,

# Inverse Proximity Effects at Spin-Triplet Superconductor-Ferromagnet
Interface

O. Maistrenko∗ Max Planck Institute for Solid State Research, D-70569
Stuttgart, Germany  C. Autieri∗ International Research Centre MagTop,
Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46,
PL-02668 Warsaw, Poland  SPIN-CNR, c/o Università di Salerno, IT-84084
Fisciano (SA), Italy  G. Livanas  SPIN-CNR, c/o Università di Salerno,
IT-84084 Fisciano (SA), Italy  Department of Physics, National Technical
University of Athens, GR-15780 Athens, Greece  P. Gentile  SPIN-CNR, c/o
Università di Salerno, IT-84084 Fisciano (SA), Italy  Dipartimento di Fisica
“E. R. Caianiello”, Università di Salerno, IT-84084 Fisciano (SA), Italy  A.
Romano  Dipartimento di Fisica “E. R. Caianiello”, Università di Salerno,
IT-84084 Fisciano (SA), Italy  SPIN-CNR, c/o Università di Salerno, IT-84084
Fisciano (SA), Italy  C. Noce  Dipartimento di Fisica “E. R. Caianiello”,
Università di Salerno, IT-84084 Fisciano (SA), Italy  SPIN-CNR, c/o Università
di Salerno, IT-84084 Fisciano (SA), Italy  D. Manske  Max Planck Institute for
Solid State Research, D-70569 Stuttgart, Germany  M. Cuoco  SPIN-CNR, c/o
Università di Salerno, IT-84084 Fisciano (SA), Italy  Dipartimento di Fisica
“E. R. Caianiello”, Università di Salerno, IT-84084 Fisciano (SA), Italy

###### Abstract

We investigate inverse proximity effects in a spin-triplet superconductor
(TSC) interfaced with a ferromagnet (FM), assuming different types of magnetic
profiles and chiral or helical pairings. The region of the coexistence of
spin-triplet superconductivity and magnetism is significantly influenced by
the orientation and spatial extension of the magnetization with respect to the
spin configuration of the Cooper pairs, resulting into clearcut anisotropy
signatures. A characteristic mark of the inverse proximity effect arises in
the induced spin-polarization at the TSC interface. This is unexpectedly
stronger when the magnetic proximity is weaker, thus unveiling immediate
detection signatures for spin-triplet pairs. We show that an anomalous
magnetic proximity can occur at the interface between the itinerant
ferromagnet, SrRuO3, and the unconventional superconductor Sr2RuO4. Such
scenario indicates the potential to design characteristic inverse proximity
effects in experimentally available SrRuO3-Sr2RuO4 heterostructures and to
assess the occurrence of spin-triplet pairs in the highly debated
superconducting phase of Sr2RuO4.

Introduction. Spin-triplet superconductors can be odd-parity in momentum, e.g.
$p$-wave Sigrist ; TanakaPWave ; ReadGreen ; Ivanov ; Kitaev01 ; VolovikBook ;
Maeno2012 ; SatoAndo , or even parity for multi-orbital configurations
Spalek2001 ; Puetter2012 ; Han2004 ; Fukaya2018 ; Suh2020  Apart from the
variety of orbital pairings, the spin-triplet part of the superconducting (SC)
order parameter (OP), typically encoded in a ${\bf d}$-vector, has a
substantial imprint on the superconducting properties. In fact, the spin
degree of freedom in superconductors naturally enriches the physical scenario
resulting into anomalous response to Zeeman or ferromagnetic fields Murakami ;
Dumi1 ; Dumi2 ; Wright ; Mercaldo2016 ; Mercaldo2017 ; Mercaldo2018 , spin-
sensitive Josephson transport Sengupta ; Yakovenko ; Yakovenko1 ; Cuoco1 ;
Cuoco2 ; BrydonJunction ; Gentile ; Brydon1 ; Brydon2 , magnetic topological
reconstructions Mercaldo2016 ; Mercaldo2017 ; Mercaldo2018 , and, on a general
ground, may lead to energy efficient superconducting spintronics Linder2015 .
Notably, intrinsic or engineered spin-triplet superconductors can host
Majorana bound states and thus are particularly impactful for topological
quantum computing Ivanov ; Kitaev03 ; Nayak ; Alicea .

While more rare in nature than the canonical spin-singlet one, several
experimental observations have led to evidences for spin-triplet
superconductivity in a large variety of materials Stewart ; Saxena ; Kyogaku ;
Jiao ; Ran ; Bauer ; Nishiyama ; Lebed ; Lee ; Shinagawa ; Bao15 ; Cuono19 ;
Noce2020 . A notable case is Sr2RuO4 whose superconducting nature is under
intense debate because recent measurements Pustogow2019 ; Ishida2020 ;
Petsch2020  pose serious constraints on the long thought spin-triplet chiral
$p$-wave superconductivity Mackenzie2017 . Such case underlines the remarkable
challenges that are typically encountered in assessing the spin-triplet
pairing in unconventional superconductors.

