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Axion Mie theory for a spherical topological insulator
by Johannes Schultz, Flavio S. Nogueira, Bernd Büchner, Jeroen van den Brink, Axel Lubk
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Submission summary
| Authors (as registered SciPost users): | Axel Lubk · Flavio Nogueira · Johannes Schultz · Jeroen van den Brink |
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| Preprint Link: | https://arxiv.org/abs/2002.03804v2 (pdf) |
| Date submitted: | 2020-10-08 12:13 |
| Submitted by: | Schultz, Johannes |
| Submitted to: | SciPost Physics |
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| Academic field: | Physics |
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| 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 describing the electromagnetic response of spherical particles including an axion electromagnetic coupling at the surface of a particle. The approach includes 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.
Author comments upon resubmission
Thank you, Johannes Schultz on behalf of the authors
Referee 1:
In the revised version, we extended the discussion of the range of validity of the axion theory in terms of topological insulators (TIs). We particularly elaborated on the significance of the gap and the magnitude of α.
Referee 2:
Before addressing the individual points, we would like to thank the referee for putting our work in the context of previous works, in particular for pointing out the important paper of Lakhtakia and Mackay.
Point 1: The solution of the homogeneous counterpart of Eq. (10) with spatially constant ϵ and Θ in the spherical coordinate system was first given by Bohren [ “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458-462 (1974)]. Just set α=β in the 1974 paper.
and
Point 2: The Mie theory provided by the authors was completely formulated by Bohren in 1974. Just set α=β in the 1974 paper. This landmark paper went unnoticed by Ge et al. [“Electromagnetic scattering by sphere of topological insulators,” Opt. Commun. 354, 225-230 (2015)] and well as by the authors of the present manuscript.
Our comment:
Thank you for pointing us at this important reference (Bohren in 1974), which is now referenced in the paper (Ref. [11] in the revised version). Similarly to our manuscript, Bohren introduced modified constitutive equations involving magnetic and electric field components (additional polarization depending on the magnetic flux density and additional magnetization depending on the electric field strength). However, only in the case of vanishing external sources do the constitutive equations used by us (Eqs. (7) and (8) in the revised version) reduce to those used by Bohren. In other words, in Bohren’s paper only the homogeneous case was treated, whereas our theory allows for external charges and currents, like the ones occurring in electron energy loss spectroscopy (EELS) experiments. As a consequence, our work provides an extension of Bohren’s 1974 paper.
Point 3: There is another way to solve the same problem, whereby the axionic term does not enter constitutive relations but enters boundary conditions. That way for the Mie theory was adopted by Lakhtakia and Mackay ["Electromagnetic scattering by homogeneous, isotropic, dielectric-magnetic sphere with topologically insulating surface states,” J. Opt. Soc. Am. B 33, 603-609 (2016)]. Section 2.F of the 2016 paper provides a comparison of the two ways.
Our comment:
The referee mentions an alternative way to formulate the problem, where the conducting surface states of the TI result in modified boundary conditions. Indeed, we also solved the problem using modified boundary conditions (Eqs. (21) - (24) in the revised version) derived from constitutive equations following from the axion-coupling. The modifications of the BCs are induced by the axionic terms in the constitutive relations (Eqs. (7) and (8) in the revised version). Consequently, the two approaches are identical and entail each other. Axionic constitutive equations affect observables by modified BCs and vice versa modified boundary conditions are consistent with an axion term in the Lagrangian density / axion terms in the constitutive equations on a more fundamental level. Summing up, both points of view are identical with respect to observable solutions (e.g., solutions of the Mie scattering problem in our case).
Point 4: Whereas Bohren provided expressions only a for an incident plane wave, Lakhtakia and Mackay considered any source that lies outside the sphere. This is significant because the authors mention the importance of the “incorporation of arbitrary sources of fields” to justify their work in comparison to that of Ge et al. [Ref. 22].
Our comment:
In our approach external sources are considered on a more general level in comparison to the work of Lakhtakia and Mackay. Indeed, Lakhtakia and Mackay first solve the homogeneous (i.e., transverse fields only) case and take care of external charges afterwards by adding transverse fields due to external sources to the homogeneous solution. In consequence this approach cannot treat cases, where charges / currents with correspondingly large longitudinal fields rest inside or very close to the sphere. Further away from the sphere, however, the restriction is less severe and the approach represents a good approximation. By employing 3-component vector spherical harmonics, we can indeed represent any 3D vector field (i.e., transverse and longitudinal) and hence solve the inhomogeneous differential equation without any approximations. This allows us, for instance, to model EELS experiments, where external charges and currents within the sphere are present (i.e., when the electron beam passes the sphere). Notwithstanding, in the newly added section containing explicit calculations of EEL spectra and electron beam deflection angles, we also focus on the aloof scattering geometry and neglect longitudinal solutions. The reason is that our analytical approach (containing special integrals, etc.) is only valid for transverse fields. We were not able to find a fully analytical solution including longitudinal fields (which of course does not preclude that an analytical solution may exist, let alone a numerical one).
