Model for missing Shapiro steps due to bias-dependent resistance

Dear Editor,

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. 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). One aspect is that it is
indeed true that studies of Shapiro steps with LZT, Andreev etc are somewhat limited.
This in my view is because of the general complexity of the Shapiro step process.
However, this also leads to the possibility of describing Shapiro steps with models
with many parameters such as frequency dependence of current phase relation, impedance,
higher harmonics in both the current phase relation as well as resistance as considered
in the current work. In general, the Shapiro step in current bias devices are significantly
complicated to model. Broadly speaking, while it is possible that the present manuscript
makes a significant advanced to the large list of works in this complicated area
many of the statements made in the manuscript are either not precise or speculative.
I think these need to be clarified with appropriate references before the manuscript
can be properly evaluated. Below I present a list of these aspects that need
clarification:

(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.

(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".

(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.

(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.

(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.

In summary, I cannot recommend the current work for publication because of the ambiguity
in much of the presentation together with what aspect of Shapiro steps is clarified by
this model. This is specially a problem given that there are already several
theoretical and experimental works showing that the ac Josephson effect and Shapiro steps
can occur in non-topological systems. The Shapiro step is also known to be quite a
complex problem because of the feedback of the fluctuations of the current into the
phase of the superconductor. The manuscript requires clarification of the context
of the model relative to existing theory (for example, I can see it being complementary)
as well as relative to the experiments discussed currently in rather imprecise terms.