Landau levels, response functions and magnetic oscillations from a generalized Onsager relation

Dear editor,

Below we reply to the referee’s second report. All his/her remaining concerns are related to magnetic oscillations, which constitute the third part (section 5) of our article. We think that we have now correctly answered all his/her criticisms contained in the first and second reports.

Best regards, The authors

Referee’s second report The authors have addressed all but two of my concerns, which were labelled (4) and (5) in previous reports. I do not recommend publication until they are addressed.

4) The prevalent use of Landau plots does not guarantee their correctness. A Landau plot analysis (where only maxima or minima are plotted) presupposes that the fundamental harmonic is dominant. Such dominance occurs only for high temperature (kT >> cyclotron energy) and strong disorder (cyclotron frequency*lifetime << 1). In particular, it is only the phase offset of the fundamental harmonic that is equal to the

\lambda=Berry phase + correction due to orbital moment + Maslov correction.

The phase offsets of higher harmonics are generally integer multiples of \lambda, as shown in the Lifshitz-Kosevich formulae in Phys. Rev. X 8, 011027. In general, the presence of multiple harmonics prevent any naïve extraction of the phase offset from identifying maxima (or minima):

a) It is possible that the reported “nonlinearities” in Landau plots are simply because of ignored higher harmonics in the interpretation of experimental data. The importance of higher harmonics has been emphasized by Dhillon and Shoenberg in http://rsta.royalsocietypublishing.org/content/248/937/1

b) Even if the fundamental harmonic were dominant, the “nonlinearities” may be a result of Dingle damping or finite temperature; both effects result in the fundamental harmonic not being a simple sinusoidal function. The naïve method of extracting the phase offset presupposes (inaccurately) that the maxima of a sinuisoidal function is identified.

I recommend that the authors refine their Landau plot analysis and address (a-b) above. I am not against making simplifying assumptions, but in the current version of the manuscript the assumptions are not stated. One suggestion is to replace “In simple cases” with the assumptions of temperature and disorder stated above, and a direct statement that only the fundamental harmonic is assumed dominant. I would also like the authors to address point (b) above.

Our answer to (4): The main goal of our article is not to recall how a Landau plot should be obtained experimentally (which is well known). However, we hear the criticisms of the referee and reply below.

That the phase offset in the case of a higher harmonic is a multiple of gamma_0(epsilon) = Berry + orbital moment correction + Maslov [called lambda by the referee] appears in our paper as equation (74). We acknowledge the fact that this is also present in Phys. Rev. X 8, 011027 (2018) by adding a citation to this reference.

(a) Magnetic oscillations usually contain several harmonics. In our paper, we make the simplifying assumption that the oscillations are dominated by the fundamental one. This is a reasonable approximation in the low field limit, when the cyclotron frequency is small compared to the temperature or to the disorder broadening. As suggested by the referee, we now clearly state these assumptions in the paper.

(b) Even in the case of a single harmonic, the Lifshitz-Kosevich formula shows that the magnetic field dependance is not only in a sinusoid but also in reduction factors R_T and R_D that account for thermal and disorder damping of the oscillations. This is well known and the reduction factors are routinely used by experimentalists to extract the cyclotron mass and the elastic scattering time (in the case of graphene, see for example Y. Zhang et al., Nature 2005 https://www.nature.com/articles/nature04235 and M. Monteverde et al., PRL 2010 https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.104.126801). The main goal of our paper is not to recall the way a Landau plot should be obtained. However, as it seems to be an issue, we added a sentence to clarify that one should take the extrema of the oscillations after the extra magnetic-field dependence contained in the reduction factors such as R_T and R_D has been removed (i.e. one should take the extrema of the sinusoid only). Indeed, we are interested in sources of non-linearity in the Landau plot which are not related to thermal or disorder effects.

Referee: 5) It is a misconception that a contact (0D or 1D) can change the chemical potential (an intensive property) of a 2D or 3D system. Attachment of a contact results merely in the creation of a contact potential difference, which compensates for the difference in the work functions of the two contacting conductors. The chemical potential in the bulk oscillates according to the thermodynamics of an electrically neutral system (i.e. a system with fixed density - not number - of electrons). These oscillations, in turn, may result in the oscillations of the contact potential, as described in Section 4.4 of Shoenberg’s “Magnetic oscillations in metals” and in the references of

https://www.sciencedirect.com/science/article/pii/0039602896004815

Further transport evidence of the field-dependence of the chemical potential in 2DEG can be found, e.g., in

http://journals.jps.jp/doi/10.1143/JPSJ.77.064713

I recommend that the authors retract their statement that the chemical potential of 2D metals is fixed, and address directly my concerns (stated in the previous report) that derive from the field-dependence of the chemical potential.

Our answer to (5): After carefully re-reading the literature on magnetic oscillations in two-dimensional systems and interviewing colleagues (both experimentalists and theoreticians), we came to the conclusion that in 2D, nothing is simple and matter depend a lot on the precise experimental setup. Some systems are better described by a fixed density, some are better described by a fixed chemical potential, some are not described by these limits.

