# Topological superconductivity in semiconductor-superconductor-magnetic
insulator heterostructures

A. Maiani  Center for Quantum Devices, Niels Bohr Institute, University of
Copenhagen, DK-2100 Copenhagen, Denmark  R. Seoane Souto  Center for Quantum
Devices, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen,
Denmark  Division of Solid State Physics and NanoLund, Lund University,
S-22100 Lund, Sweden  M. Leijnse  Center for Quantum Devices, Niels Bohr
Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark  Division of
Solid State Physics and NanoLund, Lund University, S-22100 Lund, Sweden  K.
Flensberg  Center for Quantum Devices, Niels Bohr Institute, University of
Copenhagen, DK-2100 Copenhagen, Denmark

###### Abstract

Hybrid superconductor-semiconductor heterostructures are promising platforms
for realizing topological superconductors and exploring Majorana bound states
physics. Motivated by recent experimental progress, we theoretically study how
magnetic insulators offer an alternative to the use of external magnetic
fields for reaching the topological regime. We consider different setups,
where: (1) the magnetic insulator induces an exchange field in the
superconductor, which leads to a splitting in the semiconductor by proximity
effect, and (2) the magnetic insulator acts as a spin-filter tunnel barrier
between the superconductor and the semiconductor. We show that the spin
splitting in the superconductor alone cannot induce a topological transition
in the semiconductor. To overcome this limitation, we propose to use a spin-
filter barrier that enhances the magnetic exchange and provides a mechanism
for a topological phase transition. Moreover, the spin-dependent tunneling
introduces a strong dependence on the band alignment, which can be crucial in
quantum-confined systems. This mechanism opens up a route towards networks of
topological wires with fewer constraints on device geometry compared to
previous devices that require external magnetic fields.

##  I Introduction

Topological superconductivity has been predicted to appear in one-dimensional
spin-orbit coupled semiconductors proximitized by an s-wave superconductor. In
these systems, an external magnetic field causes the gap to close. In the
presence of a strong spin-orbit interaction, the gap reopens, leading to a
topological phase [1, 2]. The quantum phase transition brings the system into
the topological regime when

$V_{\mathrm{Z}}^{2}>\mu^{2}+\Delta^{2}\,,$ (1)

where $V_{\mathrm{Z}}$ is the Zeeman energy, $\mu$ the semiconductor chemical
potential, and $\Delta$ the superconducting pairing amplitude. In the
topological phase, the system behaves as a spinless p-wave superconductor and
Majorana bound states appear at the edges of the system. These states have
been proposed to be used for topological quantum computation [3, 4]. Following
this idea, different hybrid semiconductor-superconductor (Sm-Sc) platforms
have been proposed to exhibit topological superconductivity, such as
proximitized nanowires [5], selective-area-grown wires [6], and two-
dimensional electron gas (2DEG) systems [7, 8, 9, 10].

One limitation of these platforms is the requirement of external magnetic
fields to induce the topological phase transition. External magnetic fields
have several drawbacks as they are detrimental to superconductivity and the
topological phase is very sensitive to the relative orientation between the
magnetic and the spin-orbit fields. This problem becomes even more evident
when considering more complicated geometries featuring several nanowires that
cannot be aligned in the same direction, like the ones proposed to demonstrate
Majorana non-Abelian statistics in real space [11, 12, 13].

Magnetic materials are alternatives to the use of external magnetic fields.
This idea was discussed by some early works in the field, where magnetic
insulators (MIs) were used to induce a spin splitting by means of stray fields
[14, 15]. In addition, clean interfaces with a MI also lead to exchange fields
in the proximitized materials, which provide a more effective way to control
the local spin splitting.

Developments in the fabrication technology have enabled the integration of
thin layers of the MI EuS in the hybrid Sm-Sc InAs-Al nanowires with excellent
interface quality [16, 17, 18]. This material has also been tested in
combination with Au showing signatures consistent with the presence of
Majorana bound states [19].

In experiments with ferromagnetic hybrid nanowires, spectroscopic measurements
have shown the onset of a zero-bias conductance peak which has been
interpreted as a signature of localized Majorana bound states at their ends
[20]. This signature has been detected only when the Al and the EuS layers
overlap [Fig. 1-(a)]. Samples with non-overlapping facets, like Fig. 1-(b),
have shown no signatures of topological phases. This behavior is in contrast
with expectations that the main effect of MIs is to induce an exchange field
in the Sm [21].

In these devices, however, the induced exchange field in the Sm is weak and
short range [18, 20]. For this reason, the topological transition requires
tuning the electrostatic gates, as pointed out by recent theory works [22, 23,
24]. Another effect of the MI layer is to induce an exchange field in the Sc
due to the proximity effect taking place at the Sc-MI interface [25, 26]. Many
experimental works have verified a strong effect of MI on thin Sc layers in
terms of an induced spin splitting of the Sc density of states [27, 28, 29,
30, 31, 32]. Excluding a direct effect of the MI on the Sm, an alternative
explanation can be that the spin-split Sc induces superconductivity and
exchange field at the same time in the Sm. In this case, the relevant exchange
coupling would be directly between the MI and the Sc [20].

Figure 1: Sketch of the cross section of a Sm nanowire in proximity to a Sc
and a MI. (a) Sm nanowire with partially overlapping Sc and MI, where a zero-
bias peak has been recently reported [20]. (b) Sm nanowire with no overlap
between the Sc and MI layers, which did not show signatures of topological
superconductivity [20]. (c) Sm nanowire with completely overlapping Sc and a
thin MI layer. Measures of devices of the kind shown in (c) have not been
reported yet.

In this paper, we provide a theory for the combined superconductivity and
magnetic proximity effect in Sc-MI-Sm and MI-Sc-Sm heterostructures, as
illustrated in Fig. 2. We show that the combined magnetic and superconducting
proximity effects of a spin-split Sc cannot induce a topological transition in
the Sm by themselves. To overcome this limitation, we propose a new
heterostructure layout where a thin film of MI between the Sm and the Sc leads
to a spin-dependent tunnel barrier [Fig. 1-(c)]. Our proposal exhibits a
parameter region where topological superconductivity is present in the Sm for
strong enough spin-dependent tunneling.

##  II Model

A MI layer can induce several effects in the Sm-Sc device, depending on the
heterostructure layout, as shown in Fig. 2.

Figure 2: Sketch representing the proximity effects in the systems considered
in this work. In the illustrations, $V^{\mathrm{MI}}$ is the native exchange
splitting in the magnetic insulator and $\Delta$ and $V^{\mathrm{Sc}}$ are the
gap parameter and the MI-induced exchange field in the Sc. Here,
$\tilde{\Delta}$ and $\tilde{V}$ are the Sc-induced gap and exchange field in
the Sm. The matrix $T$ is the hopping matrix of the tunneling Hamiltonian
coupling the two materials. In panel (a), the magnetic insulator induces an
exchange field in the Sc. The spin-split Sc is coupled to the Sm by a spin-
symmetric tunneling Hamiltonian. In panel (b), a thin magnetic insulator layer
is placed between the Sc and Sm. The MI induces an exchange field in the Sc,
also providing a spin-dependent tunnel barrier between the Sc and the Sm. In
both cases, we ignore the effect of the MI on the Sm as it only increases
$\tilde{V}$ close to the interface to the MI.

One of the effects of the MI layer is the induction of an exchange field in
both the Sm and the Sc. This is due to the microscopic scattering interaction
between the electrons and the localized magnetic moments in the MI. We can
describe the effect of the interface through a Heisenberg-like term
$H_{\mathrm{int}}=-J\int\differential[3]{r}\mathbf{S}^{\mathrm{MI}}\cdot\mathbf{S}$,
which describes the coupling between the spin density of the metals
$\mathbf{S}$ and the localized spins in the insulator,
$\mathbf{S}^{\mathrm{MI}}$. The coupling strength $J$ is related to the
exchange integral between the localized orbitals and the free electron. This
coupling strength is in principle different for the conduction band electrons
of the MI and the electrons in the proximitized material [33]. These
considerations apply to ferromagnetic materials as well as antiferromagnetic
insulators [34]. We consider the Sc layer width smaller than the
superconducting coherence length, $\xi_{0}$, which is the characteristic decay
length for the induced exchange field in a Sc-MI heterostructure [26]. In the
experiments [20], the SC and MI layers have a width of few nanometers while
the typical size of a magnetic domain in the EuS is $\sim 10\xi_{0}$ for EuS-
Al heterostructures [31]. Since the dimensions of the systems under
consideration are smaller than $\xi_{0}$, we can disregard inhomogeneities and
assume that the MI induces a homogeneous exchange field in the Sc, which
couples to the spin of the electrons by the Zeeman term
$H_{\mathrm{Z}}=\mathbf{V}^{\mathrm{Sc}}\cdot{\bf\it\sigma}$.

In this work we disregard the induced exchange field in the Sm as experimental
evidence suggests that it is small in the tested devices [18, 20]. We note
that recent theoretical works suggest that it can lead to topological
superconductivity by careful control of the gate voltages [22, 23, 24]. A
direct MI-induced exchange field in the Sm would provide an enhancement of the
Zeeman energy in the Sm, enlarging the topological region in our case. We also
neglect the magnetic orbital effects induced by the stray field of the MI as
they are usually weaker than ones induced by the exchange field [16].

If the MI considered is a magnetic semiconductor like EuS, the conduction band
is not accessible at low temperatures. The modest band gap can be used to
fabricate spin filter tunnel barriers, using thin films, that allows spin-
dependent tunneling through it when placed between two metallic regions [33].
The proposed setup is illustrated in Fig. 2-(b).

In the following, we compare the two different situations represented in Fig.
2. For the case shown in Fig. 2-(a), the MI only causes a spin splitting in
the Sc density of states. This situation is relevant for the geometry shown in
Fig. 1-(a) where a thick MI layer is placed in between the two materials such
that the tunneling is strongly suppressed [20]. In Fig. 2-(b) we show a
different situation where a thin layer of MI is placed in between the Sc and
Sm working as a spin-filter tunnel barrier, having a weak effect on the Sm
density of states. This case applies to the device geometry shown in Fig.
1-(c) but could also be relevant in the case of Fig. 1-(a) if the MI layer is
sufficiently thin and, therefore, all the interfaces contribute to the
tunneling processes.

