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Majorana bound states in encapsulated bilayer graphene

by Fernando Peñaranda, Ramón Aguado, Elsa Prada, Pablo San-Jose

This is not the latest submitted version.

Submission summary

Authors (as Contributors): Pablo San-Jose
Submission information
Preprint link: scipost_202211_00049v1
Date submitted: 2022-11-25 19:10
Submitted by: San-Jose, Pablo
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational

Abstract

The search for robust topological superconductivity and Majorana bound states continues, exploring both one-dimensional (1D) systems such as semiconducting nanowires and two-dimensional (2D) platforms. In this work we study a 2D approach based on graphene bilayers encapsulated in transition metal dichalcogenides that, unlike previous proposals involving the Quantum Hall regime in graphene, requires weaker magnetic fields and does not rely on interactions. The encapsulation induces strong spin-orbit coupling on the graphene bilayer, which in turn has been shown to open a sizeable gap and stabilize fragile pairs of helical edge states. We show that, when subject to an in-plane Zeeman field, armchair edge states can be transformed into a p-wave one-dimensional topological superconductor by laterally contacting them with a conventional superconductor. We demonstrate the emergence of Majorana bound states (MBSs) at the sample corners of crystallographically perfect flakes, belonging either to the D or the BDI symmetry classes depending on parameters. We compute the phase diagram, the resilience of MBSs against imperfections, and their manifestation as a 4$\pi$-periodic effect in Josephson junction geometries, all suggesting the existence of a topological phase within experimental reach.

Current status:
Has been resubmitted


Author comments upon resubmission

Dear Editor,

Thank you for taking the time to process our submission. We have uploaded an updated manuscript and a response to each of the invited and contributed reports. All points raised have been addressed, and significant improvements and corrections have been implemented in the manuscript and figures.

An explicit argument to support publication in SciPost Physics as opposed to Core is also requested. We believe our work satisfies criteria 2 and 3 of the SciPost Expectations,

2 - Present a breakthrough on a previously-identified and long-standing research stumbling block

3 - Open a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work

As argued also by Prof. Wimmer in his report, the field of Majorana bound states is currently hindered by material-related problems (disorder in particular) associated to the use of different types of proximitized semiconductor nanowires and quantum wells. Despite strenuous efforts (see e.g. the manuscript from the recent Microsoft collaboration) we are still far from achieving robust topological protection of low-energy states in experiments. It has thus becomes necessary to find some way forward by looking at alternative implementations that may not suffer to such extent from these problems. Two dimensional crystals seem to many researchers like a promising approach. Not only are fabrication techniques improving rapidly, but the flexibility afforded by the combination of different crystals, as showcased in this work and others, opens many doors towards optimizing topological gaps.

Our work is a new proposal in this direction, in an effort to overcome the above stumbling blocks (Expectation #2). Our proposal is conceptually simple, and we do our best to characterize the extent of its validity in the presence of imperfections. It combines ingredients that have been demonstrated to behave just as one would expect from theory (e.g. strong proximity effect in lateral MoRe/graphene contacts and inverted spin-orbit gaps in bilayer graphene encapsulated in WSe2). This is already a far cry from many of the results obtained in nanowires, in particular those related to topological band inversions, which remain muddy at best. Needless to say, any new platform for Majoranas will require solving a whole new set of problems, but experimentalists are already looking at this and other implementations of Majorana physics in 2D crystals (both bound Majoranas and chiral Majoranas). In the particular case of our proposal, at least one group (S. Goswami in TUDelft) is already exploring its implementation experimentally (Expectation #3). The Contributed Report believes that our proposal goes beyond what is possible experimentally. Goswami and Invited Referee 2 seem to disagree. We argue that one cannot know until it is tried.

Apart from the Expectations, SciPost Physics also demands that candidate manuscripts satisfy six General Criteria. We have taken care to satisfy all of them.

1- Clarity: at least one referee commends the clarity of the manuscript.

2- Summary: we provide abstract, introduction and a concise summary of our main results.

3- Reproducibility: all our code is available to reproduce our results, and all the model details employed are included in the manuscript.

4- Citations: we have done our best to cite an extensive selection of the literature relevant to our work (78 works).

5- Code availability: provided through Zenodo.

6- Outlook: we include a conclusion, with a final summary and outlook.

