulating crystals. Describing the detailed functional form of the superconducting energy gap in a self-consistent multi-band framework, based on the McMillan equations, we confirm the strong inter-band coupling. Our quantitative comparison of the measured local density of states with a material-specific tight-binding model ultimately confirms the hybrid electronic structure, in both normal and superconducting states for the same interlayer hopping, thus accurately predicting the magnitude of the induced WTe₂ order parameter, $\Delta_{\rm WTe_{2}}(B=0)\simeq 0.55$ meV, stable beyond a 2 T magnetic field. Despite the strong hybridization, we find that a measurable enhancement of the measured local density of states persists at the crystal edges, detectable in both the normal state and in a slight enhancement of the order parameter in the superconducting state. We believe that our multi-band treatment of strongly hybridized van-der-Waals heterostructures will form a useful tool to mapping spatial variation of the induced superconducting order parameter in wider range proximitized atomically-thin topological materials [51].

This research is supported by National Research Foundation (NRF) Singapore, under the Competitive Research Programme ”Towards On-Chip Topological Quantum Devices” (NRF-CRP21-2018-0001), with partial support from a Singapore Ministry of Education (MOE) Academic Research Fund Tier 3 grant (MOE2018-T3-1-002). The work was supported in part by Grants-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (nos. 16H02109, 18K19013, and 19H00859). The work at Northeastern University was supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences grant number DE-SC0019275 and benefited from Northeastern University’s Advanced Scientific Computation Center (ASCC) and the NERSC supercomputing center through DOE grant number DE-AC02-05CH11231. KEJG acknowledges support from the Agency for Science, Technology and Research (A*STAR) under its A*STAR QTE Grant No. A1685b0005. H.L. acknowledges the support by the Ministry of Science and Technology (MOST) in Taiwan under grant number MOST 109-2112-M-001-014-MY3. SM would like to acknowledge the new faculty seed grant from IIT Madras under project number Project No: PHY/18-19/703/NFSC/SHAA. BW acknowledges a Singapore National Research Foundation (NRF) Fellowship (NRF-NRFF2017-11). We thank Jack T. Hellerstedt for technical advice during the early stages of crystal growth.

Van-der-Waals molecular beam epitaxy (MBE) was performed in an Omicron Lab10 MBE (base pressure $2\times 10^{-10}$ mBar) on bulk crystals of 2H-NbSe₂ (HQ graphene, Netherlands) and highly oriented pyrolytic graphite (HOPG). The substrates were mechanically cleaved in UHV after degassing overnight (300° C), following co-deposition of W (99.998$\%$) and Te (99.999$\%$) for approximately 1h at a Te:W flux ratio of roughly 260:1. We use slightly different substrate temperatures for the two heterostructures with 145° C for 1T’-WTe₂/NbSe₂, and 230° C for 1T’-WTe₂/HOPG. Typical STM images of the WTe₂/HOPG and WTe₂/NbSe₂ heterostructures, respectively, are shown in Fig. 5(a) and 5(b), alongisde measurements of the respective WTe₂ monolayer height, extracted from $z$-height histograms 5(c) and 5(d).

NbSe₂ is a type-II superconductor [52, 53] with a critical temperature $T_{C}=7.2$ K and field $B_{\rm C2}\sim 5$ T [54, 55]. The microscopic detail of its superconductivity have been investigated, intensively, both theoretically and experimentally [40, 41, 36, 56].

The NbSe₂ Fermi surface [59, 46, 60] is composed of at least two bands at the Fermi energy. Tight-binding calculations of the NbSe2₂ low-energy bandstructure are shown in Fig. 6, where black and red lines are bands due to the $d_{3z^{2}-r^{2}}$ orbitals, centered on the two Nb atoms within the unit cell (Nb bands). These form bonding and anti-bonding orbitals, giving rise to cylinders at the six $K$-points of the hexagonal Brillouin zone. A third band (predominantly Se) [40] has been neglected in our model, as it is not believed to contribute to multi-band superconductivity in NbSe₂ [40]. As a result of the two-band nature, two distinct superconducting energy gaps, $\Delta_{L}$ and $\Delta_{S}$, are usually observed at very low temperatures ($T\ll T_{C}$) [41, 38, 57] (see Table 1), each hosted within the two Nb bands [40, 41, 44]. Superconducting pairing is significantly stronger in the Nb $K$-bands [40, 41, 36], giving rise to the larger of the two gaps $\Delta_{L}\simeq 1.3$ meV. At $\Gamma$, the pairing is weaker with order parameter $\Delta_{S}$, and is usually found ranging between $0-0.3$ meV [41, 36].