To simplify the treatment, we consider a translation-invariant system along
the longitudinal direction, $z$, consisting of a single-channel Sm coupled to
a Sc. The complete Hamiltonian of the system reads

$H=H_{\mathrm{Sm}}+H_{\mathrm{Sc}}+H_{\mathrm{t}}\,.$ (2)

The Sm is described by

$\begin{split}H_{\mathrm{Sm}}=\sum_{p_{z}}\quantity(\frac{p_{z}^{2}}{2m_{\mathrm{Sm}}^{*}}-\mu_{\mathrm{Sm}})c_{p_{z}}^{\dagger}\sigma_{0}c_{p_{z}}\\\
+p_{z}c_{p_{z}}^{\dagger}\alpha_{x}\sigma_{y}c_{p_{z}}\,,\end{split}$ (3)

where we use the spinors $c_{p_{z}}=(c_{p_{z}\uparrow}$,
$c_{p_{z}\downarrow})$ and $c_{p_{z}\sigma}$ is the electron annihilation
operator in the Sm, while $\alpha_{x}$ is the spin-orbit coupling strength.

The bare Hamiltonian for the Sc is given by

$\begin{split}H_{\mathrm{Sc}}=\sum_{np_{z}}&\xi_{np_{z}}a_{np_{z}}^{\dagger}\sigma_{0}a_{np_{z}}+a^{\dagger}_{np_{z}}\mathbf{V}^{\mathrm{Sc}}\cdot{\bf\it\sigma}a_{np_{z}}+\\\
&\quantity(a^{\dagger}_{np_{z}}{\bf\it\Delta}_{np_{z}}i\sigma_{y}a^{\dagger}_{n-p_{z}}+\mathrm{H.c.})\,,\end{split}$
(4)

where $a_{np_{z}}$ is the electron annihilation operator for the mode $n$ in
the Sc and
$\xi_{np_{z}}=\frac{p_{z}^{2}}{2m_{\mathrm{Sc}}}+\varepsilon_{n}-\mu_{\mathrm{Sc}}$.
We neglect the possible superconductive interband coupling and we assume
singlet pairing in the parent superconductor gap matrix
${\bf\it\Delta}_{np_{z}}=\Delta_{0,np_{z}}\sigma_{0}$. As anticipated, our Sc
model features a homogeneous exchange field
$\mathbf{V}^{\mathrm{Sc}}=V_{z}\mathbf{e}_{z}$ induced by the nearby MI, which
we consider to be aligned to the wire. A crucial information is the
distribution of the transverse modes in the Sc, $\varepsilon_{n}$, which
strongly depends on the device geometry. The energy separation between
transverse modes can range from a value larger than the superconducting gap
for very thin Sc film to zero for a bulk superconductor, where the modes above
the gap form a continuum.

The Sc and Sm regions are coupled through a spin-dependent, momentum
conserving, tunneling Hamiltonian

$H_{\mathrm{t}}=\sum_{np_{z}}\quantity(c^{\dagger}_{p_{z}}T_{np_{z}}a_{np_{z}}+\mathrm{H.c.})\,,$
(5)

where the hopping matrix $T_{np_{z}}$ describes the electron tunneling
processes taking place at the interfaces between the two materials. We write
$T_{np_{z}}$ in the basis of Pauli matrices
$T_{np_{z}}=t_{0,np_{z}}\sigma_{0}+\sum_{i}t_{i,np_{z}}\sigma_{i}$. In the
following, we set $t_{x,np_{z}}=t_{y,np_{z}}=0$ as they are negligible for a
uniformly polarized MI layer. Moreover, we consider only the case of real and
positive $t_{0,np_{z}}$ and $t_{z,np_{z}}$. This is justified as we are
assuming that spin-orbit coupling is absent in the Sc. An extended derivation
with all the terms can be found in Appendix A.3.

The bare Sc retarded Green’s function in the basis of time-reversed pairs
reads:

$\begin{split}G^{R}_{\mathrm{Sc}}(\omega,n,p_{z})=\Big{[}(\omega+i0^{+})\tau_{0}\sigma_{0}-\xi_{np_{z}}\tau_{z}\sigma_{0}\\\
-{\bf\it\Delta}_{np_{z}}\tau_{x}\sigma_{0}-V_{z}\tau_{0}\sigma_{z}\Big{]}^{-1}\,,\end{split}$
(6)

where the $\tau_{i}$ are the Pauli matrices in the particle-hole space. In
this work we ignore any back action of the Sm on the Sc as the electron
density in the Sm is orders of magnitude smaller than the one in the Sc.

The retarded Green’s function of the Sm reads
$G^{R}_{\mathrm{Sm}}(\omega,n,p_{z})=\quantity[\omega+i0^{+}-H_{\mathrm{Sm}}(p_{z})-\Sigma(\omega,p_{z})]^{-1}$
where the self-energy
$\Sigma(\omega,n,p_{z})=\sum_{n}T_{np_{z}}G^{R}_{\mathrm{Sc}}(\omega,n,p_{z})T_{np_{z}}^{{\dagger}}$
describes the coupling to the Sc. This allows us to write a low-energy
effective model for the Sm. Since we focus on the the quantum phase transition
characterized by the gap closing, we work with the effective Hamiltonian
$H_{\mathrm{eff}}=H_{\mathrm{Sm}}+\tilde{H}_{0}$ where the induced Hamiltonian
is $\tilde{H}_{0}=\Sigma(\omega=0)$. We can split the effective Hamiltonian
into three different contributions:

$\begin{split}\tilde{H}_{0}=\sum_{p_{z}}c_{p_{z}}^{\dagger}\tilde{\mu}(p_{z})\sigma_{0}c_{p_{z}}+\\\
+c^{\dagger}_{p_{z}}\tilde{V}_{z}(p_{z})\sigma_{z}c_{p_{z}}+\\\
+\quantity(c^{\dagger}_{p_{z}}\tilde{{\bf\it\Delta}}(p_{z})i\sigma_{y}c^{\dagger}_{-p_{z}}+\mathrm{H.c.})\,,\end{split}$
(7)

with
$\tilde{{\bf\it\Delta}}(p_{z})=\tilde{\Delta}_{0}(p_{z})\sigma_{0}+\sum_{i}\tilde{\Delta}_{i}(p_{z})\sigma_{i}$.
These three terms describe the shift in the chemical potential, the induced
exchange field and the induced superconducting gap matrix in the Sm. The
explicit forms of these contributions are

$\tilde{V}^{(1)}_{z}(p_{z})=\sum_{n}\frac{V_{z}(t_{0,np_{z}}^{2}+t_{z,np_{z}}^{2})}{\xi_{np_{z}}^{2}+\Delta_{0,np_{z}}^{2}-V_{z}^{2}}\,,$
(8)

$\tilde{V}^{(2)}_{z}(p_{z})=\sum_{n}\frac{-2\xi_{np_{z}}t_{z,np_{z}}t_{0,np_{z}}}{\xi_{np_{z}}^{2}+\Delta_{0,np_{z}}^{2}-V_{z}^{2}}\,,$
(9)

$\tilde{\Delta}_{0}(p_{z})=\sum_{n}\frac{\Delta_{0,np_{z}}(t_{0,np_{z}}^{2}-t_{z,np_{z}}^{2})}{\xi_{np_{z}}^{2}+\Delta_{0,np_{z}}^{2}-V_{z}^{2}}\,,$
(10)

where we have divided the induced Zeeman term into two contributions
$\tilde{V}_{z}=\tilde{V}_{z}^{(1)}+\tilde{V}_{z}^{(2)}$. The first one is
proportional to the splitting in the parent Sc (8), while the second one is
linked to the spin-asymmetric tunneling amplitude of the barrier $t_{z}$ (9).
If we include the energy dependence, a triplet component $\tilde{\Delta}_{z}$
is also present, vanishing for $\omega=0$. Therefore, only the singlet pairing
$\tilde{\Delta}_{0}$ is important to describe the topological transition. We
assume that the parent Sc pairing potential is constant along $p_{z}$,
$\Delta_{0,np_{z}}=\Delta_{0,n}$. This is justified since the length of the
wire is comparable to the Sc coherence length. For thin films, $\Delta_{0,n}$
is approximately constant for each band, even if different from the bulk value
[35]. For this reason, we further simplify the model by assuming the gap to be
equal for each transverse mode of the Sc, i.e. $\Delta_{0,n}=\Delta_{0}$.

##  III Results

###  III.1 Semiconductor symmetrically coupled to a spin-split superconductor

In this section, we show that the combined superconducting and exchange
proximity effects induced by the coupling to a spin-split Sc cannot induce,
alone, a topological transition in the Sm. We consider a system like the one
sketched in Fig. 2-(a), where MI induces a spin splitting in the Sc. The Sc,
in turn, induces superconductivity and an exchange field in the Sm. To check
the presence of topological phases, we calculate the ratio between the induced
exchange field, $\tilde{V}_{z}$, and the gap, $\tilde{\Delta}_{0}$. The
topological phases appear when the condition in Eq. (1) is met, which leads to
a closing of the superconducting gap at $p_{z}=0$. For this reason, we focus
on this point in momentum space in the following. A necessary condition to
satisfy the inequality is having a gap polarization ratio
$\absolutevalue{V_{\mathrm{Z}}/\Delta}>1$.

Taking the ratio of Eqs. (8) and (10) for $t_{z}=0$, we see that

$\absolutevalue{\frac{\tilde{V}_{z}}{\tilde{\Delta}_{0}}}=\absolutevalue{\frac{V_{z}}{\Delta_{0}}}\,,$
(11)

so the induced gap polarization ratio in the Sm is the same as in the Sc.

The gap polarization ratio $\absolutevalue{V^{\mathrm{Sc}}/\Delta_{0}}$ has to
be less than unity in the parent Sc, otherwise superconductivity is suppressed
in the whole device. In the case of a large homogeneous Sc, this ratio is
limited by the stricter Chandrasekhar-Clogston bound, which dictates that a
finite superconducting gap can only be maintained if
$\absolutevalue{V^{\mathrm{Sc}}/\Delta_{0}}<1/\sqrt{2}$ [36, 37]. In thin Sc
films, quantum confinement causes an increase of the superconducting pairing
amplitude leading to a superconductive phase that survives under stronger
exchange fields. However, the Chandrasekhar-Clogston limit still holds in
terms of gap polarization ratio [35]. Even below this limit, the gap parameter
in a clean Sc subjected to strong exchange fields ceases to be spatially
homogeneous [38, 39]. Therefore, it is not possible to obtain topological
phases by coupling a spin-split Sc to a Sm.

