# Spin supercurrent in superconductor/ferromagnet van-der-Waals
heterostructures

G. A. Bobkov  Moscow Institute of Physics and Technology, Dolgoprudny, 141700
Moscow region, Russia  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

We study dissipationless spin transport induced by a charge supercurrent in a
monolayer van der Waals superconductor under the applied magnetic field and in
a bilayer superconductor/ferromagnet (S/F) heterostructure with no external
field. It is shown that in both cases a combined action of the Ising-type
spin-orbit coupling and the Zeeman field results in appearance of a nonunitary
superconducting triplet correlations with nonzero average Cooper pair spin,
which carry spin current in the presence of a condensate motion. Properties of
this dissipationless spin current are investigated. In particular, it is shown
that it manifests a rectification effect. In addition, in S/F heterostructures
the amplitude and the sign of the spin current are controlled by gating.

##  I Introduction

In spintronics, spin current plays an essential role to transfer the
information associated with the spin degrees of freedom and to manipulate the
magnetization. Spin currents can be carried by itinerant electrons or spin
waves. In the first case they can be generated via various methods including
the spin-polarized current injection from the ferromagnet[1, 2, 3], spin
battery[4, 5, 6, 7, 8], spin Hall effect[9, 10, 11, 12, 13, 14, 15], by
applying an electric field to non-centrosymmetric systems with spin-orbit
coupling[16, 17]. The spin wave spin currents are carried by excitations of
localized magnetic moments through exchange interactions and can be
implemented even in insulating ferromagnets [18, 19, 20, 21, 22].

The spin currents mentioned above are accompanied by energy dissipation either
through the Joule heat or through damping in the magnetic system. On the other
hand, it would be desirable to be able to generate and transport non-
dissipative or weakly dissipative spin currents over long distances. This
opportunity is provided by superconducting spintronics [23, 24]. One of the
directions of the superconducting spintronics is the study of possible ways to
generate and control spin currents, the carriers of which are Cooper pairs.
Although the spin degrees of freedom usually do not manifest themselves in
singlet superconductors, the superconducting spin current carried by triplet
Cooper pairs can be finite in $\mathrm{{}^{3}He}$ [25], triplet
superconductors [26, 27], noncentrosymmetric superconductors [28, 29, 30], as
well as in superconductor/ferromagnet (S/F) hybrids and Josephson junctions
via ferromagnetic weak links [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42,
43, 44, 45]. In particular, it was shown that in two-dimensional
superconductors with Rashba spin-orbit interaction, the generation of
dissipationless bulk spin current by charge supercurrent is possible [29]. The
charge supercurrent induces an averaged spin polarization of the pairs via the
Edelstein effect [46, 47, 48], which is linear in the supercurrent and the
spin current is just the charge current times the spin polarization.
Consequently, spin supercurrent is proportional to the charge supercurrent
squared. A very large number of important works are devoted to spin currents
in S/F heterostructures [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45]. However, they mostly study spin supercurrents, which exist only in
restricted regions of space, for example weak links of S/F/S Josephson
junctions or regions of the order of the superconducting coherence length near
the interfaces of S/F and F/S/F structures. The main interest of such spin
supercurrents is their capability to induce spin transfer torques and
magnetization dynamics of the ferromagnetic elements [49, 50, 51, 52, 53, 54,
55, 56].

From the other hand, quantum materials with a large controllable charge
current-induced spin polarization and capability to transfer spin currents
over long distances are promising for next-generation all-electrical
spintronic science and technology [57, 58]. Van der Waals (vdW) metals with
high spin-orbit coupling have attracted significant attention for an efficient
charge to spin conversion process [59, 60, 61, 62, 63, 64, 65, 66, 67, 68]. In
particular, an electrical generation of spin polarization in
$\mathrm{NbSe_{2}}$ up to room temperature has been demonstrated [69].
However, in that study the charge-spin conversion signal was only observed
with a higher bias current above the superconducting critical current,
limiting the observation of signal only to the non-superconducting state of
$\mathrm{NbSe_{2}}$.

Here we predict that charge current-induced spin supercurrents can occur in
superconducting $\mathrm{NbSe_{2}}$ monolayers under the applied magnetic
field or in bilayer $\mathrm{NbSe_{2}}$/F heterostructures, where F is a vdW
ferromagnet consisting of one or a few layers. We unveil the physical role of
the Ising-type spin-orbit coupling and the Zeeman field in the generation of
non-unitary triplet Cooper pairs possessing nonzero averaged spin, which
become spin current carriers in the presence of the condensate motion. It is
also obtained that the pair spin current has a component, which manifests a
rectification effect unlike the charge supercurrent that generates it. For the
case of S/F bilayers it is shown that the degree and sign of the spin
polarization can be controlled by gating.

The paper is organized as follows. In Sec. II we present analytical and
numerical results for the spin currrent in $\mathrm{NbSe_{2}}$ monolayer under
the applied external magnetic field. Sec. II.1 is devoted to formulation of
the model, and in Sec. II.2 we describe the Green’s function approach, which
is used further for analytical calculations. In Sec. II.3 we present
analytical results for the averaged spin of non-unitary Cooper pairs, which
arise as a result of partial conversion of singlet pairs under the influence
of the Ising spin-orbit coupling and the applied field. Analytical results for
the spin carried by these pairs are presented in Sec. II.4. Sec. II.5 is
devoted to the numerical results for the spin current, which go beyond our
analytical approximation and exhibit important additional physics including
rectification effect. In Sec. III we study spin current in a S/F
heterostructure on a specific example of $\mathrm{NbSe_{2}/VSe_{2}}$
heterostructure. Sec. III.1 is devoted to the description of the corresponding
model and in Sec. III.2 the results for the spin current in this
heterostructure are presented, including demonstration of spin current control
using gating potential. Sec. IV contains conclusions from our work.

##  II $\mathrm{NbSe_{2}}$ under the applied magnetic field

###  II.1 Model

Figure 1: Top view of the $\mathrm{NbSe_{2}}$ monolayer atomic structure. The
electronic hops involved in the tight-binding hamiltonian (1) are shown by
black arrows. Bold black arrows correspond to the nearest-neighbor hopping.
The phases of these hopping elements accounting for the Ising-type spin-orbit
coupling are also shown. It is assumed that a Zeeman field $\bm{h}$ and a
supercurrent $\bm{j}_{e}$ are applied to the system.

