# Interfacial control of vortex-limited critical current in type II
superconductor films

Marius K. Hope  Center for Quantum Spintronics, Department of Physics,
Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
Morten Amundsen  Nordita, KTH Royal Institute of Technology and Stockholm
University, Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden  Dhavala Suri
Francis Bitter Magnet Laboratory & Plasma Science and Fusion Center,
Massachusetts Institute of Technology, Cambridge, MA-02139, USA  Jagadeesh S.
Moodera  Francis Bitter Magnet Laboratory & Plasma Science and Fusion Center,
Massachusetts Institute of Technology, Cambridge, MA-02139, USA  Department of
Physics, Massachusetts Institute of Technology, Cambridge, MA-02139,USA
Akashdeep Kamra<EMAIL_ADDRESS>Condensed Matter Physics Center
(IFIMAC) and Departamento de Física Teórica de la Materia Condensada,
Universidad Autónoma de Madrid, E-28049 Madrid, Spain  Center for Quantum
Spintronics, Department of Physics, Norwegian University of Science and
Technology, NO-7491 Trondheim, Norway

###### Abstract

In a small subset of type II superconductor films, the critical current is
determined by a weakened Bean-Livingston barrier posed by the film surfaces to
vortex penetration into the sample. A film property thus depends sensitively
on the surface or interface to an adjacent material. We theoretically
investigate the dependence of vortex barrier and critical current in such
films on the Rashba spin-orbit coupling at their interfaces with adjacent
materials. Considering an interface with a magnetic insulator, we find the
spontaneous supercurrent resulting from the exchange field and interfacial
spin-orbit coupling to substantially modify the vortex surface barrier,
consistent with a previous prediction. Thus, we show that the critical
currents in superconductor-magnet heterostructures can be controlled, and even
enhanced, via the interfacial spin-orbit coupling. Since the latter can be
controlled via a gate voltage, our analysis predicts a class of
heterostructures amenable to gate-voltage modulation of superconducting
critical currents. It also sheds light on the recently observed gate-voltage
enhancement of critical current in NbN superconducting films.

##  I Introduction

The superconducting phase emerges when pairs of conduction electrons
experiencing an attraction condense into a macroscopic quantum coherent state
[1, 2]. Below a critical temperature, often abbreviated to Tc, the
superconducting phase is lower in energy than the normal metallic state. This
macroscopic coherence supports dissipationless flow of charge up to a maximum
critical current density above which superconductivity is destroyed and the
system reverts to its normal metal state [1, 2]. The virtually absent
dissipation and macroscopic coherence underlie the central role
superconductors are playing in various emerging quantum technologies [3, 4].
The major fraction of these employ the so-called conventional superconductors
that emerge from metallic states with very high charge carrier densities and
are well-understood within the Bardeen-Cooper-Schrieffer (BCS) theory [5].

In contrast, a large variety of superconductors, broadly called
“unconventional”, emerge from insulating or low electron density normal states
[6]. The physics of these varies from one system to another and only a few
similarities can be noted. Among them is their demonstrated tunability via an
applied gate voltage or an electric field [7, 8, 9, 10, 11], which can
significantly modify the carrier concentration and thus, various
superconducting properties such as Tc. The low carrier density is of
fundamental importance in this gate voltage control for two reasons. First, an
electric field generated by an adjacent gate electrode can only exist in a
semiconducting or insulating system. In typical metals, an applied electric
field is screened out by the charge carriers on atomic length scales [12, 13,
14, 15]. Second, the relative change in carrier concentration induced by an
applied gate voltage is negligible for metallic systems due to their very high
pre-existing carrier densities [16, 12].

Hence, it came as a (pleasant) surprise when the critical current in a range
of metallic conventional superconductor films responded significantly to gate
voltages [17, 18, 19, 20, 21, 22, 23, 24]. Two key features should be noted.
First, the modification observed in critical current is even in the gate
voltage, i.e., it depends only on the electric field strength. Second, Tc
remained unchanged with the gate voltage. Several potential mechanisms such as
quasiparticle injection [21, 24], disorder [18], Sauter-Schwinger effect [25],
and others [26, 27, 28] have been considered to explain the observed
suppression of the critical current with the gate voltage. However,
explanations consistent with all the key experimental features and a broad
consensus for each material system are still being pursued [23]. Furthermore,
the recently observed enhancement in the critical current of type II
superconductor NbN films [22] stands out. While it shares the even-in-gate-
voltage dependence and no-Tc-change feature with other observations, none of
the mechanisms speculated above can justify an increase in the critical
current, which has been attributed to a vortex surface barrier based mechanism
[22, 29, 30]. A systematic theory of these observations should aim to explain
the specific gate voltage dependencies of Tc and critical current. At the same
time, it should clarify what system property (e.g., carrier density or
interfacial electric field) is primarily influenced by the gate voltage and
how this change affects the superconductor properties [31].

In this article, taking inspiration from the above mentioned experiments, we
theoretically investigate the vortex mechanism of critical current control,
put forth as a possible origin of the observed enhancement in NbN films [22].
In this process, we identify the key physics at play, finding it to be
consistent with the main experimental features mentioned above. However, our
focus is on investigating superconductor$|$magnetic insulator (S$|$MI) hybrids
because they are found to follow similar physics, be simpler to analyze
theoretically, and manifest a potentially stronger gate-voltage control or
enhancement of superconducting critical currents. Our choice of investigating
such hybrids is additionally inspired by a recent work due to Mironov and
Buzdin, which found superconductor$|$ferromagnetic insulator interfaces to
host spontaneous supercurrents [32]. They further commented on a consequent
renormalization of the vortex surface barrier by the induced spontaneous
supercurrent. Here, following similar considerations [32, 33], we
theoretically demonstrate a control of vortex surface barriers in a range of
S$|$MI hybrids, where MI can be a ferro- or an antiferromagnet [34]. More
significantly, we delineate the critical current dependence on these vortex
barriers generalizing the work of Shmidt [29, 30] in several crucial aspects.
Such S$|$MI hybrids have already become the workhorse for superconducting
spintronics (e.g., see Refs. [35, 36, 37, 38, 39, 40, 41, 42]) and constitute
a mature conveniently-fabricated experimental platform. We exclusively
consider metallic conventional superconductors that are well described via BCS
and Ginzburg-Landau (GL) theories [1, 2]. Furthermore, as we focus on the
vortex mechanism of critical current [29, 30], we consider type II
superconductors.

Our first key premise, which could have a broader relevance for the gate
voltage control and experiments mentioned above, is that the gate voltage
primarily influences the interfacial Rashba spin-orbit coupling (SOC) [43,
44]. It induces a strong electric field that is screened within atomic length
scales from the superconductor surface [31, 14]. However, the finite voltage
drop across the infinitesimal interface causes a significant gate voltage-
induced interfacial Rashba SOC [45]. This assumption is supported by the
recent observation of gate voltage-controlled interfacial spin-orbit torques
in a metallic ferromagnet/semiconductor heterostructure [46]. The gate voltage
was found to induce a strong Rashba SOC at such an interface. Furthermore,
gate voltage modulation of Rashba SOC in superconducting phases living on
oxide interfaces has previously been shown to underlie superconductivity
modulation [8, 47, 48]. Our second key premise is that the critical current in
our type II superconductor films is the value at which the Lorentz force on
vortices spontaneously nucleated at one side overcomes the Bean-Livingston
surface barrier [49]. At this current, these vortices are able to traverse the
film thereby causing dissipation and destroying superconductivity [29, 30], in
the sense that a nonzero voltage develops. Further, the dissipation causes
heating and may subsequently kill the superconductivity thermally. Thus, the
critical current is determined and controlled directly by the surfaces. This
assumption is also supported by several theoretical and experimental works
consistent with this vortex mechanism of critical current in some type II
superconducting films [50, 51, 52, 53, 54, 55]. However, this is only one of
the mechanisms that can determine critical current in a film, as discussed
further below.

Thus, employing the GL phenomenology, we analytically evaluate the vortex
surface barrier and the ensuing critical current in type II superconductor
films interfaced with magnetic insulators. In the presence of interfacial
Rashba SOC and effective exchange field induced by the adjacent magnet, a
spontaneous supercurrent flows at the interface, as shown recently by Mironov
and Buzdin [32]. It substantially modifies the vortex surface barrier thereby
increasing or decreasing the critical current, depending on the relative
orientations of induced exchange field, applied magnetic field, and transport
current through the film. The modification in vortex barrier and critical
current is proportional to the Rashba SOC strength, which can be tuned by a
gate voltage. The critical temperature Tc of the superconducting state remains
unaltered as per this mechanism. For superconductor films without adjacent
magnets [22], the critical current change is second order in the Rashba SOC.
It is thus expected to be weaker than in superconductor-magnet hybrids and
even in the applied gate voltage.

##  II Overview and Model

In this section, we provide a qualitative discussion of the key phenomena at
play and introduce the general mathematical framework for the problem. Our
goal here is to provide an intuitive understanding and an overview of the
methodology rather than the full details.

Different mechanisms can determine the critical current in superconducting
films [30, 56, 57, 1]. The most well-known is the critical depairing
mechanism, which causes a rapid destruction of the Cooper pairs when their
kinetic energy begins to exceed the superconducting condensation energy [56,
1]. Other mechanisms, such as weakest-link [57], begin to dominate films with
some disorder and granular structure. In type II superconductor films with
thicknesses larger than the coherence length, there is an additional mechanism
which can become dominant in clean films [29, 30]. Vortices [58, 59] have a
tendency to be nucleated and annihilated at the surfaces of a type II
superconductor film. These are unable to penetrate the film due to the so-
called Bean-Livingston surface barrier [49, 1, 60]. In the presence of a
transport current through the film, these vortices additionally experience a
Lorentz force which tends to pull them from one side to the other. At large
enough current, this Lorentz force exceeds its counterpart due to the Bean-
Livingston barrier. Consequently, vortices nucleated at one surface move
across the superconducting film to the other side causing dissipation and loss
of superconductivity [29]. The dominance of this mechanism requires two main
properties - a weak Bean-Livingston surface barrier and an absence of vortex
(or flux) pinning centers in the superconducting film. This situation is
likely to happen when the interface is somewhat disordered, thereby lowering
the barrier, and the “bulk” is clean, thereby preventing disorder that could
pin the vortices. In this work, we consider such films and assume the vortex
instability to determine the critical current. Further discussion on which
films may be dominated by this mechanism is provided in Sec. VIII.