CDW order on the pristine NbSe₂ surface also plays an important role in scanning tunneling microscopy measurements of the superconducting gap. In pristine NbSe₂, the 3Q charge density wave (CDW) gives rise to an anisotropic Fermi surface and a reduced Brillouin zone with hexagonal symmetry [41]. Possible mechanism for a suppression of $T_{C}$ have been argued to result from a reduction of the highly anisotropic superconducting gap due to disorder averaging and/or from the suppression of assistive short range CDW correlations [61]. From a multi-band perspective, the CDW wave vectors $g_{i,\rm CDW}=1/3G_{i}$ link states at the $\Gamma$ point (associated with the NbSe₂ small gap) to the Nb cylinders at $K$ (associated with the large gap). This allows for the observation of the large gap on pristine NbSe₂ surfaces in scanning tunnelling experiments [40, 41], which are most sensitive to states at $\Gamma$ (tunnelling of electrons with finite transverse momentum is strongly suppressed in STM due to the vertical tunneling path). Once CDW order is suppressed, STM will predominantly probe states at $\Gamma$, which would explain the reduced gap observed [41]. Figure 7 shows a comparison of the measured gap on both the pristine and the post-growth NbSe₂ surface, with a reduction in gap size clearly visible.

Low-temperature scanning tunneling spectroscopy (STM/STS) measurements were carried out in an Omicron low-temperature STM (junction temperature $T\simeq 4.5$ K) and a Unisoku USM1300 He-3 STM (junction temperature $T\simeq 500$ mK), respectively. Spectroscopic measurements were obtained using standard lock-in techniques, with an AC excitation of amplitude of 1.5 mV at 831 Hz for measurements of the normal-state electronic structure, and $60~{}\mu$V at 931 Hz for measurements in the superconducting state. In all measurements, we fix the current set point 200 pA at 10 mV, ahead of spectroscopy. All spectra of superconducting states through the paper have been spatially averaged, symmetrised about zero bias, and normalised with respect to the normal state conductance, unless otherwise specified.

In the limit of low bias and low temperature ($k_{B}T\sim eV$), the tunneling differential conductance, measured by scanning probe spectroscopy, is usually expressed as a convolution of the tunneling density of states $N(E)$ and the derivative of the Fermi-Dirac distribution,

For a bulk superconductor with a single isotropic band, a Bardeen-Cooper-Schrieffer (BCS) density of states $N(E)=E/\sqrt{E^{2}-\Delta^{2}}$ is often assumed, and can result in acceptable fits at moderate temperature $T\lesssim T_{C}$. However, as the derivative of the Fermi-Dirac distribution is a peaked function with half-width $\sim 3.5~{}k_{B}T$, it is responsible for thermal smearing of the measured superconducting tunneling density of states. An additional phenomenological broadening parameter, $\gamma$ (Dynes parameter [43]) is often considered for tunneling experiments, accounting for additional pair-breaking mechanisms.

At low temperature ($T\ll T_{C}$), fine detail of the functional form of the superconducting energy gap can be observed [40, 44, 56]. Indeed, in Fig. 1(h) we see that a BCS/Dynes model provides a good fit only at $T\simeq 4.5$ K, but fails to describe the detail of the measured superconducting density of states at lower temperature. A more detailed model is needed to describe the low-temperature data, taking the multi-band nature of superconductivity in NbSe₂ into account.

In the presence of multiple (non-interacting) bands $i$ at the Fermi energy, the total tunneling density of states may be expressed as a sum over two or more partial densities of states,

The partial densities of states can further be expressed as

where it is assumed that the order parameter $\Delta(E,\theta)$ may not be constant, but itself be a complex energy-dependent and (potentially anisotropic [56]) function. The prefactor, $\tilde{N_{i}}(E_{F})=N_{i}(E_{F})T_{i}$, can be interpreted [41] as an effective partial density of states weight, renormalized from the true density of states $N_{i}(E_{F})$ by the tunneling probability $T_{i}$ into band $i$.

Figure 8 shows a comparison of different models to fit our data, based on Eq. (1), including a single-band BCS/Dynes model ($i=L,\Delta_{L}$), a two-band BCS/Dynes model ($i=L,S,\Delta_{L,S}$), and an anisotropic single-band model ($i=1$). The second derivative of the tunneling current and close-ups of the coherence peaks are shown alongside to clarify differences in the models. For the anisotropic model we have assumed an empirical anisotropy function, as previously employed for NbSe₂ [56], reflecting the sixfold symmetry of the NbSe₂ Fermi surface,

Among the different models, only the two-band BCS and the anisotropic band model provide a somewhat satisfying fit to the measured NbSe₂ data [56]. All models, including the anisotropic band fit underestimate the width of the coherence peaks (Fig. 8(c)).

