# Generation of spin-triplet Cooper pairs via a canted antiferromagnet

Simran Chourasia<EMAIL_ADDRESS>Condensed Matter Physics Center
(IFIMAC) and Departamento de Física Teórica de la Materia Condensada,
Universidad Autónoma de Madrid, E-28049 Madrid, Spain  Lina Johnsen Kamra
Center for Quantum Spintronics, Department of Physics, Norwegian University of
Science and Technology, NO-7491 Trondheim, Norway  Irina V. Bobkova  Moscow
Institute of Physics and Technology, Dolgoprudny, 141700 Moscow, Russia
National Research University Higher School of Economics, Moscow, 101000 Russia
Akashdeep Kamra  Condensed Matter Physics Center (IFIMAC) and Departamento de
Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid,
E-28049 Madrid, Spain

###### Abstract

Spinful triplet Cooper pairs can be generated from their singlet counterparts
available in a conventional superconductor (S) using two or more noncollinear
magnetic moments, typically contributed by different magnets in a multilayered
heterostructure. Here, we theoretically demonstrate that an S interfaced with
a canted antiferromagnet (AF) harbors spinful triplet Cooper pairs
capitalizing on the intrinsic noncollinearity between the two AF sublattice
magnetizations. As the AF canting can be controlled by an applied field, our
work proposes a simple bilayer structure that admits controllable generation
of spin-triplet Cooper pairs. Employing the Bogoliubov-de Gennes framework, we
delineate the spatial dependence of the spin-triplet correlations. We further
evaluate the superconducting critical temperature as a function of the AF
canting, which provides one experimental observable associated with the
emergence of these triplet correlations.

##  I Introduction

The dissipationless flow of charge in superconductors is partly responsible
for their central role in the various emerging quantum technologies [1, 2].
The widely available conventional superconductors, such as Al and Nb, are made
of spin-singlet Cooper pairs, which harbor no net spin [3]. A superconductor
hosting spin-triplet Cooper pairs can support dissipationless spin currents
[4, 5, 6, 7, 8], deemed valuable for switching magnetic memories [9, 10, 11,
12, 13], as well as exotic excitations, such as Majorana bound states [14,
15]. Such an unconventional superconductor can be engineered from its
conventional counterpart employing heterostructures incorporating magnetic
multilayers [5, 16, 7, 8, 6]. The basic requirement for achieving spinful
triplets from spin-singlets is exposing the latter to two or more noncollinear
spin-splitting fields. A wide variety of multilayered hybrids comprising
conventional superconductors (S) and ferromagnets (F) has been employed to
achieve the desired spinful triplets [17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33], coming a long way from the initial critical
temperature studies [34, 35].

Since a Néel ordered antiferromagnet (AF) bears no net spin or magnetic
moment, for some time it was considered inert at causing spin-splitting in an
adjacent superconductor. Indeed, early experiments found a metallic AF to
behave just like a normal metal when considering its effect on an adjacent
superconductor [36]. More recent experiments, on the other hand, found the AF
to substantially affect the adjacent S with intriguing dependencies [37, 38,
39, 40, 41, 42]. From the theory perspective, Josephson junctions [43, 44] and
interfaces [45, 46] involving itinerant AFs were shown to exhibit non-trivial
properties due to quasiparticle reflections. Moreover, a recent work
demonstrates that an uncompensated interface of an insulating AF with an
adjacent S induces a strong spin-splitting as well as spin-flip scattering
thereby strongly influencing the S [47]. Subsequent work found even the fully
compensated interface between the AF and S to be spin-active [48]. This has
been understood as due to the AF inducing Néel triplets whose pairing
amplitude has an alternating sign in space similar to the AF spin [49].
Altogether, the potential usefulness of AFs [50] in engineering novel
superconducting effects and devices, such as a filter [51], is starting to be
understood.

Pekar and Rashba [52] already recognized long ago that even though the net
spin vanishes in an AF, at the lattice constant length scale, the AF harbors a
spin or magnetization profile that rapidly varies in space changing its sign
from one lattice site to the next, which should manifest itself in physical
observables. As a result, the AF with its two sublattice magnetizations
antiparallel to each other generates zero-spin Néel triplets [49]. Proceeding
further along this line of thought, a homogeneous canted AF with its
sublattice magnetizations deviating from an antiparallel alignment effectively
harbors a noncollinear spin texture capable of generating spinful triplet
Cooper pairs in an adjacent S. This exciting possibility is theoretically
examined in the present work.  Furthermore, a recent experiment [53]
demonstrates generation and use of spinful triplet correlations employing the
intrinsically noncollinear ground state of a kagome AF.

Here, employing the Bogoliubov-de Gennes (BdG) framework [54], we
theoretically investigate a bilayer structure consisting of an insulating AF
exchange coupled via a compensated interface to an adjacent S. We examine the
critical temperature and spin-triplet correlations in the S as a function of
the canting in the AF, which allows us to continuously tune the AF from its
collinear antiparallel state to it effectively becoming an F. We find that
Néel triplets are generated in the S both from the interband pairing channel
considered recently [49], and from the conventional intraband pairing.  The
former channel is dominant at half-filling in which case states from two
different electronic bands are energetically close to the chemical potential
and can participate in forming these unconventional Néel triplet Cooper pairs
[49]. This can be compared with pairing at finite energies in other
multisublattice systems [55]. The conventional intraband pairing channel
dominates away from half-filling when only states within the same electronic
band are energetically close to each other and the chemical potential. We find
that the  intraband pairing channel results in Néel triplets formation due to
an imprinting of the Néel character by the AF on the normal state electronic
wavefunctions in the S. Although this channel of Néel triplet generation is
found to be much weaker than the interband pairing channel, it admits
qualitatively new effects. We show that it is only in the intraband pairing
channel that spinful Néel triplets are generated due to the intrinsic
noncollinearity of a canted AF. The S critical temperature variation as a
function of the AF canting angle is found to be consistent with the intraband
Néel triplets being weaker than their interband counterparts, and may offer a
convenient experimental signature of this interplay.

The paper is organized as follows. Section II introduces the model and BdG
framework employed in our analysis. The dependence of spin-triplet
correlations on space and AF canting is discussed in Sec. III, while the
variation of superconducting critical temperature is discussed in Sec. IV.
Until this point, we consider a one-dimensional model that allows a simple and
semianalytic understanding of the essential physics. In Sec. V, we employ a
two-dimensional  (2-D) model for the S to validate our prior results and
further examine the spatial dependence of the  s-wave and p-wave triplet
correlations. We conclude with discussion and summary of the key points in
Sec. VI. The appendices provide details of the BdG framework, analytic
evaluation of the normal state electronic properties in the bilayer,  a
discussion of the decay length of the spin-triplet correlations inside the S,
critical temperature results for the 2-D system, triplet correlations for a
different configuration of the AF sublattice magnetizations, and the
parameters employed in our numerical routines.

