# Beyond the standard model of topological Josephson junctions:
From crystalline anisotropy to finite-size and diode effects

Barış Pekerten  0000-0002-5794-9706<EMAIL_ADDRESS>Department of
Physics, University at Buffalo, State University of New York, Buffalo, New
York 14260, USA  David Brandão  0000-0002-5772-1832 Department of Physics,
University at Buffalo, State University of New York, Buffalo, New York 14260,
USA  Bailey Bussiere  0009-0009-7117-5784 Department of Physics, University at
Buffalo, State University of New York, Buffalo, New York 14260, USA  David
Monroe  0000-0002-4640-8912 Department of Physics, University at Buffalo,
State University of New York, Buffalo, New York 14260, USA  Tong Zhou
0000-0003-4588-5263 Department of Physics, University at Buffalo, State
University of New York, Buffalo, New York 14260, USA  Eastern Institute for
Advanced Study, Eastern Institute of Technology, Ningbo, Zhejiang 315200,
China  Jong E. Han  0000-0002-5518-2986 Department of Physics, University at
Buffalo, State University of New York, Buffalo, New York 14260, USA  Javad
Shabani  0000-0002-0812-2809 Center for Quantum Phenomena, Department of
Physics, New York University, New York, New York 10003, USA  Alex Matos-
Abiague  Department of Physics and Astronomy, Wayne State University, Detroit,
MI 48201, USA  Igor Žutić  0000-0003-2485-226X<EMAIL_ADDRESS>Department of
Physics, University at Buffalo, State University of New York, Buffalo, New
York 14260, USA

###### Abstract

A planar Josephson junction is a versatile platform to realize topological
superconductivity over a large parameter space and host Majorana bound states.
With a change in Zeeman field, this system undergoes a transition from trivial
to topological superconductivity accompanied by a jump in the superconducting
phase difference between the two superconductors. A standard model of these
Josephson junctions, which can be fabricated to have a nearly perfect
interfacial transparency, predicts a simple universal behavior. In that model,
at the same value of Zeeman field for the topological transition, there is a
$\pi$ phase jump and a minimum in the critical superconducting current, while
applying a controllable phase difference yields a diamond-shaped topological
region as a function of that phase difference and a Zeeman field. In contrast,
even for a perfect interfacial transparency, we find a much richer and
nonuniversal behavior as the width of the superconductor is varied or the
Dresselhaus spin-orbit coupling is considered. The Zeeman field for the phase
jump, not necessarily $\pi$, is different from the value for the minimum
critical current, while there is a strong deviation from the diamond-like
topological region. These Josephson junctions show a striking example of a
nonreciprocal transport and superconducting diode effect, revealing the
importance of our findings not only for topological superconductivity and
fault-tolerant quantum computing, but also for superconducting spintronics.

††preprint: AIP/123-QED

Josephson junctions (JJs), with two superconductors (S) separated by a
nonsuperconducting (normal) region (N) provide a fascinating manifestation of
proximity effects, where a given material is transformed by its neighbors.[1,
2] The superconductivity leaking from the two S regions can merge,
establishing a relative phase difference, $\phi$, transforming the whole N
region into a superconductor.[3] Decades before Josephson’s prediction,[4]
this behavior was realized experimentally.[5, 1] Even in conventional
materials, through proximity effects, JJs reveal an exotic emergent behavior,
absent in any of their constituent regions. With the interplay between spin-
orbit coupling (SOC) and Zeeman field (applied or due to exchange coupling),
the proximity-induced superconductivity could be transformed from a
conventional spin-singlet into spin-triplet $p$-wave superconductivity. This
system could support topological superconductivity over an enhanced parameter
range by the applied relative phase difference.[6, 7, 8, 9, *Kotetes2020:PRB,
11, 12, 13]