Taking the multi-band nature of superconductivity in NbSe₂ into account, a compelling model to describe the superconducting density of states was employed by Noat et al. [41], based on the McMillan equations [14]. The McMillan model was originally developed to describe superconductivity between two proximity-coupled materials $i=1,2$, in which the intrinsic order parameters $\Delta_{i}^{0}$ are renormalised by an inter-layer electronic coupling of strength $\Gamma_{ij}$. In the context of multi-band superconductivity [42, 40, 41] the inter-layer coupling has since been interpreted as an inter-band scattering rate from band $i$ to band $j$ (“proximity-effect in reciprocal space”). Different from the other models discussed above, the order parameter $\Delta_{i}(E)$ in the McMillan model is a complex, energy-dependent gap function

Inclusion of these order parameters in Eq. (1) gives rise to a set of self-consistent equations to describe the multi-band system, that can be solved numerically to fit the measured tunneling spectra.

As highlighted in Ref. [41], in addition to the directly measured partial density of states (Eq. (1)), and independent estimate of the partial densities of states ratio at the Fermi-energy $\frac{N_{j}(E_{F})}{N_{i}(E_{F})}=\frac{\Gamma_{ij}}{\Gamma_{ji}}$ can be obtained from a ratio of the inter-band coupling parameters. Simultaneous extraction of both during the self-consistent numerical fits thus allow for an accurate extraction.

To describe multi-band superconductivity in the hybrid WTe₂/NbSe₂ heterostructure, we consider a third order parameter, $i=3$, in Eq. 1. To limit the number of free parameters, we consider coupling $\Gamma_{13(31)}$ only between WTe₂ and the Nb $K$-band (large gap $\Delta_{L}^{0}$), which is expected to have the strongest contribution to the proximity coupling [41], and fix $\Gamma_{23(32)}=0$. We further assume that WTe₂ does not carry any intrinsic superconductivity, such that $\Delta_{\rm WTe_{2}}^{0}=0$. Finally, all NbSe₂ related parameters – determined independently from two-band fits to the substrate – can be fixed in the three-band model, in particular, $\Delta_{L,S}^{0}$ and $\Gamma_{12(21)}$, leaving only three independent fitting parameters in the three-band model.

Figures 8(d)-8(f) shows a comparison our three band McMillan model (solid red line) and the previous two-band model (solid blue line), applied to the WTe₂/NbSe₂ heterostructure. Importantly, a two-band model cannot represent the data well. From the three band model of the heterostructure, we are able to extract the intrinsic NbSe₂ order parameters $\Delta_{L}^{0}=1.13\pm 0.05$ meV and $\Delta_{S}^{0}=0.43\pm 0.08$ meV, in excellent agreement those extracted from the two-band model of NbSe₂ [44, 41]. Consistency of the NbSe₂ order parameter, extracted separately from two-band and three band models, confirm that the properties of the NbSe₂ parent superconductor remain unaffected by the presence of the WTe₂ epilayer. The renormalised complex order parameters extracted from the three-band fits are plotted in Figs. 9(c) and 9(d). Here, we observe only a minor re-balancing of the inter-band tunneling rates $\Gamma_{ij}$ with a concomitant small decrease in the Re($\Delta_{L,S}(E_{F})$), as a result of the additional coupling to the third band. The value of the induced order parameter, $\Delta_{\rm WTe_{2}}(E_{F})=0.58\pm 0.02$ meV agrees well with recent local probe [39] and transport spectroscopy [38] of non-epitaxial heterostructures.

The 1T’-WTe₂/2H-NbSe₂ heterostructure was modelled using a real-space mean field tight-binding Hamiltonian that is solved using a Bogoliubov-De-Gennes (BdG) formalism. The Hamiltonian contains four separate parts given by,

The individual terms of the Hamiltonian are given by,

In $H^{N}_{0}$ operators $(c^{{\dagger}}_{il\sigma},c_{il\sigma})$ represent the creation and annihilation operators, respectively, at the site $i$ with a single Nb $d_{z^{2}}$ orbital contributing per site (see below for details). The layer index $l=(1,2)$ corresponds to the two layers of 2H-NbSe₂, respectively, taking spin $\sigma=(\uparrow,\downarrow)$ into account. The individual real-space tight-binding hopping matrix elements for 2H-NbSe₂ are generated by a basis transformation to orbital basis from a two-band Hamiltonian that has been studied previously to match ARPES [62] and STM experiments [63] on 2H-NbSe₂.

