Jan 10, 2018 - Hybrid superconductor-semiconductor devices are currently one of the most promising platforms for realizing robust Majorana zero modes...

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arXiv:1801.03439v2 [cond-mat.mes-hall] 21 Jan 2018

Center for Quantum Devices and Station Q Copenhagen, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark (Dated: January 23, 2018) Hybrid superconductor-semiconductor devices are currently one of the most promising platforms for realizing Majorana zero modes. We address the role of band bending and superconductorsemiconductor hybridization in such devices by analyzing a gated single Al-InAs interface using a self-consistent Schr¨ odinger-Poisson approach. Our numerical analysis shows that the band bending leads to an interface quantum well, which localizes the charge in the system near the superconductorsemiconductor interface. We investigate the hybrid band structure and analyze its response to gate voltage and thickness of the Al layer, and we find that one can obtain states with strong superconductor-semiconductor hybridization at the Fermi energy. We analyze the relative metal and semiconductor characters of the hybridized bands which are important for the induced pairing and effective g-factors. In addition, we obtain approximate analytical expressions to further back our numerical results and clarify key aspects of the band structure. We conclude by discussing the consequences of our findings for the realization of Majorana zero modes in nanowire-based systems.

I.

INTRODUCTION

Metal-semiconductor interfaces is a well-established field in the context of semiconducting electronics for implementing either Schottky diodes or Ohmic contacts, cf Refs. [1–3]. In the former case, the interface coupling forces the semiconductor to experience band bending leading to a depletion of charge near the semiconductor side of the interface, while the latter situation gives an accumulation of charges. The separation between the two behaviors is mainly determined by the sign of the difference of the metal work function (WM ) and the electron affinity of the semiconductor (χSM ), which we denote as Φbulk = χSM −WM . However, in practise other interfacial effect influence the band offset between the two materials. Here we work with an effective offset which we denote by Φ. Recently, metal-semiconductor interfaces have attracted renewed attention in the context of engineered topological superconductivity where conventional s-wave Cooper pairing can be induced into the semiconductor. It has been theoretically predicted that the induced superconducting pairing combined with spin-orbit coupling (SOC) and an applied magnetic field can drive the system into the topological superconducting (p-wave) [4– 7] state. This topologically non-trivial state supports charge-neutral zero-energy end states which obey nonAbelian exchange statistics. These so-called Majorana zero modes have interest for implementation of topological quantum computing [8–14]. The first experimental spectroscopic signatures of Majorana zero modes in superconductor-semiconductor hybrid devices were reported in Ref. [15] and have since then been refined via the fabrication of ultra-clean epitaxial Al-InAs hybrids [16, 17]. These improved fabrication methods have led to promising reports of Majorana fingerprints in both epitaxial nanowire hybrids [18–21] and 2D epitaxial superconductor-semiconductor hybrids with lithographically defined 1D channels [22, 23]. However, many microscopic details of the interfaces

and in particular the degree of hybridization between the metal and semiconductor, which determines the induced superconducting pairing, the SOC strength and the effective g-factor, are not well-understood. Neither are the number of subbands occupied in the nanowires and the resulting semiconductor electron density. In fact, there seems to be a large spread in critical magnetic fields and gate voltages required to induce the topological state. A recent study showed a systematic dependence of the effective g-factor on gate voltage, when measured by the slope of the induced gap with applied field [19]. Some theoretical progress in understanding the variation of the experimental results has already been made. For example in Ref. [24] it was shown that orbital contributions might help to understand the large measured g-factor values, while the authors of Ref. [25] have assessed the role of the electrostatic environment in the nanowire devices. The effect of hybridization on the induced pairing has also been considered [26, 27]. However, the interplay between hybridization and the electrostatically determined self-consistent potential has not been explored and a better understanding of this could potentially guide future experimental designs and theoretical modelling [28–30] of Majorana devices. In this paper we address the above-mentioned open issues by employing a self-consistent continuum model for the semiconductor-metal hybridization which incorporates band-bending effects due to electrostatics. Motivated by the existing epitaxial Majorana devices, we focus on an Al-InAs interface. Our analysis relies on applying the Thomas-Fermi and Schr¨odinger-Poisson methods to calculate band edge and charge density profiles for the hybrid Al-InAs device shown in Fig. 1(a). The main conclusion of both approaches is that the band bending leads to a quantum well, which confines the electrons in the system in the vicinity of the superconductorsemiconductor interface. From the Schr¨odinger-Poisson approach we obtain the reconstructed band structure and investigate its response to varying the gate voltage as well

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FIG. 1. (Color online) (a) Schematic of the hybrid device consisting of a layer of metal, semiconductor and dielectric with translational invariance in the plane. The rightmost edge of the dielectric is in contact with a gate electrode which keeps it at a voltage VG with respect to the grounded metallic layer. This translates into a voltage difference between the grounded metallic layer and the lower edge of the semiconductor, which we denote VD . (b) Band diagram for the metal and semiconductor region before contact. (c) Band diagram of the metal and semiconductor region after contact. (d) Self-consistent band edge profiles in the semiconductor region for several values of VD . Solid lines indicate the results obtained via the self-consistent Schr¨ odinger-Poisson method and dashed lines indicate the results obtained from the Thomas-Fermi approximation. (e) Selfconsistent charge density profiles in the semiconductor region for several values of VD . Solid lines indicate the results obtained via the self-consistent Schr¨ odinger-Poisson method and dashed lines indicate the results obtained from the Thomas-Fermi approximation.

as the thickness of the Al layer. The degree of hybridization turns out to be very sensitive to the thickness of the Al layer and native band offset Φ, while only a weak dependence on the gate voltage is found. These parameters may be tuned such that a single band with strong superconductor-semiconductor hybridization crosses the Fermi level while leaving out bands with negligible coupling to the superconductor. Our numerical findings are further understood via employing an analytical approach. Based on our numerical investigation and approximate analytical expressions, we discuss the hybridization and

the resulting g-factor, SOC strength and induced superconducting gap for the interfacial bands crossing the Fermi energy.

II.

SELF-CONSISTENT BAND BENDING A.

