# Tunable proximity effects and topological superconductivity in ferromagnetic
hybrid nanowires

Samuel D. Escribano  Departamento de Física Teórica de la Materia Condensada
C5, Condensed Matter Physics Center (IFIMAC) and Instituto Nicolás Cabrera,
Universidad Autónoma de Madrid, E-28049 Madrid, Spain  Elsa Prada  Instituto
de Ciencia de Materiales de Madrid (ICMM), Consejo Superior de Investigaciones
Científicas (CSIC), E-28049 Madrid, Spain  Yuval Oreg  Department of Condensed
Matter Physics, Weizmann Institute of Science, Rehovot, Israel 7610  Alfredo
Levy Yeyati  Departamento de Física Teórica de la Materia Condensada C5,
Condensed Matter Physics Center (IFIMAC) and Instituto Nicolás Cabrera,
Universidad Autónoma de Madrid, E-28049 Madrid, Spain<EMAIL_ADDRESS>
###### Abstract

Hybrid semiconducting nanowire devices combining epitaxial superconductor and
ferromagnetic insulator layers have been recently explored experimentally as
an alternative platform for topological superconductivity at zero applied
magnetic field. In this proof-of-principle work we show that the topological
regime can be reached in actual devices depending on some geometrical
constraints. To this end, we perform numerical simulations of InAs wires in
which we explicitly include the superconducting Al and magnetic EuS shells, as
well as the interaction with the electrostatic environment at a self-
consistent mean-field level. Our calculations show that both the magnetic and
the superconducting proximity effects on the nanowire can be tuned by nearby
gates thanks to their ability to move the wavefunction across the wire
section. We find that the topological phase is achieved in significant
portions of the phase diagram only in configurations where the Al and EuS
layers overlap on some wire facet, due to the rather local direct induced spin
polarization and the appearance of an extra indirect exchange field through
the superconductor. While of obvious relevance for the explanation of recent
experiments, tunable proximity effects are of interest in the broader field of
superconducting spintronics.

Introduction.— Engineering topological superconductivity in hybrid
superconductor/semiconductor nanostructures where Majorana zero modes may be
generated and manipulated has emerged as a great challenge for condensed
matter physics in the last decade Aguado (2017); Lutchyn  _et al._ (2018);
Prada  _et al._ (2020). While Rashba-coupled proximitized semiconducting
nanowires appears as one of the most successful platforms Lutchyn  _et al._
(2010); Oreg  _et al._ (2010), reaching the topological regime in these
devices requires applying large magnetic fields. This turns out to be not only
detrimental to superconductivity, but it also imposes some constraints in the
design of quantum information processing devices Plugge  _et al._ (2017);
Karzig  _et al._ (2017).

Figure 1: Hybrid nanowire geometries. (a,b) Sketches of the devices studied in
this work: a hexagonal cross section InAs nanowire (green) is simultaneously
proximitized by an Al superconductor layer (light grey) and an EuS magnetic
insulator layer (blue). Two side gates and one back gate (dark grey) allow to
tune the chemical potential and control the position of the wavefunction
inside the heterostructure. Different dielectrics are used in the experiments
Vaitiekėnas  _et al._ (2021) to allow gating (SiO2, in purple) and to avoid
the oxidation of the EuS layer (HfO2 and AlO2, in orange and yellow,
respectively). In the overlapping device, (a), the Al and EuS layers overlap
on one facet, while in the non-overlapping device, (b), they are grown on
different facets. (c) Diagram of the conduction and valence-band edge
positions (in red and blue, respectively) across the heterostructure, spanning
the three different materials. In the Al and InAs, the Fermi energy is located
in the conduction band (close to the band bottom in InAs), whereas in the EuS
it is inside its insulating gap. The EuS conduction band is spin splitted
(being $h_{\rm{ex}}$ the exchange coupling). For this simulation we fix all
gate voltages to zero.

Recent experiments Liu  _et al._ (2020a, b); Vaitiekėnas  _et al._ (2021) have
been exploring an alternative route in which an epitaxial layer of a
ferromagnetic insulator is also added to the superconductor/semiconductor
nanowire system. While the idea of replacing the external magnetic field by
the ferromagnetic layer appears as rather straightforward in simplified models
Sau  _et al._ (2010), there are open questions when applied to realistic
systems. Microscopic calculations are required to demonstrate whether or not
the topological regime could be reached for the actual geometrical and
material parameters, as well as gating conditions. Moreover, understanding the
interplay of magnetic and superconducting proximity effects in such devices is
of relevance in the broader field of superconducting spintronics Wolf  _et
al._ (2001); Linder and Robinson (2015) and quantum thermodevices Giazotto
_et al._ (2006); Machon  _et al._ (2013).

To address this problem we perform comprehensive numerical simulations of the
ferromagnetic hybrid nanowire devices (see Fig. 1). Related studies have been
performed concurrently Woods and Stanescu (2020); Maiani  _et al._ (2021); Liu
_et al._ (2020c); Langbehn  _et al._ (2021). We include the interaction with
the electrostatic environment that typically surrounds the hybrid
nanostructures by solving the Schrödinger-Poisson equations self-consistently
in the Thomas-Fermi approximation.

We show that topological superconductivity can indeed arise in these systems
provided that certain geometrical and electrostatic conditions are met. We
find that, for realistic values of the external gates, device layouts where
the Al and EuS layers that partially cover the wire overlap on one facet
develop extended topological regions in parameter space with significant
topological gaps. This is in contrast to devices where the superconducting and
magnetic layers are grown on adjacent facets. This could explain why recent
experiments find zero bias peaks in bias spectroscopy experiments –compatible
in principle with the existence of a Majorana zero mode at the wire’s end
–only in the former geometry but not in the latter.

Concerning the magnetization process, an open issue is whether the spin
polarization is directly induced by the ferromagnet in the semiconducting
nanowire electrons, or indirectly through a more elaborate process where it is
first induced in the superconducting layer (at the regions where the Al and
EuS shells overlap) and then in the wire. For instance, Ref. Vaitiekėnas  _et
al._ , 2021 suggested that the hysteretic behavior found in some devices could
be in agreement with the indirect mechanism. We find that there is indeed an
indirect induced magnetization through the Al layer, but this cannot drive a
topological phase transition by itself. Conversely, there is strong direct
magnetization from the EuS into the InAs, but only over a very thin region
close to the ferromagnet. Interestingly, both mechanisms –direct and indirect
–contribute to achieve robust and sizable topological regions in the phase
diagram.

Finally, as a guide for future experiments, we elucidate the role of external
potential gates in current device layouts. We show that the topological phase
depends critically on the nanowire wavefunction location, a property that can
be controlled by tuning appropriately those gates.

Device geometries and model.— Following closely the experiments of Ref.
Vaitiekėnas  _et al._ , 2021, we consider the two types of device geometries
depicted in Fig. 1(a) and (b). In both cases, a hexagonal cross-section InAs
nanowire is partially covered by epitaxial Al and EuS layers. The main
difference between them is that, in the overlapping device [Fig. 1(a)], the Al
and EuS layers partially overlap on one facet, while in the non-overlapping
one [Fig. 1(b)] they lie on adjacent facets. Various dielectrics surrounding
the hybrid wires are included in our electrostatic simulations although we
find that they play a minor role. Last, there are three gate electrodes used
in the experiments to tune the electrostatic potential inside the devices: one
back-gate and two side-gates. We analyze other geometries in the Supplemental
Material (SM) Sup, .

In this work we address the bulk electronic properties of these hybrid
nanowires, which we assume are translational invariant along the $z$
direction. A schematic band diagram of the three different materials in the
transverse directions, $x,y$, can be seen in Fig. 1(c). The Al layer is a
metal whose conduction band lies at $-11.7$ eV below the Fermi level Ashcroft
and Mermin (1976). Despite the fact that the conduction band of the InAs is
typically at the Fermi level, experimental angle-resolved photoemission
spectroscopy (ARPES) Schuwalow  _et al._ (2019) and scanning tunneling
microscopy (STM) Reiner  _et al._ (2020) measurements on epitaxial Al/InAs
structures show that there is a band offset of $\sim 0.2$ eV between the Al
and the InAs. This imposes an electron doping of the InAs conduction band
close to the Al/InAs interface. On the other hand, soft x-ray ARPES (SX-ARPES)
experiments on the EuS/InAs interface Liu  _et al._ (2020a) indicate that the
InAs conduction band lies well within the EuS band gap, which is of the order
of $1.7$ eV Eastman  _et al._ (1969). In particular, the EuS conduction band
is located 0.7 eV above the Fermi level and the $4f$ valence bands $1$ eV
below Eastman  _et al._ (1969); Liu  _et al._ (2020a). The EuS conduction band
is characterized by an exchange field $h_{\rm ex}$ that shifts the spin-up and
spin-down energies by roughly $\pm 100$ meV Mauger and Godart (1986); Moodera
_et al._ (2007); Alphenaar  _et al._ (2009). In addition to this, and
similarly to the InAs/Al interface, SX-ARPES experiments Liu  _et al._ (2020a)
also revealed a band bending of the order of $0.1$ eV at the InAs/EuS
interface, which imposes a smaller charge accumulation at this junction as
well. All these band alignments are further distorted by the electric fields
defined by the gate electrodes. However, for sufficiently small fields one can
assume that only the InAs conduction band moves and can neglect its
hybridization with the EuS valence bands (see the SM Sup  for further
details).

