Selective Area Grown Semiconductor-Superconductor Hybrids: A Basis for Topological Networks

We introduce selective area grown hybrid InAs/Al nanowires based on molecular beam epitaxy, allowing arbitrary semiconductor-superconductor networks containing loops and branches. Transport reveals a hard induced gap and unpoisoned 2e-periodic Coulomb blockade, with temperature dependent 1e features in agreement with theory. Coulomb peak spacing in parallel magnetic field displays overshoot, indicating an oscillating discrete near-zero subgap state consistent with device length. Finally, we investigate a loop network, finding strong spin-orbit coupling and a coherence length of several microns. These results demonstrate the potential of this platform for scalable topological networks among other applications.

We introduce selective area grown hybrid InAs/Al nanowires based on molecular beam epitaxy, allowing arbitrary semiconductor-superconductor networks containing loops and branches. Transport reveals a hard induced gap and unpoisoned 2e-periodic Coulomb blockade, with temperature dependent 1e features in agreement with theory. Coulomb peak spacing in parallel magnetic field displays overshoot, indicating an oscillating discrete near-zero subgap state consistent with device length. Finally, we investigate a loop network, finding strong spin-orbit coupling and a coherence length of several microns. These results demonstrate the potential of this platform for scalable topological networks among other applications.
In this Letter, we investigate a novel approach to the growth of semiconductor-superconductor hybrids that allows deterministic on-chip patterning of topological superconducting networks based on SAG. We characterize key physical properties required for building Majorana networks, including a hard superconducting gap, induced in the semiconductor, phase-coherence length of several microns, strong spin-orbit coupling, and Coulomb block-ade peak motion compatible with interacting Majoranas. Overall, these properties show great promise for SAGbased topological networks.
Selective area growth was realized on a semi-insulating (001) InP substrate. PECVD grown SiO x was patterned using electron beam lithography and wet etching. InAs wires with triangular cross-sections were grown by molecular beam epitaxy (MBE). The Al was grown in-situ by MBE using angled deposition covering one of the facets. The excess Al was removed by wet etching [ Fig. 1(a-c)]. The details of the semiconductor growth are given in Ref. [17], while it is superconductor-semiconductor proximity effects that are emphasized in the present study. Data from four devices are presented. Device 1 [ Fig. 1 consists of a single barrier at the end of a 4 µm wire, defined by a lithographically patterned gate adjacent to a Ti/Au contact where the Al has been removed by wet etching. This device allowed density of states measurement at the end of the wire by means of bias spectroscopy, to investigate the superconducting proximity effect in the InAs. Evolution of Coulomb blockade in temperature and magnetic field was studied in Device 2 [ Fig. 2(b)]-a hybrid quantum dot with length of 1.1 µm defined by two Ti/Au gates adjacent to etched-Al regions. The barrier voltages V B were used to create tunneling barriers. The chemical-potential in the wires was tuned with gate voltage V G . Device 3 was a micron-size square loop [ Fig. 4 [30] for more detailed description of the growth, fabrication and measurement setup.
Differential conductance, G, in the tunneling regime, as a function of source-drain bias, V SD , for Device 1 [ Fig. 1(e)] at V G = −9.2 V reveals a gapped density of states with two peaks at V SD = 110 µeV and 280 µeV. We tentatively identify the two peaks with two populations of carriers in the semiconductor, the one with a larger gap residing at the InAs-Al interface and with a smaller at the InAs-InP. The magnitude of the larger superconducting gap is consistent with enhanced energy gaps of 290 µeV for 7 nm Al film [31]. The zero-bias conductance is ∼ 400 times lower than the above-gap conductance, a ratio exceeding VLS nanowire [12,32,33] and 2DEG devices [14], indicating a hard induced gap. We note, however, that co-tunneling through a quantumdot or multichannel tunneling can enhance this ratio [34]. The spectrum evolution with V G from enhanced to suppressed conductance around V SD = 0 mV is shown in Fig. S 1 in Supplemental Material [30].
Transport through a Coulomb island geometry [ Fig. 2] at low temperatures shows 2e-periodic peak spacing as a function of V G . Coulomb diamonds at finite bias yield a charging energy E C = 60 µeV (see Fig. S1 in Supplemental Material [30]), smaller than the induced gap, ∆ * ∼ 100 µeV, as seen in Fig. 1(e). The zero-bias Coulomb blockade spacing evolves to even-odd and finally to 1eperiodic peaks with increasing temperature, T . The 2e to 1e transition in temperature does not result from the destruction of superconductivity, but rather arises due to the thermal excitation of quasiparticles on the island, as investigated previously in metallic islands [35,36] and semiconductor-superconductor VLS nanowires [37].
