1. INTRODUCTION

The discovery of topological insulators [1], such as the chalcogenides ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, and ${\mathrm{Sb}}_2{\mathrm{Te}}_3$, has prompted a flurry of research activity, much of which has been directed toward revealing their optical properties [2–4]. To this end, the theory underpinning optical scattering from spheres made of topological insulators was recently developed [5]. Classically, a topological insulator may be modeled as an achiral biisotropic material whose nonreciprocity is captured by a magnetoelectric pseudoscalar $\gamma$ [6]. Alternatively, a topological insulator may be regarded as an isotropic dielectric–magnetic material whose surface is endowed with a surface admittance $\gamma$. These two different models give rise to identical results in terms of optical scattering [7]. Macroscopically, topological insulation is a surface phenomenon manifesting as protected conducting states that exist at the surface, but not in the bulk, of a topological insulator [1,2]. Therefore, we adopted the latter model in this paper.

Electromagnetic plane waves bound to the surface of a topological insulator are investigated here. Previous studies in this area have focused upon surface-plasmon-polariton waves [8–12]. In contrast, we investigate Dyakonov waves guided by the planar interface of a topological insulator and an anisotropic dielectric material [13,14]. While Dyakonov-wave propagation guided by the planar interface of two homogeneous dielectric materials, one isotropic and the other anisotropic, is possible only for a very small range of propagation directions, these surface waves offer considerable potential for long-range on-chip communication [15]. Parenthetically, let us note the dramatic enlargement of the range of propagation directions if the anisotropic partnering material is either a hyperbolic material [16,17] or a periodically nonhomogeneous material [18], but the range-enlargement issue lies outside the scope of this paper.

The anisotropic dielectric material is taken to be a columnar thin film (CTF) [19,20] here. CTFs may be effectively regarded as orthorhombic biaxial materials for optical purposes. Their optical properties and porosity may be engineered through judicious control of the vapor deposition technique used for their fabrication. Dyakonov waves are studied by solving the corresponding canonical boundary-value problem [21] in which the topological insulator occupies the half-space $z<0$ while the CTF occupies the half-space $z>0$. We contrast the characteristics of Dyakonov waves at the CTF/topological insulator interface with the characteristics of Dyakonov waves at the interface of a CTF and an isotropic dielectric material whose bulk properties are the same as the topological insulator but whose surface is endowed by a surface conductivity $\tilde{\sigma }$ [22] instead of the surface admittance $\gamma$.

2. BOUNDARY-VALUE PROBLEM FOR DYAKONOV-WAVE PROPAGATION

A schematic diagram illustrating the growth of a CTF by vapor deposition is provided in Fig. 1. The parallel columns grow on a substrate that is oriented parallel to the $xy$ plane. The angle between the growing columns and the substrate plane is $\chi$, while the angle between the incident vapor flux and the $xy$ plane is ${\chi }_v\le \chi$. Without loss of generality, Dyakonov-wave propagation parallel to the $x$ axis in the $xy$ plane is considered. The orientation of the CTF’s morphologically significant plane relative to the direction of Dyakonov-wave propagation is specified by the angle $\psi$, as is schematically illustrated in Fig. 2. Accordingly, the relative permittivity dyadic of the CTF is expressed as

 

Fig. 1. Schematic representation of the growth of a CTF.

Download Full Size | PDF

 

Fig. 2. Orientation of the CTF’s morphologically significant plane.

Download Full Size | PDF

Let ${\underset{\_}{\mathcal{E}}}_{\ell }$ and ${\underset{\_}{\mathcal{H}}}_{\ell }$ denote the (complex-valued) electric and magnetic field phasors, respectively, of angular frequency $\omega$, with $\ell =c$ for the region $z>0$ and $\ell =s$ for the region $z<0$. According to the Maxwell curl postulates, the phasors satisfy

According to the Maxwell curl postulates, the phasors satisfy

3. NUMERICAL STUDIES

For our computations, the CTF was taken to be made from titanium dioxide. The following are experimentally determined values of the principal refractive indexes for such a CTF at a free-space wavelength of 633 nm [19]:

