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Delocalized states in three-terminal superconductor-semiconductor nanowire devices

by P. Yu, B. D. Woods, J. Chen, G. Badawy, E. P. A. M. Bakkers, T. D. Stanescu, S. M. Frolov

Submission summary

As Contributors: Sergey Frolov
Arxiv Link: (pdf)
Data repository:
Date submitted: 2021-08-21 18:05
Submitted by: Frolov, Sergey
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Experiment
Approaches: Theoretical, Experimental, Computational


We fabricate three-terminal hybrid devices with a nanowire segment proximitized by a superconductor, and with two tunnel probe contacts on either side of that segment. We perform simultaneous tunneling measurements on both sides. We identify some states as delocalized above-gap states observed on both ends, and some states as localized near one of the tunnel barriers. Delocalized states can be traced from zero to finite magnetic fields beyond 0.5 T. In the parameter regime of delocalized states, we search for correlated subgap resonances required by the Majorana zero mode hypothesis. While both sides exhibit ubiquitous low-energy features at high fields, no correlation is inferred. Simulations using a one-dimensional effective model suggest that delocalized states may belong to lower one-dimensional subbands, while the localized states originate from higher subbands. To avoid localization in higher subbands, disorder may need to be further reduced to realize Majorana zero modes.

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Reports on this Submission

Anonymous Report 4 on 2021-9-27 (Invited Report)


The manuscript under review aims to be a systematic study of localized and delocalized states in short proximitized nanowires. To this end, the authors perform local and non-local conductance measurements to identify those states, and show a comparison to numerical simulations.

This paper is follow-up study to an earlier Nature Physics paper of the same experimental group, but focuses on a different parameter regime (different super-gate voltage) and different measurements (non-local conductance measurements), and hence is a sufficiently different study to be considered as a stand-alone research.

However, the current version of the manuscript feels rather uninspired and it is not clear to me what we can really learn from these measurements. In addition, it contains several claims that are in my opinion unfounded and not supported by the research. I will list specific points below.

In summary, I believe that the measurements in this paper definitely deserve publication somewhere, and after the controversial statements I list below have been amended, I think the paper could be published for example in SciPost Physics Core. Given that a clear breakthrough/new idea is missing in this paper (local and non-local measurements have been used before), I currently don't think that publication in SciPost Physics is feasible.

Specific questions/criticism:

1. The authors write "In most cases, to identify non-Majorana states
it is sufficient to analyze two-terminal measurements on
just one nanowire in a self-consistent fashion." What do they mean by that? That one can identify trivial ABS always by two-terminal mesaurements? What do they mean by "self-consistent" in this context?

2. On page 2, the authors speculate on the origin of the delocalized states at zero field. They argue that because the states are inside and outside the gap, it is unlikely that they arise from Andreev reflection, and thus need to be far away from the superconductor. The authors then later take this premise as granted. I am missing a more critical treatment, also discussing other possibilities. As the authors write themselves, this is a speculation from their side.

3. On page 3, the authors write "What is more surprising is that in the same segment we observe wavefunctions significantly localized either on the left or on the right end (L and R states)." First, do the authors know what states those are? Is there a strong coupling to the superconductor, or are these mainly localized in a dot and weakly coupled to the superconductor? Second, why is the presence of these states surprising? Many experiments have seen quantum dot states near tunnel barriers.

4. Further on the same page, the authors write "In particular, we explore the hypothesis that correlated subgap resonances can be found in the regime that shows delocalized wavefunctions at zero field.". I do not understand this statement, and I strongly believe it to be wrong. First, ideally one should look for Majorana zero modes (MZMs) in a regime where there is a good superconducting gap in the absence of a magnetic field, and hence there should be *no* delocalized states. Furthermore, the authors essentially speculated before that the delocalized states have little induced superconductivity - again detrimental for MZMs.