##  II AF/S bilayer model

We consider a bilayer structure comprising an insulating AF exchange coupled
via a compensated interface to the S layer, as depicted schematically in Fig.
1(a). We anticipate that all three kinds of spin-triplet correlations will be
generated in the S when the AF sublattice magnetizations are canted [Fig.
1(a)]. We employ the Bogoliubov-de Gennes method and numerically evaluate the
superconducting properties self-consistently [54]. The canted-AF is taken to
be an ideal insulator with a large band-gap. Consequently, the Hamiltonian is
formulated only for the itinerant electrons in S  as they never enter the
insulating AF. The AF’s influence on the S is accounted for by incorporating a
spatially dependent spin-splitting caused by the AF spins [47, 56, 57].
Furthermore, with the aim of allowing a semianalytic understanding to the
essential physics, we consider a one-dimensional model as depicted
schematically in Fig. 1(b). The resulting Hamiltonian is given by

$\displaystyle H=$
$\displaystyle-\mu\sum_{j,\sigma}c_{j,\sigma}^{\dagger}c_{j,\sigma}-t\sum_{\left<i,j\right>}\sum_{\sigma}c_{i,\sigma}^{\dagger}c_{j,\sigma}-\frac{J}{2}\sum_{j}\vec{M}_{j}\cdot\vec{S}_{j}$
$\displaystyle+\sum_{j}\left(\frac{|\Delta_{j}|^{2}}{U}+\Delta_{j}^{*}c_{j,\downarrow}c_{j,\uparrow}+\Delta_{j}c_{j,\uparrow}^{\dagger}c_{j,\downarrow}^{\dagger}\right),$
(1)

where $c_{j,\sigma}^{\dagger}$ ($c_{j,\sigma}$) is the creation (annihilation)
operator of an electron with spin $\sigma$ at site $j$ of the S layer, with
$j=1,2,\dots,N$ as the site index. Here, the spin quantization axis is taken
along the $z$-axis. We further consider periodic boundary conditions by
allowing electrons to hop between sites $j=1$ and $j=N$.

Figure 1: (a) Schematic depiction of the system and key physics under
investigation. Equal-spin and zero-spin triplet correlations are generated in
a conventional $s$-wave spin-singlet superconductor when it is interfaced with
a canted antiferromagnet (canted-AF). This results from the intrinsic
noncollinearity between the two AF sublattice magnetizations. (b) Schematic
depiction of our model investigated using the Bogoliubov-de Gennes method. The
black circles represent lattice sites of the superconductor  hosting itinerant
electrons and Cooper pairs. Blue (red) circles represent A (B) sublattice
sites of the  electrically insulating canted-AF. The blue and red arrows
denote the local AF magnetic moments. The canting angle $\theta_{t}$ allows us
to vary the magnet from being a collinear AF ($\theta_{t}=0$) to a ferromagnet
($\theta_{t}=\pi/2$).

In Eq. (II), $\mu$ is the chemical potential. Within our theoretical method,
it is determined via the filling factor $f$, which is the fraction of filled
electronic states in the system. For $f=0.5$ (half-filled band), we obtain
$\mu=0$. This corresponds to the Fermi wave-vector $k_{F}=\pi/2a$ located at
the AF Brillouin zone boundary, where $a$ is the lattice constant. The normal
state electronic dispersion and properties for the case of an AF in its
collinear anti-parallel state have been discussed in Appendix A. For $f\neq
0.5$, $\mu$ is non-zero and the Fermi level is away from the AF Brillouin zone
boundary. In the following analysis, we will consider the two qualitatively
distinct cases of $f=0.5$ and $f\neq 0.5$, corresponding to $\mu=0$ and
$\mu\neq 0$.

The second term in Eq. (II) is the kinetic energy term, describing hopping
between nearest neighboring sites $\left<i,j\right>$ with $t>0$ as the hopping
parameter.

The third term in the Hamiltonian [Eq. (II)] accounts for the spin-splitting
due to the localized magnetic moments in the canted-AF [58, 6, 59, 47].
$J\vec{M}_{j}/2$ is the local spin-splitting field which causes an energy
shift of spin-up and spin-down electrons by $\mp J/2$, with respect to the
local spin-quantization axis along $\vec{M}_{j}$.  The strength of this spin-
splitting, and thus the value of $J$ in our model, depends on the S thickness
and several other parameters [47]. It can thus be tuned in a broad range via
appropriate thin films fabrication [47] as per the experimental requirements.
Here,
$\vec{M}_{j}=[(-1)^{j+1}\cos\theta_{t}\,\hat{x}+\sin\theta_{t}\,\hat{y}]$ is
the unit vector along the direction of local magnetic moment at the $j^{th}$
canted-AF site. The AF sublattice magnetizations are taken to be in the
$x$-$y$ plane to examine the spin-triplet correlation with the quantization
axis ($z$) perpendicular to this plane. A different magnetic configuration has
also been investigated and discussed in the appendix.
$\vec{S}_{j}=[(c_{j,\uparrow}^{\dagger}c_{j,\uparrow}-c_{j,\downarrow}^{\dagger}c_{j,\downarrow})\hat{z}+(c_{j,\uparrow}^{\dagger}c_{j,\downarrow}+c_{j,\downarrow}^{\dagger}c_{j,\uparrow})\hat{x}+(-ic_{j,\uparrow}^{\dagger}c_{j,\downarrow}+ic_{j,\downarrow}^{\dagger}c_{j,\uparrow})\hat{y}]$
is the spin operator of an electron at site $j$ of the superconductor.

The last term in Eq. (II) accounts for the conventional $s$-wave spin-singlet
superconducting correlations. It is obtained by mean field approximation of
the pairing interaction $-U\sum_{j}n_{j,\uparrow}n_{j,\downarrow}$, where
$U>0$ is the attractive pairing potential and
$n_{j,\sigma}=c_{j,\sigma}^{\dagger}c_{j,\sigma}$ is the number operator [54].
$\Delta_{j}=-U\langle c_{j,\downarrow}c_{j,\uparrow}\rangle$ is the resulting
superconducting order parameter.

The total Hamiltonian Eq. (II) is numerically diagonalized and the
superconducting state is determined self-consistently as detailed in Appendix
B. The exact parameters employed in our numerical routines have been specified
in Appendix E.

##  III Triplet correlations

In this section, we quantify and investigate the different spin-triplet
correlations in the S. Consider the anomalous Matsubara Green’s function
$F_{jj,\sigma\sigma^{\prime}}(\tau)=-\langle
T_{\tau}c_{j,\sigma}(\tau)c_{j,\sigma^{\prime}}(0)\rangle$, where
$\tau=i\tilde{t}$ is the imaginary time with $\tilde{t}$ as the time [60, 61].
Further, $T_{\tau}$ is the ordering operator for imaginary time $\tau$. In the
Fourier space, we obtain

$\displaystyle
F_{jj,\sigma\sigma^{\prime}}(i\omega_{l})=\int_{0}^{\beta}e^{i\omega_{l}\tau}F_{jj,\sigma\sigma^{\prime}}(\tau)~{}d\tau,$
(2)

where $\beta=\hbar/k_{B}T$, $k_{B}$ is the Boltzmann constant, $T$ is the
temperature, and $\omega_{l}=(2l+1)\pi/\beta$ are the fermionic Matsubara
frequencies with integer $l$. See Appendix B for further calculation details.
Since the spin-triplet correlations are odd in frequency [5], we take a sum
over all positive Matsubara frequencies to define an appropriate dimensionless
quantity that would allow us to quantify the correlations

$\displaystyle
F_{j,\sigma\sigma^{\prime}}=\frac{1}{\beta}\sum_{\omega_{l}>0}F_{jj,\sigma\sigma^{\prime}}(i\omega_{l}).$
(3)

Employing this notation, we express the relevant superconducting correlations

$\displaystyle F^{s}_{j}$
$\displaystyle=\frac{1}{2}\left(F_{j,\downarrow\uparrow}-F_{j,\uparrow\downarrow}\right),$
(4) $\displaystyle F^{t,z}_{j}$
$\displaystyle=-\frac{1}{2}\left(F_{j,\downarrow\uparrow}+F_{j,\uparrow\downarrow}\right),$
(5) $\displaystyle F^{t,x}_{j}$
$\displaystyle=\frac{1}{2}\left(F_{j,\uparrow\uparrow}-F_{j,\downarrow\downarrow}\right),$
(6) $\displaystyle F^{t,y}_{j}$
$\displaystyle=\frac{i}{2}\left(F_{j,\uparrow\uparrow}+F_{j,\downarrow\downarrow}\right).$
(7)

$F^{s}$ is the spin-singlet correlation. $F^{t,z}$, $F^{t,x}$, and $F^{t,y}$
are the zero-spin triplet correlations when spin is measured along $z$-, $x$-,
and $y$-axis respectively. Together, the latter three [Eqs. (5)-(7)] allow us
to express all three kinds of the spin-triplet correlations with $z$
quantization axis. We evaluate the quantities defined in Eqs. (4)-(7) to
investigate the different superconducting correlations in our system.