We consider a $\mathrm{NbSe_{2}}$ monolayer under the applied magnetic field.
The atomic structure of the monolayer is illustrated in Fig. 1. The electronic
structure of the system is modelled by the following tight-binding
Hamiltonian:

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

Here $c_{\bm{i},\sigma}$ is an electron annihilation operator at site $\bm{i}$
in plane of the $\mathrm{NbSe_{2}}$ layer and for spin
$\sigma=\uparrow,\downarrow$. $\mu_{S}$ is a chemical potential of the
$\mathrm{NbSe_{2}}$ layer. To correctly describe electronic structure of the
$\mathrm{NbSe_{2}}$ monolayer we assume not only nearest-neighbor hopping, but
several hops $\bm{i}\to\bm{j}$ with complex hopping elements
$t^{\bm{i}\bm{j},\sigma}_{S}$. $\langle\bm{i}\bm{j}\rangle$ means summation
over all involved neighbors. To find the values of these hopping elements the
DFT-calculated low-energy electron spectra were fitted by a single-band tight-
binding model in a triangular lattice [70], taking into account the complex
hopping elements between the first to the sixth neighbours
$t_{S}^{0}e^{i\varphi_{0}},...,t_{S}^{5}e^{i\varphi_{5}}$, where $t_{S}^{i}$
is the corresponding hopping energy and $\varphi_{i}$ accounts for the spin-
orbit coupling. Only $\varphi_{0}\equiv\varphi\neq 0$ for the case of the
Ising-type spin-orbit coupling. All the involved hops are schematically shown
in Fig. 1 by black arrows. The parameters extracted from the fits are listed
in Table 1. They are in good agreement with the data reported earlier [71].

$\mu_{S}$ | $t_{S}^{0}$ | $t_{S}^{1}$ | $t_{S}^{2}$ | $t_{S}^{3}$ | $t_{S}^{4}$ | $t_{S}^{5}$ | $\varphi$
---|---|---|---|---|---|---|---
31.4 | 17.5 | 99.8 | -7.8 | -3.6 | -14.3 | 0.5 | 1.48
Table 1:  Parameters of the one-band tight-binding model fitted to the DFT-
calculated electron spectra of $\mathrm{NbSe_{2}}$. All values of the hopping
amplitudes and other energies are given in meV.

$\bm{h}$ is the Zeeman field produced by the applied magnetic field. The
orbital effect of the applied magnetic field is small in the monolayer limit
and can be neglected. $\Delta$ is the superconducting order parameter, which
is to be calculated self-consistently as $\Delta=\lambda\langle
c_{\bm{i},\downarrow}c_{\bm{i},\uparrow}\rangle$, where $\lambda$ is the
pairing constant. Then we apply a supercurrent $\bm{j}_{e}$ along the
$\mathrm{NbSe_{2}}$ layer and study the spin supercurrent generated by this
charge supercurrent.

###  II.2 Green’s functions technique

For analytical calculation of the amplitudes of triplet Cooper pair
correlations, average spin of the Cooper pair and the spin current we use the
Green’s functions technique. The Matsubara Green’s function is $4\times 4$
matrix in the direct product of spin and particle-hole spaces. Introducing the
Nambu spinor
$\check{\psi}_{\bm{i}}=(c_{{\bm{i}},\uparrow},c_{\bm{i},\downarrow},c_{\bm{i},\uparrow}^{\dagger},c_{\bm{i},\downarrow}^{\dagger},)^{T}$
we define the Green’s function as follows:

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

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

$\displaystyle\hat{j}c_{\bm{i},\sigma}=\sum\limits_{<\bm{i}\bm{j}>}t_{S}^{ij,\sigma}c_{\bm{j},\sigma}$
(3)

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. 72. The resulting Gor’kov equation takes the form:

$\displaystyle
G_{\bm{i}}^{-1}\check{G}_{\bm{i}\bm{j}}(\omega_{m})=\delta_{\bm{i}\bm{j}},$
(4) $\displaystyle
G_{\bm{i}}^{-1}=\tau_{z}\left(\hat{j}+\mu_{S}-\check{\Delta}_{\bm{i}}i\sigma_{y}-\bm{h}\check{\bm{\sigma}}\right)+i\omega_{m}.$
(5)

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:

$\displaystyle\check{G}(\bm{R},\bm{p})=F(\check{G}_{\bm{i}\bm{j}})=\int
d^{2}re^{-i\bm{p}(\bm{i}-\bm{j})}\check{G}_{\bm{i}\bm{j}},$ (6)

where $\bm{R}=(\bm{i}+\bm{j})/2$ and the integration is over $\bm{i}-\bm{j}$.
The presence of a homogeneous supercurrent flowing in the $\mathrm{NbSe_{2}}$
layer is taken into account via the phase gradient of the superconducting
order parameter $\check{\Delta}(\bm{R})=\Delta\tau_{x}e^{i\bm{q}\bm{R}}$,
where $\Delta$ is the absolute value of the order parameter, which does not
depend on coordinates, and $\bm{q}$ is the superconducting phase gradient
(total momentum of the Cooper pair) aligned with the supercurrent
$\bm{j}_{e}$. Making the gauge transformation

$\displaystyle\check{G}(\bm{R},\bm{p})=e^{i\bm{q}\bm{R}\tau_{z}/2}\check{G}_{q}(\bm{p})e^{-i\bm{q}\bm{R}\tau_{z}/2},$
(7)

we turn to the Green’s function $\check{G}_{q}(\bm{p})$, which does not depend
on the spatial coordinates. We also define the following transformed Green’s
function to simplify further calculations and to present the Gor’kov equation
in a more common form:

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

where subscript $\tau$ means that the explicit matrix structure corresponds to
the particle-hole space. Then we obtain the following Gor’kov equation for
$\check{\tilde{G}}_{q}(\bm{p})$:

$\displaystyle\left(i\omega_{m}\tau_{z}+i\tau_{y}\check{\Delta}-\bm{h}\bm{\sigma}\tau_{z}\right)\check{\tilde{G}}_{q}(\bm{p})+\Xi(\bm{p})=1,$
(13)

where introducing the explicit structure of the Green’s function
$\check{\tilde{G}}_{q}(\bm{p})$ in the particle-hole space