Under these assumptions, we describe the wires using the following continuous
model Hamiltonian

$\displaystyle H=\left[\vec{k}^{T}\frac{\hbar^{2}}{2m_{\rm
eff}(\vec{r})}\vec{k}-E_{\rm F}(\vec{r})+e\phi(\vec{r})+h_{\rm
ex}(\vec{r})\sigma_{z}\right]\tau_{z}$
$\displaystyle+\frac{1}{2}\left[\vec{\alpha}(\vec{r})\cdot\left(\vec{\sigma}\times\vec{k}\right)+\left(\vec{\sigma}\times\vec{k}\right)\cdot\vec{\alpha}(\vec{r})\right]\tau_{z}+\Delta(\vec{r})\sigma_{y}\tau_{y},\
\ \ $ (1)

where $\vec{r}=(x,y)$, $\vec{k}=(-i\vec{\nabla}_{r},k_{z})$, and
$\sigma_{\alpha}$ and $\tau_{\alpha}$ denote Pauli matrices in spin and Nambu
spaces, respectively. The parameters $m_{\rm eff}$, $E_{\rm F}$, $h_{\rm ex}$
and $\Delta$, corresponding to the effective mass, Fermi energy, exchange
field and superconducting pairing amplitude, are taken differently for each
region according to estimations from the literature, as summarized in Table I
of the SM Sup . To simulate the disordered outer surface of the Al layer and
the irregular EuS/Al interface we introduce a random Gaussian noise in
$E_{F}(\vec{r})$ Sup . The other parameters, i.e., the electrostatic potential
$\phi(\vec{r})$ and the spin-orbit coupling (SOC) inside the wire
$\alpha(\vec{r})$, are determined in a self-consistent way. For this purpose,
we obtain $\phi(\vec{r})$ by solving the Schrödinger-Poisson equations within
the Thomas-Fermi approximation Mikkelsen  _et al._ (2018); Escribano  _et al._
(2019). The SOC $\alpha(\vec{r})$ varies locally with the electric field and
is accurately calculated using the procedure described in Ref. Escribano  _et
al._ , 2020, see also Sup, . Notice that the exchange field does not give rise
to a magnetic orbital term in the Hamiltonian, as opposed to what happens in
wires under an external magnetic field Nijholt and Akhmerov (2016); Kiczek and
Ptok (2017); Dmytruk and Klinovaja (2018); Winkler  _et al._ (2019).

To obtain the electronic properties we diagonalize Eq. (1). To this end, we
discretize it into a tight-binding Hamiltonian using an appropriate mesh,
which is dictated by the Al Fermi wavelength Sup . Notice that a description
of the three material regions (Al, InAs and EuS) on the same footing
constitutes a demanding computational task. In the second part of this work we
build a simplified model in which we integrate out the Al and EuS layers and
include their proximity effects over the InAs wire as position-dependent
effective parameters. This allows us to explore the system’s phase diagram for
a broader range of parameters.

Figure 2: Full model results. (a-c) Spin-resolved partial density of states
(pDOS) for the overlapping device integrated over the InAs wire volume (top
row) and the Al layer volume (bottom row) when (a) the exchange field
$h_{\rm{ex}}$ in the EuS layer and the Rashba SOC $\alpha_{\rm{R}}$ in the
InAs wire are set to zero, (b) only the exchange field is turned on, and (c)
both are present. Red and blue correspond to the pDOS for different spin
orientations along the $z$ axis (wire’s direction). (d) Low-energy band
structure versus $k_{z}$ for the hybrid-wire parameters in (c). The colorbar
represents the relative weight $W$ of a given state in the Al layer (black)
and in the InAs wire (yellow). The wavefunction weight in the EuS layer is
negligible since it is an insulating material. The $\mathbb{Z}_{2}$
topological invariant is $\mathcal{Q}=-1$, signaling a topological phase. We
take here $V_{\rm bg}=-0.95$ V and $V_{\rm sg}^{\rm(L,R)}=0$ V. Other
parameters can be found in Table I of the SM. Figure 3: Same as in Fig. 2 but
for the non-overlapping device. The $\mathbb{Z}_{2}$ topological invariant in
(d) is now $\mathcal{Q}=1$, signaling a trivial phase.

Full model results.— We first focus on the density of states (DOS) and
dispersion relation of the overlapping geometry (see Fig. 2) fixing the side-
gate voltages to zero and the back-gate to $\sim-1$ V. In order to identify
the separate effect of the magnetic and superconducting terms, we perform
three different calculations: in the first one we switch off the exchange
field in the EuS and the Rashba SOC in the InAs [Fig. 2(a)]; then we switch on
$h_{\rm{ex}}$ [Fig. 2(b)] and finally we also connect $\alpha_{\rm{R}}$ [Fig.
2(c)].

In the top panel of Fig. 2(a) we show the partial DOS integrated over the InAs
volume. It exhibits a well-defined induced superconducting gap, although
halved with respect to the $\sim 0.2$ meV gap observed in the DOS integrated
over the Al shell volume, Fig. 2(a) bottom panel. This is in accordance with
what one expects from a conventional superconducting proximity effect Chang
_et al._ (2018); de Moor  _et al._ (2018); Winkler  _et al._ (2019).

Figure 4: Simplified-model results. The Al and EuS layers are integrated out
and their respective effective induced pairing amplitude and exchange field on
the InAs wire are included within the streaked regions shown in the sketches
of (a) the overlapping device and (b) the non-overlapping one. We take
$\Delta^{\rm(Al)}=0.23$ meV and $h_{\rm ex}^{\rm(Al)}=0.06$ meV over a wide
region of 45 nm near the Al interface. $h_{\rm ex}^{\rm(Al)}$ is only present
in (a) where the Al and the EuS are in contact. We include $h_{\rm
ex}^{\rm(EuS)}$ over a thin region of $1$ nm close to the EuS layer. (c)
Topological phase diagram of the overlapping device versus back-gate
potential, $V_{\rm bg}$, and exchange field at the EuS-InAs interface, $h_{\rm
ex}^{\rm(EuS)}$, for $V_{\rm sg}^{\rm(L,R)}=0$. In the topological regions, we
show with colors the expectation value of the induced exchange field (left
panel), the topological minigap (middle panel), and the wavefunction profile
(right panel), all of them for the transverse subband closest to the Fermi
energy. The parameters for which the wavefunctions are plotted are indicated
with arrows. (d) Same as (c) but for the non-overlapping device and fixing
$V_{\rm sg}^{\rm(R)}=2$ V and $V_{\rm sg}^{\rm(L)}=0$. The values of $V_{\rm
sg}^{\rm(L,R)}$ in (c) and (d) are taken to maximize the topological regions
in each case. Other parameters can be found in Table I of the SM Sup . The
extension of the topological phase is very much reduced with respect to (c)
(almost negligible for some subbands) both in the $V_{\rm{bg}}$ and $h_{\rm
ex}^{\rm(EuS)}$ axes. Moreover, in the regions where it is present, the
topological gap is small. In contrast, large topological regions with stronger
minigaps are found for the overlapping device. The reason is twofold: (i) the
wavefunction can be pushed simultaneously close to the Al and EuS layers due
to the electrostatics of the overlapped shells, which increases the
superconducting and magnetic proximity effects; (ii) the induced exchange
field feeds both from the direct and indirect contributions in this case.

When $h_{\rm{ex}}\neq 0$ (but $\alpha_{\rm{R}}=0$) we observe two main
features in the spin-resolved partial DOS. First, an energy splitting of the
superconducting coherence peak appears in the Al [Fig. 2(b) bottom panel],
which is of the order of $\sim 0.06$ meV, in agreement with recent theoretical
and experimental results on Al/EuS junctions Hao  _et al._ (1991); Strambini
_et al._ (2017); Rouco  _et al._ (2019); Zhang  _et al._ (2020). This
agreement without any fine tuning of the parameters in our model is
encouraging about its validity. Second, there is a complete closing of the
induced gap in the InAs [Fig. 2(b) top panel]. This points to an induced
exchange field larger than $\sim 0.1$ meV, the induced gap in the
semiconductor, and therefore, larger than in the Al layer. In contrast to
previous proposals Liu  _et al._ (2020b); Vaitiekėnas  _et al._ (2021) our
results suggest that in the current case topological superconductivity is
achieved below the Chandrasekar-Clogston limit Chandrasekhar (1962); Clogston
(1962) for the Al ($h_{\rm ex}^{\rm(Al)}<\Delta/\sqrt{2}$).

