SciPost Submission Page
Engineered Josephson diode effect in kinked Rashba nanochannels
by Alfonso Maiellaro, Mattia Trama, Jacopo Settino, Claudio Guarcello, Francesco Romeo, and Roberta Citro
This is not the latest submitted version.
Submission summary
Authors (as registered SciPost users):  Claudio Guarcello · Alfonso Maiellaro · Jacopo Settino · Mattia Trama 
Submission information  

Preprint Link:  scipost_202405_00043v1 (pdf) 
Date submitted:  20240527 14:07 
Submitted by:  Maiellaro, Alfonso 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
The superconducting diode effect, reminiscent of the unidirectional charge transport in semiconductor diodes, is characterized by a nonreciprocal, dissipationless flow of Cooper pairs. This remarkable phenomenon arises from the interplay between symmetry constraints and the inherent quantum behavior of superconductors. Here, we explore the geometric control of the diode effect in a kinked nanostrip Josephson junction based on a twodimensional electron gas (2DEGs) with Rashba spinorbit interaction. We provide a comprehensive analysis of the diode effect as a function of the kink angle and the outofplane magnetic field. Our analysis reveals a rich phase diagram, showcasing a geometry and fieldcontrolled diode effect. The phase diagram also reveals the presence of an anomalous Josephson effect related to the emergence of trivial zeroenergy Andreev bound states, which can evolve into Majorana bound states. Our findings indicate that the exceptional synergy between geometric control of the diode effect and topological phases can be effectively leveraged to design and optimize superconducting devices with tailored transport properties.
Author indications on fulfilling journal expectations
 Provide a novel and synergetic link between different research areas.
 Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work
 Detail a groundbreaking theoretical/experimental/computational discovery
 Present a breakthrough on a previouslyidentified and longstanding research stumbling block
Current status:
Reports on this Submission
Strengths
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.
Weaknesses
See the report.
Report
This manuscript contains a detailed numerical study of the superconducting diode effect (SDE) in short SNS junctions using tightbinding lattice models of short Josephson junctions with kinked geometries and in the presence of an outofplane 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 Rashbabased superconducting junctions. Overall, the exposition is clear and wellstructured.
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 engineeringoriented, 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 subgap characteristics of these systems.
Requested changes
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).
Recommendation
Accept in alternative Journal (see Report)
Strengths
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 insitu control knob to switch these devices between the conventional reciprocal junction regime to the nonreciprocal 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 kinkangle $\theta$ 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, nontrivial 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 CurrentPhase relation $I(\phi)$.
Weaknesses
1  A thorough explanation of the role played by the junction kink angle and by Rashba spinorbit coupling in producing the Josephson Diode Effect is lacking.
2  The authors do not mention Refs. https://iopscience.iop.org/article/10.1088/09538984/27/20/205301 and https://journals.aps.org/prb/pdf/10.1103/PhysRevB.98.144510 where the Anomalous Josephson Effect in Rashba spinorbit 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 spinorbit 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).
Report
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, nontrivial 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 currentphase relation $I(\phi)$, contributing to ongoing research in this area.
However, the article lacks a detailed explanation of the role played by the junction kinkangle and Rashba spinorbit 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 SciPost, thus deserving the publication.
Requested changes
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/10.1088/09538984/27/20/205301 and https://journals.aps.org/prb/pdf/10.1103/PhysRevB.98.144510
3  Discuss the choice of $M_z$ parallel to the Rashba spinorbit 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).
Recommendation
Publish (meets expectations and criteria for this Journal)
Author: Alfonso Maiellaro on 20240723 [id 4645]
(in reply to Report 4 on 20240715)
We thank the referee for their careful reading of the manuscript and for the positive assessment of our results. We also appreciate the referee for highlighting some critical points requiring clarification. Below, we provide a detailed response to the referee’s comments and critiques: 1) The transport properties of a kinked nanostrip, influenced by Rashba spinorbit coupling, are impacted by nonAbelian phases in the electron motion, which are tied to the specific sequence of paths followed by the current. The nonAbelian phase accumulated along each path depends on its orientation. To elucidate, let us assume a nonAbelian phase A ̂ is accumulated along the first branch, and a second phase B ̂ is accumulated along the second branch. The total nonAbelian phase A ̂B ̂ affects the quantum transport for a specific direction. Conversely, for the reversed path, the phase B ̂A ̂ is accumulated. Since A ̂B ̂≠B ̂A ̂, this results in nonreciprocal transport properties of the junction. Furthermore, due to the symmetry properties of the model, reversing the direction of the Zeeman term results in a reversal of the current sign.
2) We acknowledge the referee for signalling out these missing references. We will add to the bibliography of the manuscript.
3) Our choice of M_z is motivated by the fact that the junction is not aligned along a single direction. So that, in order to obtain a homogeneous effect of the magnetic field on the device, we choose an out of plane configuration.