Setup and Electrostatics

We consider the hybrid device depicted in Fig. 1(a), showing a layered structure consisting of metal, semicon-

3 ductor and dielectric with translational invariance in the xy plane. The device is characterized by two boundary conditions for the electrostatic potential φ: (i) the metal layer is assumed to be a grounded conductor with φ = 0, (ii) the rightmost edge of the dielectric is in contact to a back gate with φ = VG . For our self-consistent modeling we focus only on the metal and semiconductor region of the device, and in this case (ii) is replaced by a scaled back gate voltage φ = VD at the semiconductor-dielectric interface. We wish to determine the conduction band edge and charge density profile in our device. The idea behind our approach is shown in the band diagrams of Figs. 1(b) and (c). We assume that the Fermi level of the metal layer (dashed line) sets the chemical potential of the hybrid system and choose this as our reference energy. The distance between the Fermi level of the metal and its conduction band edge is determined by its Fermi energy, EF , which we set to the bulk Al value, i.e. EF = 11.7 eV [31]. Before contact (Fig. 1(b)), the conduction band edge of the semiconductor is assumed to be below the Fermi level of the metal corresponding to a positive difference between the electron affinity of the semiconductor and work function of the metal, that is Φ > 0. The results turn out to be very sensitive to this value and here we start by studying the case when the value of this experimentally not yet fully resolved parameter is Φ = 0.1 eV. When the metal and semiconductor layer are contacted (Fig. 1(c)), the band edges of the semiconductor will bend due to the presence of charges that are transferred from the metal into the semiconductor conduction band. We assume that only the conduction band electrons contribute to the band bending in the semiconductor, thus disregarding the presence of the valence band electrons. This approach is valid for the low-temperature range of interest, where kB T Eg ≈ 0.418 eV [32]. Furthermore, we must restrict ourselves to back gate voltages where the valence band edge of the semiconductor stays below the Fermi level of the metal; otherwise the band bending would lead to the formation of an unwanted hole pocket near the semiconductor dielectric interface. As indicated in Fig. 1(c), this condition is satisfied as long as eVD > −Φ − Eg = −0.518 eV, which thus defines a lower bound for values of VD that we can apply. We determine the band edge and charge density profiles for our device using both a Thomas-Fermi approximation and a self-consistent Schr¨ odinger-Poisson method. Both of these methods rely on determining φ(z) through Poisson’s equation, which is solved only in the semiconductor region of our device d dφ ρ(z) εr =− . (1) dz dz ε0 Here εr denotes the dielectric constant of the semiconductor, which we set to εr = 15.15 corresponding to InAs, while ρ(z) denotes the charge density of conduction band electrons. The boundary conditions for Eq. (1) are the previously described electrostatic boundary conditions,

i.e. φ(0) = 0 and φ(L2 ) = VD .

B.

The Thomas-Fermi approach

The Thomas-Fermi approximation relies on the assumption that the electronic charge density is given by the same expression as the standard result for a homogenous 3D electron gas ρ(z) = −

i3 e hp 2m (z)/~ . InAs F 3π 2

(2)

Here F (z) denotes the local Fermi energy in the semiconductor, which is determined by the offset between the conduction band edge and Fermi level of the metal, i.e. F (z) = Φ+eφ(z), while mInAs denotes the effective mass of the semiconductor, which we set to mInAs = 0.023me corresponding to zincblende InAs [32]. The Thomas-Fermi approach further combines Eq. (2) with Poisson’s equation (1). This is most conveniently done via the introduction of the following rescaled quantites: (i) electrostatic potential ϕ ≡ eφ/Φ and (ii) coordinate ζ ≡ z/`TF . Here p we defined the Thomas−1 Fermi length-scale `TF = e|ρ(φ = 0)|/(εInAs Φ), where ρ(φ = 0) denotes the charge density obtained in Eq. (2) for φ = 0. With the help of these rescaled quantities, Poisson’s equation (1) may be rewritten in the following dimensionless form 3/2 d2 ϕ(ζ) = 1 + ϕ(ζ) . 2 dζ

(3)

The boundary conditions for ϕ follows from the boundary conditions for φ, i.e. ϕ(0) = 0 and ϕ(L2 /`TF ) = eVD /Φ. We solve Eq. (3) using standard numerical techniques for non-linear differential equations. Figs. 1(d) and (e) show the result of a Thomas-Fermi calculation (dashed lines) of the conduction band edge and charge density profile in an Al-InAs device for several values of VD . The calculation was done with thicknesses of L1 = 5 nm and L2 = 100 nm for the Al and InAs layers, respectively. We find that a triangular well forms near the Al-InAs interface due to band bending, which leads to charge accumulation close to the superconductor. Furthermore, for a positive value of VD , we find that electrons also accumulate near the dielectric away from the superconductor, thus leading to puddles with reduced superconductor-semiconductor hybridization. This situation is undesired since the hole puddle would give an ungapped region in the semiconductor and introduce quasiparticle poisoning to the Majorana device.

C.

The Schr¨ odinger-Poisson approach

The Schr¨odinger-Poisson approach is based on selfconsistently solving Poisson’s equation (1) together with

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FIG. 2. (Color online) Hybrid band structure of Al-InAs obtained with parameters L1 = 5 nm, L2 = 100 nm, Φ = 0.1 eV and VD = −0.5 V. (a) Large-scale zoom of the band structure showing that the bands are predominately Al-like with a narrow region of band segments, which have a large InAs weight. (b) Zoom-in showing that band segments with strong hybridization appear only for narrow ranges of k-values. (c) Zoom-in at the Fermi level revealing that this specific choice of parameters does not lead to states with strong hybridization at the Fermi energy.

the following Schr¨ odinger equation 2 d ~ dψn,k ~2 k 2 − + Ec (z) + ψn,k = En,k ψn,k . dz 2m(z) dz 2m(z) (4) The xy plane translational invariance allows us to consider a fixed in-plane wavevector k = (kx , ky ) of magnitude k = |k|. The quantities m(z) and Ec (z) denote the effective mass and band edge of the hybrid system with the latter given by (see Fig. 1(c)) −EF , −L1 ≤ z ≤ 0, Ec (z) = (5) −Φ − eφ(z), 0 < z ≤ L2 . The boundary conditions for the differential equation (4) are the hard-wall boundary conditions, i.e. ψn,k (−L1 ) = ψn,k (L2 ) = 0. We solve it using a standard finite difference approach, explained in Appendix A. To obtain the self-consistent solution of Eqs. (1) and (4) we need the electronic charge density which is found by integrating over the occupied eigenstates of Eq. (4), according to Z X −e ∞ ρ(z) = dk k |ψn,k (z)|2 Θ(−En,k ), (6) π 0 n with Θ denoting the Heaviside step function. We calculate ρ(z) from the above by solving Eq. (4) for many

different values of k and subsequently evaluating the integral numerically. Figs. 1(d) and (e) show the results of a Schr¨ odingerPoisson (solid lines) calculation of the conduction band edge and charge density profile with the same parameters as the previously described Thomas-Fermi calculation. The two approaches yield remarkably similar results for the band edge and quite similar results for the charge density profile. The strongest deviation for the latter appears at the metal-semiconductor interface, where the Thomas-Fermi result approaches the value predicted by Eq. (2), while the value obtained using the Schr¨ odinger method rises steeply due to the hybridization with the Al layer.

D.