$\displaystyle\check{\tilde{G}}_{q}(\bm{p})=\left(\begin{array}[]{cc}G_{q}(\bm{p})&F_{q}(\bm{p})\\\
\bar{F}_{q}(\bm{p})&\bar{G}_{q}(\bm{p})\end{array}\right)_{\tau},$ (16)

we can write the term $\Xi(\bm{p})$ as follows:

$\displaystyle\Xi=\left(\begin{array}[]{cc}\hat{\xi}_{S}(\bm{p})G_{q}(\bm{p})&\hat{\xi}_{S}(\bm{p}+\bm{q}/2)F_{q}(\bm{p})\\\
\hat{\xi}_{S}(\bm{p}-\bm{q}/2)\bar{F}_{q}(\bm{p})&\hat{\xi}_{S}(\bm{p})\bar{G}_{q}(\bm{p})\end{array}\right)_{\tau}.$
(19)

Here $\hat{\xi}_{S}$ is a diagonal matrix in spin space, describing normal
state electron spectrum of the $\mathrm{NbSe_{2}}$, with elements
$\xi_{S}^{\sigma}(\bm{p})=\mu_{S}+\sum\limits_{<\bm{0}\bm{j}>,\sigma}t_{S}^{\bm{0}\bm{j},\sigma}e^{i\bm{p}\bm{j}}$.

Figure 2: (a) Fermi surface of the electronic system described by hamiltonian
(1) with parameters listed in Tab. 1. The valley index corresponding to each
of the contours around $K$-points $\nu=\pm 1$ is indicated. Red contours are
related to spin $\uparrow$ ($s=+1$), and blue contours correspond to spin
$\downarrow$ ($s=-1$). (b) DFT-calculated low-energy band structures of the
1H-$\mathrm{NbSe_{2}}$ monolayer. See Ref. 70 for parameters and details of
calculations.

The superconducting order parameter is calculated from the self-consistency
equation

$\displaystyle\Delta=\lambda
T\sum\limits_{\omega_{m}}\int\frac{d^{2}p}{(2\pi)^{2}}\frac{{\rm
Tr}[\check{\tilde{G}}_{q}\sigma_{0}(\tau_{x}-i\tau_{y})]}{4}.$ (20)

Further we work near the critical temperature and expand the Green’s function
$\check{\tilde{G}}_{q}(\bm{p})$ in powers of the superconducting order
parameter $\Delta$:

$\displaystyle
G_{q}(\bm{p})=G_{q}^{0}(\bm{p})+\Delta^{2}G_{q}^{2}(\bm{p})+...,$ (21)
$\displaystyle\bar{F}_{q}(\bm{p})=\Delta\bar{F}_{q}^{1}(\bm{p})+...~{}~{}~{}~{}~{}~{}$

Then the solution of Eq. (13) for the leading contribution
$\bar{F}_{q}^{1}(\bm{p})$ to the anomalous Green’s function takes the form:

$\displaystyle\bar{F}_{q}^{1}(\bm{p})=(-i\omega_{m}+\bm{h}\bm{\sigma}-\hat{\xi}_{S}(\bm{p}-\bm{q}/2))^{-1}\times$
(22)
$\displaystyle\times(i\omega_{m}-\bm{h}\bm{\sigma}-\hat{\xi}_{S}(\bm{p}))^{-1}.~{}~{}~{}~{}~{}~{}~{}~{}$

For the leading superconducting contribution to the normal Green’s function
one obtains:

$\displaystyle
G_{q}^{2}(\bm{p})=-(i\omega_{m}-\bm{h}\bm{\sigma}-\hat{\xi}_{S}(\bm{p}))^{-1}\times$
$\displaystyle(-i\omega_{m}+\bm{h}\bm{\sigma}-\hat{\xi}_{S}(\bm{p}-\bm{q}/2))^{-1}\times$
$\displaystyle\times(i\omega_{m}-\bm{h}\bm{\sigma}-\hat{\xi}_{S}(\bm{p}))^{-1}~{}~{}~{}~{}~{}~{}$
(23)

The electric $\bm{j}_{e}$ and spin $\bm{j}_{s}^{i}$ ($i=x,y,z$ means the
$i$-th component in spin space) currents can be calculated from the matrix
current $\hat{\bm{j}}$ with the following elements in spin space
($\alpha,\beta=\uparrow,\downarrow$):

$\displaystyle\bm{j}^{\alpha\beta}=T\sum_{\omega_{m}}\int\frac{d^{2}p}{(2\pi)^{2}}\bm{v}_{F}^{\alpha\beta}G_{q,\alpha\beta}(\bm{p})$
(24)

where

$\displaystyle\bm{v}_{F}^{\uparrow\uparrow,\downarrow\downarrow}(\bm{p})=\frac{d\xi_{S}^{\uparrow,\downarrow}}{d\bm{p}}$
(25)

and

$\displaystyle\bm{v}_{F}^{\uparrow\downarrow,\downarrow\uparrow}(\bm{p})=\frac{d}{d\bm{p}}(\sum\limits_{<\bm{0}\bm{j}>,\sigma}\frac{t_{S}^{\bm{0}\bm{j},\sigma}+t_{S}^{\bm{0}\bm{j},-\sigma}}{2}e^{i\bm{p}\bm{j}}).$
(26)

Then

$\displaystyle\bm{j}_{e}=\bm{j}^{\uparrow\uparrow}+\bm{j}^{\downarrow\downarrow},~{}~{}~{}\bm{j}_{s}^{i}={\rm
Tr}[\hat{\bm{j}}\sigma_{i}].$ (27)

The electric and spin current are normalized to the value of the electric
charge $e$ and the electron spin $\hbar/2$, respectively.

