Axion Mie Theory for Electron Energy Loss Spectroscopy in Topological Insulators

We are again thankful for the referee’s valuable comments and criticism. The dispute surrounding the manuscript mainly circles around our claim to treat the inhomogeneous Mie problem in a way not presented elsewhere. Studying the references given in the referee's previous response, we still believe our results do represent a novel contribution. That pertains to the explicit derivation of the coupled radial differential equations (18)-(20) and the derivation of EELS cross sections and deflection angles, but also the explicit notation of the scattering matrix (34-37) of the homogeneous problem. The references given by the referee almost exclusively refer to the homogeneous problem and the projection of the inhomogeneous solution to transverse fields only (i.e., solutions of the homogeneous problem). In particular, they do not provide solutions at source points crucial for our aim to model EELS experiments being routinely carried out in our lab. As we did not and do not want to give the impression to rewrite Mie theory for topological insulators (in particular since the homogeneous problem has been discussed previously), however, we make the title of our paper more specific to the problem we are interested in: “Axion Mie Theory for Electron Energy Loss Spectroscopy in Topological Insulators”, in line with the comments of the referee.

Following are detailed comments pertaining to the points raised by the referee.

A.: Thank you for pointing out that sign mismatch. It has been corrected in the manuscript.

B.: Eq. (16) holds for spatially constant $\epsilon$ and $\Theta$, which resolves the issue (and was already noted a few lines below eq. (16) in the original version).

C.: We have indeed not proved the completeness of the vector spherical harmonics, which we, however, only use at the surface of the sphere (and not for decomposing fields in radial direction). However, there is a relatively straightforward way of obtaining this result via well known results from group theory. First note that the vector spherical harmonics (VSHs) we are using are simultaneous eigenfunctions of the operators ${\vec{J}}^2$ and $J_z$, where $\vec{J}=\vec{L}+\vec{S}$ is the total angular momentum adding the orbital angular momentum $\vec{L}$ to the spin operator $\vec{S}$ associated to a spin 1 particle (the photon). One of the main results of the general theory of angular momentum is that the simultaneous eigenstates of ${\vec{J}}^2$ and $J_z$ form a complete basis. Indeed, this basis is easily constructed using the addition theorems for angular momenta via the so called Clebsch-Gordan coefficients. The resulting VSHs have been used for more than eight decades in nuclear physics and other quantum mechanical problems [see, for instance, Appendix B of J. M. Blatt and V. F. Weisskopf, “Theoretical Nuclear Physics”, Springer-Verlag (1979), and V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, “Quantum Electrodynamics”, 2nd edition, Butterworth-Heinemann (2012), § 6 and § 7] . There is no issue regarding the completeness here, as it follows from group representation theory associated as applied to the construction of general angular momenta in quantum mechanics. The version used by Barrera et al. that we use is a mere rewriting of the VSH derived in the way described above. Indeed, these are precisely the same (up to proportionality factors), as the ones used in the textbook by Berestetskii et al., where a very lucid discussion is given.
Furthermore, they are specifically designed in such a way as to carry no radial dependency, i.e., the radial dependency in the expansion (17) is completely given by the expansion coefficients $E_{lm}^\perp$, $E_{lm}^{(1)}$, and $E_{lm}^{(2)}$. We hope that this resolves the issue.

D.: See point C., the spatial dependency of the expansion coefficients already stems from the fact that the expansion (17) is over the sphere and not full space, i.e., the radial dependency must be in the expansion coefficients (VSH are independent of $r$). Of course, source terms will also affect the spatial dependency, ultimately given by solution of (18-20).

D.1 and 2: We agree with the referee that the solution at any point in space (including source) may be found by appropriate Green’s functions. The crucial point is then to have explicit VSH expansions of the appropriate Green’s function / dyad. However, Chap. 2 of Faryad and Lakhtakia [Infinite-Space Dyadic Green Functions in Electromagnetism, IoP Books, 2018] as well as Wood [Marconi Rev. 34, 149 (1971)] / Wood [Reflector Antenna Analysis and Design, IEE Peter Peregrinus, 1980]) use only transverse functions (transverse VHS) in the expansion of the Green’s function, which then corresponds to the free space propagator of the fields. That is perfectly fine for the homogeneous Mie problem. However, if sources are present in the problem and we need to evaluate the fields at the sources (as is the case of electron energy loss spectroscopy), this type of approach amounts to an approximation at best. The accuracy of the latter ultimately depends on how close the sources (i.e., the electron beam) are to the sphere/nanoparticle. In EELS the beam can be also on the particle, which is the reason for writing (18-24) derived from Barrera VSH is the discussion of the fully inhomogeneous problem. The solution of (18-24) for a general source distribution represents the general solution to that problem. Alternatively, a complete expansion of the Green’s dyadics including the full set of vector spherical harmonics (i.e., including longitudinal fields) would also work. Yaghjian et al. indeed discusses general forms of Green’s dyadics but no expansion into vector spherical harmonics. Both Wood references (Wood [Marconi Rev. 34, 149 (1971)], Wood [Reflector Antenna Analysis and Design, IEE Peter Peregrinus, 1980]) also project out the longitudinal part. These references lack the complete expansion as required for the general solution of the inhomogeneous Mie scattering problem (which would certainly be especially useful for us).