We employ the GL phenomenology to describe our superconductor film with
thickness $d$, coherence length $\xi$, and London penetration depth $\lambda$,
under the assumption $\xi\ll d\ll\lambda$. As depicted in Fig. 1, an
externally injected transport current $I_{\mathrm{ext}}$ flows parallel to the
surfaces. An external magnetic field $\boldsymbol{H}_{e}$ is applied
tangential to the superconductor surfaces. The adjacent magnets and interfaces
to them are incorporated in our model by including Rashba SOC and proximity-
induced exchange field in a thin layer with thickness $l_{so}$ adjoining the
interface (Fig. 1).

Figure 1: Schematic depiction of the system under consideration. A type II
superconductor film S (pink) is sandwiched between two insulators (yellow). An
in-plane magnetic field $\boldsymbol{H}_{e}$ is applied and a current
$I_{\mathrm{ext}}$ is driven through the film. Interfacial Rashba spin-orbit
coupling (SOC) is assumed to exist in thin layers (blue) adjoining the
interfaces. Its strength can be tuned via an applied gate voltage $V_{G}$.
When the adjacent insulators are magnetic, exchange field $\boldsymbol{h}$ is
induced in the interfacial layer (blue) of S. The critical current in such
films depends on Rashba SOC and the induced exchange field. These can be
controlled via the gate voltage $V_{G}$ and the applied magnetic field
$\boldsymbol{H}_{e}$ providing multiple handles on critical current in the
superconductor. The coordinate system on the right depicts the orthogonal
relation between the symmetry-breaking vector $\hat{\boldsymbol{n}}$ on the
upper S surface, the exchange field $\boldsymbol{h}$, and the effective field
$\boldsymbol{\alpha}$ experienced by the S [Eq. (1)].

The GL free energy density $f(\boldsymbol{r})$ at a position $\boldsymbol{r}$
within the superconductor film is [32, 61, 62, 33, 63]:

$\displaystyle f(\boldsymbol{r})$
$\displaystyle=a|\psi|^{2}+\frac{b}{2}|\psi|^{4}+\frac{\left|\hat{\boldsymbol{D}}\psi\right|^{2}}{2m}+\frac{\boldsymbol{B}^{2}}{2\mu_{0}}+\frac{\boldsymbol{\alpha}}{2m}\cdot\left(\psi^{*}\left(\hat{\boldsymbol{D}}\psi\right)+\mathrm{h.c.}\right),$
(1)

where $\psi$ is the superconducting order parameter, $a$ and $b$ are the
standard GL coefficients describing a homogeneous superconductor,
$\hat{\boldsymbol{D}}\equiv-\textit{i}\hbar\nabla-2e\boldsymbol{A}$ is the
gauge-invariant momentum operator ($e<0$), $m$ is the effective mass of a
Cooper pair,
$\boldsymbol{B}=\boldsymbol{B}(\boldsymbol{r})=\nabla\times\boldsymbol{A}$ is
the spatially resolved magnetic flux density with $\boldsymbol{A}$ the
magnetic vector potential, and $\mu_{0}$ is the vacuum permeability.

The last term in Eq. (1) above captures the free energy contribution due to
the combined effect of Rashba SOC and a proximity-induced exchange field
$\boldsymbol{h}$ [61, 63, 62], and is parameterized via
$\boldsymbol{\alpha}=\boldsymbol{\alpha}(\boldsymbol{r})=\beta(\boldsymbol{r})\left(\hat{\boldsymbol{n}}\times\boldsymbol{h}\right)$
[32, 33]. Here, $\beta(\boldsymbol{r})$ is proportional to the Rashba SOC
parameter $\alpha_{\mathrm{Ra}}$ that may depend on the position, and
$\hat{\boldsymbol{n}}$ is the unit vector along the direction of the spatial
symmetry-breaking that causes Rashba SOC. $\beta$ further depends on the
properties of the superconductor and will be detailed in the context of
estimating effects using experimentally obtained parameters (Sec. VII). In our
consideration of magnet-superconductor interfaces, we assume
$\alpha_{\mathrm{Ra}}$ to be a nonzero constant in a thin layer with thickness
$l_{so}$ next to the film surface (Fig. 1), and $\hat{\boldsymbol{n}}$ is
normal to the interface directed outwards from the superconductor. This free
energy contribution can be motivated as follows. Due to Rashba SOC, electrons
with a given spin have lower energy if they move along a preferred direction.
Thus, in the presence of an exchange field, which lowers the energy of
electrons with a certain spin, the orbital motion of the Cooper pairs also
develops a preference resulting in spontaneous supercurrents [32]. The field
$\boldsymbol{h}$ needs to come from an exchange-induced spin-splitting via
proximity to a magnet, and not an applied magnetic field.

We work here in the London limit, i.e., the superconducting order parameter is
assumed to have a constant magnitude:
$\psi(\boldsymbol{r})\equiv\psi_{0}e^{i\varphi(\boldsymbol{r})}$. In this
limit, our spatial resolution is assumed to be much larger than $\xi$ and
vortices are treated as point objects via delta functions. Using Eq. (1), the
total free energy $F$ is thus expressed as [33]

$\displaystyle F$ $\displaystyle=\int
d^{3}r\left[f_{0}+\frac{1}{2\mu_{0}\lambda^{2}}\left\\{\left(\boldsymbol{\Phi}-\boldsymbol{A}\right)^{2}+\frac{\boldsymbol{\alpha}}{e}\cdot\left(\boldsymbol{\Phi}-\boldsymbol{A}\right)+\lambda^{2}\boldsymbol{B}^{2}\right\\}\right],$
(2)

where the integral runs over the superconductor volume $V_{S}$, $f_{0}\equiv
a\psi_{0}^{2}+b\psi_{0}^{4}/2$ is a constant,
$\boldsymbol{\Phi}\equiv(\hbar/2e)\boldsymbol{\nabla}\varphi$ is the vorticity
[1], and $\lambda\equiv\sqrt{m/(4e^{2}\mu_{0}\psi_{0}^{2})}$ is the London
penetration depth.

The equation governing the superconducting state is obtained by minimizing the
total free energy [Eq. (2)] with respect to $\boldsymbol{A}$ 111In the
considered London limit, minimization of the free energy with respect to
$\psi$ becomes superfluous., which yields:

$\displaystyle\mu_{0}\lambda^{2}\boldsymbol{j}$
$\displaystyle=-\boldsymbol{A}+\boldsymbol{\Phi}+\frac{\boldsymbol{\alpha}}{2e},$
(3)

where we have additionally employed the Maxwell’s equation for current density
$\mu_{0}\boldsymbol{j}=\nabla\times\boldsymbol{B}$. Considering vortices with
flux along $\hat{\boldsymbol{z}}$ at positions $\boldsymbol{\rho}_{v}$ in the
x-y plane, we assume translational invariance along the z direction. With
these assumptions, taking curl on both sides of Eq. (3) and employing
Maxwell’s equations, we obtain:

$\displaystyle\lambda^{2}\nabla^{2}\boldsymbol{B}-\boldsymbol{B}$
$\displaystyle=-\phi_{0}\hat{\boldsymbol{z}}~{}\sum_{v}\delta_{2}(\boldsymbol{\rho}-\boldsymbol{\rho}_{v})-\frac{\nabla\times\boldsymbol{\alpha}}{2e},$
(4)

where $\phi_{0}\equiv h/(2e)$ is the magnetic flux quantum associated with
each vortex and $\delta_{2}(\boldsymbol{\rho})$ is the two-dimensional Dirac
delta function. Employing Eq. (4) together with the boundary conditions
resulting from the external magnetic field and the injected transport current,
we can determine the magnetic flux density and supercurrent distribution in
the superconductor for any assumed vortex configuration. These can then be
substituted in Eq. (2) to obtain the Gibbs free energy density $G/V_{S}$ in a
constant applied field $\boldsymbol{H}_{e}$ averaged over the superconductor:

$\displaystyle\frac{G}{V_{S}}$ $\displaystyle=\frac{1}{V_{S}}\int
d^{3}r\left[\frac{1}{2\mu_{0}}\left[\boldsymbol{B}^{2}+\lambda^{2}(\boldsymbol{\nabla}\times\boldsymbol{B})^{2}-\frac{\boldsymbol{\alpha}^{2}}{4e^{2}\lambda^{2}}\right]-\boldsymbol{B}\cdot\boldsymbol{H}_{e}\right],$
(5)

where we have employed Eq. (4) and dropped the constant contribution $f_{0}$
in obtaining the expression above.

We are now ready to summarize our methodology. Assuming an applied magnetic
field, externally injected transport supercurrent, and vortices located at
positions $\boldsymbol{\rho}_{v}$ in the superconductor, we evaluate the
magnetic flux density $\boldsymbol{B}(\boldsymbol{r})$ by solving Eq. (4).
This solution is substituted in Eq. (5) to obtain the superconductor Gibbs
free energy density as a function of the vortex positions
$\boldsymbol{\rho}_{v}$. The force per unit volume $\boldsymbol{F}_{s}$
experienced by a vortex is given by the negative gradient of the Gibbs free
energy density with respect to the vortex position $\boldsymbol{\rho}_{v}$
222Shmidt accounted for the Lorentz force on the vortices by incorporating an
effective term in the Gibbs free energy density [29]. Such an incorporation
does not seem possible in the general case of arbitrary vortex locations.
Thus, we treat Lorentz force in its direct form and separately from the force
evaluated via the Gibbs free energy density gradient.. To this, we need to add
the Lorentz force per unit volume exerted by the externally injected transport
current density $\boldsymbol{j}_{\mathrm{ext}}$ making the net force per unit
volume on a vortex:

$\displaystyle\boldsymbol{F}_{\mathrm{net}}$
$\displaystyle=\boldsymbol{F}_{s}+\frac{\phi_{0}\boldsymbol{j}_{\mathrm{ext}}(\boldsymbol{\rho}_{v})\times\hat{\boldsymbol{z}}}{\mathcal{A}},$
(6)

where $\mathcal{A}$ is the area per vortex. Up to the critical current,
$\boldsymbol{F}_{\mathrm{net}}$ evaluated at the surface is directed outwards
from the superconductor and vortices are not allowed to enter. Just above the
critical current, $\boldsymbol{F}_{\mathrm{net}}$ drives vortices nucleated at
one surface to the other resulting in dissipation and loss of
superconductivity.

