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

Wei Tao  Zheng Jue Tong  Division of Physics and Applied Physics, School of
Physical and Mathematical Sciences, Nanyang Technological University,
Singapore 637371, Singapore  Anirban Das  Department of Physics, Indian
Institute Of Technology Madras, Chennai, Tamil Nadu 600036, India
Computational Materials Science Group IIT Madras Chennai, Tamil Nadu 600036,
India  Duc-Quan Ho  Division of Physics and Applied Physics, School of
Physical and Mathematical Sciences, Nanyang Technological University,
Singapore 637371, Singapore  Yudai Sato  Masahiro Haze  The Institute for
Solid State Physics, The University of Tokyo, 5-1-5 Kashiwa-no-ha, Kashiwa
277-8581, Japan  Junxiang Jia  Division of Physics and Applied Physics, School
of Physical and Mathematical Sciences, Nanyang Technological University,
Singapore 637371, Singapore  K. E. Johnson Goh  Institute of Materials
Research and Engineering (IMRE), Agency for Science, Technology and Research,
Singapore  BaoKai Wang  Department of Physics, Northeastern University,
Boston, Massachusetts 02115, USA  Hsin Lin  Institute of Physics, Academia
Sinica, Taipei, Taiwan  Arun Bansil  Department of Physics, Northeastern
University, Boston, Massachusetts 02115, USA  Shantanu Mukherjee  Department
of Physics, Indian Institute Of Technology Madras, Chennai, Tamil Nadu 600036,
India  Computational Materials Science Group IIT Madras Chennai, Tamil Nadu
600036, India  Quantum Centres in Diamond and Emergent Materials
(QuCenDiem)-Group IIT Madras Chennai, Tamil Nadu 600036, India  Yukio Hasegawa
The Institute for Solid State Physics, The University of Tokyo, 5-1-5 Kashiwa-
no-ha, Kashiwa 277-8581, Japan  Bent Weber<EMAIL_ADDRESS>Division of
Physics and Applied Physics, School of Physical and Mathematical Sciences,
Nanyang Technological University, Singapore 637371, Singapore  ARC Centre of
Excellence for Future Low-Energy Electronics Technologies (FLEET), School of
Physics, Monash University, Clayton VIC 3800 Australia

###### Abstract

The interplay of topology and superconductivity has become a subject of
intense research in condensed matter physics for the pursuit of topologically
non-trivial forms of superconducting pairing. An intrinsically normal-
conducting material can inherit superconductivity via electrical contact to a
parent superconductor via the proximity effect, usually understood as Andreev
reflection at the interface between the distinct electronic structures of two
separate conductors. However, at high interface transparency, strong coupling
inevitably leads to changes in the band structure, locally, owing to
hybridization of electronic states. Here, we investigate such strongly
proximity-coupled heterostructures of monolayer 1T’-WTe2, grown on NbSe2 by
van-der-Waals epitaxy. The superconducting local density of states (LDOS),
resolved in scanning tunneling spectroscopy down to 500 mK, reflects a hybrid
electronic structure, well-described by a multi-band framework based on the
McMillan equations which captures the multi-band superconductivity inherent to
the NbSe2 substrate and that induced by proximity in WTe2, self-consistently.
Our material-specific tight-binding model captures the hybridized
heterostructure quantitatively, and confirms that strong inter-layer hopping
gives rise to a semi-metallic density of states in the 2D WTe2 bulk, even for
nominally band-insulating crystals. The model further accurately predicts the
measured order parameter $\Delta\simeq 0.6$ meV induced in the WTe2 monolayer
bulk, stable beyond a 2 T magnetic field. We believe that our detailed multi-
band analysis of the hybrid electronic structure provides a useful tool for
sensitive spatial mapping of induced order parameters in proximitized
atomically thin topological materials.

Proximity induced superconductivity, quantum spin Hall, Scanning tunneling
microscopy, topological insulator

††preprint: APS/123-QED††preprint: APS/123-QED

##  I Introduction

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].