After Kitaev’s pioneering work on spin-triplet topological superconductivity
and Majorana bound states (MBS),[14, 15] an early work suggested planar JJs as
materials realization that supports non-Abelian properties.[6] However, most
of the subsequent studies have focused on proximitized semiconductor
nanowires[16, 17, 18, 19, 20, 21, 22] and MBS detection through the quantized
zero-bias conductance peak,[23, 24, 25, 26] which can also arise from
extrinsic effects.[27, 28, 29] Experiments have realized a high-transparency
epitaxially grown planar JJs with proximity-induced superconductivity from Al
into the InAs two-dimensional electron gas (2DEG).[30] This advance has
stimulated important theoretical efforts recognizing that topological
superconductivity can be supported over a larger parameter range than in
semiconductor nanowires, while also providing phase-sensitive signatures,[11,
12, 31] as well other opportunities in 2D platforms.[32, 33, 13, 34, 35, 36,
37, 38, 39]

Figure 1: (a) Planar JJ formed by two superconducting (S) regions with
superconducting gap $\Delta$ and phase difference $\phi$ covering a 2DEG
normal (N) region. Coordinate axes and JJ dimensions are indicated. The
supercurrent $I$ flows along $\pm\hat{x}$, perpendicular to the applied
magnetic field $\mathbf{B}$. (b) The misalignment of the crystallographic
direction $[010]$ and $\mathbf{B}$ is represented by the angle $\theta_{c}$
which, with Dresselhaus SOC, leads to crystalline anisotropy.

Experiments on planar JJs,[40, 41, 42, 43, 44, 45] including detecting the
phase signature of topological superconductivity[42] and a superconducting
diode effect,[42, 46, 47, 48] reveal a much richer physics than the standard
model. Such a model predicts a universal behavior for highly-transparent
interfaces, with the diamond-like topological region as a function of $\phi$
and in-plane Zeeman field.[12] In this work, we provide a more general
description of planar JJs for the geometry illustrated in Fig. 1(a), where
$W_{S}$ ($W_{N}$) the width of the S (N, uncovered 2DEG) region and $L$ is the
length. In Fig. 1(b), the presence of Dresselhaus SOC in the 2DEG[49, 50] and
a related crystalline anisotropy is represented by the angle $\theta_{c}$.[51,
52, 53] We show that a finite $W_{S}$ and Dresselhaus SOC each strongly modify
the properties of planar JJs implied by the standard model.

We solve the discretized Bogoliubov-de Gennes (BdG) equation using a finite-
difference method for BdG Hamiltonian, $H_{\text{BdG}}$, in the Nambu basis
$(\psi_{\uparrow},\psi_{\downarrow},\psi^{\dagger}_{\downarrow},-\psi^{\dagger}_{\uparrow})$
with $\psi$ ($\psi^{\dagger}$) destruction (creation) operators of the given
spin[51, 53]

$\displaystyle H_{\text{BdG}}$
$\displaystyle=\left[\frac{\mathbf{p}^{2}}{2m^{\ast}}-\mu\left(x,y\right)+H_{\text{SO}}\right]\tau_{z}-\frac{g^{\ast}\mu_{B}}{2}\mathbf{B}\cdot\bm{\sigma}$
$\displaystyle\quad+\Delta\left(x,y\right)\tau_{+}+\Delta^{\ast}\left(x,y\right)\tau_{-}\;,$
(1)

where $H_{\text{SO}}$ contains Rashba and Dresselhaus SOC terms

$\displaystyle H_{\text{SO}}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{\hbar}(p_{y}\sigma_{x}-p_{x}\sigma_{y})+\frac{\beta}{\hbar}\big{[}(p_{x}\sigma_{x}-p_{y}\sigma_{y})\cos{2\theta_{c}}$
(2) $\displaystyle-$
$\displaystyle(p_{x}\sigma_{y}+p_{y}\sigma_{x})\sin{2\theta_{c}}\big{]}.$