Thus, we formulate the following generalization of Shmidt’s criterion [29] to
define the critical current:

$\displaystyle I_{c}$
$\displaystyle=\mathrm{min}_{\mathrm{vortex~{}axes}}\left[\left.\boldsymbol{F}_{\mathrm{net}}(I_{\mathrm{ext}})\right|_{\mathrm{surface}}\cdot\hat{\boldsymbol{n}}=0\right],$
(7)

where the minimum value over all possible configurations for the vortex axes
has to be chosen. The requirement of minimizing over all possible vortex axes
arises when there is no preferred direction set by, for example, a large
external applied field and is crucial in the above general criterion. For ease
of calculations, we find it convenient to choose our coordinate frame with the
vortex axis along $\hat{\boldsymbol{z}}$ [Eq. (4)]. Nevertheless, our analysis
puts no restrictions on the vortex axis direction in general.

Following the analysis as outlined above and described in detail for certain
cases of interest below, we obtain the following expression for the critical
current in the superconducting film valid for a wide range of parameters:

$\displaystyle I_{c}=$
$\displaystyle\frac{2Wd^{2}}{4\lambda^{2}}\mathrm{min}\left\\{\left(H_{sL}-H_{e}\right),\left(H_{sR}+H_{e}\right)\right\\},$
(8)

where $W$ is the superconductor film width perpendicular to the direction of
external current injection, and $H_{sL,sR}$ represent the vortex surface
barrier fields on the left and right surfaces. The physics behind this
expression will be clarified in the next sections. For now, we note that this
is one of our main results which shows that the critical current in the
superconductor can be influenced by controlling the vortex surface barrier
fields $H_{sL,sR}$ via various interfacial effects.

With the methodology outlined above, we describe a superconductor film without
any SOC in Sec. III. This corresponds to assuming
$\boldsymbol{\alpha}=\boldsymbol{0}$ and provides a preliminary understanding
of the critical current for the vortex mechanism considered here. In Sec. IV,
we analyze a superconductor bearing Rashba SOC and exchange field resulting in
a finite and general $\boldsymbol{\alpha}(\boldsymbol{r})$. The next section V
evaluates critical current in different superconductor-magnet hybrids. Here,
interfaces with ferro- or antiferromagnets are assumed to provide exchange
field and interfacial Rashba SOC giving rise to finite $\boldsymbol{\alpha}$
in a thin layer next to the interface. As a result, the critical current is
found to depend on the SOC strength, controllable via a gate voltage, and the
exchange field, tuned via magnetic order in the adjacent materials. A robust
control of critical current is thus predicted. In Sec. VI, we consider a
superconductor film interfaced with nonmagnetic insulators such that it bears
a gate voltage-tunable Rashba SOC but no exchange field. We argue from
symmetry that the Rashba SOC alone gives a correction to critical current that
scales as $\alpha_{\mathrm{Ra}}^{2}$. This dependence is compared with the
experiments observing critical current enhancement in such superconducting
films [22]. We estimate the degree of critical current modulation using
experimentally measured parameters in Sec. VII. Some of the assumptions and
weaknesses of our theoretical model are discussed in Sec. VIII, where comments
on future efforts are also made. We conclude by summarizing our work in Sec.
IX.

##  III Superconductor film without SOC

Figure 2: (a) Schematic depiction of the system under consideration. An
external magnetic field $H_{e}~{}\hat{\boldsymbol{z}}$ is applied and a
current $I_{\mathrm{ext}}~{}\hat{\boldsymbol{y}}$ is injected. We consider the
superconductor film to host a chain of vortices with flux
$\phi_{0}~{}\hat{\boldsymbol{z}}$, with $\phi_{0}$ the flux quantum. (b)
Normalized Gibbs free energy density $\tilde{G}\equiv G_{1}4\pi
ad\mu_{0}\lambda^{2}/(V_{S}\phi_{0}^{2})$ [Eq. (13)] vs. the vortex position
$\tilde{x}_{v}\equiv 2x_{v}/d$. In this work, we focus on low fields
$H_{e}<H^{\prime}$ for which the energy profile does not harbor stable or
metastable vortex states in the superconductor. (c) Schematic depiction of the
forces experienced by the upper vortex with flux along $\hat{\boldsymbol{z}}$
situated close to the left surface and the lower vortex with flux along
$-\hat{\boldsymbol{z}}$ located near the right surface. The force on each
vortex is comprised of attraction towards an image antivortex (purple),
interaction with the Meissner currents resulting from the applied field
(blue), and the Lorentz force due to externally injected current (green). The
applied field along $\hat{\boldsymbol{z}}$ lowers the surface barrier
experienced by the upper vortex close to the left, while it increases the
surface barrier experienced by the lower vortex on the right.

We begin by analyzing the vortex instability and critical current in a
superconductor film without SOC in this section. We reproduce the key aspects
of Shmidt’s analysis [29] and extend it to include vortices with arbitrarily
oriented axes, thereby formulating the generalized criterion introduced above
[Eq. (7)]. This generalization is crucial for adequately describing the
critical current.

We consider an infinite superconductor film with thickness $d$ and the y-z
plane situated in its middle, as depicted in Fig. 2 (a). An external magnetic
field $H_{e}\hat{\boldsymbol{z}}$ is applied and a current $I_{\mathrm{ext}}$
is injected into the film along $\hat{\boldsymbol{y}}$ via an external
circuit. Vortices form regular patterns in their stable states in type II
superconductors [58, 59]. Thus, we assume the superconductor film to host
vortices with flux along $\hat{\boldsymbol{z}}$ located at $x=x_{v}$ and
$y=ma$, where $m$ takes integer values and $a$ is the inter-vortex spacing
[see Fig. 2 (a)].

The magnetic flux density in the superconductor is evaluated by solving Eq.
(4) with the appropriate boundary conditions and
$\boldsymbol{\alpha}=\boldsymbol{0}$. As Eq. (4) is a linear differential
equation, its solution is simply the sum of individual contributions. Thus,
the magnetic flux densities contributed by the applied field, the injected
current, and the vortices one at a time add up to yield the total magnetic
flux density in the superconductor. The former two contributions are
conveniently described by imposing the boundary conditions:
$\boldsymbol{B}(x=\pm d/2)=\mu_{0}(H_{e}\mp H_{I})~{}\hat{\boldsymbol{z}}$
[see Fig. 2 (c)], where $H_{I}$ is the magnetic field generated by the
injected current at the superconductor surfaces. The magnetic flux density due
to the vortices stems directly from the source terms on the right hand side of
Eq. (4). Relegating details to appendix A, the resulting magnetic flux density
is evaluated as:

$\displaystyle\boldsymbol{B}(x,y)$
$\displaystyle=\left[\mu_{0}H_{e}\frac{\mathrm{cosh}\left(\frac{x}{\lambda}\right)}{\mathrm{cosh}\left(\frac{d}{2\lambda}\right)}-\mu_{0}H_{I}\frac{\mathrm{sinh}\left(\frac{x}{\lambda}\right)}{\mathrm{sinh}\left(\frac{d}{2\lambda}\right)}+B_{v}(x,y)\right]\hat{\boldsymbol{z}},$
(9)

where $B_{v}(x,y)$ is the vortex chain contribution detailed in appendix A.
$H_{I}$ is determined by requiring that the injected current density
$\boldsymbol{j}_{\mathrm{ext}}(x)$ obtained by employing Maxwell’s equation on
the corresponding contribution to the magnetic flux density [Eq. (9)]:

$\displaystyle\boldsymbol{j}_{\mathrm{ext}}(x)$
$\displaystyle=-\frac{\partial}{\partial
x}\left(-H_{I}\frac{\mathrm{sinh}\left(\frac{x}{\lambda}\right)}{\mathrm{sinh}\left(\frac{d}{2\lambda}\right)}\right)~{}\hat{\boldsymbol{y}}=\frac{H_{I}}{\lambda}\frac{\mathrm{cosh}\left(\frac{x}{\lambda}\right)}{\mathrm{sinh}\left(\frac{d}{2\lambda}\right)}~{}\hat{\boldsymbol{y}},$
(10)

integrated over area becomes the total injected current:

$\displaystyle\int_{-\frac{d}{2}}^{\frac{d}{2}}dx~{}\boldsymbol{j}_{\mathrm{ext}}(x)\cdot\hat{\boldsymbol{y}}~{}W$
$\displaystyle=I_{\mathrm{ext}},$ (11) $\displaystyle\implies H_{I}$
$\displaystyle=\frac{I_{\mathrm{ext}}}{2W},$ (12)

where $W$ is the film width along the z-direction. It is convenient to analyze
our system in terms of $H_{I}$ keeping in mind its one-to-one relation with
the externally injected current [Eq. (12)].

Employing Eq. (9) in Eq. (5), we obtain the Gibbs free energy density for the
configuration under consideration:

$\displaystyle\frac{G_{1}}{V_{S}}$
$\displaystyle=\frac{G_{0}}{V_{S}}+\frac{\phi_{0}H_{e}}{ad}\frac{\mathrm{cosh}\left(\frac{x_{v}}{\lambda}\right)}{\mathrm{cosh}\left(\frac{d}{2\lambda}\right)}-\frac{1}{ad}\frac{\phi_{0}^{2}}{4\pi\mu_{0}\lambda^{2}}\left[I_{1}(x_{v})-I_{2}(x_{v})\right],$
(13)

where $G_{0}/V_{S}$ is a constant detailed in appendix A and does not depend
on the vortex positions. The term involving $H_{e}$ captures the interaction
between vortices and the Meissner currents generated in order to screen the
external field. The term involving $I_{1}(x_{v})$, detailed in appendix A,
captures the interaction of vortices with the surfaces. It is responsible for
Bean-Livingston surface barrier posed to the vortices. The term involving
$I_{2}(x_{v})$, also detailed in appendix A, accounts for interaction between
the vortices which weakly depends on their distance from the surfaces. As
detailed in appendix A, this contribution becomes negligible when $a\gtrsim
d$, which is the case of our interest as discussed below. Hence, inter-vortex
interaction is disregarded in the rest of our analysis and conveniently, the
stability criterion determining the critical current can be formulated
considering forces on an individual vortex.

