pnasresearcharticle Lesser The practical realization of Majorana zero modes in
quasi-one-dimensional topological superconductors is greatly hindered by the
need to apply strong magnetic fields. This study proposes a novel way to
engineer these exotic states using only superconducting phase bias, which
requires negligible magnetic fields or currents. The proposed device is
experimentally accessible and robust, as we show by comprehensive theoretical
modeling. Furthermore, it has the potential of providing substantially cleaner
experimental signatures of Majorana zero modes than the currently available
platforms, paving the way to building a topological qubit. All authors
conceived the model. O.L. performed the simulations. A.S. and M.W. provided
considerations on the experimental feasibility and performed preliminary
simulations which informed the work. All authors analyzed the results and
wrote the manuscript. The authors declare no competing interests. 1To whom
correspondence should be addressed. E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Phase-induced topological superconductivity in a planar heterostructure

Omri Lesser  Department of Condensed Matter Physics, Weizmann Institute of
Science, Rehovot, Israel 7610001  Andrew Saydjari  Department of Physics,
Harvard University, Cambridge, Massachusetts 02138  Marie Wesson  John A.
Paulson School of Engineering and Applied Sciences, Harvard University,
Cambridge, Massachusetts 02138, USA  Amir Yacoby  Department of Physics,
Harvard University, Cambridge, Massachusetts 02138  John A. Paulson School of
Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts
02138, USA  Yuval Oreg  Department of Condensed Matter Physics, Weizmann
Institute of Science, Rehovot, Israel 7610001

###### Abstract

Topological superconductivity in quasi-one-dimensional systems is a novel
phase of matter with possible implications for quantum computation. Despite
years of effort, a definitive signature of this phase in experiments is still
debated. A major cause of this ambiguity is the side effects of applying a
magnetic field: induced in-gap states, vortices, and alignment issues. Here we
propose a planar semiconductor-superconductor heterostructure as a platform
for realizing topological superconductivity without applying a magnetic field
to the two-dimensional electron gas hosting the topological state. Time-
reversal symmetry is broken only by phase-biasing the proximitizing
superconductors, which can be achieved using extremely small fluxes or bias
currents far from the quasi-one-dimensional channel. Our platform is based on
interference between this phase biasing and the phase arising from strong
spin-orbit coupling in closed electron trajectories. The principle is
demonstrated analytically using a simple model, and then shown numerically for
realistic devices. We show a robust topological phase diagram, as well as
explicit wavefunctions of Majorana zero modes. We discuss experimental issues
regarding the practical implementation of our proposal, establishing it as an
accessible scheme with contemporary experimental techniques.

###### keywords:

topological phases of matter $|$ majorana zero modes $|$ topological
superconductivity

The quest for discovering novel phases of matter has seen rapid development in
the past few decades, with the advent of topological quantum matter (1, 2, 3,
4, 5). Phases that would be equivalent under Landau’s order-parameter paradigm
were found to be distinguished by topological properties. The earliest and
perhaps most salient phase characterized by its topology is the quantum Hall
effect (6), where the chiral edge modes signal the non-trivial bulk topology
(7, 8). It was later realized that a similar phenomenon can take place with a
zero net magnetic flux (9), which opened the door to the field of Chern
insulators.

Shortly after these important discoveries, the connection to superconductivity
(SC) was made, by the concept of topological superconductivity. This novel
phase of matter supports Majorana zero modes (MZMs), which are predicted to
have non-Abelian exchange statistics (10, 11). It has been vigorously studied
in recent years, following early ideas in one (12) and two (13) spatial
dimensions. Further theoretical studies advanced the field from toy models to
practical implementations (14), fuelled by the potential application of the
exotic MZMs in quantum computation (11, 15, 16).

A prominent Majorana platform is hybrid superconductor-semiconductor nanowires
(17, 18), where a Zeeman field combined with strong spin-orbit coupling (SOC)
drive a proximitized nanowire into a topologically non-trivial phase.
Following the theoretical proposals, several tunneling experiments (19, 20)
observed zero-bias conductance peaks as possible signatures of MZMs. The early
results triggered many subsequent studies, including on current-biased (21)
and disordered (22, 23) nanowires and carbon nanotubes (24, 25, 26) as the
one-dimensional platform. Other important Majorana realization platforms
include quantum wells subjected to an in-plane magnetic field (27),
topological insulator-superconductor heterostructures (13), semiconductor-
ferromagnet heterostructures (28, 29), chains of magnetic adatoms on spin-
orbit-coupled superconductors (30, 31), iron-based superconductors (32, 33),
and most recently full-shell proximitized nanowires (34, 35).

A different route, first taken in two important theoretical works (36, 37), is
based on planar semiconductor-superconductor heterostructures. These studies
proposed utilizing a phase-biased Josephson junction, forming a quasi-1D
geometry, with an applied in-plane magnetic field. Such a planar device is a
highly attractive platform for the experimental realization of topological
superconductivity, as it is less delicate than nanowires, does not require
strict magnetic field alignment, and is easier to fabricate. The theoretical
proposals triggered two recent experiments that indeed reported zero-bias
conductance peaks which may indicate the existence of MZMs (38, 39). However,
a drawback of this platform is the need to apply an appreciable magnetic field
to the device. The applied magnetic field diminishes superconductivity and
could give rise to magnetic-impurity states, hindering the detection of MZMs.

In this manuscript, we propose a route to realizing topological
superconductivity in a planar geometry without applying a magnetic field in
the Josephson junction. Previous proposals achieved this by using
supercurrents to break time-reversal symmetry; however, the large currents
required and fringing fields in the junction complicate implementation (40).
Our proposal relies solely on phase-biasing the proximitizing superconductor,
a possibility that was recently pointed out (41). The approach is based on
engineering the geometry such that the topological Aharonov-Casher phase (42),
induced by the spin-orbit coupling, constructively interferes with the SC
phase winding (13, 43, 41), giving rise to a zero-energy state. We would like
to emphasize that the role of the phase biases in our proposal and in Fu and
Kane’s proposal (13) is different. In Fu and Kane’s proposal, the topological
phase results from the interplay between the surface of a 3D topological
insulator and a superconductor, even in the absence phase bias. The role of
the phase bias is to form a discrete vortex supporting an MZM. Here the phases
themselves drive the system into a topological phase in a conventional
material platform with strong spin-orbit coupling. Our geometry also benefits
from eliminating unbound trajectories as in the case of zig-zag junctions
(44). Not only does this proposal eliminate the complications of magnetic
fields in the junction coexisting with the MZMs, but it is also accessible
with current materials and fabrication techniques.

## Minimal model and “sweet spot"

We shall now demonstrate the possibility of realizing MZMs in a magnetic-
field-free planar geometry. To this end, we develop a toy model that supports
a “sweet spot" with perfectly localized MZMs (45). This is analogous to tuning
the Kitaev chain to zero chemical potential and equal hopping and pairing
amplitudes, which makes the edge modes perfectly localized in a single site
(12).