5. On page 4, the authors argue that for localized states strong disorder is necessary. Why is this the case? Isn't this due to the limitations of the model, i.e. the choice of potentials, etc.? I can imagine to get quantum dot like states even in the absence of disorder if potentials are chosen accordingly.

6. My biggest concern is the applicability of the numerical simulations to the experiment: The authors argued before that the delocalized states are states separated from the superconductor. However, in the 1D numerical model there is just one mode that is well-proximitized. That is, in my opinion, why the numerical simulations need strong disorder to get agreement with experiment. But this conclusion is weak, having started from orthogonal assumptions!

7. I have a general question: How general are the experimental results for hybrid nanowire systems? In particular, the experiments are performed with sputtered NbTiN which is presumably very disordered. What would you expect to change with e.g. epitaxial superconductors?

  • validity: low
  • significance: ok
  • originality: ok
  • clarity: low
  • formatting: reasonable
  • grammar: good

Anonymous Report 3 on 2021-9-25 (Invited Report)


In this experimental-theoretical work, the authors report three-terminal conductance measurements in a semiconductor-superconductor nanowire device. They observe quantum states that they appear as peaks in conductance and they aim at distinguishing between localized and delocalized states. They ascribe those states with the presence of disorder in the nanowire and develop a theoretical model to test the plausibility of their hypothesis.
This manuscript is a follow-up of their previous work [1]. They have measured the same device where they already distinguish localized and delocalized states in Figure 4 of [1]. In the present work, more data has been added in order to tackle the problem and another type of measurement has been performed, i.e. fig.1b.

Delocalized states might play an important role for the formation of MZMs in hybrid nanowires. Therefore, being able to detect them and to understand their origin is definitely important. However, I am not sure this follow-up work brings new insights to deserve publication in SciPost. Before this, I have some comments and questions for the authors.

In the following, I present my comments and questions.

Experimental set-up:
1. Why did the authors use the gates T3 to make the nanowire segments “highly electrostatically n-doped”? Would it not be easier/better to position the normal leads (NL and NR) closer to the S-lead? Is there a risk to create unwanted quantum dots (QDs) in the T3 segment?

2. Do the authors know if QDs are formed in the tunnel barriers? If yes, would it be possible to remove them by shrinking the size of the tunnel barrier as it has been shown in [2,3]? It is a known fact that QDs in the tunnel barrier affect the result of tunneling spectroscopy experiments. Therefore, getting rid of them would make the work stronger. Could the authors comment on this point?

General comments and questions:
1. Fig.1b represents the novelty of this work, however I am not fully sure that it is needed to reach the conclusion of this work. Why R1 and L1 are also present in this measurement even though they are localized on the right and left parts, respectively?

Could the authors reach the same conclusion without the result of Fig.1b?

Why is the S-lead floating and not grounded for this measurement? This is in contrast with what has been previously studied in the literature of three-terminal device, as an example see ref. [4]. Could the authors motivate and comment their choice? Do the authors expect differences if the s-lead would have been grounded instead of floating? In addition, a protocol explaining how to correct data properly in a three-terminal measurement has been recently published [5].

2. The abstract reads “We identify some states as delocalized above-gap states”, however this is not commented in the rest of the manuscript. One can understand that the delocalized states are above-gap states only looking at Fig.S3, is this statement correct? If so, could the authors explicitly mention it in the text?

3. MZMs are defined as delocalized. “Delocalized” means that the wave function is spread throughout all the semiconductor-superconductor system, whereas MZMs are more often described as “non-local” because the wave function is localized at the ends of the hybridized nanowire but there is no wave function in the middle. Could the authors explain more precisely what they define as delocalized?

Furthermore, the authors state “Since Majorana modes are themselves delocalized states and require a uniform density along the nanowire, we focus on a regime with delocalized states to search for MZMs.” Could the authors explain why MZMs should appear in the regime where there are delocalized states? Can the presence of delocalized states be detrimental for the formation of MZMs?