###  III.1 Numerical results

In an isolated conventional superconductor, only $F^{s}$ is non-zero while
$F^{t,x}$, $F^{t,y}$, and $F^{t,z}$ are zero. Now, if we consider an F/S
bilayer,  then the propagating electronic wavefunctions in the normal state of
the S film acquire a spin-dependent phase associated with their spin-dependent
energies resulting from the spin-splitting induced by the F [62]. The result
is a relative phase difference between the opposite spin electrons as they
propagate. In the superconducting state of the S layer, this causes the zero-
spin triplet correlation with spin-quantization axis along the F magnetization
to become non-zero [62, 5]. For example, if the magnetization of the
ferromagnet is along the $z$-axis, then $F^{t,z}$ becomes non-zero while
$F^{t,x}$ and $F^{t,y}$ remain zero. Similarly, $F^{t,x}$ and $F^{t,y}$ become
non-zero when magnetization of the ferromagnet is along $x$\- and $y$-axis,
respectively.

Quasiclassical theory [60, 61] shows that the triplet vector
$\vec{F^{t}_{j}}=F^{t,x}_{j}\hat{x}+F^{t,y}_{j}\hat{y}+F^{t,z}_{j}\hat{z}$
always has a component aligning with the local exchange field $J\vec{M}_{j}/2$
whether the magnetization of the ferromagnet in a F/S bilayer is homogeneous
or inhomogeneous [63, 4]. We, therefore, study these correlations in order to
decompose the contribution of antiferromagnetic and ferromagnetic components
of the canted-AF.

Figure 2: Spatial variation of the normalized triplet correlations for 10
lattice sites considering maximal noncollinearity corresponding to the canting
angle $\theta_{t}=\pi/4$ and filling factor $f=0.6$ ($\mu/\Delta_{0}\approx
37$, where $2\Delta_{0}$ is the zero-temperature superconducting gap without
the adjacent AF). We plot the real part of the zero-spin triplet correlation
$F^{t,z}_{j}$ [panel (a)], and the imaginary parts of the spin-triplet
correlations $F^{t,x}_{j}$ [panel (b)] and $F^{t,y}_{j}$ [panel (c)]. The
imaginary part of the former and the real parts of the latter two are zero.
All the correlations are normalized by the spatially averaged magnitude of the
singlet correlation $\overline{|F^{s}|}$. The detailed parameters employed for
the numerical evaluation are specified in Appendix E.

For $\theta_{t}=0$ (Fig. 1), our considered AF becomes a collinear
antiferromagnet with the axis of magnetic moments along the $x$-direction. As
we increase the value of $\theta_{t}$ by a small amount, the canted-AF
acquires a net magnetization along the $y$-direction. So the canted-AF can be
decomposed into an antiferromagnetic component (along the $x$-axis) and a
ferromagnetic component (along the $y$-axis). For a collinear AF, we obtain
Néel triplets. This means the component of $\vec{F^{t}_{j}}$ parallel to the
axis of the Néel vector modulates with the Néel order of the AF [49]. For
$\theta_{t}=\pi/2$, we effectively obtain a F/S structure with the F
magnetization along the $y$-axis.

We now investigate the case of $\theta_{t}=\pi/4$ that produces maximum
noncollinearity between the two AF sublattice magnetization. In Fig. 2,
triplet correlations for a canted-AF/S bilayer have been plotted as a function
of space for filling factor $f=0.6$. We see that $F^{t,z}$, the component of
$\vec{F^{t}}$ perpendicular to the canted-AF sublattice magnetizations plane,
is also being generated along with the in-plane components $F^{t,x}$ and
$F^{t,y}$. It oscillates from a constant positive to negative value with Néel
order.  This oscillation can be understood in terms of the noncollinearity
that generates $F^{t,z}$. From one lattice site to the next, the angle between
the spin-splitting at the site and its direct neighbors is changing sign.
However, $F^{t,z}$ only appears for non-zero $\mu$, and is zero at half-
filling ($f=0.5$) where $\mu=0$. $F^{t,x}$ too oscillates between a constant
positive and negative value with Néel order, while $F^{t,y}$ is constant in
space. Both $F^{t,x}$ and $F^{t,y}$ are imaginary, consistent with previous
theoretical results for F/S bilayers [63, 62]. The AF sublattice magnetization
configuration has been so chosen here to obtain and focus on the non-zero and
spatially constant sum of the equal-spin triplets $F_{\uparrow\uparrow}$ and
$F_{\downarrow\downarrow}$. In Appendix F, we present the correlations for a
configuration when the AF Néel order is aligned with the $z$-axis [49]. To
conclude this discussion, the intrinsic noncollinearity of the canted AF
successfully generates all three components of the spin-triplet correlations.
The generation of spinful triplet Cooper pairs has conventionally been
accomplished via the noncollinearity between the magnetizations of different
ferromagnetic layers [33].

Figure 3: Variation of triplet correlations with canting angle $\theta_{t}$
for filling factors $f=0.5$ $(\mu=0)$ and $f=0.6$ $(\mu/\Delta_{0}\approx
37)$, where $\Delta_{0}$ represents the zero-temperature value of the
superconducting order parameter in absence of the canted-AF layer. (a) The
average magnitude of the normalized spin-triplet correlation $F^{t,z}$ is
maximum when the noncollinearity between the two sublattices is maximum, i.e.
at $\theta_{t}=\pi/4$. However, it is zero at half-filling ($f=0.5$). (b)
Average magnitude of the normalized spin-triplet correlation
$F^{t,x}=[F_{\uparrow\uparrow}-F_{\downarrow\downarrow}]/2$ decreases as the
effective antiferromagnetism becomes weaker with increasing $\theta_{t}$. (c)
The average magnitude of the normalized spin-triplet correlation
$F^{t,y}=[i(F_{\uparrow\uparrow}+F_{\downarrow\downarrow})]/2$ increases as
the effective ferromagnetism becomes stronger with increasing $\theta_{t}$.
The averages are taken over all the sites and are denoted via an overhead bar.
The detailed parameters employed for the numerical evaluation are specified in
Appendix E.

