# Gate-controlled superconducting proximity effect of
superconductor/ferromagnet van der Waals heterostructures

G. A. Bobkov  Moscow Institute of Physics and Technology, Dolgoprudny, 141700
Moscow region, Russia  K. A. Bokai  St. Petersburg State University, 7/9
Universitetskaya nab., 199034 St. Petersburg, Russia  Moscow Institute of
Physics and Technology, Dolgoprudny, 141700 Moscow region, Russia  M. M.
Otrokov  Instituto de Nanociencia y Materiales de Aragon (INMA), CSIC-
Universidad de Zaragoza, Zaragoza 50009, Spain  A. M. Bobkov  Moscow Institute
of Physics and Technology, Dolgoprudny, 141700 Moscow region, Russia  I.V.
Bobkova  Moscow Institute of Physics and Technology, Dolgoprudny, 141700
Moscow region, Russia  National Research University Higher School of
Economics, 101000 Moscow, Russia

###### Abstract

The discovery of 2D materials opens up unprecedented opportunities to design
new materials with specified properties. In many cases, the design guiding
principle is based on one or another proximity effect, i.e. the nanoscale-
penetration of electronic correlations from one material to another. In a few
layer van der Waals (vdW) heterostructures the proximity regions occupy the
entire system. Here we demonstrate that the physics of magnetic and
superconducting proximity effects in 2D superconductor/ferromagnet vdW
heterostructures is determined by the effects of interface hybridization of
the electronic spectra of both materials. The degree of hybridization can be
adjusted by gating, which makes it possible to achieve a high degree of
controllability of the proximity effect. In particular, we show that this
allows for electrical switching of superconductivity in such structures on and
off, as well as for control of the amplitude and sign of the Zeeman splitting
of superconducting spectra.

Introduction. — It it well-known that in superconductor/ferromagnet (S/F)
heterostructures the exchange field of the F layer suppresses the
superconductivity in the S layer near the S/F interface via the proximity
effect[1, 2]. The mechanism of the suppression is twofold. First, the F
induces triplet correlations in the S at the expense of singlet ones thus
suppressing the singlet superconducting order parameter. Second, if the F is
metallic the Cooper pairs can penetrate into it and be destroyed there by the
ferromagnetic exchange field. In a thin-film S/F bilayer, where the
thicknesses of the S and F layers, $d_{S}$ and $d_{F}$, are small with respect
to the superconducting and ferromagnetic coherence lengths $\xi_{S}$ and
$\xi_{F}$, respectively, the suppressing action of these two factors can be
described by modelling the bilayer by a homogeneous superconducting film with
a reduced effective order parameter
$\Delta_{eff}=\nu_{S}d_{S}\Delta/(\nu_{S}d_{S}+\nu_{F}d_{F})$ in an effective
reduced exchange field $h_{eff}=\nu_{F}d_{F}h/(\nu_{S}d_{S}+\nu_{F}d_{F})$.
Here $\Delta$ is the superconducting order parameter of the isolated S layer
and $h$ is the exchange field of the isolated ferromagnet, $\nu_{S(F)}$ is the
density of states at the Fermi level in the S (F) layer [3]. With increasing
of the effective exchange field the superconductivity is monotonically
suppressed [4]. The behavior of the local density of states (LDOS) in the
effective exchange field $h_{eff}$ is also well-studied and has a typical BCS-
like shape with Zeeman splitting of the coherence peaks [5]. The splitting of
the peaks is $2h_{eff}$.

All physics discussed above is related to bilayers, which are thin with
respect to the coherence lengths, but still contain large number of atomic
layers, such that the properties of the S and F layers are close to the
properties of the corresponding bulk materials and effects of an interface
hybridization of electronic spectra [6] or quantum-size effects [7, 8, 9] are
not important. However, the advent of 2D materials has provided unprecedented
opportunities for novel heterostructures in the form of van der Waals stacks,
laterally stitched 2D layers and more complex architectures[10, 11, 12].
Nowadays a lot of van der Waals materials, which retain their superconducting
[13, 14, 15, 16, 17] and ferromagnetic [18] properties up to the monolayer
thickness are being discovered, thus enabling exploration of the proximity
effects in the S/F heterostructures in the truly 2D limit.

