Topological superconductivity in semiconductor-superconductor-magnetic-insulator heterostructures

Hybrid superconductor-semiconductor heterostructures are promising platforms for realizing topological superconductors and exploring Majorana bound state 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 strong spin-orbit interaction, the gap reopens, leading to a topological phase [1,2].The quantum phase transition can bring the system into the topological regime when where V Z is the Zeeman energy, µ the semiconductor chemical potential, and ∆ the superconducting gap.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-areagrown (SAG) 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 where all nanowires 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 (MI) were used to induce a spin splitting by means of stray fields [14,15].In addition, clean interfaces with a MI lead also 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 of 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].If a direct effect of the MI on the Sm is excluded, a proposed alternative hypothesis is that the induction of an exchange field in the Sm could be medi-ated by the spin-split Sc which induces superconductivity and exchange field at the same time.In this case the relevant exchange coupling would be directly between the MI and the Sc [20].
FIG. 1. Sketch of the cross section of a Sm nanowire in proximity to a Sc and a MI.(a) shows 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 figure (c) have not been reported yet.
In this article, 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, alone, topological phases in the Sm.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.
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 of the electrons with the localized magnetic moments in the MI.We can describe the effect of the interface by means of a Heisenberg-like term H int = − d 3 r JS MI • S, which describes the coupling between the spin density of the metals S, with the localized spins in the insulator, S MI .The coupling strength J is related to the exchange integral between the localized orbitals and the free electron and could be 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 width of the Sc layer to be smaller than the superconducting coherence length, ξ 0 , which is FIG. 2. Sketch representing the proximity effects in the systems considered in this work.In the illustrations, V MI is the native exchange splitting in the magnetic insulator, ∆ and V Sc are the gap parameter and the MI-induced exchange filed in the Sc.Here, ∆ and Ṽ are the Sc-induced gap and exchange field in the Sm.The matrix T is the hopping matrix of the tunneling Hamiltonian which couple the two materials.In panel (a), the magnetic insulator is inducing 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 Ṽ close to the interface to the MI.
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 ∼ 10ξ 0 for EuS-Al heterostructures [31].Since the dimensions of the systems under consideration are smaller than ξ 0 , we can disregard inhomogeneities and assume that the MI induces a homogeneous exchange field in the Sc, which couples with the spin degree of freedom of the electrons by the Zeeman term In this work we disregard the induced exchange field in the Sm as experimental evidence suggests that it is small in 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 directly MI-induced exchange field in the Sm would provide an enhancement of the Zeeman energy in the Sm, enlarging the topological region.We also neglect the magnetic orbital effects induced by the stray field of the MI as this effect is usually weaker than 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 translationinvariant system along the longitudinal direction, z, consisting of a single-channel Sm coupled to a Sc.The complete Hamiltonian of the system reads ( The Sm is described by where we use the spinors c pz = (c pz↑ , c pz↓ ) and c pzσ is the electron annihilation operator in the Sm, while α x is the spin-orbit coupling strength.
The bare Hamiltonian for the Sc is given by where a npz is the electron annihilation operator for the mode n in the Sc and ξ npz = We neglect possible superconductive interband coupling and we assume singlet pairing in the parent superconductor gap matrix ∆ npz = ∆ 0,npz σ 0 .As anticipated, our Sc model features a homogeneous exchange field V Sc = V z e z induced by the nearby MI, which we consider to be aligned to the wire.An important quantity is the distribution of the transverse modes in the Sc, ε n , which strongly depends on the device geometry.It ranges from a value larger than the superconducting gap for very thin Scs, to zero in bulk materials.
The Sc and Sm regions are coupled through a spindependent, momentum conserving, tunneling Hamiltonian where the hopping matrix T npz describes the electron tunneling processes taking place at the interfaces between the two materials.We can write T npz in the basis of Pauli matrices T npz = t 0,npz σ 0 + i t i,npz σ i .In the following, we set t x,npz = t y,npz = 0 as they are negligible for an uniformly polarized MI layer.Moreover, we consider only the case of real and positive t 0,npz and t z,npz .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 the Appendix A 3.
The bare Sc retarded Green function in the basis of time-reversed pairs reads: where the τ 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 function of the Sm reads npz describes the coupling to the Sc.This allow 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 will work with the effective Hamiltonian H eff = H Sm + H0 where the induced Hamiltonian is H0 = Σ(ω = 0).Indeed, we can split the effective Hamiltonian in three different contributions: with ∆(p z ) = ∆0 (p z )σ 0 + i ∆i (p z )σ 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 form of these contributions are: where we have divided the induced Zeeman term into two contribution Ṽz = Ṽ (1) z .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 ∆z is also present, vanishing for ω = 0. Therefore, only the singlet pairing ∆0 is important to describe the topological transition.We will assume on that the parent Sc gap is homogeneous and equal for each transverse mode of the Sc, i.e. ∆ 0,npz = ∆ 0 from now.

