Transition to the quantum hall regime in InAs nanowire cross-junctions

We present a low-temperature electrical transport study on four-terminal ballistic InAs nanowire cross-junctions in magnetic fields aligned perpendicular to the cross-plane. Two-terminal longitudinal conductance measurements between opposing contact terminals reveal typical 1D conductance quantization at zero magnetic field. As the magnetic field is applied, the 1D bands evolve into hybrid magneto-electric sub-levels that eventually transform into Landau levels for the widest nanowire devices investigated (width = 100 nm). Hall measurements in a four-terminal configuration on these devices show plateaus in the transverse Hall resistance at high magnetic fields that scale with (ve2/h)−1. e is the elementary charge, h denotes Planck’s constant and v is an integer that coincides with the Landau level index determined from the longitudinal conductance measurements. While the 1D conductance quantization in zero magnetic field is fragile against disorder at the NW surface, the plateaus in the Hall resistance at high fields remain robust as expected for a topologically protected Quantum Hall phase.

Recently, such magnetic field-dependent transport experiments have revealed the crossover from a one-dimensional transport in zero magnetic field towards the selective population of spatially separated edge channels at high magnetic fields [21]. The hallmark of ballistic 1D quantum transport is a step-wise increase of the electrical conductance in units of the conductance quantum G 0 =2e 2 /h as a function of gate voltage when the temperature is so low that thermal excitations across the one-dimensional sub-band gaps at the Fermi level are not possible [18,34,35]. Subjecting a 1D quantum wire to an out-of-plane magnetic field B leads to the emergence of Landau quantization and Zeemann-splitting, which causes the degeneration of the Landau levels. When the magnetic length l m =(h/2πeB) becomes larger than the width of the quantum wire, spatially separated counter-propagating edge states are formed, where scattering in a single channel is forbidden and scattering between edge states is exponentially suppressed [21]. Here, e denotes the elementary charge and h is Planck's constant. In principle, these counterpropagating edge states can be observed as quantized Plateaus in the Hall resistance of R xy =(ve 2 /h) −1 that only depend on natural constants and the integer filling factor v=1, 2, 3, K of the system. However, the short NW channel lengths used in the experiment described in [21] did not allow for the attachment of lateral Hall probes. In this regard, III-V semiconductor NW cross-junction devices are particularly interesting because their four-terminal geometry naturally facilitates lateral Hall measurement probes [2,4,[36][37][38] and progress in material growth has enabled the observation of ballistic 1D quantum transport in crossed NW devices [35,39].

Nanowire device fabrication
For our experiment, we have grown InAs NW cross-junction by template-assisted selective epitaxy (TASE) on SOI within pre-patterned SiO 2 nanotube templates that determine the length, cross-section, position and axial direction of the NW devices. Metal-organic chemical vapor deposition is used to epitaxially grow InAs on Si within the templates. The template design is chosen such that each individual branch is pointing along an 〈110〉 axis of the InAs crystal.
Trimethylindium and tertiarybutylarsine are used as precursors in H 2 carrier gas at a total pressure of 60 Torr for the InAs growth at 550°C. Further details on the growth can be found elsewhere [35,[39][40][41]. Using electron-beam lithography and lift-off technique, electrical contacts were defined at each NW terminal of the cross-junction. The protective SiO 2 template has been selectively removed in the contact area, using a combination of reactive ion etching and buffered oxide wet etch. Prior to thermal evaporation of the Ni/Au (50 nm/30 nm) contacts, the contact area was immersed in an ammonium sulfide solution for passivation. Importantly, the SiO 2 -growth-template protects the InAs NW structures from oxidation and other environmental influences. However, to investigate the effect of scattering at the NW surface, on some of the devices the protective SiO 2 template was removed on the entire device.
To ensure a homogeneous electric field distribution in our devices when tuning the Fermi level, we prepared solid electrolyte gates [13]. Therefore, following the recipe of Liang et al [42], a mixture of Polyethyleneoxide(PEO)/LiClO 4 (8:1 in weight ratio) dissolved in methanol was spun on the chips at 1500 rpm, followed by a baking process at 90°C. A sketch of the device is shown in figure 1(a).