This result is independent of the mode distribution in the Sc within the
constant $\Delta_{0}$ and $V_{z}$ approximation 111 If we allow the pairing
potential to be different in each subband, the induced gap polarization ratio
could take any value in the range of
$\quantity[\min(V_{z}/\Delta_{0,n}),\max(V_{z}/\Delta_{0,n})]$ depending on
the coupling terms $T_{n}$. For a gaped superconductor all the subbands close
to the Fermi level fulfill $\Delta_{0,n}>V_{z}$. With this constrain, it is
still not possible to get topological superconductivity in a semiconductor
coupled to a spin-split superconductor through a spin-symmetric barrier. . The
same result is found for a continuous flat density of transverse modes in the
Sc where $\Delta_{0}$ is taken finite and constant for a wide range of
energies being zero otherwise, as shown in Appendix B. In this case, the
Zeeman term can be weakly enhanced, but this effect is totally negligible in
realistic systems. This result can be generalized to the case where a
multimode Sm is coupled to a multimode Sc as discussed in Appendix A.2.

###  III.2 Spin-dependent tunneling

We consider now the case of a spin-asymmetric tunneling between the Sc and the
Sm, taken as momentum independent for simplicity and described by
$T=t_{0}\sigma_{0}+t_{z}\sigma_{z}$. As Eqs. (8)–(10) show, the induced terms
in the effective Hamiltonian are dependent on the distribution of the
transverse modes in the Sc with respect to the chemical potential. In
particular, they decay with the energy difference between the bottom of the Sc
subband and the chemical potential. This means that the modes closer to the Sm
Fermi energy give the dominant contribution to the induced superconducting
pairing and exchange fields at $p_{z}=0$.

We first analyze the contribution of an isolated Sc mode to the effective
Hamiltonian, as the ones from the different modes add up. The behavior of the
induced term in the effective Hamiltonian is illustrated in Fig. 3. The two
terms of the induced exchange field $V^{(1)}_{z}$ and $V^{(2)}_{z}$ sum
constructively for $\mu_{\mathrm{Sc}}>\varepsilon_{n}$. Both $\tilde{V}_{z}$
and $\tilde{\Delta}_{0}$ decay for $\absolutevalue{\xi_{n,0}}\to\infty$,
leading to the existence of an optimal regime where both the induced exchange
field and superconducting pairing are maximal and
$\absolutevalue{\tilde{V}_{z}/\tilde{\Delta}_{0}}>1$. This is the ideal region
for searching for topological superconductivity.

Figure 3: Induced gap $\tilde{\Delta}_{0}$ (blue line), exchange field
contributions $\tilde{V}_{z}^{(1)}$ and $\tilde{V}_{z}^{(2)}$ (dashed lines),
and total spin splitting
$\tilde{V}_{z}=\tilde{V}_{z}^{(1)}+\tilde{V}_{z}^{(2)}$ (red line) at
$p_{z}=0$ induced in the Sm. We show results for a single transverse mode of a
spin-split Sc as a function of the chemical potential in the Sc,
$\mu_{\mathrm{Sc}}-\varepsilon_{0}$. Parameters: $\Delta_{0}=1$, $t_{0}=1$,
$t_{z}=0.4$, $V_{z}=0.5$. The area with gray background is topologically
trivial while the one with white background satisfies the condition
$\absolutevalue{\tilde{V}_{z}}>\tilde{\Delta}_{0}$

.

In general, a spin-dependent component $t_{i}$ in the tunneling matrix
enhances the collinear component in the exchange field $\tilde{V}_{i}$,
suppressing the non collinear components in $\tilde{\mathbf{V}}$ and the
superconducting gap in the Sm. From Eqs. (8)–(10) we see that $\tilde{\Delta}$
and $\tilde{V}_{z}^{(1)}$ share the same dependence on the Sc band structure
(they decay as $\sim\xi^{-2}$) while having a different prefactor which
depends on the tunneling matrix. Therefore, the spin splitting in the Sc leads
to an enhancement of $\absolutevalue{\tilde{V}_{z}}$ and provides a first
mechanism to induce topological superconductivity in the Sm.

In contrast to $\tilde{V}_{z}^{(1)}$, the second contribution
$\tilde{V}_{z}^{(2)}$ to the induced exchange field in the semiconductor is
totally independent of the spin polarization in the parent superconductor.
This term depends solely on the spin-asymmetric component of the tunneling
Hamiltonian. Moreover, $\tilde{V}_{z}^{(2)}$ has an energy dependence
$\sim\xi^{-1}$, with a sign that depends on the relative position of the mode
to the Sc Fermi energy. Since $\tilde{\Delta}_{0}$ and $\tilde{V}_{z}^{(1)}$
decay as $\sim\xi^{-2}$, $\tilde{V}_{z}^{(2)}$ dominates as the energy
difference between the Sm and Sc modes increases. This contribution exhibits a
sign change, which leads to a cancellation of $\tilde{V}_{z}^{(2)}$ in the
limit of small energy separation between modes. However, in thin Scs, the
transverse modes can exhibit a large energy separation because of quantum
confinement effects.

This result suggests that there are two different mechanisms that can drive
the system to the topological phase. First, the spin-dependent tunneling
enhances the induced gap-polarization ratio of spin-split superconductor.
Alternatively, the spin-dependent tunneling combined with the quantum
confinement in the Sc can lead to the appearance of the topological phase in
regimes where, otherwise, the phase diagram would be globally trivial.

###  III.3 Combined proximity effect dominated by one transverse mode

If the separation between transverse modes in the Sc is large enough, the
contribution of the mode closest to the chemical potential dominates over the
other ones. We can visualize this system like two coupled one-dimensional
wires, where one features both spin splitting and superconductivity while the
other features only spin-orbit coupling. In the rest of this section, we set
the energy of the bottom of the single Sc band to zero, $\varepsilon_{0}=0$,
as it only causes a shift of the energy scale. In this way
$\mu_{\mathrm{Sc}}=-\xi_{0,p_{z}=0}$.

The topological transition can occur when the induced exchange field is larger
than the induced gap. In this specific case, both the spin splitting and the
superconducting pairing are inherited by the parent Sc, while the effective
chemical potential is the sum of the bare nanowire electrochemical potential
and the renormalization term induced by the proximity effect. To identify the
critical lines, we impose the condition
$\tilde{V}_{z}^{2}=(\mu_{\mathrm{Sm}}+\tilde{\mu})^{2}+\tilde{\Delta}_{0}^{2}$.
To simplify the calculation, we take the limit $\Delta_{0}\to 0$ as the
chemical potential of the Sc is usually much bigger than the gap parameter. In
this way, we find the analytical conditions

$\begin{split}t_{z}=+t_{0}\pm\sqrt{+V_{z}\mu_{\mathrm{Sm}}+\mu_{\mathrm{Sm}}\mu_{\mathrm{Sc}}}\,,\\\
t_{z}=-t_{0}\pm\sqrt{-V_{z}\mu_{\mathrm{Sm}}+\mu_{\mathrm{Sm}}\mu_{\mathrm{Sc}}}\,.\end{split}$
(12)

This result allows us to identify the regions in the parameter space where the
topological phases exist. The same result can be obtained by calculating
analytically the Pfaffian invariant of this system, as shown in Appendix C. An
illustration of the phase diagram of this simple system is shown in Fig. 4,
where topological superconductivity appears for a relatively wide range of
parameters. Except for very low tunneling amplitudes, topological
superconductivity is present in the region where $t_{z}/t_{0}\sim 1$, which is
the point corresponding to one spin tunneling component being completely
suppressed (perfect spin filter). A strong polarization of the barrier,
however, tends to suppress the induced gap, closing it for $t_{z}/t_{0}=1$.
The topological region can be enlarged by controlling the chemical potential
of the Sm and Sc. The size of the topological region depends on the product
$\mu_{\mathrm{Sm}}\mu_{\mathrm{Sc}}$, as shown in Eqs. (12).

Figure 4: Phase diagram for a single mode Sm coupled to a single mode Sc. We
represent the effective gap in the Sm
$\absolutevalue{\Delta_{\mathrm{eff}}}=\min_{n,p_{z}}E_{n,p_{z}}$, where
$E_{n,p_{z}}$ are the Hamiltonian eigenvalues and the sign is given by the
topological invariant $W_{\mathbb{Z}_{2}}$ (Appendix C). The black dashed
lines demarcate the boundaries of the topological region (red color),
described by Eqs. (12). We have set $V_{z}=0$, which leads to
$\tilde{V}_{z}^{(1)}=0$. The remaining parameters are
$\mu_{\mathrm{Sc}}/\Delta_{0}=10$, $m^{*}_{\mathrm{Sc}}=1$,
$m^{*}_{\mathrm{Sm}}=0.25$, $\alpha_{x}/\Delta_{0}=5$. For the left panel we
take $t_{0}=5$ while for the right panel $\mu_{\mathrm{Sm}}/\Delta_{0}=1$.

The first conclusion is that the effect of the spin splitting of the Sc is
negligible for realistic parameters, as it appears summed to the Sc Fermi
energy which is many orders of magnitude bigger [Eqs. (12)]. Analyzing the
behavior with respect to the tunneling parameters, it can be noticed that
there exists a region corresponding to
$t_{0}\sim\pm\sqrt{-V_{z}\mu_{\mathrm{Sm}}+\mu_{\mathrm{Sm}}\mu_{\mathrm{Sc}}}$,
where a small polarization of the barrier can induce a topological phase with
a relatively big gap. Since the chemical potential in the nanowire is tunable
using electrostatic gates, it is, in principle, possible to set the operating
point of the device near this optimal point.

###  III.4 Superconductor with several relevant transverse modes

In a more realistic scenario, the Sc features many transverse modes. In this
case, Eqs. (8)–(10) explain how the induced terms in the Sm effective
Hamiltonian are determined by the sum of the contributions of each mode. The
relative energy of the transverse modes in the Sc strongly affects the gap
polarization in the Sm, becoming a critical factor for the appearance of the
topological phase. For the systems of interest, the dimensions of the Sc
section vary from a few to hundreds of nanometers. For small length scales,
the effect of quantum confinement separates the Sc subbands such that it is
not possible to treat the density of state like a continuum. Both
$\tilde{\Delta}_{0}$ and $\tilde{V}_{z}^{(1)}$ have a Lorentzian shape with a
full width at half maximum $\Gamma=\sqrt{\Delta_{0}^{2}-V_{z}^{2}}$. We can
use $\Gamma$ as a reference to distinguish between discrete modes,
$\delta\varepsilon\gg\Gamma$, and a continuum distribution of modes,
$\delta\varepsilon\ll\Gamma$, where $\delta\varepsilon$ is the average
separation of modes. Indeed, if the average energy separation between the
transverse modes is $\delta\varepsilon\gtrsim\Gamma$, the resonance peaks do
not overlap entirely and the resonance peaks of $V_{z}^{(2)}$ do not
completely cancel out. In this case, the net exchange field experienced in the
Sm will be due to the sum of the contribution of each transverse mode.