4. The localized and delocalized states seem to arise because of the disorder in the hybrid nanowire. Can other mechanisms give similar results? What is the difference between QD states and disorder states? Can the authors provide measurements in which superconductivity is suppressed, with a high perpendicular magnetic field for instance? I believe that such measurement would help to understand if there are QDs in the system.

5. In this work, quantum states appear in the tunneling spectroscopy experiment also in absence of magnetic fields. However, this is not the case for other work in the literature, see refs. [2,3,6,7]. Could the authors comment on the difference between their result and the result of refs. [2,3,6,7]? Please, see Fig.2b of ref. [2], Fig.1c of ref. [3], Fig.3b of ref. [6] and Fig.1b of ref. [7]. Is it a correct statement that in refs. [2,3,6,7] the level of disorder is much smaller?

6. It would be helpful to add the theoretical plot of Fig.S12 in Fig.3 of the main text.

7. Could the authors suggest how one could reduce the disorder in hybrid nanowires?

1. How do the authors estimate the amplitude of the disorder potential Emin?
2. Does the tunnel barrier length play a role in the results of simulations? If yes, could the authors explain how?

Minor comments:
1. Refs. 4, 30 and 31 has been published in peer-reviewed journals, please cite them properly.

2. In some cases, the measurements are plotted in normalized units. Could the authors give an explanation about this choice? Could they add the plots not in normalized units in the Supplementary Information?

3. I ask the authors to rephrase the following sentence “We fabricate three-terminal nanowire devices which nominally fulfill requirements for Majorana zero modes: they are built around InSb nanowires that have significant intrinsic spin-orbit coupling, with superconductivity induced by a NbTiN superconductor.” Coulomb blockade NbTiN island experiment never showed 2e transport at zero field and NbTiN exhibits a soft gap. Therefore, I am not sure that InSb/NbTiN fulfills the requirements for realizing MZMs. Furthermore, there are still many open questions, like what is the effective spin-orbit coupling of InSb coupled to NbTiN. For these reasons, I kindly ask the authors to rephrase the sentence.

4. Colorbar is missing for figure 5a,b,c,d. Y-axis label is missing for figure 5e.

5. It would be helpful to know the values of all gate voltages, for instance what is the value of TL in Fig.1b,c and d?

In conclusion, I believe that being able to distinguish localized and delocalized states via three-terminal conductance measurement will be important for future experiments. In this work, experiments and simulations give a strong suggestion that this is feasible and that there are delocalized states coming from disorder. However, I am not yet sure that this work in the present form deserves a publication in SciPost because I am not yet convinced that it adds valuable information hidden in their previous work [1]. In particular, the manuscript in the present form does not meet the General Acceptance Criteria 1,3 and 4 of SciPost.

[1] Nat. Phys. 17, 482-488 (2021).
[2] Nat. Nano. 10, 232-236 (2015).
[3] Science 373, 82-88 (2021).
[4] Phys. Rev. Lett. 124, 036802 (2020).
[5] arXiv:2104.02671v1 (2021).
[6] Nat. Comm. 12, 4914 (2021).
[7] Nat. Nano. 13, 192-197 (2018).

Requested changes

See report.

  • validity: ok
  • significance: good
  • originality: low
  • clarity: ok
  • formatting: good
  • grammar: good

Anonymous Report 2 on 2021-9-9 (Invited Report)


In this joint experimental-theoretical work, the Authors discuss conductance measurements in a three-terminal InSb/NbTiN device. They identify bound states in the device via NS tunnelling conductance measurements and try to determine their spatial extent using non-local measurements between the normal terminals. These measurements, when compared to simulations of a toy model of the nanostructure, yield qualitative information about the inhomogeneity in the nanowire, which is deemed too high to allow for Majorana zero modes to occur in this material platform, at least in the current generation of devices.

This conclusion is sensible, and not surprising: NbTiN is sputtered on the InSb nanowire and devices thus fabricated consistently show a soft sub-gap density of states at B=0. This has been long identified as a non-starter for a gapped topological phase to occur. This conclusion is also not completely new: e.g. it has been previously reached by the same group in Phys. Rev. Lett. 123, 107703 (2019), based on NS measurements alone.