In this work we take an alternative exploratory path to access the spin-
triplet nature of the superconducting pairing by focusing on its spin degree
of freedom and investigating the inverse proximity effects that can occur at
the spin-triplet superconductor (TSC)-ferromagnet (FM) interface, assuming
that spin polarization may leak into the TSC. Inverse proximity effects have
been studied in heterostructures involving spin-singlet superconductors
Tokuyasu1988 ; Bergeret2004 ; Tollis2015 ; Bergeret2005  with unexpected
consequences when nodal excitations occur Dibernardo2019 . Instead, while it
is well consolidated that the physics of TSC-FM heterostructures is richer
than their singlet counterparts because the orientation of the FM moment
relative to the TSC d-vector introduces extra channels of coupling, to the
best of our knowledge, the inverse proximity effects both for the magnetic and
superconducting components in TSC-FM have not yet been fully addressed. There
are various fundamental physical aspects to be accounted for once the spin
polarization of the FM penetrates into the TSC regarding the reconstruction of
the SC-OP and the modification of the magnetic proximity (MP) due to the
presence of the spin-triplet pairs. The analysis is performed by considering
two prototypical spin-triplet pairings with uniaxial and planar d-vector and
various characteristic spatial profiles of the magnetization in the TSC with
inequivalent character of the MP (Fig. 1). We find that the TSC order
parameter in the region of the magnetic-superconducting coexistence is
significantly sensitive to the orientation of the magnetization, thus
unveiling clearcut anisotropy signatures. Similarly, the behavior of the
induced spin-polarization in the TSC depends on the character of the d-vector
and on the inverse MP pattern. Then, motivated by the timely debate on the
nature of the superconducting phase in Sr2RuO4 Pustogow2019 ; Petsch2020  and
on the anomalous magneto-transport properties of SrRuO3-Sr2RuO4
heterostructure Anwar2016 ; Anwar2019a ; Anwar2019b , we evaluate the MP of
the Sr2RuO4 when interfaced with SrRuO3. By ab-initio methods we find that the
SrRuO3 induces a large magnetic moment in the Sr2RuO4 interface layers with
characteristic profiles that depend on the strength of the Coulomb
interaction. This magnetic scenario supports the general investigation of the
inverse proximity effects in the presence of different types of magnetic
leakage into the TSC. The recent developments in the fabrication of Sr2RuO4
thin films Krock2010 ; Marshall2017 ; Uchida2017 ; Palomares2020  and hetero-
structures allow to experimentally investigate the properties of the inverse
proximity effect, which we discuss in our work. This can provide relevant
experimental paths to address the debate about the nature of the
superconductivity in Sr2RuO4.

Figure 1: Illustration of the magnetic and superconducting configurations for
the TSC-FM heterostructure with chiral (a),(b) and helical (c),(d) order
parameters, respectively. The magnetization is out-of-plane (a),(c) or in-
plane (b),(d) oriented. (e) the examined magnetic profiles can have (A) a
step-like magnetization at the interface with substantial absence of magnetic
proximity, (B) a monotonous decaying of the spin polarization within the TSC
on the scale of the superconducting coherence length, or (C) a sign changing
penetration in the TSC, respectively. The blue region stands for the
ferromagnetic layers. In the gray area the spin-polarization induced by the FM
into the TSC has vanishing amplitude. Figure 2: Evolution of the pairing
amplitude for chiral TSC heterostructure near the TSC-FM interface assuming
different magnetization profiles as reported in Fig. 1(e). (a),(b) and (c)
correspond to step-like, monotonous decay, and sign-changing spatial
dependence of the magnetization, respectively. The black circles (red squares)
refers to a configuration with magnetization oriented along the out-of-plane
$z$ -direction (in-plane $x$-direction), respectively. The $z$-($x$-) spin
polarization is collinear (transverse) to the ${\mathbf{d}}$-vector. In the
panels (d),(e),(f) we show the spatial dependence of the difference, $\Delta
M$, between the magnetization in the superconducting ($M^{SC}$) and normal
state ($M^{N}$) for the parallel (circles) and transverse (squares)
configurations, respectively. The dashed line in (d),(e),(f) indicates the
amplitude of the magnetization in the normal metal state. The green area
indicates the layers where the magnetization penetrates into the TSC and there
is a coexistence of non-vanishing magnetization and superconductivity. The
blue region stands for the ferromagnetic layers. In the gray area the spin-
polarization induced by the FM is substantially zero.

Model and methodology. We consider an FM-TSC heterostructure with a layered
geometry described by a single-band tight-binding model with a spin dependent
nearest-neighbor attractive interaction that can stabilize spin-triplet
pairing with chiral or helical d-vectors (Fig. 1) Cuoco1 ; Cuoco2 ;
Terrade2014 ; Terrade2013 . The total Hamiltonian of the system is then
defined on a lattice with size $L_{x}\times L_{y}\times L_{z}$
($L_{x}=L_{y}=L$) assuming periodic boundary conditions along $x$ and $y$. The
simulation is performed for a system having $L=100$ and $L_{z}=40$. Since the
out-of-plane superconducting coherence length is of the order of about six
unit cells, the size of the system is adequate for tracking the inverse and
direct proximity effects on the characteristic superconducting length scales.
We indicate each site by a vector $\vec{i}\equiv(i,i_{z})$, with $i$ labeling
the $(xy)$ in-plane atomic positions and $i_{z}$ the different layers along
the $z$-direction. The FM region corresponds to layers with $i_{z}\geq 0$ and
the TSC one to layers with $i_{z}<0$, respectively. The Hamiltonian can be
expressed as

