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Bloch-Lorentz magnetoresistance oscillations in delafossites

by Kostas Vilkelis, Lin Wang, Anton Akhmerov

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Anton Akhmerov · Kostas Vilkelis
Submission information
Preprint Link:  (pdf)
Code repository:
Date accepted: 2023-05-22
Date submitted: 2023-03-31 14:01
Submitted by: Vilkelis, Kostas
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


Recent measurements of the out-of-plane magnetoresistance of delafossites (PdCoO$_2$ and PtCoO$_2$) observed oscillations closely resembling the Aharonov-Bohm effect. Here, we show that the magnetoresistance oscillations are explained by the Bloch-like oscillations of the out-of-plane electron trajectories. We develop a semiclassical theory of these Bloch-Lorentz oscillations and show that they are a consequence of the ballistic motion and quasi-2D dispersion of delafossites. Our model identifies the sample wall scattering to be the most likely factor limiting the visibility of these Bloch-Lorentz oscillations in existing experiments.

Published as SciPost Phys. 15, 019 (2023)

Author comments upon resubmission

We thank the referees for their feedback. The summary of the revisions is in the list of changes below.

We would also like to directly respond to some of the referees' objections. While the referees raise no concerns regarding the technical validity of our work, they question the impact of our work and its suitability to SciPost physics. Before responding to the detailed points raised by the referees, we would like to address this general evaluation.

The manuscript by Putzke et al. reports an "extraordinary" long range and high temperature quantum phase coherence in delafossites. This claim is central to the manuscript: it is the focus of the popular summary, abstract, and the conclusion. For example:

  • The abstract states "These results demonstrate extraordinary single-particle quantum coherence lengths in delafossites."
  • In explaining the new finding of the manuscript, the second paragraph states "Here, we report an unexpected robust manifestation of phase coherence intrinsic to the out-of-plane transport" and lists no other new observations.
  • The final sentence of the paper is "As quantum coherence emerges as its own subject in technology, it will be interesting to explore whether applications can exploit the rare macroscopic, single-particle phase coherence in the delafossites."

That this claim is indeed viewed as extraordinary by the community is confirmed by the manuscript being published in Science (the other properties of delafossites reported in the manuscript were known before). The reasoning of the manuscript critically relies on semiclassical explanations of the phenomenon not existing. In fact, the authors examine several possible explanations and rule them out, concluding that no semiclassical explanations exist.

Because our work proposes such a semiclassical explanation, it removes the need to explain the unexpected behavior. Our model requires no additional assumptions beyond established facts about delafossites, namely their high ballistic mean free path. This is the main impact of our work, and the reason why we consider it relevant to the community.

In the resubmitted version we have clarified this relation between our result and the Ref. 12.

Response to specific referee remarks

I have little of substance to add to the report of the other referee. At the same time, it strikes me that the present manuscript overstates the contrast to the publication by Putzke, and that there may be an issue of wording which differs between communities.

Concretely, the authors state that "our formalism does not rely on phase coherence". I suspect that many physicists would consider any interference phenomenon to be based on phase coherence on some level.

Our statement was indeed not technically accurate: we do assume that the band structure of the material exists, and that is a phase coherent phenomenon. We, however, disagree with the referee's assessment that we overstate the contrast: our model requires being able to resolve the bandwidth of the interlayer hopping $t_z \approx 10 \textrm{ meV} \sim 100 \textrm{ K}$, as opposed to assuming phase coherence over the scale of 10 microns at similar temperatures. In the updated version of the manuscript we stated specifically that our explanation does not rely on phase coherence of trajectories traversing the sample.

I had a similar feeling of unease concerning the statement that “Our result also bears physical interest because it is a rare example of a case where ℏ appears without interfering paths.” On some level, I can think of many instances where hbar appears without the need to invoke interfering paths, at least (and perhaps at most) superficially.

While we do not see a problem with the statement—in most cases $\hbar$ does appear in presence of interfering paths, we have removed the statement from the updated manuscript because it is not central to our claim.

In a similar vein, I am not sure I can make any real sense of the authors' statement that "The possible semiclassical origin of the oscillations was overlooked in the previous analysis. "

In the work by Putzke et al, the central long-range phase coherence claim relies on the absence of semiclassical theories which would explain the observed phenomenon. Below we quote verbatim the paragraph from the manuscript, where the authors rule out possible semiclassical origins of the observed phenomenon.

The few known B-linear oscillatory phenomena in singly connected solids are semiclassical [Sondheimer resonances (16), Azbel-Kaner cyclotron motion (17), or geometric resonances in the presence of acoustic waves]; rely on a superconducting order parameter to establish macroscopic phase coherence [Fraunhoffer interference in Josephson junctions (18)] or exploit artificially introduced nanometric length scales or interference effects in tunneling between parallel quantum wires or wells (19). None of these can explain our data because the observed h/e periodicity clearly indicates long-range, single-particle-phase coherence as their origin.

This quote fully substantiates our claim.