Point 5: The vector spherical wave functions of Barrera et al. [Ref. 23] are more cumbersome than those of Stratton [Electromagnetic Theory, McGraw-Hill, 1941] and Morse and Feshbach [Methods of Theoretical Physics Vol. II, McGraw-Hill, 1953]. Many of the manipulations presented by the authors in this manuscript would be eased if the vector spherical wave functions of Stratton and Morse&Feshbach are employed. Their orthogonalities on a unit sphere are exceedingly simple and lead to very simple manipulations, as the authors would find on consulting their Ref. 20. Neither Appendix A nor Appendix B are needed in 2020.,
Point 6a: Instead or TE and TM classification which is strictly valid as r→∞, a classification in terms of toroidal and poloidal fields is much better. because it applies everywhere. The electric field is poloidal if the magnetic field is toroidal (in isotropic achiral medium) and vice versa. This classification follows immediately from the vector spherical wave functions of Stratton and Morse&Feshbach; see Chandrasekhar and Kendall [“On force-free magnetic fields,” Astrophys. J. 126, 457-461 (1957)].
and
Point 6b: The toroidal and poloidal classification will impact the dubious straightness of arrows drawn in Fig. 2 at distance of just 0.1 radius from the surface of the sphere.
Our comment:
The referee suggests to use vector spherical harmonics defined by Stratton [Electromagnetic Theory, McGraw-Hill, 1941] including poloidal and toroidal classification to simplify derivations. Indeed, the 2-component vector spherical harmonics of Stratton including the poloidal-toroidal classification is only valid for divergence free, i.e., transverse / solenoidal vector fields. The general hierarchy for the general decomposition of 3D vector fields (on compact domains) consists first of the Helmholtz decomposition into longitudinal / irrotational fields and transverse / solenoidal fields. The former may be represented by a scalar field, whereas the latter can be further decomposed into a poloidal and a toroidal field component (both can be represented by a scalar field). Thus (in a handwaiving way), we have a complete decomposition of a general 3D vector field into 3 scalar fields, which may be truncated to 2 components in the case of transverse fields only. In the present work, we explicitly allow for external charge densities in the Maxwell Equations, and thus neither the dielectric displacement nor the electric field strength is divergence free, and we have to resort to the complete 3-component decomposition by Barrera et al. (Ref. [28] in the revised version). This also explains why we used the TE and TM classification based on the general applicable Helmholtz-decomposition. In the current version, we make this point clearer and employ yet another popular naming convention from literature, where magnetic (electric) modes refer to TE (TM) modes respectively.
Point 7a: Since the distance r is specified in nm in Figs. 1 and 2, this reviewer was surprised at not being able to find the frequency or the free-space wavelength used for the calculations. Neither was the relative permittivity reported for the material of the sphere nor the axionic constant.
and
Point 7b: The same lacuna is evident for Fig. 3.
Our comment:
Thank you for bringing up that point. We have added the details of the calculations and apologize for any sloppiness.
Point 8: The axionic admixing of TE and TM modes (as the authors put it) is trivial, all the more so because it is already known from the 1974 and the 2016 papers mentioned above. What is needed is a numerical estimate of a measurable quantity at a realizable frequency for a sphere made of a real material that has axionic properties. That is missing.
Our comment:
As already pointed out, Bohren as well as Lakhtakia and Mackay solved only the homogeneous problem exactly. The effect of the axionic term can be derived neither from the solution of Bohren nor from Lakhtakia and Mackays work, if arbitrary external charges and currents are present. We, however, fully agree with the wish of the referee for explicit results on measurable quantities. In order to measure the signature of the axionic coupling, we propose EELS experiments. The high spatial and spectral resolution of modern EELS machines allows for the study of many different aspects (shape dependence, coupling, etc.) of localized surface resonances occurring in nanoparticles. In order to describe those EELS experiments theoretically, we explicitely considered arbitrary external charges and currents. For topologically non-trivial materials we found a unique deflection of the electron beam potentially measurable. In the revised version we derived analytic solutions of the EEL spectra and the electron beam deflection angles for a real scenario.
