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Bloch-Lorentz magnetoresistance oscillations in delafossites
by Kostas Vilkelis, Lin Wang, Anton Akhmerov
|As Contributors:||Anton Akhmerov · Kostas Vilkelis|
|Arxiv Link:||https://arxiv.org/abs/2012.08552v2 (pdf)|
|Date submitted:||2021-07-12 11:30|
|Submitted by:||Vilkelis, Kostas|
|Submitted to:||SciPost Physics|
Recent measurements of the out-of-plane magnetoresistance of delafossites (PdCoO$_2$ and PtCoO$_2$) observed oscillations which closely resemble the Aharanov-Bohm effect. We develop a semiclassical theory of these oscillations and show that they are a consequence of the quasi-2D dispersion of delafossites. We observe that the Lorentz force created by an in-plane magnetic field makes the out-of-plane motion of electrons oscillatory, similarly to Bloch oscillations. The visibility of these Bloch-Lorentz oscillations is limited by sample wall scattering. Therefore, the aspect ratio of the sample controls the intensity of scattering. Our theory offers a way to design an experimental geometry that is better suited for probing the phenomenon and to investigate the out-of-plane dynamics of ballistic quasi-two-dimensional materials.
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Anonymous Report 1 on 2021-8-29 (Invited Report)
Vilkelis et al. develop a semi-classical theory for the the B-linear oscillations of magnetoresistance observed in the delafossites. This is primarily motivated by the very interesting recent work on PdCoO2 by Putzke et al., Science 368 1234-1238 (Ref. 12 in the manuscript). The basic mechanism postulated by Vilkelis et al. is explained in Fig. 2 a-c: Due to the low disorder in delafossites, electron trajectories out of the plane x-y plane can be commensurate with the sample width (A in Fig. 2) such that a minimum in z-conductivity occurs or in the opposite case of half-integer period (B in Fig. 2) the maximum of conductivity occurs. The authors then also discuss various scattering mechanisms that they claim can more accurately reproduce the experimental data. They also discuss the impact of tilting the magnetic field.
I find the manuscript is mostly well written (with a few minor typos e.g. lenght) and the method of calculation is clear. The calculation is relatively straightforward and appears technically correct for the assumptions they make but I have several concerns whether it actually applies to the system in question (or any other system), I outline these below. I must say from the start that — whilst I do believe that the work is potentially publishable in some form — unfortunately I do not think that it meets the high quality, originality, or interest standards that I would expect of SciPost Physics. I find the ultimate message of the current manuscript rather confused. I am therefore very cautious to recommend publication in any SciPost journal until the authors respond to my queries and much more clearly explain the points of agreement and disagreement with theory and experiments of Putzke et al. in order for the reader to understand what is gained by this study.
Outline of my main questions/concerns:
1) I find the motivation for the manuscript rather confusing. The authors claim that Putzke et al.’s theory “reproduces the results but offers limited insight into the nature of the phenomenon”. To be quite frank I think this statement misleading. Putzke et al. explain clearly the underlying physics of their theory in terms of a diffraction-like grating effect of the hopping between adjacent layers, based on this picture they then perform theoretical calculations using the Kubo formula and it fits very well with their experimental findings (Fig. 4 of Ref. 12). Putzke et al. do admit they are open to alternative explanations at the end of their manuscript, however, the current authors should clearly explain:
i) What additional mechanistic insights are gained from their semi-classical explanation?
ii) What experimental observations can their calculation explain which the mechanism of Putzke et al. cannot?
iii) Conversely, are there aspects of the experiment that Putzke et al.’s theory can better explain?
iv) Do the two theoretical pictures connect with each other or are they contradictory?
2) The authors admit that a ballistic mean free path e.g. the ~ 20 um found in Ref.  would rule out their mechanism since the oscillations suggested by the authors would become “fully visible” in such a case. The authors then propose a workaround which is scattering from the sample walls. I think the authors should explain the basis on which they claim this is the “dominant mechanism” for scattering. To be precise, do they believe there is any experimental data in Putzke et al. to support this assertion? Further, how do the authors reconcile their claim with the experimental observation (pg. 4 end - start pg. 5 of Ref. 12) that in samples where the mfp was reduced from 20 um to 1 um by irradiation oscillations were not visible in 8 um wide samples but became visible only when the sample was narrowed to 1 um (i.e. ballistic)? (If boundary scattering is dominant then why does a narrower sample exhibit the oscillatory behaviour but a wider one does not?)
3) The authors have a discussion on titled magnetic fields which concludes that the fact that oscillations disappear when the cyclotron orbits fit into the sample. I was surprised to see that at no point in section 3.3 do the authors mention that Putzke et al. also discuss this and indeed show very nice data of the transition from linear in B oscillations to SdH like oscillations when the field angle is tilted. The authors should outline why their discussion on this matter differs from Putzke et al. or, if the mechanism is the same (which is appears to me they are), they should acknowledge this fact.
4) At no point is the reader shown what “full visibility” of the purported oscillations in conductance look like for a clean system, how the oscillations looks with decreasing mfp, or the difference in appearance of the oscillations with/without the dominant boundary scattering. All of these are key messages of the paper.
5) The authors state: "The extension of the theory to a realistic sample geometry is straightforward... but it strongly increases the computational costs and therefore we consider it unjustified for the purposes of our study." I do not understand why the numerical integration of Eq. (29) becomes so costly for a realistic sample geometry and, since the whole message of the manuscript is that the the authors can example the experimental data, it seems strange to claim it is unjustified to show their calculation for the correct experimental geometry.
6) At the end of the abstract the authors state: "Our theory
offers a way to design an experimental geometry that is better suited for probing
the phenomenon and to investigate the out-of-plane dynamics of ballistic quasitwo-dimensional materials." However, I do not see any discussion of this in the main text. Do the authors simply mean that one should change the aspect ratio L/W to see clearer oscillations, as in Fig. 3?
To conclude: The authors make strong claims about their mechanisms relation to Putzke et al.'s experiments but it is unclear to me what is gained from their study compared to the previous proposed mechanism or how these relate. I do not see the significant advance in understanding of the Putzke et al. experiment which would warrant publication in SciPost Physics. Indeed it is not entirely clear to me whether this (relatively) simple calculation is even able to explain the experiment equally as well as the proposed mechanism by Putzke et al. without resorting to a scattering mechanism which seems unsupported by the actual experiment.