The hybrid band structure

Having determined the band bending profile, we proceed to investigate the hybrid band structure En,k obtained from solving the Schr¨odinger equation (4). Here we focus on results where Ec (z) was obtained using the Schr¨odinger-Poisson approach, but in Appendix B, we also compare these to results based on simpler approximations including the Thomas-Fermi approach. We first address the overall character of the bands based on the results displayed in Fig. 2 showing both En,k as well as the weights in the InAs region of the corre-

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FIG. 3. (Color online) Band structure dependence on Al thickness and VD . Horizontal direction indicates changing Al thickness and vertical indicates changing VD . The semiconductor weights at the Fermi level of crossing bands are displayed in the top right corner of each subfigure.

sponding wave functions ψn,k . We have chosen to include negative values of k, such that the presented band structure corresponds to a cut through the 2D paraboloidal band structure of the system. From the large-scale zoom provided by Fig. 2(a) it is evident that the band structure is composed mainly of segments with negligible InAs weight corresponding to states localized in the Al region. It however also contains a dense region of band segments with high InAs weight corresponding to states which are predominantly localized in the InAs region. Strikingly, band segments with strong superconductorsemiconductor hybridization, i.e. with substantial weight of both InAs and Al, occur rarely as shown in Fig. 2(b). We present in the next section an analytical approach for predicting when bands have substantial hybridization.

experimentally. Our results are summarized in Fig. 3, where the top right corner of each subfigure shows the weight at the Fermi level of the crossing bands. We focus on negative values of VD for which all states are located close to the superconductor-semiconductor interface and restrict ourselves to gate voltages above −0.518 V which, as previously discussed, defines a lower bound for values of VD .

As discussed in Sec. IV, band segments with mixed weights are in fact crucial for the realization of robust Majorana zero modes. They correspond to states with both a strong SOC strength and sizeable superconducting gap, proportional to the Al character of the band, given by the weight wAl . Below we therefore investigate whether it is possible to obtain states of this character at the Fermi level. Furthermore, other bands that cross the Fermi level should not have large InAs weight (wInAs ) at the Fermi energy since this would give rise to a soft gap, and we must therefore also make sure that this is satisfied.

Figs. 3(b) and (e) show that it is indeed possible to obtain a situation where a band with strong hybridization crosses the Fermi level, while InAs-like bands are kept above the Fermi level. Evidently this is obtained by tuning the Al layer thickness, which delicately determines the position and hybridization of the lowest Al-like band. In contrast to this, the hybridization of the bands responds far more weakly to changes in VD . From Fig. 3 one can verify that lowering the gate voltage has the expected effect of pushing up the bands, but the response is weak and most importantly does not significantly affect the degree of hybridization. Furthermore, the gate only affects the position of the InAs-like bands, while the band exhibiting strong hybridization appears practically unchanged. The reason for this is that the semiconductor component of the wavefunction of a strongly hybridized state is concentrated near the superconductor-semiconductor interface, where the superconductor screens the presence of the gate electrode.

To investigate the conditions for having states with mixed weights at the Fermi energy, we address the effects of varying the effective gate voltage VD and Al layer thickness L1 , which are both parameters that can be tuned

To conclude this section, we investigate the effects of varying the parameter Φ, which so far has been set to the value Φ = 0.1 eV. Our results are summarized in Fig. 4, which displays the band structure at the Fermi level for

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FIG. 4. (Color online) Band structure dependence on Φ obtained with parameters L1 = 4.75 nm, L2 = 100 nm and VD = −0.5 V. Increasing the value of Φ leads to a deeper well at the superconductor-semiconductor interface, which gives rises to more InAslike bands below the Fermi level. In (a,b,c) we show the band structure for Φ = 0.1 eV, Φ = 0.3 eV and Φ = 0.5 eV, respectively.

different values of Φ using the parameters L1 = 4.75 nm and VD = −0.5 V (chosen to achieve strong hybridization at the Fermi level). The results show that increasing values of Φ lead to the emergence of more InAslike bands below the Fermi level, thereby leaving several InAs-like states at the Fermi energy with negligible coupling to the superconductor. This is not surprising, since Φ effectively determines the depth of the quantum well at the superconductor-semiconductor interface, but it does appear contradictory to reported experimental measurements on hybrid Al-InAs structures which suggest a regime in which all states at the Fermi energy are strongly coupled to the superconductor. In our framework such a regime is achievable only with a small value of e.g. Φ = 0.1 eV, which thus explains our motivation for originally choosing this value for Φ. It should, however, be emphasized that this chosen value is somewhat smaller than found by recent angle resolved photoemission spectroscopy (ARPES) experiments Φ ∼ 0.23 eV [33]. The ARPES measurement were done for a bulk zincblende structure, but the relevant structure for the nanowire systems is wurtzite where the electron affinity is known to be ∼ 0.1 eV smaller. Therefore, using Φ ≈ 0.1 eV for nanowire systems could be consistent with these experiments. Moreover, it should be noted that these values significantly differ from the bulk value for the difference between the work function of Al and electron affinity of InAs [34, 35] Φbulk ∼ 0.7 eV.

III.

A.

ANALYTICAL APPROACH TO HYBRIDIZATION

Effective-square versus triangular well model

gular well. This is applicable in the case of a large negative VD and the situation corresponds to the one depicted in Fig. 1(b), with the only difference that the physical width of InAs L2 is replaced by an effective width L02 , roughly given by the length for which the triangular potential crosses the Fermi level. In fact, by comparing the reconstructed band structures of the two models, we find that they share the same qualitative features. To illustrate this connection, we compare in Fig. 5 (6) the InAs weights obtained via the Schr¨odinger-Poisson method for parameters Φ = 0.1 eV, L2 = 100 nm and L1 = 5 nm (L1 = 4.75 nm), with the ones calculated using the square potential for L02 = 16 nm. In the same plots we also include the weights calculated via employing the approximate analytic expressions to be discussed in the next paragraph. One observes that, given the way L02 is chosen, the weights obtained using these two models are in good agreement for energies near the Fermi level and begin to deviate for energies which lie above the Fermi level. This deviation mainly happens for small k since the discrepancy is related to the sensitivity of the InAslike bands to the electric field. The agreement allows us to extract approximate analytical expressions describing the hybridization characteristics using the square-well model.

B.

Square-well model and hybrid band structure

The wave functions in both regions are given by sinusoidal functions with wave numbers kAl and kInAs . For a fixed k they have the form C1 sin[kAl (L1 + z)], z ∈ [−L1 , 0], (7) N C2 ψInAs (z) = sin[kInAs (L02 − z)], z ∈ (0, L02 ]. (8) N ψAl (z) =

To shed light on the factors determining the degree of the superconductor-semiconductor hybridization, we proceed with studying a simpler and analytically tractable model. This model is obtained by replacing the triangular potential in the semiconductor with a rectan-

Due to the broken translational invariance along the z direction, the wave numbers are determined by the ener-

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FIG. 5. (Color online) InAs weights for L1 = 5 nm and Φ = 0.1 eV. Solid purple line: Weights obtained numerically using the self-consistent Schr¨ odinger-Poisson method for L2 = 100 nm. Dotted blue line: Weights obtained numerically using the Schr¨ odinger equation within the square-well model for L02 = 16 nm. Dashed red line: Weights obtained within the square-well model using the approximate analytical expressions of Eqs. (14), (17) and (22) for L02 = 16 nm. The energetically-highest weakly modified Al level is the one with n∗ = 27. We observe that the numerical methods are in good agreement up to n = 30. For n > 30 the numerical results obtained using the square-well show significant deviations from the ones calculated with the actual triangular potential. The analytical results follow the numerical ones and manage to capture the qualitative features of the weights. However, the approximate analytical approach is inadequate to describe bands with n ≥ 31.