Figure 1: (Color online) Superconducting WTe2/NbSe2 heterostructures. (a), (b)
STM topography (1.6 V, 100 pA) of monolayer 1T’-WTe2 islands grown on a single
crystal of 2H-NbSe2 by van-der-Waals epitaxy. (c) STM measured height profile
corresponding to the dashed white line in (d), showing a monolayer height of
8.2 Å. The insert shows the respective 1T’-WTe2 and 2H-NbSe2 crystal
structures. (d) Atomic-resolution close-up (100 mV, 2.2 nA) of the 1T’-WTe2
edge, showing detail of the atomic alignment. The respective chalcogen
sublattices are seen to align along the $y$-direction of the 1T’-WTe2 crystal
(scale bar: 6.2 Å). (e)-(g) 2D FFTs of the 1T’-WTe2 lattice (e), an area with
46% WTe2 monolayer coverage (f), and the pristine NbSe2 surface (g) after
cleaving. (all scale bars: 2 nm-1). We estimate $a\simeq a_{1}=3.5\pm 0.2$ Å,
and $b\simeq 2a_{2}\textrm{cos}(30\degree)=6.2\pm 0.3$ Å for the lattice
constants. (h) Temperature dependence of the superconducting local density of
states (LDOS), measured in the WTe2/NbSe2 2D bulk, comparing measurements at
$T=4.5$ K (black) and 500 mK (blue). Solid blue and black lines are fits to a
conventional single-band BCS model. (i) High-resolution tunneling spectra,
comparing the WTe2/NbSe2 bulk spectrum with that measured on the bare NbSe2
substrate. Solid red and blue lines are fits to our two-band (NbSe2) and
three-band (WTe2) McMillan models, respectively (see main text for detail).
(j) Second derivative of the tunnel current, corresponding to the data shown
in (i). Dashed lines and arrows indicate the positions of the respective
superconducting energy gaps.

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.

##  II Results and Discussion

###  II.1 Crystal Growth and Atomic Structure

Figure 1 shows scanning tunneling microscopy (STM) data of 1T’-WTe2/2H-NbSe2
heterostructures, grown by bufferless low-temperature van-der-Waals epitaxy.
In agreement with previous studies of WTe2 on bilayer graphene (BLG) [23, 24],
we observe Volmer-Weber growth of islands with disordered boundaries (Figs.
1(a) and 1(b)) and size up to a few tens of nanometre in diameter, that are
poly-crystalline on NbSe2 substrates. For this work, we focus on the
intermediate coverage limit ($\sim 50$%) in order to maximize the density of
QSH edges, although nearly complete monolayers with coverage of $>$95% can be
achieved with reduced crystal quality.

Atomic-level detail of the WTe2 lattice alignment with the substrate is shown
in Figs. 1(d)-1(g) where we find that the respective chalcogen (Te and Se)
sublattices align along the atomic rows of 1T’-WTe2 ($y$-direction). A 2D fast
Fourier transform (Fig. 1(f)) of an area with $\sim 50$% WTe2 monolayer
coverage indeed confirms that the Bragg peaks of the WTe2 and NbSe2 lattices
coincide within the accuracy of the measurement. We extract $a\simeq
a_{1}=3.5\pm 0.2$ Å, and $b\simeq 2a_{2}\textrm{cos}(30\degree)=6.2\pm 0.3$ Å
, in good agreement with the lattice parameters of WTe2 ($a=3.48$ Å, and
$b=6.28$ Å ) [20]. $a_{1}=a_{2}=3.45$ Åare the lattice parameters of NbSe2,
local lattice matching would imply a 5% compressive lattice strain along $b$,
which has previously been shown to further stabilize the WTe2 bulk gap [26].

A first indication that we may expect strong hybridization in WTe2/NbSe2
heterostructures comes from a measurement of the monolayer height $h=0.9\pm
0.1$ nm, extracted from $z$-height distributions of large-scale topographic
STM images (see Fig. 5, Appendix A), and is comparable to measurements of van-
der-Waals stacked WTe2/NbSe2 ($h\simeq 0.7$ nm) [39]. The significantly lower
layer height, compared to WTe2/BLG ($h\simeq 1.2$ nm) [24] and WTe2/HOPG
($h=1.2\pm 0.1$ nm) (this work), suggests a smaller van-der-Waals gap,
resulting in stronger interlayer coupling and hybridization of electronic
states.

We observe some variations in the electronic structure across different
monolayer crystals, reflecting varying doping levels and the presence of in-
gap impurity states around the visible adatom disorder (Figs. 1(a)-(b). We
therefore focus on spectra obtained on clean monolayer regions which display a
clear suppression of the LDOS over an energy range of several tens of meV
around the Fermi energy, and with signatures of an edge state in the normal
state. As shown in the comparison of Fig. 2 below, such spectra agree well
with tight-binding calculations of the normal state electronic structure, as
well as with those published on WTe2/BLG [23, 24, 26, 27]. Given the metallic
nature of the two substrates used in this work, we do not expect that exitonic
insulating effects [29, 30] play a dominant role here, as Coulomb interaction
would be expected to be strongly screened.

