The classical method of images gives the electrostatic potential of a point charge near an isotropic dielectric half-space in closed form. E. J. Mele (Am. J. Phys. 69, 557 (2001)) recast the anisotropic version of this problem for a diagonal susceptibility tensor using an 'anamorphosis' (coordinate-stretching) transformation, and noted that the construction was not developed for general tensors or general geometries. The planar half-space family with a general symmetric positive-definite permittivity tensor is already solved by the exact-image theory of Lindell and coworkers (1990-2005), where the image of a point charge is a mirror point charge plus a continuous charge distribution on a conical surface, and Deng (2020) solved the finite-thickness anisotropic film for diagonal tensors. What remains open is the completion of this program: an exact image-theoretic and anamorphic characterization of the screening of a point charge by anisotropic dielectrics for non-diagonal tensors in layered and film geometries, for curved interfaces, and for spatially varying anisotropy — that is, the analytic mapping between the structure of the susceptibility tensor and the geometry and strength of the image system, together with the induced surface and volume bound charge.

Background

In electrostatics, the method of images replaces a boundary-value problem — for example, a point charge qq placed a distance dd in vacuum in front of a semi-infinite dielectric — by an equivalent free-space problem in which fictitious 'image' charges, placed outside the physical region, reproduce the true potential. For an isotropic dielectric half-space with relative permittivity ϵ\epsilon, the exterior potential is reproduced by a single image charge of magnitude q(ϵ−1)/(ϵ+1)q(\epsilon-1)/(\epsilon+1) at the mirror point, and the induced bound charge (the polarization charge of the medium) resides entirely on the interface.

An anisotropic dielectric is characterized by a symmetric positive-definite permittivity (or susceptibility) tensor ϵij\epsilon_{ij} rather than a scalar; its polarization Pi=∑jχijEjP_i=\sum_j\chi_{ij}E_j then need not be parallel to the electric field EE. The method of images does not extend to this case in any obvious way: the scalar reflection coefficient becomes direction-dependent, and, as Mele showed, a nonzero volume bound charge density appears inside the medium (which vanishes in the isotropic case). Mele (2001) solved the problem for a diagonal tensor ϵ=diag(ϵ∥,ϵ∥,ϵ⊥)\epsilon=\mathrm{diag}(\epsilon_\parallel,\epsilon_\parallel,\epsilon_\perp) — the case of layered crystals — by an 'anamorphosis', a stretching of coordinates that maps the anisotropic problem locally onto an isotropic one; the exterior potential is then a solution of an unconventional image problem, while the interior potential is obtained by transforming an image solution rather than being an image solution itself.

The exact-image literature for anisotropic media is older and more complete than the diagonal-case discussion suggests. Lindell, Sihvola and Viitanen (1990) gave an exact image theory for the uniaxially anisotropic half-space; Lindell, Nikoskinen and Flykt (1996) treated small deviations from transverse isotropy; and Lindell, Nikoskinen and Viljanen (1997), using Heaviside operational calculus (a symbolic operator algebra for differential operators), solved the half-space problem for a general symmetric positive-definite permittivity tensor, finding that the image consists of a mirror point charge plus a continuous charge distribution on a conical surface whose opening angle is an analytic function of the transverse anisotropy. Hänninen, Lindell and Nikoskinen (2004) extended this to an anisotropic boundary of an anisotropic half-space, and Hänninen, Nikoskinen and Lindell (2005) to the interface between two anisotropic half-spaces. For finite-thickness media, Deng (2020) derived the complete electrostatic response of a film with a diagonal anisotropic susceptibility to an arbitrary external potential — charges inside or outside — in closed Fourier-integral form, showing the induced polarization charge is a continuous distribution, with a point-like image emerging only in the semi-infinite limit. For isotropic curved boundaries an image theory also exists (e.g., Sten and Ilmoniemi's layered dielectric sphere), and Wait (1990) analyzed DC current flow into a three-dimensionally anisotropic conductor, the conduction dual of the dielectric problem.