Figure 1: (a) Model S/F bilayer with square atomic lattice. $t_{SF}$ is the
hopping element between the S and F layers. (b) $\Delta(h)$ for two different
values of $\mu_{F}$ calculated for the model bilayer from panel (a). The red
curve corresponds to the case of strong hybridization of electronic spectra of
S and F layers, and the blue curve illustrates the case of weak hybridization.
$t_{S}=70\Delta_{0}$, $t_{F}=1.25t_{S}$, $\mu_{S}=23\Delta_{0}$,
$t_{SF}=1.5\Delta_{0}$, $T=0.35\Delta_{0}$. $\Delta_{0}$ is the
superconducting order parameter of the isolated S layer at the same
temperature. (c) Atomic structure of the $\mathrm{NbSe_{2}/VSe_{2}}$ bilayer.
Figure 2: All the results are obtained in the model of the square lattice
bilayer. (a) $h_{eff}(\mu_{F})$ extracted from the Zeeman splitting of the
DOS. In the grey regions we are not able to extract reliable data. Insert: DOS
as a function of quasiparticle energy. Blue and red curves represent the DOS
in the S and the F layers, respectively. The Zeeman splitting of the DOS
$2h_{eff}$ is shown. F.E.G. are finite energy gaps appearing as a result of
the S and F layers spectra hybridization. (b) $\Delta(\mu_{F})$ for different
values of the exchange field $h$ of the F layer. (c) Spectral function of the
S/F bilayer. Quasiparticle spectra $\varepsilon(\xi)$ for spin-up (yellow) and
spin-down (red) quasiparticles are seen. Left picture corresponds to
$\mu_{F}=18.8\Delta_{0}$ and for the right picture $\mu_{F}=28.3\Delta_{0}$.
Other parameters are the same as in Fig. 1.

Works on proximity effects in van der Waals S/F heterostructures are already
actively appearing, and interest in such heterostructures is growing. In
particular, in a F/$\mathrm{NbSe_{2}}$/F heterostructure, the interaction
between the Ising superconductivity in $\mathrm{NbSe_{2}}$ and magnetism was
predicted to control the magnetic ordering of the heterostructure [19], and a
giant anisotropy of magnetoresistance was observed in Ising
superconductor/ferromagnetic insulator junctions [20]. The interplay between
spin-orbit coupling, ferromagnetism and superconductivity in
$\mathrm{NbSe_{2}/CrBr_{3}}$ heterostructures was studied [21]. Infinite
magnetoresistance and nonreciprocal charge transport in
$\mathrm{NbSe_{2}/CrSBr}$ superconducting spin valves was investigated [22],
reemergent superconductivity in $\mathrm{NbSe_{2}/CrCl_{3}}$ structures was
found [23]. Topological superconductivity was observed in van der Waals
heterostructures combining a two-dimensional ferromagnet with a
$\mathrm{NbSe_{2}}$ superconductor [24]. Also, Josephson effect in S/F/S van
der Waals heterostructures was investigated [25, 26].

2D materials possess unique features, which allow for high degree of
controllability of superconductivity [27, 28] and ferromagnetism [29, 30] by
gating. Here we predict that 2D S/F heterostructures also allow for high
controllability of the proximity effects. By considering a simplest possible
S/F bilayer composed of a 2D ferromagnet and a 2D superconductor we
demonstrate that the proximity influence of the F layer on the
superconductivity is strongly sensitive to the difference between the filling
factors of the S and F layers. This sensitivity is due to the fact that in
such 2D heterostructures the particular value of $h_{eff}$, which is induced
in the S layer, and the particular degree of a Cooper pair destruction due to
the leakage in the F are determined by the degree of hybridization of
electronic spectra of the individual materials. The degree of hybridization
can be adjusted by changing the relative filling factors by gating one of the
materials. Thus the strength of the superconductivity suppression can be
controlled by gating. In particular, the triplet superconductivity and the
Zeeman splitting of the DOS can be switched on/off by gating and nonmonotonic
dependencies of the supercondicting order parameter on the gating potential
can be obtained. We begin by considering the basic physics in the framework of
a simplified tight-binding model and then demonstrate the same physics in the
heterostructure composed of real van der Waals materials.

Basic physics: bilayer in the framework of a tight-binding hamiltonian on a
square lattice. — System under consideration is shown in Fig. 1(a). It
consists of two 2D layers: a 2D F and a 2D S. The system is modelled by the
following tight-binding Hamiltonian on a square lattice:

$\displaystyle\hat{H}=\sum\limits_{i,\alpha,\beta}\hat{c}^{\dagger}_{i,\alpha}\left(\begin{matrix}0&0\\\
0&(\bm{h}\bm{\sigma})_{\alpha,\beta}\\\
\end{matrix}\right)\hat{c}_{i,\beta}-~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle-\sum\limits_{i,\sigma}\hat{c}^{\dagger}_{i,\sigma}\left(\begin{matrix}\mu_{S}&0\\\
0&\mu_{F}\\\
\end{matrix}\right)\hat{c}_{i,\sigma}+\sum\limits_{i}\left[\hat{c}_{i,\uparrow}\left(\begin{matrix}\Delta&0\\\
0&0\\\ \end{matrix}\right)\hat{c}_{i,\downarrow}+H.c.\right]-$
$\displaystyle-\sum\limits_{<ij>,\sigma}\hat{c}^{\dagger}_{i,\sigma}\left(\begin{matrix}t_{S}&0\\\
0&t_{F}\\\
\end{matrix}\right)\hat{c}_{j,\sigma}-\sum\limits_{i,\sigma}\hat{c}^{\dagger}_{i}\left(\begin{matrix}0&t_{SF}\\\
t_{SF}&0\\\ \end{matrix}\right)\hat{c}_{i,\sigma}~{}~{}~{}~{}~{}~{}~{}$ (1)

