Correction to: Nature Nanotechnologyhttps://doi.org/10.1038/s41565-017-0032-8, published online 15 January 2018. The Letter reports Majorana signatures in hybrid InSb semiconductor nanowire–NbTiN superconductor devices. The devices exhibit a conductance plateau near the conductance quantum 2e²/h at bias voltages above the superconducting gap (normal conductance), accompanied by an enhanced Andreev conductance at bias voltages below the superconducting gap (subgap conductance). We have attributed these experimental observations to ballistic transport as supported by a theoretical analysis¹, finding mean free paths on the order of or larger than the effective wire segment (the segment covered by the superconducting electrode). Here, we correct errors discovered on reanalysis of the original data², following concerns raised by readers. Due to the age of the paper, it cannot be corrected directly in the original publication, thus the updates are provided via this amendment. We provide additional discussion on the claim of ballistic transport so as to avoid misinterpretations. External peer review of the reanalysis concluded that the claims in the Letter remain. An extended public repository including data obtained from nanowire devices that were not included in the publication can be found in ref. ². We note the lack of a series of flat and precisely quantized conductance plateaus (a staircase), a clear ballistic transport characteristic (see the two newly included Supplementary Figs. 1 and 7 showing larger voltage ranges of Fig. 1 and the original Supplementary Fig. 5). Our earlier studies on ballistic transport in nanowire devices^3,4 indicate that vapour–liquid–solid nanowires do not have the proper geometry for observing a conductance staircase without the application of a magnetic field perpendicular to the wire axis, which requires ideal (Landauer) reservoirs interfacing the ballistic region, absorbing charge carriers with near-unit probability. Similar to our earlier studies, ohmic contacts in the present nanowire devices do not satisfy the conditions of Landauer reservoirs. However, the transport in the effective wire segment can nevertheless be ballistic whose characteristic is a plateau feature near 2e²/h in normal conductance together with an enhanced Andreev conductance. Importantly, precise quantization is not realistic, prevented by the two-terminal device geometry, inevitably decreasing the conductance. In summary, a plateau feature with an enhanced Andreev conductance together with our theoretical analyses indicate that a large fraction of transport is ballistic over distances of the order of our device length. We add a discussion to the main text of the Letter as follows: “… followed by a dip in conductance due to channel mixing²⁰ [ref. ¹ below]. We do not observe higher plateaus (Supplementary Figs. 1 and 7), which we attribute to the contacts not satisfying the conditions of Landauer reservoirs, resulting in residual scattering more effective at larger conductance. This is in line with our earlier studies^25,39 [refs. ^4,5 below] which indicated that vapour–liquid–solid nanowires do not have the proper geometry for observing a conductance staircase without the application of a perpendicular magnetic field. From the absence of quantum dots, the observed induced gap …”. The following text should also have been included in the abstract: “… exhibiting clear ballistic transport properties manifested by a conductance plateau with an Andreev enhancement, albeit lacking a quantized conductance staircase hindered by the device geometry.” The conductance values reported in the publication are ~8% lower (near 2e²/h) than the actual value (corrected Fig. 1). This deviation is due to a drop in the gain of the current-to-voltage amplifier at an ac excitation frequency of 67 Hz⁵. As a result, there is a slight change in the Andreev conductance enhancement factor and the superconducting contact transparency extracted from the enhancement (a comparison between the values quoted in the publication and the corrected ones is given below in B). The general conclusions do not rely on the exact value of the conductance as precise quantization is not expected due to the two-terminal device geometry. The subtracted series resistance of 3 kΩ in the original Fig. 1 was an overestimation (see corrected Fig. 1 in the Supplementary Data file). The subtraction of 3 kΩ was not mentioned in the original publication. A comparison of the original and corrected Fig. 1 is presented in a Supplementary Data file accompanying this correction. For all the figures in the original publication except Fig. 1, we either subtracted a contact resistance value of 0.5 kΩ, which is an underestimation¹, or no resistance at all. We note that in tunneling measurements the overall resistance is significantly higher than the normal metal contact resistance whose contribution can therefore be neglected. Figure 1, however, was used to estimate the superconducting contact transparency