# Harnessing the superconducting diode effect through inhomogeneous magnetic
fields

Leonardo Rodrigues Cadorim  Edson Sardella  Departamento de Física, Faculdade
de Ciências, Universidade Estadual Paulista (UNESP), Caixa Postal 473,
17033-360, Bauru-SP, Brazil  Clécio C. de Souza Silva  Departamento de Física,
Centro de Ciências Exatas e da Natureza, Universidade Federal de Pernambuco,
Recife–PE, 50670-901, Brazil  Corresponding author<EMAIL_ADDRESS>
###### Abstract

We propose a superconducting diode device comprising a central superconducting
film flanked by two wires carrying an applied DC bias, suitably chosen so as
to generate different asymmetric field profiles. Through numerical simulations
of the coupled Ginzburg-Landau and heat-diffusion equations, we show that this
design is capable of efficiently breaking the reciprocity of the critical
current in the central superconductor, thus promoting the diode effect in
response to an applied AC current. By adjusting the DC bias in the wires, we
find the optimum inhomogeneous field profile that facilitates the entrance of
vortices and antivortices in a given polarity of the applied AC current and
impede their entrance in the other polarity. This way, the system behaves as
an ideal superconducting half-wave rectifier, with diode efficiencies
surpassing 70%. Furthermore, we detail the behavior and diode efficiency of
the system under different experimental conditions, such as the substrate heat
transfer coefficient and the sweep rate of the external current.

##  I Introduction

In modern electronics, the search for faster and efficient devices is a
permanent practice aiming for enhanced performance. In the midst of this
search, the superconducting diode has played a prominent role [1, 2]. Such
devices combine all the functionality of semiconductor diodes, e.g., the
conversion of alternate $(AC)$ currents into direct currents $(DC)$, with the
vantage of carrying electricity without resistance.

At the core of the superconducting diode lies a fundamental characteristic,
the absence of symmetry in current flow, expressed as $I_{c}^{+}\neq
I_{c}^{-}$, where $I_{c}^{\pm}$ is the critical current leading to the
resistive state at a given current direction. This concept originated decades
ago as a mean of manipulating the density and motion of magnetic flux quanta,
known as vortices, within superconductors [3, 4]. The primary goal was to
disrupt the inversion symmetry of the pinning potential acting on vortices.
This would enable an alternating current to induce vortex motion solely along
the potential’s easy direction, resulting in a resistive flux-flow state
during one half cycle and purely superconducting current flow during the other
one. Subsequently, experimental realization of these flux quanta diodes
swiftly followed, with diverse research groups employing various
nanolithographic techniques to selectively break spatial inversion symmetry
for vortex motion, such as regular arrays of asymmetric dots and antidots [5,
6, 7], asymmetric distributions of symmetric dots and antidots [8, 9, 10],
regular arrays of nanosized magnetic dipoles [11, 12], and asymmetric weak-
pinning channels [13].

Recently, superconducting diodes have gained renewed interest, fueled in part
by the discovery of novel intrinsic mechanisms governing nonreciprocal current
flow. This resurgence encompasses diverse systems, ranging from
heterostructures composed of superconducting materials and topological
insulators or ferromagnetic materials [14, 15, 16, 17, 18, 19, 20], different
superconducting systems with an applied magnetic field [21, 22, 23, 24, 25,
26, 27, 28], and even superconducting systems with no external field [29, 30,
31, 32]. Yet another possibility comes from Josephson junction based systems,
where the diode effect is achieved due to the asymmetric current-phase
relation of the junction [33, 34, 35, 36, 37, 38]. This diverse exploration
marks an intriguing phase in the study of superconducting diodes and their
various applications.

Intrinsic superconducting diode effects (SDE) rely on non-reciprocal pair-
breaking mechanisms inherent to the material [39, 40, 41]. Although these
mechanisms are of fundamental importance, they lack the high degree of
controllability of vortex-based diode effects (VDE), where the non-reciprocal
critical currents can be carefully designed by e.g. nanostructuring.
Furthermore, in typical experimental situations, the superconducting sample
enters the resistive state at currents considerably lower than the theoretical
depairing current, either as a result of vortex motion or the nucleation of
supercritical hot spots. This makes it challenging to discern experimentally
intrinsic and non-intrinsic effects [42].

During the resistive state of a VDE, the material is still superconducting and
dissipation is induced by vortex motion (flux-flow). Since the flux-flow
resistance is necessarily lower than the normal state resistance, the measured
DC signals in VDE are relatively small. This is a clear disadvantage when
compared to the strict superconducting diode, where the dissipationless
superconducting state for current applied in one direction is expected to
transition directly to the fully normal state when the current is reversed,
leading to a considerably larger DC signal. Remarkably, Hou et al. [16]
recently reported on the observation of highly efficient SDE on conventional
superconducting films where the AC switching between superconducting and
normal states is induced by non-reciprocal penetration of vortices due to
asymmetrical conditions at the sample edges, that is, by an extrinsic, vortex-
based mechanism. However, the reason why the flux-flow state was suppressed in
favor of a sudden jump to the full normal state, thus making this VDE into a
SDE, is unclear. A possible explanation can be inferred from Ref. [43], where
the authors investigated the diode effect in a superconducting stripe with a
conformal array of nanoholes of broken inversion symmetry. This work suggests
that hot posts induced by fast vortex motion can drive the superconductor
directly into the normal state thus bridging the gap between VDE and SDE.