Figure 5: Induced gap $\tilde{\Delta}_{0}$ (blue line), exchange field
contributions $\tilde{V}^{(1)}_{z}$ and $\tilde{V}^{(2)}_{z}$ (dashed lines)
and total spin splitting
$\tilde{V}_{z}=\tilde{V}_{z}^{(1)}+\tilde{V}_{z}^{(2)}$ (red line) at
$p_{z}=0$ in the Sm in the case of a multimode Sc. Parameters: $\Delta_{0}=1$,
$t_{0}=1$, $t_{z}=0.4$, $V_{z}=0.5$, $\delta\varepsilon=10$. The area with
gray background is topologically trivial while the one with white background
satisfies the condition $\absolutevalue{\tilde{V}_{z}}>\tilde{\Delta}_{0}$.
The effect of the resonance peaks in $\tilde{V}_{z}^{(2)}$ cause an
oscillation between topologically trivial and nontrivial phases while varying
the chemical potential in the Sc.

To provide a clearer picture, we consider the case where the band structure of
the Sc can be described by perfectly equidistant transverse modes. In this
case, $\delta\varepsilon$ is the relative separation between the Sc modes.
This approximation recovers the continuum limit for a 2D system, where we
expect a constant density of transverse modes
$g_{m}=(\delta\varepsilon)^{-1}$. We also define $\delta\mu_{\mathrm{Sc}}$ as
the relative position of the Fermi energy in the Sc from the middle of the
band. We proceed by summing up the modes contributions following Eqs. (8)–(10)
to calculate the value of the induced terms in the effective Hamiltonian
(calculation details can be found in Appendix D). The behavior of the induced
terms for $p_{z}=0$ is illustrated in Fig. 5.

By varying the chemical potential in the Sc, as the subbands cross the Fermi
energy we see an alternation of trivial regions (gray background) and regions
where topological superconductivity can appear by tuning $\mu_{\mathrm{Sm}}$
(white background). This behavior is more clearly illustrated by the
topological invariant and the effective gap, shown in Fig. 6. In this figure,
the effective gap is given by the smallest eigenvalue of the effective
Hamiltonian (7).

Figure 6:  Phase diagram for a single mode Sm coupled to a multimode Sc,
showing the effective gap in the Sm
$\absolutevalue{\Delta_{\mathrm{eff}}}=\min_{n,p_{z}}E_{n,p_{z}}$, where
$E_{n,p_{z}}$ are the Hamiltonian eigenvalues and sign given by the
topological invariant $W_{\mathbb{Z}_{2}}$. We observe an oscillation between
the topological (red) and trivial (blue) phases as we vary the chemical
potential measured from the middle of the band $\delta\mu_{\mathrm{Sc}}$. As
the density of transverse modes $g_{m}$ increases, the peaks get closer and
overlap until they merge in the continuous limit $g_{m}\gg(\Delta_{0})^{-1}$.
In the continuum limit, the system is in a globally trivial or topological
phase, depending on the condition in Eq. (14). We assume that the chemical
potential in the Sm is tuned such that $\mu_{\mathrm{Sm}}=-\tilde{\mu}$.
Parameters used: $m^{*}_{\mathrm{Sc}}=1$, $m^{*}_{\mathrm{Sm}}=0.2$,
$\alpha_{x}/\Delta_{0}=20$, $V_{z}/\Delta_{0}=0.5$, $t_{0}/\Delta_{0}=1$. For
the left panel we use $t_{z}/\Delta_{0}=0.4$, while for the right panel we use
$t_{z}/\Delta_{0}=0.7$.

The observation of an alternation of topological and trivial phases while
varying $\delta\mu_{\mathrm{Sc}}$ is consistent with Fig. 5. The limit
$g_{m}\ll 1/\Delta_{0}$ corresponds to a large Sc where the effect of quantum
confinement becomes completely negligible. This behavior can be realized if
the average separation of the transverse modes is such that
$\delta\varepsilon\ll\Gamma$. In this limit, the contributions from each mode
overlap leading to a globally trivial or topological phase, depending on the
polarization of the spin-filter barrier $t_{z}/t_{0}$. A quantitative
criterion can be obtained by integrating Eqs. (8)–(10) over a constant density
of transverse modes as shown in Appendix B. In this limit, the term in Eq. (9)
vanishes and we get for the gap polarization

$\absolutevalue{\frac{\tilde{V}_{z}}{\tilde{\Delta}_{0}}}=\absolutevalue{\frac{V_{z}}{\Delta_{0}}}\absolutevalue{\frac{t_{0}^{2}+t_{z}^{2}}{t_{0}^{2}-t_{z}^{2}}}\,.$
(13)

We note that the spin-dependent tunneling leads to an enhancement of the
induced exchange field, while reducing the induced superconducting gap. This
effect can be used to bring the induced gap polarization ratio above one,
closing the gap at $p_{z}=0$ and inducing a quantum phase transition to the
topological phase. Therefore, the spin-asymmetric tunneling provides a way to
overcome the limitation of Eq. (11) and to induce a phase transition to the
topological phase, Fig. 2-(b). The topological phase appears for a barrier
polarization

$\frac{t_{z}}{t_{0}}>\sqrt{\frac{1-\absolutevalue{V_{z}/\Delta_{0}}}{1+\absolutevalue{V_{z}/\Delta_{0}}}}\,.$
(14)

##  IV Discussion

In this section, we discuss the applicability of the spin tunneling mechanism
for generating topological superconductivity to the case of EuS-Al-InAs
platforms.

EuS is a magnetic semiconductor with an optical bandgap of
$E_{g}=$1.65\text{\,}\mathrm{eV}$$, which can be effectively tuned by quantum
confinement and, therefore, it depends on the film thickness [41]. The
magnetic properties of this material are attributed to the Eu atoms which are
characterized by strongly localized half-filled $4f$ shells. They behave as
localized spins with a magnetic moment of around $7\mu_{B}$. For this reason,
the EuS can be well described as a Heisenberg ferromagnet with a Curie
temperature of $T_{m}=$16.6\text{\,}\mathrm{K}$$ [42, 43]. The magnetization
induces a splitting of the conduction band of around $\Delta
E_{c}=$0.36\text{\,}\mathrm{eV}$$ [44]. These properties are very promising in
the view of fabricating spin-filter tunnel barriers.

As discussed in the previous section, the behavior of the system is strongly
dependent on the band structure of the Sc, which is controlled by the
dimension of the Al region. If a wide Al shell is used, the transverse modes
of the nanowire will be densely distributed in the energy spectrum. In this
case, $\tilde{V}_{z}^{(2)}$ vanishes and topological superconductivity can
only be induced using the spin-asymmetric tunneling to enhance
$\tilde{V}_{z}^{(1)}$ and suppress $\tilde{\Delta}_{0}$, Eq. (13).

To check if the ferromagnetic hybrid nanowire is in this regime, we can
estimate the average mode separation by a simple particle in a box model
$\delta\varepsilon\simeq\frac{\hbar^{2}}{2m^{*}_{\mathrm{Sc}}}\frac{\pi^{2}}{L_{\mathrm{Sc}}^{2}}$,
where $L_{\mathrm{Sc}}$ refers to the largest dimension of the cross section
of the Sc shell. Assuming that
$\Gamma\sim$100\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$, which is in line
with the experimental measurements of Al-EuS heterostructures [31, 20], we
estimate that, in order to observe well-separated modes, the maximum dimension
of the shell should be in the order of $60\text{\,}\mathrm{nm}$. In
experiments, the facet length is around $60\text{\,}\mathrm{nm}$ [20]. For
this reason, we expect the contributions of the modes in the Sc to overlap
significantly.

Previous measurements performed on EuS-Al heterostructures have shown a
polarization of around 50% of the gap [31]. In this case, applying (14), we
can estimate that a spin-dependent barrier with a 58% polarization is enough
to bring the Sm to the topological regime.

Estimating the optimal magnetic barrier length to achieve this polarization is
a complicated task as strong spin-polarized band bending is expected at the
interfaces. As the barrier gets thicker, we expect a stronger polarization,
but at the same time, the coupling between the two systems gets strongly
suppressed. Therefore, the optimal barrier length would be determined by the
trade-off between a strongly polarizing thick barrier, which, however, suffers
low transparency, and a thin transparent barrier with low polarization.

The introduction of MI in 2DEG is more challenging as the geometry and
fabrication constraints make the introduction of Sc-Sm, MI-Sm, and MI-Sc
interfaces at the same time particularly complicated. The use of MI as a spin-
filter tunnel barrier simplifies the design of topological quantum devices by
eliminating the need of the Sc-Sm interface. Using this new operating
principle, the geometry sketched in Fig. 2-(b) can provide a viable
alternative to achieve zero-field topological superconductivity in 2DEG
devices.

Finally, we note that both nanowires Sc shell and Sc layers in 2DEG systems
are around $5\text{\,}\mathrm{nm}$ thick, so we can already expect a
measurable effect of quantum confinement in this direction. For this reason,
we expect that the resonance effect discussed previously can be measured by
scaling down the width of the devices.

##  V Conclusions

In this work, we have demonstrated that a spin-split superconductor cannot
induce, alone, topological superconductivity in a spin-orbit coupled
semiconductor by the combined superconducting and magnetic proximity effect.
This is in contradiction with the hypothesis that an exchange field in the
superconductor can induce a topological transition in the semiconductor. We
have shown how a spin-filter tunnel barrier, provided by a thin magnetic
insulating layer between the semiconductor and the superconductor, can be used
to overcome this limitation. The spin-dependent tunneling suppresses the
induced superconducting pairing potential while enhancing the spin splitting,
thus providing a way to bring the system to the topological phase. In the case
of a distribution of discrete modes in the superconductor, the phase diagram
exhibits an alternation of trivial and topological regions as a function of
the chemical potential of the superconductor due to a change on the sign in
the contributions to the exchange field. The total induced exchange field
strongly depends on the relative energy difference between the closest levels
to the Fermi energy in the superconductor. While this mechanism is unlikely to
explain the results in Ref. [20] because of the relatively thick EuS layer
used, the concept of spin-dependent coupling can be exploited in the next
generation of topological superconducting devices without magnetic field. Spin
filter tunnel barriers can be achieved by depositing a thin film of few
nanometers of a ferromagnetic insulator at the interface between the
superconductor and the semiconductor. The proposed mechanism is compatible
with the currently used hybrid superconductor-semiconductor platforms,
including nanowires and 2DEG systems.