Electrical transport experiments
The transport experiments were performed in a temperature variable cryostat (Physical Property Measurement System from Quantum Design) with a base temperature of 2 K in helium atmosphere and equipped with a 9 T magnet. Two electrical contact configurations have been used in our experiments: first, two-terminal transport measurements across two opposed terminals of the NW cross junctions. We refer to measurements in this kind of contact configuration as 'longitudinal' and mark the corresponding transport parameters with the sub-index xx. Bias spectroscopy at zero magnetic field was performed, measuring the current I xx through the devices in response to an applied DC bias voltage V xx . Moreover, the longitudinal magnetoresistance (R xx (B)= V xx (B)/I xx ) has been measured in this contact configuration. Second, four-terminal transport measurements are performed. In this configuration, a current bias I xx is applied between two opposed terminals of the NW cross and the voltage response V yy on the two opposed terminals transverse to the currentinjection terminals is measured. We refer to measurements in this kind of contact configuration as 'transverse' and mark the corresponding transport parameters with the sub-index xy. In this contact configuration the Hall-resistance R xy (B)= V xy (B)/I xx is measured. For the current-bias measurements, a current of 100 nA with standard low frequency ( f=6 Hz) lock-in technique is applied. The magnetic field is always applied perpendicular to the x and y-direction (B=B z ). Because the liquid ion gate freezes at low temperatures (T<250 K), each measurement is performed at a fixed gate voltage in an individual cool down, taking approximately 90 min each. To avoid hysteresis effects during the temperature cycles, the gate voltage was increased stepwise from −2 to 1 V between the cycles. To solely obtain the longitudinal resistance of the InAs cross-junctions, contact resistances of (6.7±0.4) kΩ were subtracted on all low temperature data shown [35,39]. NW cross-junctions (figure 1(b)) with widths (w=30, 50 and 100 nm) and a height (h=25 nm) in the range of the Fermi wavelength λ of InAs (typically λ Inas ≈ 35 nm) [27] are investigated. Hall measurements on our devices at 300 K (supplemental material (SM) is available online at stacks.iop.org/SST/34/035028/mmedia) reveal an electron concentration of about´-5 10 cm 17 3 and a mobility between 3700 and 6000 cm 2 V −1 s −1 , decreasing with shrinking width, which we attribute to increased surface scattering. We note that all obtained results are independent of contact terminal iteration (see SM for details).

Scanning transmission electron microscopy analysis
For structural and compositional analysis of the NW devices, Scanning Transmission Electron Microscopy (STEM) is performed. Cross-sectional lamellas are prepared by means of an FEI Helios Nanolab 450 S focused ion beam instrument operated at accelerating voltages of 30 and 5 kV. STEM imaging and STEM-energy dispersive x-ray spectroscopy (EDX) analysis are performed using a double spherical aberration-corrected microscope (JEOL JEM-ARM200F), operated at 200 kV, and equipped with a silicon drift detector (JEOL Dry SD100GV) with 100 mm 2 detection area for EDX analysis.