0 0.0

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2 2 ~2 k 2 ~2 kInAs ~2 k 2 ~2 kAl + −EF = + −Φ = E. 2mAl 2mAl 2mInAs 2mInAs

(9)

Here N and C1,2 denote constants that will be found via the normalization and appropriate matching conditions at the interface z = 0, respectively. The wave function matching yields the transcendental equation: mAl mInAs tan(kAl L1 ) = − tan(kInAs L02 ). kAl kInAs

(10)

The above equation supports two types of solutions characterized by: (i) kAl ∈ R and kInAs ≡ i|kInAs | ∈ I and (ii) kAl,InAs ∈ R. The first type describes dispersive solutions in Al which leak inside InAs within a width ξInAs = 1/|kInAs |. The second type of solutions correspond to states which disperse in z ∈ [−L1 , L02 ].

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FIG. 6. (Color online) InAs weights for L1 = 4.75 nm and Φ = 0.1 eV. Solid purple line: Weights obtained numerically using the self-consistent Schr¨ odinger-Poisson method for L2 = 100 nm. Dotted blue line: Weights obtained numerically using the Schr¨ odinger equation within the square-well model for L02 = 16 nm. Dashed red line: Weights obtained within the square-well model using the approximate analytical expressions of Eqs. (14), (17) and (24) for L02 = 16 nm. The energetically-highest weakly modified Al level is the one with n∗ = 26. The two numerical methods yield quite similar results up to n = 28. For n > 28 the numerical results obtained using the square-well show significant deviations from the ones calculated using the Schr¨ odinger-Poisson method. The analytical results follow the numerical ones and manage to capture the qualitative features of the weights. The approximate analytical expression fail to describe weights corresponding to n ≥ 29.

C.

gy

1.0

0.5

The band structure features

Let us first discuss the bands which originate mainly from the pure Al bands. These become very weakly modified by the InAs conduction band considered here, and belong to the first type of solutions mentioned earlier possessing an imaginary kAl . In fact, wave functions of this type will penetrate only a very small distance into the InAs region. For these deep bands, one has kAl ≈ nπ/L1 , corresponding to the Al layer being an infinite square well with energies 2 ~2 nπ ~2 k 2 Al + − EF . (11) En,k = 2mAl 2mAl L1 For the approximate energy dispersions of these bands see Appendix C. There are n∗ such bands where n∗ is defined by EnAl∗ + 1 ,k=0 . −Φ. 4

(12)

8 For Φ = 0.1 eV and L1 = 5 nm (L1 = 4.75 nm) we find n∗ = 27 (n∗ = 26) which corresponds to EnAl∗ ,k=0 ≈ −0.76 eV (EnAl∗ ,k=0 ≈ −0.46 eV). For these weakly modified Al-like bands, the penetration depth into the semiconductor layer is approximately given by r −1 ξInAs ≈

Al + Φ /~2 . k 2 − 2mInAs En,k

(13)

For the above parameters and n∗ = 27 (n∗ = 26), we find that ξInAs ≈ 1.6 nm (ξInAs ≈ 2.1 nm) at k = 0. At finite k the penetration length becomes even smaller which becomes evident from the equation above. Finally, the InAs weight of such a En≤n∗ ,k band is approximately given by the expression 2

(nπ)

wInAs ≈ (nπ)2 +

mAl mInAs

2

L1

3 .

(14)

ξInAs

The numerical methods employed earlier for retrieving Figs. 5 and 6 confirm that the InAs weight increases with n, and for n∗ it attains values of the order of 5 − 10%. In the same figures we have also calculated the weight via the approximate Eq. (14). We find that the InAs weight, while still reasonably low, is overestimated by this approximate formula. The same happens for ξInAs . In contrast to the low degree of hybridization achieved for the typical device parameters considered here when n ≤ n∗ , for n > n∗ it is possible to find bands that depending on the value of k exhibit strong hybridization. We have studied the structure of the solutions of the transcendental equation (10) and found that also the bands above n∗ have a one-to-one correspondence to the pure Al bands, in accordance with Ref. [3]. However, they significantly differ compared to the pure Al bands, since they become strongly modified in the presence of InAs, especially for small k. Therefore, a band with n > n∗ is generally divided into three k-space regions for which kInAs ∈ R or kInAs ∈ I. See for instance Fig. 7. For large k the new bands resemble the weakly modified Al bands with n ≤ n∗ , since the pure InAs bands are concentrated in the small k region. In most of the cases, the n > n∗ bands are thus Al-like as k → ∞ and become InAs-like as k → 0. In between, new band structure segments appear upon hybridization, which glue the pure Al and InAs bands together. These are characterized by k2 ∈ I. Such segments are depicted in Fig. 2(b,c) and illustrated with dashed (cyan) ellipses in Figs. 7(a,b). In general, they appear for k` ≤ k ≤ kh , with k`,h given by the inequalities: Al En− 1 ,k ≤ 2

~2 k 2 Al − Φ . En+ 1 4 ,k 2mInAs

for n > n∗ .

(15)

It is possible that k` = 0 and one obtains the situation of Figs. 2(c), 3(c) and 7(a) of an Al-like band with additionally strong InAs character for small k. Essentially, the n∗ +1th pure Al band is pushed downwards after contact with InAs, so that a new band now appears below

the location of the pure InAs bands. See band n = 28 in Fig. 7(a). This occurs if the following condition is satisfied for the last Al-like band (meaning for n = n∗ ) EnAl∗ + 1 ,k=0 ≤ 2

~2 k 2 − Φ. 2mInAs

(16)

This type of gluing band segments, appearing only for k ∈ [k` , kh ], possesses an InAs weight given approximately by (see Appendix C) wInAs ≈

1 . 1 + L1 /ξInAs

(17)