_Note added._ Recently, several independent papers on the same subject have
been made publicly available as preprints.

The result concerning the impossibility of topological phases by
superconductor-mediated exchange field has been extended to generic systems in
terms of a more general requirement about the minimal Zeeman field required
for a topological phase transition [45].

The ferromagnetic hybrid nanowire physics has been explored using a self-
consistent treatment of a diffusive superconductor in Ref. [46]. The results
are in agreement with the ones presented in our work. Another independent work
has found that perfectly clean systems with an extremely thin superconductive
layer could be used to obtain topological phases in the MI-Sc-Sm stack [47].

##  VI Acknowledgments

This research was supported by the Danish National Research Foundation, the
Danish Council for Independent Research — Natural Sciences, and the Microsoft
Corporation. This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme under Grant Agreement No. 856526. R.S.S. and M.L. acknowledge
funding from QuantERA project “2D hybrid materials as a platform for
topological quantum computing” and from NanoLund.

## References

  * Oreg  _et al._ [2010] Y. Oreg, G. Refael, and F. Von Oppen, Phys. Rev. Lett. 105, 177002 (2010), arXiv:1003.1145 .
  * Lutchyn  _et al._ [2010] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010).
  * Kitaev [2001] A. Y. Kitaev, Physics-Uspekhi 44, 131 (2001), arXiv:0010440 [cond-mat] .
  * Nayak  _et al._ [2008] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008), arXiv:0707.1889 .
  * Lutchyn  _et al._ [2018] R. M. Lutchyn, E. P. Bakkers, L. P. Kouwenhoven, P. Krogstrup, C. M. Marcus, and Y. Oreg, Nat. Rev. Mater. 3, 52 (2018), arXiv:1707.04899 .
  * Vaitiekenas  _et al._ [2018] S. Vaitiekenas, A. M. Whiticar, M. T. Deng, F. Krizek, J. E. Sestoft, C. J. Palmstrøm, S. Marti-Sanchez, J. Arbiol, P. Krogstrup, L. Casparis, and C. M. Marcus, Phys. Rev. Lett. 121, 1 (2018), arXiv:1802.04210 .
  * Shabani  _et al._ [2016] J. Shabani, M. Kjaergaard, H. J. Suominen, Y. Kim, F. Nichele, K. Pakrouski, T. Stankevic, R. M. Lutchyn, P. Krogstrup, R. Feidenhans’L, S. Kraemer, C. Nayak, M. Troyer, C. M. Marcus, and C. J. Palmstrøm, Phys. Rev. B 93, 1 (2016), arXiv:1511.01127 .
  * Pientka  _et al._ [2017] F. Pientka, A. Keselman, E. Berg, A. Yacoby, A. Stern, and B. I. Halperin, Phys. Rev. X 7, 021032 (2017).
  * Hell  _et al._ [2017] M. Hell, M. Leijnse, and K. Flensberg, Phys. Rev. Lett. 118 (2017), 10.1103/PhysRevLett.118.107701, arXiv:1608.08769 .
  * Fornieri  _et al._ [2019] A. Fornieri, A. M. Whiticar, F. Setiawan, E. Portolés, A. C. C. Drachmann, A. Keselman, S. Gronin, C. Thomas, T. Wang, R. Kallaher, G. C. Gardner, E. Berg, M. J. Manfra, A. Stern, C. M. Marcus, and F. Nichele, Nature 569, 89 (2019).
  * Alicea  _et al._ [2011] J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, Nature Physics 7, 412 (2011).
  * Aasen  _et al._ [2016] D. Aasen, M. Hell, R. V. Mishmash, A. Higginbotham, J. Danon, M. Leijnse, T. S. Jespersen, J. A. Folk, C. M. Marcus, K. Flensberg, and J. Alicea, Phys. Rev. X 6, 1 (2016), arXiv:1511.05153 .
  * Karzig  _et al._ [2017] T. Karzig, C. Knapp, R. M. Lutchyn, P. Bonderson, M. B. Hastings, C. Nayak, J. Alicea, K. Flensberg, S. Plugge, Y. Oreg, C. M. Marcus, and M. H. Freedman, Phys. Rev. B 95, 1 (2017), arXiv:1610.05289 .
  * Fu and Kane [2008] L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008), arXiv:0707.1692 .
  * Sau  _et al._ [2010] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, Phys. Rev. Lett. 104, 1 (2010), arXiv:0907.2239 .
  * Liu  _et al._ [2020a] Y. Liu, S. Vaitiekėnas, S. Martí-Sánchez, C. Koch, S. Hart, Z. Cui, T. Kanne, S. A. Khan, R. Tanta, S. Upadhyay, M. E. Cachaza, C. M. Marcus, J. Arbiol, K. A. Moler, and P. Krogstrup, Nano Lett. 20, 456 (2020a), arXiv:1910.03364 .
  * Blümel  _et al._ [2019] N. V. Blümel, A. Goschew, Y. A. Shokr, and P. Fumagalli, AIP Adv. 9, 035016 (2019).
  * Liu  _et al._ [2020b] Y. Liu, A. Luchini, S. Martí-Sánchez, C. Koch, S. Schuwalow, S. A. Khan, T. Stankevič, S. Francoual, J. R. Mardegan, J. A. Krieger, V. N. Strocov, J. Stahn, C. A. Vaz, M. Ramakrishnan, U. Staub, K. Lefmann, G. Aeppli, J. Arbiol, and P. Krogstrup, ACS Appl. Mater. Interfaces 12, 8780 (2020b), arXiv:1908.07096 .
  * Manna  _et al._ [2020] S. Manna, P. Wei, Y. Xie, K. T. Law, P. A. Lee, and J. S. Moodera, Proc. Natl. Acad. Sci. , 201919753 (2020), arXiv:1911.03802 .
  * Vaitiekėnas  _et al._ [2020] S. Vaitiekėnas, Y. Liu, P. Krogstrup, and C. M. Marcus, Nature Physics (2020), 10.1038/s41567-020-1017-3.
  * Zhang  _et al._ [2020] X. P. Zhang, V. N. Golovach, F. Giazotto, and F. S. Bergeret, Phys. Rev. B 101, 180502 (2020), arXiv:2002.08230 .
  * Woods and Stanescu [2020] B. D. Woods and T. D. Stanescu, arXiv:2011.01933 (2020).
  * Escribano  _et al._ [2020] S. D. Escribano, E. Prada, Y. Oreg, and A. L. Yeyati, arXiv:2011.06566 (2020).
  * Liu  _et al._ [2020c] C.-X. Liu, S. Schuwalow, Y. Liu, K. Vilkelis, A. L. R. Manesco, P. Krogstrup, and M. Wimmer, arXiv:2011.06567 (2020c).
  * Millis  _et al._ [1988] A. Millis, D. Rainer, and J. A. Sauls, Phys. Rev. B 38, 4504 (1988).
  * Tokuyasu  _et al._ [1988] T. Tokuyasu, J. A. Sauls, and D. Rainer, Phys. Rev. B 38, 8823 (1988).
  * Tedrow  _et al._ [1986] P. M. Tedrow, J. E. Tkaczyk, and A. Kumar, Phys. Rev. Lett. 56, 1746 (1986).
  * Hao  _et al._ [1990] X. Hao, J. S. Moodera, and R. Meservey, Phys. Rev. B 42, 8235 (1990).
  * Ren  _et al._ [2007] C. Ren, J. Trbovic, R. L. Kallaher, J. G. Braden, J. S. Parker, S. Von Molnár, and P. Xiong, Phys. Rev. B - Condens. Matter Mater. Phys. 75, 1 (2007).
  * Wolf  _et al._ [2014] M. J. Wolf, C. Sürgers, G. Fischer, and D. Beckmann, Phys. Rev. B - Condens. Matter Mater. Phys. 90, 1 (2014), arXiv:1408.0176 .
  * Strambini  _et al._ [2017] E. Strambini, V. N. Golovach, G. De Simoni, J. S. Moodera, F. S. Bergeret, and F. Giazotto, Phys. Rev. Mater. 1, 054402 (2017), arXiv:1705.04795 .
  * Bergeret  _et al._ [2018] F. S. Bergeret, M. Silaev, P. Virtanen, and T. T. Heikkilä, Rev. Mod. Phys. 90, 041001 (2018).
  * Miao and Moodera [2015] G. X. Miao and J. S. Moodera, Phys. Chem. Chem. Phys. 17, 751 (2015).
  * Kamra  _et al._ [2018] A. Kamra, A. Rezaei, and W. Belzig, Phys. Rev. Lett. 121, 247702 (2018), arXiv:1806.10356 .
  * Wójcik [2015] P. Wójcik, J. Supercond. Nov. Magn. 28, 339 (2015).
  * Chandrasekhar [1962] B. S. Chandrasekhar, Appl. Phys. Lett. 1, 7 (1962).
  * Clogston [1962] A. M. Clogston, Phys. Rev. Lett. 9, 266 (1962).
  * Fulde and Ferrell [1964] P. Fulde and R. A. Ferrell, Phys. Rev. 135 (1964), 10.1103/PhysRev.135.A550.
  * Larkin and Ovchinnikov [1964] A. I. Larkin and Y. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 47, 1136 (1964).
  * Note [1] If we allow the pairing potential to be different in each subband, the induced gap polarization ratio could take any value in the range of $\quantity[\mathop{min}\displaylimits(V_{z}/\Delta_{0,n}),\mathop{max}\displaylimits(V_{z}/\Delta_{0,n})]$ depending on the coupling terms $T_{n}$. For a gaped superconductor all the subbands close to the Fermi level fulfill $\Delta_{0,n}>V_{z}$. With this constrain, it is still not possible to get topological superconductivity in a semiconductor coupled to a spin-split superconductor through a spin-symmetric barrier.
  * Poulopoulos  _et al._ [2012] P. Poulopoulos, B. Lewitz, A. Straub, S. D. Pappas, S. A. Droulias, S. Baskoutas, and P. Fumagalli, Appl. Phys. Lett. 100, 211910 (2012).
  * Müller and Nolting [2002] W. Müller and W. Nolting, Phys. Rev. B - Condens. Matter Mater. Phys. 66, 1 (2002), arXiv:0206449 [cond-mat] .
  * Müller and Nolting [2004] W. Müller and W. Nolting, Phys. Rev. B - Condens. Matter Mater. Phys. 69, 1 (2004), arXiv:0403075 [cond-mat] .
  * Wachter [1972] P. Wachter, C R C Crit. Rev. Solid State Sci. 3, 189 (1972).
  * Pöyhönen  _et al._ [2020] K. Pöyhönen, D. Varjas, M. Wimmer, and A. R. Akhmerov, arXiv:2011.08263 , 1 (2020), arXiv:2011.08263 .
  * Khindanov  _et al._ [2020] A. Khindanov, J. Alicea, P. Lee, W. S. Cole, and A. E. Antipov, arXiv:2012.12934 , 1 (2020), arXiv:2012.12934 .
  * Langbehn  _et al._ [2020] J. Langbehn, S. A. Gonzalez, P. W. Brouwer, and F. von Oppen, arXiv:2012.00055 (2020), arXiv:2012.00055 .
  * Stanescu and Das Sarma [2017] T. D. Stanescu and S. Das Sarma, Phys. Rev. B 96, 1 (2017), arXiv:1702.03976 .
  * Tewari and Sau [2012] S. Tewari and J. D. Sau, Phys. Rev. Lett. 109, 1 (2012), arXiv:1111.6592 .

