Mechanisms of Andreev reflection in quantum Hall graphene

Submission summary

 As Contributors: Anton Akhmerov · Antonio Manesco Arxiv Link: https://arxiv.org/abs/2103.06722v2 (pdf) Code repository: https://doi.org/10.5281/zenodo.4597080 Data repository: https://doi.org/10.5281/zenodo.4597080 Date submitted: 2021-10-01 11:00 Submitted by: Manesco, Antonio Submitted to: SciPost Physics Core Academic field: Physics Specialties: Condensed Matter Physics - Theory Condensed Matter Physics - Computational Approaches: Theoretical, Computational

Abstract

We simulate a hybrid superconductor-graphene device in the quantum Hall regime to identify the origin of downstream resistance oscillations in a recent experiment [Zhao et. al. Nature Physics 16, (2020)]. In addition to the previously studied Mach-Zehnder interference between the valley-polarized edge states, we consider disorder-induced scattering, and the previously overlooked appearance of the counter-propagating states generated by the interface density mismatch. Comparing our results with the experiment, we conclude that the observed oscillations are induced by the interfacial disorder, and that lattice-matched superconductors are necessary to observe the alternative ballistic effects.

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Submission 2103.06722v2 on 1 October 2021

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Report

This paper addresses the interesting problem of interference between chiral Andreev edge states,
the subject of several recent experiments. The main advance made here is to study the effect of
disorder. The authors argue that the signal seen in one of the recent experiments [Zhou et al., Ref.
16] is intrinsically due to disorder. The authors are certainly correct that disorder has a large effect
on the experimental results, though I am not persuaded that disorder is intrinsically necessary to
produce the experimental signal. Nevertheless, I recommend that this work be published by
SciPost. Before publication, however, there are a number of issues that should be addressed, as
follows.

1. A key aspect of the authors' argument is that the interface between the graphene and the
superconductor is smooth (i.e., in the absence of disorder). The authors should highlight this more
clearly in their text to ensure that the reader is fully aware that they are working in this regime.
They change the chemical potential smoothly over 50 nm which corresponds to 2-4 magnetic
lengths or about 35 tight-binding lattice constants. The authors give only one reference for this 50
nm number--Ref. 26. Since this is such a key part of the argument, it would be good to support
this point with more evidence or references, if possible.

2. The argument in the second paragraph of Sec. 2 is weak. The authors estimate the magnitude of
intervalley coupling using $m\sim vk_F$ and use the $k_F$ corresponding to the graphene
envelope function--that is, the deviation from the Dirac point. (Note that this is not a value for
$k$ in Eq. (2), which denotes the total crystal momentum, and so is somewhat confusing.) But
why not use the $k_{F}$ in the superconductor, which is order $1/a$ ? Because the gap in the
superconductor is much less than the cyclotron gap in the graphene, most of the quasiparticle
wavefunction is in the superconductor, so it seems to me that $k_{F,\text{super}}$ is more appropriate.
Using their argument, one immediately concludes $m\sim v k_0$, in contrast to
what they say.

3. When the superconductor is disordered (the most interesting case), are the results sensitive to
the magnitude of the disorder? How many different disorder configurations did the authors
investigate (approximately)?

4. Labeling the additional modes in Sec. 4 and Fig. 4 as "non-chiral" is premature. For a mode to
be non-chiral, the transverse wavefunction of the mode should be the same for the right-moving
and left-moving excitations. So, to make the point that the additional modes are non-chiral, the
authors should show that the transverse wavefunction is (approximately) independent of the
direction.

In the quantum Hall regime, the longitudinal wavevector of the excitation is directly connected to
its transverse position--in the semiclassical regime, the transverse position of the guiding center
is proportional to the longitudinal wavevector. In the dispersion in Fig. 4(a), the difference in $k$
between the various modes at the chemical potential seems to be pretty substantial-- $\delta k \,\ell_B$ does
not appear to be small. Thus, I would have thought that they are separated by a distance
of order $\ell_B$, and so are still chiral modes. Is this not the case?

