In-plane selective area InSb-Al nanowire quantum networks

Strong spin-orbit semiconductor nanowires coupled to a superconductor are predicted to host Majorana zero modes. Exchange (braiding) operations of Majorana modes form the logical gates of a topological quantum computer and require a network of nanowires. Here, we develop an in-plane selective-area growth technique for InSb-Al semiconductor-superconductor nanowire networks with excellent quantum transport properties. Defect-free transport channels in InSb nanowire networks are realized on insulating, but heavily mismatched InP substrates by 1) full relaxation of the lattice mismatch at the nanowire/substrate interface on a (111)B substrate orientation, 2) nucleation of a complete network from a single nucleation site, which is accomplished by optimizing the surface diffusion length of the adatoms. Essential quantum transport phenomena for topological quantum computing are demonstrated in these structures including phase-coherent transport up to 10 $\mu$m and a hard superconducting gap accompanied by 2$e$-periodic Coulomb oscillations with an Al-based Cooper pair island integrated in the nanowire network.

Indium-Antimonide (InSb) and Indium-Arsenide (InAs) nanowires are promising candidates for realizing Majorana-based topological quantum computers. [1] InSb, in particular, is interesting for its high electron mobility, strong spin-orbit coupling and large Landé gfactor. [2][3][4][5] Over the past years, extensive efforts have been made to improve the quality of InSb nanowires, grown by the vapor-liquid-solid (VLS) mechanism. [6,7] A major achievement is the clean and epitaxial semiconductor-superconductor interface relying on insitu metal evaporation and complex substrate processing. [8,9] This has been critical for showing the first quantized conductance of Majorana zero-modes, a milestone towards braiding experiments. [10] To braid these Majorana states, scalable nanowire networks with a high degree of interconnectivity are required. [11] Recently, out-of-plane growth of InSb nanowire networks has been demonstrated by merging multiple wires during VLS growth.
Phase-coherent and ballistic transport have been observed, demonstrating the high quality of these network structures. [8,12] This technique, however, requires predefined positioning of the nanowires with nanometre accuracy in order to form networks, leading to a drastic decrease in yield with increasing network complexity. Moreover, merging of VLS nanowires inevitably forms a widening of the nanowire diameter at and around the junction with a 75% chance of a defect forming at the junction, which negatively affects the one-dimensionality of the system. [8] Here, we demonstrate a more scalable approach, using an in-plane selective-area growth (SAG) technique (i.e. parallel to the substrate surface), that relies on a template or mask to selectively grow one semiconductor material on top of another. [13][14][15][16][17][18][19] Our technique has several advantages over out-of-plane growth. First, the flexibility of network designs is significantly enhanced, as the preferred design can be written and etched directly into a mask enabling complex structures. Second, the growth can be confined within the mask, keeping the entire structure one-dimensional with an easily controllable and constant cross-section size.
Finally, the technique excels in scalability, readily allowing for the growth of complex structures suggested for Majorana braiding experiments in a variety of theoretical proposals. [11,20,21] The large lattice mismatch between InSb and any other III-V semiconductor substrate material, [22] however, poses an important challenge making it difficult to grow defect-free InSb nanowires on large-bandgap or insulating substrates.
Furthermore, the disorder created by the lattice mismatch can be detrimental to the topological protection of Majorana states. [23] Most of the previous SAG studies have focused on an InAsbased material system for nanowire networks [14][15][16][17]19] which has a smaller lattice mismatch with InP or GaAs substrates. InSb nanowires have been grown by SAG on (100) GaAs substrates but the electron transport demonstrated is diffusive. [18] Here, we demonstrate an inplane SAG technique for scalable and high quality InSb nanowire networks, which shows all the relevant quantum transport properties (e.g. long coherence length and excellent induced superconducting properties) necessary for topological qubits.