4) We acknowledge the referee for signalling this weak point. We will add the mentioned lines to Fig.2 (b) and (f).
5) According to the referee suggestions, we will provide a new version of the figure with updated caption and labels.
6) According to the referee suggestions, we will add to the manuscript a short paragraph to provide the reader with more context about Majorana polarization, which has been defined in ref.s [67, 68] of the manuscript.
Attachment:
Strengths
1. The paper presents a comprehensive study of the superconducting diode effect in a kinked Josephson Junction between two superconducting nano strips with Rashba spinorbit 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.
Weaknesses
1. A discussion of a possible experimental realisation is lacking.
Report
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 subgap Bogoliubovde 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 subgap quasiparticle excitations that the system can sustain (trivial Andreev bound states, Majorana bound states and Zeroenergy 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.
Requested changes
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.
Recommendation
Ask for minor revision
Author: Alfonso Maiellaro on 20240723 [id 4646]
(in reply to Report 5 on 20240715)
We thank the referee for their careful reading of the manuscript and for the positive assessment of our results.
Among the experimental platforms, we repute that the oxides interfaces between LAO and STO are suitable candidates. Here it’s possible to realize the kinked junctions with typical Rashba interaction of the order of magnitude considered in our manuscript (see Refs. [5561]). Moreover, different techniques are experimentally available to engineer the junctions with desired geometry, like the energy ion irradiation technique (see Ref. 38 and.https://doi.org/10.3390/nano11020398; https://doi.org/10.1103/PhysRevB.96.020504) or ebeam lithography with lift off technique (see Ref. 37). The realistic length of the weak link can be approximatively down to 30 nm and the applied magnetic field, compatible with superconductivity, can be up to 200 mT (Ref. 38). The energy scale that we used in our manuscript is the hopping amplitude t whose benchmark values are of the order of 100 meV in oxides interfaces (see Ref. 60).
Another suitable platform is represented by the flakes of 2HNbSe2, a layered twodimensional (2D) transition metal dichalcogenide, a prototypical typeII superconductor that exhibits strong anisotropic responses to external magnetic field (Ref. 7).
Strengths
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.
Weaknesses
Not complete list of described applications and references.
Report
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 hightemperature 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 velocitymatching steps when changing the sign of external magnetic field, have been observed in experiment https://doi.org/10.1088/13616668/aaacc3. While the work of the author continues general ideology of the cited above papers, the authors show the possibility of geometry and magnetic fieldcontrolled diode effect in a nanostrip Josephson junction based on a twodimensional 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.
Requested changes
See the report
Recommendation
Ask for minor revision
Author: Alfonso Maiellaro on 20240723 [id 4647]
(in reply to Report 3 on 20240712)We thank the referee for their careful reading of the manuscript and for the positive assessment of our results. We also appreciate the referee for signalling out the missing references. We will add them to a new version of the manuscript. With this changes we hope that the manuscript can be accepted for publication.
Strengths
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 currentphase relations
Weaknesses
1 An estimate of the superconducting coherence length is not provided.
2 NonAbelian 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.
Report
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 followup works. I think the manuscript can be published once the requested changes are addressed.
Requested changes
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 nonAbelian 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(cd), top panels, what is the difference between thick grey and black curves? Why are the black curves virtually independent of \phi?
Recommendation
Ask for minor revision
Author: Alfonso Maiellaro on 20240723 [id 4648]
(in reply to Report 2 on 20240708)
We thank the referee for their careful reading of the manuscript and for the positive assessment of our results. We also appreciate the referee for highlighting some critical points requiring clarification. Below, we provide a detailed response to the referee’s comments and critiques:
1) The coherence length can be derived by the model parameters as \xi=\hbar^2/(a m^\ast \Delta) where a, m^\ast and \Delta are respectively the lattice constant, the effective mass and the superconducting gap. Accordingly, we can estimate a length of 20 a, the latter being much smaller than the electrode length.
2) The transport properties of a kinked nanostrip, influenced by Rashba spinorbit coupling, are impacted by nonAbelian phases in the electron motion, which are tied to the specific sequence of paths followed by the current. The nonAbelian phase accumulated along each path depends on its orientation. To elucidate, let us assume a nonAbelian phase A ̂ is accumulated along the first branch, and a second phase B ̂ is accumulated along the second branch. The total nonAbelian phase A ̂B ̂ affects the quantum transport for a specific direction. Conversely, for the reversed path, the phase B ̂A ̂ is accumulated. Since A ̂B ̂≠B ̂A ̂, this results in nonreciprocal transport properties of the junction. Furthermore, due to the symmetry properties of the model, reversing the direction of the Zeeman term results in a reversal of the current sign.
3) For every θ and M_z values, we numerically identify μ_0 as the bottom of the lowest energy band in the normal phase. The energy offset measured from μ_0 is indicated as e_0. The diode is induced by both θ and M_z, as it is explained at point 2. The anomalous Josephson effect only requires the presence of M_z as discussed in ref. [39] of the manuscript.