and Andreev enhancement in the high conductance regime, requiring a realistic exclusion of the contact resistance. Following our previous paper⁴, which found normal metal contact resistance values between 1.5–3.25 kΩ per contact and was based on fitting the measured conductance using theory (single mode interfacing a superconductor), which provided reasonable agreement after excluding 3 kΩ, we subtracted 3 kΩ to exclude the resistance of the normal metal contact. During our reanalysis, we have discovered that the minimum resistance of this device at the largest applied gate voltages is 2.9 kΩ, a value providing an upper bound on the contact resistance. Here, 2.9 kΩ would be the contact resistance under the assumption that the nanowire itself has zero resistance at largest gate voltages. The contact resistance can be estimated with an alternative method by subtracting a series resistance to match the observed conductance plateau at bias voltages above the superconducting gap to the expected quantized value, a procedure not done in the original publication. By taking the conductance averaged at positive and negative |V| ~ 1.7 mV (around the largest bias voltages available for this analysis) we find that the quantized value is reached for a contact resistance of 0.77 kΩ. (Considering only the positive bias and separately only the negative bias results in a range of 0–2.13 kΩ for the contact resistance.) In our corrected estimate of the contact resistance, we have applied the calibration procedure⁵ that corrects for ac circuit effects, uses calibrated values for the series resistance of the setup where Fig. 1 was measured and directly corrects the error listed in A above. Upon reanalysis we estimate the following contact resistance values, enhancement factors and transparencies: (Table presented.) Contact resistance Enhancement factor Transparency Lower bound 0 kΩ 1.26 0.88 Conservative estimation¹ (used in corrected Fig. 1) 0.5 kΩ 1.32 0.90 Current best estimate 0.77 kΩ 1.36 0.90 Original estimate in paper 3 kΩ >1.5 >0.93 The corrected superconducting contact transparency value of 0.9 does not affect the claim of high transparency. The claim of ballistic transport does not rest on the exact value of the conductance plateau and hence is also unaffected. The original Methods section omits the indication of subtracted series resistances which account for the normal metal contact resistance in each figure. The following is included here for the corrected Methods: The original Methods section omits the indication of subtracted series resistances which account for the normal metal contact resistance in each figure. The following is included here for the corrected Methods: “Contact resistance treatment. A fixed-value series resistance of 0.5 kΩ has been subtracted in Figs. 1 and 4, Supplementary Figs. 1, 2b,c and 4–9 to account for the contact resistance of the normal metal lead. This value is smaller than the lowest contact resistance we have obtained for InSb nanowire devices²⁵ (ref. ⁴ below), which makes the interface transparency estimated from Fig. 1 a lower bound. For the remaining figures, no series resistance has been subtracted to account for the normal metal contact resistance.” In the original Supplementary Fig. 5 (now Supplementary Fig. 6), a charge jump was corrected by removal of 12 line traces (corresponding to +0.15 V to +0.04 V in gate voltage in the measured data) and offset of the gate voltage axis by 0.12 V after the charge jump (–1 V to +0.03 V) to maintain continuity of the axis. This processing was not mentioned in the original publication. The corrected Supplementary Fig. 6 excludes this processing and represents the data as measured. In the original Supplementary Fig. 5 (now Supplementary Fig. 6), a charge jump was corrected by removal of 12 line traces (corresponding to +0.15 V to +0.04 V in gate voltage in the measured data) and offset of the gate voltage axis by 0.12 V after the charge jump (–1 V to +0.03 V) to maintain continuity of the axis. This processing was not mentioned in the original publication. The corrected Supplementary Fig. 6 excludes this processing and represents the data as measured. A comparison of the original and corrected Fig. SI5 (now Fig. SI6) is presented in a Supplementary Data file accompanying this correction. Original Supplementary Fig. 1f (now Supplementary Fig. 2f): The offset mentioned in the caption is erroneously given as 0.006 × 2e²/h but is 0.01 × 2e²/h. Original Supplementary Fig. 4a,b (now Supplementary Fig. 5a,b) were indicated to present data from Fig. 2a (or original Supplementary Fig. 1a). This is incorrect. The data used are from the original Supplementary Fig. 1b (now Supplementary Fig. 2b) which has the same measurement settings as in Fig. 2a except the barrier gate is –1.5 V (the barrier gate is –1.4 V in Fig. 2a or original Supplementary Fig. 1a). In the original panels c–e of Supplementary Fig. 7 (now Supplementary Fig. 9c–e) the bias polarity is mistakenly inverted.