These recent advances emphasize the necessity of better understanding the role
of heating effects in superconducting diodes. Additionally, the suggestion
that the VDE can be made into a fully SDE redirects the research on vortex
diode systems towards the optimization of the diode efficiency, defined as
$\epsilon=|I_{c}^{-}-I_{c}^{+}|/(I_{c}^{-}+I_{c}^{+})$. Tackling these
challenges is a key point for obtaining more efficient and reliable
superconducting diodes.

In the present work, we investigate the interplay between the superconducting
and the vortex diode effects (SDE and VDE) in thin superconducting films
exposed to external asymmetric fields. By integrating the coupled time-
dependent Ginzburg-Landau and heat diffusion equations, we explore simple
inhomogeneous field landscapes, from asymmetric fully positive to
antisymmetric (half-positive, half-negative) flux profiles, induced either by
ferromagnetic films or DC currents in nearby wires as illustrated in Fig. 1.
Our findings reveal that both profile types promote VDE, with the
antisymmetric one exhibiting the highest diode efficiencies, up to 70%.
Furthermore, at large enough AC amplitudes, VDE is replaced by SDE as a result
of vortex-antivortex collisions and subsequent hotspot proliferation. We
further demonstrate that reducing the substrate’s heat transfer coefficient
accelerates the VDE-to-SDE transition, thereby enhancing the superconducting
diode efficiency. These insights illuminate recent discoveries of highly
efficient SDE and emphasize the role of vortex-antivortex collisions and the
importance of optimizing substrate designs and cooling environments for future
superconducting diode platforms.

Figure 1: Schematic view of a superconducting film (green) subjected to
asymmetric field profiles $B_{z}(y)$ induced by: (a) an asymmetrically lying
ferromagnetic film (orange) with off-plane magnetization ${\bf M}$; (b) the
same as (a) but with in-plane magnetization and symmetric arrangement of the
bilayer; and (c) currents applied onto lateral superconducting stripes (blue).
${\bf J}_{1}$ and ${\bf J}_{2}$ can be adjusted to generate different field
profiles. Here, setting ${\bf J}_{2}={\bf J}_{1}=-J\hat{\bf x}$ emulates
$B_{z}(y)$ similar to that induced by the magnet in (b). In all cases, an
alternating current ${\bf J}_{ac}$ applied parallel to $x$ induces
nonreciprocal vortex penetration and motion.

The outline of this work is as follows. In Sec. II we present our system and
the theoretical formalism used to investigate it. Our results and their
discussion are presented in Sec. III. Finally, we present our concluding
remarks in Sec. IV.

##  II Theoretical Formalism

The resistive state of superconductors in the presence of a driving current
can be described by the time-dependent Ginzburg-Landau equations. In reduced
units, these equations can be written as: [44, 45]

$\displaystyle\frac{u}{\sqrt{1+\gamma^{2}|\psi|^{2}}}\left[\frac{\partial}{\partial
t}+\frac{1}{2}\gamma^{2}\frac{\partial|\psi|^{2}}{\partial t}\right]\psi=$
$\displaystyle\left(\mbox{\boldmath$\nabla$}-i\textbf{A}\right)^{2}\psi+\psi(1-T-|\psi|^{2}),$
(1) $\sigma\frac{\partial\textbf{A}}{\partial
t}=\textrm{Im}\left[\bar{\psi}{(\mbox{\boldmath$\nabla$}-i\textbf{A})}\psi\right]-\kappa^{2}\mbox{\boldmath$\nabla$}\times\mbox{\boldmath$\nabla$}\times\textbf{A},$
(2)

where $\psi$ is the superconducting order parameter, A is the vector
potential, $\sigma$ is the normal conductivity, and
$\kappa=\lambda(0)/\xi(0)$, with $\lambda(0)$ the London penetration depth and
$\xi(0)$ the coherence length at zero temperature. Here we fix $u=5.79$, and
$\gamma=20$. In Eqs. (1) and (2) we express lengths in units of $\xi(0)$, $T$
in units of the critical temperature, $T_{c}$, time in units of
$t_{GL}=\pi\hbar/8uk_{B}T_{c}$, $\psi$ in units of the field-free order
parameter at $T=0$, $\psi_{\infty}(0)$, and A in units of $\xi(0)H_{c2}(0)$,
with $H_{c2}(0)$ the upper critical field at $T=0$.

To account for the dissipation effects caused e.g. by v-av collisions, Eqs.
(1) and (2) must be coupled with the heat diffusion equation: [46, 47]

$\nu\frac{\partial T}{\partial
t}=\zeta\nabla^{2}T+\sigma\left(\frac{\partial\textbf{A}}{\partial
t}\right)^{2}-\eta(T-T_{0}),$ (3)

where $\nu$, $\zeta$, and $\eta$ are the thermal capacity, thermal
conductivity of the material, and the heat transfer coefficient of the
substrate, respectively, and $T_{0}$ is the temperature of the thermal bath.
In this work, we have used $\nu=0.03$, $\zeta=0.06$, and varied $\eta$ from
$2.0\times 10^{-5}$ to $2.0\times 10^{-3}$.