For n > n∗ and k ∈ [k` , kh ] the length ξInAs satisfies the approximate relation v u ~2 k 2 Al u mInAs 2mInAs − Φ − En− 12 ,k t 2 . 1 + L1 /ξInAs ≈ 1 + π Al mAl + EF E1,k=0 (18) Based on Figs. 5 and 6 one infers that it is precisely these band segments which have mixed Al-InAs character and exhibit strong hybridization. Thus, in order to ensure the robustness of the Majorana device one should maximize the difference kh −k` and ensure that these segments cross the Fermi level. Strikingly, at the level of approximation considered here, the penetration depth, InAs weight and energy of these segments do not depend on L02 . This also implies a very weak dependence on the back-gate potential and VD , thus, explaining the findings of Fig. 3. The above discussion has already covered the bands with n ≤ n∗ for all k, as well as the band segments for n > n∗ for k > k` . We proceed with investigating the properties of the band segments appearing for k < k` which primarily possess InAs character. At this point we distinguish two scenarios, corresponding to Figs. 7(a,b), depending on whether a new band (scenario I) appears below the pure InAs bands or not (scenario II). Scenario I When the condition of Eq. (16) is satisfied, a band appears below the pure InAs levels, as in Fig. 7(a). In this case we find at higher energies hybridized InAslike bands with n > n∗ + 1, k < k` and modified energies given by InAs (n∗ + 1)2 En−n + λ(n − n∗ − 1)2 EnAl∗ +1,k ∗ −1,k , (n∗ + 1)2 + λ(n − n∗ − 1)2 (19) where we introduced the InAs energy levels

En,k ≈

InAs Es,k

~2 k 2 ~2 = + 2mInAs 2mInAs

sπ L02

2 −Φ

(20)

and defined a hybridization coefficient λ=

mAl mInAs

2

L1 L02

3 .

(21)

9

FIG. 7. (Color online) Figures (a) and (b) depict the two possible hybrid band structure scenarios for energies in the vicinity of the isolated InAs conduction band edge. The band structures shown are obtained via the self-consistent Schr¨ odinger-Poisson method for Φ = 0.1 eV and L2 = 100 nm. In (a) ((b)) we have L1 = 5 nm (L1 = 4.75 nm) and we have additionally shown the relevant pure Al and InAs bands for L02 = 16 nm. The dashed (cyan) ellipses illustrate band segments of strong superconductorsemiconductor hybridization which glue the pure Al and InAs bands together and generally appear for k` ≤ k ≤ kh , see Eq. (15). In (a) we encounter scenario I in which k` = 0 for n = n∗ + 1 and a band appears below the pure InAs due to the hybridizationinduced downward bending of the n∗ + 1th pure Al band. This scenario is realized when the condition of Eq. (16) is satisfied. In contrast, in (b) this condition is not satisfied and a separated band does not appear below the pure InAs levels. However, gluing segments with k` > 0 appear so to connect the pure InAs and Al bands.

The above approximation holds for bands satisfying InAs En−n . EnAl∗ +5/4,k , while for higher energies diffe∗ −1,k rent approximations apply. See Appendix C. The InAs weights for these bands are approximately given by wInAs

(n∗ + 1)2 ≈ . (n∗ + 1)2 + λ(n − n∗ − 1)2

InAs modified. We find that for En−n . EnAl∗ +3/4,k the ∗ −1/2,k energy of the segments defined for k < k` and n > n∗ read (see Appendix C for details)

En,k ≈

(22)

As it can be seen from Figs. 5 and 6, the Al-content of these bands is practically negligible, at least for the parameter values considered here. It is desirable that these band segments acquire a sizeable superconducting gap and get pushed to higher energies, and thus enhance the device protection against quasiparticle poisoning. To achieve this goal one could use a metal with a smaller Fermi energy in order to reduce n∗ + 1. Scenario II So far we have examined the situation in which the n∗ + 1th pure Al level does not get glued to a pure InAs level for k = 0, but instead it is pushed downwards in energy yielding a band below the pure InAs ones. This is the case of Fig. 7(a). However, within the present model a slight modification of L1 by 2.5 ˚ A can lead to a different situation in which the condition of Eq. (16) is not satisfied. As a result one finds a different approximate expression for the InAs weight of the levels above n∗ and small k, i.e. k < k` . In this case, corresponding to Figs. 3(b,e), 4(a) and 7(b), the pure Al levels become glued with the pure InAs ones via appropriate segments of mixed character. While the InAs weight and ξInAs of these segments appearing for k` ≤ k ≤ kh are given by the equations discussed earlier, the expressions describing the InAs-like parts living in k < k` become

InAs L02 En−n + L1 EnAl∗ + 1 ,k 1 ∗ − ,k 2

L1 + L02

2

,

(23)

and the InAs weight is given by the simple (n, k)independent formula wInAs =

1 . 1 + L1 /L02

(24)

We note that for L1 /L02 ≈ 1/3 the weight is wInAs ≈ 75%. From Fig. 6 we find that this result agrees well with the corresponding Schr¨odinger-Poisson calculation for n = 27. As n increases one finds stronger deviations because for higher energies the differences between the square and triangular wells become more pronounced. In the next paragraph we show how to extend our approach and obtain an improved agreement with the Schr¨ odingerPoisson results.

D.

Extended square-well model and fit to the Schr¨ odinger-Poisson solution

The above conclusions can help us understand the obtained band structure when the electrostatic effects, introduced by a non-zero φ, are taken into account. The strongly hybridized bands have a relatively short decay

10 100

100

80

80

60

60

40

40

20

20

0 0.0

0.5

1.0

1.5

0 0.0

100

100

80

80

60

60

40

40

20

20

0 0.0

0.5

1.0

1.5

0 0.0

0.5

0.5

1.0

1.0

terms due to the Zeeman coupling, Rashba SOC and spin-singlet superconductivity. Using that the problem is translational invariant in the xy plane, we write the Hamiltonian in k-space as 1 ~2 k 2 Hk (z) = pˆz pˆz + + Ec (z) τz (26) 2m(z) 2m(z) g(z)µB + B · σ + α(z) (zˆ × ~k) · στz + ∆(z)τx , 2

1.5

1.5

FIG. 8. (Color online) InAs weights shown in the upper (lower) panel for L1 = 5 nm (L1 = 4.75 nm) and Φ = 0.1 eV. Solid purple line: Weights obtained numerically using the self-consistent Schr¨ odinger-Poisson method for L2 = 100 nm. Dashed red line: Weights obtained within the square-well model using L02 = 16 nm. Dotted green line: Weights obtained using the extended square-well model which assumes an energy dependent L02 for bands above the Fermi level. Here we have used L02 = 40.8 nm (L02 = 91.6 nm) for the left (right) panel. In both cases, the extended analytical model yields a significantly improved agreement with the Schr¨ odinger-Poisson approach.

length inside InAs and weakly feel the electrostatic potential. On the other hand, the InAs-like bands are extended over a larger region and are prone to the gateinduced electric fields. The effect of the triangular well is to broaden the effective width L02 for energies above the Fermi level. In fact, for a band with index n consisting of InAs an InAs band with energy Es,k ≥ 0 (s = n − n∗ − 1 or s = n − n∗ ) one can define an effective energy dependent L02,s , given by L02,s =