###  II.3 Spin polarization of Cooper pairs

In this subsection we present approximate analytical results for the spin
structure, which is acquired by Cooper pairs in the $\mathrm{NbSe_{2}}$
monolayer as a result of the simultaneous presence of the Ising spin-orbit
interaction and the applied magnetic field. The Fermi surface of the
$\mathrm{NbSe_{2}}$ monolayer is presented in Fig. 2(a). It consists of the
central contour surrounding the $\Gamma$-point and six contours around $K$ and
$K^{\prime}$-points, which are spin-split due to the Ising-type spin orbit
coupling [73]. In this section we assume that the characteristic energy value
of the spin splitting of the $K$-valleys $V_{SO}$ is much larger than any
other energy scales in our system, $h$ and $\Delta$. This assumption is in
good agreement with results of the DFT calculations for the
$\mathrm{NbSe_{2}}$ monolayer, where the Ising spin splitting of the order of
$100$meV was reported [73]. For our analytical consideration we also neglect
the spin splitting of the Fermi countours around the $\Gamma$-point because it
is much smaller that the spin splitting around $K$-points. Under such
assumptions we can integrate the anomalous Green’s function
$\bar{F}_{q}^{1}(\bm{p})$ over the absolute value of the electron momentum in
the vicinity of each of the Fermi contour (the so-called $\xi$-integration).
As a result we obtain the anomalous Green’s function

$\displaystyle\bar{f}_{q}^{1,\nu,s}(\bm{n})=-\frac{1}{i\pi}\int
d\xi\bar{F}_{q}^{1}(\bm{p}),$ (28)

which only depends on the direction $\bm{n}=\bm{p}/p$ on the Fermi surface,
but not on the absolute value of the electron momentum $p$. This is the same
procedure that is used to go to the quasiclassical approximation in the theory
of superconductivity[74]. However, here we do not derive any quasiclassical
equations for $\bar{f}_{q}^{1,\nu,s}(\bm{n})$, since our system is spatially
homogeneous. The $\xi$-integrated Green’s function is determined at each of
the Fermi surface contours around $K$-points, which are marked by the valley
index $\nu=\pm 1$ and spin index $s=\pm 1$ [see Fig. 2(a)]. To obtain the
$\xi$-integrated Green’s function at the countour corresponding to spin $s$
and valley $\nu$ we should take

$\displaystyle\xi_{S}^{s}(\bm{p})=\xi,~{}~{}~{}\xi_{S}^{-s}(\bm{p})=s\nu
V_{SO}.$ (29)

Also expanding the normal state electron dispersion up to the leading order
with respect to the condensate momentum $\bm{q}$

$\displaystyle\xi_{S}^{s}(\bm{p}-\bm{q}/2)=\xi_{S}^{s}(\bm{p})-{\bm{v}}_{F}^{s,\nu}\bm{q}/2,$
(30)

where ${\bm{v}}_{F}^{s,\nu}=\bm{v}_{F}^{ss}(\bm{p}=\bm{p}_{F}^{s,\nu})$ is the
Fermi velocity at the corresponding Fermi-surface contour, we obtain the
following expression for the $\xi$-integrated anomalous Green’s function:

$\displaystyle\bar{f}_{q}^{1,\nu,s}(\bm{n})={\rm
sgn}\omega_{m}\frac{-(\sigma_{0}+s\sigma_{z})/2+i((\bm{z}\times\bm{h})\bm{\sigma})\nu
V_{SO}^{-1}}{i\omega_{m}-s(\bm{z}\bm{h})-\bm{v}_{F}^{s,\nu}\bm{q}/4},$ (31)

The anomalous Green’s function can be always represented in the form

$\displaystyle\bar{f}_{q}^{1,\nu,s}(\bm{n})=f_{s}^{\nu,s}(\bm{n})+\bm{d}^{\nu,s}(\bm{n})\bm{\sigma},$
(32)

where $f_{s}^{\nu,s}$ accounts for singlet superconducting correlations and
$\bm{d}^{\nu,s}$ describes triplet correlations. The averaged spin of a Cooper
pair

$\displaystyle\bm{S}\propto i\hbar
T\sum\limits_{s,\nu,\omega_{m}}\int\frac{d\Omega}{2\pi}\bm{d}^{\nu,s*}\times\bm{d}^{\nu,s}.$
(33)

From Eqs. (31) and (32) we obtain

$\displaystyle\bm{d}^{\nu,s*}\times\bm{d}^{\nu,s}=\frac{-is\nu
V_{SO}^{-1}}{\omega_{m}^{2}+(s(\bm{z}\bm{h})-\bm{v}_{F}^{s,\nu}\bm{q}/4)^{2}}(\bm{z}\times(\bm{z}\times\bm{h})).$
(34)

It is worth noting that the non-unitary triplet pairing [25, 75] arises here
due to the fact that there are nonzero $\bm{z}$ and $\bm{z}\times\bm{h}$
components of the $\bm{d}^{\nu,s}$-vector with the $\pi/2$-shift between them.
This does not occur if one considers the Ising-type spin-orbit coupling in the
quasiclassical approximation, which implies small spin-orbit splitting of the
Fermi surface countours around the $K$-points [76]. In this consideration we
do not calculate the $\xi$-integrated anomalous Green’s function at the Fermi-
surface contour around the $\Gamma$-point since in the framework of our
approximation the non-unitary triplet is absent there due to the absence of
the Ising-type spin-orbit splitting.

The total average spin of the Cooper pair is determined by the integration of
$\bm{d}^{\nu,s*}\times\bm{d}^{\nu,s}$ over all the Brillouin zone, see Eq.
(33). However, the pairs formed by electrons belonging to outer and inner
(with respect to the corresponding $K$-points) Fermi contours contribute to
the spin supercurrent with different condensate velocities, see the next
subsection. For this reason it is useful to calculate the average spin of the
Cooper pairs integrated over outer and inner Fermi surface contours
separately. These quantities are denoted by $\bm{S}_{out}$ and $\bm{S}_{in}$
and take the following form:

$\displaystyle\bm{S}_{out(in)}\propto T\sum_{\omega_{m}}(\pm
N_{out(in)})\frac{(\bm{z}\times(\bm{z}\times\bm{h}))}{(\omega_{m}^{2}+(\bm{z}\bm{h})^{2})V_{SO}},$
(35)

where $N_{out(in)}$ is the normal state density of states at the outer (inner)
Fermi surface contours. The sign $\pm$ originates from the factor $s\nu$,
which is $+1(-1)$ for outer (inner) Fermi surface contours.