Finally, in Fig. 2(c) top panel we observe that a gap is opened again in the
presence of SOC. This sequence of gap closing and reopening at a high-symmetry
$k_{z}$-point is a signature of a topological phase transition. The band
structure shown in Fig. 2(d) further illustrates the spatial distribution of
the low-energy states in this last case (i.e., with $h_{\rm ex}\neq 0$ and
$\alpha_{\rm R}\neq 0$). The weight $W$ of each state in the different
materials is represented with colors, from a state completely located in the
Al layer (black) to one completely located in the InAs wire (yellow). The
lowest-energy states close to $k_{z}=0$ have significant weight both in the Al
and in the InAs, as expected for a topological superconducting phase Winkler
_et al._ (2019). We prove that the system in Figs. 2(c) and 2(d) is indeed in
the topological regime by calculating the $\mathbb{Z}_{2}$ topological
invariant $\mathcal{Q}$. For large Hamiltonian matrices, this can be achieved
by computing the Chern number from the eigenvalues of the Wilson matrix, which
only involves the lowest-energy eigenstates at the Brillouin zone borders Sup
. We find $\mathcal{Q}=-1$, which actually corresponds to the nontrivial case.

Strikingly, the same analysis for the non-overlapping geometry (Fig. 3)
reveals that the magnetic proximity effect in this case is not strong enough
to close and reopen the superconducting gap in the wire. The reason for this
behavior can be traced to the limited spin polarization induced in the
nanowire for this geometry. Hence, there is no topological phase in this case,
at least for this choice of gate voltages.

Simplified model and phase diagram.— We consider now the Hamiltonian of Eq.
(1) restricted to the InAs wire, where we include an effective pairing
amplitude $\Delta^{(\rm{Al})}=0.23$ meV and an exchange field $h_{\rm
ex}^{(\rm{EuS})}=100$ meV on the cross-section regions closer to the Al and
the EuS shells, respectively, as schematically depicted in Figs. 4(a) and (b).
We also include a smaller exchange coupling $h_{\rm ex}^{(\rm{Al})}=0.06$ meV
in the Al-proximitized region of the overlapping device. The magnitude of
these parameters and the extension of the corresponding regions are extracted
by adjusting to the behavior of the full model results, as shown in the SM Sup
.

In Fig. 4(c) we present the topological phase diagram of the overlapping
device as a function of the back-gate voltage and the exchange field of the
EuS. Notice that $h_{\rm ex}^{\rm(EuS)}$ should be $100$ meV according to our
full model. However, departures from the idealized model of Eq. (1) might
reduce the value of the induced magnetic exchange. For instance, the mismatch
between the minima of the InAs conduction band (at the $\Gamma$ point) and the
EuS one (at the $\mathrm{X}$ point Nirpendra  _et al._ (2007)) could suppress
their hybridization depending on material growth directions or other details
(see the SM Sup ), leading to a smaller $h_{\rm ex}^{\rm(EuS)}$ value. Thus we
allow this parameter to vary between $0$ and $100$ meV to evaluate the
robustness of our results with respect to this value. With colors, in the left
panel of Fig. 4(c) we show the induced exchange coupling, $h_{\rm
ex}^{\rm(ind)}=\left<h_{\rm ex}(\vec{r})\sigma_{z}\right>$ 111Note that the
induced exchange coupling in the wire, $h_{\rm ex}^{\rm(ind)}$, is much
smaller (approximately one per cent Sup ) than the EuS parent Zeeman field,
$h_{\rm ex}^{(\rm{EuS})}$., and in the middle panel the induced minigap,
$\Delta_{\rm min}=\left|E(k_{z}=k_{\rm F})\right|$, for the energy state
closest to the Fermi energy in both cases. In these plots, white means trivial
(i.e., $\mathcal{Q}=1$), while the colored regions correspond to the
topological phase. There are several topological regions against $V_{\rm bg}$
corresponding to different transverse subbands. In those regions, the
condition that $h_{\rm ex}^{\rm(ind)}$ is larger than the square root of the
induced gap squared plus the chemical potential squared is fulfilled, as
expected Lutchyn  _et al._ (2010); Oreg  _et al._ (2010). To the right in Fig.
4(c) we show the probability density of the transverse subband closer to the
Fermi level at $k_{z}=0$ across the wire section for the parameters indicated
with arrows. In the three cases exhibited, the wavefunction concentrates both
around the left-upper facet covered by Al, and the top facet where the Al and
EuS layers overlap. This is consistent with the requirement of maximizing
simultaneously the superconducting and magnetic proximity effects.

The phase diagram for the non-overlapping device is shown in Fig. 4(d). The
extension of the topological phase is very much reduced, almost negligible for
some subbands. It is interesting to observe that, for realistic gate potential
values, the wavefunction needs to be very spread across the wire section in
order to acquire the superconducting and magnetic correlations for the
topological phase to develop. This in turn translates into narrow back-gate
voltage ranges for which this is possible and small topological minigaps.

In the SM Sup  we further analyze the previous phase diagrams as a function of
the right-side gate voltage, obtaining similar results. We also consider
alternative geometries, nevertheless finding that the overlapping
configuration of Fig. 1(a) gives rise to more extended, robust and tunable
topological regions for realistic parameters. In particular, we find that the
best way to optimize the topological state (i.e., increase its minigap) is by
fixing a large negative back-gate potential and a small positive right-side
gate potential. In doing so, the wavefunction is pushed towards the
superconductor-ferromagnet corner of the wire and thus the superconducting and
magnetic proximity effects are maximized.

Conclusions.— From calculations of the DOS, band structure, topological
invariant and the phase diagram, we conclude that the hybrid InAs/Al/EuS
nanowires studied in Ref. Vaitiekėnas  _et al._ , 2021 can exhibit topological
superconductivity under certain geometrical and gating conditions. For a
topological phase to exist, the nanowire wavefunction must acquire both
superconducting and magnetic correlations such that the induced exchange field
exceeds the induced pairing. Since the proximity effects occur only in wire
cross-section regions close to the Al and EuS layers, the wavefunction needs
to be pushed simultaneously close to both materials by means of nearby gates.
Our numerical simulations demonstrate that this is electrostatically favorable
in device geometries where the Al and EuS shells overlap over some wire facet.
This configuration is further advantageous in that, apart from a direct
magnetization from the EuS layer in contact to the wire, there is an indirect
one through the Al layer, which favors reaching the topological condition.
While our model includes the effect of disorder at the Al layer surface and at
the EuS/Al interface, we have not considered other sources of disorder, like
,e.g., the presence of magnetic domains. However, these domains can be aligned
by the application of a small field that is then switched to zero, as done in
Ref. Vaitiekėnas  _et al._ (2021). A subsequent study which considers a fully
diffusive Al layer Khindanov  _et al._ (2021) reaches similar conclusions as
our work (although disorder increases the induced exchange field required to
achieve a topological phase).

Finally, as a side outcome, our microscopic analysis demonstrates the
tunability of the magnetic and superconducting induced couplings in the
nanowire. This opens up the possibility of engineering the material
geometries, their disposition, and the electrostatic environment to enter and
abandon the topological regime at will and, thus, the appearance of Majorana
modes. These ideas can be applied to other materials and experimental
arrangements, which surely will favor more experiments in the field, as well
as in other fields such as superconducting spintronics.

Acknowledgments.— We would like to thank S. Vaitiekėnas and C. M. Marcus for
illuminating discussions on their experimental results. We also thank M.
Alvarado and O. Lesser for valuable help with the topological invariant
calculation. A.L.Y. and E.P. acknowledge support from the Spanish MICINN
through Grants No. FIS2016-80434-P (AEI/FEDER, EU) and No. FIS2017-84860-R, by
EU through Grant No. 828948 (AndQC), and through the “María de Maeztu”
Programme for Units of Excellence in R&D (Grant No. MDM-2014-0377). Y.O.
acknowledges partial support through the ERC under the European Union’s
Horizon 2020 research and innovation programme (Grant Agreement LEGOTOP No.
788715), the ISF Quantum Science and Technology (2074/19), the BSF and NSF
(2018643), and the CRC/Transregio 183.

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## Supplemental Material

## Numerical implementation

In this section we describe the specific details of the two geometries
(overlapping and non-overlapping) considered in the main text, as well as
their characteristic material parameters. In addition, we explain how we
perform the numerical calculations for their electronic structure including
the effect of the electrostatic potential.