###  II.2 Multi-Band Superconductivity

Figure 2: (Color online) Hybrid electronic structure of strongly-coupled
1T’-WTe2 heterostructures. (a)-(b) Spatial profiles of the measured normal-
state local density of states (LDOS) across the monolayer edge for (a)
WTe2/HOPG and (b) WTe2/NbSe2, alongside corresponding STM height profiles.
(c),(d) individual point spectra of the local density of states (LDOS) in 2D
bulk (black) and edge (red), measured at $T=4.2$ K, and compared to tight-
binding calculation (e, f), respectively. (g)-(j) Tight-binding band
structures of free-standing WTe2 monolayers in $\Gamma-Y$ (g) and $\Gamma-X$
(i) direction, respectively, and compared to the spin-resolved (blue/red) WTe2
orbital weight of a WTe2/NbSe2 heterostructure in $\Gamma-Y$ (h) and
$\Gamma-X$ (i). The measured LDOS in (a) and (b) was taken along
$y$-direction, across the $\Gamma-X$ edge. Red dashed lines in (a) and (b)
indicate the positions at which the edge state point spectra in (c) and (d)
were measured. Black arrows indicate spectral features resolved on WTe2/NbSe2,
only, arising from the NbSe2 substrate. Red arrows indicate the enhanced LDOS
at the crystal edges. (k) Lattice model and first Brillouin zone of 1T’-WTe2,
indicating high-symmetry points and directions. (l)-(n) Spatial profiles of
the integrated edge-state LDOS (-50 meV to +20 meV) for (l) WTe2/HOPG and (m)
WTe2/NbSe2, compared to tight-binding calculations (n). We define the edge
position, electronically, where the integrated LDOS peaks and exponentially
decays into the 2D bulk gap. Red lines in (l)-(n) are exponential fits to
extract the edge state decay lengths, $\xi=(0.7\pm 0.1)$ nm (WTe2/HOPG),
$\xi=(0.7\pm 0.2)$ nm (WTe2/NbSe2), and $\xi=(1.0\pm 0.0)$ nm (theory).

The clear signature of a superconducting energy gap, measured in the 2D bulk
of monolayer WTe2, is shown in Figs. 1(h), comparing the superconducting LDOS
at 4.5 K with that measured at 500 mK. Fits to conventional BCS theory (solid
lines) describe the data well only at 4.5 K, but fail to describe the details
of the energy gap resolved at 500 mK. This becomes particularly apparent by an
underestimation of the width and height of the coherence peaks at $\pm 1$ meV.
A comparison of high-resolution spectra measured, respectively, on the WTe2
and NbSe2 surfaces, are shown in Fig. 1(i). We observe a slightly reduced gap
size on NbSe2 post MBE growth, compared to the pristine NbSe2 surface, which
is likely due to a partial quenching of the 3Q charge density wave (CDW) order
[40, 41] leading to disorder averaging of the Fermi surface anisotropy and
reduced inter-band coupling (see Fig. 7 and Appendix B for further detail). A
further reduction of the superconducting gap measured on WTe2/NbSe2
corroborates that superconductivity is induced into the monolayer via the
substrate.

As detailed below, we are able to accurately describe the measured
superconducting LDOS within a self-consistent multi-band framework based on
the McMillan equations [14, 42, 41] (blue/red lines). In the presence of
multiple non-interacting bands $i$, the superconducting density of states can
be expressed as a simple sum $N(E)=\sum\limits_{i}N_{i}(E)$ of two or more
partial densities of states,

$N_{i}(E)=\tilde{N_{i}}(E_{F})\operatorname{Re}\left[\frac{\lvert{E-i\gamma_{i}}\rvert}{\sqrt{(E-i\gamma_{i})^{2}-\Delta_{i}(E)^{2}}}\right]$
(1)

Within conventional BCS theory, the order parameter $\Delta$ is a constant and
the prefactor. The prefactor, $\tilde{N_{i}}(E_{F})$), denotes the partial
state density of states of band $i$ in the normal state and, in the
superconducting state, takes the role [41] of an effective partial density of
states weight to the total density of states (Eq. (6)). An additional
broadening parameter $\gamma_{i}$ is usually included to account for energy
broadening during the tunneling process (Dynes parameter) [43].

