# Magnetic anisotropy of superconducting transition in S/AF heterostructures
with spin-orbit coupling

G. A. 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  A.A. Golubov
Faculty of Science and Technology and MESA+ Institute for Nanotechnology,
University of Twente, 7500 AE Enschede, The Netherlands

###### Abstract

The influence of Rashba spin-orbit coupling (SOC) on superconducting
correlations in thin-film superconductor/antiferromagnet (S/AF) structures
with compensated interfaces is studied. A unique effect of anisotropic
enhancement of proximity-induced triplet correlations by the SOC is predicted.
It manifests itself in the anisotropy of the superconducting critical
temperature $T_{c}$ with respect to orientation of the Néel vector relative to
the S/AF interface, which is opposite to the behaviour of $T_{c}$ in
superconductor/ferromagnet structures. We show that the anisotropy is
controlled by the chemical potential of the superconductor and, therefore, can
be adjusted in (quasi)2D structures.

Introduction.— The interplay between superconductivity and ferromagnetism in
thin film superconductor/ferromagnet (S/F) heterostructures usually manifests
itself as a change in superconductivity of the S layer due to the proximity to
the F layer. The most well-known and studied effect is induced by the magnetic
proximity triplet superconducting correlations[1, 2]. Further studies [3, 4,
5, 6, 7, 8, 9, 10, 11] have predicted and observed that spin-orbit coupling
(SOC) in S/F bilayers can produce an anisotropic depairing effect on triplets.
One of the manifestations of the anisotropic depairing is that the critical
temperature $T_{c}$ of the bilayer depends on the orientation of the F layer
magnetization with respect to the S/F interface[9, 12, 13, 14].

One of consequences of a SOC-driven modulation of superconductivity is the
possibility for a reciprocal effect i.e., a reorientation of the F layer
magnetization due to superconductivity[15, 16, 17]. For sufficiently thin
ferromagnetic layers, a change from in-plane (IP) to out-of-plane (OOP)
magnetization has been predicted [15] and realised [17] in magnetic tunnel
junctions. The possibility to control magnetic anisotropies using
superconductivity - a key step in designing future cryogenic magnetic memories
and spintronics applications.

However, the finite net magnetization of ferromagnets presents a significant
drawback for applications in nanoscale devices. On the other hand,
antiferromagnets (AFs) are magnetically ordered materials with zero net
magnetization and negligible stray fields, as well as intrinsic high-frequency
dynamics. Due to these advantages they are being actively studied as
alternatives to ferromagnets for spintronics applications[18, 19, 20]. For AF-
based superconducting spintronics it is of crucial importance to study
proximity effects in superconductor/antiferromagnet (S/AF) heterostructures
[21, 22, 23, 24, 25, 26, 27, 28, 29]. It has been reported that in S/AF
structures superconducting triplet correlations also arise due to the
proximity effect. Depending on the particular antiferromagnetic order, system
geometry and the interface properties they can be of different types. In
particular, if the S/AF interface possesses nonzero net magnetization
(uncompensated interface), it induces Zeeman splitting and conventional
triplet correlations in the adjacent superconductor[30]. The compensated S/AF
interfaces also induce triplet correlations, but they are of the Néel type[31,
32], that is their amplitude flips sign from one lattice site to the next,
just like the Néel spin order in antiferromagnets. For S/AF heterostructures
with canted AFs the mixture of conventional and Néel triplet correlations has
been predicted [33].

At the same time effects of SOC on the proximity effect in S/AF hybrids are
much less explored. In particular, the anomalous phase shift in S/AF/S
Josephson junctions with SOC[34], anisotropy of the critical current [35] and
topological superconductivity in S/AF hybrids [36] have been predicted, also
the anisotropic magnetoresistance has been calculated[37]. Here we study
anisotropic effect of Rashba SOC on triplets in S/AF thin film bilayers with
fully compensated AFs. It is found that in addition to the anisotropic
depairing of triplet correlations known in S/F hybrids, a unique effect of
anisotropic enhancement of the triplets by the SOC occurs in the S/AF case. We
unveil the physical mechanism of the effect and demonstrate that it can
manifest itself in opposite trend in the anisotropy of the superconducting
transition as compared to S/F heterostructures. Namely, in S/F thin film
bilayers the critical temperature is higher for OOP magnetization orientation
than for the IP magnetization[14] due to the fact that SOC suppresses triplets
oriented OOP more than triplets oriented IP. Here we demonstrate the
possibility of the opposite effect for S/AF thin-film bilayers with SOC, which
occurs due to the anisotropic enhancement of triplets by SOC.