In this study we use InP as a substrate because it has a type I band alignment with InSb and becomes semi-insulating at low temperatures. The large band gap of 1.34 eV compared to 0.17 eV of InSb ensures good confinement. [24] The lattice mismatch between these materials is 10.4%, making large scale defect-free growth of InSb a challenge. [25] It is important to have defect-free single crystalline material to obtain a high carrier mobility by reducing electron scattering. [26] On a (100) substrate, many stacking faults are expected to form inclined to the substrate (along the {111} planes) to relax the lattice strain. [13] To avoid the formation of these defects, we use InP (111)B as growth substrate. We will show that strain from the lattice mismatch is relieved directly at the substrate/nanowire interface. Moreover, atomically flat twin planes can form parallel to the substrate, after which the nanowires grow completely defect-free, separating the mismatch-induced disorder from the nanowire top part. On a (111)B substrate, growth nuclei with an odd and even number of horizontal twins will have a 180ºrotated crystallographic orientation with respect to each other. When these nuclei merge by lateral growth, a defect is formed at the interface; details of the atomic structure of such a defect can be found in the supplementary figure S1. These defects may act as scattering sites for electrons. [26] Therefore, it is important to enable growth of a complete, in-plane network structure from a single nucleation site. For this, a large surface diffusion length of the precursor material is required as well as a low nucleation probability. Here, we show that metalorganic vapour-phase epitaxy (MOVPE) grown InSb in-plane selective area networks (InSANe) can indeed generate complicated 1D networks from a single nucleation site.  1a). These four orientations are all three-fold symmetric, such that networks can be grown with angles of 30, 60 and 90 degrees between two nanowires (see figure   S3). The growth starts by forming a nucleus in one of the lines, which develops into an InSb island over time ( figure S4). All nuclei are terminated with a {111}B top facet and {110} side facets ( figure 1b and 1d), regardless of the line orientation, implying that surface growth kinetics and lateral growth rates are identical for all <110> and <112> growth directions. For longer growth times, the growth continues in the lateral direction following the mask opening by growing {110} facets, as evidenced by atomic force microscopy (AFM) and transmission electron microscopy (TEM) (see supplementary information). When the structure is fully grown in the in-plane direction, the growth continues in the vertical <111>B direction. The height of the InSb network can be precisely tuned by the growth time (see figure S5). When the InSb grows higher than the mask, it also starts to expand in the lateral direction, especially at acute corners of the structure (figure S6), which is not ideal for transport measurements as the one-dimensional confinement is lost. An example of a network is shown in figure 1c (and figure S7), whose structure corresponds to the proposed geometry of a four-topological-qubit device. [20] The {110} planes of the original growth fronts are visible on the convex corners of this structure (figure 1d). These facets do not form on the concave corners due to the connection with another branch of the network (figure S6). Our platform provides freedom of design and scalability for a plethora of device structures (figure S8).
In order to minimize the formation of inclined defects, the number of nucleation sites in the mask openings per unit length (n) is investigated as a function of the input V/III ratio of the Tri-Methyl-Indium (TMIn) and Tri-Methyl-Antimony (TMSb) precursors. For this purpose, InSb is grown for a short time (1 minute) to observe the early stages of nucleation and understand the nucleation probability as a function of V/III ratio. n is determined for growth in the <112> and <110> oriented trenches for different V/III ratios (figure 2). The results show a clear decrease of n, and thus an increase of the diffusion length of the adatoms on the InP surface, with increasing V/III ratio. At the highest TMSb pressures, the nucleation of InSb islands is completely inhibited. The inset in figure 2a is a logarithmic plot of the same data up to a V/III ratio of 20,000, showing a decrease of n with increasing V/III ratio. We note that there is no parasitic growth on the mask under any of these conditions. The red-circle and black-square (blue and green triangle) data points are all taken with the same TMSb (TMIn) partial pressure and varying TMIn (TMSb) partial pressure respectively. Figure 2a shows that these datapoints all follow the same trend, indicating that not the total flow but the V/III ratio is important in the studied range. Figure 2b shows a scanning electron microscopy (SEM) image of nuclei grown using a low V/III ratio. The InSb islands are only tens of nanometres long and approximately 20 nm thick. Figure 2c shows an SEM image of a representative nucleus grown with a very high V/III ratio. Here, the InSb island is much longer (300 nm) and less than 20 nm high. By integrating over the total volume of all nuclei in a given structure, we find that the lateral growth rate is determined by the TMIn flux (with higher flux giving faster growth rates) and the V/III ratio (with higher V/III ratio giving slower growth rates). From these results, we conclude that Sb changes the surface