The inhomogeneous field that allows the emergence of the diode effect can be
introduced through different routes. Panels $(a)$ and $(b)$ of Fig. 1 show two
possible scenarios using a superconductor/ferromagnet heterostructure, where
the inhomogeneous field comes from the ferromagnet. The system depicted in
panel $(b)$ was investigated in Ref. 16. In the present work, we introduce a
new possible configuration $(c)$ comprising a central superconducting film
carrying the AC current and flanked by two superconducting wires carrying DC
currents suitably chosen to produce the desired inhomogeneous field profile.
Configurations depicted in panels $(b)$ and$(c)$ have the advantage that the
AC current self-field matches the profile of the inhomogeneous field for one
AC current polarity and opposes it for the other one. As we shall see later
on, this scenario increases the diode efficiency. We focus our detailed
investigation on configuration (c), due to its versatility in producing
different profiles, from fully positive to antisymmetric, by simply adjusting
the lateral currents $J_{1}$ and $J_{2}$. In a practical perspective, this
configuration offers the possibility of easily reversing the diode effect by
changing the polarity of the side currents.

In what follows, the side stripes have width $w=50\xi(0)$, thickness
$d_{s}=5\xi(0)$ and their separation from the edges of the central
superconductor is $s=5\xi(0)$. As it follows from elementary magnetostatics,
the $z$ component of the total field produced by the stripes is:

$\displaystyle H_{nh}(y)$ $\displaystyle=$
$\displaystyle\frac{J_{1}d_{s}}{2\pi\kappa^{2}}\ln\left(\frac{-y+w+s+L_{y}/2}{-y+s+L_{y}/2}\right)$
(4) $\displaystyle-$
$\displaystyle\frac{J_{2}d_{s}}{2\pi\kappa^{2}}\ln\left(\frac{y+w+s+L_{y}/2}{y+s+L_{y}/2}\right)$

The central superconductor has width $L_{y}=200\xi(0)$, thickness $d=\xi(0)$,
and is driven by an applied AC current given by $J_{ac}(t)=J_{a}\sin(2\pi
t/\tau)$, where $\tau$ is the period of each oscillation.

Eqs. (1)-(3) are solved numerically only for the central superconductor,
supplemented by periodic boundary conditions along $x$, with period
$L_{x}=400\xi(0)$. At the edges $y=\pm L_{y}/2$, $\psi$ and $T$ satisfy the
boundary conditions
$\hat{\textbf{y}}\cdot(\mbox{\boldmath$\nabla$}-i\textbf{A})\psi=0$ and
$\hat{\textbf{y}}\cdot\mbox{\boldmath$\nabla$}T=0$ [46, 47]. Additionally, the
local field $\textbf{h}=\bm{\nabla}\times\textbf{A}$ at the sample edges
satisfies $h_{z}(x,\pm L_{y}/2)=H_{nh}(\pm L_{y}/2)\pm L_{y}J(t)/2\kappa^{2}$,
which formally accounts for the applied current $J(t)$ in the sample [48].

##  III Results and Discussion

To analyze the conditions for the occurrence of the superconducting diode
effect or the vortex diode effect in our system, we have varied both the
amplitude of the applied AC current and the values of $J_{1}$ and $J_{2}$
carried by each side stripe. We limit $J_{1}$ and $J_{2}$ to be lower than
$0.52~{}\sigma\hbar/2e\xi(0)t_{GL}$. With the thermal bath temperature fixed
at $T_{0}=0.96T_{c}$, such currents would destroy superconductivity in the
central stripe. Therefore, the side stripes must be composed of a
superconducting material with larger critical temperature and depairing
current density. In what follows, voltages are expressed in units of
$V_{0}=\hbar/2et_{GL}$ and current densities in units of $j_{0}=\sigma
V_{0}/\xi(0)$. Further, we fix $\kappa=5$ and, unless stated otherwise, the
period of the AC current is $\tau=\tau_{0}=10^{5}t_{GL}$.

Figure 2: Phase-diagram displaying the color plot of the output voltage
$V_{DC}$ (in units of $V_{0}=\hbar/2et_{GL}$) as a function of $J_{1}$ and the
amplitude of the AC current (both in units of $j_{0}=\sigma V_{0}/\xi(0)$). We
fix $J_{2}=0$.

###  III.1 Fully positive asymmetric flux profiles
($J_{1}\neq 0$, $J_{2}=0$)

We start our analysis by considering the case where the asymmetric flux
profile is positive everywhere in the superconducting film. This situation is
accomplished by choosing $J_{1}>0$ and $J_{2}=0$ in Eq. (4). Fig. 2 shows the
color plot of the time averaged voltage $V_{DC}$ as a function of the
amplitude of the AC current and the value of $J_{1}$. As can be seen, if we
increase the amplitude of the applied current with a fixed value of $J_{1}$,
$V_{DC}$ starts from very small values and then gradually increases until it
jumps to a region with a larger value (red region in the figure). Inside this
region, the voltage continues to increase with the current amplitude. When
$V_{DC}$ reaches its maximum value, the voltage output suddenly drops to
another low value region and then gradually decreases as the current is
further increased. In the large voltage region, $V_{DC}$ is almost ten times
larger than the values obtained outside this domain. The appearance of two
separate islands of $V_{DC}$ at small $J_{1}$ is probably an artifact of the
finite step used to vary $J_{a}$, which might hinder finer details.

Figure 3: (Color online) The voltage signal as a function of time (solid blue
line) for $J_{1}=0.26$ and three current values, $J_{a}=2.20\times 10^{-3}$
(panel $(a)$), $J_{a}=2.45\times 10^{-3}$ (panel $(b)$) and $J_{a}=2.55\times
10^{-3}$ (panel $(c)$). In each panel, the red dashed lines represent the
voltage if the system were in the normal state.