It seems to me that much of the distinction built up between Putzke et al and the present manuscript is semantic rather than substantial.

The claim of "extraordinary" long range phase coherence due to the observations that cannot be explained by a semiclassical mechanism and a semiclassical explanation of the same observation are substantially different.

I think the question of the relevance of this semiclassical mechanism in the delafossites considered in Putzke et al. is still open, but would likely require further experiments to properly distinguish between the Kubo mechanism and semiclassical description set out in this paper.

We agree that a further experiments would be required to determine whether delafossites have such long range phase coherence. For example, we argue in our latest work (arXiv:2303.04310) that if delafossites have long-range coherence then diamagnetic oscillations with the same Aharonov-Bohm period should appear.

As the authors pointed out, the mechanism is also somewhat similar to a paper by Pippard from 1966 (Ref. 13). Although I think the manuscript will be of some interest to those working on the delafossites, I do not see significant interest for a general audience that would normally be expected of a manuscript in the flagship SciPost Physics journal. I therefore think the manuscript would be better suited to SciPost Physics Core.

The claim of unusual phase coherence is interesting to a broad physics community. The work by Putzke et al. was clearly evaluated to be of broad interest because it is published in Science. Our work invalidates a critical logical step supporting this claim, which is the main reason why we believe it fulfills the requirements of SciPost Physics.

List of changes

* In the introduction, we add Ref[13] to show how unusual high-temperature long-range coherence claimed by Ref[12] is.
* Section 2 added the temperature upper bound for which the derived semiclassical result is valid.
* "Bloch-Lorentz oscillations" term defined at the end of section 3.1.

Reports on this Submission

Anonymous Report 2 on 2023-4-6 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2012.08552v4, delivered 2023-04-06, doi: 10.21468/SciPost.Report.7016


In their response arguing for publication in Scipost Physics the authors’ put a heavy emphasis on the fact Putzke et al. was published in Science. Stating their “work invalidates a critical logical step supporting this claim, which is the main reason why we believe it fulfills the requirements of SciPost Physics.” If the authors truly believe that the claims of the Science paper are absolutely not correct, then the appropriate mechanism would be to write a comment on that paper.

In my mind the authors’ simply provide a (technically correct) semiclassical calculation, very similar to Pippard 1966, which may or may not have anything to do with the Putzke experiment. If the authors wanted to make such a strong contrast with the experiment, a constructive course of debate would have been to state the advantages/disadvantages of each explanation and then to suggest experiments to distinguish between the mechanisms. In contrast, the authors now stating that the existence of their explanation “invalidates” the original explanation seems premature without further experimental evidence. (I note that the authors have a new preprint in this direction, but I can only judge the paper that I was first sent to review almost 2 years ago).

The one piece of evidence the authors claim supports their explanation is that the coherence length measured using SdH oscillations is an order of magnitude shorter than observed in B-periodic oscillations. However, Putzke et al. already explain that the relevant coherence length for SdH oscillations is averaged over the full hexagonal FS whereas the B-periodic oscillations only arise from the flat portions of the FS. This seems a quite reasonable explanation to me that is backed up by the results of Nandi et al. 2018.

To conclude: I do not think that this paper “invalidates” the previous Science paper. It does provide an explanation that requires further experiments to distinguish between the original explanation and this one. Therefore, as previously stated, it is certainly of interest to the Delafossite community, but until such experiments are performed it is of limited interest to those outside of the community (especially as the mechanism was already discussed by Pippard). I stick with my original suggestion that this paper should be published in SciPost Physics Core.

A final thing, I note in the latest version the final sentence has a typo “. modelingling”.

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Anonymous Report 1 on 2023-4-5 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2012.08552v4, delivered 2023-04-05, doi: 10.21468/SciPost.Report.7006


Of the three statements I criticised, one has been withdrawn, the other corrected, and the third is now emphasized with increased vigour.

The latter point strikes me as reasonable, in that the authors take up a form of words utilised in the publication which motivated their study. I had not properly appreciated this aspect in my preceding report.

In my preceding report, I had made the following suggestion: "It seems to me that much of the distinction built up between Putzke et al and the present manuscript is semantic rather than substantial. I would leave it up to the authors how to deal with this. 1- But my recommendation would be to desist from building up expectations about deep insights which dissolve upon closer inspection."

I think how the authors have dealt with the first statement has addressed that point. Regarding the second, their response states:

"our model requires being able to resolve the bandwidth of the interlayer hopping tz≈10 meV∼100 K, as opposed to assuming phase coherence over the scale of 10 microns at similar temperatures. "

I am unable to process the logical status of the 'as opposed' in the middle of the sentence, in as far as I do not see an opposition between the two statements. This reinforces my impression that the thinking of the authors contains further assumptions and/or semantics which are not shared by me, and perhaps other members of the community.

But I think this refereeing process has now run its course, and I am happy to let the readers make up their own mind, as the strong points of the manuscript in my eyes are clearly in favor of publication.

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