The Gibbs free energy density [Eq. (13)] is plotted against vortex position
$x_{v}$ in Fig. 2 (b) for different values of the external applied field. At
low fields, the energy profile is dominated by the $I_{1}(x_{v})$ term, and
thus the Bean-Livingston barrier, which prevents the vortices from entering
the superconductor. This can also be visualized as an attraction between the
vortex and an imaginary image vortex with reverse polarity, as depicted in
Fig. 2 (c). For $H_{e}>H^{\prime}$, the energy profile develops a minimum at
$x_{v}=0$ [Fig. 2 (b)]. Vortices are still prevented from entering by a finite
surface barrier. But if they do manage to enter somehow, they could find a
metastable position in this minimum at $x_{v}=0$. $H^{\prime}$ is determined
by the condition:

$\displaystyle\left.\frac{\partial^{2}G_{1}}{\partial
x_{v}^{2}}\right|_{x_{v}=0}$ $\displaystyle=0~{}\implies
H^{\prime}\approx\frac{2\phi_{0}}{\pi\mu_{0}d^{2}}.$ (14)

When the external field reaches a larger value $H_{s}$, the surface barrier is
eliminated [Fig. 2 (b)]. Vortices can freely enter the sample and organize
into a regular stable pattern in the middle. $H_{s}$ is thus defined via:

$\displaystyle\left.\frac{\partial G_{1}}{\partial x_{v}}\right|_{x_{v}=\pm
d/2}$ $\displaystyle=0~{}\implies
H_{s}\approx\frac{\phi_{0}}{2\pi\mu_{0}d\xi}.$ (15)

It is the high-field regime that Shmidt was primarily interested in. He
predicted a “peak effect” in the critical current vs. applied field curve that
results from an increased stabilization of the vortices at higher fields [29].
The prediction has been verified experimentally (see Refs. [54, 50] and the
references therein). Because of this interest in the high field regime, Shmidt
only considered vortices whose flux was parallel to the applied field.

In contrast, we limit ourselves to the low-field regime $H_{e}<H^{\prime}$
such that no stable or metastable states of vortices exist in the
superconductor. As a result, our formulated criterion for critical current
[Eq. (7)] needs to consider forces at the surfaces only, and not within the
film. However, we now need to consider all possible directions for the vortex
axis since it is not fixed by the external field. Thus, in our considerations,
the critical current is the lowest value at which the net force on a vortex at
any of the surfaces vanishes for any direction of the vortex axis [Eq. (7)].
Furthermore, as we consider low fields, the inter-vortex spacing $a$ between
an assumed vortex chain is large and we may safely ignore the inter-vortex
interactions.

Including the Lorentz force due to injected current in our ongoing analysis of
vortices with flux along $\hat{\boldsymbol{z}}$, the critical current is
determined using Eq. (6):

$\displaystyle-\frac{1}{V_{S}}\left.\frac{\partial G_{1}}{\partial
x_{v}}\right|_{x_{v}=-d/2}+\frac{\phi_{0}j_{\mathrm{ext}}(-d/2)}{ad}$
$\displaystyle=0,$ (16)

where $j_{\mathrm{ext}}(x)$ is given by Eq. (10) and $G_{1}$ by Eq. (13). As
detailed in appendix A, the equation above yields the critical current
expressed via the magnetic field it produces at the surface [Eq. (12)]:

$\displaystyle H_{I_{c}}$
$\displaystyle=\frac{d^{2}}{4\lambda^{2}}\left(H_{s}-H_{e}\right),$ (17)

where $H_{s}$ is the field at which the Bean-Livingston barrier would be
annihilated and is given by Eq. (15) for the case under consideration.

Consistent with Shmidt’s analysis for relatively large applied magnetic field
along $\hat{\boldsymbol{z}}$, we have so far considered vortex axis to be
along $\hat{\boldsymbol{z}}$. In this case, the Lorentz force drives the
vortices along $\hat{\boldsymbol{x}}$. Thus, just above the critical current,
vortices nucleated on the left surface traverse through the film and
annihilate on the right edge [Fig. 2 (c)]. Consequently, the vortex surface
barrier on the left determines critical current. As noted above, since we
consider the external field to be weak, we need to consider instability for
any vortex axis direction.

To appreciate this point, let us first consider $H_{e}=0$. As depicted in Fig.
2 (c), the field generated by the injected current points along
$\hat{\boldsymbol{z}}$ ($-\hat{\boldsymbol{z}}$) at $x=-d/2$ ($x=d/2$). Thus,
two independent vortex instabilities start to develop at each of the surfaces
as injected transport current is increased. Vortices with flux along
$\hat{\boldsymbol{z}}$ ($-\hat{\boldsymbol{z}}$) are being pushed from left
(right) to right (left) and the corresponding instability will be determined
by the vortex barrier on the left (right) surface. For equivalent surfaces,
the two instabilities will set off at the same critical current. However, two
surfaces are never really identical. In practice, one of the two surfaces will
have a lower vortex barrier and will determine the critical current. A nonzero
$H_{e}$ applied along $\hat{\boldsymbol{z}}$, helps to reach the instability
on the left surface sooner, as clarified via the forces depicted in Fig. 2
(c). Altogether, the critical current ($\geq 0$) is determined via:

$\displaystyle H_{I_{c}}$
$\displaystyle=\frac{d^{2}}{4\lambda^{2}}\mathrm{min}\left\\{\left(H_{sL}-H_{e}\right),\left(H_{sR}+H_{e}\right)\right\\},$
(18)

where $H_{sL}$ ($H_{sR}$) is the magnetic field at which the vortex surface
barrier on the left (right) surface is extinguished. While within our simple
model it was evaluated as given in Eq. (15), the critical current expression
above [Eq. (18)] is general even when $H_{sL,sR}$ are no longer given via Eq.
(15). Due to various reasons, such as disorder and localized enhanced field
strengths around corners, the effective value of $H_{s}$ is expected to be
lower than what is evaluated in Eq. (15) for an ideal surface [29, 1].

Thus, generalizing Shmidt’s analysis, we have understood the key factors
underlying the vortex instability and critical current in the type II
superconductor film. From the expression obtained [Eq. (18)], we can see that
the critical current is determined by the weaker of the two surface barriers.
We can also anticipate that altering the barriers $H_{sL,sR}$ via a gate
voltage, for example, would enable a control over the critical current.

##  IV Superconductor film with SOC and exchange field

Now we include a finite $\boldsymbol{\alpha}$ [Eq. 1] in our considerations.
As noted before, this contribution to the free energy results from the
combined effect of Rashba SOC and an exchange spin-splitting field. It is
important that the latter is not caused by an external magnetic field, that
would necessarily cause an additional orbital contribution, but by an exchange
interaction [32, 35, 66, 67, 68, 69, 70]. Thus, we consider
$\boldsymbol{\alpha}$ in our superconductor to originate in the interfaces
with magnetic insulators (MIs), which could be ferromagnets [35, 67, 36, 69]
or antiferromagnets with uncompensated surfaces [34]. Furthermore, we assume
$\boldsymbol{\alpha}$ to be small and evaluate its influence on the system up
to the first order.

Figure 3: (a) Schematic depiction of spontaneous supercurrent density
$\boldsymbol{j}_{L}$ (blue arrows) and the resulting forces (green arrows) on
a vortex located at different places in the superconductor. The combined
effect of SOC and exchange field results in a finite $\boldsymbol{\alpha}$ and
consequently, a spontaneous supercurrent directed parallel to
$\boldsymbol{\alpha}$ [Eq. (3)] in the interfacial layer, shaded light blue. A
much smaller oppositely directed supercurrent density throughout the film
(depicted via small blue arrows pointing upwards) ensures zero net current in
equilibrium. (b) The normalized finite-$\boldsymbol{\alpha}$ Gibbs energy
density contributions $\tilde{G}_{\alpha}=G_{\alpha}4e\lambda
da\mu_{0}/(V_{S}\phi_{0}\alpha_{L,R}\cos\theta_{L,R})$ vs. the normalized
vortex position $\tilde{x}_{v}=2x_{v}/d$. The blue and red curves respectively
depict the contributions from left and right interfaces.

As detailed in Sec. II, we model the two interfaces with MIs via effective
interfacial layers of thicknesses $l_{so}$ that harbor finite
$\boldsymbol{\alpha}$ [see Fig. 3 (a)]:

$\displaystyle\boldsymbol{\alpha}(x)=$
$\displaystyle~{}\alpha_{L}(\mathrm{cos}\theta_{L}\hat{\boldsymbol{y}}+\mathrm{sin}\theta_{L}\hat{\boldsymbol{z}})~{}\Theta(-x-x_{so})$
$\displaystyle-\alpha_{R}(\mathrm{cos}\theta_{R}\hat{\boldsymbol{y}}+\mathrm{sin}\theta_{R}\hat{\boldsymbol{z}})~{}\Theta(x-x_{so});\quad
x\in[-d/2,d/2],$ (19)

where $\alpha_{L,R}$ parameterize $\boldsymbol{\alpha}$ magnitude at the left
and right interfaces, $\theta_{L,R}$ correspondingly allow for their general
in-plane directions, $x_{so}\equiv d/2-l_{so}$, and $\Theta(x)$ is the
heaviside step function. We now need to solve Eq. (4) to obtain the magnetic
flux density in the superconductor. Since this is a linear equation, the
contribution of finite $\boldsymbol{\alpha}$ simply adds to the magnetic flux
density presented in the previous section for
$\boldsymbol{\alpha}=\boldsymbol{0}$ [Eq. (9)], and is evaluated as:

$\displaystyle\boldsymbol{B}_{\alpha}(x)=$
$\displaystyle-[\mathrm{cos}\theta_{L}B_{L}(x)+\mathrm{cos}\theta_{R}B_{R}(x)]~{}\boldsymbol{\hat{z}}+[\mathrm{sin}\theta_{L}B_{L}(x)+\mathrm{sin}\theta_{R}B_{R}(x)]~{}\boldsymbol{\hat{y}},$
(20)

where

$\displaystyle B_{L}(x)$
$\displaystyle=\frac{\alpha_{L}}{4e\lambda}\frac{\mathrm{cosh}\left(\frac{d-|x+x_{so}|}{\lambda}\right)-\mathrm{cosh}\left(\frac{x-x_{so}}{\lambda}\right)}{\mathrm{sinh}\left(\frac{d}{\lambda}\right)},$
(21) $\displaystyle B_{R}(x)$
$\displaystyle=\frac{\alpha_{R}}{4e\lambda}\frac{\mathrm{cosh}\left(\frac{d-|x-x_{so}|}{\lambda}\right)-\mathrm{cosh}\left(\frac{x+x_{so}}{\lambda}\right)}{\mathrm{sinh}\left(\frac{d}{\lambda}\right)}.$
(22)

Employing the total magnetic flux density in Eq. (5), we obtain the total
average Gibbs free energy density:

$\displaystyle\frac{G_{2}}{V_{S}}$
$\displaystyle=\frac{G_{1}}{V_{S}}+\frac{G_{\alpha
0}}{V_{S}}+\frac{G_{\alpha}}{V_{S}},$ (23)

where $G_{1}$ is the contribution without SOC [Eq. (13)], $G_{\alpha 0}$ does
not depend on the vortex position and has been detailed in appendix B, and

$\displaystyle\frac{G_{\alpha}}{V_{S}}$
$\displaystyle=-\frac{\phi_{0}}{da\mu_{0}}\left[B_{L}(x_{v})\mathrm{cos}\theta_{L}+B_{R}(x_{v})\mathrm{cos}\theta_{R}\right],$
(24)

constitutes the Gibbs energy contribution resulting from finite
$\boldsymbol{\alpha}$, that exerts additional forces on the vortices,
evaluated via Eq. (6).