4) We acknowledge the referee for signalling out the typo. We will substitute “green curves” with “magenta and dark magenta curves” in the manuscript.
5) According to the referee suggestions, we will add to the manuscript a short paragraph to provide the reader with more context about Majorana polarization, which has been defined in ref.s [67, 68] of the manuscript.
6) Both grey and black curves represent the BdG subgap eigenenergies. The black ones correspond to the lowest energy states that in the topological regime are insensitive to the phase difference ϕ, the latter being a signature of Majorana character. A sentence will be added to App C to elucidate this property.
Strengths
1it 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.
Weaknesses
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.
Report
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.
Requested changes
1 Inclusion of a simple interpretation
2 improved reference list
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Author: Alfonso Maiellaro on 20240723 [id 4649]
(in reply to Report 1 on 20240624)
We thank the referee for their careful reading of the manuscript and for the positive assessment of our results. We also appreciate the referee for highlighting some critical points requiring clarification. Below, we provide a detailed response to the referee’s comments and critiques:
1) The transport properties of a kinked nanostrip, influenced by Rashba spinorbit coupling, are impacted by nonAbelian phases in the electron motion, which are tied to the specific sequence of paths followed by the current. The nonAbelian phase accumulated along each path depends on its orientation. To elucidate, let us assume a nonAbelian phase A ̂ is accumulated along the first branch, and a second phase B ̂ is accumulated along the second branch. The total nonAbelian phase A ̂B ̂ affects the quantum transport for a specific direction. Conversely, for the reversed path, the phase B ̂A ̂ is accumulated. Since A ̂B ̂≠B ̂A ̂, this results in nonreciprocal transport properties of the junction. Furthermore, due to the symmetry properties of the model, reversing the direction of the Zeeman term results in a reversal of the current sign.
2) We appreciate the referee comment about the bibliography. We will improve the references list to provide the reader with more context.
Author: Alfonso Maiellaro on 20240730 [id 4666]
(in reply to Report 6 on 20240726)We thank the referee for his/her careful reading of the manuscript and for recognizing that our work represents a step forward in the characterization and optimization of SDE systems. Below, we provide a detailed response to the referee’s comments and critiques:
1) We partially agree with the referee that Rashba nanowires forming Josephson junctions with kinked/bent geometries have been previously considered, but we disagree with the statement that the novelty of our results is moderate. Indeed, we believe that our system is already a deviation from the pristine nanochannel case, and our study represents a significant advancement, both in the methodology and in the analysis, compared to the the previous papers. In the physical picture emerging in our system, the kink angle of the Josephson junction affects transport properties by modifying the energy profile, altering the conductance and showing a rich phenomenology of bound states. It influences the critical current and introduces local variations in hopping connectivity. The latter aspect, i.e. a bendingcontrolled Josephson junction transparency, is completely missing in the Phys Rev. B 103, 144520 (2021), which is based on a onedimensional quantum wave guide approach. Furthermore, in our system, we introduce a transversal width L_w, playing a relevant role in the emerging physical picture. Indeed, for every kinked realization θ, the system experiences a surface roughness where some neighboring sites are connected by hopping and others are not (see Fig.1(b)). The latter effects simulate the detrimental effect of disorder, which is extremely relevant in experiments. The effect of random weak disorder (like oxygen vacancies) doesn’t affect our picture as long as the disorder potential, which couples to the density, is lower then the relevant energy scales in our system (hopping and superconductiong gap). For these reasons, we believe that our work properly addresses a physical condition presenting a relevant deviation from the pristine nanochannel case. We will add a sentence to the manuscript to further elucidate these aspects.
2) The breaking of timereversal and inversion symmetry are believed to be crucial ingredients for the manifestation of the SDE effect. When translational invariance along the transport direction is broken (θ≠θ_n) under a magnetic field M_z, both the requirements are satisfied and a relevant SDE is observed. Conversely when the inversion symmetry along the transport direction is preserved SDE is not expected. A sentence will be added to the manuscript in order to clarify this aspect.
3) Among the experimental platforms, we repute that the oxides interfaces between LAO and STO are suitable candidates. Here it’s possible to realize the kinked junctions with typical Rashba interaction of the order of magnitude considered in our manuscript (see Refs. [5561]). Moreover, different techniques are experimentally available to engineer the junctions with desired geometry, like the energy ion irradiation technique (see Ref. 38 and.https://doi.org/10.3390/nano11020398; https://doi.org/10.1103/PhysRevB.96.020504) or ebeam lithography with lift off technique (see Ref. 37). Another suitable platform is represented by the flakes of 2HNbSe2, a layered twodimensional (2D) transition metal dichalcogenide, a prototypical typeII superconductor that exhibits strong anisotropic responses to external magnetic field (Ref. 7). We will also add a paragraph to the manuscript to discuss these experimental platforms, meeting also the requirement of Referee 5.