We now examine the dependence of these spin-triplets on the canting angle. To
this end, we plot the average magnitudes of the three spin-triplet
correlations vs. the canting angle $\theta_{t}$ in Fig. 3. As we change
$\theta_{t}$ from $0$ to $\pi/2$ the system changes from a collinear AF (along
$x$-axis) to a collinear F (along $y$-axis). As discussed above, $F^{t,z}$ is
found to vanish identically at $\mu=0$ ($f=0.5$) for all canting angles. This
will be explained further below. However, for non-zero $\mu$ (away from the
half-filling case), it increases from 0 to a finite value as we go from a
collinear antiferromagnetic alignment to maximal noncollinearity between the
sublattice magnetic moments, and decreases back to zero in the ferromagnetic
alignment [Fig. 3(a)]. This component thus results directly due to the AF
sublattice magnetization noncollinearity.

The spatially averaged value of $|F^{t,x}|$ decreases as $\theta_{t}$ goes
from 0 to $\pi/2$ [Fig. 3(b)]. This component, therefore, essentially follows
the Néel vector magnitude and appears to stem directly from the
antiferromagnetism [49]. It has been understood as being due to the interband
pairing, which is feasible when $\mu$ is smaller or comparable to the
superconducting gap. However, we also find such Néel triplets to be present
for $\mu\approx 37\Delta_{0}$ ($f=0.6$), although they are significantly
weaker than for the case of $\mu=0$ [Fig. 3(b)]. We attribute this observation
to a modification of the normal state electronic wavefunctions by the AF, so
that even the conventional intraband pairing causes a finite generation of the
Néel triplets. Finally, the average value of $|F^{t,y}|$ increases with the
canting angle and it appears to be caused primarily by the net magnetization.

###  III.2 Insights from simplified analytics

In order to understand the difference between $\mu=0$ and $\mu\neq 0$ cases,
we examine the electronic properties of a bilayer comprising a normal metal
and an AF, as detailed in Appendix A. For $\mu=0$, the Fermi wave-vector is
$k_{F}=\pi/2a$. This means that the electrons participating in the formation
of Cooper pairs have $|k|\sim\pi/2a$. The eigenfunctions with $|k|\sim\pi/2a$
are such that the probability of finding an electron is non-zero on one
sublattice while it is zero on the other sublattice. Now, the electrons near
the Fermi level of the superconductor which interact with sublattice A do not
see sublattice B and vice-versa. Therefore the triplet correlations are
generated independently by sublattices A and B. The resultant correlations we
obtain are a sum of correlations generated by the two sublattices. So, the
only non-zero components are $F^{t,x}$ and $F^{t,y}$. This separation of the
two sublattices at a special value of the electronic chemical potential is
reminiscent of a similar result obtained for spin pumping via AFs [64, 65, 66,
67].

In contrast, for $\mu\neq 0$, the Fermi level is within one of the bands and
the wavefunctions of the states near the Fermi energy are such that the
electrons on a site of the A sublattice also have a non-zero probability at
the sites of the B sublattice. There is no way for the electrons to arrange
themselves to decouple the two sublattices [66, 67]. The electrons experience
a spin-splitting field in one direction at site 1 (of sublattice A) and in
another direction at site 2 (of sublattice B), then again the first
orientation at site 3 (of sublattice A). So, the electrons see the
noncollinearity between the magnetic moments on the adjacent lattice sites.
Therefore, the correlation $F^{t,z}$ along the direction perpendicular to the
plane of magnetic moments of the canted-AF becomes non-zero along with the in-
plane components $F^{t,x}$ and $F^{t,y}$ when $\mu\neq 0$. At the same time,
the electronic amplitudes at the two sublattices are different due to the
adjacent AF, as detailed in Appendix A. This lends the normal electronic
states a weak Néel character which manifests itself in the emergence of Néel
triplets even for the conventional intraband pairing.

##  IV Critical temperature

The formation of spin-triplet Cooper pairs comes at the cost of destroying
their spin-singlet counterparts that are originally produced in and stabilize
the superconducting state [68, 69, 59]. Hence, the critical temperature is
reduced with the formation of spin-triplets, which may offer a convenient
experimental signature. Thus, we investigate the critical temperature of our
AF/S bilayer now via numerical self-consistent solution of the BdG equation
(II).

Figure 4: (a) Normalized critical temperature $T_{c}$ vs. canting angle
$\theta_{t}$ for filling factor (a) $f=0.5$ and (b) $f=0.6$ considering
different values of the spin-splitting $J$. Here, $T_{c0}$ and $2\Delta_{0}$
are respectively the critical temperature and the zero-temperature
superconducting gap of the same superconductor without the AF layer. The
detailed parameters employed for the numerical evaluation are specified in
Appendix E.

Critical temperature $T_{c}$ vs. canting angle $\theta_{t}$ is plotted in Fig.
4 for (a) $f=0.5$ ($\mu=0$) and (b) $f=0.6$ ($\mu\approx 37\Delta_{0}$). We
find that for $\mu=0$, $T_{c}$ increases with $\theta_{t}$ while it manifests
the opposite dependence for $\mu\neq 0$. These intriguing and distinct
dependencies can be understood based on our analysis of spin-triplets
generation above.

Let us first consider the $f=0.5$, corresponding to $\mu=0$, case presented in
Fig. 4(a). In this case, a strong generation of spin-zero Néel triplets [Fig.
3(b)] due to interband pairing leads to maximal $T_{c}$ suppression at
$\theta_{t}=0$. Hence, the $T_{c}$ increases with $\theta_{t}$ since the
$T_{c}$ suppression is stronger for the collinear AF case ($\theta_{t}=0$)
than for the F case ($\theta_{t}=\pi/2$). Further, when the exchange field is
large enough, we find a complete suppression of superconductivity at
$\theta_{t}=0$ corresponding to a vanishing $T_{c}$. The abrupt change in
$T_{c}$ with $\theta_{t}$ here is attributed to the additional contribution to
superconductivity suppression by the opening of a normal dispersion bandgap by
the AF, as described in Appendix A. This normal state bandgap predominantly
affects the superconducting pairing at half-filling when $\mu=0$.

For the case of $f=0.6$, the Néel spin-triplets generation by the
antiferromagnetic order is much weaker (Fig. 3). On the other hand, the
ordinary spin-triplets generation by a ferromagnet remain of the same order of
magnitude as for $f=0.5$. Thus, $T_{c}$ is largest for $\theta_{t}=0$ and it
decreases with $\theta_{t}$. The amplitude of spin-triplets generated due to
the noncollinearity [Fig. 3(a)] remains small and does not seem to affect the
$T_{c}$ dependence substantially. Contrary to the $\mu=0$ case, the variation
of $T_{c}$ with $\theta_{t}$ is smooth even for large values of the exchange
field $J$.

##  V Correlations in 2-D

In our discussion above, we have considered a one-dimensional (1-D)
superconductor with the aim of examining essential physics employing analytic
results discussed in Appendix A. We now validate these results using a two-
dimensional (2-D) model for the S. This further allows us to examine how the
spin-triplet correlations vary with space as we move away from the S/AF
interface.

In Fig. 1(b), the superconducting lattice is along the $x$-axis. We add more
such 1-D layers in the $y$-direction to create our 2-D model for the S. Each
site of this 2-D sheet is indexed as $(j_{x},j_{y})$ so that $j_{x}$ takes a
value between 1 to $N_{x}$, and $j_{y}$ takes a value between 1 to $N_{y}$,
where $N_{x}$ and $N_{y}$ are the number of sites along $x$\- and
$y$-direction, respectively. The spin-splitting effect is experienced only by
the electrons at the sites next to the AF/S interface. Thus, it suffices to
treat the AF via the same 1-D model as before. Overall, the Hamiltonian of Eq.
(II) is modified by letting all site indices $j$ take the form
$(j_{x},j_{y})$. Summation in the spin-splitting term is now over the sites
with indices of the form $(j_{x},1)$. We continue to consider periodic
boundary condition along the $x$-axis, like in the 1-D case.