The basic building block of our model is a ring composed of $N\geq 3$ sites,
see Fig. 1A. Electrons can hop from each site to its nearest neighbors through
a spin-orbit-coupled medium, and each site $n$ is proximitized by a SC with a
different phase $\phi_{n}$. The ring is described by the tight-binding
Bogoliubov-de-Gennes (BdG) Hamiltonian

$\displaystyle H$
$\displaystyle=\sum_{n=1}^{N}\left(te^{i\lambda\sigma_{z}}c_{n}^{\dagger}c_{n+1}-\mu
c_{n}^{\dagger}c_{n}\right)$ (1) $\displaystyle+\sum_{n=1}^{N}\left(\Delta
e^{i\phi_{n}}c_{n,\uparrow}^{\dagger}c_{n,\downarrow}^{\dagger}+\text{H.c.}\right),$

where $\sigma$ labels the spin degree of freedom, $t$ is the hopping
amplitude, $\lambda$ is the SOC angle, $\mu$ is the chemical potential, and
$\Delta$ is the SC pairing potential. The gauge transformation
$c_{n}\rightarrow c_{n}e^{i\phi_{n}/2}$ eliminates the phases from the SC
terms, in exchange for adding the complex amplitude
$e^{i\left(\phi_{n+1}-\phi_{n}\right)/2}$ to the hopping term from site $n+1$
to site $n$.

Figure 1: Toy model for realizing a “sweet spot" relying on SC phase bias
only. (A) Illustration of the ring described by (1) for $N=3$ sites, each
connected to its nearest neighbors by a hopping amplitude $t$ through a spin-
orbit-coupled medium characterized by the angle $\lambda$. Each site is
proximitized by a superconductor with pair potential $\Delta$ and a different
phase $\phi_{n}$. The parameters controlling the ring’s behavior are tuned
such that each ring contains two overlapping Majorana zero modes, $\alpha$ and
$\beta$. (B) Two rings are coupled at two points (top). If only $\beta_{1}$
and $\alpha_{2}$ are coupled, those states are gapped out, leaving the zero-
energy states $\alpha_{1}$, $\beta_{2}$ intact and localized at the left and
right ring, respectively (bottom). (C) Energies of the lowest-lying (blue) and
second-lowest-lying (orange) states in the system composed of two coupled
rings. For certain values of $\lambda^{\prime}$ a zero-energy state appears,
with a finite gap to the second-lowest state. At $\lambda^{\prime}=0,\pm\pi$
we observe spin degeneracy. (D) Zero-energy Majorana wavefunctions at one of
the special values of $\lambda^{\prime}$, each state localized in a different
ring.

It is known that SC phase winding is necessary in order to obtain a zero-
energy state in such a model (43). For simplicity, we assume that
$\phi_{n}=n\varphi$ where $\varphi=2\pi m/N$ with $m\in\mathbb{Z}$, i.e., an
integer number of vortices. With this choice $\phi_{n+1}-\phi_{n}=\varphi$ is
constant, so the model becomes translationally invariant and can therefore be
analyzed in momentum space. The Bloch Hamiltonian reads

$\displaystyle H$
$\displaystyle=\sum_{k,\sigma}\left[2t\cos\left(k+\frac{\varphi}{2}+\lambda\sigma\right)-\mu\right]c_{k\sigma}^{\dagger}c_{k\sigma}$
(2) $\displaystyle+\sum_{k}\left(\Delta
c_{k\uparrow}^{\dagger}c_{-k\downarrow}^{\dagger}+\text{H.c.}\right),$

where $k=\frac{2\pi}{N}q,\,q=0,1,\ldots,N-1$. As we show in Appendix S.I, it
is possible to tune the model’s parameters such that two zero-energy states
appear: a pair of MZMs, delocalized around the ring. The key ingredient of the
model is the interplay between the gauge-invariant topological Aharonov-Casher
phase, $3\lambda$ (arising due to electron and hole trajectories circulating
the ring), and the SC phase winding.

Having established the possibility of realizing MZMs in a ring, the next step
is coupling two such rings, as shown in Fig. 1B. Each ring hosts two MZMs, and
our goal is to couple the rings such that one MZM in each ring remains
uncoupled, and the other two MZMs are gapped out, as illustrated at the bottom
of Fig. 1B. Since the bare MZMs are delocalized throughout their rings, the
only way to achieve this goal is by interference of trajectories (46), and
therefore the two rings must be coupled via more than one link. In the
coupling form suggested in Fig. 1B, the rings are connected at two points via
spin-orbit-coupled links of amplitude
$t^{\prime}e^{i\lambda^{\prime}\sigma_{x}}$ 111The $\sigma_{x}$ Pauli matrix
appears in the inter-ring SOC because of the choice made in the original
Hamiltonian (1) of taking all intra-ring SOC terms proportional to
$\sigma_{z}$, which is a local rotation of the spin..

By choosing the SC phases at the two rings and controlling the inter-ring SOC
angle $\lambda^{\prime}$, it is possible to tune into a “sweet spot" where one
MZM in each ring remains intact. We show this explicitly in Appendix S.I by
projecting the coupling Hamiltonian to the low-energy subspace and making sure
that one MZM in each ring is left intact. This situation is demonstrated in
Fig. 1C, where we plot the energies of the lowest and second-lowest states as
a function of $\lambda^{\prime}$. At special values of $\lambda^{\prime}$, the
lowest energy is zero and the second-lowest energy is finite, implying the
existence of two MZMs, each localized in a different ring, with a gap to
excitations. If we now concatenate these double-ring building blocks on the
plane, the localization length of the Majorana edge modes will be one ring
independent of the overall length of the chain. We have therefore achieved a
topological superconducting phase by tuning the three superconducting phases,
without the Zeeman effect. A sweet spot is obtained by setting $\mu$,
$\Delta$, $\lambda$, $\lambda^{\prime}$ to proper values.

## Realistic model

How can one realize a phase similar to the “sweet spot" in a realistic
experiment? The key point is using a geometry that supports closed orbits with
a non-zero Aharonov-Casher phase (42) and SC phase winding (41). Here we
demonstrate this idea using a planar semiconductor-superconductor
heterostructure and analyze the topological phase diagram as a function of
easily controlled parameters—the SC phases and the chemical potential.

We consider the unit-cell geometry depicted in Fig. 2. The system is made of a
two-dimensional electron gas (2DEG) with Rashba SOC, partially covered by SCs
with fixed phases. The main difference between this configuration and the
original proposals that rely on a Zeeman field (36, 37) is the addition of a
third SC with phase-control. The non-straight features, characterized by the
lengths $\left(W_{\rm B},L_{\rm B}\right)$, will help stabilize the
topological phase by increasing the energy gap (along the lines of Ref. (44)).
In what follows we will show that when the SC phases wind, the system can be
driven into a topological SC phase akin to the one realized in the toy model,
even at zero applied Zeeman field.