Fig. 3 shows the voltage output (solid blue line) as a function of time for
three different AC current amplitudes and $J_{1}=0.26$. In each panel, the
dashed red line shows the voltage that would be obtained if the system were
completely normal, i.e., $V_{n}(t)=(J_{a}L_{x}/\sigma)\sin\left(2\pi
t/\tau\right)$. Panel $(a)$ corresponds to a scenario in the low $V_{DC}$
region which occurs before the region with maximum voltage. In this case, the
vortex dynamics caused by the current is never sufficient to send the system
to the normal state and so it stays superconducting throughout the whole
period, although a finite resistance appears due to the vortex motion.
Therefore, the system displays a small but finite $V_{DC}$. This occurs due to
the presence of the inhomogeneous field, which is positive throughout the
whole width of the superconductor, but is stronger at the top edge and weaker
at the bottom one. In the absence of the AC current, the inhomogeneous field
nucleates vortices near the top edge of the film. For positive AC polarity,
the self-field profile, that is, the field produced by the current itself, is
positive at the top edge and negative at the bottom one, thus matching the
external field profile where it is stronger and opposing it at its weaker
side. In contrast, for the negative AC current polarity, the AC current self-
field opposes the inhomogeneous field at the top edge and matches it at the
bottom one. This discrepancy leads to an asymmetric vortex dynamics for each
current polarity, thus leading to the finite $V_{DC}$.

In panel $(b)$, we show the voltage as a function of time for a case inside
the large $V_{DC}$ region. The reason for the jump in the output voltage now
becomes clear. For the positive polarity of the AC current, the vortex
dynamics produces enough heat to completely destroy the superconducting state
for a certain interval of time. A similar behavior was found in Ref. 43.
Superconductivity is recovered as the magnitude of the current decreases and,
for the negative polarity, the normal state is never achieved. The complete
destruction of superconductivity only for one polarity causes a large
discrepancy between the calculated voltage in the two halves of the period,
resulting then in a large $V_{DC}$. The reason for the asymmetry between
positive and negative polarities will be discussed below.

Finally, panel $(c)$ shows the voltage for a case of low $V_{DC}$ values that
occurs after the large $V_{DC}$ region. Here, as we have also encountered for
panel $(b)$, the system goes to the normal state for the positive polarity.
The difference now is that the current is sufficiently strong to completely
destroy the superconducting state in both polarities. As a result, the
asymmetry between the two halves of the period decreases, so decreasing
$V_{DC}$. Nevertheless, the average voltage continues to be finite, once the
time lapse that the system remains at the normal state is larger for the
positive polarity. Animations of the evolution of the order parameter for the
three cases presented in Fig. 3 can be found as Supplementary Material.

As can be seen, the region of the phase-diagram depicted by panel $(a)$ of
Fig. 3 presents the characteristic behavior of the vortex diode effect, with
the DC voltage arising from the asymmetric vortex motion for each current
polarity. In contrast, in panel (c), $V_{DC}$ comes from the superconducting
diode effect, with the major difference that the superconducting state is
completely destroyed for both current polarities inside the region with
enhanced output voltage.

Figure 4: (Color online) Color plot of the order parameter and local
temperature difference $T-T_{0}$ at four different times for parameters
corresponding to panel $(b)$ in Fig. 3. Each line of the figure corresponds to
a black circle in the positive current branch.

Once panel $(b)$ represents the more interesting case of the superconducting
diode effect with enhanced $V_{DC}$, let us take a closer look at the vortex
dynamics that leads to this physics. Figs. 4 and 5 depict the time evolution
of the order parameter and local temperature for positive and negative AC
current polarity, respectively. From top to bottom, the panels follow a
sequence of time instants identified by black circles in Fig. 3$(b)$. Panels
$(a),(b)$ in Fig. 4 shows the vortex configuration at $t=0$, where the applied
current vanishes. As we can see, vortices penetrate the superconductor at the
top edge, near the lateral stripe with nonzero current. The configuration we
encountered resembles the characteristic pattern of a conformal vortex crystal
[49, 50]. As the time evolves and the applied AC current becomes positive, the
vortices begin to move towards the opposite side of the sample $(c),(d)$, at
the same time new vortices nucleate at the top edge. The nucleation and
annihilation of vortices dissipate energy and thus locally increase the system
temperature (see the animations for the evolution of the local temperature in
the Supplementary Material). This local heating leads to the partial
suppression of the superconducting state, but, for this small vortex velocity,
the heat is quickly removed from the film and the system remains at the
superconducting state.

As the current is further increased, the vortex velocity increases and,
simultaneously, the magnitude of the AC current becomes sufficiently large to
allow the penetration of antivortices in the opposite side of the
superconductor, as depicted in panels $(e)$ and $(f)$. The vortices and
antivortices annihilate in pairs near the bottom edge of the superconductor,
in a process that dissipates enough heat to induce hot spots. In this panel,
to help the dynamics visualization, a few vortices and antivortices are
indicated by the labels $v$ and $av$, respectively. Finally, for a
sufficiently large magnitude of the AC current (see panels $(g)$ and $(h)$),
the nucleation and annihilation of vortex pairs become more frequent and the
hot spots are able to spread across the sample. This ultimately leads to the
complete suppression of the superconducting state, which is only reestablished
at a later time after the applied current is decreased.