Against this background, the unresolved scientific objective is the completion of exact image theory for anisotropic screening beyond the single planar interface: general (non-diagonal) tensors, multilayer and film geometries, curved boundaries, and spatially nonuniform anisotropy.

Problem Statement

Characterize and construct the exact image system for the screening of an external point charge by an anisotropic dielectric medium, beyond the already-solved cases, and read off from it the induced surface and volume bound charge. The already-solved baseline is the planar half-space family: a diagonal tensor (Mele 2001, via anamorphosis) and a general symmetric positive-definite permittivity tensor (Lindell and coworkers 1990-2005, where the image is a mirror point charge plus a continuous conical charge sheet), as well as diagonal-tensor films (Deng 2020, in integral form). The open target is (1) the completion of the analytic mapping between the structure of the susceptibility tensor and the geometry and strength of the image system (point images and continuous image distributions), including the corresponding induced bound-charge densities, for general non-diagonal tensors in two mandatory planar media — a single finite-thickness anisotropic film and a multilayer (stacked) anisotropic medium; and (2) image (or anamorphic) representations for the canonical cases where no exact image construction is currently known, comprising three mandatory targets — an anisotropic dielectric sphere facing a point charge, an anisotropic dielectric cylinder facing a point charge, and a medium with spatially varying anisotropy — where for each target either an explicit construction or a proof that no exact image construction exists resolves that target, including convergence properties of any series or integral representation that arises. A separate, optional application direction — not part of the mandatory target — is the quantitative incorporation of the resulting screening formulas into models of realistic heterogeneous anisotropic materials (e.g., van der Waals heterostructures — layered stacks of two-dimensional crystals such as graphene on hexagonal boron nitride — and other engineered anisotropic dielectrics) to predict interaction energies and gate screening.

References

  1. E. J. Mele, 'Screening of a point charge by an anisotropic medium: Anamorphoses in the method of images', American Journal of Physics 69(5), 557-562 (2001), DOI 10.1119/1.1341252, https://doi.org/10.1119/1.1341252
  2. I. V. Lindell, A. H. Sihvola, A. J. Viitanen, 'Exact image theory for uniaxially anisotropic dielectric half-space', Journal of Electromagnetic Waves and Applications 4(2), 129-143 (1990), DOI 10.1163/156939390X00032, https://doi.org/10.1163/156939390X00032
  3. I. V. Lindell, K. I. Nikoskinen, M. J. Flykt, 'Electrostatic image theory for an anisotropic half-space slightly deviating from transverse isotropy', Radio Science 31(6), 1361-1368 (1996), DOI 10.1029/96RS01727, https://doi.org/10.1029/96RS01727
  4. I. V. Lindell, K. I. Nikoskinen, A. Viljanen, 'Electrostatic image method for the anisotropic half space', IEE Proceedings - Science, Measurement and Technology 144(4), 156-162 (1997), DOI 10.1049/ip-smt:19971317, https://doi.org/10.1049/ip-smt:19971317
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  6. J. J. Hänninen, K. I. Nikoskinen, I. V. Lindell, 'Electrostatic image theory for two anisotropic half-spaces', Electrical Engineering (Archiv für Elektrotechnik) 88(1), 1-12 (2005), DOI 10.1007/s00202-004-0255-1, https://doi.org/10.1007/s00202-004-0255-1
  7. Hai-Yao Deng, 'Electrostatic responses of anisotropic dielectric films', European Journal of Physics 41(3), 035203 (2020), DOI 10.1088/1361-6404/ab806c, arXiv:1906.03654 (preprint posted 2019), https://doi.org/10.1088/1361-6404/ab806c
  8. James R. Wait, 'Current flow into a three-dimensionally anisotropic conductor', Radio Science 25(5), 689-694 (1990), DOI 10.1029/RS025i005p00689, https://doi.org/10.1029/RS025i005p00689
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