### 1\. Geometry

A sketch of the two devices analyzed in this work is shown in Fig. 1 of the
main text. In both cases a hexagonal cross-section InAs nanowire (in green) of
radius $60$ nm is partially covered by two different thin layers. Both are
grown epitaxially over certain facets of the wire. The highly transparent
epitaxial interfaces improve the hybridization between the nanowire and the
layers. One of these layers is a 6 nm-thick shell made of superconducting Al
(light grey), whose outer surface (2 nm-thick) is oxidized. The other layer is
made of EuS (blue), a well-known ferromagnetic insulator, and its width is 8
nm in the overlapping device and 5 nm in the non-overlapping one. Despite the
difference between these two values, we find that the precise thickness of the
EuS, for the ferromagnetic exchange coupling mechanism studied here, plays a
negligible role because it acts as a mere (magnetic) insulator. We note that,
in the overlapping device, the interface between the Al and the EuS is also
grown epitaxially in the experiments but the interface is not completely flat.
This roughness is characterized by a corrugation width of $\sim 1$ nm.

This hybrid system is surrounded by three gates (dark grey), one at the bottom
and two at the left and right sides, which allow to tune the wire doping level
and the charge distribution inside the heterostructure simultaneously. The
side gates are placed symmetrically from the center of the wire at $30$ nm
distance from the outer corners of the wire, and their height is half the
wire’s width. On the other hand, the wire is placed on top of a SiO2
dielectric substrate (violet) with a thickness of $200$ nm. This substrate
fixes the distance between the back gate and the bottom of the wire.

There are other dielectrics in the devices that are necessary to avoid the
unwanted oxidation of the EuS layer. They are a 8 nm-thick layer of HfO2
(orange) surrounding the whole heterostructure in the overlpaping device, and
a $10$ nm-thick layer of AlO2 (yellow) covering the EuS layer in the non-
overlapping one. We include them in our electrostatic simulations to obtain a
better qualitative agreement with the experiment of Ref. Vaitiekėnas  _et al._
, 2021 although they play a minor role in our results. Finally, we assume that
the rest of the environment (white) is vacuum. We summarize all these
geometrical parameters in Table 1.

Table 1: Parameters used for the realistic calculations of the main text. All widths are extracted from Ref. Vaitiekėnas  _et al._ , 2021. Temperature is fixed to $100$ mK in all our simulations. Material |  Parameter |  Value |  Refs.
---|---|---|---
InAs |  width |  120 nm |  -
|  height |  104 nm |
|  $m_{\rm eff}$ |  0.023$m_{0}$ |  Vurgaftman  _et al._ ,2001
|  $E_{\rm F}$ |  0 |  Liu  _et al._ ,2020a
|  $h_{\rm ex}$ |  0 |  -
|  $\Delta$ |  0 |  -
|  $\alpha$ | (see text for details)
|  $\epsilon_{\rm InAs}$ |  15.5$\epsilon_{0}$ |  Levinshtein  _et al._ ,2000
|  $\rho_{\rm surf}$ |  $2\cdot 10^{-3}\mathrm{\left(\frac{e}{nm^{3}}\right)}$ |  Thelander  _et al._ , 2010, Reiner  _et al._ ,2020
Al |  width |  6 nm |  -
|  oxidation width |  2 nm |
|  $m_{\rm eff}$ |  $m_{0}$ |  Segall,1961
|  $E_{\rm F}$ |  -11.7 eV |  Ashcroft and Mermin,1976
|  $h_{\rm ex}$ |  0 |  -
|  $\Delta$ |  0.23 meV |  Chang  _et al._ ,2018
|  $\alpha$ |  0 |  Chiang and Dzyuba,2017
|  $\epsilon_{\rm Al}$ |  $\infty$ |  -
|  $V_{\rm Al}$ |  0.2 eV |  Reiner  _et al._ , 2020, Schuwalow  _et al._ , 2019, Liu  _et al._ ,2020c
EuS |  width222Note that in the experiments, the width of the EuS layer is $8$ nm in the overlapping device while it is $5$ nm in the non-overlapping one. |  8/5 nm |  -
|  corrugation width |  1nm |  Liu  _et al._ ,2020b
|  $m_{\rm eff}$ |  0.3$m_{0}$ |  Xavier,1967
|  $E_{\rm F}$ |  0.8 eV |  Alphenaar  _et al._ , 2009, Liu  _et al._ ,2020a
|  $h_{\rm ex}$ |  0.1 eV |  Mauger and Godart, 1986, Alphenaar  _et al._ ,2009
|  $\Delta$ |  0 |  -
|  $\alpha$333We have found no explicit information about the SO coupling in EuS materials. We therefore assume that it is negligibly small. |  0 |  -
|  $\epsilon_{\rm EuS}$ |  10$\epsilon_{0}$ |  Axe,1969
Dielectrics |  SiO2 width |  200 nm |  -
|  HfO2 width |  80 nm |
|  Al2O3 width |  8 nm |
|  $\epsilon_{\rm SiO_{2}}$ |  3.9$\epsilon_{0}$ |  Robertson,2004
|  $\epsilon_{\rm HfO_{2}}$ |  25$\epsilon_{0}$ |  Levinshtein  _et al._ ,2000
|  $\epsilon_{\rm Al_{2}O_{3}}$ |  10$\epsilon_{0}$ |  Levinshtein  _et al._ ,2000
|  $\epsilon_{\rm vacuum}$ |  $\epsilon_{0}$ |  -

### 2\. Model Hamiltonian

The basic arguments leading to our continuum model Hamiltonian have been
presented in the main text. As explained there, we focus on the InAs, Al and
EuS conduction bands and neglect the InAs and EuS valence bands. Notice that
the EuS conduction band has its minimum at the $\mathrm{X}$-point Nirpendra
_et al._ (2007) rather than at $\Gamma$, as InAs and Al do. When electrons
tunnel from one material to the other, for instance from InAs to EuS, this
could lead to momentum mismatch depending on the growth direction between both
materials and thus to a decreased coupling. Moreover, the EuS could exhibit
grains (or other forms of disorder), details which are beyond the qualitative
analysis in this work. For this reason, we consider an effective continuum
model for the different materials where the EuS conduction band minimum is
also taken at the $\Gamma$-point. We note that other sources of momentum
mismatch coming from different effective masses, finite size materials,
certain sources of disorder explained below, etc., are taken into account.
This model, which we call full model in the main text, gives an optimistic
hybridization between EuS and InAs. Since the boundary between the EuS and Al
is disordered (see below), our model essentially describes correctly this
hybridization as momentum conservation along that interface breaks under such
disorder.

Under the assumptions presented above, we arrive to the following Bogoliubov-
de Gennes Hamiltonian to describe the heterostructure (Eq. (1) in the main
text)

$\displaystyle H$ $\displaystyle=$
$\displaystyle\left[\vec{k}^{T}\frac{\hbar^{2}}{2m_{\rm
eff}(\vec{r})}\vec{k}+e\phi(\vec{r})-E_{\rm
F}(\vec{r})\right]\sigma_{0}\tau_{z}$ (2) $\displaystyle+$ $\displaystyle
h_{\rm ex}(\vec{r})\sigma_{z}\tau_{z}+\Delta(\vec{r})\sigma_{y}\tau_{y}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\left[\vec{\alpha}(\vec{r})\cdot\left(\vec{\sigma}\times\vec{k}\right)+\left(\vec{\sigma}\times\vec{k}\right)\cdot\vec{\alpha}(\vec{r})\right]\tau_{z},$

where $\vec{k}=(k_{x},k_{y},k_{z})$ is the momentum operator vector,
$\vec{r}=(x,y)$ is the radial spatial operator vector (for a translational-
invariant wire along $z$), and $\sigma$ and $\tau$ are the Pauli matrices in
spin and Nambu space, respectively. The first term in Eq. (2) is the kinetic
energy, where $m_{\rm eff}(\vec{r})$ is the effective mass that takes
different constant values inside the different parts of the heterostructure.
The function $\phi(\vec{r})$ is the spatial-dependent electrostatic potential
across the section of the hybrid nanowire. In the following subsection we
explain in detail how we compute it. The quantity $E_{\rm F}(\vec{r})$ is the
energy difference at $\phi=0$ between the band bottom of the conduction band
(in each region) and the Fermi level of the entire heterostructure, which we
fix to zero in this work. To simulate the disordered outer surface of the Al
(the oxidized region) and the irregular EuS/Al interface, we assume that
$E_{F}(\vec{r})$ in these disordered regions is characterized by a random
Gaussian noise with a standard deviation of 10% of the Fermi energy in the
material. This accidental disorder breaks the parallel momentum conservation
leading to an enhanced hybridization between the wire’s and the layers’
electronic states Winkler  _et al._ (2019). The operators $h_{\rm
ex}(\vec{r})$ and $\Delta(\vec{r})$ are the exchange field and the
superconducting pairing amplitude, which take non-zero values only in the EuS
and Al layers, respectively. Finally, the last term in Eq. (2) is the spin-
orbit (SO) interaction, where $\alpha$ is the so-called SO coupling. In the
last subsection we discuss separately how we describe it for the whole
heterostructure. The different values for the parameters that we use in our
simulations, together with the references from where they are extracted, are
shown in Table 1.