InAs Es,k=0 , e|Ez |

(25)

since these bands appear for small k. Here Ez (z) = −dφ/dz denotes the electric field in the system, which is non-zero only in the semiconductor’s region. For VD = −0.5 V we find that |Ez | ≈ 6.5 meVnm−1 . By appropriately varying L02 depending on the band, we obtain a very good agreement with the Schr¨ odinger-Poisson results as shown in Fig. 8.

where σx,y,z (τx,y,z ) are Pauli matrices operating in spin (electron-hole space). Furthermore, g(z) denotes the gfactor of the hybrid system, which is set to g = +2 in the Al region and g = −14.9 [32] in the InAs region. The magnetic field B is in-plane such that orbital effects play little role. The term α(z) is the Rashba SOC strength, which is given by [32] ~α(z) =

EFFECTIVE PARAMETERS FOR MAJORANA DEVICES

Having solved the electrostatics and studied the metalsemiconductor hybridization in detail, we now discuss its consequences for the realization of Majorana zero modes. The starting point for our analysis is the Bogoliubov-de Gennes Hamiltonian in the presence of the self-consistent potential determined above, see Eq. (5), with added

(27)

2me /mInAs + gInAs /2 ' −23.3. me /mInAs − gInAs /2

(28)

with g¯InAs = gInAs

The final component entering Eq. (26) is the superconducting order parameter ∆(z) which is non-zero only in Al. For bulk Al, ∆Al ≈ 370 µeV. If we assume large negative back gate voltages leading to a constant electric field in the semiconductor of the order of Ez (z) ≈ 20 meVnm−1 , as estimated from the results in Fig. 1(d), the strength of the Rashba SOC in Eq. (27) becomes ~|αeff | ≈ 0.2wInAs eV˚ A. For wInAs ≈ 0.5 we find ~|αeff | ≈ 0.1 eV˚ A, ∆eff = wAl ∆Al ≈ 170 µeV and geff = gAl wAl + gInAs wInAs ≈ −6.5. In our self-consistent electrostatics analysis we neglected the three above-mentioned additional contributions. Nevertheless, we have verified that the inclusion of the SOC term in the self-consistent Schr¨odinger-Poisson problem does not significantly modify the obtained results. This also holds for both magnetic field and superconducting energy scales which constitute the smallest energy scales in the problem. Therefore, one could use the presented results for the band edge profiles and hybridization degrees for further modeling the physics of experimentally realized Majorana devices.

V. IV.

g¯InAs ~2 eEz (z) ~¯ gInAs µB Ez (z) ≡ , 2Eg 2 2me Eg

DISCUSSION AND CONCLUSIONS

We have assessed the role of band bending and superconductor-semiconductor hybridization in Majorana devices by studying a planar, gated Al-InAs interface. Our results were based on a self-consistent effective-mass Schr¨odinger-Poisson approach, which revealed that the band bending leads to an approximately triangular quantum well along with a charge accumulation layer at the Al-InAs interface. We also compared the

11 Schr¨ odinger-Poisson calculation with a Thomas-Fermi approach which ignores the hybridization and found remarkably similar results for the band bending. This can be useful for future calculations since one can use the computationally faster Thomas-Fermi approach to determine the self-consistent potential and then solve the Schr¨ odinger equation in this potential. The character of the superconductor-semiconductor hybridization was addressed by calculating the band structure of the hybrid system and investigating its response to varying the Al layer thickness, gate voltage and native band offset. Our main finding is that the system parameters may be tuned to a situation as shown in Fig. 3(e) where a band with strong superconductorsemiconductor hybridization crosses the Fermi level, while higher levels of predominately InAs character stay above it. Such a situation is ideal for inducing superconductivity in the InAs region, which requires strong hybridization with the Al region, while simultaneously keeping out the InAs-like bands, which would give rise to a soft superconducting gap. The conditions for having the ideal situation shown in Fig. 3(e) turns out to be extremely sensitive to the Al layer thickness, while a far weaker dependence on the gate voltage was found. Furthermore, another important parameter is how the metal Fermi level aligns with the semiconductor conduction band, see Fig. 1. Here we have used Φ in the range 0.1-0.3 eV, which is supported by recent experiments [33], but not by known bulk values. Therefore, it could be that there is some surface chemistry that still needs to be resolved before a more complete understanding of these structures can be reached. We have seen that the hybridization and the number of bands below the Fermi energy depend strongly on the thickness of the Al layer and therefore one expects that disorder such as variations by even a single monolayer of the deposited Al layer could have strong effects. Likewise, if a nanowire structure is formed out of the quasi-2D system studied here, a non-regular cross section could give qualitatively different results from what we found for the 2D translationally invariant setup. Moreover, the strong dependence on Al thickness also raises the question whether a more microscopic description (for example a tight-binding model of the Aluminum) would give a different result. We speculate that the details will naturally be different but also that the sensitivity to thickness remains because of the large mismatch of energy scales between the two materials. Both the effect of disorder

−~2 zi+1 − zi−1

1

and more detailed band structure are natural questions for further research. To back our numerical findings, we analyzed the superconductor-semiconductor hybridization using an analytical approach showing that the hybridization is only sizable when there is resonance between the uncoupled Al and InAs bands. This behavior might seem surprising given the absence of a barrier between the materials, and it appears to be a result of mismatch of the wavefunctions in the metal square well and triangular semiconductor well. In conclusion, devices based on Al-InAs or similar material combinations are indeed promising candidates for Majorana physics and several experiments have already shown signatures of Majorana zero modes. However, based on the analysis here it seems to require a fine balance between several parameters, such as the metal thickness and band alignments. In our simulations the effect on gate voltage is very limited when it comes to the degree of hybridization and position of the bands. Disorder effects might help in relaxing these conditions. Studying these effects would require a 2D simulation, which we intend to pursue in future works. Experimentally, there seems to be a stronger dependence on gate voltage which could be due to the gate coupling inhomogeneously to the structure. A better understanding of this would require a full 3D self-consistent simulation. At the time of submission two other works addressing Schr¨odinger-Poisson calculation for superconductorsemiconductor hybrid structures appeared [36]. ACKNOWLEDGEMENTS

We would like to thank A. Akhmerov, A. Antipov, K. Bj¨ornson, M. Hell, R. Lutchyn, S. Schuwalow, S. Vaitiek´enas, A. Vuik, G. Winkler, and M. Wimmer for useful discussions. This work was supported by The Danish National Research Foundation and by the Microsoft Station Q Program. Appendix A: Numerical methods 1.

The Schr¨odinger equation (4) is solved on a 1D grid using the finite difference approximation:

ψi+1 − ψi 1 ψi − ψi−1 − ∗ m∗i+1/2 zi+1 − zi mi−1/2 zi − zi−1

Here m∗i+1/2 denotes the average value of the effective

The Schr¨ odinger equation

! +

~2 k 2 2m∗i

+ Ec,i ψi = Eψi .