$\displaystyle H$
$\displaystyle=\sum_{{\mathbf{k}},i_{z},\sigma}(\epsilon_{{\mathbf{k}}\sigma}-\mu)c^{\dagger}_{{\mathbf{k}}\sigma}(i_{z})c_{{\mathbf{k}}\sigma}(i_{z})-\sum_{i_{z}}{\mathbf{h}}(i_{z})\cdot{\mathbf{M}}(i_{z})+$
(1) $\displaystyle-$
$\displaystyle\sum_{<ij>,i_{z}<0}V_{\sigma,\sigma^{\prime}}n_{i\sigma}(i_{z})n_{j\sigma^{\prime}}(i_{z})$
$\displaystyle+$ $\displaystyle\sum_{\delta=\pm
1}\sum_{{\mathbf{k}}\sigma}t_{\perp}(c^{\dagger}_{{\mathbf{k}}\sigma}(i_{z}+\delta)c_{{\mathbf{k}}\sigma}(i_{z})+{\text{h.c.}})$

with $c_{{\bf k}\sigma}(i_{z})$ being the annihilation operator of an electron
with planar momentum ${\bf k}$, spin $\sigma$ at the layer $i_{z}$ and
$\langle ij\rangle$ denotes the in-plane nearest-neighbor sites, $\mu$ is the
chemical potential, $V_{\sigma,\sigma^{\prime}}$ is the spin-dependent in-
plane pairing strength,
${\mathbf{M}}(i_{z})=\sum_{i,s,s^{\prime}}c^{\dagger}_{s}(i){\vec{\sigma}}_{s,s^{\prime}}c_{s^{\prime}}(i)$
is the spin density of the layer $i_{z}$, and
$\epsilon_{k\sigma}=-2t[\rm{cos}(k_{x})+\rm{cos}(k_{y})]$ is the in-plane
electronic spectrum. ${\mathbf{h}}(i_{z})$ is the layer dependent exchange
field that sets the amplitude and orientation of the magnetization within the
heterostructure (Fig. 1). We assume a good electronic matching at the
interface expressed by having $t_{\perp}=0.5t$. The selection of the three
profiles in Fig. 1 is general enough to include the phenomenology which is
expected when interfacing a correlated paramagnetic metal with a ferromagnet.
In fact, the breaking of the translational symmetry due to the interface
allows to get a reconstruction of the MP with monotonous or oscillating
spatial behavior. Such occurrence is also expected to be most likely to occur
in paramagnetic electronic systems having a peak at finite momentum in the
magnetic suceptibility. Furthermore, since the Fermi length is of the order of
the unit cell dimension, the MP is taken as more significant in the layers
nearby the interface.

For the present study the quartic term of the model Hamiltonian in Eq. 1 is
decoupled in the canonical Hartree-Fock approximation within the pairing
channel. We introduce the layer dependent superconducting pairing amplitude
for each spin-triplet channel to self-consistently determine the corresponding
order parameters with chiral and helical symmetry as:

$\displaystyle F^{0}_{\theta}$ $\displaystyle=$
$\displaystyle\frac{1}{L^{2}}\sum_{\sigma\neq\sigma^{\prime},{\mathbf{k}}}(\sin
k_{x}-i\theta\sin k_{y})\langle
c_{-{\mathbf{k}}\sigma}c_{{\mathbf{k}}\sigma^{\prime}}\rangle$ $\displaystyle
F^{\sigma}_{\theta}$ $\displaystyle=$
$\displaystyle\frac{1}{L^{2}}\sum_{\mathbf{k}}(\sin k_{x}-i\theta\sin
k_{y})\langle c_{-{\mathbf{k}}\sigma}c_{{\mathbf{k}}\sigma}\rangle$

where $\theta=\pm$ is employed to set the winding of the spin dependent order
parameter, while $\langle...\rangle$ is the average over the ground state.
Depending on the ampltitudes of $V_{\sigma,\sigma^{\prime}}$ the chiral or
helical spin-triplet superconductivity can be stabilized Terrade2013 ; Cuoco1
; Gentile .

Figure 3: Evolution of the pairing amplitude for the helical-TSC
heterostructure near the TSC-FM interface, assuming different magnetization
profiles in the TSC, as reported in Fig. 1(e). (a),(b) and (c) corresponds to
the step-like, monotonous decay, and sign-changing spatial dependence of the
magnetization, respectively. Circles (squares) stand for a magnetization which
is oriented along the out-of-plane $z$-direction (in-plane $x$-direction),
respectively. The $z$-($x$-) oriented spin polarization is transverse
(coplanar) to the ${\mathbf{d}}$-vector. In the panels (d),(e),(f) we report
the spatial dependence of the difference, $\Delta M$, between the
magnetization in the superconducting ($M^{SC}$) and in the normal ($M^{N}$)
state with coplanar and transverse configurations. The dashed gray line in
(d),(e),(f) indicates the amplitude of the magnetization in the normal metal
for the different types of examined magnetic proximity.

Inverse superconducting and magnetic proximity effects. We start by discussing
the superconducting proximity effects for the chiral $d_{z}$-TSC (Fig. 2),
assuming three different magnetization leakage into the TSC (Fig. 1 (e)). The
chiral amplitude $F^{0}_{+}$ depends on the orientation of the FM spin-
polarization (Fig. 2(a)), exhibiting a suppression of its value when the
magnetization is parallel ($M_{z}$) to the ${\mathbf{d}}$-vector compared to
the transverse configuration ($M_{x}$), as schematically shown in Figs. 1
(a,b).

Here, the use of different MP profiles allows us to assess how the leakage of
spin polarization in the TSC affects the SC-OP and the resulting spin
polarizability of the TSC at the interface.