We perform a numerical diagonalization of the Hamiltonian (2) to obtain the
eigenspectrum. To this end, we discretize this continuum model into a tight-
binding Hamiltonian. We perform the discretization only in the $x$ and $y$
directions (the section of the wire), and we assume that in the other
direction (the wire’s direction $z$) the nanowire is translationally
invariant. In the directions $x$ and $y$ the momentum must be substituted by
their corresponding partial derivatives, i.e.,
$k_{x}\rightarrow-i\partial_{x}$ and $k_{y}\rightarrow-i\partial_{y}$, and
then the partial derivatives can be discretized using the central difference
method, e.g.

$k_{x}\rightarrow-i\frac{\partial}{\partial
x}\rightarrow-i\frac{c_{i+1}^{\dagger}c_{i}-c_{i-1}^{\dagger}c_{i}}{2h},$ (3)

where $h$ is the mesh spacing. We choose $h$ such that the grid can
accommodate the Fermi wavelength of the eigenstates. Otherwise, a poor spacing
would lead to spurious solutions. As Al is a metal with a large Fermi energy,
it is characterized by a large Fermi momentum, which implies that a small mesh
spacing must be used. We find that $h\sim 0.1$ nm is a reasonable mesh size in
our simulations. Once the Hamiltonian is transformed into a tight-binding one,
we diagonalize it numerically (for each $k_{z}$ and $\phi(\vec{r})$ values)
using the standard ARPACK tools provided by the package Scipy of Python. This
gives as a result the energy subbands $E^{(j)}(k_{z})$ for the different
transverse subbands index $j$, and their corresponding wavefunctions
$\Psi^{(j)}(x,y,k_{z})$, which are actually four-component spinors. We perform
this diagonalization using the Python package MajoranaNanowires Escribano
(2020).

As the three materials that comprise the hybrid wire are coupled, the energy
subbands correspond to wavefunctions that spread all across the
heterostructure. To characterize the weight $W_{R}$ of the wavefunction on a
certain material $R$, we project the probability density of any state $j$ onto
this region,

$W_{R}^{(j)}(k_{z})=\int_{\vec{r}\in
R}dx\;dy\left|\Psi^{(j)}(x,y,k_{z})\right|^{2}.$ (4)

In close relation to this quantity, we also compute the spin-polarized partial
density of states (pDOS), i.e., the DOS integrated over different regions $R$,
using the following expression

$\displaystyle\mathrm{pDOS_{\uparrow,\downarrow}^{(R)}}(E)=\sum_{j}\int_{\vec{r}\in
R}dx\;dy\int_{k_{z}}dk_{z}$
$\displaystyle\cdot\delta(E-E^{(j)}(k_{z}))\left|\Psi^{(j)}_{\uparrow,\downarrow}(x,y,k_{z})\right|^{2}.\>\>$
(5)

If $R$ includes the three materials in this equation, then the resulting
expression is the total spin-polarized $\mathrm{DOS}_{\uparrow,\downarrow}$ in
the device. Then, one can compute the total DOS by summing both spin
directions, $\mathrm{DOS_{T}=DOS_{\uparrow}+DOS_{\downarrow}}$.

Lastly, to characterize the $D$-class topological phase Altland and Zirnbauer
(1997); Schnyder  _et al._ (2008) (to which the Majorana nanowires belong), we
compute the Chern number

$C=\frac{1}{2\pi
i}\int_{R_{k_{z}}}dk_{z}\left<\nabla_{k_{z}}\Phi_{k_{z}}\left|\times\right|\nabla_{k_{z}}\Phi_{k_{z}}\right>,$
(6)

where $R_{k_{z}}$ denotes the first Brillouin zone ($k_{z}\in(-\pi,\pi]$ in
our system), and $\Phi_{k_{z}}$ is the ground state of the system, i.e., the
Slater determinant of all the $j$-th eigenstates whose energies $E^{(j)}$ are
below the Fermi energy. To easily compute numerically the Chern number we
perform some transformations following the ideas of Refs. Loring and Hastings,
2010; Zhang  _et al._ , 2013; Lesser and Oreg, 2020. First, we apply the
Stokes theorem

$C=\frac{1}{2\pi i}\oint_{\partial
R_{k_{z}}}dk_{z}\left<\Phi_{k_{z}}|\nabla_{k_{z}}\Phi_{k_{z}}\right>,$ (7)

in such a way that the (line) integral is now performed through a closed
contour around the reciprocal unit cell. Then, we discretize the momentum
space $k_{z}\rightarrow k_{l}$, what allows us to replace the derivatives by
finite differences and the integral by a summation over the reciprocal space
points

$\displaystyle C=\frac{1}{2\pi
i}\sum_{l}\frac{\left<\Phi_{k_{l}}|\Phi_{k_{l+1}}\right>-\left<\Phi_{k_{l}}|\Phi_{k_{l-1}}\right>}{2(k_{l+1}-k_{l})}=$
$\displaystyle\frac{1}{2\pi}\mathrm{Arg}\left\\{\prod_{l}\left<\Phi_{k_{l}}|\Phi_{k_{l+1}}\right>\right\\},$
(8)

where
$\mathrm{Arg}\left\\{z\right\\}\equiv\Im\left\\{\log\left(\frac{z}{|z|}\right)\right\\}$
is the complex argument function of a complex number $z$. The product of
Slater determinants $\left<\Phi_{k_{l}}|\Phi_{k_{l+1}}\right>$ in the above
equation can be rewritten as a determinant of a matrix
$\tilde{C}_{k_{l},k_{l+1}}$, whose matrix element $(i,j)$ can be directly
written in terms of the eigenfunctions of the Hamiltonian,

$\tilde{C}_{k_{l},k_{l+1}}^{i,j}=\left<\Psi^{(i)}(k_{l})|\Psi^{(j)}(k_{l+1})\right>.$
(9)

From this, the Chern number is simply given by

$\displaystyle
C=\frac{1}{2\pi}\mathrm{Arg}\left\\{\det\left(\prod_{l}\tilde{C}_{k_{l},k_{l+1}}\right)\right\\}\equiv$
$\displaystyle\frac{1}{2\pi}\mathrm{Arg}\left\\{\det(\mathcal{W})\right\\},$
(10)

where $\mathcal{W}$ is the so-called Wilson matrix in the literature Yu  _et
al._ (2011); Alexandradinata  _et al._ (2016); Bradlyn  _et al._ (2019).
Finally, we use that the determinant of any square matrix can be written as
the product of its eigenvalues to rewrite the expression as

$C=\frac{1}{2\pi}\sum_{l}\mathrm{Arg}\left\\{\lambda_{l}\right\\},$ (11)

where $\lambda_{l}$ are all the eigenvalues of the Wilson matrix
$\mathcal{W}$. In practice, it is enough to include only the high-symmetry
$k$-points in the product inside the Wilson matrix. In our case, these points
are $k_{z}=\left\\{-\pi,0,\pi\right\\}$, which gives as a result the Wilson
matrix

$\mathcal{W}=\tilde{C}_{-\pi,0}\tilde{C}_{0,\pi}\tilde{C}_{\pi,-\pi}.$ (12)

Additionally, notice that in principle the matrix $\tilde{C}_{k_{l},k_{l+1}}$
involves all the eigenstates of the Hamiltonian (below the Fermi level) at two
different $k_{z}$ points, $k_{l}$ and $k_{l+1}$, as shown in Eq. (9). But
since only the non-trivial topological eigenstates provide a non-zero
contribution to the Chern number, and these states can only emerge close to
the Fermi level in the studied system, it is therefore enough to include the
closest states to the Fermi level in the Wilson matrix. This assumption allows
to significantly alleviate the computational effort in our calculations.

To summarize this last part, the procedure that we follow in order to compute
the topological invariant is the following: We first calculate the eigenstates
$\Psi^{(j)}(k_{z})$ of the Hamiltonian at $k_{z}=\left\\{0,\pi,-\pi\right\\}$.
We need only those $j$-th states that are below and close to the Fermi energy.
We find that $200$ eigenstates are enough in our system to properly
characterize the topological phase. We then compute the coupling matrices
$\tilde{C}_{k_{l},k_{l+1}}$ of Eq. (9) and, from them, the Wilson matrix
$\mathcal{W}$ using Eq. (12). We calculate its eigenvalues $\lambda_{i}$, and
from them we compute the Chern number using Eq. (11), which is a positive
integer. We finally identify the $\mathbb{Z}_{2}$ topological invariant

$\mathcal{Q}=(-1)^{C},$ (13)

which is positive (+1) for the trivial topological phase and negative (-1) for
the non-trivial one.