Different from BCS theory, the McMillan model [14] considers coupled bands
with order parameters $\Delta_{i}(E)$, that are energy-dependent complex
functions of the form

$\Delta_{i}(E)=\frac{\Delta_{i}^{0}+\Gamma_{ij}\Delta_{j}(E)/\sqrt{\Delta_{j}^{2}(E)-(E-i\gamma_{j})^{2}}}{1+\Gamma_{ij}/\sqrt{\Delta_{j}^{2}(E)-(E-i\gamma_{j})^{2}}}$
(2)

Here, an inter-band Cooper pair tunneling across the van-der-Waals gap at rate
$\Gamma_{ij}$ renormalizes the intrinsic order parameters of the separate
materials $\Delta_{i}^{0}$, leading to a set of self-consistent equations,
that can be solved numerically to fit our data.

Although originally developed to describe the superconducting proximity effect
in single-band metallic systems, the McMillan model has more recently been
successfully applied to describe intrinsic multi-band superconductivity in
layered two-band superconductors [42], including NbSe2 [41, 44]. For this
work, we further extend the McMillan model to describe the hybrid electronic
structure of 1T’-WTe2/NbSe2 heterostructures, thus treating multi-band
superconductivity and proximity-coupling within the same theoretical
framework. To this end, we consider a third order parameter $\Delta_{\rm
WTe_{2}}$, coupled to the NbSe2 $K$-band (see supporting information),
assuming $\Delta_{\rm WTe_{2}}^{0}=0$. Fits to both a two-band (NbSe2, red
line) and a three-band (WTe2/NbSe2, blue line) model describe the data well
(Fig. 1(i)), and reproduce the known NbSe2 order parameters (see Table 1.
Indeed, we find that simpler BCS-based models are unable to represent the data
well at 500 mK (Appendix B), further confirming the multi-band hybrid
electronic structure. For the real part of the induced order parameter, we
find $\Delta_{\rm WTe_{2}}(E_{F})=(0.57\pm 0.02)$ meV, reflecting induced
pairing in the strongly coupled heterostructure, in reasonable agreement with
recent transport [38] and local probe [39] spectroscopy of non-epitaxial
WTe2/NbSe2 hetero-junctions.

###  II.3 Hybrid Electronic Structure

As inferred from our three-band model, we understand that superconductivity is
induced in the WTe2 bulk as a result of strong inter-band coupling to the
substrate giving rise to a WTe2/NbSe2 hybrid electronic structure. We confirm
this notion in Fig. 2 by directly comparing spectra of the normal-state LDOS
with tight-binding band structure calculations [45] (Appendix F). In Figs.
2(a) and 2(b) we show the spatial evolution of the measured normal-state LDOS
across a monolayer edge, comparing the WTe2/HOPG (a) and the WTe2/NbSe2 (b)
heterostructure, alongside corresponding height-profiles. Individual point
spectra of the 2D bulk (black) and 1D edge (red) are furthermore shown in the
inserts to Figs. 2(c) and (d). In both heterostructures, the LDOS shows clear
signs of a suppression over $\simeq 70$ meV around the Fermi energy,
corresponding to a soft gap. This gap is significantly more developed in
WTe2/HOPG, but shows with comparable magnitude and position also on
WTe2/NbSe2. A residual 2D bulk LDOS within the gap is measurable on both
substrates, but is much more pronounced in WTe2/NbSe2, attesting to strong
hybridization in the heterostructure.

Tight-binding band structure calculations of a freestanding WTe2 monolayer are
shown in Fig. 2(c), and compared with the spin-resolved WTe2 orbital weight of
the hybridized WTe2/NbSe2 heterostructure (Fig. 2(d). We find best agreement
for an interlayer hopping of 0.15 eV (see Appendix), similar in magnitude to
the Nb-Nb hopping strength within the 2H-NbSe2 cell [46]. A slightly larger
hopping strength would be expected given the shorter Nb-Te separation. The
direct comparison with our measured spectra (inserts) shows reasonable
agreement with regard to the position of the band edges (horizontal dashed
lines), the Fermi energy ($E=E_{F}$), as well as edge state features (red
arrows). Indeed, the WTe2/HOPG spectra resemble closely those previously
reported for monolayer WTe2 grown on bilayer graphene, [23, 24] indicating
that heterostructures with graphitic substrates are only weakly hybridized.
The additional spectral features (black arrows), however, only observed in the
WTe2/NbSe2 heterostructure, need to be attributed the presence of a NbSe2
partial DOS (reference spectrum in the left-hand insert to Fig. 2(d)),
confirming the hybridization picture.

