Engineered Josephson diode effect in kinked Rashba nanochannels

1- It discusses a timely topic in mesoscopic condensed matter physics with potential technological application in dissipationless electronics.
2- Logical and clear exposition of the results
3- Interesting phase diagram $M_z$ vs $\theta$ and identification of the sweet spot enhancing the superconducting diode effect (SDE), with remarkably large diode efficiencies.
4- Meticulous characterisation as a function of the channel bending of the sub gap spectrum and CPR characteristics an its impact on the SDE.

See the report.

This manuscript contains a detailed numerical study of the superconducting diode effect (SDE) in short SNS junctions using tight-binding lattice models of short Josephson junctions with kinked geometries and in the presence of an out-of-plane magnetic field. Exploring the phase diagram as a function of these two experimentally accesible knobs the authors found large diode efficiencies, setting a starting point for further studies and the eventual optimisation of bended Rashba-based superconducting junctions. Overall, the exposition is clear and well-structured.

Although I consider this work to be of general interest to the condensed matter community, I have several concerns that incline my recommendation toward publication in a more specialized journal. These concerns are as follows:

Regarding the novelty of the results, strong Rashba nanowires forming Josephson junctions in kinked/bent geometries have previously been considered as a means to study the SDE. In particular, in Phys. Rev. B 103, 144520 (2021), the authors previously discussed the impact of the topological phase transition on the nonreciprocity of their supercurrents and also discussed the CPR characteristics in similar devices. Although this work represents a step forward in the characterization and optimization of these systems, I consider the novelty of the results obtained to be moderate.

Despite the fact that to some extent the manuscript is engineering-oriented, I find it lacking a deeper connection with the physical system under study. Assessing the applicability of the results found using the toy model to describe the experimental complexity will substantially increase the relevance of the work and might change my consideration toward publication in SciPost. This includes addressing topics such as the estimation of the MCA coefficient (\gamma_s), analyzing the deviation from the pristine nanochannel case, assessing the detrimental impact of a noisy electrostatic potential environment, and considering the presence of lattice disorder—effects known to be present in experiments and to have a significant impact on the topological phases and sub-gap characteristics of these systems.

1- A paragraph on the SDE and spatial symmetries of Rashba nanowires, I believe it would be of benefit to the general reader to understand, for instance, why at commensurate angles ($\theta_n$) there is no anisotropy in the supercurrent.

2- Assess the applicability of the results found into a realistic model of Rashba nanochannels (see Report).

Accept in alternative Journal (see Report)

1 - Josephson Diode Effect and Anomalous Josephson Effect represent interesting up to date topics in the active research field of superconducting electronics.

2 - Tuning the Diode Effect efficiency by changing the junction kink angle provides a new intriguing way of engineering these kind of devices.

3 - Magnetic field represents an in-situ control knob to switch these devices between the conventional reciprocal junction regime to the non-reciprocal case (showing Josephson Diode Effect and Anomalous Josephson Effect).

4 - High values of diode effect efficiency $\eta$, up to $\sim 46 \%$, are predicted.

5 - The authors discuss properly and extensively the diode effect efficiency $\eta$ and the $I(\phi=0)$ behavior when tuning the kink-angle $\theta$ and the magnetic field $M_{z}$.

6 - The authors analyze in depth the difference between the three transport regimes and the diode efficiency behavior wrt to $M_{z}$, in terms of the emergence of Andreev Bound States (ABS), Zero energy ABS and Majorana bound states (MBS).

7 - Interesting link between Josephson Diode Effect, non-trivial topological phases, and the presence of Majorana Zero Modes

8 - The authors suggest a different benchmark for testing the presence of MBS from the junction Current-Phase relation $I(\phi)$.

1 - A thorough explanation of the role played by the junction kink angle and by Rashba spin-orbit coupling in producing the Josephson Diode Effect is lacking.

2 - The authors do not mention Refs. https://iopscience.iop.org/article/10.1088/0953-8984/27/20/205301 and https://journals.aps.org/prb/pdf/10.1103/PhysRevB.98.144510 where the Anomalous Josephson Effect in Rashba spin-orbit junctions is analyzed.

3 - Results in Ref. https://journals.aps.org/prb/pdf/10.1103/PhysRevB.98.144510 point out that the maximum Diode Effect efficiency (the maximum current asymmetry) can be reached when the magnetic field lies in the junction plane. Why do the authors choose the magnetic field $M_{z}$ parallel to the Rashba spin-orbit coupling (SOC) (i.e. out of the junction plane)? Did the authors consider investigating other magnetic field - SOC configurations?