### 3\. Electrostatic potential

We obtain the electrostatic potential $\phi(\vec{r})$ by solving the Poisson
equation across the section of the wire,

$\vec{\nabla}\left(\epsilon(\vec{r})\vec{\nabla}\phi(\vec{r})\right)=-\rho(\vec{r}),$
(14)

where $\epsilon(\vec{r})$ is the dielectric permitivity, and $\rho(\vec{r})$
is the charge inside the heterostructure. On the one hand, $\epsilon(\vec{r})$
is characterized by a different constant value inside each material (see Table
1), leading to abrupt jumps at the interfaces. Note that the Poisson equation
must be solved on the entire system, including the dielectric environment. On
the other hand, the charge inside the hybrid nanowire is given by,

$\rho(\vec{r})=\rho_{\mathrm{surf}}(\vec{r})+\rho_{\mathrm{mobile}}(\vec{r}).$
(15)

The term $\rho_{\mathrm{surf}}$ models the surface charge that is typically
present in the uncovered nanowire facets Thelander  _et al._ (2010); Reiner
_et al._ (2020). We model it as a positive charge-density layer of thickness 1
nm in every nanowire facet not covered by the Al Winkler  _et al._ (2019);
Escribano  _et al._ (2020). This leads to a band bending at these facets and,
thus, to an accumulation of charge in the wire in those regions [see Fig.
1(c)]. In principle, the wire facets covered by EuS should not display this
type of charge accumulation layer because the EuS is grown epitaxially.
However, recent experiments Liu  _et al._ (2020a) showed that there is a band
bending at the EuS/InAs interface of the same magnitude as the one between
InAs and air. Hence, we model both phenomena with the same
$\rho_{\mathrm{surf}}$. We find that $\rho_{\mathrm{surf}}=2\cdot
10^{-3}\mathrm{\left(\frac{e}{nm^{3}}\right)}$ creates a band bending of 0.1
eV, which is the value reported in experiments for InAs/vacuum Thelander  _et
al._ (2010); Reiner  _et al._ (2020) and InAs/EuS Liu  _et al._ (2020a)
interfaces.

Figure 5: Electrostatic potential across the heterostructure section in the
overlapping device (a) and in the non-overlapping one (b). (c,d) Mobile charge
density in the wire computed within the Thomas-Fermi approximation using the
upper electrostatic potentials. Same parameters as in Figs. 2 and 3 of the
main text, respectively.

The second term in Eq. (15) corresponds to the mobile charges inside the
hybrid structure. In principle, they should be computed from the solutions of
the Schrödinger equation using the Hamiltonian of Eq. (2). Since this equation
depends in turn on the electrostatic potential given by the Poisson equation
of Eq. (14), the Schrödinger-Poisson equations have to be solved together in a
self-consistent way. To reduce the computational effort, we use the Thomas-
Fermi approximation. Within this approximation, one assumes that the charge
density of the wire is given by the one of a non-interacting 3D electron-gas
444We ignore the contribution coming from the valence bands as they could only
play a role for $V_{\rm bg}\lesssim-4$ V in our simulations.

$\displaystyle\rho_{\rm mobile}\simeq\rho_{\rm mobile}^{\rm(TF)}=\>\>\>\>$
$\displaystyle-\frac{e}{3\pi^{2}}\left[\frac{2m_{\rm
eff}\left|e\phi(\vec{r})-E_{F}\right|f\left(-(e\phi(\vec{r})-E_{F})\right)}{\hbar^{2}}\right]^{\frac{3}{2}},\>\>\>\>\>\>$
(16)

where $f(E)$ is the Fermi-Dirac distribution for a given energy $E$ and
temperature $T$. This approximation has been used in previous works Mikkelsen
_et al._ (2018); Escribano  _et al._ (2019, 2020) on InAs nanowires and
Al/InAs heterostructures, where it is demonstrated that it provides an
excellent agreement with the results of the full Schrödinger-Poisson
calculations. In this way, both equations are decoupled and, therefore, one
only needs to diagonalize the Hamiltonian once for each potential
$\phi(\vec{r})$ (and momentum $k_{z}$).

Nonetheless, the Poisson equation still needs to be solved self-consistently
[note that $\rho_{\rm mobile}^{\rm(TF)}$ depends on $\phi(\vec{r})$], although
at a reduced computational cost. To carry out this self-consistency we use an
Anderson iterative mixing,

$\rho_{\rm mobile}^{(n)}=\beta\tilde{\rho}_{\rm
mobile}^{(n)}+(1-\beta)\rho_{\rm mobile}^{(n-1)},$ (17)

where $n$ is a certain iteration in the self-consistent process, and $\beta$
is the so-called Anderson coefficient which is a constant in the range
$[0,1]$. In the first step ($n=0$) we take $\rho^{(0)}=0$ and we compute the
electrostatic potential using the Poisson equation. Then, at any other
arbitrary step $n$, we compute the charge density $\tilde{\rho}_{\rm
mobile}^{(n)}$ using Eq. (16) with the electrostatic potential found using the
$\rho_{\rm mobile}^{(n-1)}$ of the previous iteration $n-1$. We finally find
the charge density at the $n^{th}$ step (i.e., $\rho_{\rm mobile}^{(n)}$)
using the Anderson mixing between these two charge densities shown in the
above equation. This mixing between two different steps ensures the
convergence to the solution. We keep this iterative procedure until the
cumulative error between $\rho_{\rm mobile}^{(n)}$ and $\rho_{\rm
mobile}^{(n-1)}$ is below $0.5\%$. To facilitate an efficient convergence, we
choose as Anderson coefficient a self-adaptive one given by,

$\beta=\beta^{(max)}\exp{\left(\frac{\max{\left(\left||\rho_{\rm
mobile}^{(n)}|-|\rho_{\rm mobile}^{(n-1)}|\right|\right)}}{\max(|\rho_{\rm
mobile}^{(n)}|,|\rho_{\rm mobile}^{(n-1)}|)}\right)},$ (18)

where $\beta^{(max)}$ is the maximum value that we allow the Anderson
coefficient to take in order to ensure converge. We set this value to 0.1 in
all our simulations.

To solve the Poisson equation, we also need to impose boundary conditions. We
fix different potentials at the boundaries of the gates for the different
simulations performed in this work. In addition to this, we fix a constant
potential of $V_{\rm Al}=0.2$ eV at the boundaries with the Al shell. The
electrostatic potential resulting from this last condition will create a band
bending of the same magnitude at the InAs/Al interface [see Fig. 1(c)]. This
band bending, reported in experiments Reiner  _et al._ (2020); Schuwalow  _et
al._ (2019); Liu  _et al._ (2020c), has been previously described in other
theoretical works Mikkelsen  _et al._ (2018); Antipov  _et al._ (2018);
Escribano  _et al._ (2018); Winkler  _et al._ (2019); Escribano  _et al._
(2019); Woods  _et al._ (2020) on similar InAs/Al heterostructures.

To solve the Poisson equation with the above ingredients we use Fenics Logg
and Wells (2010); Logg  _et al._ (2012), a finite element solver for Python.
We use a mesh discretization of $1$ nm.

As we show in the main text, the precise geometries of the devices, the
position of the Al and EuS shells on the InAs facets and the gates, play a key
role in the appearance of a topological phase. To better illustrate this fact,
we show in Fig. 5 the electrostatic potential in the overlapping (a) and the
non-overlapping (b) devices. In these simulations, the back-gate voltage is
set to a negative value, particularly $V_{\rm bg}=-1$ V. This is a typical
situation in our simulations and in the experiments, as one needs to deplete
the wire in order to populate it with just a few bands. Hence, the
electrostatic potential is negative at the bottom of the wire, while it is
positive and maximum close to the Al/InAs interface due to the Al/InAs band
bending. This is translated into an accumulation of (mobile) charges
$\rho_{\rm mobile}$ in this interface, quantity that we show in Fig. 5(c) and
(d) for the same devices. Despite the similarity between the electrostatic
profiles of both devices, we show in the main text that they give rise to a
very different energy spectrum. This can be understood by noticing the
position of the the charge density with respect to the EuS. The better part of
the charge density is located at the Al/InAs interface. In the overlapping
device [Fig. 5(c)], this charge is thus also close to the left part of the EuS
layer (the top facet). On the contrary, the EuS layer in the non-overlapping
device [Fig. 5(d)] is far apart from the charge density. This apparently minor
difference is what enhances the hybridization with the EuS in the overlapping
device, and what suppresses the same hybridization in the non-overlapping one.
We remark that this hybridization is necessary to induce a strong enough
exchange field in the wire, and therefore, to create a topological phase.

### 4\. Spin-orbit coupling

The SO interaction included in the Hamiltonian of Eq. (2) arises whenever an
spatial symmetry is broken. In this Hamiltonian, only linear terms with $k$
are included as they are known to be the dominant ones. The strength of the SO
interaction is given by the SO coupling $\vec{\alpha}(\vec{r})$ and its value
depends on the precise material. For Al, the SO coupling is negligible Chiang
and Dzyuba (2017), while for the EuS we have found no information about its
value in the literature. We thus set to zero the SO coupling of both materials
in our simulations. By contrast, the SO coupling in InAs nanowires can take
large values Campos  _et al._ (2018); Escribano  _et al._ (2020). Because the
topological protection of the Majorana nanowires depends strongly on the
precise value of this interaction, a proper description is crucial to reach a
good qualitative agreement with the experiments.