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The electrical characterisation of classical and quantum devices is a critical step in the development cycle of heterogeneous material stacks for semiconductor spin qubits. In the case of silicon, properties such as disorder and energy separation of conduction band valleys are commonly investigated individually upon modifications in selected parameters of the material stack. However, this reductionist approach fails to consider the interdependence between different structural and electronic properties at the danger of optimising one metric at the expense of the others. Here, we achieve a significant improvement in both disorder and valley splitting by taking a co-design approach to the material stack. We demonstrate isotopically purified, strained quantum wells with high mobility of 3.14(8) × 10⁵ cm² V⁻¹ s⁻¹ and low percolation density of 6.9(1) × 10¹⁰ cm⁻². These low disorder quantum wells support quantum dots with low charge noise of 0.9(3) μeV Hz^−1/2 and large mean valley splitting energy of 0.24(7) meV, measured in qubit devices. By striking the delicate balance between disorder, charge noise, and valley splitting, these findings provide a benchmark for silicon as a host semiconductor for quantum dot qubits. We foresee the application of these heterostructures in larger, high-performance quantum processors.

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Spin-orbit interaction (SOI) plays a key role in creating Majorana zero modes in semiconductor nanowires proximity coupled to a superconductor. We track the evolution of the induced superconducting gap in InSb nanowires coupled to a NbTiN superconductor in a large range of magnetic field strengths and orientations. Based on realistic simulations of our devices, we reveal SOI with a strength of 0.15-0.35 eV Å. Our approach identifies the direction of the spin-orbit field, which is strongly affected by the superconductor geometry and electrostatic gates.

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Majorana modes are zero-energy excitations of a topological superconductor that exhibit non-Abelian statistics^1–3. Following proposals for their detection in a semiconductor nanowire coupled to an s-wave superconductor^4,5, several tunnelling experiments reported characteristic Majorana signatures^6–11. Reducing disorder has been a prime challenge for these experiments because disorder can mimic the zero-energy signatures of Majoranas^12–16, and renders the topological properties inaccessible^17–20. Here, we show characteristic Majorana signatures in InSb nanowire devices exhibiting clear ballistic transport properties. Application of a magnetic field and spatial control of carrier density using local gates generates a zero bias peak that is rigid over a large region in the parameter space of chemical potential, Zeeman energy and tunnel barrier potential. The reduction of disorder allows us to resolve separate regions in the parameter space with and without a zero bias peak, indicating topologically distinct phases. These observations are consistent with the Majorana theory in a ballistic system²¹, and exclude the known alternative explanations that invoke disorder^12–16 or a nonuniform chemical potential^22,23.

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Topological superconductivity is a state of matter that can host Majorana modes, the building blocks of a topological quantum computer. Many experimental platforms predicted to show such a topological state rely on proximity-induced superconductivity. However, accessing the topological properties requires an induced hard superconducting gap, which is challenging to achieve for most material systems. We have systematically studied how the interface between an InSb semiconductor nanowire and a NbTiN superconductor affects the induced superconducting properties. Step by step, we improve the homogeneity of the interface while ensuring a barrier-free electrical contact to the superconductor and obtain a hard gap in the InSb nanowire. The magnetic field stability of NbTiN allows the InSb nanowire to maintain a hard gap and a supercurrent in the presence of magnetic fields (∼0.5 T), a requirement for topological superconductivity in one-dimensional systems. Our study provides a guideline to induce superconductivity in various experimental platforms such as semiconductor nanowires, two-dimensional electron gases, and topological insulators and holds relevance for topological superconductivity and quantum computation.@en

The dissertation reports a series of electron transport experiments on semiconductor nanowires towards realizing the hypothesized topological quantum computation. A topological quantum computer manipulates information that is stored nonlocally in the topology of a physical system. Such an operation possesses advantages over the current quantum computation platforms due to its robustness against local sources of decoherence, offering a natural fault-tolerance. Among various candidate platforms to realize topological quantum computation, semiconductor nanowires with strong spin-orbit coupling attached to conventional superconductors have emerged as a prime contender. The predicted topological properties of such a system is associated with the emergence of Majorana modes. The presence of disorder has been considered to be the main obstacle towards the realization of a topological quantum computer based on semiconductor nanowires. Disorder can mimic the experimentally measurable properties of Majoranas, or can render the promise of fault-tolerance ineffective. The experiments in the dissertation aim for eliminating the disorder on the surface of the nanowire, and in the interface between the nanowire and the superconductor. Following a series of investigations demonstrating materials improvements, ballistic Majorana nanowire devices are realized.@en