4 - In Fig.2 (b) and (f) the case with $\theta=0$ and $M_{z}\neq 0$, which is the reference case, is not reported.

5 - The plots in Figs. 3 (a), (d), (g) and (c), (f), (i) lack measurement units.

6 -Majorana Polarization could be better discussed in the three different cases analyzed in Figs. 3 (b)-(e)-(h).

The article presents significant advancements in superconducting electronics, focusing on the Josephson Diode Effect and Anomalous Josephson Effect. It highlights the intriguing possibility to tune diode effect efficiency through the junction kink angle and magnetic field, predicting high efficiencies up to 46%. The authors provide a comprehensive analysis of diode effect efficiency with respect to variations in the kink angle $\theta$ and the magnetic field $M_{z}$.
This paper establishes an interesting link between the Josephson Diode Effect, non-trivial topological phases, and the presence of Majorana bound states (MBS). Additionally, the authors propose a novel benchmark for testing the presence of MBS based on the junction's current-phase relation $I(\phi)$, contributing to ongoing research in this area.
However, the article lacks a detailed explanation of the role played by the junction kink-angle and Rashba spin-orbit coupling (SOC) in producing the Josephson Diode Effect. The authors do not mention some references on the Anomalous Josephson Effect and do not discuss the choice of magnetic field orientation with respect to the SOC and junction plane. Some figures lack measurement units.
Despite these small weaknesses, the paper is well written and offers valuable insights and advancements in the field. In conclusion, this work definitely matches criteria for Sci-Post, thus deserving the publication.

1 - Explain the role of kink angle and Rashba SOC (and their interplay) in producing the Josephson Diode Effect

2 -Mention results in Refs. https://iopscience.iop.org/article/10.1088/0953-8984/27/20/205301 and https://journals.aps.org/prb/pdf/10.1103/PhysRevB.98.144510

3 - Discuss the choice of $M_z$ parallel to the Rashba spin-orbit coupling (SOC)

4 -Report the $\theta=0$ and $M_{z}\neq 0$ cases in Fig.2 (b) and (f)

5 - Add measurement units in Figs. 3 (a), (d), (g) and (c), (f), (i).

6 - Add a brief comment on the Majorana Polarization in the three cases analyzed in Figs. 3 (b)-(e)-(h).

Publish (meets expectations and criteria for this Journal)

1. The paper presents a comprehensive study of the superconducting diode effect in a kinked Josephson Junction between two superconducting nano strips with Rashba spin-orbit coupling and Zeeman splitting;
2. The paper is well written and easy to understand;
3. The system exhibits a rich phenomenology as a function of the different parameters (angle of the junction and magnetic field);
4. The figures in the main text are clear and well explained.

1. A discussion of a possible experimental realisation is lacking.

The manuscript presents a detailed study of the superconducting Josephson effect in a junction between two superconducting nanostrips connected though a kinked weak link. The Josephson current is calculated from the sub-gap Bogoliubov-de Gennes spectrum. The calculation is standard and most likely correct. The critical current shows a rich phenomenology in terms of the Zeeman field and the angle of the kink. In a certain range of parameters, the system also exhibits the anomalous Josephson effect.

The rich physics of this setup is explained in terms of the different types of sub-gap quasi-particle excitations that the system can sustain (trivial Andreev bound states, Majorana bound states and Zero-energy Andreev bound states).

The material presented in the manuscript meets the criteria for acceptance in SciPost Physics as it opens a new direction of research in the field of anomalous/topological superconducting effects.

The only shortcoming is that it does not discuss a possible experimental platform where the predicted physics could be observed. In particular, it would be nice to have realistic values of the material parameters, geometrical transverse dimensions, and energy scales.

1. Add a small paragraph on a possible experimental platform where to observe the physics discussed in the manuscript, possibly with a discussion of realistic values of the material parameters, geometrical transverse dimensions, and energy scales.

Ask for minor revision

1. Important subject
2. Applications to nanoelectronic devices.
3. High value of achieved diod effect reaching 40%.
4. Prediction of anomalous Josephson effect related to the Andreev bound states.

Not complete list of described applications and references.