Figure 2: Unit cell of the proposed geometry for realizing topological
superconductivity in a planar heterostructure using phase bias only. Blue
regions are covered by SCs, whereas the green region is an uncovered 2DEG. The
SC phases are $0$ at the middle SC, $\phi_{1}$ and the bottom SC, and
$\phi_{2}$ at the top SC. The geometric parameters are indicated on the
schematic. When $W_{\rm B}>W_{\rm N}$ straight trajectories that do not
encounter SCs at all are eliminated, thus facilitating a large topological gap
(44). The unit cell is repeated along the $x$ direction.

The heterostructure is described by the Hamiltonian

$\displaystyle H=$
$\displaystyle\left[\frac{\hbar^{2}}{2m^{*}}\left(k_{x}^{2}+k_{y}^{2}\right)+\hbar\alpha\left(\sigma_{y}k_{x}-\sigma_{x}k_{y}\right)-\mu\right]\tau_{z}$
(3) $\displaystyle+{\rm Re}\Delta(x,y)\tau_{x}+{\rm
Im}\Delta^{*}(x,y)\tau_{y},$

where $k_{x},k_{y}$ are the momenta along the $x,y$ directions, $m^{*}$ is the
effective electron mass, $\alpha$ is the Rashba SOC parameter, $\mu$ is the
chemical potential, $\Delta$ is the local superconducting pairing potential,
and $\tau$, $\sigma$ Pauli matrices act in particle-hole and spin spaces,
respectively. The magnitude of $\Delta(x,y)$ is taken to be a constant
$\Delta$ in the proximitized regions, and zero in the non-proximitized
regions. The SC phases are 0 at the middle SC, $\phi_{1}$ at the bottom SC,
and $\phi_{2}$ at the top SC, as indicated in Fig. 2. The corresponding Nambu
spinor is
$\vec{\Psi}=\left(\psi_{\uparrow},\psi_{\downarrow},\psi_{\downarrow}^{\dagger},-\psi_{\uparrow}^{\dagger}\right)^{\rm
T}$, where $\psi_{s}$ annihilates an electron of spin $s$ along the $z$ axis.

The system is assumed be finite along the $y$ direction, and the unit cell is
repeated along the $x$ direction. For the numerical simulations, the
Hamiltonian (3) is discretized on a square lattice of (unless specified
otherwise, the lattice spacing is $a=10\,{\rm nm}$). Further details on the
tight-binding model are given in Appendix S.II. We note that while the
geometry proposed in Fig. 2 is the focus of this study, it is merely an
example of the general concept. As long as the geometry supports trajectories
with a discrete vortex and a spin-orbit phase, it is expected to work as well.
We have numerically demonstrated a topological phase transition in several
similar geometries.

## Topological phase diagram

In this section we present the full topological phase diagram of the proposed
system. The Hamiltonian (3) belongs to symmetry class D (47, 48, 49), with the
particle-hole operator being $\mathcal{P}=\tau_{y}\sigma_{y}\mathcal{K}$
($\mathcal{K}$ is complex conjugation). Enhancement to the BDI class, as in
the case of a single planar Josephson junction (50), is generically prevented
by the existence of two distinct phase differences, and only occurs at fine-
tuned conditions. In (quasi) one dimension, systems in symmetry class D
possess a $\mathbb{Z}_{2}$ topological invariant that we label $\mathcal{Q}$,
which takes the values $+1,-1$ for the trivial and topological phases,
respectively. We calculate the invariant by attaching two leads to a finite-
size system and invoking the scattering formula $\mathcal{Q}={\rm det}r$,
where $r$ is the reflection matrix from one of the leads to itself (51). The
scattering calculations are performed using the Kwant software package (52).

Apart from the topological invariant, the key property of the system is the
energy gap. To find the gap, we impose periodic boundary conditions along $x$,
such that the momentum $k_{x}$ is a good quantum number. We then densely scan
the Brillouin zone and calculate the lowest energy at each value of $k_{x}$.

Fig. 3 shows the resulting phase diagram as a function of the phase
differences $\theta=\left(\phi_{1}-\phi_{2}\right)/2$ and
$\phi=\left(\phi_{1}+\phi_{2}\right)/2$. The system only supports a
topological phase when the SC phases wind, which is evident from the phase
diagram: $Q=-1$ only in certain regions inside the triangle
$\pi-\theta<\phi<\pi+\theta$ (for $0\leq\theta\leq\pi/2$). This triangle sets
the optimal phase boundaries we can expect in such a system (43, 41), and the
topological region we obtain here is not far from that. Furthermore, we get a
substantial topological gap of over $10\%$ of the induced SC gap $\Delta$.
Such a large energy gap has important implications on making the MZM
localization length short, and on enabling reliable experimental detection.
The maximal value of the gap is obtained near the center of the topological
region.

Figure 3: Topological phase diagram of the model (3), showing the topological
invariant $\mathcal{Q}$ which is $-1$ ($+1$) in the topological (trivial)
phase, multiplied by the energy gap. The phase diagram is shown as a function
of $\theta=\left(\phi_{1}-\phi_{2}\right)/2$ and
$\phi=\left(\phi_{1}+\phi_{2}\right)/2$. The physical parameters are
$\mu=0.26\,{\rm meV}$, $\alpha=23\,{\rm meVnm}$, and the geometric parameters
are $L_{\rm SC}=320\,{\rm nm}$, $W_{\rm SC}=50\,{\rm nm}$, $L_{\rm B}=50\,{\rm
nm}$, $W_{\rm B}=80\,{\rm nm}$, $L_{\rm M}=240\,{\rm nm}$, $W_{\rm
M}=120\,{\rm nm}$, $W_{\rm N}=40\,{\rm nm}$. See (53) for an interactive
version of this figure.

In this structure, closed trajectories in which the electron scatters off all
three SCs seem vital for the existence of the topological phase (41). They are
present also when the discontinuous rectangles of the middle SC are connected
by a thin SC, thus becoming continuous. Then, normal tunneling or crossed
Andreev tunneling through the thin segment will enable such trajectories. We
have verified numerically that this is indeed possible, but in our checks this
came at price of lowering the topological gap. Notice though that in a
completely straight geometry, i.e., $W_{\rm B}=0$, it is impossible to obtain
a topological phase—the required closed orbits do not exist.

To demonstrate the robustness of the topological phase, we examine its
stability to variations in the chemical potential and phase bias. In Fig. 4 we
show the topological phase diagram as a function of $\mu$ and $\theta$, fixing
$\phi=\pi$. We find that the topological phase persists for an appreciable
range of parameters, indicating that no delicate fine-tuning is necessary for
the system to support MZMs. Stability to such variations is of paramount
practical importance, as elaborated in the experimental considerations
section.

Figure 4: Topological phase diagram of the model (3), as a function of the
chemical potential $\mu$ and the phase difference $\theta$, for $\phi=\pi$.
The parameters are the same as in Fig. 3.

The geometric parameters are chosen here according to the rule of thumb
$L\Delta\approx\alpha$ (41), where $L$ is a typical length in the uncovered
2DEG region. Given different 2DEG and SC material, one should tune the lengths
to approximately match this condition (see Appendix S.III and the interactive
figure (53)).