Semiconductor nanowires have opened new research avenues in quantum transport owing to their confined geometry and electrostatic tunability. They have offered an exceptional testbed for superconductivity, leading to the realization of hybrid systems combining the macroscopic quantum properties of superconductors with the possibility to control charges down to a single electron. These advances brought semiconductor nanowires to the forefront of efforts to realize topological superconductivity and Majorana modes. A prime challenge to benefit from the topological properties of Majoranas is to reduce the disorder in hybrid nanowire devices. Here we show ballistic superconductivity in InSb semiconductor nanowires. Our structural and chemical analyses demonstrate a high-quality interface between the nanowire and a NbTiN superconductor that enables ballistic transport. This is manifested by a quantized conductance for normal carriers, a strongly enhanced conductance for Andreev-reflecting carriers, and an induced hard gap with a significantly reduced density of states. These results pave the way for disorder-free Majorana devices.

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Majorana zero modes (MZMs) are prime candidates for robust topological quantum bits, holding a great promise for quantum computing. Semiconducting nanowires with strong spin orbit coupling offer a promising platform to harness one-dimensional electron transport for Majorana physics. Demonstrating the topological nature of MZMs relies on braiding, accomplished by moving MZMs around each other in a certain sequence. Most of the proposed Majorana braiding circuits require nanowire networks with minimal disorder. Here, the electronic transport across a junction between two merged InSb nanowires is studied to investigate how disordered these nanowire networks are. Conductance quantization plateaus are observed in most of the contact pairs of the epitaxial InSb nanowire networks: the hallmark of ballistic transport behavior.

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Majorana zero modes (MZMs), prime candidates for topological quantum bits, are detected as zero bias conductance peaks (ZBPs) in tunneling spectroscopy measurements. Implementation of a narrow and high tunnel barrier in the next generation of Majorana devices can help to achieve the theoretically predicted quantized height of the ZBP. We propose a material-oriented approach to engineer a sharp and narrow tunnel barrier by synthesizing a thin axial segment of Ga_xIn_1-xSb within an InSb nanowire. By varying the precursor molar fraction and the growth time, we accurately control the composition and the length of the barriers. The height and the width of the Ga_xIn_1-xSb tunnel barrier are extracted from the Wentzel-Kramers-Brillouin (WKB) fits to the experimental I-V traces.

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The motion of an electron and its spin are generally not coupled. However in a one-dimensional material with strong spin-orbit interaction (SOI) a helical state may emerge at finite magnetic fields, where electrons of opposite spin will have opposite momentum. The existence of this helical state has applications for spin filtering and cooper pair splitter devices and is an essential ingredient for realizing topologically protected quantum computing using Majorana zero modes. Here, we report measurements of a quantum point contact in an indium antimonide nanowire. At magnetic fields exceeding 3 T, the 2 e2/h conductance plateau shows a re-entrant feature toward 1 e2/h which increases linearly in width with magnetic field. Rotating the magnetic field clearly attributes this experimental signature to SOI and by comparing our observations with a numerical model we extract a spin-orbit energy of approximately 6.5 meV, which is stronger than the spin-orbit energy obtained by other methods.@en

The charge transport properties of individual InSb nanowires based transistors are studied at 4.2 K in the quasiballistic regime. The energy level separations at zero magnetic field are extracted from a bias voltage spectroscopy. The magnetoconductance under a magnetic field applied perpendicularly to the nanowire axis is investigated up to 50 T. Owing to the magnetic reduction of the backscattering, the electronic states of the quasi-one-dimensional electron gas are revealed by Landauer-Büttiker conductance quantization. The results are compared to theoretical predictions revealing the spin and orbital degeneracy lifting. At sufficiently high magnetic field the measurements show the evolution to the quantum Hall effect regime with the formation of Landau orbits and conducting edge states.@en

Ballistic electron transport is a key requirement for existence of a topological phase transition in proximitized InSb nanowires. However, measurements of quantized conductance as direct evidence of ballistic transport have so far been obscured due to the increased chance of backscattering in one-dimensional nanowires. We show that by improving the nanowire-metal interface as well as the dielectric environment we can consistently achieve conductance quantization at zero magnetic field. Additionally we study the contribution of orbital effects to the sub-band dispersion for different orientation of the magnetic field, observing a near-degeneracy between the second and third sub-bands.

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