Since the interference pattern produced by the additional modes shown in Fig. 4 is not very
strong, is it possible that the weak interference pattern is due to the modes remaining chiral?

5. In connection with Fig. 3(b), the authors state that they "observe that conductance in the
presence of edge disorder varies only slowly [Fig. 3(b)] because the edge states near the NS
interface still maintain a definite valley polarization, and ...". While this seems like a very plausible
explanation, the authors haven't actually shown that valley polarization at a disordered edge is
maintained. The sentence should either be rephrased as a likely explanation, or the authors should
show the valley-density for a disordered edge--it seems that they have this readily available
since they show it for a clean case in App. A.

6. The use of the word "exhaustive" in the conclusion is not warranted and should be removed:
"We believe that our analysis of the Andreev edge state scattering mechanisms is exhaustive, ...".
This is certainly not an exhaustive study; for instance, different disorder strengths have not been
studied, and the authors have not varied the smoothness of the interface. There are clearly quite a
few ways in which the work could be extended.

7. The conclusion about Ref. 16, Zhou et al., is, in my view, somewhat misguided. The authors are
accurate in their portrayal of the theoretical work in Ref. 16, which presents results only for the
clean (not disordered) case. However, the paper is primarily experimental, and the authors make
clear that disorder has a large effect on their experimental results. Note, for instance, these two
sentences from Zhou et al.:

"Experimentally, the beating pattern between the two CAESs is likely to be affected by multiple
parameters, such as the $\textit{interface roughness, disorder potential}$ electron density profile
near the contact and even positions of vortices in the superconducting contact (Supplementary
Figs. 5 and 6). As a result, the downstream resistance measured as a function of the gate voltage
acquires a pattern of $\textit{random but highly reproducible fluctuations}$ (Fig. 1c), in which the
signal is positive or negative depending on whether the super-conductor emits predominantly an
electron or a hole." (emphasis added)

From the presentation in the current manuscript, a reader would get the impression that there is
no awareness of disorder at all in Ref. 16, which is far from the case. I suggest that the authors
modify their references to Zhou et al. to correct this imbalance.

8. A clarification about labeling in two figures is needed. How does the region labeled "N" differ
from that labeled "QH" in Figs. 1 and 6? Does this mean that $B=0$ in the region labeled N? If so,
have the authors checked that the resistance arising from the change in magnetic field doesn't
influence the results?

9. Throughout the paper, insufficient information is given about the parameters used for the
numerical results. First, I did not see information about the geometry. The widths of the leads,
length of the interface, width of the disordered region in the superconductor, etc. should be give.

Second, are the parameters used for the conductance in Fig. 4 the same as for Fig. 2?

What is the value of $\Delta$ used in Fig. 5?

I assume the results in Fig. 3 are for a zigzag interface (with zigzag vacuum edges), is that right? In
that case the sketch in Fig. 3(a) is highly misleading and should be changed.

In Appendix C, what is the width of the superconducting region before the metallic lead is attached?

10. In several places there seems to be a typo in that both $B$ and $\Delta$ are nonzero for $x>0$. The
authors presumably mean that $B$ is nonzero in the QH graphene while $\Delta$ is nonzero in the
superconductor.

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Report

This work investigates the origin of Andreev interference in a superconductor-graphene heterostructure in the quantum Hall regime. The authors identify three mechanisms leading to deviations from constant conductance. They found that while lattice and Fermi level mismatch at perfect interfaces generate a regular interference pattern in nonlocal conductance, an irregular interference pattern occurs only in the presence of a strong disorder. This latter effect allows them to explain the origin of irregular interference patter in a recent experiment, Ref.[16].

I find the paper very interesting, clearly written, and timely. For these reasons I recommend its acceptance for publication in the current form.

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