## Real-space analysis and Majorana wavefunctions

In the previous section, we have established the existence of a topological
phase by calculating the $\mathbb{Z}_{2}$ invariant. We now turn our focus to
the real-space hallmark of the topological phase, which is the existence of
localized zero-energy Majorana end states. These states are the truly tangible
manifestation of the topological nature of the phase, and experimentally they
reveal themselves as zero-bias conductance peaks in tunneling measurements
(27, 19, 20, 38, 39).

A typical MZM wavefunction is shown in Fig. 5, both as a 2D heat map and as a
1D curve (integrated along the transverse direction $y$). The parameters are
chosen well within the topological phase, as inferred from Fig. 3. As we
expect, two states of near zero energy appear in this regime, localized at
opposite edges of the device. The MZM wavefunctions are concentrated a bit
more on the normal 2DEG regions than on the SC regions, though its noticeable
leakage into the SC region indicates a strong proximity effect.

Remarkably, the Majorana states are localized over one unit cell, which is
about $0.5\,\mu{\rm m}$ long. A smaller unit cell for the same spin-orbit
energy would deviate from the optimal geometry, leading to a smaller
topological gap. This localization length also agrees with the continuum
approximation $\xi\approx\hbar v_{\rm F}/E_{\rm gap}$, where $v_{\rm F}$ is
the Fermi velocity. The relation between the localization length and the
energy gap is further demonstrated in Fig. 6A, by calculating both of them as
a function of $\theta$ inside the topological phase.

Figure 5: Real-space simulation of Hamiltonian (3) with geometry proposed in
Fig. 2, in the topological phase. Top: wavefunction of the zero-energy
Majorana state localized at the left edge. The wavefunction is almost
completely localized at the first unit cell. The black borders indicate the
boundary between the SC and the 2DEG. Notice that the MZM wavefunction has
significant support beneath the SCs; we find that such behavior occurs for a
wide range of parameters but not always. Bottom: the above wavefunction
integrated over $y$ shows exponential decay along $x$. An exponential fit
yields a localization length of $\xi_{\rm M}\approx 0.5\,\mu{\rm m}$. The
simulation included 10 unit cells, and for clarity only the 4 left-most ones
are shown. The other MZM is localized at the right edge of the device. The
parameters used are the same as in Fig. 3, with $\phi=\pi$, $\theta=\pi/4$.
Figure 6: (A) Localization length (green) and energy gap (blue) in the
topological phase. The localization length is roughly inversely related to the
energy gap, as expected by the continuum approximation $\xi\approx\hbar v_{\rm
F}/E_{\rm gap}$, where $v_{\rm F}$ is the Fermi velocity. (B) Lowest-lying BdG
energies as a function of the phase difference $\theta$ in a finite system.
The topological phase transition is signaled by a closing and re-opening of
the energy gap, while leaving two Majorana states (red) at zero energy. The
parameters used are the same as in Fig. 3, with $\phi=\pi$ and 6 unit cells.

We close this section with another explicit demonstration of the topological
phase transition. Fig. 6B shows the evolution of the lowest-lying energies as
a function of $\theta$ in a finite system of 6 units cells 222Notice that
$\theta$ is chosen for convenience, and a similar plot as a function of $\mu$
or $\phi$ may also be obtained.. The energy gap closes and re-opens, leaving a
pair of MZMs bound to zero energy. This signals the transition from the
trivial to the topological phase. The topological gap is maximized near
$\theta=\pi/4$, and then closes and re-opens again near $\theta=3\pi/8$, this
time leaving no in-gap states, indicating that the system returns to the
topologically trivial phase. Notice the agreement between this real-space
calculation and the $\phi=\pi$ cut of the phase diagram shown in Fig. 3.

## Experimental considerations

This proposal for phase-only topological superconductivity can apply to a
general class of material platforms: large spin-orbit semiconducting quantum
wells such as HgTe, InAs, and InSb which have well developed fabrication
procedures (54, 55, 56). While induced-superconductivity has been demonstrated
in all three platforms, we focus on HgTe below due to its high mobility ($\mu$
= 100–800 $\times$103 cm2V-1s-1) and lack of surface states.

Figure 7: Feasible device fabrication layout. Light blue regions represent
superconductors which are deposited after etching to make contact to the
quantum well (QW). Superconducting connectors between center SC islands are
deposited in the same step using an airbridge technique or a separate step
milling through native oxides which form after the first SC deposition. The
two phase degrees of freedom are set independently by current loops which
eliminates even fringing fields from the junction region which hosts the MZMs.
2DEG is removed outside the junction region. The whole device is covered in a
metal on oxide electrostatic gate for tuning the density/chemical potential
(not shown in top view). Conductance tunnel probes, shown in orange, are
included for detecting MZMs at the ends of the device.

The minimum feature sizes of the proposed geometry (Fig. 2) can be achieved by
standard electron beam lithography ($\geq$ 30 nm). See caption of Fig. 7 for
brief outline of the proposed fabrication procedure. Practical geometry
constraints are also placed by the material. The width $W$ between
superconducting contacts must be smaller than the coherence length of the
induced superconductivity in the Josephson junction which is limited to
$\zeta_{T}\sim\hbar v_{\rm F}$/$2\pi k_{\rm B}T$. While the induced coherence
length provides a loose bound on the junction width to $\leq$ 50 $\mu$m at 20
mK, this bound quickly becomes strict at higher temperatures. Further, in
order to work in the quasi-ballistic regime, the mean free path $l_{e}$ must
be at least on the order of $W$. For HgTe, we have $l_{e}\sim 1.6\,\mu\rm{m}$
for $\mu\sim 100\times 10^{3}\,{\rm cm}^{2}{\rm V}^{-1}{\rm s}^{-1}$ and
$n\sim 1\times 10^{12}\,{\rm cm}^{-2}$. Since there is no competing drive for
larger junction widths, as in the case of the in-plane field proposal (37,
36), this upper bound need not be saturated thus suggesting our proposal will
be more robust to lower-quality semiconducting materials.

With these upper and lower bounds from the material in mind, we still have to
determine the optimal operational dimensions of the device. Dimensional
analysis suggests that a typical coherence length set by $\alpha/\Delta$
should be of the order of the typical lengths of the device $L$. Ref. (41)
analyzed a simplified tight-binding model of a related (non-planar) system,
and showed that this rule of thumb indeed leads to an optimal situation, with
a maximal topological region in the $\theta$–$\phi$ plane of Fig. 3. For fixed
$\Delta$, this rule of thumb, $L\Delta\approx\alpha$, implies that for larger
spin-orbit coupling, larger unit cells are required to achieve the largest
topological gap. In contrast to fully 1D proposals, here the confinement of
the MZM is set by the size of the unit cell, rather than just the topological
gap as is expected in a continuum setting. For moderate $\alpha=23$ meV nm
with the superconducting gap of aluminium, this condition fixes $L\sim$ 1
$\mu$m. For fixed $\alpha$, increasing the superconducting gap shortens this
length scale. Advantageously, HgTe growth can be tuned from negligible toward
the very large spin-orbit splitting up to 30 meV (57, 58), with $\alpha$ as
large as 120 meV nm, which can be matched to the superconductor used to
balance fabrication limitations on dimension and optimize confinement. Notice
that even for the case of Al, we saturate lithographic precision at
$\alpha\sim 10$ meV nm, where $E_{\rm SO}\sim 1.5\,{\rm meV}\geq\Delta$,
suggesting extraordinary confinement is possible.

