Model for missing Shapiro steps due to bias-dependent resistance

The manuscript titled "Model for missing Shapiro steps due to bias-dependent resistance" presents "a phenomenological model that displays suppression of the odd Shapiro steps. We perform resistively-shunted junction model calculations and introduce peak in differential resistance as function of the bias current." The manuscript is motivated by the fact that "missing steps were observed not only in non-trivial Josephson junctions but also in trivial Josephson junctions." As summarized in the motivation, the state of the literature maybe used to conclude "these developments indicate that missing Shapiro steps, or even 4pi periodic Josephson effects themselves, may not unambiguously identify a Majorana regime,". This in my view somewhat weakens the motivation for why the community should be interested in modeling Shapiro step experiments in more detail as presented in this work.

Just because Shapiro step experiments cannot be used for uniquely identifying a Majorana regime does not mean that it does not call for further investigation. On the contrary, it is important that we model Shapiro steps to understand why they may not be used as an indicator for 4pi periodicity as has been used in several previous experiments. It is also important to do so to understand the different reasons (other than 4 pi periodicity) that missing steps may show up in data obtained from (trivial) systems.

In fact, I would think that the more experimentally difficult, but simpler to interpret Josephson radiation in Refs 9 and 10 has also been subject to the possibility of a LZT interpretation, which can arise from Andreev bound states as shown in Physical Review B 95.6 (2017): 060501. Incidentally, Andreev states are listed as part of the motivation for the phenomenological model (Eq. 1) presented in this work. The motivation mentions that the possibility of ruling out LZT (and I imagine associated Andreev state based mechanisms) based on power and rf dependence is the motivation of the phenomenological model considered in this work. However, I was unable to find evidence in the manuscript that the present model (Eq 1) could explain the general power and frequency dependence of Shapiro step experiments better (i.e. without introducing additional parameters).

Since we posted this numerical simulation paper, we have also done experiments where we experimentally mimic the frequency dependence of missing Shapiro steps under trivial conditions. https://scipost.org/10.21468/SciPostPhys.18.6.203 The referee is correct, what is needed is fine-tuning, i.e. the introduction of additional parameters. With the help of this, we can make any steps disappear, not just the odd ones. We can then select data such that the reader sees the pattern that matches the most interesting case. Since our work has been public for some time, the group that originally reported missing odd Shapiro steps have also come up with a paper indicating a trivial explanation for those phenomena. https://www.nature.com/articles/s41467-025-58299-z Given this realization, there is no need to invoke MSLZ physics to explain any of the experiments done so far. But it is an interesting piece of physics which can perhaps be experimentally proven more conclusively in the future.

We invoke ABS in a very narrow context that they can produce peaks in current-voltage characteristics of superconducting devices, and we model such peaks as peaks in differential resistance without looking into the microscopics of it.

(1) In the paragraph "In this paper, we are considering nonlinearities at finite bias which 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." it is unclear what resonances are. The dynamical current response of a generic Josephson junction is a complex function of both frequency as well as current, which contains MARs and ABSs. Why these are captured even qualitatively by the model in Eq. 1 is unclear. For example MARs and ABSs are associated with changes in occupation of ABSs. I would expect these to be modeled as changes in the supercurrent as in Ref 17.

We are using “resonances” as any peaks or dips in differential resistance that often show up in data obtained from Josephson junctions (we have added this clarification now). Such sharp and non-monotonic features in transport data are commonly referred to as “resonances” because they often correspond to resonantly-enhanced transmission, be it resonant tunneling or a process assisted by a photon or a phonon.

In our paper, as we have mentioned in the motivation section, we are using a phenomenological model that doesn’t look at the origin of these peaks, rather how these resistance peaks would affect Shapiro steps if they were present.

In typical DC measurements of Josephson junctions, where the differential resistance of the junction is measured as a function of the current or voltage bias, MARs or ABSs can show up as peaks or dips in resistance. Therefore, it is justified to model them as per eqn 1.

(2) The model in this work in Eq. 1 effectively introduces a multi-parameter model by introducing multiple peaks in the function R(I) to match the sizes of the Shapiro steps. The motivation for this presumably comes from comments such as "we analyze data from several experiments where current-voltage characteristics of Josephson junctions exhibit sharp features in differential resistance" in the discussion of relevant experiments. Unfortunately, this and the discussion that follows does not clarify the situation. For example it is not explicitly stated what the connection between the R(i) and "resonance voltages" or "resonances above the switching current".