System and theoretical approach.—We consider a thin-film S/AF bilayer, where
the antiferromagnet is assumed to be an insulator, see Fig. 1. The magnetism
is staggered. We assume that the S/AF interface is fully compensated, that is
the interface magnetization has zero average value. The sites in the
superconductor are marked by the radius-vector
$\bm{i}=(i_{x},i_{y},i_{z})^{T}$, the interface is in the $(x,z)$-plane. The
influence of the antiferromagnetic insulator on the superconductor is
described by the exchange field
$\bm{h}_{\bm{i}}=\bm{(}-1)^{i_{x}+i_{z}}\bm{h}$ [30]. The superconductor S is
assumed to be homogeneous along the $y$-direction and is described by the
lattice Hamiltonian:

$\displaystyle\hat{H}=-$ $\displaystyle
t\sum\limits_{\langle\bm{i}\bm{j}\rangle,\sigma}\hat{c}_{\bm{i}\sigma}^{\dagger}\hat{c}_{\bm{j}\sigma}+\sum\limits_{\bm{i}}(\Delta_{\bm{i}}\hat{c}_{\bm{i}\uparrow}^{\dagger}\hat{c}_{\bm{i}\downarrow}^{\dagger}+H.c.)-$
$\displaystyle\mu\sum\limits_{\bm{i},\sigma}\hat{n}_{\bm{i}\sigma}+\sum\limits_{\bm{i},\alpha\beta}\hat{c}_{\bm{i}\alpha}^{\dagger}(\bm{h}_{\bm{i}}\bm{\sigma})_{\alpha\beta}\hat{c}_{\bm{i}\beta}+$
$\displaystyle
iV_{R}\sum\limits_{i}(\hat{c}_{\bm{i}}^{\dagger}\sigma_{z}\hat{c}_{\bm{i}+\bm{e}_{x}}-\hat{c}_{\bm{i}}^{\dagger}\sigma_{x}\hat{c}_{\bm{i}+\bm{e}_{z}}-H.c.),$
(1)

where $\langle\bm{i}\bm{j}\rangle$ means summation over the nearest neighbors,
$\hat{c}_{\bm{i}\sigma}^{\dagger}(\hat{c}_{\bm{i}\sigma})$ is the creation
(annihilation) operator for an electron with spin $\sigma$ at site $\bm{i}$.
$t$ parameterizes the hopping between adjacent sites, $\Delta_{\bm{i}}$
accounts for on-site s-wave pairing, $\mu$ is the electron chemical potential,
and the last term describes the Rashba SOC with Rashba constant $V_{R}$.
$\hat{n}_{\bm{i}\sigma}=\hat{c}_{\bm{i}\sigma}^{\dagger}\hat{c}_{\bm{i}\sigma}$
is the particle number operator. $\bm{e}_{k}$ with $k=x,y,z$ are unit vectors
along the corresponding axis. The lattice constant is denoted by $a$. Here we
define Pauli matrices $\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})^{T}$ in
spin space.

Figure 1: Sketch of the thin-film AF/S bilayer with SOC. The SOC is induced in
the S layer by proximity to a heavy metal layer like Pt (shown as the SO
layer). The SOC can also be due to inversion inversion symmetry breaking in
the S film by itself. The Néel vector of the AF makes angle $\theta$ with the
plane of the structure. $\theta=0$ corresponds to the IP and $\theta=\pi/2$
accounts for the OOP orientations. Unit cell with two sites A and B is also
shown.

Anisotropy of triplets and $T_{c}$.—The numerical calculations are performed
in the formalism of the Gor’kov Green’s functions in two-sublattice
framework[31, 32], generalized to take into account the SOC. Relegating
technical details of the Green’s functions formalism to the Supplemental
material[38], here we present and discuss the dependencies $T_{c}(h)$ for S/AF
structures with IP and OOP orientations of the Néel vector. They have been
compared to $T_{c}(h)$ of the S/F system with the same absolute value of the
induced exchange field $h$. The numerical results are presented in Fig. 2.
First, it is seen that while for S/F heterostructures $T_{c}$ is always higher
in the presence of SOC (dashed curves), for AF/S heterostructures the trends
are opposite for large $\mu\gg T_{c0}$ and for small $\mu\lesssim T_{c0}$,
where $T_{c0}=T_{c}(h=0)$. At small $\mu$ the behavior of $T_{c}$ is
qualitatively similar to the case of S/F bilayers, and at large $\mu$ it is
opposite - the presence of SOC suppresses $T_{c}$.