Here, ${\bf p}$ is the momentum, $\mu\left(x,y\right)$ is the chemical
potential, $\mathbf{B}=B\,\hat{y}$ is the in-plane magnetic field, $\mu_{B}$
is the Bohr magneton, while $m^{\ast}$ and $g^{\ast}$ are the effective mass
and $g$-factor. The proximity-induced s-wave superconducting gap term is
$\Delta(x,y)=\Delta_{0}\,f_{\mathrm{BCS}}(B,T)\,\Theta(|x-W_{N}/2|)\,\exp{[i\,\phi(x,y)/2}]$,
where $\Theta$ is the step function and
$\phi(x,y)=\phi_{0}\,\mathrm{sgn}{(x)}$, with $\phi_{0}$ the phase difference
across the JJ, while $\Delta_{0}$ is real and uniform in the S regions. The
usual BCS dependence of $\Delta(x,y)$ on $B$ and temperature, $T$, is given by
$f_{\mathrm{BCS}}(B,T)$ with $f_{\mathrm{BCS}}(B=0,T=0)\,=\,1$.[54] $\alpha$
($\beta$) is the Rashba (Dresselhaus) SOC strength. We consider that $y$
direction is translationally invariant in Eq. (1), such that wavevector
$k_{y}=p_{y}\,/\,\hbar$ is a good quantum number, and numerically obtain the
energy spectrum and topological properties of its discretized version.[55, 53]

Planar JJs can be phase biased; for example, $\phi$ can be externally set by
embedding a JJ in a superconducting loop threading the magnetic flux through
the loop. Alternatively, a phase-unbiased JJ attains its ground state value,
$\phi_{GS}$, at which the free energy of the system, $F(\phi,B)$, is minimized
and satisfies $\partial_{\phi}F(\phi_{GS},B)=0$. In phase-unbiased topological
JJs the system self-tunes into a topological state as the in-plane $B$ is
varied by causing a phase jump in $\phi_{GS}\sim\pi$. We obtain[54]

$\displaystyle F(\phi,B)$
$\displaystyle=-k_{B}T\sum_{E_{n},k_{y}}\mathrm{ln}\left[2\,\mathrm{cosh}\left(E_{n}(k_{y},B)/2k_{B}T\right)\right],$
(3)

from the calculated spectrum $\\{E_{n}(k_{y},B)\\}$, where $k_{B}$ is the
Boltzmann constant. The current-phase relation (CPR),
$I(\phi,B)=(2e/\hbar)\,\partial F/\partial\phi$, is then

$\displaystyle I(\phi,B)$
$\displaystyle=-\frac{e}{\hbar}\,\sum_{E_{n},k_{y}}\,\frac{\partial
E_{n}(k_{y})}{\partial\phi}\,\mathrm{tanh}\left(\frac{E_{n}(k_{y},B)}{2k_{B}T}\right).$
(4)

The extrema of $F(\phi,B)$ corresponds to the zeros of the supercurrent,
$I(\phi,B)\propto\partial_{\phi}F(\phi,B)=0$, hence $\phi_{GS}$ follows one of
the zeros of the CPR.[53] The forward (reverse) critical current,
$I_{c}^{\pm}(B)$, is
$I_{c}^{\pm}(B)=\left|\min_{\phi}\,(\max_{\phi})\,I(\phi,B)\right|$, and the
total critical current is $I_{c}(B)=\max(I_{c}^{+},I_{c}^{-})$.We normalize
critical currents, $\widetilde{I\,}_{c}(B)\equiv
I_{c}(B)\,/\,I_{c,\mathrm{max}}$, $\widetilde{I\,}_{c}^{\pm}(B)\equiv
I_{c}^{\pm}(B)\,/\,I_{c,\mathrm{max}}$, to the maximum critical current,
$I_{c,\mathrm{max}}\equiv\max_{B}I_{c}(B)$. Similarly, $I_{c,\mathrm{min}}$,
and $\widetilde{I\,}_{c,\mathrm{min}}\equiv
I_{c,\mathrm{min}}\,/\,I_{c,\mathrm{max}}$, denote the critical current at its
dip near the topological transition and its normalized value.