The most difficult aspect of implementing the phase-only proposal is tuning
stably into the topological phase. Our parameter space consists of geometrical
lengths, 2DEG density, spin-orbit strength, and superconducting phases.
Geometrical lengths are set during the fabrication and can only be made
uniform to within $10$ nm. Thus, for a fixed geometry, we require our tunable
parameters to bring us to a topological regime. In addition, the existence of
a topological phase cannot be sensitive to small variations in lengths from
unit cell to unit cell, or even within a unit cell. We have demonstrated this
insensitivity in (53).

While back-gates on semiconductor quantum wells are possible (59), none of the
platforms discussed above have implemented this technology while maintaining
high mobility. As a result, control over the 2DEG density and spin-orbit
strength are both coupled to the potential of the top gate. Gated, the mean
density of HgTe can be tuned from 4$\times$1011 cm-2 to 1$\times$1012 cm-2
while the spin-orbit splitting can be tuned up to 30 meV (57, 60). In Figs. 4
and 6, the simulated density is approximately $1-6\times 10^{9}$ and
$0.6-4\times 10^{10}$ cm-2 respectively, with stable regions corresponding to
approximately $1\times 10^{9}$ and $1.4\times 10^{10}$ cm-2 are found. Since
these densities are much less than typical semiconductor densities and the
stable regions are on the order of the fluctuations of the gate potential, we
simulate more realistic conditions in Fig. S3, with a density of $5\times
10^{11}$cm-2 showing stable regions of width $5\times 10^{10}$ cm-2 which
should be achievable with current experimental platforms.

The superconducting phases are easily tuned during operation of this device.
The size of the phase-biasing loops and the current limits set the phase
resolution and stability. A 10–20 $\mu{\rm m}^{2}$ loop can be fabricated with
ease, is robust to sub-Gauss offsets in the ambient magnetic field, and can be
biased with currents $\leq$ 150 $\mu{\rm A}$ without causing significant Joule
heating. With these parameters, a current resolution of 10 nA corresponds to a
flux precision of $2\times 10^{-4}\phi_{0}$. As shown in Figs. 4, S2, S3, this
precision is more than sufficient to tune stably into the topological regime.

As shown in Fig. 7, conductance tunnel probes can be used to detect MZMs at
the ends of the junction. In many MZM tunneling experiments, the topological
gap is close to the energy resolution of the tunnel probe ($10$ $\mu$eV). The
large gaps in our proposal, 100 $\mu$eV for Nb, should mitigate this issue.
The type-II nature of large-gap superconductors often complicates measurements
due to small lower critical magnetic fields $H_{c1}$ which allow flux
penetration, motion, and trapping during device operation in external magnetic
fields. However, the lack of critical magnetic fields should be an ideal use
case for large-gap superconductors 333Type-II superconductors also have the
disadvantage for phase-based proposals of short-coherence lengths. Here the
relevant coherence length should be that in the semiconductor,
$\zeta_{T}\sim\hbar v_{\rm F}$/$2\pi k_{\rm B}T$, and thus avoid this
problem.. The fact that we do not have an external field decreasing the parent
gap of the superconductor should further improve the MZM visibility during
tunneling experiments to the end of the junction.

## Conclusion

We have shown that MZMs can arise in a phase-controlled planar semiconductor-
superconductor device without applying a Zeeman field. At the heart of our
scheme lies interference between Aharonov-Casher phase (42), stemming from the
spin-orbit interaction, and SC phase winding. As shown exactly by means of the
toy model and then numerically with a realistic model, proper tuning of these
two phases drives the system into a topological superconducting state.

Several important advantages of our proposal are worth mentioning. It enjoys
all the benefits of planar geometry compared to nanowires, in particular
easier fabrication and the ability to generalize to a Majorana network.
Moreover, since the proposal is interference-based rather than Zeeman-based,
various limitations related to magnetic fields are lifted. In the original
planar Josephson junction proposals (37, 36), the Zeeman field required to
make the system topological is of the order of the Thouless energy, which
scales like the inverse of the junction’s width. Therefore making the junction
narrower – which is desirable to achieve more ballistic transport – comes at
the price of applying a stronger magnetic field. Here the magnetic field is
not needed, making the above consideration irrelevant and facilitating the use
of extremely narrow junctions. Furthermore, MZM-related experiments typically
favor the use of Al due to it being a type-I SC that does not trap flux. Our
proposal enables the use of larger-gap type-II SCs, such as Nb, since flux
trapping is no longer a problem, thus opening the door to robust large-gap
MZMs.

The discussion in the previous section shows that our proposal is realistic
under existing experimental technologies. The entire class of large spin-orbit
semiconductors are viable platforms, and importantly no topological materials
are required. As we have shown, the MZM can be localized to within a single
unit cell, whose size is set by the device geometry. The smallest physical
dimension is set by fabrication limitations and the SOC strength (by the
$L\Delta\approx\alpha$ criterion). Thus, in the limit of large SOC, the
maximal topological gap can coincide with an extremely short localization
length.

In the future, it will be interesting to explore possible extensions of the
design principles we suggest and investigate here in quasi one-dimensional
systems to two dimensions, e.g., via a coupled-wire construction. This is
expected to lead to the formation of a two-dimensional chiral superconducting
state.

We are grateful to A. Akhmerov, A. Melo, and B. A. Bernevig for illuminating
discussions. The work at Weizmann was supported by the European Union’s
Horizon 2020 research and innovation programme (Grant Agreement LEGOTOP No.
788715), the DFG (CRC/Transregio 183, EI 519/7-1), ISF Quantum Science and
Technology (2074/19), the BSF and NSF (2018643). A.S. gratefully acknowledges
support by a National Science Foundation Graduate Research Fellowship
(DGE-1745303). M.W. is supported by the DoD NDSEG Fellowship. A.Y., A.S., and
M.W. are supported by the Quantum Science Center (QSC), a National Quantum
Information Science Research Center of the U.S. Department of Energy (DOE),
the Gordon and Betty Moore Foundation through Grant GBMF 9468, the National
Science Foundation under Grant No. DMR-1708688, and the STC Center for
Integrated Quantum Materials, NSF Grant No. DMR-1231319.

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## Supporting Information

###  S.I Full analysis of the sweet spot

Here we show the complete derivation of the conditions for a zero-energy state
in an $N$-sites ring, find their wavefunctions, and calculate the coupling
between zero modes of adjacent rings.