Figure 5: (Color online) Color plot of the order parameter and local
temperature difference $T-T_{0}$ at four different times for parameters
corresponding to panel $(b)$ in Fig. 3. Each line of the figure corresponds to
a black circle in the negative current branch.

The vortex dynamics for the negative current branch is rather different. At
$t=\tau/2$, the vortices again arrange themselves in the conformal crystal
like pattern, as shown in Fig.5, panels $(a)$ and $(b)$. At later times, the
magnitude of the applied current increases and the vortices at the top exit
the sample (see panels $(c)$ and $(d)$), but at a lower rate than in the case
depicted in Fig. 4, once the top edge displays a larger surface barrier.
Simultaneously, a few vortices enter the superconductor through the bottom
edge. Though both of these processes are enhanced in panels $(e)$ and $(f)$,
the lower field at the bottom edge sustains a lower vortex density than in the
positive branch. Since the dissipated power is proportional to the number of
moving vortices and antivortices, hot spots are considerably weaker in this
case. This is further confirmed in panels $(g)$ and $(h)$, where antivortices
nucleate at the top of the superconductor and annihilation of vortex pairs
occurs just as in the positive current branch. However, the dissipation in
this case is never sufficient to suppress the superconducting state and the
system never transitions to the normal state.

###  III.2 Antisymmetric flux profiles ($J_{1}=J_{2}$)

Although the scenario depicted in Figs. 4 and 5 produces a large output
voltage $V_{DC}$, it suffers from the problem of a finite voltage in the
negative polarity branch of the AC current induced by flux-flow. This nonzero
resistance is a shortcoming to the very purpose of a superconducting diode,
i.e., to have all the functionalities of a regular diode with the benefit of
dissipationless transport of current in half-wave of the AC signal. A possible
way to overcome this issue is to introduce a finite current $J_{2}$ in the
second stripe, such that the surface barrier to vortex entry through the
bottom edge increases up to a point where it can not be surpassed by the self-
field of the AC current during the negative half cycle.

Figure 6: Phase-diagram displaying the color plot of the output voltage
$V_{DC}$ (in units of $V_{0}=\hbar/2et_{GL}$) as a function of $J_{2}$ and the
amplitude of the AC current (both in units of $j_{0}=\sigma V_{0}/\xi(0)$). We
fix $J_{1}=0.26$.

In Fig. 6 we show the output average voltage $V_{DC}$ as a function of $J_{a}$
and $J_{2}$, with $J_{1}=0.26$ fixed. As compared to the case $J_{2}=0$
depicted in Fig. 2, the large DC voltage region becomes considerably wider and
the values of $V_{DC}$ gets considerably higher as $J_{2}$ increases. In
particular, for $J_{2}=J_{1}=0.26$, representing a perfect antisymmetric flux
profile as illustrated in Fig. 1$(c)$, these enhancements are maximized: the
high-$V_{DC}$ region is about three times wider and the maximum output voltage
twice as high as the values observed in Fig. 2. To understand the reason
behind the improved diode effect exhibited by the antisymmetric flux profile,
in Fig. 7 we show

Figure 7: The voltage signal as a function of time (solid blue line) for
$J_{1}=J_{2}=0.26$ and three current values, $J_{a}=2.10\times 10^{-3}$ (panel
$(a)$), $J_{a}=2.40\times 10^{-3}$ (panel $(b)$) and $J_{a}=2.80\times
10^{-3}$ (panel $(c)$). In each panel, the red dashed lines represent the
voltage if the system was in the normal state.

the voltage output as a function of time for three different values of $J_{a}$
and $J_{1}=J_{2}=0.26$. As shown in panel $(a)$, for the positive branch,
shortly after $t=0$, when the magnitude of the AC current is sufficient to put
in motion the vortices induced by the inhomogeneous field, the voltage becomes
finite and gradually increases with current. In this case, though, the
dissipation process is never large enough to totally suppress the
superconducting state and we get a behaviour that resembles the positive
branch of panel $(a)$ in Fig. 3. On the other hand, the negative current
branch is significantly different from the previous studied cases. As can be
seen, after a small voltage signal corresponding to the vortex induced by the
inhomogeneous field leaving the sample, the voltage becomes zero for the
remaining half period of negative current. In other words, the system behaves
as a perfect half-wave rectifier.

In panel $(b)$, while the negative current branch continues to present a null
voltage signal, vortex creation and annihilation process is now sufficient to
force the transition of the system to the normal state in the positive current
branch. This is also the case for the positive current branch in panel $(c)$,
which also presents a new behavior for the negative branch. Here, for a
certain time interval, the AC current magnitude is large enough to overcome
the surface energy barrier imposed by the inhomogeneous field and vortices
begin to flow in the superconductor, inducing a finite voltage. Animations of
the evolution of the order parameter for each panel in Fig. 7 can be found as
Supplementary Material. In these three cases, we can clearly see the reason
behind the increment in $V_{DC}$, once the asymmetry between the two current
polarities significantly increases in comparison to the voltage-time curves
shown in Fig. 3. Furthermore, we note that the desired scenario for a
superconducting diode is obtained for the parameters shown in panel $(b)$,
once we are able to obtain a large rectified voltage while simultaneously
having no dissipation for the second half of the AC current signal.