The class D topological charge, $Q_{D}$,[56, 14, 57, 58, 59, 60] for the JJ is

$\displaystyle Q_{D}$ $\displaystyle={\rm sgn}\left[{\rm
Pf}\\{H(k_{y}=\pi)\sigma_{y}\tau_{y}\\}/{\rm
Pf}\\{H(k_{y}=0)\sigma_{y}\tau_{y}\\}\right],$ (5)

where ${\rm Pf}\\{...\\}$ denotes the Pfaffian, with $Q_{D}=1\,(-1)$
determining the system in the trivial (topological) phase. We employ
experimentally relevant parameters for Al/InAs planar JJs,[42] with
$\Delta_{0}=0.23$meV, $m^{\ast}=0.027m_{0}$, where $m_{0}$ is the bare
electron rest mass, and $g^{\ast}=10$. We parametrize the strengths of Rashba
and Dresselhaus SOC through an overall strength, $\lambda_{\mathrm{SO}}$, and
an angle, $\theta_{\mathrm{SO}}$, defined as

$\lambda_{\mathrm{SO}}\equiv\sqrt{\alpha^{2}+\beta^{2}}\;,\;\theta_{\mathrm{SO}}\equiv{\rm
arccot}(\alpha/\beta)\;,$ (6)

with $\alpha=\lambda_{\mathrm{SO}}\,\cos\theta_{\mathrm{SO}}$ and
$\beta=\lambda_{\mathrm{SO}}\,\sin\theta_{\mathrm{SO}}$.[53, 51] We take
$\mu(x,y)=\mu_{0}-\epsilon(\lambda_{\mathrm{SO}},\theta_{c})$ constant
throughout the system, with $\mu_{0}=1\;$meV and
$\epsilon(\lambda_{\mathrm{SO}},\theta_{c})=(2m^{*}\lambda_{\mathrm{SO}}^{2}/\hbar^{2})\,\left[1+(\sin{2\theta_{c}})^{2}\right]$
denoting the bottom of the single particle energy band.

Figure 2: Finite size effects with $W_{N}=160,\,600\;$nm (solid, dashed) and
$\alpha=5,\,15\;$meVnm (blue, red). (a) Oscillating minimum topological Zeeman
field $B_{T}(W_{S})$ at $\phi=\pi$. (b), (c) Topological phase diagram at a
dip, peak in $B_{T}$ at $W_{S}=152,\,176\;$nm [vertical lines and circle,
rectangle in (a)] showing topological charge $Q_{D}(\phi,B)=1,\,-1$ (hatched,
unhatched) and normalized topological gap
$\widetilde{\Delta}_{\mathrm{top}}(\phi,B)$ (grayscale), with superimposed
ground state phase $\phi_{\mathrm{GS}}(B)$ (violet, left axis) and normalized
critical current $\widetilde{I}_{c}^{\pm}(B)$ (red, right axis).

The standard model of topological JJs provides important ideas for topological
superconductivity and its phase control.[12] By assuming Rashba SOC and a
perfect transparency, $\tau=1$,[12] at N/S interfaces, which is realized in
Al/InAs JJs with $\tau=0.98$,[61, 62] that model leads to a universal behavior
for topological superconductivity. The topological region with
$Q_{D}(\phi,B)=-1$ has a diamond shape,[12] formed by closing of a topological
gap for $\phi=\pi$, at $B=0$ and
$B^{*}=\pi\sqrt{2m^{*}\mu}/(g^{*}\mu_{B}W_{N})$. For $\phi=0,2\pi$ and
$Q_{D}=1$ at any $B$, the topological region shrinks to zero. For a phase-
unbiased JJ, at $B^{*}/4$, a topological transition is accompanied by: (i) dip
(local minimum) in $\widetilde{I}_{c}$ and (ii) $0-\pi$ jump in $\phi_{GS}$.