Firstly, we observe that in the magnetized region of the TSC, the amplitude of
the chiral order parameters decreases for the $M_{z}$ orientation, while it is
substantially unaffected for the transverse $M_{x}$ case. This is consistent
with the spin orientation of the Cooper pairs. The behavior with transverse
spin-polarization $M_{x}$-FM (Fig. 2(b),(c)) unveils a weak enhancement when
the inverse MP increases in length penetration. Such outcome substantially
indicates that the spin-polarization entering into the TSC yields more spin-
triplet pairs for the proximity into the FM. For the $M_{z}$-state we notice
that the expected damped oscillation of the pairing amplitude does not exhibit
variation as a function of the MP.

Let us now focus on the behavior of the induced spin polarization due to the
leakage or lack of magnetization within the TSC region of the heterostructure.
From the behavior of the SC order parameter it is immediate to observe that
the effects in the TSC are tied to the orientation of the magnetization with
respect to the ${\mathbf{d}}$-vector. Then, in order to relate the behavior of
the induced magnetization to the character of the MP, the strategy is to
compute the magnetization difference for each layer of the superconducting
state with respect to that of the normal metal configuration, i.e. that one
corresponding to a vanishing pairing interaction $V$. This physical quantity
gives indication of the response of the superconductor close to the interface
not only for the presence of a non-vanishing exchange field in the region
where the order parameter is typically reduced in amplitude, but also to the
gradient of the spin-polarization which is particularly enhanced at the TSC-FM
boundary. In Figs. 2(d),(e),(f) we follow the spatial evolution of the induced
magnetization for the various MP profiles. As a general trend, we observe that
the magnetization change between the superconducting and the normal states is
substantially negligible for the $M_{x}$ orientation, i.e. transverse to the
${\mathbf{d}}$-vector with the exception of the layers very close to the
interface. There, the induced magnetization typically changes sign
irrespective of the MP profile, thus indicating that the TSC tends to screen
the MP by spatially modulating the spin-polarization. For the transverse
magnetization, it is interesting to link the enhancement of the SC proximity
effect (Fig. 2(c)) with that of the spin polarization in Fig. 2(f). The result
indicates that an increase of the magnetization can be observed at distance of
several layers from the interface due to the penetration of the spin-triplet
Cooper pairs with spin polarization collinear to $M$. On the other hand, the
response of the TSC to the $M_{z}$ leakage, i.e. parallel to the
${\mathbf{d}}$-vector, is typically larger in amplitude and significantly
sensitive to the character of the MP. We find that for the case of a step-like
magnetization at the interface the response of the TSC is paramagnetic (Fig.
2(d), i.e. the spin polarization is enhanced with respect to the normal
state), with a penetration in the superconductor over a spatial range of the
order of the out-of-plane coherence length. The impact of the MP is
significantly different when considering the case of a longer range
penetration of the magnetization in the TSC with monotonous decay (Fig. 2(e)).
In this regime, the amplitude of the induced spin-polarization spatially
oscillates around the one in the normal state configuration. Thus, the TSC
yields a vanishing net spin polarization in average. When considering the
sign-changing MP (Fig. 2(f)), we find that the magnetic reconstruction due to
the TSC leads to an overall paramagnetic response with an enhancement of the
spin polarization distribution in all layers close to the interface. We argue
that the variation of the induced magnetic effects is mainly due to the
gradient of the magnetization close to the interface because the amplitude is
comparable for the two configurations. In fact, the character of the observed
induced spin-polarization nearby the TSC-FM interface can be grasped by
analyzing a simplified effective model with only two superconducting layers in
the presence of an inhomogeneous exchange field. The details of the outcome of
such effective model are reported in the Appendix. The changeover from
paramagnetic to diamagnetic of the TSC response as a function of the layer
position can be ascribed to the gradient of the exchange field rather than to
the effective strength of the magnetization with respect to the SC gap (see
Appendix).

At this stage it is worth asking whether and how the magnetic reconstruction
at the TSC-FM gets modified if one is considering a TSC with helical pairing,
i.e. $F^{0}=0$ and $F^{\uparrow}_{+}=F^{\downarrow}_{-}$ thus marked by a two-
components ${\mathbf{d}}$-vector lying in the $xy$ plane, as shown in Fig. 1
(c),(d). For the helical TSC the superconducting proximity does not exhibit an
oscillating behavior because the ${\mathbf{d}}$-vector lies in a plane and
thus one cannot find a magnetization direction that is fully collinear to it
across the whole Brillouin zone. The behavior of the TSC pairing amplitude is
not much affected by the change of the MP (Fig. 3(a),(b),(c)). Moreover, for
the coplanar configuration the presence of the magnetic interface induces a
pairing amplitude in each spin-channel with opposite winding compared to that
of the SC order parameter, and this is irrespective of the character of the
MP. Such component can yield a transverse contribution in the spin
polarization with respect to the magnetization orientation. More interesting
is the behavior of the spin polarization in the TSC. One finds that for the
step-like magnetization profile, the response of the TSC is paramagnetic-like
for the coplanar configuration while it is diamagnetic-like for the transverse
$M_{z}$ state (see Fig. 3(d)). The effect typically extends over a distance of
the order of the coherence length. The amplitude of the induced spin
polarization is approximately isotropic. This aspect is a consequence of the
multi-component ${\mathbf{d}}$-vector. A similar isotropic behavior is also
observed for the case of longer range penetration of the magnetization with a
tendency to exhibit a sign change both in the TSC and in the FM side of the
heterostructure (Fig. 3(e),(f)).