Results and discussion
One-dimensional quantized conductance at zero magnetic field In a first transport experiment, we verify that the InAs nanowire cross-junctions are ballistic 1D quantum wire systems at zero magnetic field. Therefore, low temperature (T=2 K) bias spectroscopy is performed with no magnetic field applied (B=0 T). Consistent with our previous studies, [35,39] the NW cross-junctions exhibit plateaus at integer multiples n of 2e 2 /h shown in the zero-bias (V bias =0 mV) line cuts of the differential longitudinal conductance G xx = dI/dV bias (figures 1(c)-(e))-the hallmark of ballistic 1D quantum transport [17,18,22]. As expected from the simple model of an infinite 1D quantum well, the energy level spacing DE , i j , given by the tips of the diamond-like structures in the 2D G xx -(V gate , V bias ) colour plots of figures 1(f)-(h), resembles the 1D sub-band splitting of the previously investigated single InAs NWs [35] and InAs NW cross-junctions [39]. Also, consistently, the smaller the NW channel width, the larger the size quantization and thus the gapping of the 1D sub-bands (w=30 nm: ΔE 3,4 =35 meV; w=50 nm: ΔE 5,6 =12 meV; w=50 nm: ΔE 6,7 =8 meV). We note that despite the similar carrier concentrations for zero gate voltage at 300 K, the sub-band population at 2 K depends on the NW channel width. This is, because the thermal activation energy becomes smaller than the size of the sub-band spacing and thus the point in V G where transport through the 1D states commences is linked to the strength of the size quantization at low temperatures. In comparison to previously investigated back-gated devices, no full pinch-off is reached in any device, because the 20 nm-thick template oxide weakens the effect of the liquid ion gate.
One-dimensional sub-band depopulation in low magnetic field Next, we investigate the evolution of the 1D features within a magnetic field. When applying a magnetic field B perpendicular to the plane of the cross-junction, we observe a stepwise increase of R xx (B) in multiples of G 0 -1 at low B (figures 2(a)-(c)). This is a signature of Landau quantization and corresponds to the magnetic depopulation of the 1D subbands with increasing field [43,44]. Compared to previous studies on InSb [5,17] and InAs [22] NWs at mK temperatures, where indications for Zeeman-split sub-bands of lifted spin-degeneracy are observed, which is reflected by halfinteger (G 0 /2) −1 -features between the resistance plateaus, such features are not obvious from our raw data. This is probably due to the finite current bias as well as the elevated base temperature used in our experiment. However, as shown later, signatures of Zeeman splitting can be obtained by further data processing. At magnetic fields above 5 T, the plateau-like 1D levels in the magnetoresistance of the widest cross-junction (w=100 nm) evolve into peaks. Such transition in the Landau level spectrum is a first indication for a transformation of the electronic structure in these NW structures [45].

Magnetic field-dependence of the sub-bands
To investigate this transition in more detail, we calculate the derivative of the inverse longitudinal resistance with respect to the gate voltage g=d(1/R xx )/dV G , which relates to the magnetoelectric band structure of the NW devices. The results for the 30 nm-wide device is shown in figure 2(d)), for the 50 nm-wide device in figure 2(e) and for the 100 nm-wide device in figure 2(f)) [5,17,22]. Because in 1D, the transition between the longitudinal resistance plateaus is linked with the onset of transport in an additional sub-band, the maxima of g can be associated with the sub-band energies. For all investigated NW samples, the sub-bands are indexed with n, starting from the zero-field plateau values. n scales non-linear with inverse magnetic field, consistent with a continuous depopulation of the 1D states. Zeeman splitting of the subbands is reflected by branching of the individual maximumtraces at high magnetic field as indicated by arrows (spin up and spin down) [20][21][22]. In addition, plotting the filling factor ν=2n into fan diagrams (figures 2(g)-(i)), provides fingerprints of a transition from size quantization at zero B-field to Landau quantization in high fields as indicated by the grey dotted lines: ij , ) at around B=7 T, 4 T and 1.2 T for the 30 nm-wide, the 50 nm-wide and the 100 nm-wide NW channels, respectively. These results agree reasonable well with a size-dependent, smooth crossover from 1D conductance quantization to hybrid magnetoelectric sub-bands in our InAs NW cross-junction data, eventually transforming into edge states for the 100 nm-wide devices in high B [47].