With the same assumption of the perfect N/S transparency, in Fig. 2 we find a
striking difference from the predicted universal $B^{*}\propto 1/W_{N}$
behavior and a diamond-shape topological region. Finite-size effects in
$W_{S}$ lead to the oscillation of the minimum in-plane Zeeman field for the
transition into a topological state, $B_{T}$, at $\phi=\pi$, where $Q_{D}$
switches from $1$ to $-1$.[12, 60] In Fig. 2(a), for the considered N-region
widths, $W_{N}=160$ $(600)\;$nm, solid (dashed) line, as well the Rashba SOC
strengths, $\alpha=5$ $(15)\;$meVnm, blue (red) line, used throughout our
calculations, we see strong oscillations in $B_{T}(W_{S})$. These oscillations
also deform the diamond-shaped topological region. This means that, at
$\phi=\pi$, a suitable choice of $W_{S}$, reducing $B_{T}$, can relax the
requirements for the onset of topological superconductivity in various
materials.

These decaying oscillations in $B_{T}$, defined at $\phi=\pi$, are also
translated to changes in the whole topological region. For a fixed
$W_{N}=160\;$nm, even a small change in $W_{S}=152,176\;$nm [vertical lines
mark the corresponding dip and peak in $B_{T}$ from Fig. 2(a)], leads to a
large change in the related size and shape of the topological region in Figs.
2(b) and 2(c), respectively. For example, in Fig. 2(b), we see a topological
phase diagram, bounded by a hatched region and $Q_{D}=1$. The topological
region is large and has a small $B_{T}$, even though there is an overall
distortion of the diamond shape. However, while the topological region with
$Q_{D}=-1$ implies that MBS can be supported, it is also important to know the
corresponding (normalized) topological gap,[11, 51, 52, 53]
$\widetilde{\Delta}_{\mathrm{top}}\equiv\Delta_{\mathrm{top}}\,/\,\Delta_{0}$,
shown as the gray scale background. Specifically, MBS are not well protected
in the regions with a small
$\Delta_{\mathrm{top}}(\phi,B)\equiv\min\left(\\{E_{n}(k_{y},\phi,B)>0\\}\right)$,
where two end-MBS hybridize with each other and move away from zero energy. In
contrast, for a slightly wider $W_{S}$ in Fig. 2(c), we see a much larger
$B_{T}$ and a smaller topological region than in Fig. 2(b).

In previous studies,[12, 31, 63] changes from the diamond-shape topological
region were attributed to the imperfect transparency which leads to the normal
reflection at N/S interfaces, not just Andreev reflections (responsible for
the formation of Andreev bounds states in JJs). However, even for perfect N/S
interfaces, we note that with a finite $W_{S}$ there are normal reflections at
the ends of the S regions. The quasiparticle wavefunctions entering the S
regions are reflected back from their ends into the N region and contribute to
the normal reflection at the N/S interfaces. The $B_{T}(W_{S})$ oscillations
in Fig. 2(a) can be understood from the resonant transmission condition, in
which the quasiparticle wavefunction forms a standing wave in the S
region.[54] This can lead to the transition into the topological phase at a
smaller in-plane $B$ and offer a wider topological region for experimentally
relevant parameters. The decay of the $B_{T}(W_{S})$ oscillations is
consistent with the semiclassical picture of the decaying quasiparticle
wavefunction in the S region of width $W_{S}$, until it reenters the N region
after a travel of $\sim 2W_{S}$ in the superconductor. With
$W_{S}\rightarrow\infty$, this source of the normal reflection vanishes and
thus minimizes $B_{T}$.