Figure 5: Spatial variation of the spin-triplet correlations for a 2-D
superconductor. The sites are indexed as $(j_{x},j_{y})$. The sites with index
$(j_{x},1)$ form the layer adjacent to the AF. A small section of size
$14\times 14$ of a superconducting sheet of size $102\times 14$ has been
plotted here for clarity. The real part of the normalized zero-spin triplet
correlation $F^{t,z}_{j}$ [panel (a)], and the imaginary parts of the
normalized spin-triplet correlations $F^{t,x}_{j}$ [panel (b)] and
$F^{t,y}_{j}$ [panel (c)] have been plotted as colormaps. The imaginary part
of the former and the real parts of the latter two are zero. We consider
maximal noncollinearity ($\theta_{t}=\pi/4$) and filling fraction $f=0.6$
($\mu=18.5\overline{|\Delta_{0}|}$). Here, $\overline{|\Delta_{0}|}$ is the
average magnitude of the superconducting order parameter in absence of the
canted-AF layer and all the correlations are normalized with respect to the
magnitude of the singlet correlation $|F^{s}_{j}|$. The detailed parameters
employed for the numerical evaluation are specified in Appendix E.

Carrying out the BdG diagonalization self-consistently and numerically, we
evaluate the spatially-resolved spin-triplet correlations [Eqs. (4)-(7)] for
this system and plot them in Fig. 5. We have the same observations in the
first layer (along the AF/S interface) of the 2-D sheet as the 1-D case
discussed in section III. Additional calculations not presented here confirm
that $F^{t,z}$ appears only when $\mu$ is non-zero. We observe that $F^{t,x}$
and $F^{t,z}$ show modulation between positive and negative values along the
$y$-axis apart from along the $x$-axis. This is interesting because the system
has nothing imposing Néel order along the $y$-axis on the correlations. In
addition, we find some layers in which the alternating pattern of positive and
negative values is skipped. This is attributed to Friedel-like oscillations
[54, 49]. We found that these skipping of pattern only appears for non-zero
$\mu$ while we get a perfect alternating pattern along the $y$-axis for
$\mu=0$ case.  Although these Néel triplet correlations flip sign at the
length scale of a lattice constant, their magnitude decays at the coherence
length scale (shown in Appendix C). The correlation component $F^{t,y}$ [Fig.
5(c)] is constant in each layer along the interface (along the $x$-axis) but
decays as we move away from the interface along the $y$-direction. So, the 2-D
case is consistent with and corroborates our 1-D results,  resulting in a
similar trend in $T_{c}$ suppression (Appendix D). We find that the spatial
pattern (oscillating with Néel order or being constant in space) imposed on
superconducting correlations along the interfacial direction also manifests
itself perpendicular to the AF/S interface.

Figure 6:  Spatial variation of the $p_{y}$-wave triplet correlations in a 2-D
superconductor. The sites are indexed as $(j_{x},j_{y})$. The sites with index
$(j_{x},1)$ form the layer adjacent to the AF. A small section of size
$12\times 12$ of a superconducting sheet of size $202\times 14$ has been
plotted here for clarity. The imaginary part of the normalized zero-spin
$p_{y}$-wave triplet correlation $P^{t,z}_{j}$ [panel (a)], and the real parts
of the normalized $p_{y}$-wave spin-triplet correlations $P^{t,x}_{j}$ [panel
(b)] and $P^{t,y}_{j}$ [panel (c)] have been plotted here. The real part of
the former and the imaginary parts of the latter two are zero. We consider
maximal noncollinearity ($\theta_{t}=\pi/4$) and filling factor $f=0.6$. All
the correlations are normalized with respect to the magnitude of the $s$-wave
singlet correlation $|F^{s}_{j}|$. The detailed parameters employed for the
numerical evaluation are specified in Appendix E.

In the 2-D case, where the AF/S interface breaks translational symmetry along
the $y$-axis, even-frequency $p_{y}$-wave spin-triplet correlations are
created, in addition to the odd-frequency $s$-wave ones [15, 13]. These are
quantified as [70]

$\displaystyle P^{t,z}_{j}=\frac{1}{8}\left[\langle
c_{j,\downarrow}c_{j+\hat{y},\uparrow}\rangle+\langle
c_{j,\uparrow}c_{j+\hat{y},\downarrow}\rangle-\langle
c_{j,\downarrow}c_{j-\hat{y},\uparrow}\rangle-\langle
c_{j,\uparrow}c_{j-\hat{y},\downarrow}\rangle\right],$ (8) $\displaystyle
P^{t,x}_{j}=\frac{1}{8}\left[-\langle
c_{j,\uparrow}c_{j+\hat{y},\uparrow}\rangle+\langle
c_{j,\downarrow}c_{j+\hat{y},\downarrow}\rangle+\langle
c_{j,\uparrow}c_{j-\hat{y},\uparrow}\rangle-\langle
c_{j,\downarrow}c_{j-\hat{y},\downarrow}\rangle\right],$ (9) $\displaystyle
P^{t,y}_{j}=\frac{i}{8}\left[-\langle
c_{j,\uparrow}c_{j+\hat{y},\uparrow}\rangle-\langle
c_{j,\downarrow}c_{j+\hat{y},\downarrow}\rangle+\langle
c_{j,\uparrow}c_{j-\hat{y},\uparrow}\rangle+\langle
c_{j,\downarrow}c_{j-\hat{y},\downarrow}\rangle\right],$ (10)

where the site index $j\pm\hat{y}\equiv(j_{x},j_{y}\pm 1)$ are the nearest
neighbors of site $j$ along the $y$-axis, and $P^{t,\alpha}_{j}$ represent the
zero-spin $p_{y}$-wave triplet correlation when the spin is measured along the
$\alpha$-axis ($\alpha=x,y,z$).

While the $s$-wave spin-triplets in Fig. 5(a)-(b) nearly follow a checkerboard
pattern, $P^{t,z}$ and $P^{t,x}$ tend to form stripes along the $y$-direction
[Fig. 6(a)-(b)] 111As the $p_{y}$-wave triplet correlations are not defined at
the edges, we plot these starting at the second row of lattice sites from the
edge.. Unlike the on-site $s$-wave pairing, the $p_{y}$-wave pairing cannot
take advantage of the complete decoupling of the two sublattices at half-
filling as they are defined on the link between sites in the A and B
sublattice. For $P^{t,x}$, this can be seen by realizing that $\langle
c_{j,\sigma}c_{j+\hat{y},\sigma}\rangle=-\langle
c_{(j+\hat{y}),\sigma}c_{(j+\hat{y})-\hat{y},\sigma}\rangle$. Thus, if the
link in the positive $y$-direction from site $j$ gives a positive contribution
to $P^{t,x}$, then the link in the negative $y$-direction from site
$j+\hat{y}$ also gives a positive contribution, as can be seen by insertion
into Eq. (9). Thus, a link from the A sublattice to the B sublattice and a
link from the B sublattice to the A sublattice give a contribution to
$P^{t,x}$ of the same sign. At half-filling, the sign of $P^{t,x}$ is thus
dictated by the sign of the spin-splitting induced at the lattice site closest
to the interface. Away from half-filling, where the two sublattices are no
longer perfectly decoupled, the perfect stripe pattern can be shifted along
the $x$-direction as can be seen for $P^{t,z}$ in Fig. 6(a). The ferromagnetic
component $P_{j}^{t,y}$ in Fig. 6(c) remains positive inside the whole
superconductor, similar to the $s$-wave triplets in Fig. 5(c). Although we
here observe qualitatively different behavior in the nearest-neighbor (p-wave)
pairing compared to the on-site (s-wave) one, we must note that the
antiferromagnetic component and the component resulting from canting are both
one order of magnitude larger for the latter. Thus, the AF’s influence on the
critical temperature and other key superconducting properties is dominated by
the odd-frequency s-wave Néel triplets generated in the S.