Best regards,

Pablo San-Jose (on behalf of all the authors)

List of changes

List of changes

- Corrected spin labeling of Fig. 2
- Improved clarity of Fig. 2 and its discussion
- Included characterization of bands in terms of orbital symmetries
- Clarified reason for spin structure of edge states
- Corrected imprecisions in the analysis of BDI-class invariants
- Added color bars to LDOS figures

A copy of the resubmitted PDF with all changes marked in red are available at:
https://github.com/fernandopenaranda/MBSinBLG/blob/main/manuscript/encapsulated.pdf


Reports on this Submission

Report 1 by Antonio Manesco on 2022-12-6 (Invited Report)

Strengths

1. The authors propose a new way to engineer Majorana zero modes in graphene that do not require strong magnetic fields and electronic interactions.

2. The code to reproduce the results is provided.

3. Many aspects relevant to experiments are addressed.

4. The authors provide a comparison with other platforms.

Weaknesses

1. Effects of disorder seem underestimated.

Report

In this submission, the authors revisited many of the points suggested by the referees. I believe that the changes improved the manuscript. Before accepting for publication, I however still would like a better intuition on the interplay between lattice orientation and the existence of topological phases, as well as the effects of strong disorder due to the superconductor. As I comment below, I think that these points, although do not invalidate the theoretical model, have significant consequences of interest for upcoming experiments.

1. If I understand correctly, the authors argue that the difference between the armchair and zigzag orientations is just the dispersion. Namely, the main difference is that for zigzag orientation the Zeeman field opens a gap at zero energy, whereas for armchair edges the gap opens at finite energy. As a consequence, only for armchair interfaces there is an energy window with "spinless helical" edge states. Interestingly, the topological gap closing as a function of the crystal orientation occurs at $\sim 7^{\circ}$, which agrees with the transition from armchair to zigzag-like behavior of the boundary conditions in the continuum limit (https://doi.org/10.1103/PhysRevB.77.085423).

However, I point out that the nature of my question was not related to the effect of the proximitized $s$-wave gap in the helical edge states. Instead, I am curious on what are the symmetries that allow the existence of helical edge states in the first place.

In a similar setup, some of the authors have shown that intervalley scattering opens a trivial gap in the electronic structure that prevents the existence of topological superconductivity (https://doi.org/10.1103/PhysRevX.5.041042). Moreover, armchair boundary conditions break valley conservation (https://doi.org/10.1103/PhysRevB.77.085423) and are one of the origins of intervalley scattering in this previously proposed platform (https://doi.org/10.1103/PhysRevB.100.125411).

In this new setup, the system is gapped in the presence of Rashba spin-orbit coupling if the NS interface is armchair. So I wonder if the lack of valley conservation is somewhat related to this gap opening. Particularly, that would have important consequences under the presence of disorder, which I discuss in the next point.

2. Because the SOC gap and the superconducting gap have the same order of magnitude, the wavefunction should extend to regions with similar sizes on both sides. I, therefore, agree that disorder on any of the two sides should work.

However, I am not convinced that the form with which disorder was treated is compatible with experiments. I expect that the helical edge states will experience still a nearly ballistic edge, particularly because the envelope of the wavefunctions decay on a scale much larger than the region in which disorder is added (If I correctly interpreted the code in the GitHub repository, disorder is added as vacancies in the outermost row of atoms). Does this observation seem reasonable? I would imagine that an appropriate simulation should include disorder up to distances comparable with the wavefunctions envelope, as nearly half of the wavefunction should be inside a disordered superconductor. Am I missing something?

At this point, one could argue that the system is a quantum spin Hall insulator, and therefore, one would not expect disorder to have a significant effect. However, as I stated, I am still unsure which symmetries protect the quantum spin Hall state. And I think this point is important for the following reasons:

* One can model disorder in graphene edges as a random staggered potential $m({\mathbf{n}} \cdot \mathbf{r}) \sigma_z$, where $\mathbf{n}$ indicates the interface direction (https://doi.org/10.1103/PhysRevB.77.085423). In this case, both the symmetry operators necessary to preserve the band crossings no longer commute with the Hamiltonian ($[\sigma_z, \sigma_i]\neq 0$ for $i=x, y$). Thus I expect the crossings to no longer be protected. For this reason, I wonder how strong disorder will affect the topological phase. Particularly because I would expect $m({\mathbf{n}} \cdot \mathbf{r})$ to typically be orders of magnitude larger than the SOC, Zeeman, and superconducting gaps.