For chosen parameters in Figs. 2(b) and 2(c), a closer look suggest other
deviations from the standard picture. The dip in $\widetilde{I}_{c}$ and
$0-\pi$ jump in $\phi_{GS}$ no longer take place at the same value of $B$,
which can be much more pronounced in other cases.[42, 53] The jump in
$\phi_{GS}$ is generally different from $\pi$,[64] the size of the jump
decreases with $\alpha$. Together, this suggests that, even for a perfect N/S
transparency, there is a considerably richer behavior for topological JJs than
commonly expected, warranting our further analysis and identifying
opportunities to facilitate experimental implementation of JJs.

Figure 3: Critical current dip near the first topological transition and
superconducting diode effect; solid/dashed lines and colors follow the
convention introduced in Fig. 2 for $W_{N}$ and $\alpha$. (a) Oscillating
critical current at dip, $\widetilde{I\,}_{c,\mathrm{min}}(W_{S})$. (b) Zeeman
field at dip, $B_{\mathrm{dip}}(W_{S})$. (c) - (f) Diode effect for
$W_{S}=128\;$nm [gray line in (a), (b)]: unequal forward, reverse normalized
critical current $\widetilde{I\,}_{c}^{\pm}(B)$ (thick, thin line; left axis),
diode effect efficiency $\eta(B)$ for $\alpha=15\;$meVnm (green, right axis),
and topological charge $Q_{D}=1,\,-1$ (hatched, unhatched).

With this motivation, we next investigate the size and location of the dip of
$\widetilde{I\,}_{c}$, denoted by $\widetilde{I\,}_{c,\mathrm{min}}$ and
$B_{\mathrm{dip}}$, respectively, and shown in Figs. 3(a) and 3(b). This dip
corresponds to the minimization of the singlet component of the finite-
momentum wavefunction, which vanishes near the topological phase transition in
the absence of normal reflection in the JJ.[12, 51, 53, 48] At a fixed
temperature, the magnitude of $\widetilde{I\,}_{c}$ is thus related to the
contribution of the normal reflection. Therefore, oscillations in
$\widetilde{I\,}_{c}(W_{S})$, similar to those in Fig. 2(a), are also present
in Fig. 3(a). However, unlike only a weak dependence of the magnitude of
$B_{T}(W_{S})$ oscillations with $\alpha$ in Fig. 2(a), the same change in the
Rashba SOC strongly changes the magnitude of the $\widetilde{I\,}_{c}(W_{S})$
in Figs. 3(a), for both $W_{N}=160$ and $600\;$nm. Turning to the dip location
in Fig. 3(b), $B_{\mathrm{dip}}(W_{S})$ shows decaying oscillations which are
strongly suppressed for a larger $W_{N}$ and with increasing $W_{S}$. Since a
finite $W_{S}$ changes the shape of the topological region and
$Q_{D}(\phi,B)=-1$, various predicted universal features of the standard
planar JJ model no longer hold. By comparing Figs. 3(b) and 3(c), we see that,
even at $\phi=\pi$, the location of the dip is outside of the topological
region. Furthermore, this location does not coincide with that of the phase
jump, nor with the location for the transition of the phase-unbiased system
into the topological region,[42] as seen in Figs. 2(b) and 2(c), red line.

With the recently renewed interest in nonreciprocal transport and diode
effects in superconducting junctions,[2, 47] we note that the 2019 report of
the phase signature of the transition to topological superconductivity in
planar JJs was also accompanied by the measured superconducting diode
effect,[42, 2] but was not discussed. The diode effect, while not considered
in the standard model of planar JJs, is also inherent to these systems.[65] We
parametrize the efficiency of the diode effect by the asymmetry of the
critical currents[66]