###  II.4 Analytical results for the spin current

As it follows from Eqs. (27), (24) and (23), the spin supercurrent is
determined by the $\xi$-integrated leading superconducting contribution to the
normal Green’s function:

$\displaystyle g_{q}^{2,\nu,s}(\bm{n})=-\frac{1}{i\pi}\int d\xi
G_{q}^{2}(\bm{p}),$ (36)

which after explicit $\xi$-integration of Eq. (23) takes the form:

$\displaystyle g_{q}^{2,\nu,s}(\bm{n})=-{\rm sgn}\omega_{m}\times$
$\displaystyle\frac{(\sigma_{0}+s\sigma_{z})+2i(\bm{z}\times(\bm{z}\times\bm{h}))\bm{\sigma}s\nu
V_{SO}^{-1}}{4(i\omega_{m}-s(\bm{z}\bm{h})-\bm{v}_{F}^{s,\nu}\bm{q}/4)^{2}}$
(37)

After performing the remaining integration over the directions at the Fermi
surface in Eq. (24) and making use of Eq. (27) we obtain the only nonzero in
spin space component of the spin current:

$\displaystyle j_{s}^{\bm{z}\times(\bm{z}\times\bm{h})}=\frac{\pi
T\bm{q}|z\times(\bm{z}\times\bm{h})|}{8V_{SO}}\times$
$\displaystyle\sum_{\omega_{m}}\frac{N_{out}v_{F,out}^{\uparrow\downarrow}v_{F,out}^{\uparrow\uparrow}-N_{in}v_{F,in}^{\uparrow\downarrow}v_{F,in}^{\downarrow\downarrow}}{(i\omega_{m}-(\bm{h}\bm{z}))^{3}}$
(38)

$\displaystyle
v_{F,out}^{\uparrow\uparrow(\downarrow\downarrow)}=|{\bm{v}}_{F}^{\uparrow(\downarrow),\nu=1(-1)}|$
(39)

$\displaystyle
v_{F,in}^{\uparrow\uparrow(\downarrow\downarrow)}=|{\bm{v}}_{F}^{\uparrow(\downarrow),\nu=-1(1)}|$
(40)

$\displaystyle
v_{F,out(in)}^{\uparrow\downarrow}=|\bm{v}_{F}^{\uparrow\downarrow}({\bm{p}=\bm{p}_{F}^{\uparrow,\nu=1(-1)}})|$
(41)

Comparing Eq. (38) to Eq. (35) we can conclude that it can be interpreted as a
sum of two different condensate flows of spinful Cooper pairs with spins
$\bm{S}_{out(in)}$ belonging to outer and inner Fermi surface contours. Both
flows are induced by the applied supercurrent with the condensate momentum
$\bm{q}$.

###  II.5 Numerical results for the spin current

The analytical results for the spin current presented in the previous section
are valid under the assumption $h/V_{SO}\ll 1$. In this limit the charge
supercurrent generates a spin supercurrent flowing in the same direction and
carrying spin directed along the vector $\bm{z}\times(\bm{z}\times\bm{h})$. It
means that if we apply the external magnetic field in plane of the
$\mathrm{NbSe_{2}}$ the spin current carries the spin component aligned with
$\bm{h}$. In this section we investigate the spin supercurrent numerically in
the framework of the Bogolubov-de Gennes approach beyond the assumption
$h/V_{SO}\ll 1$. As it follows from results presented in this section the
analytical expressions obtained under the assumption $h/V_{so}\ll 1$ work well
only at $h/V_{so}\lesssim 0.03$. Since superconductivity in a monolayer
$\mathrm{NbSe_{2}}$ is protected by the Ising-type spin-orbit coupling [77,
78, 79, 80, 81] and can survive up to rather strong in-plane field of the
order of $0.05-0.1~{}h/V_{so}$, such a numerical analysis is important.

Figure 3: Numerical results for the spin current. External applied magnetic
field is applied in plane of the S layer along the $x$-axis, that is
$\bm{h}=h_{x}\bm{x}$. (a) Amplitude and the direction of the black arrows show
the amplitude and the direction of the spin current
$\bm{j}_{s}^{\bm{z}\times(\bm{z}\times\bm{h})}\equiv\bm{j}_{s}^{x}$ for a
given direction of the condensate momentum $\bm{q}$ coinciding with the
direction of the charge supercurrent $\bm{j}_{e}$. (b) The same as in (a) but
for $\bm{j}_{s}^{z}$. (c) Maximal (with respect to the direction of $\bm{q}$)
value of the ratio $j_{s}/j_{e}$ for a given absolute value of the condensate
momentum $q$ close to $q_{c}$ as a function of $h_{x}$. Red and blue lines
correspond to two nonzero spin current components $j_{s}^{z}$ and $j_{s}^{x}$,
respectively. Larger values of $h_{x}$ fully suppress superconductivity.

We diagonalize the Hamiltonian (1) by the Bogoliubov transformation:

$\displaystyle
c_{\bm{i}\sigma}=\int\frac{d^{2}p}{(2\pi)^{2}}e^{i\bm{p}\bm{i}}\sum\limits_{n}u_{n\sigma}(\bm{p})\hat{b}_{n}+v^{*}_{n\sigma}(\bm{p})\hat{b}_{n}^{\dagger},$
(42)

where $\hat{b}_{n}^{\dagger}(\hat{b}_{n})$ are the creation (annihilation)
operators of Bogoliubov quasiparticles. Then the resulting Bogoliubov – de
Gennes equations take the form:

$\displaystyle\xi_{S}^{\sigma}(\bm{p}+\bm{q}/2)u_{n,\sigma}+\sigma\Delta
v_{n,-\sigma}+$ (43)
$\displaystyle+(\bm{h}\bm{\sigma})_{\sigma\alpha}u_{n,\alpha}$
$\displaystyle=\varepsilon_{n}u_{n,\sigma}$
$\displaystyle-\xi_{S}^{\sigma}(\bm{p}-\bm{q}/2)v_{n,\sigma}+\sigma\Delta^{*}u_{n,-\sigma}+$
(44) $\displaystyle+(\bm{h}\bm{\sigma}^{*})_{\sigma\alpha}v_{n,\alpha}$
$\displaystyle=-\varepsilon_{n}v_{n,\sigma},$

$\displaystyle\Delta=\lambda\sum\limits_{n}\int\frac{d^{2}p}{(2\pi)^{2}}(u_{n,\downarrow}v_{n,\uparrow}^{*}(1-f_{n})+u_{n,\uparrow}v_{n,\downarrow}^{*}f_{n}),$
(45)

where $f_{n}=1/[1+{\rm exp}(-\varepsilon_{n}/T)]$ is the Fermi distribution.

