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Axion Mie Theory of Electron Energy Loss Spectroscopy in Topological Insulators

by Johannes Schultz, Flavio S. Nogueira, Bernd Büchner, Jeroen van den Brink, Axel Lubk

Submission summary

As Contributors: Axel Lubk · Flavio Nogueira · Johannes Schultz · Jeroen van den Brink
Arxiv Link: https://arxiv.org/abs/2002.03804v4 (pdf)
Date accepted: 2021-09-06
Date submitted: 2021-08-24 10:42
Submitted by: Schultz, Johannes
Submitted to: SciPost Physics Core
Academic field: Physics
Specialties:
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical

Abstract

Electronic topological states of matter exhibit novel types of responses to electromagnetic fields. The response of strong topological insulators, for instance, is characterized by a so-called axion term in the electromagnetic Lagrangian which is ultimately due to the presence of topological surface states. Here we develop the axion Mie theory for the electromagnetic response of spherical particles including arbitrary sources of fields, i.e., charge and current distributions. We derive an axion induced mixing of transverse magnetic and transverse electric modes which are experimentally detectable through small induced rotations of the field vectors. Our results extend upon previous analyses of the problem. Our main focus is on the experimentally relevant problem of electron energy loss spectroscopy in topological insulators, a technique that has so far not yet been used to detect the axion electromagnetic response in these materials.

Published as SciPost Phys. Core 4, 023 (2021)



Author comments upon resubmission

Dear Dr. Attaccalite,

Unfortunately Prof. Javier García de Abajo has not accepted the invitation. He initially confirmed our request to review the manuscript, but he is currently not reachable. We have tried to recontact him directly multiple times, but unfortunately got no response. We have no explanation and hope he is doing well. Although we can not understand the referees criticism fully, we now tried to address the requested changes diligently (you can find a version in which changes to the manuscript have been marked red under https://seafile.ifw-dresden.de/f/9b5c16daec874984b302/). We added a more precise description of the inhomogeneous case as requested by the referee. Please note that we already referred to [1] in the previous version, which provides a detailed and unequivocal derivation of the inhomogeneous Mie problem (without axion contribution) and declaration of the applied approximations. Concerning the homogeneous problem, we have cited once more previous works on Mie scattering on TI spheres, and set our work more detailed in context to them. We have to reject, however, the request by the referee to remove the chapter containing Eqs. (25)-(39) for several reasons:

I) From our point of view the claim of the referee "(The “homogeneous TI sphere” problem has no novelty in it.)" is not correct. As pointed out already in the previous response letter there are several novelties in our treatment: - we provide an explicit and exact expression of the scattering matrix elements - we fully incorporate the axion term and do not apply a pertubation approach like e.g., Ge et al. [2] - we use a complete set of alternative vector spherical harmonics (VSH) which are, contrary to the claim of the referee, useful and necessary to solve the complete inhomogeneous problem, including arbitrary external charges and currents. The referee ignored our arguments completely and just claimed that all those aspects are not new without providing any references.

II) In condensed matter theory it is common practice to set novelties in context to previous works and recapitulate already treated points in order to make the subject more accessible to a broad audience of physicists. While the Mie theory is well known for the plasma physics and plasmonics community, it is less familiar to condensed matter physicists.

III) As already discussed in the previous version, the solution of the homogeneous equations are used to approximately solve the EELS problem ("one can directly adopt the results of the homogeneous calculations..."). Hence, the removal of Eqs. (25)-(39) would result in a less sound narrative regarding the derivation of the EELS solution.

In view of the critique by the referee on the lack of novelty, we elaborate further on the significance of the inhomogeneous solution and the role of the three-component set of VSH employed in the manuscript. The classical Mie approach considers scattering of transversal electromagnetic waves (i.e., in the absence of external charges and currents), which directly leads to a coupling of two components of the VSH. Consequently, only homogeneous transverse modes are possible and the solution space is restricted to such solutions, which can be exploited by using another set of strictly transverse two-component (poloidal and tororidal) VSHs, as advocated by the referee. To illustrate the lack of modes in case of free wave excitations, we performed boundary element simulations of the plasmonic response of a gold sphere with respect to a plane wave excitation, and an excitation by the evanescend field of a sharply focused electron beam, corresponding to the inhomogeneous case ( e.g., in EEL spectroscopy). Please see https://seafile.ifw-dresden.de/f/1cbb0bb930504473a9af/ for details and the resulting cross sections / spectra. One clearly observes that the mode below 0.5 eV only occurs in the full solution of the EELS problem and not in the homogeneous problem (i.e., plane wave excitation). Moreover, the low energy mode is also absent in the inhomogeneous Mie solution restricted to transverse mode solution space. Consequently, this results corroberates the importance of a complete three component set of VSH to solve the EELS problem completely. It also demonstrates that the restriction of the inhomogeneous case to transverse solutions correctly reproduces the transverse modes of the full solution, which we exploited in the dedicated EELS section of the paper. All in all we endeavored to address the criticized points and hope to fulfill now the criteria for publication in SciPost Core.