As a result of the strong hybridization in WTe2/NbSe2, we further observe a
substantial weakening of the topological edge state signature, compared to our
observations on WTe2/HOPG and WTe2/BLG[23, 24, 26, 27]. This is illustrated in
Figs. 2(e)-2(g) by plotting spatial profiles of the integrated LDOS within the
gap (-50 meV and +20 meV) away from the edge position. We extract roughly
consistent exponential decay lengths of $\sim 1$ nm for both heterostructures.
However, the ratio of the edge state LDOS to that in the bulk is much lower
for WTe2/NbSe2 ($\simeq 1.8$), compared to WTe2/HOPG ($\simeq 10$), given the
large residual LDOS in the bulk gap. Both edge state decay length and LDOS
ratio agree well with our tight-binding calculations of the hybrid electronic
structure for WTe2/NbSe2 as shown in Fig. 2(g).

In Fig. 3, we plot the calculated superconducting LDOS for the same inter-
layer hopping (0.15 eV) as used for both the band structure calculation in
Fig. 2(d) and the LDOS profile in Fig. 2(g). We extract $\Delta=0.58$ meV on
the top-most tellurium sublattice, which the STM is expected to be most
sensitive to, in remarkable agreement with the experiment ($\Delta_{\rm
WTe_{2}}(E_{F})=(0.57\pm 0.02)$ meV), as extracted from our three-band fits.

The magnetic field stability of superconducting pairing in the WTe2 monolayer
bulk is investigated in Fig. 3(b), where we plot the measured superconducting
LDOS for magnetic fields up to $B=2.3$ T, applied perpendicular to the sample
plane ($c$-axis).

Figure 3: (Color online) Magnetic field dependence of the WTe2/NbSe2
superconducting local density of states (LDOS). (a) Calculated LDOS at 500 mK
for an interlayer coupling of 0.15 eV with (red) and without (blue) energy
broadening (0.15 meV) included. (b) Magnetic field dependence of the measured
superconducting LDOS up to $B=2.3$ T, with magnetic field applied
perpendicular to the heterostructure ($c$-axis).

###  II.4 Spatial Profile at the Monolayer Edge

In Fig. 4, we investigate the spatial evolution of the superconducting LDOS
across the WTe2 monolayer edge. An atomic resolution STM image of a clean WTe2
edge is shown in Fig. 4(a), indicating position and direction of the STM
height profile. The spatial dependence of the superconducting energy gap,
measured at 500 mK, is shown Fig. 4(b), alongside the superconducting partial
density of states weight $\tilde{N_{i}}(E,E_{F})$ of the WTe2/NbSe2
heterostructure (Fig. 4(c)). The latter has been extracted self-consistently,
using Eqns. (1) and (9). We observe a pronounced crossover in the partial
density of states at ($x_{0}\simeq 5.3$ nm), coinciding with the edge
position, beyond which the WTe2 DOS drops to zero as the STM tip leaves the
monolayer crystal. In the monolayer bulk, we find that WTe2 dominates the
total DOS with $(63\pm 5)$%, with a contribution of $(37\pm 5)$% by the NbSe2
substrate partial DOS. Interestingly, a more pronounced enhancement in the
superconducting partial DOS at the edge appears absent, at least when compared
to the normal-state DOS in Fig. 2(f). This is likely due to the vanishing
total LDOS at the edge at $E-E_{F}\simeq 0$ seen in the normal state (compare
red curves in Figs. 2(c) and 2(d), possibly due to the presence of an
interaction driven pseudogap [47, 48, 23, 49].