###  III.3 Hot spot dynamics

Figure 8: Evolution of the order parameter (panels in the left column) and the
temperature increment $T-T_{0}$ (panels in the right column) before the
destruction of the superconducting state for parameters of panel $(b)$ in Fig.
7. The white dashed line depicted in panel $(a)$ delineates the superconductor
into two distinct halves, with the magnetic flux being composed by vortices in
the top region and antivortices in the bottom one.

To further describe our diode system, in this Subsection we detail how the
vortex response to the applied current leads to the destruction of the
superconducting state for positive current polarity. Fig.8 shows the time
evolution of the order parameter (left panels) and of the temperature
variation $T-T_{0}$ near the destruction of the superconducting state for
positive current polarity and parameters corresponding to panel $(b)$ of Fig.
7. In this figure, each panel line corresponds to the same instant of time.

From panels $(a)$ and $(b)$, we can identify that, among the processes of
vortex creation, flow and annihilation, it is the latter that dissipates more
heat, once each of the more pronounced hot spots seen in panel $(b)$
corresponds to a v-av collision observed in panel $(a)$. Here, we can note
that the perfect antissymmetric profile of the inhomogeneous field guarantees
that v-av annihilation always occurs, once both objects colide at the center
of the film. For small currents much smaller than $I_{c}^{+}$, the hot spots
formed by the dissipated heat of a v-av collision gradually disappears due to
the heat removal mechanism. Comparing panels $(b)$ and $(d)$, we can see an
example of this in the three hot spots on the left side of panel $(b)$. In
contrast, the hot spots by the right of panel $(d)$ depict the mechanism that
leads to the destruction of superconductivity at $I_{c}^{+}$. In this case,
heat removal is not strong enough to overcome the additional heat generated by
new vortex-antivortex pairs at the same location (see panel $(c)$). As a
result, the elevated temperature locally suppress the superconducting state,
which attracts yet more vortex-antivortex pairs into the region. This chain
reaction leads to the formation of stripes of depleted superconductivity
across the width of the sample, as shown in panels $(e)$ and $(f)$. Finally,
panels $(g)$ and $(h)$ shows the configuration of the system just before the
total destruction of the superconducting state. As we can see, more stripes
are formed along the sample and the hot spots spread throughout the
superconductor.

###  III.4 Diode efficiency

Figure 9: The voltage signal as a function of time (solid blue line) for a
linear current cycle with $J_{1}=J_{2}=0.26$ and total sweep time
$\tau=10^{6}t_{GL})$. Red dashed curve represents the normal state voltage.
The arrows represent the critical currents for the complete destruction of the
superconducting state (dark blue) and the onset of vortex motion (orange) at
each current polarity.

Let us now investigate the efficiency of our superconducting diode device,
which is defined as
$\epsilon_{SDE}=\left(I_{c}^{-}-I_{c}^{+}\right)/\left(I_{c}^{-}+I_{c}^{+}\right)$,
where $I_{c}^{\pm}$ is the critical current for the complete destruction of
the superconducting state for a given current polarity. To do this, we replace
the sine wave AC current by a linear current cycle, with total sweep time
$\tau=10\tau_{0}=10^{5}t_{GL}$, thereby mimicking the usual experimental
procedure of measuring the voltage in current-sweep mode. We keep
$J_{1}=J_{2}=0.26$ fixed, once these parameters make possible the occurrence
of the ideal superconducting diode scenario. Fig. 9 shows the voltage signal
as a function of time for these parameters. $I_{c}^{\pm}$ is highlighted in
this figure by the two blue arrows and their values give us
$\epsilon_{SDE}=0.124$.

From Fig. 9 we can define another useful critical current, namely the current
which marks the onset of vortex motion, $I_{cv}$, which are marked by the
orange arrows. If we now define
$\epsilon_{VDE}=\left(I_{cv}^{-}-I_{cv}^{+}\right)/\left(I_{cv}^{-}+I_{cv}^{+}\right)$
as the efficiency of the diode effect caused by the finite voltage associated
with the flux-flow, we have $\epsilon_{VDE}=0.619$ for this set of parameters.
The much larger value of $\epsilon_{VDE}$ compared to $\epsilon_{SDE}$ can be
promptly understood if we notice that, for positive current polarity, the
profile of the self-field produced by the AC current matches the profile of
the external inhomogeneous field. As a result, a large amount of vortices is
present in the superconductor, meaning that a small current amplitude is
sufficient to start the flux-flow. In contrast, for negative current polarity,
the profiles of the inhomogeneous field and the AC current self-field are
opposites. This means that there are no vortices in the system, unless the
current amplitude is large enough to overcome the Bean-Livingston barrier.
This is depicted at the second orange arrow in Fig. 9 which occurs at a large
current amplitude, very close to $I_{c}^{-}$. This discrepancy is the origin
of the large value of $\epsilon_{VDE}$.

Figure 10: Top panel shows $|V(t)|$ as a function of $|I_{a}(t)|$. Blue and
red curves represent the half period with positive and negative $J_{a}(t)$,
respectively. Solid and dashed lines represent the regions where $|I_{a}(t)|$
is being increased and decreased, respectively. Yellow and blue background
marks the current region of vortex and diode rectification, with green
background depicting the region where they coexist. Bottom panel shows $R(t)$
as a function of $|I_{a}(t)|$, with the same definitions of the top panel
following.