We pause to physically interpret our mathematical results. For simplicity, let
us focus on the effect of the left interface by assuming $\alpha_{R}=0$. The
contribution of the right interface can be understood in an analogous manner.
On account of finite $\alpha_{L}$, a spontaneous supercurrent density is
generated in the interfacial layer with thickness $l_{so}$ next to the
interface. Since the total equilibrium current through the film must vanish,
this causes a weaker oppositely directed supercurrent density in the entire
superconductor layer [see Fig. 3 (a)]. Consequently, this spatially dependent
spontaneous supercurrent density exerts a relatively large force on a vortex
located at the interface and a weak force if it is located in the bulk. This
can also be seen via the slope of the Gibbs free energy density contribution
due to $\boldsymbol{\alpha}$ plotted in Fig. 3 (b). Thus, the spontaneous
supercurrents at the interfaces significantly alter the vortex surface barrier
fields and consequently the critical current.

To determine the critical current, we should employ the instability condition
Eq. (7) for all possible vortex axes. However, assuming $\boldsymbol{\alpha}$
to be small, we expect the vortex axis for the instability at each of the
interface to be determined primarily by the magnetic flux density generated by
the externally injected current. This is analogous to our considerations
without SOC in the previous section, where we found the critical current to be
determined by the vortex surface barrier at each interface, as given by Eq.
(18). Following the analogous analysis, the critical current including the
effect of SOC can still be expressed via Eq. (18) with $\boldsymbol{\alpha}$
contributing corrections to the surface barrier fields evaluated to be:

$\displaystyle\Delta H_{sL}$
$\displaystyle=-\frac{1}{\mu_{0}ed}\left[\alpha_{L}\mathrm{cos}\theta_{L}\left(1-\frac{l_{so}}{d}\right)+\alpha_{R}\mathrm{cos}\theta_{R}\left(\frac{l_{so}}{d}-\frac{l_{so}d}{4\lambda^{2}}\right)\right]\approx-\frac{\alpha_{L}\mathrm{cos}\theta_{L}}{\mu_{0}ed},$
(25) $\displaystyle\Delta H_{sR}$
$\displaystyle=\frac{1}{\mu_{0}ed}\left[\alpha_{R}\mathrm{cos}\theta_{R}\left(1-\frac{l_{so}}{d}\right)+\alpha_{L}\mathrm{cos}\theta_{L}\left(\frac{l_{so}}{d}-\frac{l_{so}d}{4\lambda^{2}}\right)\right]\approx\frac{\alpha_{R}\mathrm{cos}\theta_{R}}{\mu_{0}ed}.$
(26)

Hence, the vortex surface barrier, or equivalently the critical current, may
be increased or decreased due to $\alpha_{L,R}$. They may further be
controlled via angles $\theta_{L,R}$, which are determined by the direction of
magnetic or Néel order in the adjacent MIs.

##  V Superconductor-magnet hybrids

We now examine the critical current in experimentally relevant
heterostructures formed by a type II superconductor (S) film adjoining
magnetic insulator (MI) film(s) on one or both sides. The latter’s role is to
provide an effective exchange field $\boldsymbol{h}$ that determines
$\boldsymbol{\alpha}\propto\alpha_{Ra}\hat{\boldsymbol{n}}\times\boldsymbol{h}$
as discussed in Sec. II. The field $\boldsymbol{h}$ results from interfacial
exchange interaction and is collinear with the magnetization of the adjacent
ferromagnet or the sublattice-magnetization of the uncompensated
antiferromagnet [34]. Furthermore, recent considerations show that a range of
metallic antiferromagnets harbor a strong spin-splitting of their electronic
states [71, 72, 73, 74, 75, 76]. These can further provide an effective
exchange field collinear with the antiferromagnetic Néel vector even with a
compensated interface [77]. Hence, the direction of $\boldsymbol{h}$, and thus
$\boldsymbol{\alpha}$, can be conveniently controlled by a weak external
magnetic field. In case of a ferromagnet, $\boldsymbol{h}$ simply becomes
collinear with the applied field. For a typical easy-plane antiferromagnet,
the Néel order and $\boldsymbol{h}$ orient perpendicular to the applied field
(e.g., see Ref. [78]). For both cases, the magnitude of $\boldsymbol{h}$ is
independent of the external field and is determined by the MI and interface
properties. Hence, the critical current in heterostructures of interest can be
controlled via the orientation of a weak applied magnetic field. Furthermore,
a gate-voltage may control the Rashba SOC parameter $\alpha_{Ra}$ [43, 44,
46].

In our considerations, we continue to assume the effects of applied magnetic
field and SOC to be weak. We evaluate the critical current up to the first
order in both these variables. This also implies that the vortex axes for the
instability that determines the critical current is governed primarily by the
magnetic field generated by the externally injected current at the surfaces of
S. The situation is thus the same as our analyses in the previous sections and
we may express critical current using Eqs. (12) and (18):

$\displaystyle I_{c}=$
$\displaystyle\frac{2Wd^{2}}{4\lambda^{2}}\mathrm{min}\left\\{\left(H_{sL}-H_{e}\right),\left(H_{sR}+H_{e}\right)\right\\},$
(27)

where $W$ is the film width along z direction (perpendicular to the injected
current flow). We further split the surface barrier fields $H_{sL,sR}\equiv
H_{sL,sR}^{0}+\Delta H_{sL,sR}$ into their values without interfacial SOC
($H_{sL,sR}^{0}$) plus the difference that can be controlled via external
parameters. Similarly, we express the total critical current as $I_{c}\equiv
I_{c}^{0}+\Delta I_{c}$. Equation (27), together with the results evaluated in
the previous section [Eqs. (25) and (26)], forms the basis for our critical
current analysis below.

###  V.1 MI-S bilayer

Figure 4: (a) Schematic of the magnetic insulator (MI)- superconductor (S)
bilayer that enables a control of the critical supercurrent in S layer. The
external current is injected along $\hat{y}$ and the MI magnetic order
determines the exchange field $\boldsymbol{h}$, which can be controlled by,
for example, an external magnetic field. (b) and (c) Change in the
superconducting critical current $\Delta I_{c}$ due to the combined action of
exchange field and interfacial SOC resulting in a finite $\boldsymbol{\alpha}$
vs. orientation of the exchange field $\boldsymbol{h}$. $I_{c}^{0}$ is the
critical current without interfacial SOC. We assume
$\alpha_{L}/e\mu_{0}dH_{sL}^{0}=0.1$. In (c), we consider
$H_{e}/H_{sL}^{0}=0.02$.

We first consider a MI-S heterostructure as depicted in Fig. 4 (a), which
corresponds to $\alpha_{R}=0$. The magnetic order parameter, and thus
$\boldsymbol{h}$, is considered to make an angle $\theta_{L}$ with the z axis,
which is defined as perpendicular to the injected current flow. Such a bilayer
structure is likely to have unequal vortex surface barrier fields at the two S
surfaces due to their different qualities. For example, disorder at the S$|$MI
interface is likely to lower $H_{sL}^{0}$ compared to the right interface
terminated in vacuum (or a capping layer). Under such a condition
$H_{sL}^{0}<H_{sR}^{0}$, the critical current is primarily determined by the
left interface [Eq. (25)] and is depicted via solid purple line in Fig. 4 (b).
Thus, SOC-mediated change in the critical current can be positive or negative,
depending on $\theta_{L}$ and the sign of $\alpha_{L}$. When we consider
identical surface barriers such that $H_{sL}^{0}=H_{sR}^{0}$, the critical
current varies as depicted by the green dashed curve in Fig. 4 (b). In this
case, for a range of $\theta_{L}$, the vortex instability is determined by the
right surface [Eq. (26)]. It depends weakly on $\alpha_{L}$ due to the
supercurrent density ($\sim l_{so}/d$) generated throughout S to cancel the
spontaneous supercurrent induced at the left interface. The range of
$\theta_{L}$ for which the critical current is determined by the left
interface shows a decrease compared to the value without SOC. Thus, SOC
contribution in this case may only lower the critical current compared to
$I_{c}^{0}$. In Fig. 4 (c), we depict the case of $H_{sL}^{0}=H_{sR}^{0}$
accounting for the small correction that results from a weak but finite
external magnetic field along the z axis. This quantifies how the applied
field lowers the vortex barrier at one surface while raising it on the other,
as detailed in Sec. III and adequately captured in Eq. (27).

When we consider $H_{sL}^{0}>H_{sR}^{0}$, critical current is determined by
the vortex instability at the right surface and is practically unaffected by
external parameters that can influence $\alpha_{L}$. Thus, in order to
accomplish MI-S heterostructures admitting a control of the critical
supercurrent, we must pay special attention to having a high quality of the
other S surface, such that $H_{sL}^{0}<H_{sR}^{0}$. This might occur naturally
if S is deposited on a MI layer and left exposed to vacuum or a capping layer.