Quantum hall regime at high magnetic field
Having established the transformation from one-dimensional bulk states in low B to edge-channels in high B using longitudinal transport parameters, we now turn to investigate the Hall resistance R xy of the NW cross-junctions. For the 30 nm-wide and the 50 nm-wide NW channels, the Hall response under applied magnetic field is strongly suppressed and no significant voltage signal could be measured. This may be explained by the fact that for the magnetic field range applied, the electron-trajectories cannot enter a side probe directly, since their cyclotron radii are larger than the widths of the side arms. In the case of the 100 nm-wide device, however, we measure a Hall response above 0.5 T and observe well-defined integer quantum Hall plateaus R xy =(ve 2 /h) −1 for B>5 T ( figure 3(a)), which position in B matches the position of the above the minima of the longitudinal resistance data. In the extreme quantum limit, the R xx is expected to vanish in the quantum Hall state. R xx in the InAs NW system, however, remains finite, which could be explained by the finite temperature and limited magnetic field range investigated or parallel conduction paths of surface and bulk. For clarity reasons, a smooth 2nd order polynomial background is subtracted from R xx to obtain ΔR xx ( figure 3(b)) as is usually done to distinguish the Shubnikov-de Haas oscillations on the surface of topological insulators from the bulk. The 1D sub-bands, which carry the current at zero field seem to be smoothly transformed into edge channels when exposed to an out-of-plane magnetic field. Our observation is concurrent with the recent interpretation of Landau level formation in InAs NW QPCs [21]. Further, as the B field is decreased, R xy exhibits the tendency of a last plateau near B=0 T (see SM for details). This is a feature of 1D confinement, [46,48] which originates from the fact that at small magnetic fields the Landau level splitting does not become arbitrary small, but evolves into the discrete 1D sub-band splitting.

Robustness of the quantum hall state
Finally, we investigate the robustness of the QH state in the cross-junction. The edge channels are predicted to be topologically protected and therefore expected to be immune to disorder. In a 100 nm-wide NW sample, both ballistic 1D transport at zero field and the QH state at high magnetic fields  are observable. By introducing random surface charges, we disturb the ballistic condition along the nanowire axis at zero magnetic field; thus, the quantized resistance plateaus of (n2e 2 /h) −1 should be suppressed, in contrast to the edge channels of the QH state which are not localized, due to the topological protection, as long as the disorder does not create inelastic scattering events. To carry out this test, we compare the devices protected by the SiO 2 template with devices, on which scattering is purposely induced at the surface of the NW cross-junction by removing the protective SiO 2 template in diluted hydrofluoric acid and subsequently storing the samples for 6 d in air. It is well-known that the electronic transport properties of III-V NW surfaces are modified when subjected to ambient conditions, [49] with the dominant mechanism being carrier scattering at the surface. This is due to the creation of dangling bonds, trap states, surface dipoles and surface oxides that not only induce a morphological disorder, but also impact the electronic band structure of the surface [49].
To address the effect of template-removal in more detail, the surface of both types of devices was investigated by energy dispersive x-ray (EDX) spectroscopy analysis in STEM mode. The line-cut data, taken as indicated in figures 4(a) and (b), reveals that compared to the protected NW ( figure 4(c)), the template-removing procedure leads to the formation of a 6 nm-thick non-stochiometric phase at the NW surface ( figure 4(d)) that modifies the electronic properties. In magneto-transport experiments (figures (e), (f)), we confirm that the quantized conductance steps at low magnetic fields are clearly suppressed in cross-junction with the disordered surface. Around zero magnetic field, a resistance peak is observed, which is a signature of weak localization [42]-a quantum mechanical interference effect that is caused by closed-loop elastic scattering paths. As the magnetic field is increased, the magnetoresistance exhibits a maximum at around B c =1.3 T, marking the chambers field, which is explained by surface wall scattering: [50] this arises when bulk scattering is small and low magnetic fields force the electrons against the sidewalls of the NW channel. Disorder at the surface will then cause scattering and the magnetoresistance increases as the magnetic field is enhanced. However, when B is increased further, such that magnetic confinement in the NW channel becomes smaller than the spatial confinement, surface scattering is reduced and the magnetoresistance decreases again. Consistent with this interpretation, the determined B c =5 T agrees well with the centre of the transition from the 1D conductance quantization to the QH regime in the template-protected samples. At higher magnetic fields, the minima on the smoothly varying background resemble the minima, indexed by ν in the longitudinal magnetoresistance of the protected InAs NW cross-junction (figure 4(e)). This indicates that at high B the sub-band structure of the as-grown NW is recovered, despite the disorder at the surface. Consistently we observed that, while ballistic 1D transport in zero magnetic is fragile against disorder at the NW surface, the QH plateaus at high magnetic fields remain robust. Consequently, R xy , measured on the disordered cross-junction shows distinct plateaus, centred on the minima of the R xx -B data. The height of the plateaus agrees well with the Hall data obtained for the protected NW structure R xy =(νe 2 /h) −1 for B>B c =5 T and is therefore consistent with the Landau-level-interpretation of the maximum in R xx above. This concludes that the observed QH state is robust against disorder, which is strong enough to supress the transmission of 1D modes at zero magnetic field. A similar test has recently been carried out on Bi 2 Se 3 NWs, [51] demonstrating topological protection of helical surface modes. We therefore believe that the robustness of the QH state observed in here, is a strong signature for topologically protected edge channel transport in InAs NW cross-junctions at high magnetic fields. However, we note that at high magnetic fields, the smooth background of the R xx -B has For B>5 T, plateaus scaling with (ve 2 /h) −1 occur. (b) Residual resistance ΔR xx for the same device at 2 K for various gate voltages V G as a function of magnetic field B. To obtain ΔR xx , a smooth polynomial background is subtracted from R xx . The minima denote the Landau level v, indexed by grey numbers. changed, the reason for which is not clear and requires further investigation.