##  VI Concluding remarks

We have theoretically demonstrated the generation of all, including the
spinful, spin-triplet Cooper pairs in a conventional superconductor by an
adjacent canted antiferromagnet. Our proposal leverages the intrinsic
noncollinearity between the two sublattice magnetizations in a canted
antiferromagnet for applications in superconducting hybrids. This canting can
be induced intrinsically by Dzyaloshinskii-Moriya interaction, such as in
hematite [72]. Additionally, it can be induced and controlled using an applied
magnetic field [73]. The resulting spin-triplets have a predominantly Néel
character, i.e., their amplitude oscillates in space on the lattice length
scale similar to the Néel spin order.  While we have considered a lattice
matched interface for concreteness and simplicity, the essential physics
remains the same even in the presence of disorder with a gradual suppression
of the Néel triplets with increasing disorder [49]. Furthermore, a metallic AF
would lead to similar phenomena as discussed here, with the additional
complication of Cooper pairs leaking into the AF. The superconducting critical
temperature is more strongly suppressed by the interband Néel spin-triplets
than by a ferromagnetic spin-splitting field of similar magnitude, thereby
offering an experimental signature of their generation. Altogether, our
analysis highlights the noncollinear nature of homogeneous canted
antiferromagnets by employing them for generating spin-triplet Cooper pairs in
a simple superconducting bilayer. This manner of generating noncollinearity
using a homogeneous antiferromagnet is expected to find use in other phenomena
that have traditionally relied on magnetic multilayers or spin textures.

##  Appendix A Normal metal interfaced with an antiferromagnet

In order to understand when interband and intraband pairing is favored, we
consider the normal-state wave functions of an AF/normal metal bilayer. The
Hamiltonian for the conducting electrons of a normal metal interfaced with an
insulating AF is modeled as [74]

$\displaystyle H=$
$\displaystyle-\mu\sum_{j,\sigma}c_{j,\sigma}^{\dagger}c_{j,\sigma}-t\sum_{\left<i,j\right>}\sum_{\sigma}c_{i,\sigma}^{\dagger}c_{j,\sigma}-\frac{J}{2}\sum_{j}\vec{M}_{j}\cdot\vec{S}_{j},$
(11)

where $\vec{M_{j}}=(-1)^{j+1}\hat{z}$ gives the magnetic texture of a
collinear AF. Other symbols have the same meaning as in section II of the main
text.

In order to calculate the eigenenergies and eigenvectors, the Hamiltonian is
written in terms of creation and annihilation operators for electrons at
sublattices A and B. Creation operators for electrons at sublattices A and B
are defined as $a_{j,\sigma}^{\dagger}=c^{\dagger}_{2j-1,\sigma}$, and
$b_{j,\sigma}^{\dagger}=c^{\dagger}_{2j,\sigma}$, where $j=1,2,\dots,N/2$.
Then, the basis is changed from Wannier wavefunctions to Bloch wavefunctions
using the relation

$\displaystyle a^{\dagger}_{j,\sigma}=\sum_{k\in
FBZ}\frac{1}{\sqrt{N/2}}e^{-ik(2j-2)a}a_{k,\sigma}^{\dagger},\quad\text{and}\quad
b_{j,\sigma}^{\dagger}=\sum_{k\in
FBZ}\frac{1}{\sqrt{N/2}}e^{-ik(2j-1)a}b_{k,\sigma}^{\dagger},$ (12)

where $a$ is the lattice constant, and $k$ is a reciprocal lattice vector in
the first Brillouin zone (FBZ). Now, the Hamiltonian can be written in the
form

$\displaystyle
H=\sum_{k}\begin{pmatrix}a_{k,\uparrow}^{\dagger}&a_{k,\downarrow}^{\dagger}&b_{k,\uparrow}^{\dagger}&b_{k,\downarrow}^{\dagger}\end{pmatrix}H(k)\begin{pmatrix}a_{k,\uparrow}&a_{k,\downarrow}&b_{k,\uparrow}&b_{k,\downarrow}\end{pmatrix}^{T},$
(13)

giving the dispersion relation

$\displaystyle E_{\pm}(k)=-\mu\pm\sqrt{4t^{2}\cos^{2}(ka)+(J/2)^{2}},$ (14)

where the two energy bands $E_{\pm}(k)$ are two-fold degenerate. The
eigenvectors for energy band $E_{-}(k)$ are

$\displaystyle\psi_{1}(k)=$
$\displaystyle\mathcal{N}_{2}\begin{pmatrix}2t\cos(ka)\\\ 0\\\
-J/2+\sqrt{4t^{2}\cos^{2}(ka)+(J/2)^{2}}\\\
0\end{pmatrix}=\begin{pmatrix}u_{Ak\uparrow}^{(1)}\\\
u_{Ak\downarrow}^{(1)}\\\ u_{Bk\uparrow}^{(1)}\\\
u_{Bk\downarrow}^{(1)}\end{pmatrix},\text{ and}$ (15)
$\displaystyle\psi_{2}(k)=$ $\displaystyle\mathcal{N}_{1}\begin{pmatrix}0\\\
2t\cos(ka)\\\ 0\\\
J/2+\sqrt{4t^{2}\cos^{2}(ka)+(J/2)^{2}}\end{pmatrix}=\begin{pmatrix}u_{Ak\uparrow}^{(2)}\\\
u_{Ak\downarrow}^{(2)}\\\ u_{Bk\uparrow}^{(2)}\\\
u_{Bk\downarrow}^{(2)}\end{pmatrix}.$ (16)

Whereas the eigenvectors for the energy band $E_{+}(k)$ are

$\displaystyle\psi_{3}(k)=$
$\displaystyle\mathcal{N}_{1}\begin{pmatrix}2t\cos(ka)\\\ 0\\\
-J/2-\sqrt{4t^{2}\cos^{2}(ka)+(J/2)^{2}}\\\
0\end{pmatrix}=\begin{pmatrix}u_{Ak\uparrow}^{(3)}\\\
u_{Ak\downarrow}^{(3)}\\\ u_{Bk\uparrow}^{(3)}\\\
u_{Bk\downarrow}^{(3)}\end{pmatrix},\text{ and}$ (17)
$\displaystyle\psi_{4}(k)=$ $\displaystyle\mathcal{N}_{2}\begin{pmatrix}0\\\
2t\cos(ka)\\\ 0\\\
J/2-\sqrt{4t^{2}\cos^{2}(ka)+(J/2)^{2}}\end{pmatrix}=\begin{pmatrix}u_{Ak\uparrow}^{(4)}\\\
u_{Ak\downarrow}^{(4)}\\\ u_{Bk\uparrow}^{(4)}\\\
u_{Bk\downarrow}^{(4)}\end{pmatrix}.$ (18)

Here, $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ are the normalization factors of
the eigenvectors. $\psi_{1}$ and $\psi_{3}$ correspond to the wavefunctions of
spin $\uparrow$ electrons, whereas $\psi_{2}$ and $\psi_{4}$ correspond to
spin $\downarrow$ electrons. Fig. 7 shows a plot of the non-zero components of
$\psi_{1}$ for positive $k$ values . Let us compare the case of $J=0$ and
$J\neq 0$ to study how the states of a metal are modified when it is brought
in contact with an AF.