Magnetic proximity at the Sr2RuO4-SrRuO3 interface. Here, in order to have an
experimentally relevant case of inverse magnetic effects, we consider the
Sr2RuO4-SrRuO3 heterostructure. SrRuO3 is an itinerant ferromagnet, which over
the past years has been the subject of intense research Matsuno2016 ;
Ohuchi2018 ; Li2020 ; Kan2020 ; Groenendijk2020 . By means of the DFT approach
(see Appendix for details) we demonstrate that the magnetic behavior of the
Sr2RuO4-SrRuO3 is non-standard if compared to conventional metal-ferromagnet
interface and can lead to magnetization profiles resembling those that have
been employed in the present study. Starting from the result that the bulk
Sr2RuO4 (see Appendix for details) is paramagnetic when the Coulomb repulsion
$U_{214}$ is below about 0.7 eV, we find that for the interface layers
proximized to SrRuO3 the Coulomb threshold for the transition into the
ferromagnetic state is significantly reduced (see Appendix). Hence, to
evaluate how the ground state of the Sr2RuO4 is affected by the ferromagnetism
in SrRuO3 we investigate a superlattice made of five Ru-O layers in each side
of the heterostructure (Fig. 4(a)). We find that the ground state of the
supercell is generally ferromagnetic in SrRuO3 with a magnetization that
slightly decreases at the interface layers. Sr2RuO4 can also lower its energy
by developing a non-vanishing magnetization that is collinear to that of the
SrRuO3 close to the interface and can change sign away from it depending on
the strength of the Coulomb interaction (Fig. 4(b)). The interface layer has
always the largest magnetization within Sr2RuO4. We tested several different
magnetic orderings for the SrRuO3-Sr2RuO4 supercell, finding that the
ferromagnetic coupling between SrRuO3 and Sr2RuO4 turns out to be the most
stable for the supercell. For U214 in the energy range [0.4-0.6] eV, we have a
sign-changing magnetization in the layers of Sr2RuO4 while for U214=0.7 eV all
layers have a spin polarization which is aligned to that of the SrRuO3 and
progressively decay in the Sr2RuO4 side of the heterostructure. The local
magnetic moment increases with U214, as expected, while the magnetic moment of
SrRuO3 stays substantially unaltered. Although the computational time limits
the size of the supercell which can be simulated, the resulting behavior
provides clear trends of the way magnetization gets reconstructed nearby the
SrRuO3-Sr2RuO4 interface.

Figure 4: (a) Schematic illustration of the SrRuO3-Sr2RuO4 superlattice with
five Ru-O layers having cubic (SrRuO3) and tetragonal structure (Sr2RuO4),
respectively. (b) evolution of the magnetization across the interface for
various values of the Coulomb interaction $U_{214}$ in Sr2RuO4. Red, green,
blue and magenta points are for $U_{214}$=0.4,0.5, 0.6 and 0.7 eV,
respectively. z${}_{113}(n)$ and z${}_{214}(n)$ are the n-th layer from the
interface for the corresponding Sr2RuO4 and SrRuO3 sides of the superlattice.

Conclusions. We have demonstrated that inverse proximity effects in TSC-FM
heterostructure exhibit clearcut signatures for both magnetic and
superconducting interface reconstruction with respect to the nature of the
pairing and the character of the magnetization leakage in the TSC. The
uniaxial ${\bf d}$-vector for the chiral TSC shows a magnetic response which
is strongly anisotropic, with a tendency to a significant spatial modulation
in amplitude and sign for the case of a long-range MP (Fig. 2). The helical
TSC with planar ${\bf d}$-vector tends instead to yield a more isotropic spin-
polarization, except for the step-like magnetic profile, which is smaller
(larger) in amplitude than that of the chiral state in the TSC (FM) side of
the junction (Fig. 3). These features can be relevant to account for the
observed anisotropy of the transport properties in Sr2RuO4-SrRuO3 Anwar2019a ;
Anwar2019b . In such a system, by means of ab-initio approaches we demonstrate
that there occurs a magnetic instability near the interface for the electronic
states of Sr2RuO4. We argue that the tendency to undergo a changeover from
paramagnetic to ferromagnetic in the Sr2RuO4 can set out the underlying
scenario for the anomalous enhancement of the magnetic moments in
Sr2RuO4-SrRuO3 heterostructure Sugimoto2015 . Our results indicate that the
presence of spin-triplet pairs can be accessed through magnetic measurements
that are spatially resolved with respect to the TSC-FM interface and performed
across the superconducting transition. By tracking the relation of the
spatially resolved magnetization in the normal and the superconducting state
one can get unique hallmarks of the d-vector structure in the superconductor.

###### Acknowledgements.

∗These authors contributed equally to this work. C. A. is supported by the
Foundation for Polish Science through the International Research Agendas
program co-financed by the European Union within the Smart Growth Operational
Programme. We acknowledge the CINECA award under the ISCRA initiatives IsC76
”MEPBI” and IsC81 ”DISTANCE” Grant, for the availability of high-performance
computing resources and support.

##  Appendix A Effective model for magnetic proximity in the spin-triplet
superconductor

In this Appendix, we compare the impact of the exchange field on both mono-
layer and bilayer geometries with chiral and helical spin-triplet
superconductivity in order to evaluate in a simplified configuration the local
role of an inhomogeneous magnetic proximity close to the interface.