Figure 2: Critical temperature of S/AF (solid curves) and S/F bilayers
(dashed) as a function of the induced exchange field $h$. Panels (a) and (b)
correspond to $\mu=t$ and $\mu=0$, respectively. Green curves represent the
results with no SOC, red and blue curves are for OOP and IP orientations,
respectively and $V_{R}=0.4t$. $T_{c0}=t/12$ for the both panels.

Furthermore, in the presence of SOC $T_{c}$ is anisotropic depending on the
angle $\theta$ between the magnetization and the interface plane. For S/F
heterostructures $T_{c}$ is always higher for OOP orientation (dashed curves)
[9, 12, 13, 14]. At the same time for AF/S heterostructures the ratio between
the values of $T_{c}$ for IP and OOP orientations is again opposite for large
$\mu\gg T_{c0}$ and for small $\mu\lesssim T_{c0}$. At $\mu\lesssim T_{c0}$
for S/AF heterostructures the ratio between $T_{c}$ of IP and OPP is the same
as for the S/F case. It is explained by the fact that triplet superconducting
correlations, induced by the proximity effect with the magnet are suppressed
stronger for OOP configuration and, consequently, have a less damaging effect
on the singlet superconductivity. At $\mu\gg T_{c0}$ the anisotropy of the
critical temperature, that is the difference between the IP and OOP $T_{c}$ is
opposite. This is due to the existence of a unique mechanism of enhancement of
the Néel-type triplet correlations in S/AF structures, which is more effective
for OOP orientation of the Néel vector.

The physical description of the both mechanisms is provided below. There is a
crossover between the opposite anisotropy regimes at some intermediate value
of $\mu\sim\pi T_{c0}$. For the chosen parameters of the the system this
crossover value $\mu_{c}\approx 0.25t\approx\pi T_{c0}$. It is interesting
that the ratio between $\mu$ and superconduting energy scale $T_{c0}$ is
crucial for very different aspects of the proximity physics of S/AF
heterostructures. For example, this parameter controls the relative importance
of different mechanisms of superconductivity suppression in S/AF hybrids [32].
The superconductivity suppression is dominated by the Néel triplets at
$\mu\lesssim T_{c0}$. On the contrary, if $\mu\gg T_{c0}$ the
superconductivity suppression is dominated by nonmagnetic disorder. For S/AF
heterostructures with canted AFs the opposite dependencies of $T_{c}$ on the
canting angle were also predicted at small and large values of $\mu$ [33].

Discussion of the mechanisms of $T_{c}$ anisotropy.—Now we discuss the
physical reasons of $T_{c}$ anisotropy in S/AF structures. In order to unveil
them let us consider quasiclassical Eilenberger equations, developed recently
for treating the proximity effect in S/AF heterostructures [31] and their
analytical solutions in the presence of Rashba SOC. The general form of the
Eilenberger equation, generalized for treating the SOC, is provided in the
Supplemental material[38]. In the vicinity of the critical temperature the
Eilenberger equation can be linearized with respect to the anomalous Green’s
function, which for the problem under consideration can be written as
$\check{f}=f_{s}\sigma_{0}\rho_{x}+\bm{f}^{i}\bm{\sigma}\rho_{i}$. Here
$f_{s}$ is its singlet component in spin space, $i=0,y,z$ and $\bm{f}^{i}$ is
the vector triplet component in spin space, corresponding to the $0,y,z$
component in sublattice space. Recall that the $y$-component in sublattice
space accounts for the on-site Néel-type triplet correlations
$\bm{f}^{AA}=-\bm{f}^{BB}$, while the $z(0)$-component describes nonlocal Néel
(conventional) correlations $\bm{f}^{AB}=-(+)\bm{f}^{BA}$. The linearized
equations take the form:

$\displaystyle 2i\omega f_{s}-2i\bm{h}\bm{f}^{y}=\Delta(g_{N}-\tilde{g}_{N}),$
(2) $\displaystyle
2i\omega\bm{f}^{0}+2i\bm{h}\times\bm{f}^{z}=\Delta(\bm{g}_{N}^{0}-\tilde{\bm{g}}_{N}^{0}),$
(3) $\displaystyle\pm 2i\mu\bm{f}^{y(z)}+2i\bm{f}^{z(y)}\times\bm{h}_{R}=\mp
i\Delta(\bm{g}_{N}^{y(z)}+\tilde{\bm{g}}_{N}^{y(z)}),$ (4)

where $\bm{h}_{R}=(V_{R}/2ta)(\bm{e}_{y}\times\bm{v}_{F})$ is the effective
Rashba pseudomagnetic field seen by an electron moving along the trajectory
determined by the Fermi velocity $\bm{v}_{F}$ and $\omega$ is the fermionic
Matsubara frequency. The right-hand side contains the electron (hole) normal
Green’s function
$\check{g}_{N}(\check{\tilde{g}}_{N})=g_{N}(\tilde{g}_{N})\sigma_{0}\rho_{x}+\bm{g}_{N}^{i}\bm{\sigma}(\tilde{\bm{g}}_{N}^{i}\bm{\sigma})\rho_{i}$.
The vector part of the on-site normal state quasiclassical Green’s function
accounting for the Néel order up to the leading order with respect to
$(h,h_{R})/|i\omega+\mu|$ takes the form:

$\displaystyle\bm{g}_{N}^{y}=-\frac{i\bm{h}{\rm
sgn}\omega}{i\omega+\mu}-\frac{i[\bm{h}_{R}\times(\bm{h}\times\bm{h}_{R})]}{(i\omega+\mu)^{3}}{\rm
sgn}\omega$ (5)
$\displaystyle\bm{g}_{N}^{z}=\frac{i\bm{h}_{R}\times\bm{h}}{(i\omega+\mu)^{2}}{\rm
sgn}\omega.$ (6)

The vector component $\bm{g}_{N}^{y}$ accounts for the Néel-type spin
polarization of the on-site DOS $P_{\bm{N}}^{A}=-P_{\bm{N}}^{B}\equiv
P_{\bm{N}}$ along the direction $\bm{N}$ ($|\bm{N}|=1$) in spin space.

$\displaystyle P_{\bm{N}}(\varepsilon)=2N_{F}{\rm
Re}[i\bm{N}\bm{g}_{N}^{R,y}(\varepsilon)],$ (7)

where $\bm{g}^{R,y}(\varepsilon)$ is obtained from $\bm{g}_{N}^{y}(\omega>0)$
by substitution $i\omega\to\varepsilon+i\delta$ and $N_{F}$ is the normal
state DOS at the Fermi surface and at $h=0$.

It is important to compare Eqs. (2)-(4) to the analogous equations for S/F
heterostructures. In this case the anomalous Green’s functions has no Néel-
type $\rho_{y(z)}$ components and the corresponding Eilenberger equations can
be written without the sublattice structure, but we write them in terms of the
same two-sublattices formalism in order to be directly compared with the AF
case. In this case
$\hat{f}_{F}=f_{F,s}\sigma_{0}\rho_{x}+f_{F,0}\sigma_{0}\rho_{0}+\bm{f}_{F}^{x}\bm{\sigma}\rho_{x}+\bm{f}_{F}^{0}\bm{\sigma}\rho_{0}$.
It obeys the following Eilenberger equations:

$\displaystyle 2i\omega f_{F,s}-2\bm{h}\bm{f}_{F}^{x}=2\Delta{\rm sgn}\omega,$
(8) $\displaystyle 2i\omega f_{F,0}-2\bm{h}\bm{f}_{F}^{0}=0,$ (9)
$\displaystyle
i\omega\bm{f}_{F}^{x(0)}-\bm{h}f_{F,s(0)}+i\bm{f}_{F}^{0(x)}\times\bm{h}_{R}=0.$
(10)