We begin from the Bloch Hamiltonian describing the ring, (2) of the main text.
Labeling
$\epsilon_{k\sigma}=-\mu+2t\cos\left(k+\frac{\varphi}{2}+\lambda\sigma\right)$,
we write the BdG Hamiltonian in matrix form:

$\displaystyle
H=\frac{1}{2}\sum_{k}\vec{\Psi}_{k}^{\dagger}\mathcal{H}\left(k\right)\vec{\Psi}_{k},$
(S1)
$\displaystyle\mathcal{H}\left(k\right)=\begin{pmatrix}\epsilon_{k\uparrow}&0&\Delta&0\\\
0&\epsilon_{k\downarrow}&0&\Delta\\\ \Delta&0&-\epsilon_{-k\downarrow}&0\\\
0&\Delta&0&-\epsilon_{-k\uparrow}\end{pmatrix},$ (S2)

where the Nambu spinor is
$\vec{\Psi}_{k}=\left(c_{k\uparrow},c_{k\downarrow},c_{-k\downarrow}^{\dagger},-c_{-k\uparrow}^{\dagger}\right)^{\rm
T}$. The condition for a zero-energy state is

$\det\mathcal{H}\left(k\right)=\left(\Delta^{2}+\epsilon_{k\uparrow}\epsilon_{-k\downarrow}\right)\left(\Delta^{2}+\epsilon_{k\downarrow}\epsilon_{-k\uparrow}\right)=0,$
(S3)

for some value of $k$. Choosing a specific pair $\left(k,\sigma\right)$, there
are many choices of parameters fulfilling the requirement
$\Delta^{2}+\epsilon_{k\sigma}\epsilon_{-k,-\sigma}=0$. For convenience, let
us consider the simple choice
$\epsilon_{k\sigma}=-\epsilon_{-k,-\sigma}=\Delta$:

$-\mu+2t\cos\left(k+\lambda\sigma+\frac{\varphi}{2}\right)=\Delta$ (S4a)
$-\mu+2t\cos\left(k+\lambda\sigma-\frac{\varphi}{2}\right)=-\Delta.$ (S4b)

Adding the two equations, we get an expression for the chemical potential,

$\mu=2t\cos\left(\frac{\varphi}{2}\right)\sqrt{1-\left(\frac{\Delta}{2t\sin\left(\frac{\varphi}{2}\right)}\right)^{2}}.$
(S5)

For simplicity let us set $\mu=0$, thus requiring $\Delta=\pm
2t\sin\left(\frac{\varphi}{2}\right)$.

As in the main text we focus on $N=3$, so the two non-trivial choices for
$\varphi$ are $\pm\frac{2\pi}{3}$. We therefore obtain $\Delta=\pm\sqrt{3}t$.
We still need to tune the SOC parameter $\lambda$ in order to guarantee the
existence of a zero-energy state. This is done by explicitly writing

$\epsilon_{k\sigma}=\Delta\ \Rightarrow\
\cos\left(k+\lambda\sigma+\frac{\varphi}{2}\right)=\frac{\Delta}{2t}=\frac{\sqrt{3}}{2}=\cos\left(\frac{\pi}{6}\right).$
(S6)

Focusing on the case of a single vortex $\varphi=\frac{2\pi}{3}$ and choosing
$k=0$, $\sigma=\uparrow$ for our zero-energy state, we find
$\lambda=-\frac{\pi}{2}$ (up to integer multiples of $2\pi$).

We have ensured the existence of a zero-energy state in the ring. To find its
wavefunction, we need to solve the equation

$\begin{pmatrix}\epsilon_{k,\sigma}&\Delta\\\
\Delta&-\epsilon_{-k-\sigma}\end{pmatrix}\begin{pmatrix}c_{k\sigma}\\\ \sigma
c_{-k-\sigma}^{\dagger}\end{pmatrix}=0.$ (S7)

But since we set $\epsilon_{k\sigma}=-\epsilon_{-k,-\sigma}=\Delta$, the
equation takes the simple form

$\Delta\begin{pmatrix}1&1\\\ 1&1\end{pmatrix}\begin{pmatrix}c_{k\sigma}\\\
\sigma c_{-k-\sigma}^{\dagger}\end{pmatrix}=0.$ (S8)

The eigen-energies are clearly $2\Delta$ (for the symmetric combination) and
$0$ (for the anti-symmetric combination). The zero-energy state’s wavefunction
is

$\psi_{0\uparrow}=\frac{1}{\sqrt{2}}\left(c_{0\uparrow}-c_{0\downarrow}^{\dagger}\right).$
(S9)

Having found a zero-energy state (and its particle-hole partner), we can
construct Majorana wavefunctions:

$\alpha=\frac{1}{\sqrt{2}}\left(\psi_{0\uparrow}+\psi_{0\uparrow}^{\dagger}\right)=\frac{1}{2}\left(c_{0\uparrow}-c_{0\downarrow}^{\dagger}\right)+\text{H.c.}$
(S10a)
$\beta=\frac{i}{\sqrt{2}}\left(\psi_{0\uparrow}-\psi_{0\uparrow}^{\dagger}\right)=\frac{i}{2}\left(c_{0\uparrow}+c_{0\downarrow}^{\dagger}\right)+\text{H.c.}$
(S10b)

We now discuss the coupling of two rings, as shown in Fig. 1B of the main
text. As we have shown, the parameters of each ring can be tuned such that it
hosts a pair of MZMs at a particular $k,\sigma$ combination. Flipping both $k$
and $\sigma$ corresponds to flipping the vorticity,
$\varphi\rightarrow-\varphi$. Let us then choose $k=0$ in both rings, and
$\varphi=\pm\frac{2\pi}{3}$ opposite in the two rings (vortex–anti-vortex
configuration), while keeping $\lambda=-\frac{\pi}{6}$ uniform. This leads to
opposite $\sigma$ values for the MZMs at the two rings. In the real-space
fermionic basis, we take the coupling to be

$V=\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&0\end{pmatrix}\otimes
t^{\prime}e^{i\lambda^{\prime}\sigma_{x}}e^{i\frac{\pi}{3}},$ (S11)

where the global $e^{i\frac{\pi}{3}}$ factor corresponds to a phase bias
between the rings (this phase bias is not an essential ingredient and it is
used here for analytical tractability; numerical simulations show that the
same outcome is found by simply rotating the rings with respect to each
other).

Projecting the coupling matrix to the basis of the MZM wavefunctions, we
obtain the following coupling matrix between the MZMs at the left and right
ring:

$\mathcal{M}=it^{\prime}\begin{pmatrix}0&\cos\lambda^{\prime}+\sqrt{3}\sin\lambda^{\prime}\\\
\cos\lambda^{\prime}-\sqrt{3}\sin\lambda^{\prime}&0\end{pmatrix}.$ (S12)

The rows of $\mathcal{M}$ correspond to $\alpha,\beta$ in ring 1, and its
columns correspond to $\alpha,\beta$ in ring 2, so our choice automatically
eliminates the $\alpha\alpha$, $\beta\beta$ couplings. If we now choose
$\lambda^{\prime}=\lambda=-\frac{\pi}{6}$, i.e., keep the SOC amplitude
_uniform_ throughout the system, we obtain

$\mathcal{M}=\begin{pmatrix}0&0\\\ i\sqrt{3}t^{\prime}&0\end{pmatrix}.$ (S13)

This means that we have a sweet spot, as illustrated in Fig. 1B of the main
text.