Figure 4: (Color online) Multi-band superconductivity at the WTe2 monolayer
edge. (a) STM-height profile measured across a clean edge of a WTe2 monolayer
crystal. The insert shows the corresponding STM topograph indicating position
and direction of the line profile (red arrow). (b) A series of tunneling
spectra measured $T=500$ mK for points across the edge. (c) The partial
density of states weight $\tilde{N_{i}}(E_{F})$, extracted from our three-band
analysis of (b), shows a clear crossover at the edge $x_{0}\simeq 5.3$ nm
(dashed line) beyond which the WTe2 partial density of states vanishes. (d)
The second derivative of the tunnel current close to the coherence peaks shows
clear signatures of three-band superconductivity. (e) Extracted spatial
profile of the induced order parameter $\Delta_{\rm WTe_{2}}(x)$ (blue),
compared to a single-band BCS/Dynes model (black). Solid red lines are fits to
Eq. (3).

A further confirmation of three-band superconductivity is given in Fig. 4(d)
where we plot the spatial profile of the second derivative of the tunneling
current for energies close to the coherence peak. We observe a sharp
transition from two-band to three-band signatures as soon as the probe tip
crosses the monolayer edge (dashed line). Arrows indicate the gap energies for
the WTe2 induced gap (blue) and the intrinsic NbSe2 small gap (red). The NbSe2
large gap at higher energy is not seen here, but is indicated in Figs. 8(a)
and 8(b) of the Appendix.

A spatial profile of the extracted order parameter $\Delta_{\rm WTe_{2}}(x)$
is shown in Fig. 4(e), in which we observe spatial oscillations with a period
of $\simeq 2$ nm. Oscillations of comparable period are also observed in the
order parameter extracted from a simple single-band BCS model, as well as are
visible in the energy-integrated normal-state LDOS (Fig. 2(f)). This suggests
that these arise from Friedel-like oscillations in the local density of states
due to scattering of 2D bulk states at the WTe2 edge. Indeed, we can fit the
extracted order parameter with a simple empirical model,

$\Delta(x)=\frac{A\sin{[2k_{F}(x-x_{0})]}}{(x-x_{0})}+B\exp\left(\frac{x-x_{0}}{\xi}\right)+C$
(3)

in which the first term reflects the oscillatory behaviour in the local order
parameter due to quasiparticle-interference in the 2D bulk. The second term
accounts for a residual enhancement of the exponentially decaying 1D edge
state, with decay length $\xi=(1.1\pm 0.2)$ nm comparable to our measurements
of the normal state LDOS (Fig. 2(g)). The simple model thus simultaneously
confirms the edge position ($x_{0}=5.3$ nm) and the Fermi wave vector[50]
$k_{\rm F}\simeq 1.8$ nm-1, and the edge state decay length. Upon crossing the
edge, $\Delta_{\rm WTe_{2}}(E_{F})$ increases to approach the smaller of the
two renormalized order parameters of NbSe2 ($\Delta_{S}(E_{F})=0.68\pm 0.08$
meV) as the three-band model emulates the two-band superconductivity of the
substrate. This transition coincides with a vanishing WTe2 partial DOS, which
confirms that the greyed-out data points in Fig. 4(e) do not contribute any
significant spectral weight to the overall measured density of states.

##  III Conclusions

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].

## Acknowledgements

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.

## Appendix A: 1T’-WTe2 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).

Figure 5: (Color online) Height estimates for monolayer WTe2 islands on
different substrates. (a)-(b) Large-scale STM topography images, showing a
number of 1T’-WTe2 islands, respectively, on HOPG (a) and on NbSe2 (b)
substrates. (c)-(d) Corresponding histograms of the $z$-height information
extracted. Fits to a normal distribution (red lines) allows to estimate the
monolayer height $(1.2\pm 0.1)$ nm (WTe2/HOPG) and $(0.9\pm 0.1)$ nm
(WTe2/NbSe2), respectively.

## Appendix B: Electronic Structure and Superconductivity of NbSe2

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].

Table 1: Multi-band superconductivity and Fermi-surface anisotropy. Summary of measured NbSe2 order parameters published in literature. Model |  Measured Gap (meV) | Reference | Experiment
---|---|---|---
|  $\Delta_{S}=0.50$ $\Delta_{L}=1.05$ | [39] | STS
Two-band (BCS) |  $\Delta_{S}=0.73$ $\Delta_{L}=1.26$ | [38] |  Specific Heat
|  $\Delta_{S}=0.85$ $\Delta_{L}=1.50$ | [57] |  Specific Heat
|  $\Delta_{\rm min}=0.65$ $\Delta_{\rm max}=1.62$ | [57] |  Specific Heat
Single-band (anisotropic) |  $\Delta_{\rm min}=0.56$ $\Delta_{\rm max}=1.30$ | [58] |  Penetration length
|  $\Delta_{\rm min}=0.62$ $\Delta_{\rm max}=1.30$ | [56] |  Electron transport
|  $\Delta_{S}=0$ $\Delta_{L}=1.3$ | [41] | STS
Two-band McMillan |  $\Delta_{S}=0-0.3$ $\Delta_{L}=1.26$ | [44] |  Electron transport
|  $\Delta_{S}=0.45$ $\Delta_{L}=1.10$ | this work | STS

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].

