Electromagnetic description of three-dimensional time-reversal invariant ponderable topological insulators

A general technique to analyze the classical interaction between ideal topological insulators, and electromagnetic sources and fields, has been previously elaborated. Nevertheless it is not immediately applicable in the laboratory as it fails to describe real ponderable media. In this work we provide a description of real topologically insulating materials taking into account their dielectric and magnetic properties. For inhomogeneous permittivity and permeability, the problem of finding the Green’s function must be solved in an ad hoc manner. Nevertheless, the physically feasible cases of piecewise constant $ϵ$, $\mu$ and $\theta$ make the problem tractable, where $\theta$ encodes the topological magnetoelectric polarizability properties of the medium. To this end we employ the Green’s function method to find the fields resulting from the interaction between these materials and electromagnetic sources. Furthermore we exploit the fact that in the cases here studied, the full Green’s function can be successfully found if the Green’s function of the corresponding ponderable media with $\theta =0$ is known. Our results satisfactorily reproduce previously existing ones and also generalize some others. The method here elaborated can be exploited to determine the electromagnetic fields for more general configurations aiming to measure the interaction between real 3D topological insulators and electromagnetic fields.

Figure 1

Geometry of semi-infinite time-reversal symmetric topological insulator ($\theta \ne 0$) in contact with the vacuum ($\theta =0$).

Figure 2

Illustration of the images electric charges and magnetic monopoles induced by a pointlike electric charge near to a planar topological insulator.

Figure 3

Illustration of the images electric and magnetic current-carrying wires induced by an infinitely straight current-carrying wire near a planar topological insulator.

Figure 4

Illustration of the images electric and magnetic charge densities induced by an infinitely uniformly charged wire near a planar topological insulator.

Figure 5

Illustration of the image sources induced by a pointlike electric charge near a spherical TI. In the region $r>b\gg a$ the electric and magnetic fields are due to images electric $\mathbf{p}$ and magnetic $\mathbf{m}$ dipoles, respectively. Inside the TI ($b\gg a>r$), the electromagnetic fields can be interpreted as those produced by a uniformly polarized sphere with polarization $\mathbf{P}$, while the magnetic field corresponds to the one produced by a uniformly magnetized sphere with magnetization $\mathbf{M}$.