Here $\hat{c}_{i,\sigma}=(c_{i,\sigma}^{S},c_{i,\sigma}^{F})^{T}$ is a vector
composed of annihilation operators for electrons belonging to the S and F
layers at site $i$ in the plane of each layer and for spin
$\sigma=\uparrow,\downarrow$. $\mu_{S,F}$ are chemical potentials of the S and
F layers, respectively, counted from the bottom of the corresponding
conduction band. We assume only nearest-neighbor hopping and $t_{S,F}$ is the
nearest-neighbor hopping element in the planes of the S and F layers. $\langle
ij\rangle$ means summation over nearest neighbors. $t_{SF}$ is the hopping
element between the S and F layers. $\bm{h}$ is the exchange field of the
ferromagnet. $\Delta$ is the superconducting order parameter in the
superconductor, which is to be calculated self-consistently as
$\Delta=\lambda\langle c_{i,\downarrow}^{S}c_{i,\uparrow}^{S}\rangle$, where
$\lambda$ is the pairing constant. The superconducting order parameter and the
LDOS are calculated via the Green’s functions methods, see Supplemental
material for details of the method [31].

Fig. 1(b) represents the dependence of the superconducting order parameter on
the exchange field $h=|\bm{h}|$ of the F layer for two different values of
$\mu_{F}$ and the same $\mu_{S}$. Blue curve in Fig. 1(b) corresponds to the
case when $\mu_{S}$ and $\mu_{F}$ differ strongly. This dependence $\Delta(h)$
is just a suppression of the superconductivity by the proximity to the F
layer, which is very similar to the classical result obtained for thin-film 3D
S/F bilayers for not very low temperature $T$ [4]. Turning to the red curve in
Fig. 1(b), which corresponds to closer values of $\mu_{F}$ and $\mu_{S}$, we
see that $\Delta(h)$ is qualitatively different. At first $\Delta$ is
completely suppressed by $h$ and than is restored practically to the value of
the isolated S layer. That is, in the considered case of the 2D S/F bilayer
the proximity effect can be nontrivial and the particular behavior depends
strongly on the difference between the chemical potentials of the layers.
Below we demonstrate that the physics of the proximity effect is determined by
the hybridization of the electronic spectra of the S and F layers. For the
blue curve the hybridization of the electronic spectra of S and F layers is
weak, and for the red curve it is strong. The degree of the hybridization is
controlled by the difference between the chemical potentials.

In principle, we can apply to the 2D bilayer our intuition developed by
studying the proximity effect in conventional 3D thin-film S/F bilayers. As it
was mentioned in the introduction, the proximity influence of the F layer on
superconductivity is described by an effective exchange field $h_{eff}$
induced in the S layer and by the destruction of the Cooper pairs in the F
layer. $h_{eff}$ can be extracted from the Zeeman splitting of the DOS. In the
considered 2D S/F bilayer the DOS also has typical BCS-shape with the Zeeman
splitting. An example is shown in the insert to Fig. 2(a). The effective
exchange field determined from the Zeeman splitting of the DOS is plotted in
Fig. 2(a) as a function of $\mu_{F}$. It is seen that it very strongly and
non-trivially depends on the chemical potential. Not only the amplitude but
also the sign of $h_{eff}$ can be adjusted by gating, what can be of interest
for spintronics applications. In the grey regions in Fig. 2(a) it is difficult
to correctly extract $h_{eff}$ from the DOS, see below. The dependence of the
superconducting order parameter $\Delta$ on $\mu_{F}$ is demonstrated in Fig.
2(b). We can see that $\Delta(\mu_{F})$ is fully correlated with
$h_{eff}(\mu_{F})$: the larger the absolute value of $h_{eff}$ the stronger
the superconductivity is suppressed. However, it is worth noting that not only
$h_{eff}$ determines the suppression of $\Delta$. The “leakage” of Cooper
pairs into the F layer and their subsequent destruction there also
contributes. We get back to this point below upon discussing the
$\mathrm{NbSe_{2}/VSe_{2}}$ bilayer structure.