energy on the InP substrate and enhances the surface diffusion of the In precursor material. [27] For the growth of large networks, a high The 10.34% decrease in vertical lattice planes is in good agreement with the reported value of a 10.4% lattice mismatch between InSb and InP [4], indicating that the heterostructure is fully relaxed (inelastically) at the interface in the lateral direction. Here, it should be mentioned that when imaging orthogonally to this crystal direction the InP/InSb interface cannot be imaged accurately due to the slight recess of the wire into the substrate. Most likely, along the long axis, misfit dislocation planes will also be present with the dislocation lines located at the interface, as we do not see vertically extending defects orthogonal to the long axis of the wires in TEM studies. To visualize the presence of horizontal twin planes in the InSb structures, we investigate a cross-section of a nanowire grown along the <110> direction (figure 3e). A zoomin on the bottom part (figure 3f) reveals a set of two twin boundaries a few nanometres above the InP/InSb interface (figure 3g). Horizontal twin planes have in total been observed in 12 out of 17 TEM samples (6 in perpendicular cuts of <110> grown wires and 6 in parallel cuts of <112> grown wires, the other 5 TEM samples did not show horizontal twin planes). The twin planes are always located within a few nanometres above the InSb/InP interface and a single crystalline InSb nanowire always forms above the twin. Once a twin plane is formed in a nucleus it will be extended into the rest of the network by lateral growth. The complete relaxation of lattice strain at the nanowire/substrate interface (bottom part) followed by horizontal twin planes helps to separate the electron wavefunctions in the active region of the device from the interface disorder. [28] This effect allows us to fabricate in-plane InSb nanowires on InP that have quantum transport properties comparable to free-standing structures as demonstrated by the high quality quantum transport in the next section.
We now turn to the quantum transport properties of our InSb InSANe system to demonstrate its feasibility for topological quantum information processing. The key ingredient in the measurement-based gate operation and topological qubit readout is the phase coherence, which can be revealed by the Aharonov-Bohm (AB) effect. [11,20]  the reproducibly high quality of our networks. The measured AB period matches the loop area in all measured devices, i.e. their periods equal h/ne (for the n th harmonic). We point out that this is the first time to observe higher order AB harmonics in nanowire loop structures [8,14,15,19,29], suggesting high quality material. This result corroborates that the lattice-mismatchinduced disorder at the nanowire-substrate interface has negligible effect on the phase coherent transport. Finally, we observe a sharp weak anti-localization (WAL) peak in the magnetoconductance of this device around zero magnetic field (figure 4e), indicating the strong spinorbit nature of the InSb nanowire. Fitting this WAL curve requires a new theoretical model applicable for nanowire networks, which will be developed in future studies.
The next important step is to introduce superconductivity in the InSb InSANe system.
In order to form superconducting contacts to create the semiconductor-superconductor hybrid networks needed for the formation of Majoranas, the InSANe InSb samples, after growth, are transferred from an MOVPE to an MBE system. Here, the surface oxides are removed using atomic-hydrogen cleaning under (ultra-high vacuum) UHV conditions followed by 7 nm aluminium deposition at a sample temperature of around 120K, leading to a clean and smooth InSb-Al interface (see figure S9). [8] Since in-situ shadowing methods to selectively grow superconductors on these InSb inplane structures are not yet developed, we exploit a reliable selective etching recipe to selectively etch Al on InSb. This novel fabrication recipe enables us to define the positions of tunnel barriers and the Al film by lithography, facilitating flexible device designs. Figure 5a shows such a device where part of the Al is selectively etched away, and a tunnel gate electrode is added to deplete the InSb wire locally. A super-gate is deposited on the superconducting region of the nanowire whose cross-section is shown in figure 5b. The differential conductance on this normal-nanowire-superconductor (N-NW-S) device reflects the quasi-particle densityof-states in the proximitized nanowire segment, i.e. the induced superconducting gap as shown in figure 5c with a line-cut in figure 5d. The sub-gap conductance reaches zero, indicating a hard gap, a necessary condition for topological protection. Magnetic field dependence of accidental quantum dot levels reveals an effective g-factor of 18.6 (figure S11), smaller than the bare InSb g-factor of 50 but significantly larger than that of Al (|g|=2), indicating the wavefunction hybridizes between InSb and Al. [28,30] The measured hard gap, together with the effective g-factor defined by the coupling between Al and InSb, suggest that the electron wavefunction is mainly distributed near the top (close to Al) where the wire is single crystalline with no noticeable disorder, [31][32][33] suggesting that disorder at the InP/InSb interface has a negligible effect here.

Data availability
The raw data and the data analysis codes that support the findings of this research are available at https://doi.org/10.5281/zenodo.4589484.