$\eta\equiv\left(I_{c}^{+}-I_{c}^{-}\right)\,/\,\left(I_{c}^{+}+I_{c}^{-}\right),$
(7)

shown along with the forward and reverse $\widetilde{I\,}^{\pm}_{c}$ for
$W_{S}=128\;$nm in Figs. 3(c), 3(d) for $W_{N}=160\;$nm, and in Figs.
3(e)-3(f). Red/blue curves retain our convention for $\alpha=5,15\;$eVnm. An
observed trend here, that $\widetilde{I\,}^{+}_{c}$ and
$\widetilde{I\,}^{-}_{c}$ will differ more for a larger $\alpha$ and larger
$W_{N}$, is expected from the spin-dependent phase shift of electron and hole
forming the Andreev bound states.[65] Away from the dip locations of
$\widetilde{I\,}_{c}^{\pm}$, $\eta\sim 0.1$, while near the
$\widetilde{I\,}_{c}^{\pm}$ $\eta$ becomes much larger, but also accompanied
with only a small $I_{c}$.[53, 48] The sign change locations in $\eta$
correspond to either the minimized singlet or the triplet part of the finite-
momentum pairing wavefunctions, which occur at different locations for narrow
and wide JJs.[48] These finite-size effects could guide future experiments on
both superconducting diode effect and topological phase transitions in JJs.

Most of the work on topological JJs considered only Rashba SOC. While for
certain crystallographic directions the spin texture from Dresselhaus SOC[49]
recovers the one expected from only Rashba SOC in Eq. (2), with the
simultaneous presence of these two SOC contributions, present in many 2DEGs,
the topological phase diagram can be strongly modified. In a situation from
Fig. 1, where in-plane ${\bm{B}}$ is perpendicular to the current direction
and for a narrow JJ with $\beta\neq 0$, one can find that
$\Delta_{\mathrm{top}}$ together with MBS protection are maximized for
$\theta_{c}=\pi/4$ for an arbitrary ratio $\alpha/\beta$, i.e. an arbitrary
$\theta_{\mathrm{SO}}$,[53, 51, 52] recall Eq. (6). We thus limit our results
to $\theta_{c}=\pi/4$ and consider the cases where Rashba and Dresselhaus have
equal strengths, $\theta_{\textrm{SO}}=\pi/4$, or the system has only
Dresselhaus SOC, $\theta_{\textrm{SO}}=\pi/2$, while keeping constant the
combined SOC magnitude, $\lambda_{\mathrm{SO}}$, defined in Eq. (6).

Figure 4: Effects of Dresselhaus SOC; in all plots
$\lambda_{\mathrm{SO}}=\sqrt{\alpha^{2}+\beta^{2}}=15\;$meVnm and
$\theta_{c}=\pi/4$, see Eq. (2). Solid/dashed lines indicate $W_{N}$ as in
Figs. 2, 3. (a), (b) Oscillating $B_{T}(W_{S})$; $\theta_{\mathrm{SO}}=\pi/4$
(red) [$\pi/2$, blue] corresponds to $\alpha=\beta$ ($\alpha=0$). (c)-(f)
Topological phase diagrams with $Q_{D}(\phi,B)=1,-1$ (hatched, unhatched) and
$\widetilde{\Delta}_{\mathrm{top}}(\phi,B)$ (grayscale), with $\phi_{GS}(B)$
(violet, right axis) and $\widetilde{I\,}_{c}(B)$ (red, left axis). (c), (d)
correspond to a minimum, peak in $B_{T}(W_{S})$ at $W_{S}=152,\,176\;$nm [gray
lines in (a)]. (e), (f) correspond to a minimum, peak at $W_{S}=104,\,136\;$nm
[gray lines in (b)].