A thermodynamic analysis of Coulomb blockade peak spacings is based on the difference in free energies, F = F O − F E , between even and odd occupied states. We consider a simple model that assumes a single induced gap ∆ * , not accounting for the double-peaked density of states in Fig. 1(e). At low temperatures (T E C , ∆ * ), F approaches ∆ * . Above a characteristic poisoning temperature, T p , quasiparticles become thermally activated and F decreases rapidly to zero. For F (T ) > E C , Coulomb peaks are 2e periodic with even peak spacings, S E ∝ E C , independent of T . For F (T ) < E C , odd states become occupied, and the difference in peak spacing, S E − S O , decreases roughly proportional to F . A full analysis following Ref. [37] (see Supplemental Material [30]) yields the peak spacing difference where η is the dimensionless gate lever arm measured from Coulomb diamonds.  in Device 2, along with Eq. (1). Thermodynamic analysis shows an excellent agreement with the peak spacing data across the full range of temperatures. The fit uses an independently measured E C , with the induced gap as a single fit parameter, yielding ∆ * = 190 µeV, a reasonable value that lies between the two density of states features in Fig. 1(e). The island remains unpoisoned below T p ∼ 250 mK.
The evolution of Coulomb blockade peaks with parallel magnetic field, B , is shown in Fig. 3(a). In this data set, peaks show even-odd periodicity at zero field due to a gate-dependent gap or a bound state at energy E 0 less than E C . A subgap state results in even-state spacing proportional to E C + E 0 and odd-state spacing E C − E 0 [27] (see Supplemental Material [30]), giving (2) Figure 3(b) shows the B dependence of even and odd peak spacings, S E,O /(S E + S O ), extracted from the data in Fig. 3(a), giving an effective g-factor of ∼ 13. Even and odd peak spacings become equal at B = 150 mT, then overshoot at higher fields with a maximum amplitude corresponding to (7 ± 1) µeV. At more positive gate voltages [ Fig. 3(c)], where the carrier density is higher, peaks are 2e-periodic at zero field, then transition through even-odd to 1e-periodic Coulomb blockade without an overshoot, with an effective g-factor of ∼ 31.
Overshoot of peak spacing, with S O exceeding S E , indicates a discrete subgap state crossing zero energy [27,38], consistent with interacting Majorana modes. The overshoot observed at more negative V G is quantitatively in agreement with the overshoot seen in VLS wires of comparable length [27]. The absence of the overshoot and the increase of the g-factor at positive V G is consistent with the gate-tunable carrier density in VLS wires [39].
To demonstrate fabrication and operation of a simple SAG network, we investigate the coherence of electron transport in the loop structure shown in Fig. 4(a), with the Al layer completely removed by wet etching. Conductance as a function of perpendicular magnetic field, B ⊥ , shows a peak around zero magnetic field, characteristic of WAL [ Fig. 4(b)]. A fit to a theoretical model for disordered quasi one-dimensional wires with strong spin-orbit coupling [40] yields an electron phase-coherence length l WAL φ ∼ 1.2 µm, and a spin-orbit length l SO ∼ 0.4 µm. We note that electrons propagating along [100] and [010] di-rections experience both Rashba and linear-Dresselhaus spin-orbit fields [41]. The magnitude of each field can be deduced from a combination of in-plane magnetic field angle and magnitude dependence of the conductance correction due to the weak (anti-) localization. Such study, however, is out of scope of this work.
Upon suppressing WAL with a large perpendicular field periodic conductance oscillations are observed [ Fig. 4(c)] with period ∆B ⊥ = 2.5 mT corresponding to h/e AB oscillations with area 1.7 µm 2 , matching the lithographic area of the loop. The oscillation amplitude, A h/e , measured from the power spectral band around h/e [ Fig. 4(d), inset] was observed to decrease with increasing temperature as seen in Fig. 4(d). The size of A h/e is dictated by two characteristic lengthsthermal length L T and phase-coherence length l AB φ [40]. We estimate L T at 20 mK to be around 1.5 µm using a charge carrier mobility µ = 700 cm 2 /(V s) and density n = 9 × 10 17 cm −3 (see Supplemental Material [30]). The thermal length is comparable to the loop circumference L = 5.2 µm, as a result, energy averaging is expected to have finite contribution to the size of the conductance oscillations [40]. [43], a fit of the logarithmic amplitude log(A h/e ) = log(A 0 ) − 1 2 log T − aT 1/2 yields log(A 0 ) ∼ −1.5 and a ∼ 6.7 K −1/2 (Fig. 4d), giving a base-temperature phase-coherence length l AB φ (20 mK) ∼ 5.5 µm. The discrepancy between the extracted l WAL φ and l AB φ has previously been observed in an experiment on GaAs/AlGaAs-based arrays of micron-sized loops [42]. It has been argued theoretically that WAL and AB interference processes are governed by different dephasing mechanisms [43]. As a result, l WAL φ and l AB φ have different temperature dependences.
Our results show that selective area grown hybrid nanowires are a promising platform for scalable Majorana networks exhibiting strong proximity effect. The hard induced superconducting gap and 2e-periodic Coulomb oscillations imply strongly suppressed quasiparticle poisoning. The overshoot of Coulomb peak spacing in a parallel magnetic field indicates the presence of a discrete low-energy state. Despite the relatively low charge carrier mobility, the measured SAG-based network exhibits strong spin-orbit coupling and phase-coherent transport. Furthermore, the ability to design hybrid wire planar structures containing many branches and loops-a requirement for realizing topological quantum information processing-is readily achievable in SAG. Future work on SAG-based hybrid networks will focus on spectroscopy, correlations, interferometry, and manipulation of MZMs. We