Figure 6: Rashba SO coupling across the wire section for a (111) zinc-blende
InAs nanowire (top) and a (0001) wurtzite InAs nanowire (bottom), in the
overlapping device, (a) and (c), and in the non-overlapping one, (b) and (d).
Same parameters as in Fig. 5.

The SO coupling has, in general, two components Winkler  _et al._ (2003),

$\vec{\alpha}(\vec{r})=\vec{\alpha}_{\rm D}+\vec{\alpha}_{\rm R}(\vec{r}),$
(19)

where $\alpha_{\rm D}$ is the Dresselhaus term and $\alpha_{\rm R}$ the Rashba
one. The Dresselhaus SO coupling arises when the crystal unit cell itself is
not symmetric, leading to a bulk inversion asymmetry. Hence, it is a spatial
independent value that only depends on the material and its precise
crystallographic structure. For InAs nanowires, it is known to be negligible
in (111) zinc-blende crystals, while its value is roughly $\vec{\alpha}_{\rm
D}=(0,0,30)$(meV$\cdot$nm) in (0001) wurtzite ones Gmitra and Fabian (2016);
Faria Junior  _et al._ (2016). On the other hand, the Rashba SO coupling
emerges when the mesoscopic system is not symmetric in some direction, as it
happens when there are interfaces or potential gates. It can be accurately
described for III-V compound semiconductors using the following equation

$\displaystyle\vec{\alpha}_{R}(\vec{r})=\frac{eP_{\mathrm{fit}}^{2}}{3}\left[\frac{1}{\Delta_{\rm
g}^{2}}-\frac{1}{(\Delta_{\rm{soff}}+\Delta_{\rm
g})^{2}}\right]\vec{\nabla}\phi(\vec{r}),$ (20)

where $P_{\rm fit}$ is a parameter that depends on the material and its
crystallographic structure, and $\Delta_{\rm g}$ and $\Delta_{\rm soff}$ are
the valence band and split-off bands, respectively. We show in Table 2 their
values. This equation has been derived in a previous work Escribano  _et al._
(2020), finding that it provides excellent results in comparison to both more
sophisticated theories and experimental measurements.

Table 2: Parameters used for the SO coupling in the InAs nanowire, extracted from Ref. Escribano  _et al._ , 2020 and references therein. These parameters are valid for a 120 nm wide nanowire, except for $\vec{\alpha}_{D}$ that is valid for any width. Crystal |  Parameter |  Value
---|---|---
(111) Zinc-blende |  $\vec{\alpha}_{\rm D}$ |  (0,0,0)
|  $P_{\rm fit}$ |  1300 meV$\cdot$nm
|  $\Delta_{\rm g}$ |  417 meV
|  $\Delta_{\rm soff}$ |  390 meV
(0001) Wurtzite |  $\vec{\alpha}_{\rm D}$ |  (0,0,30) meV$\cdot$nm
|  $P_{\rm fit}$ |  700 meV$\cdot$nm
|  $\Delta_{\rm g}$ |  467 meV
|  $\Delta_{\rm soff}$ |  325.7 meV

We note that the Rashba SO coupling is a position-dependent function because
it depends on the gradient of the electrostatic potential $\phi(\vec{r})$. To
better illustrate this dependence, in Fig. 6 we show calculations of the SO
coupling for the two devices studied in this work (first column and second
column) and for (111) zinc-blende (first row) and (0001) wurtzite (second row)
InAs nanowires. The Rashba SO coupling in zinc-blende nanowires (top row)
turns out to be roughly twice than in wurtzite crystals (bottom row). This has
been pointed out in other theoretical works Campos  _et al._ (2018); Escribano
_et al._ (2020) and it is intimately related to the symmetries of both
crystals. But in both cases, the spatial profile of the Rashba coefficient is
the same. Particularly, the maximum is located at the Al/InAs interface,
precisely where the charge density is maximum [see Fig. 7(c,d)], what benefits
the topological protection of the states populating the wire. We also note
that the maximum of the SO coupling is above 10 meV$\cdot$nm for both types of
crystals (even for this small $V_{\rm bg}$ value), which is large enough to
create a measurable topological gap.

In our simulations we have used the parameters that correspond to (111) zinc-
blende nanowires, which are the typical ones used in experiments. However,
(0001) wurtzite nanowires would have provided similar features (not shown),
although with smaller topological minigaps.

## Simplified model

The diagonalization of the Hamiltonian of Eq. (2) is computationally demanding
because of the small grid spacing imposed by the large Fermi momentum of the
Al layer. For this reason, we are only able to use this model to obtain the
band structure and DOS for certain gate voltages, but not to compute the phase
diagram of the studied devices in a broad range of parameters. To overcome
this problem, we use a less demanding model that includes the induced pairing
and exchange field in the wire in an heuristic way (as energy-independent but
position-dependent parameters). This simplified model only considers the wire,
and thus the effective Hamiltonian can be discretized using a larger lattice
spacing ($h=1$ nm in our simulations). In the following subsections we explain
how both phenomena (the induced superconducting and magnetic effects) can be
incorporated to reproduce qualitatively the behaviour of the full model.

Figure 7: (a) Comparison of the induced superconducting gap
$\Delta^{*}=\mathrm{min}\left(|E(k_{z})|\right)$ versus back-gate potential
between the full model (dots) and the simplified model (lines). To the left,
schematics of the device. Two facets of the wire are covered with an 8 nm wide
Al layer. The full model includes the superconducting layer in the Hamiltonian
at the same level as the nanowire. In the simplified model the Al layer is
integrated out and its superconducting proximity effect on the wire
Hamiltonian is included as a region of width $W_{\rm prox}^{\rm(Al)}$ with
pairing amplitude $\Delta^{\rm(Al)}$ (marked as a streaked region in the
schematics). To the right, $\Delta^{*}$ versus $V_{\rm{bg}}$; in the
simplified model we take $\Delta^{\rm(Al)}=0.23$ meV and consider different
$W_{\rm prox}^{\rm(Al)}$ (in colors). (b) Same as (a) but now comparing the
induced exchanged field $\bar{h}_{\rm ex}^{\rm(ind)}$ versus $V_{\rm{bg}}$.
The EuS shell covers two facets and has a width of 8 nm in the full model
calculations. In the simplified model we take $W_{\rm prox}^{\rm(EuS)}=1$ nm
with an exchange field of $h^{\rm(EuS)}_{\rm ex}=100$ meV.

### 1\. Induced superconductivity

We consider the superconducting proximity effect in the nanowire effectively
by introducing a region close to the InAs/Al interface with a finite pairing
amplitude $\Delta^{\rm(Al)}$ [see sketch of Fig. 7(a)]. The parameter
$\Delta^{\rm(Al)}$ and the width of this proximitized region, $W_{\rm
prox}^{\rm(Al)}$, are then chosen such that this simplified model reproduces
(approximately) the same behaviour for the induced superconducting gap as the
full model.

In Fig. 7(a) we show the evolution of the induced gap $\Delta^{*}$ versus the
back-gate potential. Dots correspond to calculations using the full model
while solid lines correspond to the simplified one. Different colors are used
for different $W_{\rm prox}^{\rm(Al)}$ widths. We find that $W_{\rm
prox}^{\rm(Al)}=45$ nm is the best fit to the full model results.

### 2\. Induced exchange field

Following the same reasoning as for the induced pairing, we model the induced
magnetization by introducing a region close to the InAs/EuS interface with a
finite exchange field $h_{\rm ex}^{\rm(EuS)}$ [see sketch in Fig. 7(b)]. Once
again, we choose $h_{\rm ex}^{\rm(EuS)}$ and the width of this proximitized
region, $W_{\rm prox}^{\rm(EuS)}$, so as to reproduce approximately the
results obtained with the full model. In Fig. 7(b) we show the mean induced
exchange field $\bar{h}_{\rm ex}^{\rm(ind)}=\sum_{j}f(E^{(j)})\left<h_{\rm
ex}\sigma_{x}\right>_{j}$ versus the back-gate potential (where $f$ is the
Fermi-Dirac distribution and $j$ is summed over all populated subbands ). Dots
correspond to the full model calculations and solid lines to the simplified
model using $h^{\rm(EuS)}_{\rm ex}=100$ meV and $W_{\rm prox}^{\rm(EuS)}=1$
nm. For the gate voltage range studied in this work (i.e. $V_{\rm bg}>-4$V)
the simplified model provides a very good agreement with the full model. The
characteristic penetration length of electrons into the insulating EuS is
given (roughly) by the magnetic length

$\lambda=\left(\sqrt{\left|k_{\rm F}^{2}-\frac{2m_{\rm eff}E_{\rm
F}^{\rm(EuS)}}{\hbar^{2}}\right|}\right)^{-1}\sim 1\ \mathrm{nm},$ (21)

which is the value of the proximitized width $W_{\rm prox}^{\rm(EuS)}$ that we
use in our simplified model.