To consider the influence of Dresselhaus SOC, in Figs. 4(a) and (b), we first
revisit the oscillation of the topological transition field, $B_{T}(W_{S})$,
shown previously in Fig. 2(a) for $\beta=0$. By choosing
$\lambda_{\mathrm{SO}}=15\;$meVnm in Fig. 4, we have a direct comparison with
the red curves for $\alpha=15\;$meVnm from Fig. 2 which show very similar
decaying oscillations with $W_{S}$, while the results from Figs. 4(a) and (b)
again confirm that amplitude of these oscillations is suppressed with $W_{N}$.
We also see that, it is not only the combined SOC magnitude, but also the
relative contribution of Dresselhaus and Rashba SOC which determines the
oscillations in $B_{T}(W_{S})$, shifted to the right for a larger $\beta$
($\theta_{\mathrm{SO}}=\pi/2$) that becomes more pronounced with a wider
$W_{N}$. As a result, while the oscillating trends in $B_{T}(W_{S})$ are
retained without and with Dresselhaus SOC, at a given $W_{S}$ and $W_{N}$, a
change in the relative strength of the Rashba and Dresselhaus SOC can strongly
alter the value of $B_{T}$ and thus also the topological phase diagram. For
example, at $W_{S}=104\;$nm, in Figs. 2(a) and 4(b) we can compare the three
corresponding $B_{T}$ values with the same SOC strength
$\lambda_{SO}=15\;$meVnm, but with various relative strengths of the
Dresselhaus SOC. Using a pure Dresselhaus SOC as a reference, $B_{T}$ for a
mixed Rashba and Dresselhaus SOC in Fig. 4(b) differ by $\sim 0.1\;$T, while
for a pure Rashba SOC in Fig. 2(a), the difference in is $\sim 0.08\;$T.
Correspondingly, with a change in the relative strength of Rashba and
Dresselhaus SOC, the topological phase diagram and the onset of topological
superconductivity can be noticeably different.

Another influence of Dresselhaus SOC can be seen in Fig. 4(c) and 4(d) for the
two chosen values of $W_{S}$, marked by gray lines in Figs. 4(a) and 4(b) and
matching the $W_{S}$ values in Fig. 2. Several differences from $\beta=0$ and
Figs. 2(b) and 2(c) can be inferred, where the evolution of
$\widetilde{I\,}_{c}$ with $B$ was overall smoother and the jumps in
$\phi_{GS}$ were less abrupt than seen in Fig. 4(c) and 4(d). This is
consistent with a shallower landscape for the free energy when $\beta\neq 0$,
which supports more rapid changes with $B$, including the dip in the
$\phi_{GS}$ in Fig. 4(c). We note that the choice of ${\bm{B}}$ along the y
axis in Fig. 1(a) maximized ${\Delta}_{\mathrm{top}}(\phi,B)$ for $\beta=0$
and led to its reduction in Fig. 4(c)-4(f). With a larger $W_{N}=600\;$nm in
Figs. 4(e) and 4(f), there are even more abrupt changes in
$\widetilde{I\,}_{c}$ and $\phi_{GS}$. While not shown in Fig. 4, $\beta\neq
0$ again supports the superconducting diode effect, discussed in Fig. 3.

Planar JJs provide a versatile platform for topological superconductivity as
well as for superconducting spintronics,[67, 68, 69] through spin-triplet
supercurrents and superconducting diode effect.[70, 2, 47] Even for a perfect
interfacial transparency, we find a much richer behavior than what is expected
from the common description of planar JJs. There are interesting future
generalizations where our results could offer a helpful starting point. For
example, in exploring the role of disorder in planar JJs,[71, 72] or more
complex geometries,[73, 38, 74] it is important to include the relevant form
of SOC and the finite-size effects. With a growing class of materials used in
JJs, SOC that is linear in the wave vector may not be sufficient and the cubic
SOC terms could become dominant.[75, 76, 77, 78] As a result, instead of the
spin-triplet $p$-wave, the proximity-induced topological superconductivity
could acquire the character of the spin-triplet $f$-wave.[76]

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

###### Acknowledgements.

This work is supported by US ONR under awards MURI N000142212764,
N000141712793, and NSF ECCS-2130845. Computational resources were provided by
the UB Center for Computational Research.

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