(A1)

mass on the two grid points i and i+1. Since the Fermi

12

0.4 0.3

Ec [eV]

(b)

Full SP TF Simplified SP No charge

0.1

0.2 0.1 0

(c)

0 0.48 0.49 0.50 0.49

–0.1

-0.1 0 0.2

20

40

60

80

100

c

(d)

En,k [eV]

0 –0.1 –0.2 –0.3 0

Al 4.75nm

–0.5

0

0.5

d 0

0.1

Ec [eV]

b

En,k [eV]

(a)

–0.1 0.45 0.46 0.46 0.52

–0.2 20

40

60

80

100

–0.5

0 k [nm-1]

0.5

InAs 100nm

FIG. 9. (Color online) Self-consistent band edge profiles and band structures obtained using the methods described in Appendix B with L1 = 4.75 nm and VD = −0.5 V. Results for both Φ = 0.1eV and Φ = 0.3eV are shown. (a) Band edge profiles obtained for Φ = 0.1 eV. (b) Band structures obtained for Φ = 0.1 eV. The numbers in the lower left corner are the weights at the Fermi energy of the bands marked by the black circle. (c) Band edge profiles obtained for Φ = 0.3 eV. (d) Band structures obtained for Φ = 0.3 eV. The numbers in the lower left corner are the weights at the Fermi energy of the bands marked by the black circle.

wave length of the metal is orders of magnitude smaller than that of the semiconductor it is advantageous to make the discretization more coarse in the semiconductor region. The results presented in this work were obtained using a fixed grid spacing of 0.1 ˚ A in the Al region and 2˚ A in the InAs region. To obtain the solutions to the system of Eq. (A1), we solve it as an eigenvalue equation. We enforce hard-wall boundary condition by setting the wave functions to zero at the ends of the 1D grid.

2.

Obtaining the self-consistent solution

Our Schr¨ odinger-Poisson approach relies on selfconsistently solving Eqs. (1), (4) and (6). For this we

employ a simple mixing scheme, where the input electrostatic potential used in each iteration is a simple mixing of the input and output electrostatic potential of the previous iteration i−1 φiin (z) = κφi−1 out (z) + (1 − κ)φin (z).

(A2)

In our calculations we used κ = 0.1. For the initial input, we use φ1in (z) = VD z/L2 . While the authors of Ref. [25] have shown that more sophisticated mixing schemes such as Anderson mixing leads to a faster convergence, we find that the simple mixing scheme above provides reasonably fast convergence within the first 50-100 self-consistent iterations.

13 Appendix B: Influence of band bending on the hybridization

The hybrid band structures shown in Sec. II D were obtained using Ec (z) obtained from the Schr¨odingerPoisson approach. This procedure is computationally demanding since it requires solving Eq. (4) a larger number of times within each self-consistent iteration when calculating the electronic density from Eq. (6). In this section, we explore the consequences of employing simpler and computationally faster approaches for calculating Ec (z). Specifically, we compare the results of Sec. II D to those obtained when Ec (z) is determined by: 1. The Thomas-Fermi approach described in Sec. II B. 2. A simplified Schr¨ odinger-Poisson approach, where Eq. (4) is solved only in the InAs region with boundary conditions ψn (0) = ψn (L2 ) = 0. In this case the wave functions are independent of k and the density in the semiconductor region is found by integrating over the 2D density of states for each subband yielding ρ(z) =

X mInAs |En | n

π 2 ~2

|ψn (z)|2 Θ(−En ).

(B1)

3. With ρ(z) = 0 in the semiconductor, i.e. neglecting band bending due to charge in the InAs such that Ec (z) = −Φ − eVD z/L2 .

(B2)

Our results are summarized in Fig. 9 where we show both the resulting band bending and the band structure plots from the different approaches for Φ = 0.1 eV and Φ = 0.3 eV. We have here chosen parameters such that a strongly hybridized band crosses the Fermi level, and the corresponding InAs weights at the Fermi energy (indicated by the circle) are shown in the bottom left corners of Figs. 9(b) and (d). When Φ = 0.1 eV (Figs. 9(a,b)) the density in the InAs region is low, and the band edges exhibit only a slight bending, staying close to the constant slope found without density in the semiconductor. Notably the strongest bending is found when the full Schr¨ odinger-Poisson approach is employed. This is due to the hybridization with the Al which induces a large electron density close to the Al-InAs interface (see Fig. 1(d)). In the case Φ = 0.3 eV, the band bending profiles are much stronger and deviate substantially from the solution without charge. Thus, the strongest bending is found from the full Schr¨odingerPoisson approach, but both Thomas-Fermi and simplified Schr¨ odinger-Poisson methods yield similar results. The same conclusion holds for the band structure plot of Fig. 9(d). Here the band structures obtained with band bending are reasonably similar, while the one obtained with constant slope evidently contains additional bands below the Fermi level due to the more shallow

band profile. Interestingly, it appears that the weight at the Fermi energy of the strongly hybridized band is only weakly dependent on the exact band bending profile and all approaches yield comparatively similar results. Appendix C: Analytical approach to hybridization

In this appendix, we demonstrate how to obtain approximate analytical expressions for the band structure properties of the square-well model of Sec. III B. We start from Eq. (10) and rewrite the transcendental equation as tan (kAl L1 ) kAl L1 π

=−

mInAs L02 tan (kInAs L02 ) , kInAs L02 mAl L1

(C1)

π

or bring it to its inverted form kAl L1 mAl L1 kInAs L02 cot (kAl L1 ) = − cot (kInAs L02 ) . π mInAs L02 π (C2) Fig. 10 depicts the functions of the l.h.s. (dashed blue) and r.h.s. (solid gold) of Eqs. (C1) and (C2) for Φ = 0.1 eV, L02 = 16 nm, k = 0, and Al width L1 = 5 nm and L1 = 4.75 nm, respectively. The energy eigenstates of the hybrid band structure are given by the crossing points of the two functions. For low energies we have essentially solutions corresponding to isolated Al. Instead, for energies in the vicinity of the pure InAs levels, we find solutions emerging from the hybridization of InAs and Al. In Fig. 10 the top and corresponding bottom panels, i.e. (a,c) and (b,d), lead to identical solutions. Nevertheless, we have included them both since, depending on the energy regime, it is more convenient to employ the inverted (C2) instead of the direct transcedental equation (C1). The aim is to Taylor expand the l.h.s. or/and r.h.s., of the respective transcedental equation employed, about zero. Depending on the case, one performs a linear or quadratic Taylor expansion. If we work with Eq. (C1) we can expand the l.h.s. as follows " # π δE δE tan (kAl L1 ) 3 ≈ 1− , kAl L1 Al Al 2 En,k=0 4 En,k=0 + EF + EF π (C3) Al with the energy shift δE = E − En,k being measured from a pure Al level, which yields a zero tangent since kAl L1 /π = n ∈ N∗ . A similar method is followed if we want to expand the r.h.s. of the same equation about the pure InAs levels. If it is instead preferable to employ Eq. (C2), then one should expand about a zero of the respective l.h.s. or r.h.s.. For instance, if we wish to expand the l.h.s. about a zero of the cotangent obtained for

