Tunable Josephson Current through a Semiconductor Quantum Dot Hybridized to Majorana Trijunction

Round 1

Reviewer 1 Report

 

The authors reported on the Tunable Josephson current through a quantum dot coupled to Majorana trijunction. The topic is interesting and a few comments are listed below.

1. The authors can further refine the innovation of this work in the summary and introduction sections.
2. It is suggested the authors to check their manuscript carefully and thoroughly to avoid some typical mistakes and mistypes.

3.It is suggested to update the references in recent two years.

4. The English should be polished by a native speaker to a publishable level.

 

 

Minor editing of English language required

Author Response

The authors reported on the Tunable Josephson current through a quantum dot coupled to Majorana trijunction. The topic is interesting and a few comments are listed below.

1. The authors can further refine the innovation of this work in the summary and introduction sections.

Our response: We appreciate the invaluable advice from the reviewer, and have added some sentences in both the SUMMARY and INTRODUCTION sections to refine the innovation of this work. For example, at the beginning of the fifth paragraph in the INTRODUCITON section, we added “Although the magnitude of the Josephson current in the above mentioned two-terminal Josephson junctions can be efficiently adjusted, the current's period can hardly be altered. In view of this issue and those previously published work concerning multiple-terminal Josephson junctions[14,41,42,43], here we propose a QD/Majorana-trijunction hybridized device to control both the magnitude and the period of the Josephson current”, and “We emphasize that the proposed device is within the reach of present-day nanofabrication technologies, and the unique impacts brought about by the regular junction are difficult to be realized in two-terminal setups”. In the SUMMARY section, we added “Interestingly, we find that the phase factor in the regulator junction will change both the magnitude, sign as well as the period of the Josephson current, which is quite different from the cases in those usual two-terminal Josephson junctions” and “... ...  whereas it is suppressed by the overlap between the MBSs in two-terminal setups”. 
2. It is suggested the authors to check their manuscript carefully and thoroughly to avoid some typical mistakes and mistypes.

Our response: We thank the reviewer for this kind comment, and are sorry for the poor English. We have carefully read the whole manuscript and corrected the mistakes and mistypes as best as we can. Please see the italic words and sentences in this revised version.

3. It is suggested to update the references in recent two years.

Our response: We totally accept this invaluable comment, and have added several references published in recent two years. Please see the newly added references numbered as Refs. [12],[17],[18],[37],[38], which were published in 2022 or 2023.

4. The English should be polished by a native speaker to a publishable level.

Our response: We are sorry for the poor English and have improved it as best as we can, with the help of a friend of native English speaker.

Author Response File: Author Response.docx

Reviewer 2 Report

The article is well-written and supports the theoretical claims. I would suggest to accept the present version.

What is/are the probable reason for the new peat at Fermi level at Figure 3c?

Author Response

The article is well-written and supports the theoretical claims. I would suggest to accept the present version.

He/She asked: What is/are the probable reason for the new peat at Fermi level at Figure 3c?

Our response: We thank the reviewer for his/her positive evaluations on our manuscript. We attribute the new peak at Fermi level at Figure 3c to the existence of the third (regular) Majorana junction. The Majorana bound states in it are exotic zero-energy states and then mainly change the states around the Fermi level of the superconductors, which is set to be zero in the present paper.

Author Response File: Author Response.docx

Reviewer 3 Report

Manuscript Title: Tunable Josephson current through a quantum dot coupled to
Majorana trijunction
Journal: Coatings, Manuscript ID: coatings-2566808


The manuscript deals with a theoretical study of the Josephson current flowing in a QD through left and right Majorana nanowires, whose amplitude and period are regulated by the phase factor of a  third junction coupled to a QD.

This paper is interesting. In general well written, even if I suggest a moderate English revision, by native speaker/colleague.

There are some issues that I find to be improved in the manuscript.

On page 3 line 12fth, please, detail which are the typical samples of Ref.s [17-19], and/or Ref.s [20,21] and Ref. [24].

Can the authors better describe their QDs (these semiconductor nano-wires with strong Rashba spin-orbit interaction in contact with s-wave superconductor substrate)?

Is the symmetry between L and R nano-wires and their length important?

Can the authors better specify the importance of the chemical nature of the QDs and maximum/minimun sizes, and their role (given that the size of the J-current is suppressed by the hybridisation between QDs and the third nanowire junction?

I think Figure 1 (and/or its description) is not completely clear. Is there a substrate formed by superconducting nanowires upon which the QD sits, and then there is a third nanowire, which is the third junction, where all three are the seed of MBs?

I think the manuscript can be accepted in the Coatings Journal, after this review.

Moderate English revision, by native speaker/colleague is needed.

Author Response

The manuscript deals with a theoretical study of the Josephson current flowing in a QD through left and right Majorana nanowires, whose amplitude and period are regulated by the phase factor of a third junction coupled to a QD.
This paper is interesting. In general well written, even if I suggest a moderate English revision, by native speaker/colleague. There are some issues that I find to be improved in the manuscript.

1. On page 3 line 12fth, please, detail which are the typical samples of Ref.s [17-19], and/or Ref.s [20,21] and Ref. [24].
Our response: We thank the reviewer for this invaluable advice, and have detailed the typical samples of the mentioned references, with some sentences rewritten. Please find the highlighted sentences at the beginning of the third paragraph in the INTRODUCTION section, which are (note that the references’ number are changed because we have added several references as required by other reviewers) “Recently, much work concerning Josephson effect has shifted towards junctions composing of topological superconductors realized by conventional nontopological ingredients,

such as Shiba chains realized by magnetic Fe atoms deposited on the surface of Pb, which is an s-wave superconductor having strong spin-orbit interaction of Rashba type[19,20], or an conventional s-wave superconductor sandwiched between two-dimensional electron gases (2DEGs) with strong Rashba spin-orbit interaction.[21]. There are also some works concerning hybridized junction with topological properties realized by particular design with conventional materials, for example, an InAs 2DEG proximitized by an epitaxial Al layer [22] and HgTe quantum well coupled to Al thin-film [23].” and “The MBSs are exotic quasiparticles of Majorana fermions recently prepared at the edges of topological superconductors typically realized in HgTe/CdTe quantum wells[26]”.