SAMPLE PREPARATION
The InAs nanowires with triangular cross-sections were selectively grown by MBE along the [100] and [010] directions on a semi-insulating (001) InP substrate with a pre-patterned (15 nm) SiO x mask [1]. A thin (7 nm) layer of Al was grown in-situ at low temperatures on one facet by angled deposition, forming an epitaxial interface with InAs. For the fabrication of the devices, Al was selectively removed using electron-beam lithography and wet etch (Transene Al Etchant D, 50 • C, 10 s). Normal Ti/Al (5/120 nm) ohmic contacts were deposited after insitu Ar milling (RF ion source, 15 W, 18 mTorr, 5 min). A film of HfO 2 (7 nm) was applied via atomic layer deposition at 90 • C before depositing Ti/Au (5/100 nm) gate electrodes.

MEASUREMENT SETUP
The measurements were carried out with a lock-in amplifier at f ac = 173 Hz in a dilution refrigerator with a base temperature of T base = 20 mK. For voltage bias measurements, an ac signal with an amplitude of 0.1 V was applied to a sample through a homebuilt resistive voltage-divider (1 : 17.700), resulting in V AC ∼ 6 µV excitation. For current bias, we applied 2 V ac signal to a 1 GΩ resistor in series with a sample giving I AC = 2 nA excitation.