D.3 We agree with the referee, however, that the homogeneous Mie problem is more conveniently solved by employing Morse and Feshbach VSHs, as these absorb the radial dependency of the fields into the vector spherical harmonics directly without requiring to explicitly calculate the radial dependency by solving a differential equation. Note, however, that even when considering only the homogeneous case a derivation employing a different set of bases (VSHs) can still have some additional merits as it provides another perspective on the problem.

E. Here we disagree. $A_{klm}=0$ is not correct for the described geometry. As the expansion of the source field is given by an integral over the whole space, there will be a longitudinal field in the presence of sources. One may restrict the problem to a finite sphere (excluding the source). In this particular case, however, surface terms show up in the corresponding Helmholtz decomposition, which reflect the presence of sources and modify the decomposition with respect to the homogeneous case. Moreover, we would not be able to compute the field at the source, which is, however, needed for the application case we have in mind. Only in the homogeneous case the longitudinal fields are exactly zero everywhere.

F. See above. We employed Barrera VSH to write the inhomogeneous problem as a system of 3 coupled inhomogeneous ordinary differential equations for the expansion coefficients. This would have been difficult with the Morse and Feshbach VSHs as they carry a radial dependency by themselves.

G. Following the suggestion of the referee, we calculated the field components for a 500 nm and a 5000 nm sphere without any significant increase of computation time. Note, that the calculated quantities (plotted in Fig. 1-3) are not a numerical solution of the inhomogeneous differential equations (18-20) but stemming from numerical calculations of the analytically derived expansion coefficients (eq. (25-27)) for the homogeneous case. Indeed, for the general inhomogeneous case (which also covers arbitrary external charges/currents and longitudinal field components) the differential equations (18-20) need to be solved numerically. Focusing on electron energy loss spectroscopy we explicitly need solutions for arbitrary external charges and currents (i.e., solutions at the position of the electron beam). Independent of numerical resources, the theory of Refs. 10, 11, and 14 does not provide solutions at source points, thus it cannot be applied as suggested by the referee.

H. Having plasmonic excitations on TIs in mind, we set the dielectric function negative (which is the fundamental condition for the presence of surface plasmonic excitations). One prominent TI with $\epsilon=-1$ at optical frequencies is Bi$_2$Se$_3$ (see e.g., [Talebi et al. ACS Nano (2016)]). A second example of a TI with negative dielectric function at optical frequencies is Bi$_2$Te$_3$ (see [Esslinger et al. ACS Photonics (2014)]). Both Bi$_2$Se$_3$ and Bi$_2$Te$_3$ are well known topological insulators. To make this clearer for the reader, we integrated a short discussion into the manuscript.

I. For more clarity we added additional information to the caption of Figs. 1-3. Note, that the plotted quantities are already denoted in the original manuscript (see e.g., the axis labels in Fig. 1).

J. We removed appendix A as it is not essential to the paper. Note, however, that the explicit form of the Helmholtz decomposition (and the ensuing toroidal-poloidal decomposition) depends on the space considered. If considering $R^3$ including sources, there is a non-vanishing longitudinal field. If considering some finite space, the decomposition requires simple connectivity and boundary terms must be considered.

K. We agree that the homogeneous problem has been treated previously, particularly in Ref. 10, see also our previous response. Note, however, that Refs. 10, 11 and 14 do not give the scattering matrix elements (34-37) explicitly, which to our point of view are furthermore not trivial! Indeed Ref. 14 even apply an additional approximation in neglecting all $\alpha^2$ terms. Whether such an additional approximation was also performed in Ref. 10 is not clear to us.

In conclusion, we addressed all the technicalities raised by the referee (e.g., concerning VSH or dielectric function) and hope that we could also explain better our point of view on the inhomogeneous Mie problem.

With best regards on behalf of the authors, Johannes Schultz

1. We made the title of our paper more specific to the problem we are interested in and changed it to “Axion Mie Theory for Electron Energy Loss Spectroscopy in Topological Insulators”.

2. We revised the abstract.

3. We have included a brief discussion of plasmonic excitations on which we focus in the manuscript.

4. We added additional information to the caption of Figs. 1-3.

5. We removed appendix A as it is not essential to the paper.

6. Further relevant references and corresponding discussions have been added. In addition, we addressed some stylistic issues and typos.