##  Appendix A Complete self-energy calculation

In this appendix, we discuss more in detail the self-energy model employed to
derive the effective Hamiltonian discussed in Eq. (7). We derive the model,
discussing the frequency dependence, the presence of multiple modes in the Sm,
and the effect of a general hopping matrix with arbitrary direction of the
magnetization axis. Our model consists in a translation-invariant system
composed by a Sm region coupled to a Sc in which an exchange field is induced
by the nearby MI. This system is described by the Hamiltonian

$H=H_{\mathrm{Sm}}+H_{\mathrm{Sc}}+H_{\mathrm{t}}\,,$ (15)

where the three terms read

$\displaystyle H_{\mathrm{Sm}}$
$\displaystyle=\sum_{mp_{z}}\zeta_{m}(p_{z})c_{p_{z}}^{\dagger}\sigma_{0}c_{p_{z}}+\sum_{p_{z}}p_{z}c_{p_{z}}^{\dagger}\alpha_{x}\sigma_{y}c_{p_{z}}\,,$
(16) $\displaystyle H_{\mathrm{Sc}}$
$\displaystyle=\sum_{np_{z}}\xi_{n}(p_{z})a_{np_{z}}^{\dagger}\sigma_{0}a_{np_{z}}+V_{z}a^{\dagger}_{np_{z}}\sigma_{z}a_{np_{z}}+\quantity[a^{\dagger}_{np_{z}}(i\sigma_{y}{\bf\it\Delta}_{np_{z}})a^{\dagger}_{n-p_{z}}+\mathrm{H.c.}]\,,$
(17) $\displaystyle H_{\mathrm{t}}$
$\displaystyle=\sum_{mnp_{z}}c^{\dagger}_{mp_{z}}T_{mn}(p_{z})a_{np_{z}}+\mathrm{H.c.}$
(18)

We adopt the spinor basis $c_{p_{z}}=(c_{p_{z}\uparrow}c_{p_{z}\downarrow})$,
where $c_{p_{z}\sigma}$ is the electron annihilation operator in the nanowire
for spin $\sigma$. Both the Sm and the Sc have a multimode structure. In Eq.
(16), we label the Sm modes by the index $m$ and we write the kinetic energy
as
$\zeta_{m}(p_{z})=\varepsilon_{m}+\frac{p_{z}^{2}}{2m^{*}_{\mathrm{Sm}}}-\mu_{\mathrm{Sm}}$,
where $m_{\mathrm{Sm}}^{*}$ and $\alpha_{x}$ are the electron effective mass
and the spin-orbit coupling in the Sm. The Hamiltonian in Eq. (17) describes
the Sc, where $a_{np_{z}}$ is the electron annihilation operator for mode $n$
with kinetic energy as
$\xi_{n}(p_{z})=\varepsilon_{n}+\frac{p_{z}^{2}}{2m^{*}_{\mathrm{Sc}}}-\mu_{\mathrm{Sc}}$.
Here, ${\bf\it\Delta}_{np_{z}}=\Delta_{0,np_{z}}\sigma_{0}$ is the
superconducting pairing for each band. We neglect superconducting interband
coupling and the proximity-induced exchange field by MI in the Sc, $V_{z}$.
The two regions are coupled by a spin-dependent tunneling Hamiltonian with
hopping matrix $T_{mn}(p_{z})$, which takes into account all the different
electron transmission processes taking place at the interfaces between the
different materials in the device. We can write $T_{mnp_{z}}$ in the basis of
Pauli matrices:

$T_{mn}(p_{z})=t_{0,mn}(p_{z})\sigma_{0}+\sum_{l=xyz}t_{l,mn}(p_{z})\sigma_{l}\,,$
(19)

where the $t_{l,mn}=t_{l,mn}^{\prime}(p_{z})+it_{l,mn}^{\prime\prime}(p_{z})$
parameters are, in general, complex numbers.

In the following, we will use the basis of time-reversed pairs, where
$\tau_{i}$ are Pauli matrices for the particle-hole space, being $\mathcal{H}$
the total Hamiltonian in this basis. In this basis, the bare Sc retarded
Green’s function reads

$\begin{split}G^{R}_{\mathrm{Sc}}(\omega,n,p_{z})&=\quantity[(\omega+i0^{+})\tau_{0}\sigma_{0}-\mathcal{H}_{\mathrm{Sc}}]^{-1}\\\
&=\quantity[(\omega+i0^{+})\tau_{0}\sigma_{0}-V_{z}\tau_{0}\sigma_{z}-\xi_{np_{z}}\tau_{z}\sigma_{0}-\Delta_{np_{z}}\tau_{x}\sigma_{0}]^{-1}=\\\
&=\frac{1}{(\omega+i0^{+}-V_{z}\sigma)^{2}-\xi_{np_{z}}^{2}-\Delta_{0,np_{z}}^{2}}\quantity[\quantity(\omega+i0^{+}-V_{z}\sigma)\tau_{0}+\xi_{np_{z}}\tau_{z}+\Delta_{np_{z}}\tau_{x}]\,.\end{split}$
(20)

The full Green’s function for the semiconductor subspace reads

$G^{R}_{\mathrm{Sm}}(\omega,p_{z})=\frac{1}{\omega+i0^{+}-\mathcal{H}_{\mathrm{Sm}}-\Sigma(\omega,p_{z})}$
(21)

where $\mathcal{H}_{\mathrm{Sm}}$ is the bare Sm Hamiltonian matrix and we
have defined the self-energy as

$\quantity[\Sigma(\omega,p_{z})]_{mm^{\prime}}=\sum_{n}\mathcal{H}_{\mathrm{t,mn}}(p_{z})G^{R}_{\mathrm{Sc}}(\omega,n,p_{z})\mathcal{H}_{\mathrm{t,m^{\prime}n}}^{{\dagger}}(p_{z})\,,$
(22)

where the $\mathcal{H}_{\mathrm{t}}$ matrix is the tunneling Hamiltonian
written in particle-hole space

$\mathcal{H}_{\mathrm{t,mn}}=(t_{0,mn}^{\prime}(p_{z})\tau_{z}+it_{0,mn}^{\prime\prime}(p_{z})\tau_{0})\sigma_{0}+\sum_{l}(t_{l,mn}^{\prime}(p_{z})\tau_{0}+it_{l,mn}^{\prime\prime}(p_{z})\tau_{z})\sigma_{l}\,.$
(23)

In the next sections, we will explore in more detail different situations
outlined in the main text adopting case-by-case assumptions and
approximations.

###  A.1 Frequency dependence

We now focus on the frequency dependence of the Sm Green’s function. For this
reason, we consider a single-band Sm, dropping the index $m$. We simplify the
hopping matrix by assuming that the tunneling matrix has the following form
$T_{n,p_{z}}=t_{0}\sigma_{0}+t_{z}\sigma_{z}$ with $t_{0}$ and $t_{z}$ real
and positive. We proceed by splitting the self-energy like
$\Sigma(\omega,p_{z})=\tilde{E}(\omega,p_{z})+\tilde{H}(\omega,p_{z})$ where
$\tilde{E}\propto\tau_{0}\sigma_{0}$ while the frequency-dependent effective
Hamiltonian $\tilde{H}(\omega,p_{z})$ contains all the other
$\tau_{i}\sigma_{j}$ terms. We study a low-energy model for the semiconductor
valid in the limit $\omega\to 0$. Since we have taken the tunneling amplitude
energy independent, we can directly Taylor expand the Sc Green’s function and
consider the lowest orders in $\omega$:

$\begin{split}G^{R}_{\mathrm{Sc}}(\omega,n,p_{z})=&+\frac{1}{V_{z}^{2}-\xi_{np_{z}}^{2}-\Delta_{0,np_{z}}^{2}}\quantity[-V_{z}\sigma_{z}\tau_{0}+\xi_{np_{z}}\sigma_{0}\tau_{z}+\Delta_{0,np_{z}}\sigma_{0}\tau_{x}]+\\\
&+\frac{\omega}{(V_{z}^{2}-\xi_{np_{z}}^{2}-\Delta_{0,np_{z}}^{2})^{2}}\quantity[-(V_{z}^{2}+\Delta_{0,np_{z}}^{2}+\xi_{np_{z}}^{2})\tau_{0}\sigma_{0}+2V_{z}\Delta_{0,np_{z}}\sigma_{z}\tau_{x}+2V\xi_{np_{z}}\sigma_{z}\tau_{z}]+\\\
&+O(\omega^{2})\,.\end{split}$ (24)

Introducing the expansion in the self-energy term and keeping the zeroth and
first order for each component, we see that the $\tau_{0}\sigma_{0}$ component
reads

$\tilde{E}(\omega,p_{z})=-\omega\sum_{n}\frac{(t_{0}^{2}+t_{z}^{2})(\xi_{np_{z}}^{2}+\Delta_{0,np_{z}}^{2}+V_{z}^{2})-4t_{0}t_{z}V\xi_{np_{z}}}{(\xi^{2}_{np_{z}}+\Delta_{0,np_{z}}^{2}-V_{z}^{2})^{2}}+O(\omega^{2})\,.$
(25)

We further split the effective Hamiltonian according to
$\tilde{H}(\omega,p_{Z})=\tilde{H}_{0}+\tilde{H}_{1}(\omega,p_{z})+O(\omega^{2})$.
The terms in $\tilde{H}_{0}$ have the same form as the ones in Eqs. (8)–(10)
in the main text and do not depend on the energy. The first-order terms in
$\omega$ are
$\tilde{\mathcal{H}}_{1}(\omega,p_{z})=\tilde{\Delta}_{z}(\omega,p_{z})\tau_{x}\sigma_{z}+P(\omega,p_{z})\tau_{z}\sigma_{z}$
where

$\tilde{\Delta}_{z}(\omega,p_{z})=-\sum_{n}\omega\frac{2(t_{0}^{2}-t_{z}^{2})V_{z}\Delta_{0,np_{z}}}{(\xi^{2}_{np_{z}}+\Delta_{0,np_{z}}^{2}-V_{z}^{2})^{2}}\,$
(26)

and we denoted by $P(\omega,p_{z})$ the remaining $\tau_{z}\sigma_{z}$
component.

