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Model for missing Shapiro steps due to bias-dependent resistance

by S. R. Mudi, S. M. Frolov

Submission summary

As Contributors: Sergey Frolov
Arxiv Link: (pdf)
Code repository:
Date submitted: 2021-06-02 14:42
Submitted by: Frolov, Sergey
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Computational
Approach: Computational


Majorana zero modes are predicted in several solid state systems such as hybrid superconductor-semiconductor structures and topological insulators coupled to superconductors. One of the expected signatures of Majorana modes is the fractional 4$\pi$ Josephson effect. Evidence in favor of this effect often comes from a.c. Josephson effect measurements and focuses on the observation of missing first or higher odd-numbered Shapiro steps. However, the disappearance of the odd Shapiro steps has also been reported in conventional Josephson junctions where no Majorana modes are expected. In this paper, we present a phenomenological model that displays suppression of the odd Shapiro steps. We perform resistively-shunted junction model calculations and introduce peaks in differential resistance as function of the bias current. In the presence of only the standard 2$\pi$ Josephson current, for chosen values of peak positions and amplitudes, we can suppress the odd Shapiro steps, or any steps, thus providing a possible explanation for the observation of missing Shapiro steps.

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Submission & Refereeing History

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Submission 2106.00495v1 on 2 June 2021

Reports on this Submission

Anonymous Report 1 on 2021-7-1 (Invited Report)


1) Brings attention to under-appreciated non-linearities and features in the I-V curve of topological Josephson junctions
2) Provides a possible explanation for accidentally missing Shapiro steps
3) efforts to connect the results to non-linearities observed in real experimental samples (Table 1).


1) ill-defined model, which could be advantageously replaced by previously studied model in the most significant case (Fiske steps)
2) missing Shapiro steps are purely accidental, and the model is purposefully constructed to exhibit missing steps with little physical/microscopic justification
3) simulations do not reproduce the experimental observations :
- the frequency dependence is incorrect
- the presence of resonances at frequencies corresponding to missing steps in the experimentally observed range does not seem extremely plausible.


In their manuscript, Mudi & Frolov report on numerical simulations of the dynamics of Josephson junctions, and more specifically about the observation of ‘missing Shapiro steps’ in the IV curve of Josephson junctions under microwave irradiation. Their work aims at nuancing recent experimental observations in which the disappearance of odd Shapiro steps has been attributed to the presence of 4π-periodic Josephson currents related to the existence of Majorana-Andreev bound states.
The core message of the current article is the following: an engineering model of a Josephson junction, the RSJ model also predicts the disappearance of Shapiro steps at desired positions when one makes the resistance R in this model in a specific way dependent on the dc bias current, Shapiro steps can be made to disappear at desired positions: “Our model also enables us to selectively suppress the even steps instead of the odd steps by positioning the peaks in R(I) at the even steps.” Thus, the authors challenge the validity of claims of gapless Majorana-Andreev bound states based on the observation of missing odd Shapiro steps.

The work is clearly presented, very accessible, and the general formatting and level of grammar are of good level.

The use of the RSJ model is a convenient choice for modeling of the dynamics of Josephson junctions, and had been already widely used in the context of topological Josephson junctions (Dominguez et al., PRB, 95, 195430, (2017), Pedder et al., PRB 96, 165429 (2017), Picó-Cortés et al., PRB 96,125438 (2017), Park et al., Phys. Rev. B 103, 235428 (2021)) to understand how 4π-periodic Josephson currents could manifest themselves. Assessing the role of variations of the differential resistance is a new and very relevant question to tackle. As Mudi & Frolov point out, non-linearities “are ubiquitous in mesoscopic devices used to search for Majorana modes. Resonances are often observed above the critical current in experimental data from a large variety of junctions. Possible causes of these resonances are Multiple Andreev Reflections (MAR) [17–20], Fiske steps [21], Andreev bound states [22, 23], or other unexplored effects.”
However, though this work points to relevant issues in topological devices and calls for better theoretical and experimental investigations, in the current state of the manuscript, the model and analysis appears too superficial, and does not reproduce many of the essential observations made in the relevant experimental papers to bring significant progress in the field.

1) The quotation above indiscriminately groups together a variety of possible non-linearities observed in a variety of superconducting devices of a variety of materials. The terminology « resonance » blurs the discussion considerably. For example, MAR yields features which are also when the Josephson supercurrent is quenched to zero, and are a reminder that the use of a simple R in the RSJ-model is an inadequate simplification. On the opposite, a Fiske step is associated with the Josephson supercurrent in large devices with a linear structure, and can rightfully called a resonance of the electromagnetic wave associated with the oscillating Josephson current. Andreev bound states are at the core of the Josephson effect, but it has been extremely hard to make them visible in experiments relying on current bias or voltage bias. The distinction of different types of resonances is of relevance, as it is straightforward to model Fiske steps in the RSJ model without modifying R (see below), while the other phenomena required more uncontrolled modifications of the RSJ model. Besides, though multiple Andreev reflection or Andreev bound states can lead to peaks or dips in the differential resistance, they do not yield equally-spaced (on a voltage scale) features as would be needed to affect only odd steps. Of relevance are mostly “self-induced steps” or Fiske steps, namely modifications of the I-V curve due to resonances in the electromagnetic environment.