Conclusion
In conclusion, we have implemented ballistic 1D NWs with well-controlled cross-section into a crossbar geometry and provided the first demonstration of a protected QH state in these systems. Magneto-transport experiments reveal that the ballistic 1D quantum system at zero field smoothly evolves into the QH regime in perpendicular magnetic fields. Careful material design, enabled by the TASE process, was critical to study this transition. As expected from theory, [45] the transition from 1D to the QH regime occurs around the same B-field (B c =5 T), where the magnetic confinement overcomes the spatial size confinement. The template-protected devices enabled an exploration of ballistic 1D transport and test its stability upon inducing disorder at the NW surface. While the ballistic transmission of 1D modes was supressed, the edge channel transport in the QH state remained robust, which we believe provides evidence for topological protection. Therefore, such NW quantum Hall devices can be useful for applications in fault-tolerant quantum computing and nondissipative electronic devices. However, there are still two main obstacles to overcome: the QH state only exists at low temperatures and high magnetic fields. Following the example of the quantum spin Hall effect in 2D quantum wells, [52,53] topologically protected edge channel transport without magnetic field might also be obtainable in future 1D NW devices. While the stoichiometry of the protected NW channel is InAs 1:1 at the NW/template interface, the NW with removed template exhibits an indium-rich surface layer of about 6 nm. (e) Magneto-transport data at zero gate voltage and 2 K for a 100 nmwide NW cross-junction before and (f) after removing the template. The ballistic 1D transport is fragile against the disorder at the channel's surface. A weak-localization peak evolves around B=0 and Chambers peaks of boundary scattering, marked by B c are observed when the template is removed. At high magnetic fields, the sub-band structure and the Hall plateaus of the disordered device resemble the features observed for the protected NW.

Associated content
The supporting material includes Hall measurements of all devices at 300 K, the magneto-transport data of the 100 nmwide device at 2 K and zero gate voltage with iterated terminals as well as a zoom-in to the suppressed Hall resistance around zero magnetic field and Hall traces of all devices investigated at 2 K.

Author contributions
JG and MB conceived the original idea for the study. MB, HS and SW synthesized the InAs nanowires. MB and JG fabricated the NW devices. JG carried out the transport measurements. NB and MR. carried out the STEM-EDX analysis. JG MB. and HS analyzed the data. HR, KM and KN supervised the project. All authors contributed to the interpretation of the data and to the writing of the manuscript.