Figure 6: (Color online) Tight-binding calculation of the low-energy NbSe2
band structure. The black and red lines show bands arising from the
$d_{3z^{2}-r^{2}}$ orbitals, centered on the two Nb atoms within the unit
cell.

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.

## Appendix C: Comparison of Different Models for the Superconducting Density
of States

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.

Figure 7: (Color online) Quenching of charge density wave (CDW) order on the
NbSe2 surface post-epitaxy. (a,b) STM images of the pristine NbSe2 surface
after cleaving in UHV (a), and following epitaxy of a sub-monolayer coverage
of WTe2 (b). (c,d) Corresponding FFTs show that clear signatures of the 3Q CDW
wave vector (c) cannot be discerned post-epitaxy (d). (e,f) Corresponding
measured local density of states, measured before (e) and after (f) epitaxy,
showing a slight reduction and rounding of the superconducting energy gap.

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,

$dI/dV\propto\int_{-\infty}^{+\infty}\frac{df(E-eV)}{dE}N(E)dE.$ (4)

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.

$N(E)=\operatorname{Re}\left[\frac{(E-i\gamma)}{\sqrt{(E-i\gamma)^{2}-\Delta^{2}}}\right]$
(5)

Figure 8: (Color online) High-resolution STS spectra of the NbSe2 and WTe2
superconducting gaps at 500 mK, comparing different models. (a)
Superconducting energy gap of NbSe2. Solid lines are fits, comparing a single-
band BCS/Dynes model (green line), a single anisotropic band model (cyan
line), a two-band BCS/Dynes model (magenta line), and a two-band McMillan
model (solid blue line). (b), (c) corresponding second derivative of the
tunneling current ($d^{2}I/dV^{2}$) and detail of the coherence peaks. (d)-(f)
Superconducting energy gap of WTe2, comparing a two-band McMillan model (solid
blue line) and a three-band McMillan model (solid red line). (e)-(f) same as
(b), (c). We observe that a two-band McMillan model is insufficient to
describe induced superconductivity in WTe2/NbSe2. All spectra shown have been
spatially averaged, symmetrised about zero bias, and normalised with respect
to the normal state conductance. Vertical arrows and dashed lines indicate the
renormalised/induced gaps, as determined from the two-band and three-band
McMillan models, respectively. Across all models, we find Dynes parameters
ranging from $0.02-0.1$ meV.

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,

$N(E)=\sum\limits_{i}N_{i}(E)$ (6)

The partial densities of states can further be expressed as

$N_{i}(E)=\tilde{N_{i}}(E_{F})\int_{-\infty}^{\infty}\frac{d\theta}{2\pi}\operatorname{Re}\left[\frac{\lvert{E-i\gamma_{i}}\rvert}{\sqrt{(E-i\gamma_{i})^{2}-\Delta_{i}(E,\theta)^{2}}}\right]$
(7)

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,

$\Delta(A,\theta)=\Delta_{0}(A\ \text{cos}(6\theta)+(1-A))$ (8)

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)).

## Appendix D: The McMillan Model

Table 2: Consistency across two-band and three-band McMillan models, when describing NbSe2 substrate and WTe2/NbSe2 heterostructures, respectively. Band $i$ | Model |  $\Delta_{i}^{0}$ (meV) |  $\Delta_{i}({E_{F}})$ (meV) | $\frac{N_{S}}{N_{L}}$ | $\frac{N_{\rm WTe_{2}}}{N_{L}}$ | $\tilde{N}_{i}(E_{F})$
---|---|---|---|---|---|---
$i=L$ | 3 band | $1.13\pm 0.05$ | $0.87\pm 0.05$ |  |  | $0.002\pm 0.001$
(Nb $K$-point) | 2 band | $1.11\pm 0.05$ | $1.00\pm 0.05$ |  |  |
$i$=S | 3 band | $0.43\pm 0.08$ | $0.68\pm 0.08$ | $0.36\pm 0.03$ |  | $0.30\pm 0.08$
(Nb $\Gamma$-point) | 2 band | $0.45\pm 0.09$ | $0.73\pm 0.09$ | $0.36\pm 0.01$ |  |
$i$=3 | 3 band | 0 | $0.58\pm 0.02$ |  | $0.19\pm 0.01$ | $0.69\pm 0.06$
(WTe2) | 2 band |  |  |  |  |