At temperatures below $T\sim 7.2$ K, pristine NbSe₂ undergoes a transition to a superconducting state. Assuming a dominant conventional $s$-wave superconducting instability, we calculate the superconducting gap $\Delta^{N}_{il}=V\langle c_{il\uparrow}c_{il\downarrow}\rangle$ self-consistently at each lattice site by solving the BdG equations. The dominant on-site pairing term $V$ is tuned to generate a superconducting gap of $\Delta\sim 1.1$ eV in agreement with the experimental results on pristine 2H-NbSe₂ (see Table 1 and references therein).

The electronic structure of 1T’-WTe₂ is modelled by generating a real space version of an 8-band Hamiltonian, containing 2 Te and 2 W atoms in a rectangular unit cell. As discussed in Ref [45, 64, 65], this Hamiltonian corresponds to a dominant contribution from Te $p_{x}$ orbitals, and W $d_{x^{2}-y^{2}}$ orbitals, which dominate the low-energy band structure owing to the distorted lattice structure in monolayer 1T’-WTe₂.

In $H^{W}_{0}$, the indices $i,j$ refer to the unit cell, and $\mu,\nu=(1,2,3,4)$ represent the atoms or orbitals within each unit cell. The model provides reasonable agreement with the low energy electronic structure including the QSHI observed in ARPES and STM experiments [23]. In the enclosed additional supplementary files, we provide the real space hopping matrix elements including the Rashba spin orbit coupling terms included in the real space Hamiltonian for 1T’-WTe₂. The monolayer edge was modelled using an open boundary condition and has been studied for various terminations for directions perpendicular and parallel to the direction of the atomic chains. We find that although the modelling of the edge with open boundary conditions is simplistic (the edge in as as-grown real material is rough with no consistent termination), the obtained local density of states shows reasonable agreement with the experiments (see Fig. 2 of the main text).

To represent a finite electronic interlayer coupling in the heterostructure, we assume a nearest neighbor hopping $t_{\perp}$, leading to an effective hybridization between the Nb d${}_{z^{2}}$ orbitals and Te p_x orbitals. This gives rise to a residual metallic density of states within the WTe₂ bulk band gap, in reasonable agreement with our experimental observations in both the normal state (see Fig. 2 of the main text) and the superconducting state (see Fig. 3 of the main text). We find best agreement with the experiments – simultaneously for normal and superconducting state – for $t_{\perp}\sim 0.15$ eV.

The induced superconducting gap on the Te and W atoms have been evaluated self consistently by calculating the anomalous averages $\Delta^{W}_{i\mu}\sim\langle d^{{\dagger}}_{i\mu\uparrow}d^{{\dagger}}_{i\mu\downarrow}\rangle$, where $\mu=(1,2,3,4)$ represents the 2 Te and 2 W atoms in the unit cell and $i$ is the unit cell index. The calculations involve a Hamiltonian where the electron operators are considered in momentum space for a direction perpendicular to the edge. For an edge along the $y$-direction, the Bogoliubov-de-Gennes transformations involve quasi-article operators,

Here, $\gamma_{n\sigma}$ are the quasiparticle operators corresponding to state $n$ and $u,v$ are the corresponding amplitudes. Similar transformations also hold for the $c$ operators for NbSe₂.

Diagonalizing the above BdG Hamiltonian, we self-consistently calculate the electron density $n(i_{x})$ and superconducting gap at each lattice site for the multi-orbital Hamiltonian. The mean-fields for the electron density would in general be given by,

Here, $f(E_{n})$ is the Fermi function.

As discussed above, the superconducting gap on NbSe₂ has been introduced with a pairing interaction term $V$. The self-consistent procedure leads to an induced even parity ($\Delta^{s}_{i_{x}j_{x}\mu}$) order parameter on 1T’-WTe₂. The induced superconducting gaps are obtained from the self-consistent solutions by calculating the following anomalous averages,

Here, $N_{k_{y}}$ is the number of $k_{y}$ divisions which is typically taken to be $30000-60000$ points to achieve high resolution for the small gaps observed in the system.

We see our model as a generalization from the approximation of a superconducting heterostructure by considering pristine WTe₂ with no effect of the proximal superconductor on the WTe₂ band structure, i.e. a heterostructure in which the only effect of the superconductor is to induce a gap on the QSH edges. Instead, our modelling accounts for a realistic finite interlayer coupling between NbSe₂ and WTe₂ through a self-consistent procedure, allowing for minor changes in the electronic structure such as hybridization and charge transfer. Our model is thus able to explain, simultaneously, a small but finite metallic density of states within the QSH gap, that leaves the edge state intact, and an induced superconducting order parameter of the same magnitude as resolved in the experiments.