For a given value of $\gamma$ or $\tilde{\sigma }$, Eq. (12) was solved to determine the values of $\psi$ for which Dyakonov-wave propagation is possible [23]. In fact, for $\gamma =\tilde{\sigma }=0$, four narrow ranges of $\psi$ are found to support Dyakonov-wave propagation: $\psi \in [\pm {\psi }_m-\mathrm{\Delta }\psi /2,\pm {\psi }_m+\mathrm{\Delta }\psi /2]$ and $\psi \in [180°\pm {\psi }_m-\mathrm{\Delta }\psi /2,180°\pm {\psi }_m+\mathrm{\Delta }\psi /2]$. For case (i), we found that the values of $\psi$ for which Dyakonov-wave propagation is possible depend upon the sign of $\varkappa$. In contrast, the values of $\psi$ for which Dyakonov-wave propagation is possible do not depend upon the sign of $\varkappa$ for case (ii).

The angle ${\psi }_m$, which represents the midpoint of the $\psi$ range that supports Dyakonov-wave propagation, is plotted against (i) ${\eta }_0\gamma /\tilde{\alpha }$ and (ii) ${\eta }_0\tilde{\sigma }/\tilde{\alpha }$ in Fig. 3. Here, ${\eta }_0=\sqrt{{\mu }_0/{\epsilon }_0}$ is the free-space impedance, while $\tilde{\alpha }=7.297352566\times {10}^{-3}$ is the fine structure constant [26]. The range of $\gamma$ values reflects a hopeful future for the presently infant field of topological insulators that could grow to encompass mixed materials and new material compositions. For case (i), the midpoint angle ${\psi }_m$ increases uniformly as $\gamma$ increases for $\varkappa >0$ whereas ${\psi }_m$ decreases uniformly as $\gamma$ increases for $\varkappa <0$. For case (ii), the midpoint angle ${\psi }_m$ increases uniformly as $\tilde{\sigma }$ increases, at a substantially faster rate than the corresponding rate of increase for the topological insulator case with $\varkappa >0$. The value of ${\psi }_m$ for $\gamma =0$, regardless of the sign of $\varkappa$, is the same as it is for $\tilde{\sigma }=0$, as may be anticipated from Eqs. (9) and (10).

 

Fig. 3. ${\psi }_m$ (deg) plotted against ${\eta }_0\gamma /\tilde{\alpha }$ for $\varkappa >0$ (solid, red curve) and $\varkappa <0$ (dashed, green curve), and against ${\eta }_0\tilde{\sigma }/\tilde{\alpha }$ (broken dashed, blue curve).

Download Full Size | PDF

The extent of the angular range for which Dyakonov-wave propagation is possible, namely, $\mathrm{\Delta }\psi$, is plotted against (i) ${\eta }_0\gamma /\tilde{\alpha }$ and (ii) ${\eta }_0\tilde{\sigma }/\tilde{\alpha }$ in Fig. 4. For case (i), the magnitude of $\mathrm{\Delta }\psi$ increases uniformly as $\gamma$ increases for $\varkappa >0$ whereas $\mathrm{\Delta }\psi$ decreases uniformly as $\gamma$ increases for $\varkappa <0$. For case (ii), the magnitude of $\mathrm{\Delta }\psi$ decreases uniformly as $\tilde{\sigma }$ increases, at a substantially faster rate than the corresponding rate of decrease for the topological insulator case with $\varkappa <0$.

 

Fig. 4. As Fig. 3 but with $\mathrm{\Delta }\psi$ (deg) on the vertical axis.

Download Full Size | PDF

Let $k_{0}m$ be the wavenumber for plane-wave propagation in the bulk CTF. In general, two distinct values of $m$ are possible, which arise as roots of the equation [23,27]

 

Fig. 5. As Fig. 3 but with $1000\log ({\overline{v}}_{\mathrm{ave}})$ on the vertical axes.

Download Full Size | PDF

The extent to which Dyakonov waves are bound to the CTF/substrate interface is gauged by the real parts of the decay constants $q_s$ and $q_{c1,c2}$. In the region $z<0$, the decay constant $q_s$ is real valued. Whereas $q_s\to 0$ as $\psi \to {\psi }_m-\mathrm{\Delta }\psi /2$, the maximum value $q_{s,\max }$ of $q_s$ is attained in the limit $\psi \to {\psi }_m+\mathrm{\Delta }\psi /2$. In Fig. 6, $q_{s,\max }$ is plotted against (i) ${\eta }_0\gamma /\tilde{\alpha }$ and (ii) ${\eta }_0\tilde{\sigma }/\tilde{\alpha }$. For case (i), $q_{s,\max }$ for $\varkappa >0$ increases uniformly as $\gamma$ increases, whereas $q_{s,\max }$ decreases uniformly as $\gamma$ increases for $\varkappa <0$. For case (ii), $q_{s,\max }$ decreases uniformly as $\tilde{\sigma }$ increases, at a noticeably slower rate than the corresponding rate of decrease for the topological insulator case with $\varkappa <0$.