A. 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, Ṽz , and the gap, ∆0 .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 the following.A necessary condition to satisfy the inequality is having a gap polarization ratio |V Z /∆| > 1.
Taking the ratio of Eqs. ( 8) and (10) for t z = 0, we see that so the induced gap polarization ratio in the Sm is the same as in the Sc.
The gap polarization ratio V Sc /∆ 0 has to be less than unity in the parent gap, otherwise superconductivity is suppressed in the whole device.In the case of a large homogeneous Sc, this ratio is limited by the stricter Chandrasekar-Clogston bound, which dictates that a finite superconducting gap can be only maintained if V Sc /∆ 0 < 1/ √ 2 [35,36].In thin Sc films, quantum confinement cause an increase of the superconducting gap leading to a superconductive phase that survives under stronger exchange fields.However, the Chandrasekar-Clogston limit still holds in terms of gap polarization ratio [37].Even before this limit, the gap parameter in a clean Sc subjected to strong exchange fields cease 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 ∆ 0 and V z approximation.The same result is found for a continuous flat density of transverse modes in the Sc where ∆ 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.

B. 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 σ 0 + t z σ z .As Eqs. (8)(9)(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 sub-band and the chemical potential.This means that modes close to the 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 different modes just 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 µ Sc > ε n .Both Ṽz and ∆0 decay for |ξ n,0 | → ∞, leading to the existence of an optimal regime where both the induced exchange field and superconducting pairing are maximal and Ṽz / ∆0 > 1.This is the ideal region for searching for topological superconductivity.In general, a spin-dependent component t i in the tunneling matrix enhances the collinear component in the exchange field Ṽi , suppressing the non-collinear components in Ṽ and the superconducting gap in the Sm.From Eqs. (8-10) we see that ∆ and Ṽ z share the same dependence on the Sc band structure (they decay as ∼ ξ −2 ) while having a different prefactor which depends on the tunneling matrix.Therefore, the spin splitting in the Sc leads to an enhancement of Ṽz and provides a first mechanism to induce topological superconductivity in the Sm.
In contrast to Ṽ (1) z , the second contribution Ṽ (2) z 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, Ṽ z has an energy dependence ∼ ξ −1 , with a sign that depends on the relative position of the mode to the Sc Fermi energy.Since ∆0 and Ṽ (1) z dominates as the energy difference between the Sm and Sc modes increases.This contribution exhibits a sign change, which leads to a cancellation of Ṽ (2) z 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 two different mechanisms can drive the system to the topological phase.First, a spin-dependent tunneling enhances the induced gappolarization ratio of spin-split superconductor.Alternatively, a 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.

C. 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, ε 0 = 0, as it only causes a shift of the energy scale.In this way µ Sc = −ξ 0,pz=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 Ṽ 2 z = (µ Sm + μ) 2 + ∆2 0 .To simplify the calculation, we take the limit ∆ 0 → 0 as the chemical potential of the Sc is usually much bigger than the gap parameter.In this way we find the analytical condition This result allows us to identify the regions in the parameter space where topological phases can 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 ∼ 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 also 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 µ Sm µ Sc , as shown in Eqs.(12).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 (12).Analyzing the behavior with respect to the tunneling parameters, it can be noticed that there exists a region corresponding to t 0 ∼ ± √ −V z µ Sm + µ Sm µ 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 by means of electrostatic gates, it is in principle possible to set the operating point of the device near this optimal point.

D. Superconductor with several relevant transverse modes
In a more realistic scenario, the Sc features many transverse modes.In this case, Eqs.(8-10) explains how the induced terms in the Sm effective Hamiltonian are determined by the sum of the contributions of each mode.The relative position 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.In 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 ∆0 and Ṽ z have a Lorentzian shape with a full width at half maximum Γ = ∆ 2 0 − V 2 z .We can use Γ as a reference to distinguish between discrete modes, δε Γ, and a continuum distribution of modes, δε Γ, where δε is the average separation of modes.Indeed, if the average energy separation between the transverse modes is δε Γ, the resonance peaks do not overlap entirely and the resonance peaks of V (2) z do not completely cancel out.In this case, the net exchange field experienced in the Sm will be due by the sum of the contribution of each transverse mode.
z cause an oscillation between topologically trivial and nontrivial phases while varying the chemical potential in the Sc.
To provide a more clear picture, we consider the case where the band structure of the Sc can be described by perfectly equidistant transverse modes.In this case, δε is the relative separation between the Sc modes.This approximation recovers the continuum limit for a 2D sys-tem, where we expect a constant density of transverse modes g m = (δε) −1 .We also define δµ 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)(9)(10) to derive 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 are 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 µ Sm (white background).This behavior is more clearly illustrated by the topological invariant and the superconducting gap, shown in Fig. 6.In this figure, the gap is given by the smallest eigenvalue of the effective Hamiltonian (7).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 δµSc.As the density of transverse modes gm increase, the peaks get closer and overlap until they merge in the continuous limit gm (∆0) −1 .In the continuum limit the system is in a globally trivial or topologically phase, depending on the condition in Eq. ( 14).
We assume that the chemical potential in the Sm is tuned such that µSm = −μ.The observation of an alternation of topological and trivial phases while varying δµ Sc is consistent with Fig. 5.The limit g m 1/∆ 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 transverse mode is such that δε Γ.In this limit, the contributions from each mode overlaps 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)(9)(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 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 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 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 eV which can be effectively tuned by quantum confinement and, therefore, it depends on the film thickness [40].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 7µ B .For this reason, the EuS can be well described as a Heisenberg ferromagnet with a Curie temperature of T m = 16.6 K [41,42].The magnetization induce a splitting of the conduction band of around ∆E c = 0.36 eV [43].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, Ṽz (2) vanish and topological superconductivity can be only induced using the spinasymmetric tunneling to enhance Ṽ z and suppress ∆0 , as summarized in 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 δε