2) The modification of R is a subtle issue in the RSJ model. As already pointed out by Likharev (Dynamics of Josephson junctions and Circuits (1986)), R is an ill-defined parameter in a Josephson junction, as it only takes a well-defined meaning when the junction is under strong bias and the Josephson supercurrent becomes negligible compared to the quasiparticle one. On the contrary, modifications of the current-phase relation (to include skewed current-phase relation, 4π- periodic contributions, etc) are somewhat better justified, though non-equilibrium effect cannot easily be taken into account.

3) The chosen model for the current-dependent resistance $R(i_{dc})$ is insufficiently justified. In particular, it is disturbing that R shows non-linearities even in the absence of supercurrent ($i_c=0$) The series of peaks and its (as a function of $i_{dc}$) are constructed purposefully as an eraser of odd Shapiro steps. Though I sympathize with the authors about the possibility of self-induced steps, such a physical effect and its consequences can be checked. It could be in a cavity-type environment, similar to what has been reported by Richards and Sterling APL 14, 394 (1969); Levinsen APL 24, 247 (1974) and Klapwijk et al, J. Low Temperature Physics 27, 801 (1977). Indeed, in that case in the time-averaged data a feature appears which has the shape of a peak, but it is followed by a dip. Such resonances have been modeled in various ways. It would be interesting to carry out such research in combination with an external microwave drive as a follow-up of research carried out by Ganz & Mercereau, JAP 46, 4986 (1975) on Josephson junctions coupled to a microwave stripline.

4) Such resonances raise the question whether the interaction of the Josephson junction with the electromagnetic environment could potentially be the source of the missing odd steps, a possibility which should not be ruled out a priori, but confirmed or disproved experimentally. In all reasonableness, compatible resonant conditions can hardly be envisioned given the wavelength (~30 cm) of the electromagnetic signal around 1 GHz . As pointed out in Table 1 of the Mudi & Frolov manuscript, these size considerations extend to most works as the studies were most likely performed in shielding enclosures of a few cm at most. Resonances at much higher frequencies (i.e.voltages) compatible with spatial dimensions of the measurement set-up have been correlated to features in the differential resistance of the devices (see SOM of Deacon et al., PRX 7, 021011 (2017)), but seem irrelevant to the studies of missing odd Shapiro steps.

5) An important argument in advocating in favor of Majorana modes is the frequency dependence. According to RSJ predictions, the coupled dynamics of cohabitating trivial and topological modes reveal the presence of the latter only at low frequencies. In the current model of Mudi & Frolov, it is clear that changing the frequency (hence the voltage height of the Shapiro steps) would lead to arbitrary missing steps, sometimes odd, sometimes even, sometimes a unique step, sometimes multiple steps, depending on the adopted profile of the differential resistance. Though Shapiro step patterns are sometimes messy or smooth, only missing odd steps have been reported. Besides, the suppression has been sometimes seen in a wide frequency range, wide enough to observe suppressed odd steps or even steps if there were produced by the mechanism discussed here by the authors.

The previous points have highlighted the relevance of taking into account non-linearities in the I-V curves of Josephson junctions. But they also clearly establish that the current manuscript by Mudi & Frolov is simplistic and partially inadequate. The model is in fact, without a proper physical justification, engineered to make steps disappear at desired locations, and it is thus not surprising that it succeeds. Doing, so, the authors ignore strong experimental facts which have been key to analyze the origin of missing odd steps in Josephson junctions. The model and its analysis is mathematically valid. It indeed provides « a possible explanation for the observation of missing Shapiro steps » as claimed, but it fails to connect to experimental observations. The results consequently have a limited reach, and are not sufficiently strong to disprove the possible topological origin of missing odd Shapiro steps.

Requested changes

1) None of the mentioned and well-known non-linearities are dealt with in the specific model studied in this manuscript in an adequate, microscopically justified manner. The authors should better explain, why a normal state resistance would (as a function of current) contain a series of peaks or clearly acknowledge that they are, in effect, proposing a new type of model and non-linearity, which is already present in the absence of a supercurrent.

2) The analysis of the frequency dependence of the Shapiro step suppression has been an important element of experimental and theoretical investigations. In the current model of Mudi & Frolov, it is clear that changing the frequency (hence the voltage height of the Shapiro steps) would lead to arbitrary missing steps, sometimes odd, sometimes even, sometimes a unique step, sometimes multiple steps, depending on the adopted profile of the differential resistance. I think the authors should modify the manuscript to better acknowledge this fact.

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

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