The matrix current can be calculated via the solutions of the Bogolubov-de
Gennes equation as follows:

$\displaystyle\bm{j}^{\alpha\beta}=\sum_{n}\int\frac{d^{2}p}{(2\pi)^{2}}\bm{v}_{F}^{\alpha\beta}\left(u_{n\alpha}u_{n\beta}^{*}f_{n}+v_{n\alpha}^{*}v_{n\beta}(1-f_{n})\right)$
(46)

The spin current component $\bm{j}_{s}^{\bm{z}\times(\bm{z}\times\bm{h})}$
calculated numerically in the framework of the Bogolubov-de Gennes approach is
demonstrated in Fig. 3(a). The direction and the absolute value of
$\bm{j}_{s}^{\bm{z}\times(\bm{z}\times\bm{h})}$ for a given direction of the
charge supercurrent $\bm{j}_{e}$ (or equivalently a given direction of the
condensate momentum $\bm{q}$), is shown by black arrows. It is seen that
$\bm{j}_{s}^{\bm{z}\times(\bm{z}\times\bm{h})}(\bm{q})$ has hexagonal
symmetry, which originated from the hexagonal symmetry of the Brillouin zone.
We can also see a slight deviation of the direction of the spin current
$\bm{j}_{s}^{\bm{z}\times(\bm{z}\times\bm{h})}$ flow from the direction of the
charge current $\bm{j}_{e}$, which is directed strictly radially. It is worth
noting that the perpendicular component of the spin supercurrent, which
appears in our numerical results, originates from the crystal structure of the
superconductor. It does not possess a finite chirality and is not related to
physical mechanisms of a spin Hall current generation, which would give a
vector structure $\bm{j}_{s}\propto[\bm{j}_{e}\times\bm{z}]$.

Our numerical calculation also demonstrates that the spin current also
transfers a non-zero spin projection along the $z$-axis $\bm{j}_{s}^{z}$. It
is shown in Fig. 3(b). The symmetry of this component is reduced to a three-
fold rotation axis. This is because this component is odd with respect to
$\bm{z}\to-\bm{z}$, what is equivalent to the reversal of the Ising spin
splitting. From Fig. 2(a) it is seen that accounting for the Ising spin
splitting of the Fermi contours around $K$-points reduces the hexagonal
rotational symmetry of the electronic structure to the three-fold rotation
axis. $\bm{j}_{s}^{z}$ has much stronger component perpendicular to the charge
current as compared to $\bm{j}_{s}^{\bm{z}\times(\bm{z}\times\bm{h})}$. Very
interesting property of $\bm{j}_{s}^{z}$ that it manifests a rectification
effect. $j_{s}^{z}$ conserves its sign under the sign reversal of the
condensate momentum, that is $j_{s}^{z}\to j_{s}^{z}$ at $\bm{q}\to-\bm{q}$.
The physical reason of this rectification effect is that the spin component of
the Cooper pair spin directed along the $z$-axis is zero in the absence of the
applied supercurrent. It is induced by the supercurrent, what leads to the
fact that $\bm{j}_{s}^{z}\propto q^{2}$ at small values of the condensate
momentum, as we have checked numerically.

In Fig. 3(c) we plot the maximal absolute value of the spin supercurrent (as
its ratio to the amplitude of the charge supercurrent producing it) in
dependence on the applied magnetic field $h_{x}$. At small $h_{x}$
$j_{s}^{\bm{z}\times(\bm{z}\times\bm{h})}\equiv j_{s}^{x}\propto h_{x}/V_{so}$
in agreement with our analytical results. In this region of small $h_{x}$
$j_{s}^{z}\propto(h_{x}/V_{so})^{2}$, what is beyond the accuracy of the
analytical approximation. However, from Fig. 3(c) we can see that the
expressions obtained under the assumption $h/V_{so}\ll 1$ work well only at
$h/V_{so}\lesssim 0.03$. Realistic magnetic fields achievable in laboratory
conditions correspond to $h/V_{so}\lesssim 10^{-2}$. That is, with good
accuracy they fall within the range of parameters where our analytical
description is valid. However, in Fig. 3 we present numerical results for a
much larger range of $h_{x}$ because effective exchange fields of this scale
can be achieved in S/F heterostructures, see Sec. III. Beyond the region
$h/V_{so}\ll 1$ the spin current component $j_{s}^{z}$, which grows
quadratically with $h_{x}$, becomes the most important contribution to the
spin current. The value of the ratio $j_{s}/j_{e}$ also depends on $q$.
Results presented in Fig. 3(c) are calculated at $q\approx q_{c}$, where
$q_{c}$ is the condensate momentum corresponding to the critical supercurrent.
Maximal possible values of the ratio $j_{s}/j_{e}$, which can be considered as
efficiency of the charge to spin current conversion in our system, can be
achived at $q\to q_{c}$ and according to our calculations can be estimated
$\sim 0.1$.

##  III $\mathrm{NbSe_{2}}/\mathrm{VSe_{2}}$ heterostructure

Here we present results for the superconducting spin current, which is
generated by the charge supercurrent in S/F heterostructures. As an example,
we consider a heterostructure consisting of a monolayer superconducting
$\mathrm{NbSe_{2}}$ and a monolayer ferromagnetic $\mathrm{VSe_{2}}$ [82, 83,
84, 85, 86]. The choice of specific materials is not the only possible one and
is due to the fact that the electronic spectra of a given heterostructure have
already been calculated earlier [70].