###  V.2 MI-S-MI trilayer

Figure 5: (a) and (b) Schematic depiction of the trilayer device that allows a
control of critical current in S layer via magnetic orders in the two adjacent
MIs. (c) Fractional change in the critical current vs. orientations of the
exchange fields induced by the two adjacent MIs. We assume $H_{e}=0$,
$H_{sL}^{0}=H_{sR}^{0}$, $|\alpha_{L}|=|\alpha_{R}|$, and
$\alpha_{L}/e\mu_{0}dH_{sL}^{0}=0.1$.

Considering the S layer to be sandwiched between two MI layers, as depicted in
Fig. 5, provides additional control. This trilayer structure further allows to
witness a full interplay between the vortex barriers at both the surfaces and
makes the parameter space larger with numerous possibilities. The critical
current is still governed by the relatively simple result Eq. (27) and can be
obtained for any desired parameter space. Considering $H_{e}=0$ and
symmetrical vortex surface barriers in the absence of SOC i.e.,
$H_{sL}^{0}=H_{sR}^{0}$, the critical current varies with the MI orders as
depicted in Fig. 5 (c). Thus, we find the highest critical current when the
two order parameters are antiparallel assuming the same sign of the Rashba SOC
at the two interfaces, which can further be controlled via gate voltage(s)
using additional electrode layers not shown explicitly in Fig. 5. In practice,
one might consider the magnetic order in one of the MIs to be fixed, for
example by exchange-biasing [79], and the other MI then allows $I_{c}$ to be
controlled via a weak applied magnetic field. The generality of our result Eq.
(27) should facilitate engineering different means of controlling the
superconducting critical current in such superconductor-magnet hybrids [80,
81, 82].

##  VI Superconductor film with gate voltage-controlled SOC

Since the present work is partly inspired by the recent observation of gate
voltage-induced critical current enhancement in superconducting NbN [22], we
pause to discuss the implications of our theory on these experimental results.
The experiments in question recorded the critical current $I_{c}$ of a NbN
bridge deposited on insulating Si$\mathrm{O}_{2}$ and gated via the p-doped Si
substrate. NbN layer was thus sandwiched between two non-magnetic insulators
and no magnetic field was applied. The critical current was found to increase
with the applied gate voltage (almost) irrespective of the latter’s polarity.
A small asymmetry with respect to the gate voltage sign was, however,
observed.

Our analysis above requires an MI-induced exchange field to achieve changes in
the critical current. This is because we have retained terms up to the first
order in the interfacial Rashba SOC
$\alpha_{\mathrm{Ra}}\propto\boldsymbol{\alpha}$. In the absence of exchange
field from an adjacent MI, the leading order effect of the interfacial SOC on
the vortex surface barrier is expected to scale as $\alpha_{\mathrm{Ra}}^{2}$
[83]. Evaluating this within an analytic model seems difficult, as discussed
further in Sec. VIII below. Nevertheless, we proceed to understand the
symmetries of critical current modulation induced by such an effect. Without
loss of generality, we may assume that $H_{sL}<H_{sR}$ such that the critical
current is given via [Eq. (27)]:

$\displaystyle I_{c}$ $\displaystyle=\frac{2Wd^{2}}{4\lambda^{2}}H_{sL},$ (28)
$\displaystyle\frac{\Delta I_{c}}{I_{c}^{0}}$ $\displaystyle=\frac{\Delta
H_{sL}}{H_{sL}^{0}}\propto\alpha_{\mathrm{Ra}}^{2}\
=\alpha_{0}^{2}+c_{1}^{2}V_{G}^{2}+2c_{1}\alpha_{0}V_{G},$ (29)

where $I_{c}^{0}$ ($\Delta I_{c}$) and $H_{sL}^{0}$ ($\Delta H_{sL}$) denote
the critical current and the surface barrier field without (due to) the
interfacial Rashba SOC. We have further assumed
$\alpha_{\mathrm{Ra}}=\alpha_{0}+c_{1}V_{G}$, with $V_{G}$ the gate voltage
[46]. Equation (29) therefore clarifies the predominantly bipolar (even in
$V_{G}$) nature of the critical current dependence on the applied gate voltage
with a possible polarity-dependence resulting from nonzero $\alpha_{0}$.
Further, the mechanism under investigation does not influence the
superconducting Tc, consistent with the experiments [22].

##  VII Estimation of the critical current modulation

In this section, we estimate the expected critical current modulation in the
S-MI hybrids considered above. A change in the critical current can be
accomplished via the MI order (e.g., magnetization of a ferromagnet)
orientation or an applied gate voltage that alters the interfacial Rashba SOC
strength.

As per our discussion in the previous section, the fractional change in the
critical current may be expressed as:

$\displaystyle\left|\frac{\Delta I_{c}}{I_{c}^{0}}\right|$
$\displaystyle=\left|\frac{\Delta H_{sL}}{H_{sL}^{0}}\right|,$ (30)
$\displaystyle=\left|\frac{\alpha_{L}\mathrm{cos}\theta_{L}}{\mu_{0}edH_{sL}^{0}}\right|,$
(31)

where we have employed Eq. (25) in achieving the above simplification. We can
further substitute for $H_{sL}^{0}$ using Eq. (15), but that is unlikely to
reflect the situation in samples of interest. The surface barrier field as
derived in Eq. (15) for an ideal surface is rather large. If it were to
determine the critical current, the latter will be comparable to the pair-
breaking mechanism-determined critical current and vortex instability
mechanism may not be operational at all. As discussed previously, in realistic
samples, interfacial disorder and magnetic field inhomogenieties around the
edges substantially reduce $H_{sL}^{0}$ making the vortex instability the
dominant mechanism in a range of samples [29, 1, 50]. Thus, we estimate
$H_{sL}^{0}$ using Eq. (28) and the experimentally measured value of
$I_{c}^{0}$. Expressing $\alpha_{L}$ in terms of the induced exchange field
magnitude $h$ and dropping the subscript $L$ denoting the left interface, we
obtain [Eq. (31)]:

$\displaystyle\left|\frac{\Delta I_{c}}{I_{c}^{0}}\right|$
$\displaystyle=\left|\frac{\beta
h\mathrm{cos}\theta}{\mu_{0}edH_{s}^{0}}\right|.$ (32)

The factor $\beta h$ can be expressed in terms of the Rashba parameter
$\alpha_{\mathrm{Ra}}$ and properties of the superconductor via [33, 63]:

$\displaystyle\beta h$
$\displaystyle=\left(\frac{v_{\mathrm{Ra}}}{v_{F}}\right)\left(\frac{c}{l_{so}}\right)\left(\frac{h}{k_{B}T_{c}}\right)\frac{\hbar}{\xi},$
(33)

where $v_{\mathrm{Ra}}=\alpha_{\mathrm{Ra}}/\hbar$ is the Rashba velocity,
$v_{F}$ is the Fermi velocity, $c$ is the lattice constant, $T_{c}$ is the
superconducting critical temperature, and $h$ is the exchange field in units
of energy. In the equation above, the parentheses enclose dimensionless
quantities. Employing Eq. (33) in Eq. (32) and dropping the explicit $\theta$
dependence, we obtain:

$\displaystyle\left|\frac{\Delta I_{c}}{I_{c}^{0}}\right|$
$\displaystyle=\left(\frac{v_{\mathrm{Ra}}}{v_{F}}\right)\left(\frac{c}{l_{so}}\right)\left(\frac{h}{k_{B}T_{c}}\right)\left(\frac{\phi_{0}}{\pi
d\xi\mu_{0}H_{s}^{0}}\right),$ (34)

where we continue to enclose dimensionless quantities in parentheses. Equation
(34) is our desired estimation and sheds light on the key features that we
pause to discuss.

The first parenthesis on the right hand side of Eq. (34) encloses the ratio
between SOC energy splitting at the Fermi wavevector and the Fermi energy.
This quantity can be tuned in a wide range via an applied gate voltage for
different systems [44, 47, 48, 43, 46]. The second parenthesis encloses the
inverse of the number of monolayers in the length $l_{so}$, the value of which
has been estimated differently in different works [32, 33, 63]. We expect
$l_{so}$ to be roughly the same as the coherence length $\xi$. The third
paranthesis encloses the exchange field as a fraction of the critical
temperature. All the three preceding factors are typically less than 1. The
fourth and final factor is the ratio between the magnetic flux density
obtained if the flux quantum was distributed over an area $d\xi$ to the
magnetic flux density corresponding to the vortex surface barrier field of the
superconducting film without SOC. This final factor can be much larger than 1.
Thus, to maximize the critical current modulation, we should employ
superconductors with small coherence lengths and film thicknesses.

In order to obtain a numerical estimate for the critical current modulation,
we consider the NbN device investigated in the recent experiments [22]. Thus,
we consider $d=10$ nm, $\xi=10$ nm, $\lambda=450$ nm, $I_{c}^{0}=80~{}\mu$A
and $W=1~{}\mu$m to obtain $\mu_{0}H_{s}^{0}\approx 0.4$ T. Further, assuming
the SOC splitting to be $5\%$ of the Fermi energy and a lattice constant of
$0.4$ nm [84], we estimate [Eq. 34]:

$\displaystyle\left|\frac{\Delta I_{c}}{I_{c}^{0}}\right|$
$\displaystyle=\left(0.05\right)\left(0.04\right)\left(0.2\right)\left(16\right)=0.006,$
(35)

where we considered $l_{so}=\xi$. This represents a modest modulation of about
$1\%$ that can be increased substantially via careful engineering of devices.

##  VIII Discussion and Outlook

The main contribution of our analysis can be divided into two fairly
independent parts. The first is establishing a relation between the critical
current and the vortex surface barriers. This is expressed in a rather general
form by Eq. (7), and for a somewhat specialized case by Eq. (18). The general
idea is that if we know the vortex surface barrier for all surfaces and flux
orientations, we may evaluate the critical current due to this mechanism. If
it is found to be smaller than the critical current due to all other
mechanisms, the film will be governed by this mechanism and we may control its
critical current via the surface barrier. The second main part of our work has
been evaluating the vortex surface barriers using simplified analytically
tractable models. While our considerations clearly demonstrate an SOC-enabled
control over vortex surface barriers, the latter are generally difficult to
evaluate for realistic samples [85] and should be treated as experimental
unknowns, where the theory may provide guidance. However, our finding that the
interfacial Rashba SOC should affect the vortex surface barrier is clear and
independent of the detailed modeling.