The most important difference between equations for triplet correlations for
AF/S and F/S structures, Eqs. (4) and (10) is that they contain different
mechanisms of triplet generation. While in S/F hybrids the triplets are
generated by the term $\bm{h}f_{F,s(0)}$ via the direct singlet-triplet
conversion [1], such type of triplet generator is absent for AF/S structures.
Instead, the triplets are generated via the antisymmetric with respect to
Matsubara frequencies vector component of the normal Green’s function
$\bm{P}_{N}^{M}=i[\bm{g}_{N}^{y}(\omega)+\tilde{\bm{g}}_{N}^{y}(\omega)]=i[\bm{g}_{N}^{y}(\omega)-{\bm{g}}_{N}^{y}(-\omega)]$.
It automatically provides the odd-frequency character of on-site triplets.
Turning to the real energies we obtain
$\bm{P}_{N}^{s}=i[(\bm{g}_{N}^{y,R}(\varepsilon)+\tilde{\bm{g}}_{N}^{y,R}(\varepsilon)]=i[(\bm{g}_{N}^{y,R}(\varepsilon)+{\bm{g}}_{N}^{y,R}(-\varepsilon)]$.
It is seen from Eq. (5) and (7) that this expression determines the symmetric
with respect to $\varepsilon$ part of the Néel-type spin polarization of the
DOS
$\bm{P}_{\bm{N}}^{s}\bm{N}=[P_{\bm{N}}(\varepsilon)+P_{\bm{N}}(-\varepsilon)]/2N_{F}$.
Therefore, the Néel triplets are generated by the staggered polarization of
the normal metal. An analogous term is absent in S/F heterostructures. In the
case $h\ll\varepsilon_{F}$ the spin polarization of normal state is negligible
and is considered to be zero in the framework of the quasiclassical
approximation. This limit is always relevant for S/F and S/AF heterostructures
because $h\lesssim\Delta$ in order to avoid the complete depairing of
superconductivity.

Consequently, the physical mechanisms of triplet generation are different in
S/F and S/AF structures. The effect of SOC on the triplets can also be
different. It can be seen directly from the solutions of Eqs. (4) and (10). Up
to the leading order in $h/|i\omega+\mu|$ and $h_{R}/|i\omega+\mu|$ the on-
site Néel-type triplet correlations take the form:

$\displaystyle\bm{f}^{y}={\rm
sgn}\omega\left[\frac{i\Delta\bm{h}}{\mu^{2}+\omega^{2}}+\frac{i\Delta[\bm{h}_{R}\times(\bm{h}\times\bm{h}_{R})](3\mu^{2}-\omega^{2})}{(\mu^{2}+\omega^{2})^{3}}\right]~{}$
(11)

It can be compared to the solution for on-site triplet correlation in the S/F
thin-film bilayer:

$\displaystyle\bm{f}_{F}^{x}=-{\rm
sgn}\omega\left[\frac{\Delta\bm{h}}{\omega^{2}}-\frac{\Delta[\bm{h}_{R}\times(\bm{h}\times\bm{h}_{R})]}{\omega^{4}}\right].$
(12)

Recall that $\bm{f}_{F}^{x}$ anomalous Green’s function accounts for the
conventional, not Néel, on-site triplet correlations
$\bm{f}_{F}^{x}=\bm{f}_{F}^{AA}=\bm{f}_{F}^{BB}$. By comparing Eqs. (11) and
(12) we see that in S/F heterostructures the SOC weakens triplets, the same is
valid for S/AF structures with $\mu\ll\pi T_{c0}$ when we can disregard $\mu$
with respect to $\omega$ in Eq. (11). At the same time in S/AF with $\pi
T_{c0}\lesssim\mu$ its influence is just opposite - it enhances triplets. In
both cases the influence of SOC on triplets is anisotropic. The anisotropy is
determined by the vector structure $\bm{h}_{R}\times(\bm{h}\times\bm{h}_{R})$.
The influence of SOC on triplets, that is suppression for S/F and enhancement
for S/AF bilayers, is maximal when $\bm{h}_{R}\perp\bm{h}$ for all electron
trajectories $\bm{n}_{F}$, what is realized for OOP orientation of the
magnetization. It is monotonically declines with decreasing $\theta$.

The physical mechanism for the suppression of the triplet correlations with
$\bm{f}\propto\bm{h}$ by SOC in S/F structures is well-known. The triplets
$\bm{f}=-{\rm sgn}\omega\Delta\bm{h}/\omega^{2}$ are obtained from singlets by
the process of the singlet-triplet conversion[1]. Then the SOC partially
converts these original triplets into the odd in momentum component
$\propto\bm{h}_{R}\times\bm{h}$. As a result, the amplitude of initial
triplets along $\bm{h}$ decreases. For dirty systems this process of triplet
depairing by SOC is analogous to anisotropic Dyakonov-Perel spin relaxation
[39, 40] because the odd in momentum component $\propto\bm{h}_{R}\times\bm{h}$
is zero after impurity averaging over trajectories and only the reduced
triplets along $\bm{h}$ survive [5, 6].