In addition to this direct proximity effect from the EuS into the InAs, there
is an (ungateable) induced exchange field from the EuS into the Al of the
order of 0.06 meV, see full-model calculations in main text. This proximity
effect is present at the interfaces where Al and EuS overlap in the
overlapping device (thus, it does not appear in the non-overlapping device).
This effect causes an indirect proximity effect from the EuS into the InAs
through the Al. We include this effect by adding an exchange field $h_{\rm
ex}^{\rm(Al)}=0.06$ meV into the superconducting proximitized region discussed
in the previous subsection.

Figure 8: (a) Energy spectrum at $k_{z}=0$ versus the back-gate voltage
$V_{\rm bg}$ using the full model for the overlapping device. Grey (white)
color corresponds to a trivial (non-trivial) topological phase. This can be
established by computing the topological invariant $\mathcal{Q}$, as explained
in the text. (b-d) Bandstructure versus $k_{z}$ for different gate voltages
around the last topological region (see arrows): (b) before, (c) inside, and
(d) after the topological region. With colored bars we show the weight of the
eigenstate in the Al layer (black) and in the InAs wire (yellow). In (e-f) we
show the spin-polarized DOS for the same back-gate voltages indicated with
arrows in (a). We set $V_{\rm sg}^{\rm(L,R)}=0$. Figure 9: Same as Fig. 8 but
for the non-overlapping device.

## Topological phase diagram for the full model

In the main text we study the topological phase diagram (versus $V_{\rm{bg}}$
and $h_{\rm{ex}}^{(\rm{EuS})}$) using the simplified model. In this section we
analyze the phase diagram versus the back gate potential using the full model,
which is more computationally demanding. Note that $h_{\rm{ex}}^{(\rm{EuS})}$
is analyzed as a free parameter in the phase diagram of the simplified model,
but it is a built-in quantity within the full model. This analysis complements
the calculations performed in Figs. 2 and 3 of the main text.

In Fig. 8 we consider the case of the overlapping device. In panel (a) we show
the energy spectrum at $k_{z}=0$ versus $V_{\rm{bg}}$. For the voltage range
shown there, we observe low energy pairs of states that cross zero energy at
(roughly) $V_{\rm bg}=-4.5$ V, $V_{\rm bg}=-4$ V and $V_{\rm bg}=-1$ V. These
zero-energy crossings of the bulk states are related to topological phase
transitions. This is demonstrated by the calculation of the topological
invariant shown with white-grey colors (see the first section of this
Supplemental Material for details on this calculation). In Fig. 8(b-d) we show
the dispersion relation versus the momentum along the wire’s direction $k_{z}$
for three specific $V_{\rm bg}$ values: (b) at $V_{\rm bg}=-1$ V, i.e., before
the last topological region; (c) at $V_{\rm bg}=-0.95$ V, inside the
topological region; and (d) at $V_{\rm bg}=-0.6$ V, after this region. In the
top panel of these three figures we show with colors the weight of the states
in the different materials, from a state completely located in the Al layer
(black) to completely located in the InAs wire (yellow). The states that cross
zero energy in this topological region correspond to mixed states with half of
the wavefunction located in the Al and the other half in the InAs. For
completeness, we also show in Fig. 8(e-f) the spin-polarized DOS for the
bandstructures shown in Fig. 8(b-d).

We now perform the same analysis but for the non-overlapping device (see Fig.
9). We find that there are no topological regions for any value of $V_{\rm
bg}$, positive or negative, at least for the right-side gate voltage used in
this simulation [see Fig. 9(a)]. This is because the induced exchange field in
this device is much smaller than the induced gap [see Fig. 9(b-d)] and
therefore no topological phase can be developed.

Figure 10: Phase diagram of the overlapping device (a) and the non-overlapping
one (b) versus the right-side gate potential $V_{\rm sg}^{\rm(R)}$ and the
exchange field at the EuS-InAs interface $h_{\rm ex}^{\rm(EuS)}$, for $V_{\rm
bg}=-4\mathrm{\ V}$. White means trivial (i.e., $\mathcal{Q}=+1$). With colors
we show the induced exchange field in the left panel, the minigap in the
middle one, and the wavefunction profile in the right one, all of them for the
transverse subband closest to the Fermi energy. The $V_{\rm sg}^{\rm(R)}$
values for which the wavefunctions are plotted are pointed with arrows.

## Topological phase diagrams for the simplified model versus side-gate
potential

Using the simplified model, in the main text we study the topological phase
diagram versus the back-gate potential. This gate not only tunes the chemical
potential inside the wire, but it also moves upwards and downwards (along the
$y$ axis) the charge probability density. This has a tremendous impact on this
kind of heterostructure because, depending on the wavefunction position, the
hybridization with the Al or the EuS layer dramatically differs. In the same
way, the side gates play a similar role, but they change the position of the
wavefunction along the perpendicular axis (the $x$ axis). Just by looking at
the geometries of both devices (see Fig. 1 of the main text) one can infer
that if the side gates push the wavefunction closer to the Fermi level towards
the right part, the hybridization with the EuS will be increased while the
hybridization with the Al will be suppressed.

To show this fact and to prove that the systems exhibit similar features in
their topological phase diagrams regardless of whether the back gate or the
side gates are tuned, we compute the topological phase diagram versus the
right-side gate 555Note that, because right and left side gates are placed
parallel to each other and symmetric with respect to the wire, they provide
comparable phase diagrams.. We show the results in Fig. 10(a) for the
overlapping device, and in Fig. 10(b) for the non-overlapping one. In the
first two columns, we show with colors the topological regions (white means
trivial, $\mathcal{Q}=1$). Particularly, colors in the first column correspond
to the induced exchange field $h_{\rm ex}^{\rm(ind)}$ for the (topological)
subband closest to the Fermi energy, and colors in the second column
correspond to its minigap $\Delta_{\rm min}=\left|E(k=k_{\rm F})\right|$. In
addition to this, we show on the right of each subfigure the probability
density across the wire section of the lowest transverse subband (at
$k_{z}=0$) at the $V_{\rm sg}^{\rm(R)}$ value pointed with the arrows.

Let us discuss first the results for the overlapping device [Fig. 10(a)]. This
geometry exhibits several topological regions that span across potential
ranges of the order of $250$ mV. Increasing the right-side gate voltage for a
fixed $h_{\rm{ex}}^{(\rm{EuS})}$ (see left panel) keeps the induced exchange
field almost constant, while it decreases the minigap (see right panel). This
is precisely because the lowest-energy wavefunction is being pushed towards
the EuS layer, moving away from the Al layer (see charge density plots on the
right).

In the non-overlapping device [Fig. 10(b)], we observe that there is no
topological phase for negative right-side gate potential values. However,
there are some small topological regions for positive gate potentials. This
happens because the wavefunction is located close to the EuS layer for these
gate potentials (see charge density plots on the right), acquiring a non-
negligible induced exchange field. But due to the large doping of the wire,
these states have a large kinetic energy, what implies that they spread all
across the wire. Hence, both quantities, the induced exchange field and the
minigap, are small or even negligible for these states.

## Two-Al-facet geometry

In this last section we consider a different geometry than those studied in
the main text. This geometry was analyzed experimentally but didn’t provide
zero bias peaks Vaitiekėnas  _et al._ (2021). Strikingly, this device is
identical to the overlapping device of Fig. 1(a) of the main text, but with an
additional InAs facet covered by the Al shell [see Fig. 11(a) for a sketch].
In Fig. 11(b) we show the lowest energy spectrum versus the back-gate
potential at $k_{z}=0$. With grey/white colors we show whether the
heterostructure has a trivial/non-trivial topological phase. For these gate
voltages, there is only one topological region at (roughly) $-1.2$ V. However,
its extension and the size of its minigap are negligibly small, in sharp
contrast to the overlapping device studied before. The bandstructure around
this topological region is shown in (c-e). The lowest-energy states correspond
to mixed states, but mainly located in the Al. More particularly, the lowest-
energy wavefunction is mainly located at the left corner of the wire (not
shown), where the band-bending of the superconductor leads to a larger
accumulation of charge. This large hybridization with the superconductor is
what suppresses the hybridization with the EuS, leading to a small exchange
field induced directly in the InAs. And this in turn is what suppresses the
appearance of topological states in this device.

Figure 11: (a) Sketch of the two-Al-facet device, which is equal to the
overlapping geometry in the main text but with one more facet of the nanowire
covered by the Al shell (the bottom-left one). (b) Energy spectrum at
$k_{z}=0$ versus the back-gate voltage for this device. White/gray colors
denote the non-trivial/trivial topological phases, computed with the
topological invariant $\mathcal{Q}$ (see text). (c-e) Band structure versus
the momentum $k_{z}$ for three different back-gate voltages. Their
corresponding DOS are shown in (f-h). Parameters are the same as in Fig. 2 of
the main text.