With this in mind, let us examine when our investigated mechanism of critical
current should dominate realistic samples. In a disordered type II
superconductor film, the vortices (or flux) get pinned in the bulk, thereby
eliminating the role of interfacial barriers. Thus, clean films devoid of flux
pinning centers are required. Due to the same reason, thin clean films are
better, since there is then a small distance to be traversed between the
surfaces. Thicker films will have a higher chance of vortices encountering
flux pinning centers while moving from one surface to the other. Further,
within our analytic model, the evaluated surface barrier [Eq. (15)] is too
high for it to determine the critical current. However, as noted by Shmidt
[29], realistic samples could manifest much weaker surface barriers due to
various reasons, such as interfacial disorder. Hence, clean thin films grown
on a substrate are good candidates, and the vortex surface barrier is best
treated as an experimental unknown.

Furthermore, vortices with out-of-plane flux may enter the superconducting
film from other surfaces not explicitly considered in our study. These might
even face a smaller surface barrier due to the large quasi-bulk distance
between the opposite surfaces. However, these will suffer from the challenge
of flux-pinning in the bulk since they have to travel a long distance between
the two surfaces. Also, the magnetic field generated by the current flowing
through the film favors spontaneous nucleation of vortices with in-plane flux.
A more careful consideration of the out-of-plane vortices requires a numerical
three-dimensional analysis and is left for a future study.

We now note some weaknesses of our analysis and suggest ways to address them
in future works. We have worked in the limit $\xi\ll d\ll\lambda$ and found
that the highest critical current modulation is achieved for the smallest $d$.
Thus, we estimated the modulation by considering $d=\xi$, which is strictly
speaking beyond the applicability of our theory. Simply put, we work in the
London limit corresponding to $\xi\to 0$ and often have to choose other
lengths comparable to $\xi$. Despite such transgressions, our analysis here
offers good estimates as explained further in the appendix A and also by
Shmidt, whose theory faced the same challenges [29]. While we have captured
the qualitative physics adequately, a quantitative reliance on our expressions
is not encouraged.

Furthermore, we assumed the effect of interfacial SOC to be small and
evaluated corrections to the critical current perturbatively up to the first
order in $\alpha_{\mathrm{Ra}}$. In this sense, we have assumed the critical
current modulation to be small at the outset. This allowed us to treat the
vortices as robust objects that are not affected by the SOC. If we attempt to
evaluate the Gibbs free energy up to the second order in
$\alpha_{\mathrm{Ra}}$, we should also account for a distortion in the shape
of the vortices as well as corrections to their core energy. This analysis,
although analytically tractable, seems discouragingly tedious. Furthermore,
the case of highest experimental interest, admitting a large critical current
modulation, corresponds to a strong interfacial SOC that would invalidate our
perturbative approach altogether and necessitate an exact numerical treatment.

Both the above-mentioned weaknesses of our analysis are good news. Within our
proposed mechanism, the experiments can achieve a much stronger critical
current modulation than adequately described by our analytical theory. The
goal of this work has been to clarify the physics of critical current control
by modifying the vortex surface barrier using a simple analytically tractable
model. Having understood the key ingredients and qualitative dependencies, the
experiments can engineer new hybrids capable of robust critical current
control via gate voltage and/or magnetic order orientation. Further, important
insights can be achieved using recently established methods for measuring
interfacial spin-orbit torques [86]. On the theory side, the vortex surface
barriers and corresponding critical currents should be evaluated numerically
[85] for various superconducting hybrids of interest [27, 87, 88], for example
using the quasiclassical Green’s function method [89]. The recent new insights
regarding orbital contributions to magnetization in superconductors with
Rashba SOC [90] pose another important question and could offer an avenue for
enhancement of the critical current modulation.

##  IX Summary

We have theoretically investigated the effect of interfacial Rashba spin-orbit
coupling on the vortex surface barrier in type II superconducting films
interfaced with one or more magnetic insulators. By formulating a general
criterion for vortex instability, we relate the critical current in such films
to the vortex surface barrier. Thus, we predict control of critical current in
type II superconducting films by influencing the vortex surface barrier.
Experimentally verifiable dependencies of this critical current modulation in
different superconductor-magnet hybrids have been worked out. We have also
delineated dependencies on the various properties of the hybrid thereby
providing design equations for engineering devices with maximal critical
current control. Our simple analytic theory presented herein lays the
foundation for and would benefit from more detailed numerical works in the
future.

## Acknowledgments

We thank Emmanuel I. Rashba for valuable discussions. We 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), and the Research Council of Norway through its Centers of
Excellence funding scheme, project 262633, “QuSpin”. Work at MIT is supported
by ARO (W911NF-20-2-0061), NSF (DMR 1700137 and NSF C-Accel Track C Grant No.
2040620), ONR (Nos. N00014-16-1-2657 and N00014-20-1-2306), and CIQM-NSF
DMR-1231319.

##  Appendix A Superconductor film without SOC

Here, we go through the analysis of a superconductor film without spin-orbit
coupling in some detail, providing the tedious mathematical expressions that
have been avoided in the main text. We consider a superconductor film hosting
a chain of vortices as described in Sec. III. The magnetic flux density
$\boldsymbol{B}(x,y)$ is determined by solving Eq. (4) which, for the case
under consideration, is given by:

$\displaystyle\lambda^{2}\nabla^{2}\boldsymbol{B}-\boldsymbol{B}$
$\displaystyle=-\phi_{0}\sum_{m}\delta(x-x_{v})\delta(y-ma)~{}\hat{\boldsymbol{z}};$
$\displaystyle\boldsymbol{B}(x=\pm d/2,y)$ $\displaystyle=\mu_{0}(H_{e}\mp
H_{I})~{}\hat{\boldsymbol{z}}.$ (36)

As discussed in Sec. III, the above boundary condition adequately describes
the applied field $H_{e}~{}\hat{\boldsymbol{z}}$ and the injected current
$I_{\mathrm{ext}}~{}\hat{\boldsymbol{y}}$. The solution to Eq. (A) is [59, 29]

$\boldsymbol{B}(x,y)=\left[\mu_{0}H_{e}\frac{\mathrm{cosh}\left(\frac{x}{\lambda}\right)}{\mathrm{cosh}\left(\frac{d}{2\lambda}\right)}-\mu_{0}H_{I}\frac{\mathrm{sinh}\left(\frac{x}{\lambda}\right)}{\mathrm{sinh}\left(\frac{d}{2\lambda}\right)}+\sum_{m}B_{m}(x,y)\right]\boldsymbol{\hat{z}},$
(37)

where

$\displaystyle B_{m}(x,y)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dk}{2\pi}e^{ik(y-ma)}B_{k}(x);$
$\displaystyle B_{k}(x)$
$\displaystyle=\frac{\phi_{0}}{2\lambda^{2}u}\frac{\mathrm{cosh}[u(d-|x-x_{v}|)]-\mathrm{cosh}[u(x+x_{v})]}{\mathrm{sinh}(ud)},$
(38)

and $u\equiv\sqrt{k^{2}+1/\lambda^{2}}$.

The average Gibbs free energy density can now be evaluated. To enable
comparison with literature [59, 29], we include the energy of the vortex core
in the following analysis. We further assume that the vortex core energy does
not depend on its position and consider it to be the same as in an infinite
superconductor. Thus, its inclusion in our analysis does not lead to any
additional forces on the vortex and does not influence the evaluated critical
current. The resulting average Gibbs free energy density is [Eq. (5)] [59, 29]

$\displaystyle\frac{G}{V_{S}}=$
$\displaystyle\frac{\varepsilon}{ad}-\frac{1}{d}\lim_{Y\to\infty}\frac{1}{Y}\int_{-Y/2}^{Y/2}dy\int_{-\infty}^{\infty}dx\frac{1}{2\mu_{0}}[\boldsymbol{B}_{\infty}^{2}+\lambda^{2}(\nabla\times\boldsymbol{B}_{\infty})^{2}]$
$\displaystyle+\frac{1}{d}\lim_{Y\to\infty}\frac{1}{Y}\int_{-Y/2}^{Y/2}dy\int_{-d/2}^{d/2}dx\left[\frac{1}{2\mu_{0}}[\boldsymbol{B}^{2}+\lambda^{2}(\nabla\times\boldsymbol{B})^{2}]-\boldsymbol{B}\cdot
H_{e}\hat{\boldsymbol{z}}\right],$ (39)

where $\varepsilon$ is the total energy per unit length of an isolated vortex
in an infinite superconductor, and $\boldsymbol{B}_{\infty}$ is its associated
magnetic field. The first two terms in Eq. (39) together represent the vortex
core energy of the chain, while the remaining is the magnetostatic
contribution in a constant applied field. Employing Eq. (37) in Eq. (39), the
average Gibbs free energy density is obtained as:

$\displaystyle\frac{G_{1}}{V_{S}}$
$\displaystyle=\frac{G_{0}}{V_{S}}+\frac{\phi_{0}H_{e}}{ad}\frac{\mathrm{cosh}\left(\frac{x_{v}}{\lambda}\right)}{\mathrm{cosh}\left(\frac{d}{2\lambda}\right)}-\frac{1}{ad}\frac{\phi_{0}^{2}}{4\pi\mu_{0}\lambda^{2}}\left[I_{1}(x_{v})-I_{2}(x_{v})\right],$
(40)

where

$\displaystyle\frac{G_{0}}{V_{S}}$
$\displaystyle\equiv\frac{\varepsilon}{ad}-\frac{\lambda\mu_{0}H_{e}^{2}}{d}\mathrm{tanh}\left(\frac{d}{2\lambda}\right)+\frac{\lambda\mu_{0}H_{I}^{2}}{d}\mathrm{coth}\left(\frac{d}{2\lambda}\right)-\frac{\phi_{0}H_{e}}{ad},$
(41) $\displaystyle I_{1}(x_{v})$
$\displaystyle\equiv\int_{0}^{\infty}dk\frac{1}{\sqrt{k^{2}+\frac{1}{\lambda^{2}}}}\left[1-\frac{2\mathrm{sinh}\left(\left\\{\frac{d}{2}-x_{v}\right\\}\sqrt{k^{2}+\frac{1}{\lambda^{2}}}\right)\mathrm{sinh}\left(\left\\{\frac{d}{2}+x_{v}\right\\}\sqrt{k^{2}+\frac{1}{\lambda^{2}}}\right)}{\mathrm{sinh}\left(d\sqrt{k^{2}+\frac{1}{\lambda^{2}}}\right)}\right],$
(42) $\displaystyle I_{2}(x_{v})$ $\displaystyle\equiv\sum_{n\neq
0}\int_{-\infty}^{\infty}dk\frac{e^{ikna}}{\sqrt{k^{2}+\frac{1}{\lambda^{2}}}}\left[\frac{\mathrm{sinh}\left(\left\\{\frac{d}{2}-x_{v}\right\\}\sqrt{k^{2}+\frac{1}{\lambda^{2}}}\right)\mathrm{sinh}\left(\left\\{\frac{d}{2}+x_{v}\right\\}\sqrt{k^{2}+\frac{1}{\lambda^{2}}}\right)}{\mathrm{sinh}\left(d\sqrt{k^{2}+\frac{1}{\lambda^{2}}}\right)}\right].$
(43)

In simplifying and evaluating the integrals in Eq. (39) using Eq. (37), we
performed integration by parts on the curl terms and replaced
$\nabla^{2}\boldsymbol{B}$ via Eq. (A).