To contrast with the references given by the referee in his/her second report, we give below several references to the literature which support the opinion that in transport experiments with good contacts to the drain and source, the reasonable approximation is to assume that the chemical potential is fixed, not that the number of electrons or the density is fixed:

• we cite from the discussion section in Sharapov, Gusynin and Beck, PRB 2004 https://journals.aps.org/prb/abstract/10.1103/PhysRevB.69.075104 about magnetic oscillations in graphene : “While for dHvA effect the condition [electron density] = const. is more natural, it is plausible that SdH effect can be measured under condition [chemical potential] = const.”

• in the book by Shoenberg, there are also discussions about this issue in the case of 2D systems. On page 49 (in section 2.3.4 “Application to real 2D systems”): “It is not always clear whether the system more closely resembles one in which [the chemical potential] is constant or one in which [the density] is constant, as [the magnetic field] is varied.” Also page 157 on the quantum Hall effect in silicon MOSFET and in AsGa heterostructure: “These features can be most simply understood by supposing that for these samples it is [the chemical potential] rather than [the density] which remains constant as [the magnetic field] is varied, though this is probably a considerable oversimplification, and probably the interpretation for the MOSFET and for the heterostructure are somewhat different in detail, ...”

• actually most experiments on SdH oscillations in 2D metals make the implicit assumption that the chemical potential is fixed, not the density. See again the two references on graphene that we cited above and also in our previous response to the first report: Y. Zhang et al., Nature 2005 https://www.nature.com/articles/nature04235 and M. Monteverde et al., PRL 2010 https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.104.126801

In the end, we think that both assumptions (either constant chemical potential or constant density of electrons) are idealizations. Experiments are closer to one or the other limit depending on details of the setup and could actually be far from both limits. For example, in a transport experiment, contacts (to the drain and source reservoirs) can be ohmic (good contacts, chemical potential imposed by reservoir) or tunnel (bad contacts, almost isolated system, constant density of electrons). Both idealizations are equally questionable and there is no universal answer that would adapt to all experimental situations. In our work, for simplicity, we decided to stick to the grand canonical ensemble with fixed chemical potential. The fact that in some cases the chemical potential also oscillates with the magnetic field and renders the analysis of measurements more complicated need not concern us. We removed the statement about the chemical potential being fixed when the 2D system is contacted and rewrote the corresponding paragraph to clarify these issues and cite relevant references.

Referee: Minor comment: 1. In appendix C, it is claimed “generally” that the maximum of the longitudinal magneto-conductivity coincide with the maxima of the density of states. This claim is supported solely by reference 59 by Ando. However, I am a little puzzled by this citation. Ando’s work describes oscillations in the “transverse conductivity” (a different naming convention?). As far as I can see, Ando never directly claims that the maxima coincide, though he shows some suggestive numerical calculations, based on the self-consistent Born approximation and presumed short-range scatterers. All these restrictions suggest Ando’s claim is not “generally” valid. I recommend that the authors do more to support their “general” claim.

Our answer to the minor comment: Our claim is supported not only by ref 59 (Ando, see the post-scriptum below) but also by ref 60 (Coleridge et al., see equations 11 and 13). It is actually also supported by the reference (Endo and Iye, J. Phys. Soc. Jpn 2008) provided by the referee and from which we cite: “Our result suggest the relation [relative change in resistance proportional to relative change in density of states] remains valid regardless of the magnitude of [relative change in density of states]”.

A qualitative argument (attributed to Pippard 1965) is given in the book of Shoenberg in the SdH section 4.5 page 153: “He pointed out that the probability of scattering is proportional to the number of states into which the electrons can be scattered, and so this probability, which determines the electron relaxation time tau and the resistivity, will oscillate in sympathy with the oscillations of the density of states at the Fermi energy”. As an order of magnitude estimate, Shoenberg states on page 154 that the relative oscillations in the conductivity are given by the thermal reduction factor R_T multiplied by the relative oscillations in the DoS. This is essentially the content of equation (13) in ref 60 by Coleridge et al. There are many more similar statements in the book by Shoenberg. For example, in section 2.5 page 69: “… the oscillations of the density of states are closely related to the oscillations of resistivity (Shubnikov-de Haas effect)...”.

In the end, we do not have a mathematical proof but we believe that the statement is qualitatively and generally correct. We added two citations in the paper to give further support to our claim.

Concerning Ando’s paper (ref 59), transverse conductivity indeed means xx conductivity (usually called longitudinal) but it is transverse/perpendicular to the magnetic field hence the name "transverse conductivity" in this reference. The oscillations in the DoS (i.e. ImX=X’’, see equation (2.5)) are given in equation (2.14) and that in sigma_xx in equation (2.15). The comparison between the two equations shows that the maxima of the longitudinal magneto-conductivity coincide with the maxima of the density of states.