Sc
, where L Sc refers to the largest dimension of the cross section of the Sc shell.Assuming that Γ ∼ 100 µeV, which is in line with the experimental measurements of Al-EuS heterostructures [20,31], we estimate that, in order to observe well-separated modes, the maximum dimension of the shell should be in the order of 60 nm.In experiments, the facet length is around 60 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.In general, as the barrier gets thicker we can 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 a 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 particularly complicated to introduce at the same time Sc-Sm, MI-Sm and MI-Sc interfaces.The use of MI as a spin filter tunnel barrier simplifies the design of topological quantum devices by eliminating the requirement of a 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 materials.
Finally, we note that both nanowires Sc shell and Sc layers in 2DEG system are around 5 nm thick, so we can already expect 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 topological transitions in the semiconductor are caused by a superconductor-mediated exchange field in nanowire in experiments.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 gap 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 net 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 around a nanometer of a ferromagnetic insulator at the interface between the superconductor and the semiconductor.The proposed mechanism is compatible with the currently used hybrid superconductorsemiconductor platforms, including nanowires and 2DEG systems.
where ijk is the Levi-Civita symbol.We can see that the presence of transverse components in the tunneling matrices cause a reduction of both the induced exchange field and superconducting pairing.However, if these amplitudes are taken complex, a non-trivial resonance contribution can appear in the Ṽ (2) i 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.

FIG. 4 .
FIG. 4. Phase diagram for a single mode Sm coupled to a single mode Sc.We represent the effective gap in the Sm |∆ eff | = minn,p z En,p z , where En,p z are the Hamiltonian eigenvalues and the sign is given by the topological invariant W Z 2 , Appendix C. The black dashed lines which demarcate the boundaries of the topological region (red color) are described Eq.(12).We have set Vz = 0, which leads to Ṽ (1) z = 0.The remaining parameters are µSc/∆0 = 10, m * Sc = 1, m * Sm = 0.25, αx/∆0 = 5.For the left panel we take t0 = 5 while for the right panel µSm/∆0 = 1.

FIG. 6 .
FIG. 6.Phase diagram for a single mode Sm coupled to a multimode Sc, showing the effective gap in the Sm |∆ eff | = minn,p z En,p z , where En,p z are the Hamiltonian eigenvalues and sign given by the topological invariant W 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 δµSc.As the density of transverse modes gm increase, the peaks get closer and overlap until they merge in the continuous limit gm (∆0) −1 .In the continuum limit the system is in a globally trivial or topologically phase, depending on the condition in Eq. (14).We assume that the chemical potential in the Sm is tuned such that µSm = −μ.Parameters used: m * Sc = 1, m * Sm = 0.2, αx/∆0 = 20, Vz/∆0 = 0.5, t0/∆0 = 1.For the left panel we use tz/∆0 = 0.4, while for the right panel tz/∆0 = 0.7.
FIG. 6.Phase diagram for a single mode Sm coupled to a multimode Sc, showing the effective gap in the Sm |∆ eff | = minn,p z En,p z , where En,p z are the Hamiltonian eigenvalues and sign given by the topological invariant W 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 δµSc.As the density of transverse modes gm increase, the peaks get closer and overlap until they merge in the continuous limit gm (∆0) −1 .In the continuum limit the system is in a globally trivial or topologically phase, depending on the condition in Eq. (14).We assume that the chemical potential in the Sm is tuned such that µSm = −μ.Parameters used: m * Sc = 1, m * Sm = 0.2, αx/∆0 = 20, Vz/∆0 = 0.5, t0/∆0 = 1.For the left panel we use tz/∆0 = 0.4, while for the right panel tz/∆0 = 0.7.