Figure 7: Non-zero components of eigenfunction $\psi_{1}$ (Eq. (15)) of a
normal metal/AF bilayer for different strengths of magnetic exchange
interaction as a function of positive $k$ values in the first Brillouin zone.

From Fig. 7, we see that the probabilities of finding a spin $\uparrow$
electron at sites of sublattice A and B are equal for $J=0$. However, for
$J\neq 0$, the probability on A sublattice is more than that on B sublattice.
It is important to note that the probability of finding spin $\uparrow$
electrons at sublattice A sites becomes 1 and that for sublattice B becomes 0
at $k=\pi/2a$. Similar asymmetries in the sublattices arise for states
$\psi_{2}$, $\psi_{3}$, and $\psi_{4}$ as soon as we make $J$ non-zero. For
$\psi_{2}$, the probability of finding a spin $\downarrow$ electron at
sublattice site B becomes more than that for sublattice A. For the states of
$E_{+}(k)$ band ($\psi_{3}$ and $\psi_{4}$), the roles of sublattices A and B
are interchanged with respect to spin.

Near the Brillouin zone boundary (BZB) of band $E_{-}(k)$ (i.e. $\psi_{1}$ and
$\psi_{2}$), all the spin $\uparrow$ electrons get localized to the sites of
sublattice A and all the spin $\downarrow$ electrons get localized to the
sites of sublattice B. For the band $E_{+}(k)$, sublattices A and B
interchange their roles and we find that spin $\uparrow$ electrons get
localized at sublattice B and spin $\downarrow$ electrons get localized at
sublattice A near the BZB. This property of the electronic states near the BZB
is the reason why interband pairing is the dominant mechanism for the
formation of on-site opposite-spin Cooper pairs in an AF/S bilayer when the
Fermi level lies within the antiferromagnetic band gap $E_{+}-E_{-}$. On the
other hand, intraband pairing is the dominant mechanism when the Fermi level
lies within one of the two bands.

##  Appendix B Bogoliubov-de Gennes calculation

The Hamiltonian in Eq. (II) can be written as

$\displaystyle H=-\mu
N+\sum_{j}\frac{|\Delta_{j}|^{2}}{U}+\frac{1}{2}\sum_{i,j}\Psi_{i}^{\dagger}\tilde{H}_{i,j}\Psi_{j},$
(19)

where
$\Psi_{j}^{\dagger}=\begin{pmatrix}c_{j,\uparrow}^{\dagger}&c_{j,\downarrow}^{\dagger}&c_{j,\uparrow}&c_{j,\downarrow}\end{pmatrix}$
and $N$ is the total number of sites. The matrix $\tilde{H}$ is diagonalized
by solving the BdG equations

$\displaystyle\sum_{j}\tilde{H}_{i,j}\phi_{j,n}=E_{n}\phi_{i,n},\text{ where}$
(20)
$\displaystyle\phi_{i,n}=\begin{pmatrix}u_{i,n,\uparrow}&u_{i,n,\downarrow}&v_{i,n,\uparrow}&v_{i,n,\downarrow}\end{pmatrix}^{T},$
(21)

is the eigenvector and $E_{n}$ the eigenenergies of $\tilde{H}$. Now the
Hamiltonian can be written as

$\displaystyle H=-\mu
N+\sum_{j}\frac{|\Delta_{j}|^{2}}{U}-\frac{1}{2}\tilde{\sum_{n}}E_{n}+\tilde{\sum_{n}}\Gamma_{n}^{\dagger}E_{n}\Gamma_{n},$
(22)

where $\tilde{\sum}_{n}$ represents sum over positive eigenenergies, and
$\Gamma_{n}$’s are Bogoliubov fermionic operators related to the old fermionic
operators by
$c_{j,\sigma}=\tilde{\sum}_{n}(u_{j,n,\sigma}\Gamma_{n}+v_{j,n,\sigma}^{*}\Gamma_{n}^{\dagger})$.
The superconducting order parameter is calculated self-consistently using the
relation

$\displaystyle\Delta_{j}=-U\tilde{\sum_{n}}\left[u_{j,n,\downarrow}v_{j,n,\uparrow}^{*}[1-f_{FD}(E_{n})]+v_{j,n,\downarrow}^{*}u_{j,n,\uparrow}f_{FD}(E_{n})\right],$
(23)

where
$f_{FD}(E_{n})=\langle\Gamma_{n}^{\dagger}\Gamma_{n}\rangle=1/(e^{E_{n}/k_{B}T}+1)$
is the Fermi-Dirac distribution.

In order to calculate the spin-triplet correlations, we consider the anomalous
Matsubara Green’s function $F_{jj,\sigma\sigma^{\prime}}(\tau)=-\langle
T_{\tau}c_{j,\sigma}(\tau)c_{j,\sigma^{\prime}}(0)\rangle$, where
$\tau=i\tilde{t}$ is the imaginary time, $\tilde{t}$ is the time, and
$T_{\tau}$ is the ordering operator for $\tau$. Taking its Fourier transform,
we get

$\displaystyle
F_{jj,\sigma\sigma^{\prime}}(i\omega_{l})=\int_{0}^{\beta}e^{i\omega_{l}\tau}F_{jj,\sigma\sigma^{\prime}}(\tau)\,d\tau=\tilde{\sum_{n}}\left[\frac{u_{j,n,\sigma}v_{j,n,\sigma^{\prime}}^{*}}{i\omega_{l}-E_{n}/\hbar}+\frac{v_{j,n,\sigma}^{*}u_{j,n,\sigma^{\prime}}}{i\omega_{l}+E_{n}/\hbar}\right],$
(24)

where $\beta=\hbar/k_{B}T$, $k_{B}$ is the Boltzmann constant, $T$ is the
temperature, and $\omega_{l}=(2l+1)\pi/\beta$ is a Matsubara frequency for
fermions with integer $l$. This expression is used in calculating the
correlations in Eq. (3) and their relevant components Eqs. (4)- (7).

##  Appendix C Decay of triplet correlations with distance in 2-D

Figure 8:  The average magnitude of the $s$-wave spin-triplet correlations
normalized by the singlet correlation in each $j_{y}$ layer parallel to the
canted-AF/S interface
$\left(\overline{|F^{t,\alpha}/F^{s}|}_{j_{y}},\,\,\alpha=x,y\right)$ is
plotted with respect to the distance from the interface. We consider half
filling ($f=0.5$) and maximal canting ($\theta_{t}=\pi/4$). The sites with
index $j_{y}=1$ form the layer adjacent to the AF. Ignoring the edge effects,
we show only the first $25$ layers of a superconducting sheet with $102\times
30$ sites. The $F^{t,z}$ component is zero in all the layers. The detailed
parameters employed for the numerical evaluation are specified in Appendix E.