Repetita juvant, and in principle I am strongly in favour of another paper supporting this conclusion and providing further detail. However, I am not sure that the conclusion is well supported or illustrated by the data shown here. I have doubts related to the way the measurements were carried out, the interpretation of the data, the modelling, and the clarity of the presentation, which I think need to be sorted out before this work can be published. Below I present remarks on all these aspects separately.


The Authors conduct three-terminal lock-in measurements in the two configurations shown in Fig. 1a. In the configuration shown in black, a voltage bias is applied to the superconducting terminal and the resulting current response in the two normal terminals is measured to determine the conductances G_L and G_R. In the configuration shown in red, the superconducting terminal is floating, a voltage bias is applied to the right normal terminal, and the current response is measured on the left normal terminal to determine another conductance quantity named G_LR. The Authors associate G_L and G_R with the local density of states at the two NS junctions, and G_LR with the non-local transport between the two-terminals: according to this interpretation features (peaks) appearing in G_LR must be associated with states extended along the entire nanowire segment between the two NS junctions. However, I am not sure that this can actually be inferred from the data as analysed or presented.

First, in a three-terminal measurement, proper treatment of the resistor network and voltage divider corrections are essential in order to extract the right conductance matrix elements. This is described in detail in arXiv:2104.02671. Without this data processing step, local and non-local responses get mixed in the current measured when a lock-in excitation is applied to one of the three terminals. Given that the Authors do not give information on how the analysis of the three-terminal circuit is carried out, I worry that these spurious effects may affect their measurements carried out in the configuration shown in red in Fig. 1a.

Second, in the measurement configuration shown in black in Fig. 1a, the superconducting terminal is floating rather than grounded. This choice should be motivated: I believe it is not appropriate for the purpose of the experiment. In the absence of a ground between the two normal contacts, the applied bias will now be distributed across the two NS junctions. Therefore, the measurement will reflect a complicated convolution of two local density of states, rather than the proper off-diagonal element of a three-terminal conductance matrix. This is true in particular if both junctions are close to the tunnelling regime, which seems to be the case, rather than one closed and one open. As a consequence, I do not think that information about the extent of the bound states wave functions can be reached using this measurement.

These circumstances potentially make the discussion and interpretation of the data more complicated than portrayed in the manuscript. Thus, more details should be given on the circuit and the lock-in measurements: based on the information present, I cannot conclude that the measurements were carried out and processed correctly, where by “correctly” I mean in a way that allows one to infer spatial information from the measurements as done by the Authors. The Authors should provide more details on voltage divider effects and on the way the tunnel junctions are tuned in the second measurement configuration.


In page 2, the Authors write:

"These states are common in our devices [14, 24]. We speculate that they may be located away from the semiconductor-superconductor interface and closer to the bottom of the nanowire. One caveat is that the bulk gap of NbTiN superconductor is much larger, of order 2 meV, so it is not possible to exclude Andreev reflections."

Given the supposed presence of extended states at zero energy and at zero magnetic field, the presence of semiconductor states not coupled to the superconductor make sense to me. However, per se this explanation does not require disorder: states at the nanowire bottom facets, away from the NbTiN, could be ballistic and still have zero energy. Later, in the theoretical analysis, these features are instead shown to exist in a disordered 1D model where the semiconductor states are all equally coupled to the superconductor. Thus, these two proposed scenarios seem to be inconsistent. What gives?

An alternative interpretation is not discussed: Is it possible that these conductance features are not a proxy of the density of states, but are instead weak Coulomb blockade oscillations caused by the spontaneous formation of quantum dots (density puddles) in the InSb nanowire?

In page 3, the Authors write:

"In this case the charging energy is reduced due to screening by the superconductor so that these are just wave functions with different spatial localisation within the nanowire."