Faithfully yours, Axel Lubk

[1] F. G. de Abajo, Relativistic energy loss and induced photon emission in the interaction of a dielectric sphere with an external electron beam, Physical Review B 59(4), 3095 (1999)

[2] L. Ge, D. Han and J. Zi, Electromagnetic scattering by spheres of topological insulators, Optics Communications 354, 225 (2015)

List of changes

1. We added a more precise description of the inhomogeneous case.
2. We have cited previous works on Mie scattering on TI spheres, and set our work more detailed in context to them.

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Comments

Anonymous on 2021-09-03  [id 1731]

Category:
remark
validation or rederivation

Inconsistency: Markdown and reStructuredText syntaxes are mixed:

Markdown: ('unordered_list', <re.Match object; span=(1383, 1939), match='- we provide an explicit and exact expression of >)

reStructuredText: ('header', <re.Match object; span=(232, 254), match='\r\n------------------\r\n'>)

For submitted/published material: please contact techsupport@scipost.org to point out this error


Hereafter
I publish, in agreement with the authors, the re-submission letter for the last version of the manuscript.
In such a way that this discussion becomes part of the public record.

Editor-in-charge
Claudio Attaccalite,

------------------

Dear Dr. Attaccalite,

Although we can not understand the referees criticism fully, we now tried to address the requested changes diligently (you can find a version in which changes to the manuscript have been marked red under https://seafile.ifw-dresden.de/f/9b5c16daec874984b302/). We added a more precise description of the in-homogeneous case as requested by the referee. Please note that we already referred to [1] in the previous version, which provides a detailed and unequivocal derivation of the inhomogeneous Mie problem (without axion contribution) and declaration of the applied approximations.
Concerning the homogeneous problem, we have cited once more previous works on Mie scattering on TI spheres, and set our work more detailed in context to them. We have to reject, however, the request by the referee to remove the chapter containing Eqs. (25)-(39) for several reasons:

I) From our point of view the claim of the referee "(The “homogeneous TI sphere” problem has no novelty in it.)" is not correct. As pointed out already in the previous response letter there are several novelties in our treatment:
- we provide an explicit and exact expression of the scattering matrix elements
- we fully incorporate the axion term and do not apply a pertubation approach like e.g., Ge et al. [2]
- we use a complete set of alternative vector spherical harmonics (VSH) which are, contrary to the claim of the referee, useful and necessary to solve the complete in-homogeneous problem, including arbitrary external charges and currents. The referee ignored our arguments completely and just claimed that all those aspects are not new without providing any references.

II) In condensed matter theory it is common practice to set novelties in context to previous works and recapitulate already treated points in order to make the subject more accessible to a broad audience of physicists. While the Mie theory is well known for the plasma physics and plasmonics community, it is less familiar to condensed matter physicists.

III) As already discussed in the previous version, the solution of the homogeneous equations are used to approximately solve the EELS problem ("one can directly adopt the results of the homogeneous calculations..."). Hence, the removal of Eqs. (25)-(39) would result in a less sound narrative regarding the derivation of the EELS solution.

In view of the critique by the referee on the lack of novelty, we elaborate further on the significance of the inhomogeneous solution and the role of the three-component set of VSH employed in the manuscript. The classical Mie approach considers scattering of transversal electromagnetic waves (i.e., in the absence of external charges and currents), which directly leads to a coupling of two components of the VSH. Consequently, only homogeneous transverse modes are possible and the solution space is restricted to such solutions, which can be exploited by using another set of strictly transverse two-component (poloidal and tororidal) VSHs, as advocated by the referee. To illustrate the lack of modes in case of free wave excitations, we performed boundary element simulations of the plasmonic response of a gold sphere with respect to a plane wave excitation, and an excitation by the evanescend field of a sharply focused electron beam, corresponding to the inhomogeneous case ( e.g., in EEL spectroscopy). Please see https://seafile.ifw-dresden.de/f/1cbb0bb930504473a9af/ for details and the resulting cross sections / spectra. One clearly observes that the mode below 0.5 eV only occurs in the full solution of the EELS problem and not in the homogeneous problem (i.e., plane wave excitation). Moreover, the low energy mode is also absent in the inhomogeneous Mie solution restricted to transverse mode solution space. Consequently, this results corroborates the importance of a complete three component set of VSH to solve the EELS problem completely. It also demonstrates that the restriction of the inhomogeneous case to transverse solutions correctly reproduces the transverse modes of the full solution, which we exploited in the dedicated EELS section of the paper.
All in all we endeavored to address the criticized points and hope to fulfill now the criteria for publication in SciPost Core.

Faithfully yours,
Axel Lubk

References:
[1] F. G. de Abajo, Relativistic energy loss and induced photon emission in the interaction of a dielectric sphere with an external electron beam, Physical Review B 59(4), 3095 (1999)
[2] L. Ge, D. Han and J. Zi, Electromagnetic scattering by spheres of topological insulators, Optics Communications 354, 225 (2015)


List of changes:
1. We added a more precise description of the inhomogeneous case.
2. We have cited previous works on Mie scattering on TI spheres, and set our work more detailed in context to them.