R(I) mimics the resistance peaks/resonances above the switching current in the experimental data. To study the experimental feasibility of these resonances affecting Shapiro steps (typically performed at a frequency of a few GHz), we need to determine the frequency corresponding to these resonances, i.e. resonances occurring at lower frequencies may affect the first few Shapiro steps while resonances at higher frequencies are not relevant. The relevant experiments (shown in Table1) obtained data in current bias, not voltage bias. Therefore, to estimate the frequency corresponding to resistance peaks, we first extract the current bias at which these resistance peaks occur. The ‘resonance voltage’ is the measured voltage at these resistance peaks. We then use the second Josephson relation V = nhf/2e to estimate the characteristic frequency corresponding to these resistance peaks or resonances. We have now added this explanation to the paper.

(3) Starting with the sentence "Shapiro steps data are often presented in published works in histogram view." some of the data in this work is presented in histogram view where "Each narrow voltage bin counts how many data points fall in that interval of voltage.". What is narrow is not specified. In fact what is a data point is not specified either. While this may be gleaned from some of the experimental paper appropriate references for where to find this is missing. In addition, even if this information is present in some of the other experiments - a manuscript should be self-contained for the key information needed to understand plots. As such these details need to be repeated in this manuscript.

To plot the histograms, which are V vs RF power vs bin counts plots, we look at the I-V characteristic at each rf power. At each rf power, we divide the entire voltage range of the Shapiro steps into smaller sections i.e. voltage bins. We count how many current points i.e. data points fall into each voltage bin (bin counts). We have now added this clarification to the paper.

(4) A significant part of the discussion of experiment is "speculative" for "However, we speculate that steps at low frequencies, themselves not sharp, may be susceptible to suppression even by slightly non-monotonic features in the current-voltage characteristics, so that there is no need for tall peaks in R(idc).". In this case "slightly non-monotonic features", "need for tall peaks" are not precise in any meaningful sense e.g. how are "slight" and "tall" defined. Similarly the sentence "A voltage suddenly develops across the junction which may be the equivalent of several Shapiro steps, making the low voltage regime inaccessible." - what "suddenly" is is quite unclear.

It is better to be up-front about which parts are speculative. We speculate there because there is no straightforward way to calculate exactly how much the resistance should change as it depends on several experimental factors such as Ic, Rn, HWHM of the peak, etc. However, at low frequencies, the change in voltage between consecutive steps is smaller compared to that at higher frequencies. Therefore, to suppress steps at lower frequencies, a smaller change in the background resistance could be sufficient compared to that required at higher frequencies. We have rephrased the statement which we hope will help clarify it: “However, we speculate that steps at low frequencies, themselves not sharp, may be susceptible to suppression by peaks of small height and HWHM compared to those required at higher frequencies.” We have also edited the statement regarding the inaccessibility of the low voltage regime: “As the junction switches to the normal state, the voltage across the junction jumps from zero to a finite voltage. For example, if the switching current is of the order of 1 uA and the normal state resistance is a 100 Ohms, then the voltage across the junction will be 10 uV, making the low voltage regime inaccessible.”

(5) Other examples of unclear sentences are: "the regime of the first few Shapiro steps is not accessible from the data, especially when the applied microwave frequency is low." Does regime simply mean that the first few Shapiro steps are not accessible? Probably not. The statement "Most of the related experiments do not have the resolution to identify differential resistance peaks that may be present at such low bias voltages." should come with citations to which experiments are being referred to. It is also not clear what is meant by not having enough resolution.

We have edited the corresponding sentences in the main text as follows: “First, the voltage corresponding to the first few Shapiro steps at low frequencies is very small, of the order of few microVolts. For example, at f = 1 GHz, the voltage of the first Shapiro step ~ 2 uV. To identify resistance peaks occurring at a voltage of a few microVolts, the experiments must have a voltage resolution smaller than that. This is not the case for most relevant experiments as can be seen in Table 1, where most estimated resonance voltages are much larger than a few microVolts.”