2. Can the authors better describe their QDs (these semiconductor nano-wires with strong Rashba spin-orbit interaction in contact with s-wave superconductor substrate)?
   Our response:The comment is welcome. Our theoretical model comprises of a semiconductor quantum dot (QD) coupled to three nanowires hosting Majorana bound states (Majorana nanowire). This model can be viewed as a generalization of that realized experimentally in Ref. 33, in which a QD is coupled to only one Majorana nanowire . The device in Ref. 33 is based on a InAs semiconductor nanowire with a

epitaxial Al layer deposited on three facets of the wire. The Al shell is etched on one end of the wire, leaving a bare InAs segment which is the QD whose energy levels are adjusted by external gate voltages. By applying a strong magnetic field, Majorana bound states are prepared at the ends of the InAs/Al segment (Majorana nanowire). In this revised version, we have better described our model in the caption of Fig. 1, which are “Schematic plot of the light-blue trijunction coupled to a QD denoted by the strong-orange circle. The QD can be defined at the end of a InAs nanowire by applying gate voltages. The junctions are realizable by depositing epitaxial Al layer on three facets of the InAs nanowire under a strong magnetic field, which induces the Majorana bound states ”.  

3. Is the symmetry between L and R nano-wires and their length important?
   Our response: The mutual orientation angle between the left and right nanowires, which are possibly not aligned along the same direction, describes the symmetry between the two nanowires. Previous work [38,43] showed that the current’s amplitude is generally suppressed when the angle is changing from 0 to π. In Reference [40], the authors found an abnormal enhancement of the Josephson current when the two Majorana nanowires are perpendicular to each other. Therefore, the symmetry between the two nanowires are important to the Josephson current.

The overlap strength between the Majorana bound states in certain nanowire, which is crucial to the Josephson current, depends on the length of the nanowire. Therefore, the difference between the two nanowire will also exert significant impacts on the Josephson current.

In the present manuscript, we have neglected the influences of the angle between the two nanowires as its influences were well studied in previous works [38,39,40,42,43]. The difference between the lengths of the two nanowires are also neglected here because we focus on the functions of the third nanowire. We admit the comment from the reviewer are very stimulating, and the lengths’ difference between the two nanowires deserves to be considered in future work.
4. Can the authors better specify the importance of the chemical nature of the QDs and maximum/minimun sizes, and their role (given that the size of the J-current is suppressed by the hybridisation between QDs and the third nanowire junction?

Our response: We thank the reviewer for this professional comment. It is well known that carriers (electrons or holes) are confined in both the three dimensions in QDs, resulting in quantized energy levels. The QDs can be realized in heterojunctions with different band structures, or in various quantum wells under external gate voltages. Whatever the fabrication technique, the QDs are characterized by the quantized energy levels which care controllable by the variation of gate voltages, chemical compositions, as well as their sizes.

   Therefore, we believe the above factors indicated by the reviewer changes the energy levels of the QDs. In previous works[38,39,40,42,43], it was found that maximum Josephson current emerges when the dot level is aligned to the Fermi level of the superconductor leads, and the current’s magnitude is suppressed when the dot level is tuned away from the Fermi level of the leads. Accordingly, we fix the dot level to the Fermi level in the present manuscript and focus on the impacts of the phase difference.

    In this revised version, we have added some sentences to specify the importance of the chemical nature of the QDs at the beginning of the NUMERICAL CALCULATION section, which are “As was found in previous works[38,39,40,42,43], the Josephson current generally reaches its maximum value when the junctions’ Fermi levels are aligned to the QD’s energy level, which is adjustable by gate voltages, chemical compositions, as well as its sizes. When the dot level is tuned to deeper or higher energy regimes, the current’s magnitude then is suppressed. Accordingly, here we fix the dot level $\varepsilon_d=\mu=0$, and focus the impacts of the regular junction. ”

5. I think Figure 1 (and/or its description) is not completely clear. Is there a substrate formed by superconducting nanowires upon which the QD sits, and then there is a third nanowire, which is the third junction, where all three are the seed of MBs?

Our response: We admit that the description of Fig. 1 is not completely clear. As was explained in Comment 2, our QD/Majorana-trijunction setup is very similar to that in Ref. 33, in which the QD is coupled to only one Majorana junction. The device in Ref. 33 can be realized as this: firstly, an InAs semiconductor nanowire with an epitaxial Al layer deposited on three facets of the wire is prepared on top SiO_x and Si substrate; secondly, the Al shell is etched on one end of the wire, leaving a bare InAs segment which is the QD whose energy levels are adjusted by external gate voltages; thirdly, by applying a strong magnetic field, Majorana bound states are prepared at the ends of the InAs/Al segment (Majorana nanowire).

   In the present manuscript, there are three Majorana nanowires, which are on the same substrate of the QD.

In this revised version, we have better described our model in the caption of Fig. 1, which are “Schematic plot of the light-blue trijunction coupled to a QD denoted by the strong-orange circle. The QD can be defined at the end of a InAs nanowire by applying gate voltages. The junctions are realizable by depositing epitaxial Al layer on three facets of the InAs nanowire under a strong magnetic field, which induces the Majorana bound states ”. 

Author Response File: Author Response.pdf