On the contrary, the enhancement of triplets by SOC in S/AF structures is
caused by the analogous enhancement of the normal state polarization. From
Eqs. (5) and (7) it follows that up to the leading order in
$h/|\varepsilon+\mu|$ and $h_{R}/|\varepsilon+\mu|$ the Néel-type polarization
of the normal state DOS along $\bm{h}$ takes the form

$\displaystyle
P_{\bm{h}}^{A}(\varepsilon)=-P_{\bm{h}}^{B}(\varepsilon)=2N_{F}\left[\frac{h}{\varepsilon+\mu}+\frac{hh_{R}^{2}\sin^{2}\phi}{(\varepsilon+\mu)^{3}}\right],$
(13)

where $h=|\bm{h}|$ and $\phi$ is the angle between $\bm{h}$ and $\bm{h}_{R}$.
It is seen that (i) the absolute value of the polarization is always enhanced
by the SOC and (ii) the enhancement is anisotropic. It reaches maximal
possible value for all the trajectories for OOP orientation of $\bm{h}$
because $\bm{h}_{R}$ is always in-plane. It is worth noting that Eq. (13) is
not applicable at $\mu=0$ because in this limit the most important
contribution to the superconducting properties is given by the small energies
$\varepsilon\lesssim T_{c0}$, where Eq. (13) is not valid because the
conditions $h\ll|\varepsilon+\mu|$ and $h_{R}\ll|\varepsilon+\mu|$ are
violated. Therefore, this consideration is not applicable for explanation of
the results at $\mu\lesssim\pi T_{c0}$ and we comment on this limit later.

Figure 3: Normal state electron spectra $\varepsilon(\xi)$ of the S layer in
the presence of $\bm{h}$ and SOC $\bm{h}_{R}$, where $\xi=-2t(\cos p_{x}a+\cos
p_{y}a+\cos p_{z}a)$. Blue dashed lines correspond to $h_{R}=0$. Points 1-4
mark the intersection of the spectra with the Fermi surface.

The reason for the described above enhancement is the specific reconstruction
of normal state electron spectra under the influence of the SOC. In Fig. 3 we
demonstrate the normal state electron spectra of the S layer proximitized by
the AF for $\bm{h}_{R}\parallel\bm{h}$ [(a)] and $\bm{h}_{R}\perp\bm{h}$
[(b)]. At $\bm{h}_{R}=0$ the spectra are doubly-degenerated. The eigen states
at the Fermi surface for an electron with spin $\sigma=\pm 1$ take the form

$\displaystyle\left(\begin{array}[]{c}\hat{\psi}_{\bm{i}\sigma}^{A}\\\
\hat{\psi}_{\bm{i}\sigma}^{B}\end{array}\right)(\bm{p})=\left(\begin{array}[]{c}\sqrt{1+\sigma
h/\mu}\\\ \sqrt{1-\sigma h/\mu}\end{array}\right)e^{i\bm{p}\bm{i}},$ (18)

The distribution of probability density of these states in the lattice
strongly oscillates between A and B sites. The total spin polarization at the
Fermi level caused by these states is
$P_{\bm{h}}^{A}=-P_{\bm{h}}^{B}=2N_{F}h/\mu$. The nonzero SOC splits the
spectra. The splitting is horizontal for $\bm{h}_{R}\parallel\bm{h}$, that is
analogous to the conventional action of the Rashba SOC in nonmagnetic metals.
The net spin polarization at the Fermi level is represented by the sum over
states $1-4$ in Fig. 3(a), multiplied by $N_{F}$. It is again
$P_{\bm{h}}^{A}=-P_{\bm{h}}^{B}=2N_{F}h/\mu$. On the contrary, for
$\bm{h}_{R}\perp\bm{h}$ the splitting of the spectra is vertical, which is
reminiscent of the Zeeman splitting in conventional metals. Effectively it is
described by the opposite shifts of the chemical potentials of the both
branches $\mu\to\mu\pm h_{R}$. The resulting spin polarizations of the states
$1-4$ are $P_{1,4}=h/(\mu+h_{R})$ and $P_{2,3}=h/(\mu-h_{R})$. The total on-
site spin polarization is
$P_{\bm{h}}^{A}=-P_{\bm{h}}^{B}=2N_{F}h\mu/(\mu^{2}-h_{R}^{2})$. Expanding
this expression with respect to $h_{R}/\mu$ we obtain Eq. (13) at $\phi=0$ and
$\varepsilon=0$. The enhancement of the staggered spin polarization appears as
a nonlinear effect of the opposite chemical potential shifts.