Now we discuss physical mechanism of the high and non-trivial sensitivity of
the proximity effect, presented in Figs. 2(a)-(b), to the chemical potential.
The proximity effect in the considered 2D S/F bilayers is governed by the
hybridization of the electronic spectra of the S and F layers. The electronic
spectra $\varepsilon(\xi)$ of the S/F bilayer corresponding to the parameters
of the left black point at the green curve of Fig. 2(b) are shown in the left
picture of Fig. 2(c). In the considered model the electronic spectra depend on
the momentum only via $\xi=-2(\cos k_{x}a+\cos k_{y}a)$, where $a$ is a
lattice constant. For this reason in Fig. 2(c) the energy is shown as a
function of $\xi$ instead of more convenient for first-principle calculations
representation as a function of the momentum along certain directions in the
Brillouin zone (BZ). Two spin-split electronic branches originating from the F
layer and two parabolic BCS-like electronic branches coming from the S layer
are clearly seen. The S-branches are also spin split and this is a signature
of the presence of $h_{eff}$ in the superconductor. The F and S branches are
hybridized in the form of anti-crossing at the points of intersection of the
branches belonging to the same spin. These anti-crossings result in the
finite-energy gaps, which are also seen in the DOS, see insert to Fig. 2(a).
For a chosen value of $\mu_{F}$ the F-branches and S-branches are far enough
from each other and, therefore, their hybridization is rather weak resulting
in the rather small value of the spin splitting of the S-branches and
$h_{eff}$. Upon increasing the value of $\mu_{F}$ we are moving to the right
along the green curve in Fig. 2(b). In this case, the F-branches of the
spectrum will move closer to the S-branches and the degree of their
hybridization grows. This results in the growth of the spin-splitting of the
S-branches and $h_{eff}$ and the stronger suppression of superconductivity.
When one of the F-branches comes as close as possible to the S-branches, the
hybridization is maximal, the spectra are strongly reconstructed and the
superconductivity is fully suppressed. In this region of $\mu_{F}$ the correct
determination of $h_{eff}$ is not possible due to the strong reconstruction of
the spectra.

Upon further increase of $\mu_{F}$ the spin-up F-branch already “passed
through the S-branches” and moved some distance away from them, as it is seen
from inspection of the right picture of Fig. 2(c). As a result the
hybridization becomes weaker, the absolute value of $h_{eff}$ decreases and
the superconductivity is restored. It is interesting that in this region of
$\mu_{F}$ $h_{eff}$ changes sign, as it is seen from Fig. 2(a) and also from
electronic spectra in the right picture of Fig. 2(c). The reason is that, in
fact, the F- and S-branches cannot “pass through each other” due to the nature
of the anti-crossing process. Namely, at certain minimal separation in $\xi$,
they abruptly switch around due to a strong hybridization, i.e., the leftmost
(rightmost) spin-up branch with a positive slope, which originally belonged to
F (S), now becomes the S-branch (F-branch). As a result, all the evolution of
the electronic spectra of the S/F bilayer upon varying of $\mu_{F}$ obeys the
rule: the order of alternating spin-up and spin-down branches must be
preserved. For this reason the sign of $h_{eff}$ is reversed. It is reversed
again upon further increase of $\mu_{F}$ when the second F-branch goes across
the S-branches.

Figure 3: $\mathrm{NbSe_{2}/VSe_{2}}$ bilayer. (a)-(b) Superconducting order
parameter as a function of gating potential $V$ of the F layer for (a) OOP and
(b) IP orientations of $\mathrm{VSe_{2}}$ magnetization. $T=2$K, critical
temperature of the monolayer $\mathrm{NbSe_{2}}$ was chosen $T_{c}=2.7$K,
which is similar to the experimental value. (c)-(d) Spin-split Fermi surfaces
obtained from single-band tight-binding hamiltonians of the
$\mathrm{NbSe_{2}}$ (solid) and $\mathrm{VSe_{2}}$ (dashed) monolayers with no
gating potential in $\mathrm{VSe_{2}}$ (c) and at $V=550$mV (d). Please note
that electronic structure of $\mathrm{VSe_{2}}$ contains an additional Fermi-
pocket around the $\Gamma$-point [31]. However, it is not fitted by our
single-band hamiltonian and is not shown here because it is far from the
$\mathrm{NbSe_{2}}$ Fermi-surfaces and thus is not hybridized with them.

$\mathrm{NbSe_{2}/VSe_{2}}$ bilayer structure.— Therefore, the high tunability
of the proximity effect in the 2D S/F bilayers is determined by the
hybridization of the electronic spectra of the materials composing the
bilayer. A fair question arises: to what extent does this effect depend on the
specific model describing the bilayer and whether it will be preserved for
real van der Waals materials, the electronic spectra of which, as a rule, are
not at all described by the considered model? To answer this question we chose
a monolayer 1H-$\mathrm{NbSe_{2}}$ as superconductor [15, 16, 32, 33] and a
monolayer 1T-$\mathrm{VSe_{2}}$ as a ferromagnet [34, 35, 36, 37, 38] to
investigate the proximity effect in the S/F bilayer constructed from these
materials. The S and F materials were chosen due to their close in-plane
lattice constants. The atomic structure of the heterostructure is illustrated
in Fig.1(c). We examine the electronic structure of both the isolated
$\mathrm{NbSe_{2}}$ and $\mathrm{VSe_{2}}$ monolayers and the
$\mathrm{NbSe_{2}/VSe_{2}}$ bilayer using the density functional theory (DFT)
electronic structure calculations[31] and use their results to build an
approximate normal state single-band tight-binding hamiltonian of the
$\mathrm{NbSe_{2}/VSe_{2}}$ heterostructure [31].