###  A.1 Superconducting spin polarization vs exchange field for monolayer

Let us start by considering a mono-layer spin-triplet superconductor in the
presence of a Zeeman term that mimics the induced magnetic proximity. We
investigate the behavior of the normal state magnetization $M^{\text{N}}$
compared with the spin-polarization $M^{\text{SC}}$ induced by the magnetic
exchange in the superconducting state. This analysis allows to understand how
the induced spin-polarization evolves in the spin-triplet superconductor.

In the case of chiral superconductivity, the $\bf{d}$-vector is uniaxial and
pointing along the out-of-plane direction with respect to the electron motion.
This means that Cooper pairs have $S_{z}=0$, and the equal spin configurations
are generically aligned in the $x-y$ plane. The outcome of the spin-
polarization in the superconducting state is reported in Tables 1,2. We find
that the induced spin moments are always aligned to those of the normal-state
magnetization and, as expected, there is a tendency to have a vanishing spin
polarization with the field parallel to the d-vector upon reaching an
amplitude that is comparable to the superconducting gap. Instead, if we have a
configuration of h perpendicular to the d-vector than there is always a net
spin polarization due to the Cooper pair spin moments being aligned to h.
Hence, Cooper pairs can get spin polarized without pair breaking and the
magnetization has a typical amplitude that is close to that of the normal
state.

In the case of helical superconductor, the d-vector has a planar orientation
that changes in the momentum space. This means that there are Cooper pairs
with both spin projections $S_{x}=0$ and $S_{y}=0$. This results into a
superconducting order parameter with Cooper pairs that are spin polarized
along the $z$-direction. Starting with an exchange field that is transverse to
the d-vector, we have that the spin moment of the Cooper pairs is aligned to
h. Hence, we obtain $M^{\text{SC}}=M^{\text{N}}$ as it was for the chiral
spin-triplet pairing. The case with $h_{x}$ is richer in terms of possible
outcomes. For this configuration there are $\vec{k}$ points in the reciprocal
space for which one gets h parallel to d, while for others h is noncollinear
to d. Indeed, one has an angle between the field and the superconducting
d-vector that can vary between $0$ and $\pi$. Then, for $h_{x}<\Delta$ one can
only spin polarize a portion of spin triplet pairs, and consequently
$M^{\text{SC}}<M^{\text{N}}$. Additionally, by allowing a configuration of the
order parameter with $C_{4}$ rotational symmetry breaking, we obtain that the
superconductor can gain energy by having more electron pairs polarized at the
expenses of redistributing them to the Fermi surface, making the order
parameter asymmetric, i.e. $d_{x}\neq d_{y}$. In this case, we obtain an
increase of the magnetization $M^{\text{SC}^{*}}>M^{\text{SC}}$. The resulting
findings are summarized in Table 2.

| $\textbf{h}\parallel\textbf{d}$ | $\textbf{h}\perp\textbf{d}$
---|---|---
$h/\Delta_{0}$ | $M^{\text{SC}}$ | $M^{\text{SC}}$
a) $h<\Delta$ | 0 | $0.9M^{\text{N}}$
b) $h\approx\Delta$ | 0 | $M^{\text{N}}$
c) $h>\Delta$ | $M^{\text{N}}$ | $M^{\text{N}}$
Table 1: Summary of the induced magnetization $M^{\text{SC}}$ for a monolayer with chiral superconductivity. Results are for two directions of the magnetic exchange field in three different regimes of amplitude for h. |  h coplanar $\textbf{d}(\vec{k})$ | $\textbf{h}\perp\textbf{d}$
---|---|---
$h/\Delta_{0}$ | $M^{\text{SC}}$ | $M^{\text{SC}*}$ | $M^{\text{SC}}$
a) $h<\Delta$ | $0.45M^{\text{N}}$ | $0.46M^{\text{N}}$ | $M^{\text{N}}$
b) $h\approx\Delta$ | $0.56M^{\text{N}}$ | $0.70M^{\text{N}}$ | $M^{\text{N}}$
c) $h>\Delta$ | $M^{\text{N}}$ | $M^{\text{N}}$ | $M^{\text{N}}$
Table 2: Summary of the induced magnetization $M^{\text{SC}}$ for a monolayer
with helical superconductivity. Results are for two directions of the magnetic
exchange field in three different regimes of amplitude for h with respect to
the superconducting gap energy $\Delta$, i.e. $h<\Delta$, $h\approx\Delta$,
and $h>\Delta$, respectively.

###  A.2 Layer dependent magnetic exchange

Let us consider a bilayer geometry which is the minimal configuration that
allows us to investigate the consequence of a nonuniform magnetic exchange
field. Having two distinct layers, one can introduce a layer dependent
exchange that can locally mimic the behavior of the induced spin-polarization
in a given heterostructure in close proximity to the interface. Hence, we
consider both layers to be superconducting, and the exchange magnetic fields
have amplitude $h_{1}$ and $h_{2}$, respectively. The energy scale of the
superconducting order parameters is $\Delta_{1(2)}$. Here, we focus again on
the induced magnetic response in each layer by evaluating the spin-
polarization for the ground state.