So far we specified to the case $N=3$. In the $N=4$ case (or in general any
even $N$), the quantization conditions support $k$ values for which $\sin k=0$
or $\cos k=0$, allowing some more flexibility in tuning the parameters. For
example, if we repeat the calculation from before for $N=4$ with
$k_{1}=-k_{2}=\frac{\pi}{2}$ and $\sigma_{1}=-\sigma_{2}=1$ (which is again a
vortex–anti-vortex configuration; we need to set $\Delta=\sqrt{2}t$ to get
$\mu=0$), we get the following Majorana coupling matrix:

$\mathcal{M}=it^{\prime}\begin{pmatrix}0&\cos\left(\lambda^{\prime}-\frac{\pi}{4}\right)\\\
\cos\left(\lambda^{\prime}+\frac{\pi}{4}\right)&0\end{pmatrix}.$ (S14)

By choosing $\lambda^{\prime}=\frac{\pi}{4}$, we can eliminate one of the off-
diagonal elements and arrive at a sweet spot. In this case, a phase bias of
$\frac{\pi}{4}$ is needed between the rings.

###  S.II Details of the tight-binding simulation

Consider a 2DEG with Rashba spin-orbit coupling:

$H\left(k_{x},k_{y}\right)=\frac{\hbar^{2}}{2m^{*}}\left(k_{x}^{2}+k_{y}^{2}\right)+\hbar\alpha\left(\sigma_{y}k_{x}-\sigma_{x}k_{y}\right)-\mu,$
(S15)

where $m^{*}$ is the effective electron mass, $\alpha$ is the SOC parameter,
and $\mu$ is the chemical potential. At $k_{y}=0$, this Hamiltonian describes
two shifted parabolas intersecting at $k_{x}=0$. The shift in the momentum
gives the SOC momentum and length:

$k_{\text{SO}}=\frac{m^{*}\alpha}{\hbar},\,\ell_{\text{SO}}=\frac{\hbar}{m^{*}\alpha}.$
(S16)

The Fermi momentum for spin $\sigma$ is given by

$k_{\text{F},\sigma}=k_{\text{F}}^{(0)}-\sigma
k_{\text{SO}}=\frac{\sqrt{2m^{*}\mu}}{\hbar}-\frac{\sigma
m^{*}\alpha}{\hbar},$ (S17)

and the Fermi velocity, which is unaffected by the SOC, is $v_{\rm
F}=\sqrt{2\mu/m^{*}}$.

We discretize this continuum model on a lattice of spacing $a$. The resulting
nearest-neighbors tight-binding Hamiltonian is

$\displaystyle H_{\text{TB}}\left(k_{x},k_{y}\right)=$
$\displaystyle-\mu_{\text{TB}}+2t_{\text{TB}}\left[2-\cos\left(k_{x}a\right)-\cos\left(k_{y}a\right)\right]$
(S18)
$\displaystyle+2\alpha_{\text{TB}}\left[\sigma_{y}\sin\left(k_{x}a\right)-\sigma_{x}\sin\left(k_{y}a\right)\right],$

where $t_{\text{TB}}$ is the nearest-neighbors hopping amplitude, and
$\mu_{\rm TB}$, $\alpha_{\rm TB}$ are the tight-binding analogs of $\mu$,
$\alpha$. In order to match the continuum and tight-binding descriptions, we
focus on $k_{y}=0$:

$\displaystyle H_{\text{TB}}\left(k_{x},k_{y}=0\right)$
$\displaystyle=-\mu_{\text{TB}}+2t_{\text{TB}}\left[1-\cos\left(k_{x}a\right)\right]$
(S19) $\displaystyle+2\alpha_{\text{TB}}\sigma_{y}\sin\left(k_{x}a\right).$

Our goal is to correctly describe the physics near the Fermi points, so we
require that the Fermi momenta and the Fermi velocities match in the two
descriptions. Due to the symmetry of the model, it is enough to look at a
single spin species $\sigma=+1$.

The conditions for matching the Fermi momenta are

$\displaystyle\begin{aligned}
\mu=&-\mu_{\text{TB}}+2t_{\text{TB}}\left[1-\cos\left(k_{\text{F}}^{\left(0\right)}a+k_{\text{SO}}a\right)\right]\\\
&+2\alpha_{\text{TB}}\sin\left(k_{\text{F}}^{\left(0\right)}a+k_{\text{SO}}a\right),\end{aligned}$
(S20) $\displaystyle\begin{aligned}
\mu=&-\mu_{\text{TB}}+2t_{\text{TB}}\left[1-\cos\left(-k_{\text{F}}^{\left(0\right)}a+k_{\text{SO}}a\right)\right]\\\
&+2\alpha_{\text{TB}}\sin\left(-k_{\text{F}}^{\left(0\right)}a+k_{\text{SO}}a\right).\end{aligned}$
(S21)

The matching of the Fermi velocities is imposed by (notice the signs)

$\displaystyle\begin{aligned} \frac{\hbar
v_{\text{F}}^{\left(0\right)}}{a}=&2t_{\text{TB}}\sin\left(k_{\text{F}}^{\left(0\right)}a+k_{\text{SO}}a\right)\\\
&+2\alpha_{\text{TB}}\cos\left(k_{\text{F}}^{\left(0\right)}a+k_{\text{SO}}a\right),\end{aligned}$
(S22) $\displaystyle\begin{aligned} -\frac{\hbar
v_{\text{F}}^{\left(0\right)}}{a}=&2t_{\text{TB}}\sin\left(-k_{\text{F}}^{\left(0\right)}a+k_{\text{SO}}a\right)\\\
&+2\alpha_{\text{TB}}\cos\left(-k_{\text{F}}^{\left(0\right)}a+k_{\text{SO}}a\right).\end{aligned}$
(S23)

Let us arrange these four equations in a matrix,

$\begin{gathered}\underbrace{\begin{pmatrix}-1&2\left[1-\cos\left(k_{\text{F}}^{\left(0\right)}a+k_{\text{SO}}a\right)\right]&2\sin\left(k_{\text{F}}^{\left(0\right)}a+k_{\text{SO}}a\right)\\\
-1&2\left[1-\cos\left(-k_{\text{F}}^{\left(0\right)}a+k_{\text{SO}}a\right)\right]&2\sin\left(-k_{\text{F}}^{\left(0\right)}a+k_{\text{SO}}a\right)\\\
0&2\sin\left(k_{\text{F}}^{\left(0\right)}a+k_{\text{SO}}a\right)&2\cos\left(k_{\text{F}}^{\left(0\right)}a+k_{\text{SO}}a\right)\\\
0&2\sin\left(-k_{\text{F}}^{\left(0\right)}a+k_{\text{SO}}a\right)&2\cos\left(-k_{\text{F}}^{\left(0\right)}a+k_{\text{SO}}a\right)\end{pmatrix}}_{A}\\\
\times\begin{pmatrix}\mu_{\text{TB}}\\\ t_{\text{TB}}\\\
\alpha_{\text{TB}}\end{pmatrix}=\underbrace{\begin{pmatrix}\mu\\\ \mu\\\
\frac{\hbar v_{\text{F}}^{\left(0\right)}}{a}\\\ -\frac{\hbar
v_{\text{F}}^{\left(0\right)}}{a}\end{pmatrix}}_{\vec{b}}.\end{gathered}$
(S24)