The value of the interlayer hopping $t_{SF}$, extracted by fitting the DFT-
calculated electronic spectrum of the $\mathrm{NbSe_{2}/VSe_{2}}$
heterostructure, was estimated to be $t_{SF}=40$meV. It results in extremely
strong proximity effect, which fully suppresses superconductivity. We believe
that this value is an upper theoretical limit of the interlayer hopping for
ideally matching lattices of the materials in a heterobilayer. In real
heterostructures it should probably be several times smaller. Moreover, if
materials with lattice constants that are not close to each other are chosen
for the experimental implementation of a heterostructure, this lead to a
further decrease in the effective interlayer hopping parameter. For these
reasons we demonstrate in Fig. 3 the results for smaller $t_{SF}$.

Panels (a) and (b) of Fig. 3 represent superconducting order parameter for the
$\mathrm{NbSe_{2}/VSe_{2}}$ heterostructure as a function of the gating
potential $V$ applied to the F layer. First of all, it worth noting that
unlike our toy model, for the $\mathrm{NbSe_{2}/VSe_{2}}$ bilayer we observe a
strong anisotropy of the proximity effect depending on the orientation of the
$\mathrm{VSe_{2}}$ magnetization $\bm{M}$. Panel (a) corresponds to the out-
of-plane (OOP) orientation of $\bm{M}$ and panel (b) shows results for the in-
plane (IP) orientation. The overall suppression of superconductivity is much
stronger for the OOP orientation. This fact is well-known and originates from
the Ising-type anisotropy of the $\mathrm{NbSe_{2}}$ [15, 39, 40, 41, 16]. The
point is that for IP orientation the spectra of the S layer are protected from
spin splitting by the Ising spin-orbit effective field and $h_{eff}$ is not
induced. The superconductivity is only suppressed by the leakage of the Cooper
pairs into the F layer. At the same time for OOP orientation the both
suppressing factors work.

Experimental and numerical data on the magnetic anisotropy in
$\mathrm{VSe_{2}}$ monolayer are controversial, as both types of anisotropy
were reported in the literature [35, 37]. Nevertheless, we found that the
controllability of the proximity effect is achievable in the both cases, the
basic physics being the same as in our toy model. Gating of the F layer shifts
its electronic spectrum near the Fermi surface and thus varies degree of
hybridization between NbSe2 and VSe2. Since the full picture of electronic
spectra is rather complicated in this case [31], in Figs. 3(c)-(d) we
demonstrate the Fermi surfaces of the heterostructure with no gating $V=0$ and
at $V=550$mV, respectively. The plotted Fermi surfaces correspond to
$t_{SF}=0$ (separate S and F layers), but on the scale of these figures the
interlayer hopping gives only very weak modifications of the curves. In Fig.
3(c) it is seen that at $V=0$ the Fermi surfaces originating from the F ans S
layers are far from each other, the hybridization of the spectra is weak and
the suppression of superconductivity is also minimal. In Fig. 3(d) the Fermi
surfaces intersect, the hybridization has the strongest value and the
suppression of superconductivity is maximal.

Conclusions.—In this work we unveil the microscopic physical mechanism of the
magnetic proximity effect, that is the suppression of the superconductivity by
the exchange field of the adjacent ferromagnet in 2D S/F heterostructures and
predict that it is determined by the degree of hybridization of electronic
spectra of the individual materials. The degree of hybridization can be
adjusted by changing the relative filling factors by gating one of the
materials allowing for high controllability of the magnetic proximity effect
by gating. We illustrate the underlying physics of the processes governing the
proximity effect using a minimal tight-binding hamiltonian model on a square
lattice, and then demonstrate the existence of the same effects in
heterostructures based on van der Waals materials: a monolayer
1H-$\mathrm{NbSe_{2}}$ as superconductor and a monolayer 1T-$\mathrm{VSe_{2}}$
as a ferromagnet. It is demonstrated that the Zeeman splitting of the DOS can
be switched on/off and reversed by gating and nonmonotonic dependencies of the
supercondicting order parameter on the gating potential are obtained.

###### Acknowledgements.

G.A.B. and I.V.B. acknowledge the support from Theoretical Physics and
Mathematics Advancement Foundation “BASIS” via the project No. 23-1-1-51-1.
The calculations performed for $\mathrm{NbSe_{2}/VSe_{2}}$ structures were
supported by the Russian Science Foundation via the RSF project No.
24-12-00152.

##  Appendix A Green’s functions approach to 2D S/F bilayer

Here we describe the Gor’kov Green’s function method, which we formulate and
use to calculate observables in the considered bilayer van der Waals
heterostructure. We begin with the following tight-binding Hamiltonian:

$\displaystyle\hat{H}=\sum\limits_{i,\alpha,\beta}\hat{c}^{\dagger}_{i,\alpha}\left(\begin{matrix}0&0\\\
0&(\bm{h}\bm{\sigma})_{\alpha,\beta}\\\
\end{matrix}\right)\hat{c}_{i,\beta}-\sum\limits_{i,\sigma}\hat{c}^{\dagger}_{i,\sigma}\left(\begin{matrix}\mu_{S}&0\\\
0&\mu_{F}\\\
\end{matrix}\right)\hat{c}_{i,\sigma}+~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle+\sum\limits_{i}\left[\hat{c}_{i,\uparrow}\left(\begin{matrix}\Delta&0\\\
0&0\\\
\end{matrix}\right)\hat{c}_{i,\downarrow}+H.c.\right]-\sum\limits_{<ij>,\sigma}\hat{c}^{\dagger}_{i,\sigma}\left(\begin{matrix}t_{S}^{ij,\sigma}&0\\\
0&t_{F}^{ij,\sigma}\\\
\end{matrix}\right)\hat{c}_{j,\sigma}-\sum\limits_{i,\sigma}\hat{c}^{\dagger}_{i}\left(\begin{matrix}0&t_{SF}\\\
t_{SF}&0\\\ \end{matrix}\right)\hat{c}_{i,\sigma}~{}~{}~{}~{}~{}~{}~{}$ (2)