In summary we hope that we could convince the referee that the paper does fulfill the criteria regarding novelty, in view of our treatment of the general inhomogeneous case of Mie scattering at a spherical TI. We made major revisions in the paper to address the referee’s valuable comments. We have added in particular a new section containing explicit calculations of aloof electron scattering on a TI sphere pertaining to EELS.
List of changes
1. We inserted a completely new section containing analytic calculations of electron energy loss (EEL) spectra and axion induced beam deflections pertaining to an electron beam flying aloof a TI sphere.
2. We expanded our analysis of the homogeneous scattering case, including an explicit and exact expression of the scattering matrix previously missing (see Eqs. (34-39) in the revised version).
3. We extended the discussion of the range of validity of the axion theory in terms of topological insulators (TIs). We particularly elaborated on the significance of the gap and the magnitude of α.
4. We added further relevant references and corresponding discussions at many points, we corrected a few typos and addressed stylistic issues.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2020-10-19 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2002.03804v2, delivered 2020-10-19, doi: 10.21468/SciPost.Report.2099
Strengths
1. Possibly, the treatment of EELS.
Weaknesses
1. Claim of novelty for the "axionic Mie problem" is not valid, in view of Refs. 10, 11, and 13.
2. Suboptimal field representation when the optimal is known.
3. Numerical results are provided only for electrically small spheres, but not for larger spheres that would make the suboptimality apparent.
Report
In their response, the authors have stated that the work of Bohren (Ref. 11), Lakhtakia and Mackay (Ref. 10), and Ge et al. (Ref. 13) is applicable for the "homogeneous", whereas their Sec. 2 is novel because it deals with the "inhomogeneous" case.
The title of this paper is "Axion Mie theory for a spherical topological insulator". In addition, the axionic parameter $\Theta\in\{0,\pi\}$, according to the authors. Finally, they deal with regions with spatially constant constitutive parameters. So, the sphere is made of a material with $\Theta=\pi$ and the outside medium has $\Theta=0$. The sources lie outside the sphere. So long as the sources are not on the boundary, and are assumed to be unaffected by the scattered field, the theory of Refs. 10, 11, and 13 applies and suffices for the Mie scattering problem.
Even if fields in a region with spatially constant constitutive parameters are considered when that region has sources, those can be obtained from standard treatments since the dyadic Green function for such a medium is known, its singularity is known, and its bilinear expansion is known. This knowledge is a century old.
Following are detailed comments.
A. This reviewer obtained $\nabla\times\vec{B} =-(\alpha/\pi)\nabla\times(\Theta\vec{E})+ik\epsilon\vec{E}-(\alpha/\pi)ik\Theta\vec{B}+(4\pi/c)\vec{j}$ from Eq. (4). This will lead to $\nabla\times\vec{B} =-(\alpha/\pi)(\nabla\Theta)\times\vec{E}-(\alpha/\pi)\Theta\nabla\times \vec{E}
+ik\epsilon\vec{E} -(\alpha/\pi)ik\Theta\vec{B}+(4\pi/c)\vec{j}$ $=-(\alpha/\pi)(\nabla\Theta)\times\vec{E}-(\alpha/\pi)\Theta (\nabla\times \vec{E}+ik \vec{B})
+ik\epsilon\vec{E} +(4\pi/c)\vec{j}$ $=-(\alpha/\pi)(\nabla\Theta)\times\vec{E} +ik\epsilon\vec{E} +(4\pi/c)\vec{j}$ by virtue of Eq. (5).
Now, Eq. (5) also delivers $\nabla\times(\nabla\times\vec{E}) + ik (\nabla\times\vec{B})=0$.
Substitute the previously derived equation here to get
$\nabla\times(\nabla\times\vec{E}) -ik(\alpha/\pi)(\nabla\Theta)\times\vec{E}
-k^2\epsilon\vec{E}+i(4\pi k/c) \vec{j}
=0$. Check the signs of various terms in Eq. (15).