This set of linear equations is _overdetermined_. The best least-squares
approximate solution is given by (61)

$\begin{pmatrix}\mu_{\text{TB}}\\\ t_{\text{TB}}\\\
\alpha_{\text{TB}}\end{pmatrix}=\left(A^{\text{T}}A\right)^{-1}A^{\text{T}}\vec{b}.$
(S25)

Substituting $A$ and $\vec{b}$ we obtain

$\mu_{\text{TB}}=-\mu+\frac{\hbar
v_{\text{F}}^{\left(0\right)}}{a}\left[\frac{\cos\left(k_{\text{SO}}a\right)}{\sin\left(k_{\text{F}}^{\left(0\right)}a\right)}-\cot\left(k_{\text{F}}^{\left(0\right)}a\right)\right],$
(S26a) $t_{\text{TB}}=\frac{\hbar
v_{\text{F}}^{\left(0\right)}}{2a}\frac{\cos\left(k_{\text{SO}}a\right)}{\sin\left(k_{\text{F}}^{\left(0\right)}a\right)},$
(S26b) $\alpha_{\text{TB}}=-\frac{\hbar
v_{\text{F}}^{\left(0\right)}}{2a}\frac{\sin\left(k_{\text{SO}}a\right)}{\sin\left(k_{\text{F}}^{\left(0\right)}a\right)}.$
(S26c)

Using this prescription, it is possible to match a tight-binding description
to a continuum model, with a large lattice spacing. The lattice spacing is
limited by the requirement
$\left(k_{\text{F}}^{\left(0\right)}+k_{\text{SO}}\right)a<\pi$, such that all
Fermi points lie in the first Brillouin zone. In Fig. S1 we show such a
matching for $a=5\,{\rm nm}$ and $a=8\,{\rm nm}$, where the chemical potential
is such that $a$ must be smaller than $9.7\,{\rm nm}$. It is evident that even
the model with $a=8\,{\rm nm}$, which is quite close to the upper bound on
$a$, captures the continuum spectrum within about $5\,{\rm meV}$ from the
Fermi energy, which is much larger than the scale set by the SC pairing
potential and the typical temperature in relevant experiments.

Figure S1: Continuum band structure of a HgTe 2DEG at $k_{y}=0$ (blue) matched
by tight-binding models with lattice spacings $a=5\,{\rm nm}$ (green) and
$a=8\,{\rm nm}$ (red), according to (S26). The parameters are
$m^{*}=0.038m_{e}$, $\hbar\alpha=200\,{\rm meV\,nm}$, and $\mu=50\,{\rm meV}$,
corresponding to the density $n=7.9\times 10^{11}\text{ cm}^{-2}$. Both models
correctly capture the physics near the four Fermi points, though the
$a=5\,{\rm nm}$ model does so for a larger range of energies.

###  S.III Results for Niobium

In this appendix, we show that Nb can be used as the proximitizing
superconductor. The SC gap of Nb is $\Delta\approx 1\,{\rm meV}$ – much larger
than that of Al. As we show here, the topological gap in our device when using
Nb can be a significant fraction of this gap, making the MZMs even more robust
and localized.

In Fig. S2A we show the phase diagram as a function of $\theta$ and $\phi$,
the analog of Fig. 3 which was calculated for Al. The topological region of
the phase diagram is quite large, and the maximal topological gap is
$\leavevmode\nobreak\ 0.08\Delta$ – quite similar to the fraction obtained for
Al. This shows that when the SOC is strong enough such that the spin-orbit
energy does not limit the topological gap, we can take advantage of the large
proximity-induced gap. Fig. S2B demonstrate the chemical potential dependence
(it is the analog of Fig. 4 which was calculated for Al). This diagram shows
stability over an appreciable range of electron densities, which is
experimentally important. We note that the physical dimensions of the device
should be smaller compared to Al, in line with the general rule
$L\Delta\approx\alpha$ (since $\Delta$ is larger).

Figure S2: Nb ($\Delta=1{\rm meV}$) as the proximitizing SC. (A) Phase diagram
as a function of the SC phase differences $\theta$ and $\phi$ for
$\mu=0.87\,{\rm meV}$, showing a substantial topological region and a large
topological gap. (B) Phase diagram as a function of the chemical potential
$\mu$ and the phase difference $\theta$ for $\phi=\pi$. The SOC parameter is
$\alpha=29\,{\rm meVnm}$, and the geometric parameters are $L_{\rm
SC}=108\,{\rm nm}$, $W_{\rm SC}=30\,{\rm nm}$, $L_{\rm B}=30\,{\rm nm}$,
$W_{\rm B}=48\,{\rm nm}$, $L_{\rm M}=60\,{\rm nm}$, $W_{\rm M}=72\,{\rm nm}$,
$W_{\rm N}=24\,{\rm nm}$. The lattice constant is $a=6\,{\rm nm}$.

Furthermore, to test our proposal under realistic semiconductor densities, we
performed simulations at higher $\mu$ values with Nb. As detailed in Appendix
S.II, this requires increasing the number of lattice points. The resulting
phase diagram is shown in Fig. S3. We find stable topological regions even at
these realistic densities, implying that our proposal is relevant for existing
semiconductor platforms. Notice that the maximum topological gap we obtain is
$0.03\Delta$ – a bit smaller than the gaps of Fig. S2 (though it can be
improved to about $0.045\Delta$ by deviating from $\phi=\pi$). That is because
the larger lattice makes the numerical optimization more computationally
challenging, and therefore we only performed rudimentary optimization to check
the validity of our platform.

Figure S3: High-density phase diagram as a function of the chemical potential
and the phase difference $\theta$ for $\phi=\pi$, as in Figs. 4 and S2B. The
diagram exhibits topological regions over significant ranges of density. To
emphasize the experimental accessibility, we also indicate the densities on
the figure. The SOC parameter is $\alpha=103\,{\rm meVnm}$, and the geometric
parameters are $L_{\rm SC}=320\,{\rm nm}$, $W_{\rm SC}=48\,{\rm nm}$, $L_{\rm
B}=48\,{\rm nm}$, $W_{\rm B}=80\,{\rm nm}$, $L_{\rm M}=240\,{\rm nm}$, $W_{\rm
M}=120\,{\rm nm}$, $W_{\rm N}=40\,{\rm nm}$. The lattice constant is
$a=8\,{\rm nm}$.