The authors present the results of investigation of Josephson diode effect in Rashba nanochannels. Various nonreciprocal effects play important role in physics and applications and were actively studied by nonlinear community as the ratchet effect and Brownian motors. Much after, this peculiar physics have been attacked by Josephson community in various SQUID and Josephson junction arrays. Here, both dc effect https://doi.org/10.1103/PhysRevLett.95.090603, and also nonreciprocal microwave transmission have been studied http://dx.doi.org/10.1103/PhysRevB.92.104501. More importantly, in high-temperature long Josephson junctions on bicrystal substrates, such nonreciprocal effects naturally appear due to inhomogeneous bias current distribution, which have been theoretically predicted https://doi.org/10.1134/S0021364012060069 and studied https://doi.org/10.1103/PhysRevB.66.134526. Later, this nonreciprocal behavior of velocity-matching steps when changing the sign of external magnetic field, have been observed in experiment https://doi.org/10.1088/1361-6668/aaacc3. While the work of the author continues general ideology of the cited above papers, the authors show the possibility of geometry and magnetic field-controlled diode effect in a nanostrip Josephson junction based on a two-dimensional electron gas. Here, the authors achieved significant diod effect, reaching 40%, which is important for varios applications of superconducting nanodevices. I think that the presented results are interesting and deserve publication in Scipost Physics after minor improvements of the manuscript with better description of the relevant literature.

See the report

Ask for minor revision

1- Diode effect and anomalous Josephson effect made possible by a geometrical feature (kink) in a nanostrip topological setup
2- Efficiency of the diode effect can be tuned through Mz and \theta and it is somehow related to the topological transition
3- Quasiparticle nature can be changed by varying Mz and \theta and can be distinguished by the behaviour of their current-phase relations

1- An estimate of the superconducting coherence length is not provided.
2- Non-Abelian phases are mentioned, but they are not clearly defined. Where do they come from? Why do they need to be mentioned?
3- The case \theta=0, but Mz finite, is never clearly addressed. This is the reference/known situation.
4- Majorana polarization is not defined.
5- The plots in App. C are not properly discussed.

In this paper the authors study the diode effect and the occurrence of the anomalous Josephson effect in a Josephson junction (JJ) which consists of two 2D superconducting nanostrips attached at a given angle through a normal weak link. The system is subject to Rashba coupling and to a perpendicular magnetic field, so that each individual nanostrip has a topological phase which supports Majorana bound states at its ends. The authors numerically calculate the critical current of the JJ as a function of the Zeeman energy Mz and of the kink angle \theta, finding strong asymmetry when changing the sign of Mz. In particular, the diode effect efficiency shows a peak in the vicinity of the topological transition. They also study the anomalous Josephson current as a function of Mz and \theta. Finally, the authors find that the quasiparticles in the system come in three different kinds depending on the value of Mz and \theta.

The paper is interesting and overall well written.
In my opinion the manuscript opens a new pathway on the possibility of realizing the diode effect and an anomalous current in Josephson junctions. It can lead to follow-up works. I think the manuscript can be published once the requested changes are addressed.

1- the Josephson junction is well defined when the lengths of the two superconducting nanostrips (electrodes of the junction) are larger than the superconducting coherence length. Can the authors provide an estimate of the latter?
2- in Sec. II the authors mention non-Abelian phases, but it is not clear what they are referring to. Can the authors be more explicit on this point?
3- it is not clear to me how the parameter \mu_0 is computed. Can the authors be more explicit on this point?
4- is the diode effect and anomalous Josephson effect always absent when \theta=0?
5- in the caption of Fig. 2(f), the authors mention green curves, but are not present in panel (f).
6- can the authors define the Majorana polarization?
7- in Fig. 5(c-d), top panels, what is the difference between thick grey and black curves? Why are the black curves virtually independent of \phi?

Ask for minor revision

1-it opens a new perspective on a very active and promising field.
2- it is well written and the results can be replicated on the basis of the details given.
3- the analysis is detailed so that not many open points are left.
4- although the system is rather artificial, some contact to experiments is made.

1- No simple, hand waving, argument is given to interpret and maybe generalise the interesting results found.
2- the reference list could be more exhaustive.

In their preprint, the Authors examine a Josephson junction based on a spin orbit coupled 2D, or quasi 1D, electron gas. The novelty is represented by the presence of a kink in the structure. The non reciprocal features, namely the superconducting diode effect and the anomalous Josephson current, are analysed and are found to depend on both the external magnetic field (not a new result) and the angle. This last finding is novel and extremely interesting in view of the search for the optimal nanostructure to employ as a superconducting diode. The analysis performed is exhaustive and easy to follow. The article is clearly written and inspiring. I definitely think it meets the publication criteria of SciPost physics. My only suggestion for improvement is the inclusion in the article of a simple interpretative scheme for the results.

1- Inclusion of a simple interpretation
2- improved reference list

Publish (easily meets expectations and criteria for this Journal; among top 50%)