B. The authors have written that Eq. (16) holds for "spatially constant $\epsilon$". That
is incorrect. Equation (3) yields $\nabla\cdot\vec{E}= (\alpha/\pi\epsilon)\nabla\cdot(\Theta\vec{B})+
(4\pi/\epsilon)\rho
$ when $\epsilon$ is spatially constant. This equation can be simplified to
$\nabla\cdot\vec{E}= (\alpha/\pi\epsilon)(\nabla\Theta)\cdot \vec{B}+ (\alpha\Theta/\pi\epsilon)\nabla\cdot \vec{B}+(4\pi/\epsilon)\rho $
$= (\alpha/\pi\epsilon)(\nabla\Theta)\cdot \vec{B}+(4\pi/\epsilon)\rho $. There is no way Eq. (16) can be correct.
However, if both $\epsilon$ and $\Theta$ are spatially constant, then $\nabla\cdot\vec{E}=
(4\pi/\epsilon)\rho $ which leads to $\nabla(\nabla\cdot\vec{E})=
(4\pi/\epsilon)\nabla\rho $. Because $\nabla\times(\nabla\times\vec{E})=\nabla(\nabla\cdot\vec{E})-\nabla^2\vec{E}$, we get $\nabla\times(\nabla\times\vec{E})=(4\pi/\epsilon)\nabla\rho-\nabla^2\vec{E}$. Substitute in the corrected version of Eq. (15) then.
C. The authors have provided Eq. (17) with reference to Barrerra et al. [Eur. J. Phys. 6, 287 (1985)]. The authors claim that the vector spherical harmonics $\vec{Y}_{\ell m}(\vec{r})$, $\vec{\Psi}_{\ell m}(\vec{r})$, and $\vec{\Phi}_{\ell m}(\vec{r})$ "form a complete basis for vector fields". The authors need to provide an explicit proof for their claim with respect to the homogenous version of Eq. (16), because completeness has been skirted by Barrerra et al. as well as by the authors cited in that paper for the same issue.
D. In the 2nd paragraph of Sec. 2, the authors state that they "tackle the problem by a piecewise solution in regions of spatially constant $\epsilon$ and $\Theta$." As this stipulation holds for Eq. (17), the spatial dependences of $E_{\ell m}^\perp(\vec{r})$, $E_{\ell m}^{(1)}(\vec{r})$, and $E_{\ell m}^{(2)}(\vec{r})$, must come from the spatial dependences of the source terms $\vec{j}(\vec{r})$ and $\rho(\vec{r})$.
D.1. At points other than a source point, the field can be found since the relevant dyadic Green function is known; see: Chap. 2 of Faryad and Lakhtakia [Infinite-Space Dyadic Green Functions in Electromagnetism, IoP Books, 2018]. This is commonly done in the electromagnetics literature for transmitting antennas.
D.2. At a source point, the field can be found since the singular part of the relevant dyadic Green function is known; see: Yaghjian [Proc. IEEE 68, 248 (1980)]. This would not be needed for the Mie scattering problem anyway.
D.3. A convenient alternative to Eq. (17) in a source-free region
with both $\epsilon$ and $\Theta$ spatially constant is $\vec{E}(\vec{r})=\sum_{\kappa\in\{1,3\}} \sum_{\ell=0}^\infty$
$\sum_{m=-\ell}^{\ell}$ $[A_{\ell m\kappa} \vec{L}_{\ell m\kappa}(k'\vec{r})+B_{\ell m\kappa} \vec{M}_{\ell m\kappa}(k'\vec{r}) +C_{\ell m\kappa} \vec{N}_{\ell m\kappa}(k'\vec{r})]$, with $k'=k\sqrt{\epsilon}$. The vector spherical wavefunctions $\vec{L}_{\ell m\kappa}(k'\vec{r})$, $\vec{M}_{\ell m\kappa}(k'\vec{r})$, and $\vec{N}_{\ell m\kappa}(k'\vec{r})$, are provided on pp. 1865-1866 of Morse and Feshbach [Methods of Theoretical Physics, McGraw-Hill, 1953]. These functions use the spherical Bessel function $j_\ell(k'r)$ if $\kappa=1$ and the spherical Hankel function $h_\ell^{(\nu)}(k'r)$ if $\kappa=3$, with the index $\nu$ depending on the sign convention chosen for time-harmonic fields. Finding the space-independent coefficients $A_{\ell m\kappa}$, $B_{\ell m\kappa}$, and $C_{\ell m\kappa}$ is not at all a problem, since the bilinear expansion of the relevant dyadic Green function is known; see: Wood [Marconi Rev. 34, 149 (1971)] as an example. Alternatively, see: Wood [Reflector Antenna Analysis and Design, IEE Peter Peregrinus, 1980]. (It is not difficult to see that $A_{\ell m\kappa}\equiv 0$.) The expansion provided in this Comment is particularly convenient for the Mie scattering problem.