 

Fig. 6. As Fig. 3 but with $q_{s,\max }$ on the vertical axis.

Download Full Size | PDF

In the region $z>0$, the decay constants $q_{c1,c2}$ are complex valued. Whereas $\mathrm{Re}[q_{c1}]\to 0$ as $\psi \to {\psi }_m+\mathrm{\Delta }\psi /2$, the maximum value $\mathrm{Re}{[q_{c1}]}_{\max }$ of $\mathrm{Re}[q_{c1}]$ is attained in the limit $\psi \to {\psi }_m-\mathrm{\Delta }\psi /2$. In Fig. 7, $\mathrm{Re}{[q_{c1}]}_{\max }$ is plotted against (i) ${\eta }_0\gamma /\tilde{\alpha }$ and (ii) ${\eta }_0\tilde{\sigma }/\tilde{\alpha }$. For case (i), $\mathrm{Re}{[q_{c1}]}_{\max }$ is independent of $\gamma$ regardless of the sign of $\varkappa$. For case (ii), $\mathrm{Re}{[q_{c1}]}_{\max }$ decreases uniformly as $\tilde{\sigma }$ increases.

 

Fig. 7. As Fig. 3 but with $\mathrm{Re}{[q_{c1}]}_{\max }$ on the vertical axis.

Download Full Size | PDF

Unlike the behavior of $\mathrm{Re}[q_{c1}]$, there is little variation in the magnitude of $\mathrm{Re}[q_{c2}]$ across the $\psi$ range for Dyakonov-wave propagation. Let $\mathrm{Re}{[q_{c2}]}_{\mathrm{ave}}$ denote the average of the two values of $\mathrm{Re}[q_{c2}]$ at $\psi ={\psi }_m\pm \mathrm{\Delta }\psi$. In Fig. 8, $\mathrm{Re}{[q_{c2}]}_{\mathrm{ave}}$ is plotted against (i) ${\eta }_0\gamma /\tilde{\alpha }$ and (ii) ${\eta }_0\tilde{\sigma }/\tilde{\alpha }$. For case (i), $\mathrm{Re}{[q_{c2}]}_{\mathrm{ave}}$ for $\varkappa >0$ increases uniformly as $\gamma$ increases whereas $\mathrm{Re}{[q_{c2}]}_{\mathrm{ave}}$ decreases uniformly as $\gamma$ increases for $\varkappa <0$. For case (ii), $\mathrm{Re}{[q_{c2}]}_{\mathrm{ave}}$ decreases uniformly as $\tilde{\sigma }$ increases, at a substantially faster rate than the corresponding rate of decrease for the topological insulator case with $\varkappa <0$.

 

Fig. 8. As Fig. 3 but with $\mathrm{Re}{[q_{c2}]}_{\mathrm{ave}}$ on the vertical axis.

Download Full Size | PDF

4. CLOSING REMARKS

In conclusion, the directions along which Dyakonov waves propagate at the planar interface of a CTF and a topological insulator are significantly modulated by varying the magnitude of the topological insulator’s surface admittance $\gamma$; so too are the decay constants and phase speeds of these Dyakonov waves. Values of $\gamma {\eta }_0/\tilde{\alpha }$ other than $\pm 1$ require the use of magnetic coatings and/or immersion in a magnetostatic field [6]. Most importantly, a Dyakonov wave propagating along the direction of a vector $\underset{\_}{u}$ has a different phase speed and different decay constants as compared with the Dyakonov wave which propagates along the direction of $-\underset{\_}{u}$. This nonreciprocity, with respect to interchanging the direction of Dyakonov-wave propagation, is not exhibited when the topological insulator is replaced by an isotropic dielectric material of the same refractive index but with a surface conductivity $\tilde{\sigma }$ instead of a surface admittance $\gamma$. Dyakonov-wave propagation thus provides a way to distinguish between topologically insulating surface states and conducting surface states.