If $\mu\lesssim\pi T_{c}$ the antiferromagnetic gap opens in the
superconductor in the vicinity of the normal state Fermi surface. In this case
the most important contribution to the pairing correlations is given by the
electronic states at the edge of the gap. They correspond to $\xi\approx 0$,
what means that the electrons are practically fully localized at one of the
sublattices. Consequently, they only feel the magnetization of the
corresponding sublattice and behave in the same way as in the ferromagnet. For
this reason our results at $\mu\lesssim\pi T_{c}$ demonstrate the same trends
as the corresponding results for S/F structures.

Summary.—The effect of Rashba SOC on triplets is studied in S/AF thin film
bilayers with fully compensated AFs. A unique effect of anisotropic
enhancement of the triplets by SOC is found. It can be experimentally observed
via the anisotropy of the superconducting transition. Our analysis highlights
the importance of the value of the chemical potential $\mu$ for the physics of
S/AF thin-film hybrids. At $\mu\lesssim\pi T_{c0}$ the influence of the SOC on
superconducting properties of the S/AF bilayers is the same as for the S/F
case - the SOC anisotropically suppresses triplets and enhances $T_{c}$, and
the maximal $T_{c}$ is reached for OOP orientation of the Néel vector. On the
contrary, at $\mu>\pi T_{c0}$ the SOC anisotropically enhances triplets and
suppresses $T_{c}$, and $T_{c}$ is minimal for OOP orientation. The effect can
be especially interesting for heterostructures composed of 2D magnets and
superconductors because of possibility of external control of the chemical
potential. The anisotropy of the superconducting transition opens a
perspective for a reciprocal effect, that is the reorientation of the Néel
vector due to superconductivity. In its turn, this possibility is an important
step for further developments in AF-based superconducting spintronics.

###### Acknowledgements.

The financial support from the Russian Science Foundation via the RSF project
No.22-22-00522 is acknowledged.

##  I Supplemental Material: Two-sublattice formalism of Green’s functions

The numerical calculations of the critical temperature are performed in the
formalism of the Gor’kov Green’s functions in two-sublattice framework[31,
32]. The unit cell with two sites in it is chosen as shown in Fig. 1 of the
main text. Introducing the two-sublattice Nambu spinor
$\check{c}_{\bm{i}}=(\hat{c}_{{\bm{i}},\uparrow}^{A},\hat{c}_{\bm{i},\downarrow}^{A},\hat{c}_{\bm{i},\uparrow}^{B},\hat{c}_{\bm{i},\downarrow}^{B},\hat{c}_{\bm{i},\uparrow}^{A\dagger},\hat{c}_{\bm{i},\downarrow}^{A\dagger},\hat{c}_{\bm{i},\uparrow}^{B\dagger},\hat{c}_{\bm{i},\downarrow}^{B\dagger})^{T}$
we define the Green’s function as
$\check{G}_{\bm{i}\bm{j}}(\tau_{1},\tau_{2})=-\langle
T_{\tau}\check{c}_{\bm{i}}(\tau_{1})\check{c}_{\bm{j}}^{\dagger}(\tau_{2})\rangle$,
where $\langle T_{\tau}...\rangle$ means imaginary time-ordered thermal
averaging and $\bm{i}$ is now the sublattice index. The Green’s function is a
$8\times 8$ matrix in the direct product of spin, particle-hole and sublattice
spaces. Therefore, we define the Pauli matrices
$\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})^{T}$ in spin space,
$\bm{\tau}=(\tau_{x},\tau_{y},\tau_{z})^{T}$ in particle-hole space and
$\rho=(\rho_{x},\rho_{y},\rho_{z})^{T}$ in sublattices space. Further we
assume the system to be homogeneous along the interface and consider the
Fourier-transformed Green’s function:

$\displaystyle\check{G}(\bm{p})=\int
d^{3}re^{-i\bm{p}(\bm{i}-\bm{j})}\check{G}_{\bm{i}\bm{j}},$ (19)

where the integration is over $\bm{i}-\bm{j}$. Then to make the resulting
Gor’kov equations simpler it is convenient to define the following transformed
Green’s function:

$\displaystyle\check{\tilde{G}}(\bm{p})=\left(\begin{array}[]{cc}1&0\\\
0&-i\sigma_{y}\end{array}\right)_{\tau}\rho_{x}e^{\frac{-\displaystyle
ip_{z}a_{z}\rho_{z}}{\displaystyle 2}}\times$ (22)
$\displaystyle\check{G}(\bm{p})e^{\frac{\displaystyle
ip_{z}a_{z}\rho_{z}}{\displaystyle 2}}\left(\begin{array}[]{cc}1&0\\\
0&-i\sigma_{y}\end{array}\right)_{\tau},$ (25)

where subscript $\tau$ means that the explicit matrix structure corresponds to
the particle-hole space. The Gor’kov equation for
$\check{\tilde{G}}(\bm{R},\bm{p})$ was originally derived in Ref. 31 and here
is generalized to take into account the Rashba SOC. The resulting Gor’kov
equation takes the form:

$\displaystyle[\check{H}\rho_{x}-\xi(\bm{p})-\bm{h}_{R}(\bm{p})\bm{\sigma}]\check{\tilde{G}}=1,$
(26)
$\displaystyle\check{H}=i\omega_{m}\tau_{z}+\mu+\tau_{z}\check{\Delta}-\bm{h}\bm{\sigma}\tau_{z}\rho_{z},$

where $\xi(\bm{p})=-2t(\cos p_{x}a_{x}+\cos p_{y}a_{y}+\cos p_{z}a_{z})$,
$\omega_{m}=\pi T(2m+1)$ is the Matsubara frequency,
$\bm{h}_{R}(\bm{p})=(V_{R}/2ta)(\bm{e}_{y}\times\bm{v}(\bm{p}))$ is the
effective Rashba pseudomagnetic field seen by an electron moving along the
trajectory determined by the velocity
$\bm{v}(\bm{p})=d\xi/d\bm{p}=2t(\bm{a}_{x}\sin[p_{x}a_{x}]+\bm{a}_{y}\sin[p_{y}a_{y}]+\bm{a}_{z}\sin[p_{z}a_{z}])$,
$\check{\Delta}=\Delta(\bm{R})\tau_{+}+\Delta^{*}(\bm{R})\tau_{-}$ with
$\tau_{\pm}=(\tau_{x}\pm i\tau_{y})/2$. For simplicity we assume
$a_{x}=a_{y}=a_{z}=a$. The superconducting order parameter $\Delta$ in S is
calculated self-consistently:

$\displaystyle\Delta=-T\sum\limits_{\omega_{m}}g\int\frac{\mbox{Tr}(\check{\tilde{G}}(\bm{p})\tau_{+}\sigma_{0}\rho_{x})}{8}\frac{d^{3}p}{(2\pi)^{3}},$
(27)

where $g$ is the pairing constant. The critical temperature is calculated from
the linearized with respect to $\Delta$ version of Eqs. (26) and (27).

Two-sublattice quasiclassical equations, which were derived in [31], can also
be generalized to take into the SOC. We introduce quasiclassical
$\xi$-integrated Green’s function:

$\displaystyle\check{g}(\bm{R},\bm{p}_{F})=-\frac{1}{i\pi}\int\check{\tilde{G}}(\bm{R},\bm{p})d\xi,$
(28)

where $\xi(\bm{p})=-2t(\cos p_{x}a_{x}+\cos p_{y}a_{y}+\cos p_{z}a_{z})$ is
the normal state electron dispersion counted from the Fermi energy. Here we
also allow for a possible spatial inhomogeneity in the plane of the S/AF
interface, where $\bm{R}$ is the in-plane radius-vector. Performing the
standard derivation of the quasiclassical equation [31] one obtains the
following Eilenberger equation for the quasiclassical Green’s function:

$\displaystyle\left[\left(i\omega_{m}\tau_{z}+\mu+\tau_{z}\check{\Delta}(\bm{R})-\bm{h}(\bm{R})\bm{\sigma}\tau_{z}\rho_{z}\right)\rho_{x}-\right.$
$\displaystyle\left.-\bm{h}_{R}(\bm{p}_{F})\bm{\sigma}\rho_{0},\check{g}(\bm{R},\bm{p}_{F})\right]+i\bm{v}_{F}\bm{\nabla}\check{g}(\bm{R},\bm{p}_{F})=0.$
(29)

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