This seems to be a speculative conclusion. If charging effects are present, even if with a reduced charging energy, then conductance features may be associated with Coulomb degeneracy points rather than energy levels of single-particle states.

Later in page 3, the Authors write:

"The presence of such states is expected given that the nanowire segment between the tunnel barriers is 400 nm in length, only a factor of 4 greater than the nanowire diameter. What is more surprising is that in the same segment we observe wave functions significantly localised either on the left or on the right end (L and R states)."

Can the localised wave functions actually be located not in the segment coupled to the superconductor, but only under the tunnel barriers, similar to the lead resonances identified in the theoretical modelling?

What is the expected induced coherence length of sub-gap states in this nanostructure? If the induced coherence length is not much shorter than 400 nm, there is no way to distinguish “localized” and “delocalized” states.


1. Can the Authors explain how they estimate Emin~ 5 meV from Fig. 5?

2. Can the value of Emin be estimated from experimental data? This would, of course, be very valuable.

3. I am puzzled by the presence of extended states (e.g. the one labeled by C in Fig. 5) at zero energy and zero magnetic field in this particular 1D model. Even in the presence of strong disorder, at B=0 the density of states at E=0 under the superconducting segment should vanish due to the s-wave pairing term, if the wire is long compared to the coherence length. Can the Authors explain? What is the induced coherence length in the simulation?

In general, despite the modelling effort and the extensive size of the study (9000 datasets as stated at the end of the paper), the lack of extraction of quantitative information from the experimental data is frustrating and affects the impact of this work.


1. I am confused by the use of the term “delocalized state” in this work. It seems to be used to mean both “extended” (i.e., a state whose wave function has support throughout the entire nanowire segment, such as a proximitized semiconductor state with energy above the induced gap) and “non-local” (i.e., a Majorana state with wave function support only at the two ends of the nanowire, but not in the middle of the nanowire segment). This distinction is important, because to identify well-separated MZMs one must differentiate between extended and non-local wave functions. I wish the Authors adjusted the language used in the work to make these things more clear.

2. Related to this point, the Authors write in page 1:

“Since Majorana modes are themselves delocalized states and require a uniform density along the nanowire, we focus on the regime with delocalized states to search for MZMs.“

It is not clear that one should focus on the regime with “delocalized” states to search for MZMs, as extended states may belong to different subbands and thus may not be a good proxy to find the right density range corresponding to a topological phase. A case in point, delocalised states are later associated with states located on the bottom facets of InSb, away from the superconductor. It seems to me that the logic is internally inconsistent on this.

3. I wish the Authors did not use normalised units in Fig. 1, or at least, that they stated clearly what the normalisation is. This must be corrected.

I also wish that they did not use a strongly saturated and divergent colormap to present the data. These choices distort quantitative comparisons between data points and, in my opinion, go against best practices.

4. In the introduction the Authors write:

“More generally, the three-terminal technique is a powerful method of studying the localisation of any wave functions, which is what we do in this work.”

Here, I believe that they should provide credit to the works that analysed three-terminal transport in hybrid nanostructures and elucidated this fact, such as (the list may not be exhaustive): Phys. Rev. B 97, 045421, Phys. Rev. B 103, 014513 and arXiv:2103.12217.

To conclude, the measurements and simulation shown in this work do not, in my opinion, provide useful evidence in favour of the conclusions reached. This paper focuses on specific features of the data which may not, by themselves, be related to the level of disorder in the device. Despite the abundance of data and simulations, no quantitative information regarding disorder strength (or its origin) is extracted. At the moment, referring to the criteria for publication in SciPost (, I do find that this work currently does not meet any of the Expectations contained there, and that it fails to meet General Acceptance Criteria #1, 3, 4, 5. I hope that the Authors can leverage these remarks to improve their manuscript.

Requested changes

See report.

  • validity: low
  • significance: ok
  • originality: poor
  • clarity: ok
  • formatting: excellent
  • grammar: excellent

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