###  III.1 Model

Tight-binding single-band hamiltonian, which reasonably fits the electronic
spectrum of a given heterostructure in the vicinity of the Fermi-surface takes
the form [70]:

$\displaystyle\hat{H}=\sum\limits_{\bm{i},\alpha,\beta}\hat{c}^{\dagger}_{\bm{i},\alpha}\left(\begin{matrix}0&0\\\
0&(\bm{h}\bm{\sigma})_{\alpha,\beta}\\\
\end{matrix}\right)\hat{c}_{\bm{i},\beta}-\sum\limits_{\bm{i},\sigma}\hat{c}^{\dagger}_{\bm{i},\sigma}\left(\begin{matrix}\mu_{S}&0\\\
0&\mu_{F}\\\
\end{matrix}\right)\hat{c}_{\bm{i},\sigma}+~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle+\sum\limits_{\bm{i}}\left[\hat{c}_{\bm{i},\uparrow}\left(\begin{matrix}\Delta&0\\\
0&0\\\
\end{matrix}\right)\hat{c}_{\bm{i},\downarrow}+H.c.\right]-\sum\limits_{<\bm{i}\bm{j}>,\sigma}\hat{c}^{\dagger}_{\bm{i},\sigma}\left(\begin{matrix}t_{S}^{\bm{i}\bm{j},\sigma}&0\\\
0&t_{F}^{\bm{i}\bm{j},\sigma}\\\
\end{matrix}\right)\hat{c}_{\bm{j},\sigma}-\sum\limits_{\bm{i},\sigma}\hat{c}^{\dagger}_{\bm{i},\sigma}\left(\begin{matrix}0&t_{SF}\\\
t_{SF}&0\\\ \end{matrix}\right)\hat{c}_{\bm{i},\sigma},~{}~{}~{}~{}~{}~{}~{}$
(47)

where
$\hat{c}_{\bm{i},\sigma}=(c_{\bm{i},\sigma}^{S},c_{\bm{i},\sigma}^{F})^{T}$ is
a vector composed of annihilation operators for electrons belonging to the S
and F layers at site $\bm{i}$ in plane of each layer and for spin
$\sigma=\uparrow,\downarrow$.
$t^{\bm{i}\bm{j},\sigma}_{S},t^{\bm{i}\bm{j},\sigma}_{F}$ are complex hopping
elements in the S and the F layers, respectively. $\mu_{S,F}$ are chemical
potentials of the S and F layers, respectively. $t_{SF}$ is the hopping
element between the S and F layers. $\bm{h}$ is the exchange field of the F
layer. We assume that the magnetization of the $\mathrm{VSe_{2}}$ layer and,
correspondingly, $\bm{h}$ is in its plane (IP-configuration). The parameters
extracted from the fits of the DFT data [70] for $\mathrm{VSe_{2}}$ are listed
in Table 2. The interlayer hopping was estimated to be $t_{SF}=30$ meV from
the DFT spectra of the $\mathrm{NbSe_{2}}/\mathrm{VSe_{2}}$ heterostructure
[70].

$\mu_{F}$ | $t_{F}^{0}$ | $t_{F}^{1}$ | $t_{F}^{2}$ | $t_{F}^{3}$ | $t_{F}^{4}$ | $t_{F}^{5}$ | $\varphi_{F}$ | $h$
---|---|---|---|---|---|---|---|---
-18.8 | -22.2 | 93.4 | -65.4 | 17.3 | -23.6 | 8.1 | 0.2 | 401
Table 2: Parameters of the one-band tight-binding model fitted to the DFT-
calculated electron spectra of $\mathrm{VSe_{2}}$. All values of the hopping
amplitudes and other energies are given in meV.

We diagonalize Hamiltonian (1) by the Bogoliubov transformation:

$\displaystyle
c_{\bm{i}\sigma}^{\eta}=\int\frac{d^{2}p}{(2\pi)^{2}}e^{i\bm{p}\bm{i}}\sum\limits_{n}u_{n\sigma}^{\eta}(\bm{p})\hat{b}_{n}+v^{\eta*}_{n\sigma}(\bm{p})\hat{b}_{n}^{\dagger},$
(48)

where $\hat{b}_{n}^{\dagger}(\hat{b}_{n})$ are the creation (annihilation)
operators of the Bogoliubov quasiparticles. $\eta=S,F$ is the layer index.
Then the resulting Bogoliubov – de Gennes equations take the form:

$\displaystyle\xi_{\eta}^{\sigma}(\bm{p}+\bm{q}/2)u_{n,\sigma}^{\eta}+\sigma\Delta_{\eta}v^{\eta}_{n,-\sigma}-t_{SF}u_{n,\sigma}^{\bar{\eta}}+$
(49)
$\displaystyle+(\bm{h}_{\eta}\bm{\sigma})_{\sigma\alpha}u_{n,\alpha}^{\eta}$
$\displaystyle=\varepsilon_{n}u_{n,\sigma}^{\eta}$
$\displaystyle-\xi_{\eta}^{\sigma}(\bm{p}-\bm{q}/2)v_{n,\sigma}^{\eta}+\sigma\Delta_{\eta}^{*}u^{\eta}_{n,-\sigma}-t_{SF}v_{n,\sigma}^{\bar{\eta}}+$
(50)
$\displaystyle+(\bm{h}_{\eta}\bm{\sigma}^{*})_{\sigma\alpha}v_{n,\alpha}^{\eta}$
$\displaystyle=-\varepsilon_{n}v_{n,\sigma}^{\eta},$

where where $\bar{S},\bar{F}=F,S$, $\Delta_{\eta}=(\Delta,0)^{T}$,
$\bm{h}_{\eta}=(0,\bm{h})^{T}$ are the superconducting order parameter, which
is nonzero only in the S layer, and the exchange field, which exist only in
the F layer, respectively.

$\displaystyle\Delta_{S}=\lambda\sum\limits_{n}(u_{n,\downarrow}^{S}v_{n,\uparrow}^{S*}(1-f_{n})+u_{n,\uparrow}^{S}v_{n,\downarrow}^{S*}f_{n}).$
(51)

The matrix current can be calculated via the solutions of the Bogolubov-de
Gennes equation as follows:

$\displaystyle\bm{j}^{\alpha\beta}=\sum_{n,\eta}\int\frac{d^{2}\bm{p}}{(2\pi)^{2}}\bm{v}_{F}^{\alpha\beta}\left(u_{n\alpha}^{\eta}u_{n\beta}^{\eta*}f_{n}+v_{n\alpha}^{\eta*}v_{n\beta}^{\eta}(1-f_{n})\right)$
(52)