In section V, we discussed the spatial variation of triplet correlations in a
2-D superconductor interfaced with a canted-AF and observed that the Néel
triplet correlations flip sign on adjacent sites to form a checkerboard
pattern. In Fig. 8, we study the decay of the $s$-wave spin-triplet
correlations with the distance from the AF/S interface. We consider half-
filling ($f=0.5$) to avoid the influence of Friedel oscillations. We observe
that the magnitude of both the ferromagnetic and antiferromagnetic components
of the correlations decay exponentially over a length scale determined by the
coherence length. The rapid oscillations of the Néel order of the canted-AF
over the atomic length scale thus only affects the sign of the triplet
correlations.

##  Appendix D  Critical temperature of 2-D S/canted-AF

Figure 9:  The critical temperature $T_{c}$ vs. canting angle $\theta_{t}$ is
plotted for a 2-D superconductor of size $302\times 3$ interfaced with a
canted-AF at filling factors $f=0.5$ and $f=0.6$. $T_{c0}$ and
$\overline{|\Delta_{0}|}$ are the critical temperature and the average
superconducting order parameter at zero-temperature, respectively, in the
absence of the magnetic layer. The detailed parameters employed for the
numerical evaluation are specified in Appendix E.

In section IV, we found that the critical temperature vs. canting angle curves
of a 1-D superconductor show opposite trends at half-filling ($f=0.5$) and
away from half-filling ($f=0.6$). In Fig. 9, we show that $T_{c}$ behaves
similarly for a 2-D superconductor. The change in $T_{c}$ as we change the
canting angle from the collinear AF case ($\theta_{t}=0$) to the F case
($\theta_{t}=\pi/2$) however decreases in magnitude. This is because with an
increasing S thickness, the effective spin-splitting induced by the AF
decreases.

##  Appendix E Numerical parameters

The parameters used for each of the figures are

  * •

Fig. 2: A 1-D superconductor with number of sites $N=302$, hopping parameter
$t=10$, pair potential $U/t=1$, magnetic exchange interaction strength
$J=0.018\,t\,(=1.09\Delta_{0})$, canting angle $\theta_{t}=\pi/4$, and filling
factor $f=0.6$ $(\mu/\Delta_{0}=37)$, at temperature
$T=0.1\,T_{c}=0.001\,t/k_{B}$ is taken to calculate $s$-wave triplet
correlations. Here, $2\Delta_{0}$ is the zero-temperature superconducting gap
without the adjacent AF.

  * •

Fig. 3: A 1-D superconductor with $N=302$, $t=10$, and $U/t=1$ at
$k_{B}T/t=0.001$ is taken for calculating correlations. For filling factor
$f=0.5\,(\mu/\Delta_{0}=0)$, $J=0.012\,t=1.09\Delta_{0}$ is taken, and for
$f=0.6\,(\mu/\Delta_{0}=37)$, $J=0.018\,t=1.09\Delta_{0}$, is taken. Here,
$2\Delta_{0}$ represents the superconducting gap in the absence of the canted-
AF at zero-temperature.

  * •

Fig. 4: A 1-D superconductor with $N=302$, $t=10$, $U/t=1$ is used for the
calculation of $T_{c}$ for filling factors $f=0.5\,(\mu/\Delta_{0}=0)$ and
$f=0.6\,(\mu/\Delta_{0}=37)$. The critical temperature of the isolated
superconductor $T_{c0}$ for $f=0.6$ is $0.0101\,t/k_{B}$ and for $f=0.5$ is
$0.0076\,t/k_{B}$.

  * •

Fig. 5: A 2-D superconducting sheet with $N_{x}=102$, $N_{y}=14$, $t=10$,
$J=0.1\,t=4.3\,\overline{|\Delta_{0}|}$, $\theta_{t}=\pi/4$, $U/t=1$, and
$f=0.6\,(\mu/\overline{|\Delta_{0}|}=18.5)$ at $k_{B}T/t=0.001$ is used to
calculate the $s$-wave triplet correlations. Here, $\overline{|\Delta_{0}|}$
represents the average magnitude of the superconducting order parameter in
absence of the canted-AF layer.

  * •

Fig. 6: A 2-D superconductor with $N_{x}=202$, $N_{y}=14$, $t=10$,
$J=0.1\,t=4.3\,\overline{|\Delta_{0}|}$, $\theta_{t}=\pi/4$, $U/t=1$, and
$f=0.6\,(\mu/\overline{|\Delta_{0}|}=18)$ at $k_{B}T/t=0.001$ is used to
calculate the $p_{y}$-wave triplet correlations. Here,
$\overline{|\Delta_{0}|}$ represents the average magnitude of the
superconducting order parameter in absence of the AF.

  * •

Fig. 8: A 2-D superconductor with $N_{x}=102$, $N_{y}=30$, $t=10$,
$J=0.1\,t=1.7\,\overline{|\Delta_{0}|}$, $\theta_{t}=\pi/4$, $U/t=1$, and
$f=0.5\,(\mu=0)$ at $k_{B}T/t=0.001$ is used to calculate the $s$-wave triplet
correlations, where $\overline{|\Delta_{0}|}$ represents the average magnitude
of the superconducting order parameter in absence of the AF.

  * •

Fig. 9: A 2-D superconductor with $N_{x}=302$, $N_{y}=3$, $t=10$, and $U/t=1$
is used to plot $T_{c}$ vs. $\theta_{t}$. For $f=0.5$,
$J=0.047\,t=1.34\,\overline{|\Delta_{0}|}=2.4\,T_{c0}$ is taken, and for
$f=0.6$, $J=0.07\,t=1.35\,\overline{|\Delta_{0}|}=2.4\,T_{c0}$ is taken, where
$T_{c0}$ and $\overline{|\Delta_{0}|}$ are the critical temperature and the
average magnitude of the superconducting order parameter at zero temperature
of the same superconductor without the AF layer.

##  Appendix F Canted-AF/S with rotated magnetic moments

To compare the results of this article with Ref. [49], one needs to rotate the
magnetic moments of the Hamiltonian [Eq. (II)] to
$\vec{M}_{j}=\left[(-1)^{j+1}\cos\theta_{t}\,\hat{z}+\sin\theta_{t}\,\hat{x}\right]$
(see Fig. 10(d)). The correlations for this rotated system is plotted in Fig.
10. Here, $F^{t,z}$ is the Néel triplet correlation coming from the
antiferromagnetic component of the canted-AF, $F^{t,x}$ comes from the
ferromagnetic component, and $F^{t,y}$ comes from the noncollinearity in the
canted-AF.

Figure 10: Spatial variation of normalized triplet correlations for 10 lattice
sites considering $\theta_{t}=\pi/4$ and $\mu/\Delta_{0}=37$, corresponding to
filling factor $f=0.6$. We show the imaginary part of the triplet correlations
$F^{t,z}_{j}$ [panel (a)] and $F^{t,x}_{j}$ [panel (b)], and the real part of
the triplet correlation $F^{t,y}_{j}$ [panel (c)]. The real part of the two
former and the imaginary part of the latter are zero. All the correlations are
normalized by the spatially averaged magnitude of the singlet correlation
$\overline{|F^{s}|}$. (d) Orientation of the magnetic moments of sublattices A
and B of the AF in the rotated configuration.

## Acknowledgments

SC and AK acknowledge financial support from the Spanish Ministry for Science
and Innovation – AEI Grant CEX2018-000805-M (through the “Maria de Maeztu”
Programme for Units of Excellence in R&D). LJK acknowledges support from the
Research Council of Norway through its Centres of Excellence funding scheme,
Project No. 262633 ‘QuSpin’. IVB acknowledges support from MIPT, Project
FSMG-2023-0014.

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