The semiconductor Green’s function can be rewritten like

$G_{\mathrm{Sm}}(p_{z})=\frac{Z(p_{z})}{\omega-Z(p_{z})\quantity(\mathcal{H}_{\mathrm{Sm}}(p_{z})-\tilde{\mathcal{H}}_{0}(p_{z})+\tilde{\mathcal{H}}_{1}(\omega,p_{z}))}\,,$
(27)

where $Z(p_{z})=(1+C(p_{z}))^{-1}$ with $\tilde{E}=-\omega C(p_{z})$.

We can see that the first-order terms are proportional to
$\omega/\sqrt{\xi_{np_{z}}^{2}+\Delta_{0,np_{z}}^{2}-V_{z}^{2}}<\frac{\omega}{\Gamma}$.
Therefore, provided that we are interested in the $|\omega/\Delta_{0}|\ll 1$
region, we can ignore these terms. This approximation breaks down in the
strong-coupling regime, namely when the tunneling amplitudes overcomes
$\Delta_{0}$. In this case, while the phase diagram can still be studied
within the $\omega\to 0$ limit, Eq. (26) overestimates the size of the induced
gap. For a more complete reference to the strong tunneling regime, we refer to
[48].

###  A.2 Multimode semiconductor

Our objective now is to generalize the previous study to the case of a
multimode Sm in the case of spin-symmetric tunneling ($t_{l}=0$). Since we
focus on the phase diagram, we consider the lowest order in the effective
Hamiltonian $\tilde{\mathcal{H}}_{0}=\Sigma(\omega=0,p_{z})$. The three
contributions read

$\tilde{\mu}(p_{z})=\sum_{n}\xi_{np_{z}}\frac{t_{0,mnp_{z}}t^{*}_{0,m^{\prime}np_{z}}}{\xi_{np_{z}}^{2}+\Delta_{0,np_{z}}^{2}-V_{z}^{2}}\,,$
(28)

$\tilde{V}^{(1)}_{z}(p_{z})=\sum_{n}V_{z}\frac{t_{0,mnp_{z}}t^{*}_{0,m^{\prime}np_{z}}}{\xi_{np_{z}}^{2}+\Delta_{0,np_{z}}^{2}-V_{z}^{2}}\,,$
(29)

$\begin{split}\tilde{\Delta}_{0}(p_{z})=\sum_{n}[&+(t_{0,mnp_{z}}t_{0,m^{\prime}np_{z}}-t_{0,mnp_{z}}^{\prime}t_{0,m^{\prime}np_{z}}^{\prime})\tau_{x}\\\
&-(t_{0,mnp_{z}}^{\prime}t_{0,m^{\prime}np_{z}}+t_{0,mnp_{z}}t_{0,m^{\prime}np_{z}}^{\prime})\tau_{y}]\frac{\Delta_{np_{z}}}{\xi_{np_{z}}^{2}+\Delta_{0,np_{z}}^{2}-V_{z}^{2}}\,.\end{split}$
(30)

For $p_{z}=0$, $t_{0}$ real and $\Delta_{0,np_{z}}=\Delta_{0}$, we can write
the effective Hamiltonian matrix as

$\mathcal{H}_{\mathrm{eff}}=\Xi\otimes\sigma_{0}\tau_{z}+M\otimes(V_{z}\sigma_{z}\tau_{0}+\Delta_{0}\sigma_{0}\tau_{x})$
(31)

where we have defined the $\Xi$ matrix as

$\Xi_{mm^{\prime}}=\zeta_{m}\delta_{mm^{\prime}}-\sum_{n}\xi_{np_{z}}\frac{t_{0,mnp_{z}}t_{0,m^{\prime}np_{z}}}{\xi_{np_{z}}^{2}+\Delta_{0,np_{z}}^{2}-V_{z}^{2}}\,,$
(32)

where $\zeta_{m}$ are the energies of the Sm modes. The $M$ matrix is

$M_{mm^{\prime}}=\sum_{n}\frac{t_{0,mnp_{z}}t_{0,m^{\prime}np_{z}}}{\xi_{np_{z}}^{2}+\Delta_{0,np_{z}}^{2}-V_{z}^{2}}\,.$
(33)

The $M$ matrix is Hermitian so it is diagonalizable as $M=U\Lambda
U^{\dagger}$. We can apply the unitary transformation
$U\otimes\tau_{0}\sigma_{0}$ to the Hamiltonian matrix and get to

$\begin{split}\mathcal{H}_{\mathrm{eff}}&=U^{\dagger}\Xi
U\otimes\sigma_{0}\tau_{z}+\Lambda\otimes(V_{z}\sigma_{z}\tau_{0}+\Delta_{0}\sigma_{0}\tau_{x})\\\
&=\hat{T}\otimes\sigma_{0}\tau_{z}+\sum_{m}\quantity[\hat{\zeta}_{m}\sigma_{0}\tau_{z}+\lambda_{m}(V_{z}\sigma_{z}\tau_{0}+\Delta_{0}\sigma_{0}\tau_{x})]\delta_{mm^{\prime}}\,,\end{split}$
(34)

where the last term is the diagonal element $\hat{\zeta}_{m}$. The off-
diagonal hopping terms in the matrix $\hat{T}$ come from the transformed
$U^{\dagger}\Xi U\otimes\sigma_{0}\tau_{z}$. This Hamiltonian is a combination
of independent modes
$\sum_{m}[\hat{\zeta}_{m}\tau_{z}\sigma_{0}+\lambda_{m}(V_{z}\sigma_{z}\tau_{0}+\Delta_{0}\sigma_{0}\tau_{x})]$
interacting through the spin-independent $\hat{T}$ terms which acts as a
tunneling Hamiltonian for these channels. The gap polarization ratio is given
by $\absolutevalue{V_{z}/\Delta_{0}}$ for these channels, which is the same as
in the parent Sc. For this reason, if the parent Sc is in a trivial state, all
the modes in the Sm will be in the trivial regime as well. The eigenmodes of
the full effective Hamiltonian will be given by super-positions of these
states. However, since we are mixing trivial states we can only obtain trivial
states as a result.

###  A.3 Case of general tunnel matrix with complex element

We now move to the case of a general tunneling matrix $T_{np_{z}}$ coupling a
multimode Sc to a monomodal Sm (thus we drop the index $m$). In this case we
consider only the zeroth-order term in $\omega$ which is the effective
Hamiltonian $\tilde{H}_{0}$. We can split it into three different
contributions:

$\tilde{H}_{0}=\sum_{p_{z}}\tilde{\mu}(p_{z})c_{p_{z}}^{\dagger}\sigma_{0}c_{p_{z}}+c^{\dagger}_{p_{z}}\tilde{\mathbf{V}}(p_{z})\cdot{\bf\it\sigma}c_{p_{z}}+\quantity(c^{\dagger}_{p_{z}}\tilde{{\bf\it\Delta}}(p_{z})c^{\dagger}_{-p_{z}}+\mathrm{H.c.})\,,$
(35)

with
$\tilde{{\bf\it\Delta}}(p_{z})=\tilde{\Delta}_{0}(p_{z})\sigma_{0}+\tilde{\Delta}_{i}(p_{z})\sigma_{i}$
and $\tilde{\mathbf{V}}=\tilde{\mathbf{V}}^{(1)}+\tilde{\mathbf{V}}^{(2)}$.
The three terms are given by

$\tilde{\mu}(p_{z})=\sum_{n}\frac{+\xi_{np_{z}}(\absolutevalue{t_{0,np_{z}}}^{2}+\absolutevalue{t_{x,np_{z}}}^{2}+\absolutevalue{t_{y,np_{z}}}^{2}+\absolutevalue{t_{z,np_{z}}}^{2})-V_{z}\quantity[(t_{0,np_{z}}t_{z,np_{z}}^{*}+\mathrm{H.c.})+(t_{x,np_{z}}t_{y,np_{z}}^{*}-\mathrm{H.c.})]}{\xi_{np_{z}}^{2}+\Delta_{0,np_{z}}^{2}-V_{z}^{2}}\,,$
(36)

$\tilde{V}^{(1)}_{z}(p_{z})=\sum_{n}V_{z}\frac{\absolutevalue{t_{0,np_{z}}}^{2}+\absolutevalue{t_{z,np_{z}}}^{2}-\absolutevalue{t_{x,np_{z}}}^{2}-\absolutevalue{t_{y,np_{z}}}^{2}}{\xi_{np_{z}}^{2}+\Delta_{0,np_{z}}^{2}-V_{z}^{2}}\,,$
(37)

$\tilde{V}^{(2)}_{i}(p_{z})=\sum_{n}-\xi_{np_{z}}\frac{\quantity[(t_{0,np_{z}}t_{i,np_{z}}^{*}+\mathrm{H.c.})+(t_{k,np_{z}}t_{j,np_{z}}^{*}-\mathrm{H.c.})]}{\xi_{np_{z}}^{2}+\Delta_{0,np_{z}}^{2}-V_{z}^{2}}\epsilon_{ijk}\,,$
(38)

$\tilde{\Delta}_{0}(p_{z})=\sum_{n}\frac{\Delta_{np_{z}}(t_{0,np_{z}}^{2}-t_{x,np_{z}}^{2}-t_{y,np_{z}}^{2}-t_{z,np_{z}}^{2})}{\xi_{np_{z}}^{2}+\Delta_{0,np_{z}}^{2}-V_{z}^{2}}\,.$
(39)

where $\epsilon_{ijk}$ is the Levi-Civita symbol. We can see that the presence
of transverse components in the tunneling matrices causes a reduction of both
the induced exchange field and superconducting pairing potential. However, if
these amplitudes are complex, a non trivial resonance contribution can appear
in the $\tilde{V}_{i}^{(2)}$ term enhancing the induced exchange field.
Moreover, when complex amplitudes are present the induced gap can show
interference effects due to the coupling between the various Sc bands induced
by the Sm.