SPECTRUM EVOLUTION WITH VG
Evolution of differential conductance, G, as a function of source-drain bias, V SD , with gate voltage, V G , for Device 1 [ Fig. 1(c,d), main text] is illustrated in Fig. S 1(a). Conductance enhancement in the range of V SD = −0.1 mV to 0.1 mV is measured at V G = −7.10 V [ Fig. S 1(b)]. At more negative gate voltage V G = −7.35 V the conductance gets suppressed in the same range of V SD .
We interpret these features to arise due to a different tunnel barrier strengths tunned by the capacitively crosscoupled V G : At more positive V G , the tunneling barrier is more transparent, resulting in Andreev reflection enhanced conductance; At more negative V G , the transport is dominated by the single electron tunneling, reflecting the local density of states at the end of the wire [2]. At V G = −7.10 V the zero-bias conductance is enhanced beyond the factor of 2 compared to high bias [ Fig. S 1(b)]. We speculate that this is caused by an interfering transport mediated via multiple channels or a quantum dot [3]. A similar study of V C dependence was not possible presumably due to a too high concentration of disorder in the junction.

FREE ENERGY MODEL
The theoretical fit in Fig. 2(b) of the main text is based on a free energy model given by Eq. 1 in the main text, where the difference in free energy between odd and even occupied states is given by with the effective number of continuum states N eff = 2V Al ρ Al √ 2∆ * k B T , where V Al is the volume of the island and ρ Al is the density of states at the Fermi energy [4]. The fit was obtained by using V Al = 2.2 × 10 −6 nm 3 , consistent with Fig. 2(a) in the main text, electron density of states ρ Al = 23 eV −1 nm −3 [4] and E C = 60 meV, measured from Coulomb diamonds [ Fig. S 2], with ∆ * as the single fit parameter.

DEVICE ENERGY
Energy, E, of a Coulomb blockaded device with electron occupancy, n, as a function of normalized gate voltage, N , can be defined as where E C is the charging energy, F is the relative free energy and N 0 = 0 (1) for even (odd) parity of the device, see Fig. 2(c) in the main text, inset. Charge degeneracy points can be extracted using Eq. (S2), from which we can deduce the normalized even and odd peak spacings in units of charge, e, as The even and odd peak spacing difference in gate voltage is given by with the dimensionless lever arm η = E C /eS. Note that in the limit of zero temperature, F is defined by the size of the induced gap, ∆ * , or, if present, by the energy of a subgap state, E 0 .

FIELD EFFECT MOBILITY
Conductance of a nanowire channel as a function of gate voltage is given by where µ is the mobility, C is the capacitance between the gate electrode and the wire, l is the length of the channel, V G is the gate voltage and V T is the threshold voltage [5]. By introducing transconductance, dG/dV G , the mobility can be expressed by Conductance, G DC = I/V , of Device 4 [ Fig. S 3 inset] measured at ∼ 4 K and V DC = 5 mV as a function of topgate voltage, V G , is shown in Fig. S 3. The transconductance peaks to ∼ 100 µS/V around V G = −1.5 V, corresponding to the highest change in conductance indicated by the black line. The nanowire length l = 1 µm is set by the distance between the contacts. The capacitance C = 1.5 fF was estimated using COMSOL modeling software. Using Eq. S6 results in µ = 700 cm 2 /(V s).
Measurements on a similar chemical beam epitaxy grown SAG Hall bar with comparable mobility result in charge carrier density of n ∼ 9 × 10 17 cm −3 [6]. The corresponding mean free path is l e = µ e 3π 2 n 1/3 ∼ 15 nm, where is the reduced Planck constant and e is the elementary charge.

THERMAL LENGTH
The size of the Aharonov-Bohm oscillations is dictated by two characteristic length scales, namely the phasecoherence and thermal lengths [7]. The thermal length is related to the energy averaging of conduction channels due to the finite electron temperature and is given by where D is the diffusion constant and k B is the Boltzmann constant. The diffusion constant is given by where v F = k F /m * is the Fermi velocity, with the Fermi wave vector k F = (3π 2 n) 1/3 and the effective electron mass in InAs m * = 0.026m e , yielding D = 0.006 m 2 /s, consistent with the values measured in vapor-liquidsolid (VLS) nanowires [8]. The resulting thermal length L T (20 mK) = 1.5 µm is comparable to the loop circumference L = 5.2 µm in Device 3 [ Fig. 4(a), main text].