Firstly, we discuss the regime of small exchange amplitude (see rows (a)-(c)
in Tables 3 and 4) and uniform $\Delta$. These regimes are appropriate for a
magnetic proximity with the magnetization penetrating inside the
superconductor, with the gap keeping about its bulk value. In the case of
uniform $h$ the results substantially reproduce the effects of the single-
layer system considered above. In fact, for $\textbf{h}\perp\textbf{d}$ in
both chiral and helical configurations we obtain $M^{\text{SC}}\approx
M^{\text{N}}$. Interestingly, when the exchange field is not uniform, instead
of following the $\textbf{h}(z)$ profile, the magnetization gets smoothed out,
such that it has a smaller gradient compared to that in the normal state. This
can be qualitatively ascribed to the fact that Cooper pairs can have a larger
interlayer tunneling probability compared to the single electrons in the
normal state. Another nonstandard behavior is obtained in the regime of strong
gradient and large amplitude of the exchange field. There, the magnetization
can have an amplitude that is even greater than that of the normal state, as
reported in (d)-(f) for $\textbf{h}\perp\textbf{d}$ in Tables 3 and 4. This
implies that a change in the number of electron pairs is in place when h is
larger than $\Delta$, that allows to have an enhancement of the spin
polarization.

Next, we consider the configuration $\textbf{h}\parallel\textbf{d}$. The first
relevant difference with the previous case is that a variation of the exchange
field can lead to a spin polarization with opposite signs in the two layers.
This behavior is reminiscent of the oscillation of the magnetization found in
the multi-layer system and generally indicates a tendency of the spin-triplet
superconductor to exhibit a diamagnetic response when the exchange field is
nonuniform. The helical case, however, never shows a sign change, setting a
clear difference between a d-vector lying in a plane or with a axial
configuration. Since in the weak gradient regime (row (c) Table 4) one can
reach a vanishing value of the spin-polarization, it is plausible to expect
that a weak sign changing profile of the spin-polarization can be obtained in
a multi-layered configuration.

This trend is consistent with the observation of a more pronounced enhancement
of $M^{\text{SC}}-M^{\text{N}}$ in the chiral spin-triplet superconductor.

Configurations | $\textbf{h}\parallel\textbf{d}$ | $\textbf{h}\perp\textbf{d}$
---|---|---
Case | $\left(\frac{h_{1}}{\Delta_{0}},\frac{h_{2}}{\Delta_{0}}\right)$ | $\left(\frac{\Delta_{1}}{\Delta_{0}},\frac{\Delta_{2}}{\Delta_{0}}\right)$ | $\left(\frac{M_{1}}{M^{\text{N}}},\frac{M_{2}}{M^{\text{N}}}\right)$ | $\left(\frac{M_{1}}{M^{\text{N}}},\frac{M_{2}}{M^{\text{N}}}\right)$
a) | $(0.6,0.6)$ | $(1,1)$ | $(0,0)$ | $(1,1)$
b) | $(0.4,0.6)$ | $(1,1)$ | $(-0.2,0.15)$ | $(0.98,0.99)$
c) | $(0.2,0.6)$ | $(1,1)$ | $(-0.8,0.3)$ | $(0.80,0.92)$
d) | $(0.1,5)$ | $(1,0)$ | $(0.65,1)$ | $(1.03,1)$
e) | $(0.1,20)$ | $(1,0)$ | $(1.5,1)$ | $(0.96,1)$
f) | $(0.1,5)$ | $(1,0.2)$ | $(0.53,0.98)$ | $(1.07,1)$
Table 3: Summary of the results for the bilayer with chiral superconductivity. Configurations |  h coplanar d | $\textbf{h}\perp\textbf{d}$
---|---|---
Case | $\left(\frac{h_{1}}{\Delta_{0}},\frac{h_{2}}{\Delta_{0}}\right)$ | $\left(\frac{\Delta_{1}}{\Delta_{0}},\frac{\Delta_{2}}{\Delta_{0}}\right)$ | $\left(\frac{M_{1}}{M^{\text{N}}},\frac{M_{2}}{M^{\text{N}}}\right)$ | $\left(\frac{M_{1}}{M^{\text{N}}},\frac{M_{2}}{M^{\text{N}}}\right)$
a) | $(0.6,0.6)$ | $(1,1)$ | $(0.55,0.55)$ | $(1,1)$
b) | $(0.4,0.6)$ | $(1,1)$ | $(0.43,0.60)$ | $(0.98,0.99)$
c) | $(0.2,0.6)$ | $(1,1)$ | $(0.,0.63)$ | $(0.80,0.92)$
d) | $(0.1,5)$ | $(1,0)$ | $(0.94,1)$ | $(1.03,1)$
e) | $(0.1,20)$ | $(1,0)$ | $(1.19,1)$ | $(0.96,1)$
f) | $(0.1,5)$ | $1,0.2$ | $(0.84,1)$ | $(1.07,1)$
Table 4: Summary of the results for the bilayer with helical
superconductivity.

##  Appendix B Ab-initio analysis

In this Section we provide the details of the ab-initio computation with the
evolution of the magnetic instability in Sr2RuO4 as a function of Coulomb
interaction and strain.