##  Appendix B BCS integration induced gap and exchange field

In this section we integrate the induced gap and exchange field due to the
coupling to the superconductor. We take the usual BCS approximation which
considers an attractive interaction between electrons with absolute energies
below $\omega_{D}$. Therefore, the parent gap becomes finite and equal to
$\Delta_{0}$ for energies in the range $[-\omega_{D},\omega_{D}]$, being zero
otherwise. Using this approximation, the induced gap is given by

$\tilde{\Delta}_{0}=2\Delta_{0}\,g_{m}(t_{0}^{2}-t_{z}^{2})\frac{\mbox{atan}\left(\omega_{D}/\sqrt{\Delta^{2}_{0}-V^{2}_{z}}\right)}{\sqrt{\Delta^{2}_{0}-V_{z}^{2}}}\,,$
(40)

where $g_{m}$ is the density of transverse modes, considered to be energy
independent. The induced exchange field is given by

$\tilde{V}_{z}=V_{z}\,g_{m}(\absolutevalue{t_{0}}^{2}+\absolutevalue{t_{z}}^{2})\frac{\pi}{\sqrt{\Delta^{2}_{0}-V^{2}_{z}}}$
(41)

where the superconductor bandwidth has been taken infinite for simplicity.

For $t_{z}=0$, $\tilde{V}_{z}/V_{z}\geq\tilde{\Delta}_{0}/\Delta_{0}$, where
the equal case corresponds to the limit $\omega_{D}\to\infty$. Taking
realistic numbers for bulk aluminum, where $\omega_{D}/\Delta_{0}\approx 200$,
we find
$(\tilde{V}_{z}/V_{z}-\tilde{\Delta}_{0}/\Delta_{0})/(\tilde{\Delta}_{0}/\Delta_{0})\sim
10^{-2}$. If we consider that the Sc proximitized by the MI fulfills the
Chandrasekhar-Clogston limit, $V_{z}<\Delta_{0}/\sqrt{2}$ [36, 37], the spin
splitting cannot cause a topological transition in the Sm, unless
$\omega_{D}/V_{z}\sim 1$. We do not expect that an energy-dependent tunneling
amplitude significantly changes this conclusion.

##  Appendix C Analytic derivation of the Pfaffian

In this section we analytically derive the topological invariant of the two-
wires system, namely a system composed by a single mode Sm coupled to a single
mode Sc. We can write the Hamiltonian matrix as

$\mathcal{H}=\begin{pmatrix}\mathcal{H}_{\mathrm{Sm}}&\mathcal{H}_{\mathrm{t}}\\\
\mathcal{H}^{\dagger}_{\mathrm{t}}&\mathcal{H}_{\mathrm{Sc}}\end{pmatrix}\,,$
(42)

where the three terms read:

$\mathcal{H}_{\mathrm{Sm}}=\quantity(\frac{p_{z}^{2}}{2m_{\mathrm{Sm}}^{*}}-\mu_{\mathrm{Sm}})\sigma_{0}\tau_{z}+\alpha_{x}p_{z}\sigma_{y}\tau_{z}\,,$
(43)

$\begin{split}\mathcal{H}_{\mathrm{Sc}}=\quantity(\frac{p_{z}^{2}}{2m_{\mathrm{Sc}}^{*}}-\mu_{\mathrm{Sc}})\sigma_{0}\tau_{z}+V_{z}\sigma_{z}\tau_{0}+\Delta\sigma_{0}\tau_{x}\,,\end{split}$
(44)

$\mathcal{H}_{\mathrm{t}}=t_{0}\sigma_{0}\tau_{z}+t_{z}\sigma_{z}\tau_{0}\,.$
(45)

We proceed by rewriting the Hamiltonian in Majorana basis by using a unitary
transformation $\mathcal{H}_{\mathrm{MJ}}=U\mathcal{H}U^{\dagger}$ where the
transformation matrix reads

$U=\frac{1}{\sqrt{2}}\begin{pmatrix}+1&0&0&-1\\\ i&0&0&i\\\ 0&1&1&0\\\
0&i&-i&0\end{pmatrix}\otimes\mathbb{I}_{2}\,,$ (46)

where $\mathbb{I}_{2}$ is the $2\times 2$ identity matrix.

Since the nanowire belongs to symmetry class $D$, the appearance of Majorana
zero modes is characterized by a change of the sign of the topological
invariant

$W_{\mathbb{Z}_{2}}=\mathrm{sgn}\quantity[\frac{\mathrm{Pf}A(p_{z}=0)}{\mathrm{Pf}A(p_{z}\to+\infty)}]\,,$
(47)

where $A=i\mathcal{H}_{\mathrm{MJ}}$ [49]. The matrix $A$ is real and
antisymmetric and thus the Pfaffian is well defined. Since the kinetic energy
dominates for large $p_{z}$, the Hamiltonian approaches the one for quasi-free
electrons, which have Pfaffian $\mathrm{Pf}A(p_{z}\to\infty)=1$. Therefore
evaluating Eq. (47) reduces to computing the Pfaffian for $p_{z}=0$. Taking
the limit $\Delta_{0}\to 0$ we found the same condition described by Eq. (12).

##  Appendix D Multiband superconductor coupled to a semiconductor

In this section we derive expressions for the induced parameters in the Sm due
to the coupling to a multimode Sc. We consider that, in the relevant energy
scale, the band structure of the Sc can be described by $N_{m}$ modes
distributed in an energy range $E_{B}$. For simplicity, we will take
equidistant modes with energy difference $\delta\varepsilon=E_{B}/N_{m}$. This
is also consistent with the continuum limit since, for a 2D system, we can
expect a density of transverse modes $g_{m}=(\delta\varepsilon)^{-1}$ constant
in energy. We also define $\delta\mu_{\mathrm{Sc}}$ as the relative position
of the Fermi energy in the Sc from the middle of the band. We proceed by
summing over the modes contributions following Eqs. (8)–(10) to derive the
value of the induced terms in the effective Hamiltonian. In the limit
$N_{m}=E_{B}/\delta\varepsilon\gg 1$ we get to

$\tilde{V}_{z}^{(1)}(p_{z})\simeq\frac{\pi}{2}g_{m}V_{z}(t_{0}^{2}+t_{z}^{2})\frac{\cot\quantity[\pi
g_{m}\quantity(i\Gamma+\delta\mu-\frac{p_{z}^{2}}{2m^{*}_{\mathrm{Sc}}})]+\cot\quantity[\pi
g_{m}\quantity(i\Gamma-\delta\mu+\frac{p_{z}^{2}}{2m^{*}_{\mathrm{Sc}}})]}{i\Gamma}\,,$
(48)

$\tilde{V}_{z}^{(2)}(p_{z})\simeq\pi
g_{m}t_{0}t_{z}\quantity{\cot\quantity[\pi
g_{m}\quantity(i\Gamma+\delta\mu-\frac{p_{z}^{2}}{2m^{*}_{\mathrm{Sc}}})]-\cot\quantity[\pi
g_{m}\quantity(i\Gamma-\delta\mu+\frac{p_{z}^{2}}{2m^{*}_{\mathrm{Sc}}})]}\,,$
(49)

$\tilde{\Delta}_{0}(p_{z})\simeq\frac{\pi}{2}g_{m}\Delta_{0}(t_{0}^{2}-t_{z}^{2})\frac{\cot\quantity[\pi
g_{m}\quantity(i\Gamma+\delta\mu-\frac{p_{z}^{2}}{2m^{*}_{\mathrm{Sc}}})]+\cot\quantity[\pi
g_{m}\quantity(i\Gamma-\delta\mu+\frac{p_{z}^{2}}{2m^{*}_{\mathrm{Sc}}})]}{i\Gamma}\,.$
(50)

To simplify the expression, we can define the auxiliary functions

$\begin{split}F(x,y)&=\frac{\cot(ix+y)-\cot(ix-y)}{2}=\frac{\sin(2y)}{\cosh(2x)-\cos(2y)}\,,\\\
G(x,y)&=i\frac{\cot(ix+y)+\cot(ix-y)}{2}=\frac{\sinh(2x)}{\cosh(2y)-\cos(2x)}\,.\\\
\end{split}$ (51)

We finally get to compact expressions for the effective Hamiltonian terms:

$\tilde{V}_{z}^{(1)}(p_{z})\simeq\pi
g_{m}\frac{V_{z}}{\Gamma}(t_{0}^{2}+t_{z}^{2})G\quantity[\pi g_{m}\Gamma,\pi
g_{m}\quantity(\delta\mu-\frac{p_{z}^{2}}{2m^{*}_{\mathrm{Sc}}})]$ (52)

$\tilde{V}_{z}^{(2)}(p_{z})\simeq 2\pi g_{m}t_{0}t_{z}F\quantity[\pi
g_{m}\Gamma,\pi
g_{m}\quantity(\delta\mu-\frac{p_{z}^{2}}{2m^{*}_{\mathrm{Sc}}})]$ (53)

$\tilde{\Delta}_{0}(p_{z})\simeq-\pi
g_{m}\frac{\Delta_{0}}{\Gamma}(t_{0}^{2}-t_{z}^{2})G\quantity[\pi
g_{m}\Gamma,\pi
g_{m}\quantity(\delta\mu-\frac{p_{z}^{2}}{2m^{*}_{\mathrm{Sc}}})]$ (54)

This expression can be used to derive the Sm effective Hamiltonian and, with a
simple linear search for the minimum eigenvalue, find the effective gap shown
in Fig. 6.