* (following the discussion in 1) If the presence of edge states depends on valley conservation, then disorder might have a significant effect. As we have recently shown (https://scipost.org/10.21468/SciPostPhysCore.5.3.045), MoRe side contacts with graphene likely introduce strong intervalley scattering. Thus, if related to intervalley scattering at all, the small gap opening at zero energy for armchair interfaces would be a lower bound. Realistic devices would have a much much larger scattering rate $\nu$. As a result, $h \nu \gtrsim \lambda_I$ would result in a trivial system.

Considering the two points above, I believe that disorder effects beyond what was simulated might be important to experimentalists. On the other hand, I expect that proximitized graphene, rather than side contacts, would suppress these problems both because of the interface smoothness and due to the screening of disorder by the superconductor (as recently explored in https://arxiv.org/abs/2207.02472). Therefore, I believe that the theory presented here might still be interesting for upcoming experiments. But the designed device might need some improvements that could be easily identified.

3. Since the authors investigated the effects of Fermi level mismatch, I suggest also providing the results on the GitHub repository.

Requested changes

1. Consider disorder over a larger region. For example, by adding a random onsite potential.

2. Show that time-reversal-invariant disorder can or cannot destroy the helical edge states.

  • validity: high
  • significance: good
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Author:  Pablo San-Jose  on 2022-12-29  [id 3192]

(in reply to Report 1 by Antonio Manesco on 2022-12-06)

Dear Dr. Manesco,

thank you for your time and your attentive report. Please find below our reply to your comments and criticisms. We have carefully considered each of them to improve the paper. We have added a new appendix with new simulations and an extended discussion about the stability of the helical edge states.

Best regards

The authors.

----- Reply to referee -----

The first and second points raised by the referee revolve around the stability of helical states in the encapsulated bilayer, when still decoupled from superconductors, but with irregular, disordered or otherwise imperfect edges. The concern is that valley scattering caused either by vacancies, short-range disorder or intermediate edge orientations could destroy the helical states.

We argue below that this is not the case, as long as the disorder remains spin-independent, or at most depends only on s_z. The symmetry protecting the helical edge states in the encapsulated bilayer *without* Rashba coupling is the conservation of out-of-plane spin s_z. Indeed, as long as Rashba can be neglected and disorder is spin-independent, the Hamiltonian commutes with s_z. Since edge states are exactly helical (opposite s_z for opposite group velocities, see Fig 2), no amount of spin-independent valley mixing or impurity-induced backscattering can localize them, as backscattering requires a spin flip. In truth, this is a fact that was not fully demonstrated in the current manuscript, as we presented spin-resolved bandstructures only for zigzag and armchair crystallographic directions, not for arbitrary edges. To show that helical states survive along any type of edge, we have added a new appendix, where we show the spin current along the boundary of a circular-strip sample of diameter ~1.2 µm. We fix Zeeman and Rashba to zero, and compute both the total density of states of the disk and the spatial density of spin currents around neutrality. We find that, despite the varying orientation of the edges, the edge modes never localize: the current is always finite along the edge, and is purely helical.

Another point of concern for the referee is that disorder was not taken to be sufficiently spread, spatially. The author is correct that disorder was introduced only as vacancies around the edges, not as a scalar disorder. Vacancies were added not only on the last row, however, but up to three rows from the edge. As requested by the referee, we now show in the new appendix that spin-independent Anderson disorder (which includes the suggested staggered potential), even if it is spread throughout the whole sample, is still unable to localize the edge states. They therefore behave as truly topological edge states under arbitrary spin-independent disorder. The reason is again the same: backscattering cannot occur, since inverting group velocity requires a spin flip, so s_z conservation protects edge transport.

Note however that this does not qualify as a Quantum Spin Hall phase, as s_z conservation is a stronger constraint than time-reversal symmetry. Indeed, the presence of Rashba spoils the argument above, as the eigenstates no longer have a good s_z. However, as we show in our analytic derivation and numerically in Fig 2, the low-energy Hamiltonian only depends on Rashba to cubic order in the momentum as measured from the Dirac points, so its effect on edge states, using a realistic magnitude, is very weak. It is only able to gap the edge states in a very narrow energy range around neutrality. These arguments, together with the rather clear experimental evidence of a gap inversion in Ref. [59] consistent with the development of a non-trivial SOC gap, are, to us, convincing evidence that the described helical phase is indeed realized in real samples.

We have included the above discussion in the resubmitted manuscript, in the form of a new appendix with new simulations. We have also uploaded the code for the new simulations to the Github repo, as well as the code for finite SC leads with a chemical potential mismatch, as requested in point 3. To use the latter, employ the keyword arguments `fermimismatch` = true/false, `LS` (for lead length) and `muS` (for lead Fermi energy shift).

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