###  B.1 Computational details

We perform spin polarized first-principles density functional calculations
within the LSDA (Local Spin Density Approximation) Kohn64 ; Kohn64a  by using
the plane wave VASP VASP  DFT package and the Perdew-Zunger Perdew
parametrization of the Ceperly-Alder data Ceperley  for the exchange-
correlation functional. The choice of the LSDA exchange functional is
suggested by a recent paper Etz12  where the Generalized Gradient
Approximation (GGA) is shown to be an approximation less accurate than LSDA
for SrRuO3. The core and the valence electrons were treated within the
projector augmented wave (PAW) method BlochlPAW  and a cutoff of 430 eV for
the plane wave basis. An 8$\times$8$\times$1 $k$-point grid is used for the
full relaxation of the bulk orthorhombic phase with spin orbit coupling. We
use a 10$\times$10$\times$1 $k$-point grid centered at the Gamma point for the
determination of the total energy of Sr2RuO4 and the full relaxation of the
heterostructure. In all cases the tetrahedron method with Blöchl corrections
BlochlCORR  was used for the Brillouin zone integrations. We optimize the
internal degrees of freedom by minimizing the total energy to be less than
10-5 eV and the remaining forces to be less than 5 meV/Å.

To catch the magnetic behavior, the Hubbard $U$ effects at the Ru sites for
4$d$ orbitals was included in the LSDA+$U$ Anisimov91 ; Anisimov93  approach
using the rotationally invariant scheme proposed in Ref. Anisimov95,  and
implemented in VASP VASP . We denote by $U_{113}$ and $J_{H,{113}}$ the
Coulomb repulsion and the Hund parameter of SrRuO3, while $U_{214}$ and
$J_{H,{214}}$ for the Sr2RuO4. In this study we fixed $U_{113}=1$eV for the Ru
4$d$ states of SrRuO3, while we use $J_{H,{113}}$=0.15 $U_{113}$. We keep
$J_{H,{214}}$=0 and we tune from 0 to 1.5 eV the Coulomb repulsion in Sr2RuO4.
These values are common to all the supercell calculations presented in this
work.

The study of SrRuO3 under strain is made with the in-plane lattice constant of
the Sr2RuO4. A small values of $U$ was used in the bulk to reproduce the
correct magnetization in DFT, but in extreme strain conditions one does not
have a reference of $U$ to describe the corresponding physical condition.

In order to simulate thin films of SrRuO3 grown on Sr2RuO4 substrate, we fix
the in-plane lattice parameter $a$ with the experimental value of the Sr2RuO4
bulk and the atomic positions of the Sr2RuO4 phase are fixed to the bulk
values. We use the value of $c$ such that the volume of SrRuO3 is equal to the
bulk volume because the theoretical volume underestimates the experimental
one. We discuss the structural and electronic properties of heterostructures
made of Sr2RuO4 and SrRuO3. We will investigate the supercell composed by
three Sr2RuO4 and five SrRuO3 layers. The focus is on the superlattice
structures with three inequivalent Sr2RuO4 layers and three inequivalent
layers of SrRuO3. Being the magnetic order in SrRuO3 dominating, we first
study the magnetic order in the SrRuO3 side of the heterostructure fixing
$U_{214}$=0. After that, we add the Coulomb repulsion in Sr2RuO4 to
investigate the magnetic instability of the Sr2RuO4 side of the supercell.
Structural optimization was performed in LSDA+$U$ approximation separately for
the different magnetic cases.

###  B.2 Magnetic instability in Sr2RuO4

We discuss the role of the Coulomb interaction and the strain in Sr2RuO4
system together with the possible magnetic instabilities and their character.

The Sr2RuO4 compound is paramagnetic in the bulk. We compare the energy of the
ferromagnetic (FM), paramagnetic (PM) and antiferromagnetic (AFM) phase in
terms of $U^{214}$ from 0 to 1 eV and by varying the in-plane lattice constant
from -3% to +3% . The outcome is reported in Fig. 5. The AFM phase is G-type,
as it can be also found in the ultrathin SrRuO3. The system is paramagnetic
for small values of $U$ and it is close to both AFM and FM magnetic phase for
compressive and tensile strain, respectively.

The rotations of the octahedra are not stabilized by the strain. Because there
are no rotations, the most important effect is the dependence of the hopping
parameters on the Ru-Ru distance. When the lattice constant gets larger, the
hopping decreases and the role of electron correlations becomes more relevant.
This explains the magnetic instability at larger in-plane lattice constant
($a$) as a function of $U$. If we impose the rotations of the octahedra and
the consequent reduction of the bandwidth, there is a critical value of the
rotation beyond which Sr2RuO4 becomes ferromagnetic at any small value of the
Coulomb interaction. The magnetic properties in Fig. 5 do not change
qualitatively if we apply spin-orbit or by assuming that the atomic positions
are not relaxed.

Now, we fix the atomic positions to the experimental values and the volume to
study more accurately the transition from the PM to the FM phase.

Figure 5: Magnetic phase diagram of bulk Sr2RuO4 as a function of the Coulomb
repulsion and in-plane strain.

Figure 6: (Color online) Energy as function of the magnetic moment of the bulk
Sr2RuO4 for different values of the Coulomb repulsion $U$. In the top panel we
show the presence of the ferromagnetic minimum close to the paramagnetic
minimum thanks to the VHS in the dxy band. In the bottom panel we can observe
a first order transition between two ferromagnetic minima thanks to the VHS in
the dγz bands. The energy of the paramagnetic phase is the reference point.

We calculate in detail the total energy as a function of the magnetization in
Fig. 6 for the values of $U$ close to the magnetic transition. In the top
panel we can observe the double minima curve giving rise to a first order
magnetic transition that is typical in a metamagnetic system. The spin-
polarized state becomes lower in energy when increase $U$ and becomes the
ground state at U=0.61 eV.

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