Here $\hat{c}_{i,\sigma}=(c_{i,\sigma}^{S},c_{i,\sigma}^{F})^{T}$ is a vector
composed of annihilation operators for electrons belonging to the S and F
layers at site $i$ in plane of each layer and for spin
$\sigma=\uparrow,\downarrow$. $\mu_{S,F}$ are chemical potentials of the S and
F layers, respectively. To correctly describe real materials we assume not
only nearest-neighbor hopping, but arbitrary hops $i\to j$ with complex
hopping elements $t^{ij,\sigma}_{S},t^{ij,\sigma}_{F}$ in the S and the F
layers, respectively. In general, the hopping elements can depend on spin and
their phases account for the spin-orbit interaction. $\langle ij\rangle$ means
summation over all involved neighbors. $t_{SF}$ is the hopping element between
the S and F layers. $\bm{h}$ is the exchange field of the F layer. $\Delta$ is
the superconducting order parameter in the S layer, which is to be calculated
self-consistently as $\Delta=\lambda\langle
c_{i,\downarrow}^{S}c_{i,\uparrow}^{S}\rangle$, where $\lambda$ is the pairing
constant.

The Matsubara Green’s function in the two-layer formalism is $8\times 8$
matrix in the direct product of spin, particle-hole and layer spaces.
Introducing the two-layer Nambu spinor
$\check{\psi}_{\bm{i}}=(\hat{c}_{{\bm{i}},\uparrow}^{S},\hat{c}_{\bm{i},\downarrow}^{S},\hat{c}_{\bm{i},\uparrow}^{F},\hat{c}_{\bm{i},\downarrow}^{F},\hat{c}_{\bm{i},\uparrow}^{S\dagger},\hat{c}_{\bm{i},\downarrow}^{S\dagger},\hat{c}_{\bm{i},\uparrow}^{F\dagger},\hat{c}_{\bm{i},\downarrow}^{F\dagger})^{T}$
we define the Green’s function as follows:

$\displaystyle\check{G}_{\bm{i}\bm{j}}(\tau_{1},\tau_{2})=-\langle
T_{\tau}\check{\psi}_{\bm{i}}(\tau_{1})\check{\psi}_{\bm{j}}^{\dagger}(\tau_{2})\rangle,$
(3)

where $\langle T_{\tau}...\rangle$ means imaginary time-ordered thermal
averaging. Introducing Pauli matrices in spin, particle-hole and layer spaces:
$\sigma_{k}$, $\tau_{k}$ and $\rho_{k}$ ($k=0,x,y,z$) and operator $\hat{j}$
as

$\displaystyle\hat{j}\check{c}_{\bm{i}}=\sum\limits_{<ij>,\sigma}\left(\begin{matrix}t_{S}^{ij,\sigma}&0\\\
0&t_{F}^{ij,\sigma}\\\ \end{matrix}\right)\hat{c}_{j,\sigma}$ (4)

one can obtain the Gor’kov equation for Green’s function in terms of the
Matsubara frequencies $\omega_{m}=\pi T(2m+1)$. The derivation is similar to
that described in Ref. 42. The resulting Gor’kov equation takes the form:

$\displaystyle
G_{\bm{i}}^{-1}\check{G}_{\bm{i}\bm{j}}(\omega_{m})=\delta_{\bm{i}\bm{j}},$
(5) $\displaystyle
G_{\bm{i}}^{-1}=\tau_{z}\left(\hat{j}+\left(\begin{matrix}\mu_{S}&0\\\
0&\mu_{F}\\\
\end{matrix}\right)-\frac{\rho_{0}+\rho_{z}}{2}\check{\Delta}_{\bm{i}}i\sigma_{y}-\bm{h}\check{\bm{\sigma}}\delta_{i_{x},0}\frac{\rho_{0}-\rho_{z}}{2}+\left(\begin{matrix}0&t_{SF}\\\
t_{SF}&0\\\ \end{matrix}\right)\right)+i\omega_{m}.$ (6)

where
$\check{\Delta}_{\bm{i}}=\Delta_{\bm{i}}\tau_{+}+\Delta_{\bm{i}}^{*}\tau_{-}$
with $\tau_{\pm}=(\tau_{x}\pm i\tau_{y})/2$ and
$\check{\bm{\sigma}}=\bm{\sigma}(1+\tau_{z})/2+\bm{\sigma}^{*}(1-\tau_{z})/2$
is the quasiparticle spin operator. Further we consider the Green’s function
in the mixed representation:

$\displaystyle\check{G}(\bm{R},\bm{p})=F(\check{G}_{\bm{i}\bm{j}})=\int
d^{2}re^{-i\bm{p}(\bm{i}-\bm{j})}\check{G}_{\bm{i}\bm{j}},$ (7)

where $\bm{R}=(\bm{i}+\bm{j})/2$ and the integration is over $\bm{i}-\bm{j}$.
Now we define the following transformed Green’s function to simplify further
calculations and to present the Gor’kov equation in a more common form:

$\displaystyle\check{\tilde{G}}(\bm{R},\bm{p})=\left(\begin{array}[]{cc}1&0\\\
0&-i\sigma_{y}\end{array}\right)_{\tau}\check{G}(\bm{R},\bm{p})\left(\begin{array}[]{cc}1&0\\\
0&-i\sigma_{y}\end{array}\right)_{\tau},$ (12)

where subscript $\tau$ means that the explicit matrix structure corresponds to
the particle-hole space. Then we obtain:

$\displaystyle\left(i\omega_{m}\tau_{z}+\left(\begin{matrix}\mu_{S}+\hat{\xi}_{S}(\bm{p})&0\\\
0&\mu_{F}+\hat{\xi}_{F}(\bm{p})\\\
\end{matrix}\right)_{\rho}+\left(\begin{matrix}\tau_{z}\check{\Delta}&0\\\
0&-\bm{h}\bm{\sigma}\tau_{z}\\\
\end{matrix}\right)_{\rho}+t_{SF}\rho_{x}\right)\check{\tilde{G}}(\bm{p})=1$
(13)

where $\hat{\xi}_{S,F}$ are diagonal matrices in spin space, describing normal
state electron spectra of the S and F layers, with elements
$\xi_{S,F}^{\sigma}(\bm{p})=\sum\limits_{<ij>,\sigma}t_{S,F}^{ij,\sigma}e^{-i\bm{p}(\bm{i}-\bm{j})}$.
Here we omit the spatial coordinate $\bm{R}$ in the Green’s function, the
order parameter and the exchange field due to the translational invariance
along the S/F interface.

The superconducting order parameter of the S/F bilayer is calculated from the
self-consistency equation

$\displaystyle\Delta=\lambda
T\sum\limits_{\omega_{m}}\int\frac{d^{2}\bm{p}}{(2\pi)^{2}}\frac{{\rm
Tr}[\check{\tilde{G}}\sigma_{0}(\tau_{x}-i\tau_{y})(\rho_{0}+\rho_{z})]}{8},$
(14)

Electron spectral function can be calculated as

$\displaystyle A_{S,F}(\bm{p},\bm{\varepsilon},s)=\frac{{\rm
Tr}[\check{\tilde{G}}^{R}(\sigma_{0}+s\sigma_{z})(\tau_{0}+\tau_{z})(\rho_{0}\pm\rho_{z})]}{8},$
(15)

where $s=\pm 1$ and $\varepsilon$ mean spin and energy of the electron,
respectively. $\check{\tilde{G}}^{R}$ can be obtained from $\check{\tilde{G}}$
with substituting $i\omega_{m}\to\varepsilon+i\delta$, $\delta$ is a positive
infinitesimal imaginary part. Local density of states (LDOS) is the momentum-
integrated spectral function:

$\displaystyle
N_{S,F}(\bm{\varepsilon},s)=\int\frac{d^{2}\bm{p}}{(2\pi)^{2}}A_{S,F}(\bm{p},\bm{\varepsilon},s)$
(16)

##  Appendix B DFT calculations of electronic structure of
$\mathrm{NbSe_{2}/VSe_{2}}$ heterostructure

Figure 4: (a)-(b) DFT-calculated low-energy band structures of the (a)
1H-$\mathrm{NbSe_{2}}$ monolayer in a nonsuperconducting state, (b)
1T-$\mathrm{VSe_{2}}$ monolayer in the ferromagnetic state, and (c)
$\mathrm{NbSe_{2}/VSe_{2}}$ heterostructure. Parts of the spectra, which
originate from the $\mathrm{NbSe_{2}}$($\mathrm{VSe_{2}}$) layer and are not
significantly affected by the hybridization are shown in blue (red). Parts of
the spectra significantly modified by the hybridization effects are shown in
pink. (d)-(e) Magnified views of the anticrossings, which appeared as a result
of the NbSe2/VSe2 hybridization.

This section describes the electronic structure calculations of the individual
1H-NbSe2 and 1T-VSe2 layers as well as the ${\rm NbSe_{2}/VSe_{2}}$
heterostructure using the density functional theory (DFT). All calculations
were performed using OpenMX package [43],[44] within the Perdew-Burke-
Ernzerhof (PBE) generalized gradient approximation (GGA) [45]. Electronic
spectra were calculated taking spin-orbit coupling into account. An effective
on-site Hubbard $U$ term of 1.0 eV was used to treat the V 3d electrons within
the DFT+$U$ approach [46, 47] At the specified $U$ value VSe2 is
ferromagnetic, in agreement with the available experiments. DFT-D3 method [48]
was employed to take into account vdW interactions during the structure
optimizations.