Fig. 10 helps in the visualization of the vortex and diode rectification
regions described above. In this figure, we show the absolute value of the
voltage, $|V(t)|$, and the resistance of our system as a function of the
absolute value of the total applied current, $|I_{a}(t)|$. Regions of vortex
and superconducting half-wave rectification are highlighted in the figure by
yellow and blue backgrounds, respectively, with the green background denoting
the current region where they overlap. From this figure, the two key factors
to improve the efficiency of a superconducting diode system becomes apparent.
First, for negative current polarity, one should seek to increase the Bean-
Livingston energy barrier, in such a manner as to increase $I_{c}^{-}$.
Simultaneously, for the positive current polarity, the goal is to approach
$I_{c}^{+}$ to the value of $I_{cv}^{+}$ displayed in the figure. For that,
the heat induced by flux-flow and vortex-antivortex annihilations must be high
enough to induce supercritical hot spots that can drive the entire system into
the normal state.

###  III.5 Optimizing diode efficiencies: the role of heat removal and sweep
rate

So far, when solving the heat diffusion equation, we have set $\eta=2\times
10^{-4}$, which mimics a substrate of moderate heat removal efficiency. Once
we have seen that the destruction of the superconducting state is governed by
the emergence and expansion of hot spots through the system, it is natural to
assume that the efficiency of our diode system depends on the heat removal
coefficient $\eta$. Fig. 11 shows both $\epsilon_{SDE}$ and $\epsilon_{VDE}$
as a function of $\eta$, for values raging from $2\times 10^{-5}$ (weak heat
removal) to $2\times 10^{-3}$ (strong heat removal).

Figure 11: Efficiencies of the superconducting, $\epsilon_{\text{SDE}}$ (blue
circles), and vortex, $\epsilon_{\text{VDE}}$ (red circles), diode effects as
a function of the heat transfer coefficient $\eta$. Here, we set
$\tau=10^{6}t_{GL}$.

The vortex diode efficiency presents a weak dependence on $\eta$, which is a
result of the low heat dissipation produced by vortices when driven by
currents just above $I_{cv}^{+}$ (see the low flux-flow resistance regime
shown in Fig. 12). However, as the current increases above $I_{cv}^{+}$, the
rate of v-av collisions at the equatorial line of the superconducting film
increases, leading to the formation of hot spots as discussed in the previous
subsection. For high $\eta$, weak hot spots are quickly removed by the
substrate so that a current much larger than $I_{cv}^{+}$ is necessary to
drive the system into the normal state. In contrast, for small $\eta$, even
weak hot spots, created in the early stages of the flux-flow regime, can grow
and drive the film into the normal state. This can considerably reduce the
value of $I_{c}^{+}$, bringing it closer to $I_{cv}^{+}$. On the other hand,
during the negative current branch, the reduction of $I_{c}^{-}$ induced by
weak heat removal is much less pronounced because $I_{c}^{-}$ is already very
close to $I_{cv}^{-}$, even in the strong heat removal regime. Combined, these
results lead to a pronounced enhancement of $\epsilon_{SDE}$ from 0.067 to
0.317 as $\eta$ is reduced from $2\times 10^{-3}$ to $2\times 10^{-5}$.
Therefore, we can conclude that an ideal superconducting diode system should
have the weakest possible heat removal scenario.

It is worth mentioning that our 2D simulations are limited to the assumption
that the thermal healing length is smaller than the combined thicknesses of
film and substrate. In actual experiments, this condition is not generally
fulfilled and typical values of $\eta$ can be much smaller than those used in
our simulations [46]. In those cases, hot spots can nucleate as soon as flux-
flow commences, thus aligning $I_{c}^{+}$ to $I_{cv}^{+}$ and thereby
$\epsilon_{SDE}$ to $\epsilon_{VDE}$. Bearing this in mind, we speculate that
the sharp superconductor-to-normal transitions observed in Ref. [16] are
probably triggered by hot spots induced by the fast penetration of vortices or
vortex-antivortex annihilation events in a scenario of very small $\eta$.

Another important factor that influences the efficiency of the diode device is
the total sweep time $\tau$, which determines the sweep rate along the current
cycle. Fig. 12 shows both $\epsilon_{SDE}$ and $\epsilon_{VDE}$ as a function
of $\tau$, with $\eta=2\times 10^{-5}$ fixed. As can be seen, both
efficiencies present similar behavior, rapidly increasing for small values of
$\tau$ and then approaching a constant value for large values. By increasing
$\tau$, we observe a decrease in both $I_{c}^{\pm}$ and $I_{cv}^{\pm}$. The
large efficiency occurs because $I_{c}^{+}$ ($I_{cv}^{+}$) decreases in a
larger rate than $I_{c}^{-}$ ($I_{cv}^{-}$). In this case, a faster increase
rate of the current, i.e., a lower value of $\tau$, leads to larger $I_{c}$
and $I_{cv}$, once there is not enough time for the system to reach the stable
state for each region of current values.

Figure 12: Efficiencies of the superconducting, $\epsilon_{\text{SDE}}$ (blue
circles), and vortex, $\epsilon_{\text{VDE}}$ (red circles), diode effects as
a function of total sweep time $\tau$. Here, $\eta=2\times 10^{-5}$.

###  III.6 Optimizing diode efficiencies: the role of the inhomogeneous field
amplitude

Figure 13: Panel $(a)$ shows the critical currents $I_{c}^{+}$ (blue circles),
$I_{c}^{-}$ (red circles), $I_{cv}^{+}$ (blue dots) and $I_{cv}^{-}$ (red
dots) as a function of the current density in the side stripes $J$. Panel
$(b)$ shows $\epsilon_{SDE}$ (blue circles) and $\epsilon_{VDE}$ (red circles)
as a function of $J$. Top $x$ axis relates $J$ with the value of the
inhomogeneous field at the sample edges $H_{e}$.