According to Comments D.1 and D.2, Eqs. (18)-(24) are unnecessary. They are certainly unnecessary for the Mie scattering problem.
E. So now to the Mie scattering problem. Suppose a sphere of radius $a$ is made of a homogeneous material with constitutive parameters $\epsilon\in\mathbb{C}$ and $\Theta=\pi$. The external medium extends to infinity in all directions and is the same as free space (i.e., $\epsilon=1$ and $\Theta=0$). The origin of the coordinate system lies at the center of the sphere. Suppose that all sources lie in the region $r>b>a$ and are unaffected by the scattered field. In the region $r<b$, the incident field will be given by the representation in Comment D.3 with $A_{\ell m\kappa}\equiv 0$, $B_{\ell m3}\equiv 0$, $C_{\ell m3}\equiv 0$, and $\epsilon=1$, as has been shown by Wood and is easy to show since the bilinear expansion of the free-space dyadic Green function is known. The scattered field will be given by the same representation with $A_{\ell m\kappa}\equiv 0$, $B_{\ell m1}\equiv 0$, $C_{\ell m1}\equiv 0$, and $\epsilon=1$. Representation of the field induced inside the sphere is already available from Refs. 10, 11, and 13. The boundary-value problem has already been solved in this general setting in Ref. 10. (Refs. 11 and 13 solve the problem for an incident plane wave.) Section 2 and Appendix B are unnecessary.
F. Section 2 and Appendix B are not only unnecessary, but they are also suboptimal. The functions $\vec{L}_{\ell m\kappa}(k'\vec{r})$, $\vec{M}_{\ell m\kappa}(k'\vec{r})$ and $\vec{N}_{\ell m\kappa}(k'\vec{r})$ are solutions of the vector Helmholtz equation for free space ($k'=k$) and a homogeneous isotropic dielectric medium ($k'\ne {k}$). The vector spherical harmonics $\vec{Y}_{\ell m}(\vec{r})$, $\vec{\Psi}_{\ell m}(\vec{r})$, and $\vec{\Phi}_{\ell m}(\vec{r})$ used in Eq. (17) are solutions of the vector Laplace equation, and therefore are necessarily suboptimal. Completeness of $\vec{L}_{\ell m\kappa}(k'\vec{r})$, $\vec{M}_{\ell m\kappa}(k'\vec{r})$ and $\vec{N}_{\ell m\kappa}(k'\vec{r})$ was explicitly proved by Aydin and Hizal [J. Math. Anal. Appl. 117, 428 (1986)].
G. The radius of the sphere is 50 nm in Figs. 1--3, whereas the free-space wavelength is 620 nm. As the radius is less than 10\% of the wavelength and $\vert\epsilon\vert=1$, this is an electrically small sphere. The authors need to try calculating for a sphere of radius 500 nm (or 5000 nm) and compare the computational resources needed for using Eq. (17) instead of the theory of Refs. 10, 11, and 13 to appreciate the suboptimality of Eq. (17).
H. In Figs. 1-3, the authors have set $\epsilon=-1$ at 620-nm free-space wavelength. What kind of a topological insulator has $\epsilon=-1$? Similarly, for Fig.~4, they have used $\epsilon=-0.5$. A thorough discussion is necessary along with plausible examples.
I. This reviewer was unable to find in the manuscript the sources responsible for the distributions presented in Figs. 1--3.
J. The poloidal-toroidal decomposition of applies in any source-free region in which $\epsilon$ and $\Theta$ are spatially constant. To see that look at Eq. (15). The third term on the left side and the only term on the right side will not exist. Furthermore, Eq. (3) would then simplify to $\nabla\cdot\vec{E}=0$. Appendix A is misleading.
K. The axionic admixing of TE and TM modes (as the authors put it) remains trivial, all the more so because it is already known from Refs. 10, 11, and 13 in the context of the Mie scattering problem.
Report #1 by Anonymous (Referee 3) on 2020-10-9 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2002.03804v2, delivered 2020-10-09, doi: 10.21468/SciPost.Report.2064
Report
I still think the discussion of the range of validity could contain more details, but the authors response is acceptable. Given the long lists of concerns the other referee had about this work, I believe that these issues are much more important than my more qualitative concerns. It appears that the authors addressed the other referee's comments in detail in their response, I will be curious to see what the other referee says. I have no objections anymore to publishing this article.