The term containing $I_{2}(x_{v})$ [Eq. (43)] represents the energy due to
interaction between the different vortices in the chain. Carrying out the
integrating using a complex contour and in the limit $d\ll\lambda$, $a\gtrsim
d$, we obtain [29]:

$\displaystyle I_{2}(x_{v})\approx 4e^{-\frac{\pi
a}{d}}\cos^{2}\left(\frac{\pi x_{v}}{d}\right),$ (44)

which becomes negligible due to its exponential suppression for $a\gtrsim d$.
At low fields relevant to our analysis, $a$ is expected to be much larger than
$d$. Thus, we may safely ignore this $I_{2}(x_{v})$ contribution. The
$I_{1}(x_{v})$ term [Eq. (42)] captures the interaction of the vortices with
the surfaces, as explained in the main text. Carrying out the integration
after a series expansion in terms of exponential functions [29], Eq. (42) is
evaluated as:

$\displaystyle I_{1}(x_{v})$
$\displaystyle=\sum_{n=0}^{\infty}\left[K_{0}\left(\frac{2dn+d-2x_{v}}{\lambda}\right)+K_{0}\left(\frac{2dn+d+2x_{v}}{\lambda}\right)-2K_{0}\left(\frac{2dn+2d}{\lambda}\right)\right],$
(45)

where $K_{m}(x)$ is the modified Bessel function of the second type and $m$th
order [91].

As we work in the London limit assuming vortices to be point objects, which
further implies $\xi\to 0$, our theoretical description is troublesome when
the vortices with size $\sim\xi$ come too close to the surfaces, i.e., when
$|x_{v}\pm d/2|\lesssim\xi$. Shmidt [29] addresses this issue via some
approximations and an exact cancellation of the vortex core energy, which is
also divergent in the London limit of $\xi\to 0$. We do not need to worry
about those details here, except for recognizing that this inadequacy causes
$I_{1}(x_{v})$ to unphysically diverge around $x_{v}=\pm d/2$. We resolve this
issue by simply redefining $I_{1}(x_{v})$ as:

$\displaystyle I_{1}(x_{v})=$
$\displaystyle\sum_{n=0}^{\infty}\left[K_{0}\left(\frac{2\xi+2dn+d-2x_{v}}{\lambda}\right)+K_{0}\left(\frac{2\xi+2dn+d+2x_{v}}{\lambda}\right)\right.$
$\displaystyle\left.-2K_{0}\left(\frac{2\xi+2dn+2d}{\lambda}\right)\right],$
(46)

noting that in our London limit $\xi\ll\lambda$, the above expression is
practically the same as Eq. (45) while avoiding unphysical divergences.

We can express the average Gibbs free energy density [Eq. (40)] in a
normalized form:

$\displaystyle\tilde{G}(x_{v})$ $\displaystyle=\frac{G_{1}}{V_{S}}\frac{4\pi
ad\mu_{0}\lambda^{2}}{\phi_{0}^{2}}=\mathrm{const.}+\frac{H_{e}\ln\left(\lambda/\xi\right)}{H_{c1}}\
\frac{\mathrm{cosh}\left(\frac{x_{v}}{\lambda}\right)}{\mathrm{cosh}\left(\frac{d}{2\lambda}\right)}-I_{1}(x_{v}),$
(47)

where $H_{c1}=\phi_{0}\ln\left(\lambda/\xi\right)/(4\pi\mu_{0}\lambda^{2})$ is
the first critical field in the bulk superconductor [1], we have disregarded
the $I_{2}(x_{v})$ term as discussed above, and $I_{1}(x_{v})$ is given by Eq.
(A). Equation (47) has been employed in plotting Fig. 2 (b), while the
constant offset has been adjusted to make $\tilde{G}(x_{v}=\pm d/2)=0$.

We now detail the approximations employed in obtaining the expressions for
$H^{\prime}$ [Eq. (14)], $H_{s}$ [Eq. (15)], and $H_{I_{c}}$ [Eq. (17)] in the
main text. As discussed above, we disregard the $I_{2}(x_{v})$ contribution to
the Gibbs free energy density [Eq. (40)] in these evaluations. Furthermore, we
consider only the $n=0$ term in $I_{1}(x_{v})$ [Eq. (A)] as $|K_{0}(x)|$
decreases with increasing $x$ making the $n>0$ terms smaller. This
approximation is very good close to the surface where one of the $K_{0}(x)$
terms becomes large, but not so good in the middle. Nevertheless, our
evaluated $H^{\prime}$ which is determined by a condition in the middle of the
film [Eq. (14)] differs from Shmidt’s result [29] by a factor close to 1, and
thus is a good estimate. In evaluating $H_{s}$ [Eq. (15)], and $H_{I_{c}}$
[Eq. (17)], we work close to the surface such that one of the $n=0$ terms
dominates $I_{1}(x_{v})$. Our approximation is thus better here and we employ
the following properties of the modified Bessel functions of the second type
[91]:

$\displaystyle\frac{dK_{0}(x)}{dx}$ $\displaystyle=-K_{1}(x),$ (48)
$\displaystyle\frac{dK_{1}(x)}{dx}$
$\displaystyle=\frac{K_{1}(x)}{x}-K_{2}(x),$ (49) $\displaystyle\lim_{x\to
0}K_{1}(x)$ $\displaystyle=\frac{1}{x},$ (50) $\displaystyle\lim_{x\to
0}K_{2}(x)$ $\displaystyle=\frac{2}{x^{2}},$ (51)

in evaluating the expressions [Eqs. (15) and (17)] reported in the main text.
These are identical to what Shmidt has obtained using a different method [29].

##  Appendix B Superconductor film with SOC and exchange field

In this section, we detail the evaluation of average Gibbs free energy density
when $\boldsymbol{\alpha}\neq\boldsymbol{0}$. The model and methodology have
already been described in Sec. IV and we only note certain additional details
here. Invoking linearity of the governing equation (4), the contribution of
our assumed $\boldsymbol{\alpha}$ [Eq. (19)] to the magnetic flux density is
obtained as the solution of:

$\displaystyle\lambda^{2}\nabla^{2}\boldsymbol{B}-\boldsymbol{B}=$
$\displaystyle\left[\frac{1}{2e}\left\\{\alpha_{L}\mathrm{cos}\theta_{L}\delta(x+x_{so})+\alpha_{R}\mathrm{cos}\theta_{R}\delta(x-x_{so})\right\\}\right]\boldsymbol{\hat{z}}$
$\displaystyle-\left[\frac{1}{2e}\left\\{\alpha_{L}\mathrm{sin}\theta_{L}\delta(x+x_{so})+\alpha_{R}\mathrm{sin}\theta_{R}\delta(x-x_{so})\right\\}\right]\boldsymbol{\hat{y}},$
(52)

with the boundary conditions $\boldsymbol{B}(x=\pm d/2)=\boldsymbol{0}$. The
result $\boldsymbol{B}_{\alpha}(x)$ has been reported in Eq. (20). Employing
the total magnetic flux density [Eqs. (9) and (20)] in Eq. (5), we obtain the
total average Gibbs free energy density as described in Eq. (23) with

$\displaystyle\frac{G_{\alpha 0}}{V_{S}}=$
$\displaystyle-\frac{l_{so}}{d}\frac{(\alpha_{L}^{2}+\alpha_{R}^{2})}{8\mu_{0}e^{2}\lambda^{2}}-\frac{1}{4\mu_{0}ed}\left[\alpha_{L}\mathrm{cos}\theta_{L}B_{\alpha,z}(-x_{so})+\alpha_{R}\mathrm{cos}\theta_{R}B_{\alpha,z}(x_{so})\right]$
$\displaystyle+\frac{1}{4\mu_{0}ed}\left[\alpha_{L}\mathrm{sin}\theta_{L}B_{\alpha,y}(-x_{so})+\alpha_{R}\mathrm{sin}\theta_{R}B_{\alpha,y}(x_{so})\right]$
$\displaystyle+\frac{H_{e}}{2ed}\left(1-\frac{\mathrm{cosh}(\frac{x_{so}}{\lambda})}{\mathrm{cosh}(\frac{d}{2\lambda})}\right)(\alpha_{L}\mathrm{cos}\theta_{L}+\alpha_{R}\mathrm{cos}\theta_{R}),$
(53) $\displaystyle\frac{G_{\alpha}}{V_{S}}=$
$\displaystyle-\frac{1}{4\mu_{0}eda}\left[\alpha_{L}\mathrm{cos}\theta_{L}B_{k=0}(-x_{so})+\alpha_{R}\mathrm{cos}\theta_{R}B_{k=0}(x_{so})\right]+\frac{\phi_{0}}{2\mu_{0}ad}B_{\alpha,z}(x_{v}),$
(54) $\displaystyle=$
$\displaystyle\frac{\phi_{0}}{\mu_{0}ad}B_{\alpha,z}(x_{v}),$ (55)

where $B_{\alpha,z}$ and $B_{\alpha,y}$ are the $z$ and $y$ components
respectively of $\boldsymbol{B}_{\alpha}$ [Eq. (20)], and $B_{k}$ is given in
Eq. (A).

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