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

$\Delta_{i}(E)=\frac{\Delta_{i}^{0}+\Gamma_{ij}\Delta_{j}(E)/\sqrt{\Delta_{j}(E)^{2}-(E-i\gamma_{j})^{2}}}{1+\Gamma_{ij}/\sqrt{\Delta_{j}(E)^{2}-(E-i\gamma_{j})^{2}}}$
(9)

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.

Figure 9: (Color online) Energy-dependent order parameters of NbSe2 and
monolayer 1T’-WTe2. (a), (b) Real (a) and imaginary (b) parts of the
renormalised NbSe2 order parameters, extracted from a two-band McMillan model.
(c), (d) Order parameters of 1T’-WTe2/NbSe2, extracted from our three-band
McMillan model. $\Delta_{\rm WTe_{2}}$ denotes the induced order parameter.

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.

## Appendix F: Mean Field Theoretical Modelling of the Hybrid Bandstructure

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,

$\displaystyle H$ $\displaystyle=$ $\displaystyle
H^{0}_{N}+H_{SC-N}+H^{0}_{W}+H_{W-N}$ (10)

The individual terms of the Hamiltonian are given by,

$\displaystyle H^{N}_{0}$ $\displaystyle=$
$\displaystyle\sum_{iljl^{\prime}\sigma}t^{ll^{\prime}}_{ij}c^{{\dagger}}_{il\sigma}c^{{\dagger}}_{jl^{\prime}\sigma}+h.c$
$\displaystyle H^{N}_{SC}$ $\displaystyle=$
$\displaystyle\sum_{il}\Delta^{N}_{il}c^{{\dagger}}_{il\uparrow}c^{{\dagger}}_{il\downarrow}+h.c$
$\displaystyle H_{W-N}$ $\displaystyle=$ $\displaystyle
t_{\perp}\sum_{ij\sigma}c^{{\dagger}}_{i2\sigma}d_{j1\sigma}+h.c$
$\displaystyle H^{W}_{0}$ $\displaystyle=$ $\displaystyle\sum_{\mu,\nu
ij\sigma,\sigma^{\prime}}t^{ij}_{\mu,\nu,\sigma,\sigma^{\prime}}d^{{\dagger}}_{i\mu,\sigma}d_{j\nu\sigma^{\prime}}+h.c$

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 px 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,

$\displaystyle d_{i_{x}k_{y}\mu\sigma}$ $\displaystyle=$
$\displaystyle\sum_{n}(u^{n}_{i_{x}k_{y}\mu\sigma}\gamma_{n\sigma}-\sigma
v^{n\star}_{i_{x}k_{y}\mu\sigma}\gamma^{\dagger}_{n\sigma})$ (11)
$\displaystyle d^{\dagger}_{i_{x}k_{y}\mu\sigma}$ $\displaystyle=$
$\displaystyle\sum_{n}(u^{n\star}_{i_{x}k_{y}\mu\sigma}\gamma_{n\sigma}^{\dagger}-\sigma
v^{n}_{i_{x}k_{y}\mu\sigma}\gamma_{n\sigma})$ (12)

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,

$\displaystyle
n_{i_{x}\mu\sigma}=\frac{1}{N_{k_{y}}}\sum_{k_{y},n}|u^{n}_{i_{x}k_{y}\mu\sigma}|^{2}f(E_{n})$
(13)

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,

$\displaystyle\Delta^{s}_{i_{x}j_{x}\mu}=\frac{1}{N_{k_{y}}}\sum_{n,k_{y}=0}^{2\pi}[u^{n}_{i_{x}k_{y}\mu\uparrow}v^{n\star}_{j_{x}k_{y}\mu\downarrow}+u^{n}_{j_{x}k_{y}\mu\uparrow}v^{n\star}_{i_{x}k_{y}\mu\downarrow}]f(E_{n})$

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.

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