###  III.2 Gate-controllable spin current

Figure 4: Numerical results for the spin current in
$\mathrm{NbSe_{2}}/\mathrm{VSe_{2}}$ heterostructure. Magnetization of the
$\mathrm{VSe_{2}}$ is in plane of the $\mathrm{NbSe_{2}}/\mathrm{VSe_{2}}$
interface, along the $x$-axis. Amplitude and the direction of the black arrows
show the amplitude and the direction of the spin current for a given direction
of the condensate momentum $\bm{q}$, as in Fig. 3. The amplitude scale of the
arrows is indicated in the upper part of the figure. The left column
represents $\bm{j}_{s}^{x}$, and the right column corresponds to
$\bm{j}_{s}^{z}$. Each horizontal row demonstrates the results for a given
gating potential $V$.(a)-(b) $V=0$; (c)-(d) $V=540$mV and (e)-(f) $V=-260$mV.
$t_{SF}=30$meV for all the panels. Figure 5: Spin-split Fermi surfaces
obtained from single-band tight-binding hamiltonians of the
$\mathrm{NbSe_{2}}$ (solid) and $\mathrm{VSe_{2}}$ (dashed) monolayers at
$t_{SF}=0$ with no gating potential in $\mathrm{VSe_{2}}$ (a); gating
potential $V=540$ (b) and $V=-260$mV (c). Fermi surfaces belonging to the
$\mathrm{NbSe_{2}}$ layer are shown by solid red (spin up) and solid blue
(spin down) curves, and Fermi surfaces of the $\mathrm{VSe_{2}}$ layer are
shown by dashed red (spin up) and dashed blue (spin down) curves.

The superspin current for the $\mathrm{NbSe_{2}}/\mathrm{VSe_{2}}$ is
presented in Fig. 4. The results shown in this figure demonstrate that the
qualitative physics discussed for the $\mathrm{NbSe_{2}}$ monolayer under the
applied magnetic field survives here. Namely, we observe the same nonzero
components of the spin current and $\bm{j}_{s}^{z}$ manifests the same
rectification property. One of advantages of the heterostructure is apparently
that in the case of a high-quality interface between the layers, according to
the results of our calculations, it is possible to achieve sufficiently large
values of the effective exchange field induced in the S layer by proximity
effect. In turn, this makes it possible to achieve significantly higher ratios
$j_{s}/j_{e}$ than is possible for realistic values of the magnetic fields
applied to the isolated monolayer superconductor.

However, the key result that distinguishes a heterostructure from a monolayer
superconductor under the applied field is that the amplitude and sign of the
spin current can be controlled by the gate voltage. We apply the gating
potential $V$ to the F layer. Amplitude (more precisely the ratio
$j_{s}/j_{e}$) and direction of the spin currents $\bm{j}_{s}^{x}$ and
$\bm{j}_{s}^{z}$ at zero gating $V=0$ are shown in Figs. 4(a) and (b),
respectively. It is seen that in this case the ratio $j_{s}/j_{e}$ is
typically less than $10^{-2}$. At the same time from Figs. 4(c)-(f) it is seen
that the amplitude of the spin current can be strongly enhanced by gating.
From comparison of Figs. 4(c) and (e), as well as (d) and (f), we can see an
even more striking result - the sign of the spin current can be changed by
changing the gating potential. The reason of the observed dependence of the
amplitude and sign of the spin currrent on the gating potential can be
understood from Fig. 5. In this Figure we demonstrate the Fermi surfaces of
the heterostructure with no gating $V=0$ [Fig. 5(a)] and at two different
gating potentials, corresponding to Figs. 4(c)-(d) and (e)-(f), respectively.
The plotted Fermi surfaces are calculated at $t_{SF}=0$ (separate S and F
layers). In Fig. 5(a) it is seen that at $V=0$ the Fermi surfaces originating
from the F (dashed curves) ans S layers (solid curves around $K$-points) are
too far from each other. It is a manifestation of the fact that the
corresponding branches of the electronic spectra are also too far from each
other and cannot be effectively hybridized at $t_{SF}=30$ meV. This distance
between the $\mathrm{NbSe_{2}}$ and $\mathrm{VSe_{2}}$ electronic branches is
of the order of $100$meV [70]. For this reason at nonzero interlayer hopping
$t_{SF}\lesssim 30$meV the effective exchange field, which is induced in the S
layer due to the hybridization between the $\mathrm{VSe_{2}}$ and
$\mathrm{NbSe_{2}}$ electronic spectra [70], is also weak. On the contrary, in
Figs. 5(b)-(c) one can see that the modified by the gating potential Fermi
surfaces of the $\mathrm{VSe_{2}}$ monolayer intersect with the
$\mathrm{NbSe_{2}}$ Fermi surfaces around the $K$-points. It leads to their
strong hybridization at nonzero $t_{SF}$ and, consequently, to appearance of
the higher effective exchange field in the S layer and a larger value of the
spin supercurrent. Moreover, from a comparison of Figs.5(b) and (c), one can
see that by applying a gating potential of different signs, one can force the
Fermi surfaces of $\mathrm{NbSe_{2}}$ to intersect with the Fermi surfaces of
$\mathrm{VSe_{2}}$, corresponding to different spins. In its turn, it results
in different signs of the effective exchange field induced in the
$\mathrm{NbSe_{2}}$ by the hybridization of the electronic spectra, what leads
to different signs of the spin current observed in Figs. 4(c)-(d) and (e)-(f),
respectively.

##  IV Conclusions

We develop a theory of dissipationless spin transport carried by a
supercurrent in a monolayer vdW superconductor due to the simultaneous
presence of the Ising-type spin-orbit coupling and a Zeeman field directed in
plane of the layer. We consider two different possibilities: the Zeeman field
created by an external magnetic field and the Zeeman field generated by
proximity effect with a ferromagnetic layer using the example of a bilayer
heterostructure $\mathrm{NbSe_{2}}/\mathrm{VSe_{2}}$. We analyze the spin
structure of triplet Cooper pairs and conclude that the simultaneous presence
of the Ising-type spin-orbit coupling and the Zeeman field causes a partial
conversion of the singlet pairing into the nonunitary triplet pairing with the
averaged pair spin $\bm{S}\propto\bm{z}\times(\bm{z}\times\bm{h})$. If the
supercurrent is applied to the system, such pairs carry a superspin current.
Furthermore, the triplet pairs acquire an additional spin component
$\bm{S}_{z}\propto\bm{z}$ induced by the applied supercurrent and proportional
to the condensate momentum, which also participate in the spin transport. For
this spin current component we predict a rectification effect unlike the
charge supercurrent that generates it. For the case of S/F bilayer
heterostructures it is shown that the value and sign of the spin polarization
carried by the spin current can be controlled by gating.

###### Acknowledgements.

The financial support from the Russian Science Foundation via the RSF project
No.24-22-00186 is acknowledged.

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