1 kAl L1 =n+ π 2

with n ∈ N+ ,

(C4)

14

FIG. 10. Plot of the l.h.s. (dashed blue) and r.h.s. (solid gold) of the direct and inverted transcedental Eqs. (C1) and (C2) for Φ = 0.1 eV and L02 = 16 nm. In (a,c) L1 = 5 nm and in (b,d) L1 = 4.75 nm. The eigenenergies are obtained when the l.h.s. and r.h.s. lines cross. Depending on the energy regime it is convenient to use the direct or inverted transcedental equation in order to obtain the eigenspectrum. The approximate analytical expressions for the eigenenergies are obtained via a Taylor expansion of the l.h.s. or/and the r.h.s. about zeros of the respective tangent or cotangent. These zeros are related to the InAs Al InAs Al . (a) is employed for inferring weakly and En+1/2,k , as well as En+1/2,k and En,k pure Al and InAs levels and the values En,k modified Al-like bands and InAs-like bands within scenario I. (b) is employed for inferring weakly modified Al-like bands within scenario II. (c) is employed for inferring the Al-InAs gluing segments within scenario I. (d) is employed for inferring modified InAs-like bands within scenario II.

we have kAl L1 cot (kAl L1 ) ≈ π " # π δE 1 δE − 1+ . (C5) Al Al 2 E1,k=0 4 En+ + EF + EF 1 ,k=0 2

mate analytical expression discussed in the text, let us remark that our approach generally holds for the case of small Al widths where the spacing of the pure Al levels is much larger that the spacing of the pure InAs levels. Our approximation further holds for arbitrary values of L02 . However, it can modify the number of Al-InAs bands for which our method is valid.

Al Here the energy shift δE = E − En+ is measured re1 ,k 2

lative to the n + 12 th pure Al level. In reality there is no such a pure Al level before contact with InAs, but this notation is convenient because it reflects that we are focusing on energy eigenstates which appear due to the Al-InAs hybridization. One can further expand the r.h.s. about the s + 21 th pure InAs level, with s ∈ N+ , in a similar fashion. Before proceeding with obtaining the various approxi-

1.

Solutions with kAl ∈ R and kInAs ≡ i|kInAs | ∈ I

After applying the matching conditions, we find that the wave function for such a solution with En,k has the approximate form r 2 n+1 √ ψn,k (z) ≈ (−1) wAl sin [kAl (z + L1 )] , (C6) L1

15 Al consider tanh (|kInAs |L02 ) ≈ 1, and set E = En,k on the r.h.s.. In Figs. 10(a,b) we show details for the n = n∗ level. We find that the eigenenergies approximately read

for z ∈ [−L1 , 0] and ψn,k (z) ≈

√

r wInAs

sinh [|kInAs |(L02 − z)] , ξInAs sinh(|kInAs |L02 ) 2

(C7)

Al En,k = En,k

for z ∈ [0, L02 ]. The InAs and Al weights are defined as R L02 0 wInAs = R L 0 2

dz |ψn,k (z)|2

dz |ψn,k (z)|2 −L1

−

En,k =

~ k 2mInAs

v 2 u 2 2 r 2 r Al u ~ k /(2m ) − Φ − E 1 InAs 1 mAl 1 mAl n− 2 ,k Al − − t + E1,k=0 + EF . Al π mInAs π mInAs E1,k=0 + EF

Scenario I In this case we use Eq. (C1) and linearize both sides. We linearize the l.h.s. about the EnAl∗ +1,k level InAs and the r.h.s. about the En−n level. See Fig. 10(a) ∗ −1,k for n = n∗ + 2. This approximation led to Eqs. (19) and (22), and holds as long as the linear approximation InAs of the l.h.s. is valid, i.e. En−n . EnAl∗ + 5 ,k . ∗ −1,k 4

Scenario II In this case we use Eq. (C2) and linearize both sides. We linearize the l.h.s. about the EnAl∗ + 1 ,k 2

InAs level and the r.h.s. about the En−n level. See 1 ∗ − 2 ,k Fig. 10(d) for n = n∗ + 3. This approximation led to Eqs. (23) and (24). Note that this approximation holds as long as the linear approximation of the l.h.s. is valid, InAs i.e. En−n . EnAl∗ + 3 ,k . In this case, for n = n∗ + 3 1 ∗ − 2 ,k 4 we are on the borderline of our approximation’s validity. For larger L02 , e.g. L02 ∼ 80nm we have to expand up to quadratic order the r.h.s. in order to obtain a good approximate solution, and in this case the energy reads

for z ∈ [−L1 , 0] and s wInAs

2 sin [kInAs (L02 − z)] , L02

(C12)

for z ∈ [0, L02 ] with the weights discussed in the main text. These solutions appear for k < k` defined in Sec. III. One

En,k

v 2 InAs Al u E − E u 1 1 L1 n∗ + 2 ,k L1 n−n∗ − 2 ,k InAs InAs t 1 + L1 = En−n + 2 + − 1 + E + Φ , 1 1 n−n − − ,k ,k=0 0 0 0 InAs ∗ ∗ 2 2 L2 L2 En−n L2 +Φ − 1 ,k=0 ∗

which is applicable for bands satisfying EnAl∗ +1/2,k > InAs En−n . The approach can be extended to higher ∗ −1/2,k

(C10)

distinguishes two scenarios I and II.

After applying the matching conditions, we find that the wave function for such a solution with En,k has the approximate form r 2 n+1 √ ψn,k (z) ≈ (−1) wAl sin [kAl (z + L1 )] . (C11) L1

ψn,k (z) ≈

Al En,k=0 + EF .

2

Solutions with kAl ∈ R and kInAs ∈ R

√

Al ~2 k 2 /(2mInAs ) − Φ − En,k

Al consider coth (|kInAs |L02 ) ≈ 1, and set E = En− on the 1 2 ,k r.h.s.. See also Fig. (10)(c). We find the eigenenergies

Note that the above expression does not depend on L02 at this level of approximation. Using the above, we obtained Eqs. (17) and (18). 2.

InAs + Φ E1,k=0

Using the above, we obtained Eqs. (13) and (14). We proceed with the band segments for n > n∗ and k ∈ [k` , kh ]. In this case we consider the inverted tranAl scedental Eq. (C2). We linearize the l.h.s. about En− , 1 ,k

and wAl = 1 − wInAs . (C8)

We first discuss the weakly modified pure Al levels, with n ≤ n∗ for all k and n > n∗ for k > kh . In this case Al we start from Eq. C1. We linearize the l.h.s. about En,k ,

2 2

2 mInAs L02 π mAl L1

(C9) s

(C13)

2

energies by separating the energy interval in regions where the direct or inverted transcendental equation is best.

16

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