In the above discussion, we have studied the efficiency of our diode system
considering a fixed antisymmetric field profile, given by $J_{1}=J_{2}=0.26$.
Here we investigate how the diode efficiency changes when varying the
amplitude of the external field profile. For that, we fix $J_{1}=J_{2}=J$ and
vary the value of $J$. We perform the calculations for a linear current cycle
with sweep time $\tau=10^{6}t_{GL}$ and $\eta=2\times 10^{-5}$. Panel $(a)$ of
Fig. 13 shows the critical currents $I_{c}^{\pm}$ (blue and red circles) and
$I_{cv}^{\pm}$ (blue and red dots). Both $I_{cv}^{+}$ and $I_{c}^{+}$ decrease
monotonically with $J$, which follows from the fact that the inhomogeneous
field profile adds to the self-field of the AC current in the positive current
branch. Therefore, increasing $J$ contributes to increasing the number of
vortices and antivortices as well as the rate of their collision, so that a
lower current is required to initiate the dissipative regime. In contrast,
$I_{cv}^{-}$ and $I_{c}^{-}$ present a non-monotonic dependence on $J$, which
can be understood as follows. On one hand, increasing $J$ makes it necessary a
larger value of the current magnitude to compensate the inhomogeneous field
and induce the resistive state for negative current polarities. On the other,
when $J$ becomes too strong, the inhomogeneous field depletes
superconductivity near the edges so that lower values of the current magnitude
are capable of driving the system into the normal state.

This discrepancy in the behavior of the system for each polarity leads to the
existence of an optimal value of $J$ that gives the largest efficiency. Panel
$(b)$ of Fig. 13 shows $\epsilon_{SDE}$ (blue circles) and $\epsilon_{VDE}$
(red circles) as a function of $J$. As one can see, $J\approx 0.36$ provides
the largest efficiency of the superconducting diode effect, while $J\approx
0.40$ is the optimal value for the vortex diode effect.

Although the previous discussion specifically deals with an inhomogeneous
field generated by the side stripes currents, we note that existence of a
maximum efficiency does not depend on the origin of the inhomogeneous field,
but solely on its magnitude. For instance, the external field could be
produced by a ferromagnetic material, as discussed in Fig. 1. To make our
analysis more general, the top $x$ axis in Fig. 13 relates the value of $J$
with the magnitude $H_{e}$ of the inhomogeneous field at both edges of the
superconducting film. In this case, the maximum efficiency for the
superconducting diode effect occurs for $H_{e}\approx 0.62H_{c2}(T)$ and for
the vortex diode effect at $H_{e}\approx 0.69H_{c2}(T)$. This result offers
valuable guidance for achieving optimal diode efficiency experimentally in
setups where the diode effect is induced by a inhomogeneous field,
irrespective of its source.

##  IV Conclusions

In conclusion, we have explored the interplay between the superconducting
diode effect (SDE) and the vortex diode effect (VDE) in thin superconducting
films subjected to external asymmetric flux profiles. Our findings highlight
the potential of inhomogeneous fields to create highly asymmetric conditions
for vortex entrance and exit, unveiling highly efficient vortex diode effects
in plain superconducting films. Notably, the diode effect observed in our
study does not require sophisticated nanofabrication techniques, as the
inhomogeneous field can be induced by plain ferromagnetic films underneath the
superconductor or by currents injected in microsized superconducting wires
coplanar with the main superconducting film.

Moreover, our results demonstrate that the perfect antisymmetric flux profile
represents the most efficient scenario for the diode effect. This profile
maximizes the difference between the Bean-Livingstone barriers in opposite
edges, leading to the highest vortex diode efficiency. Additionally, the
built-in coexistence of vortices and antivortices in this scenario promotes a
high rate of vortex-antivortex annihilations at the equatorial line of the
sample during the positive current branch. This induces hot spots, even in the
early stages of the flux-flow regime, bringing forward the transition to the
normal state. In contrast, during the negative current branch, the current
self-field compensates the external antisymetric field profile, eliminating
vortices and antivortices from the sample, thus inducing the system into a
perfectly superconducting state. Overall, for a wide range of parameters, the
system behaves as a perfect superconducting half-wave rectifier, presenting no
resistance or heat flow for half the current cycle.

Finally, we have demonstrated that the efficiency of the superconducting diode
effect can be significantly enhanced by reducing the heat removal capabilities
of the substrate and adjusting the sweeping rate in current sweep measurements
or the frequency of the AC excitation in DC versus AC measurements. These
insights not only contribute to the fundamental understanding of extrinsic
diode effects in superconducting systems but also offer practical strategies
for optimizing the performance of superconducting diode platforms in various
applications.

###### Acknowledgements.

LRC and ES thank the Brazilian Agency FAPESP for financial support, grant
numbers 20/03947-2 and 20/10058-0, respectively. CCSS thanks financial support
from Conselho Nacional de Desenvolvimento Científico e Tecnológico - Brasil
(CNPq), grant number 312240/2021-0, and Universidade Federal de Pernambuco,
Edital Produtividade Qualificada. This work was financed in part by
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES),
Finance Code 001.

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