To simulate both the individual monolayers and bilayer heterostructure the
supercell approach was used. The thickness of the vacuum layer was larger than
20 ${\rm\AA}$ to minimize interactions between periodic images. The energy
cutoff of 3000 eV was used for the numerical integrations. 2D Brillouin Zone
was sampled by the $\Gamma$-centered $14\times 14\times 1$ k-point grid. To
match 1H-NbSe2 and 1T-VSe2 in the $(1\times 1)$ cell the lattice parameter of
the heterostructure was set to $a=3.40{\rm\AA}$, which is the average of the
experimental lattice parameters of the single layers, $a_{\rm
NbSe_{2}}=3.45{\rm\AA}$ and $a_{\rm VSe_{2}}=3.35{\rm\AA}$ [36]. The positions
of atoms were optimized until the ionic forces were smaller than 0.005
eV/${\rm\AA}$. The electronic convergence criterion was set to $3\cdot
10^{-6}$eV.

For the calculations in the framework of the Green’s functions approach,
described in the previous section, we need the electron dispersions
$\xi_{S,F}^{\sigma}$ of separate $\mathrm{NbSe_{2}}$ and $\mathrm{VSe_{2}}$
monolayers and the interlayer hopping element $t_{SF}$. To obtain
$\xi_{S,F}^{\sigma}$ the DFT-calculated low-energy electron spectra
$\varepsilon_{\sigma}$ were fitted by a single-band tight-binding model in a
triangular lattice, taking into account the complex hopping elements between
first to sixth neighbours $t_{0}e^{i\varphi_{0}}-t_{5}e^{i\varphi_{5}}$, where
$t_{i}$ is the corresponding hopping energy and $\varphi_{i}$ accounts for the
spin-orbit coupling. Only $\varphi_{0}\neq 0$ for the case of the Ising-type
spin-orbit coupling, which occurs in the considered materials. The fitting
formula takes the form[19]:

$\displaystyle\xi_{S,F}^{\sigma}=2t_{0}[\cos(2\alpha-\varphi_{0}\sigma)+2\cos\beta\cos(\alpha+\varphi_{0}\sigma)]+2t_{1}[\cos(3\alpha-\beta)+~{}~{}~{}~{}~{}$
$\displaystyle+\cos(2\beta)+\cos(3\alpha+\beta)]+2t_{2}[\cos(2\alpha+2\beta)+\cos(2\alpha-2\beta)+\cos(4\alpha)]+~{}~{}~{}$
$\displaystyle+2t_{3}[\cos(5\alpha+\beta)+\cos(4\alpha-2\beta)+\cos(4\alpha+2\beta)+\cos(\alpha-3\beta)+\cos(5\alpha-\beta)+~{}$
(17)
$\displaystyle\cos(\alpha+3\beta)]+2t_{4}[\cos(3\alpha-3\beta)+\cos(3\alpha+3\beta)+\cos(6\alpha)]+2t_{5}[\cos(6\alpha-2\beta)+$
$\displaystyle+\cos(4\beta)+\cos(6\alpha+2\beta)]-\mu+h\sigma~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

where $\alpha=p_{x}a/2$, $\beta=\sqrt{3}p_{y}a/2$, $\mu$ is the chemical
potential, and $h$ is the exchange splitting, $\sigma=\pm 1$ means the spin of
the electron.

Fig. 4 shows the DFT-calculated electron spectra of the single layers and that
of the heterostructure. The parameters extracted from the fits of the data
presented in Fig. 4 are listed in Table 1. For the ${\rm NbSe_{2}}$ they are
in good agreement with the data reported earlier [19]. Anticrossings at the
”intersections” of the ${\rm NbSe_{2}}$ and ${\rm VSe_{2}}$ spectra allow us
to estimate the interlayer hopping element $t_{SF}$ because the gap due to an
anticrossing is equal to $2t_{SF}$. We obtain that $t_{SF}$ can reach up to
$40$ meV, which for the studied here case of the ideally aligned and matched
${\rm NbSe_{2}}$ and ${\rm VSe_{2}}$ lattices should be considered as an upper
limit estimate.

|  $\mu$, meV |  $t_{0}$, meV |  $t_{1}$, meV |  $t_{2}$, meV |  $t_{3}$, meV |  $t_{4}$, meV |  $t_{5}$, meV | $\varphi_{0}$ |  $h$, meV
---|---|---|---|---|---|---|---|---|---
${\rm NbSe_{2}}$ | 31.4 | 17.5 | 99.8 | -7.8 | -3.6 | -14.3 | 0.5 | 1.48 | 0
${\rm VSe_{2}}$ | -18.8 | -22.2 | 93.4 | -65.4 | 17.3 | -23.6 | 8.1 | 0.2 | 401
Table 1: Parameters of the one-band tight-binding model fitted to the DFT-
calculated electron spectra.

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