Dispersion properties of artificial topological insulators based on an infinite double-periodic array of elliptical quartz elements

Subject and Purpose. Special features of all-dielectric electromagnetic analogues of topological insulators in the microwave range are considered, aiming at studying the infl uence of geometrical and constitutive parameters of topological insulator elements on the dispersion properties of topological insulators based on a two-dimensional double-periodic array of dielectric elements. Methods and Methodology. Th e evaluation of dispersion properties and electromagnetic fi eld spatial distribution patterns for topological insulators is performed using numerical simulation programs. Results. Th e electromagnetic analogue of a topological insulator based on a double-periodic array of elliptical quartz cylinders has been considered. By numerical simulation, it has been demonstrated that the electromagnetic properties of the structure are controllable by changing the quartz uniaxial anisotropy direction without any changes in other parameters. A combined topological insulator made up of two adjoining ones diff ering in shapes of their unit cells has been considered with the numerical demonstration that frequencies of surface states are controllable by choosing the quartz uniaxial anisotropy direction. It has been shown that it is at the interface of two diff erent in shape unit cells that the electromagnetic fi eld concentration at a surface state frequency takes place. Conclusion. A possibility has been demonstrated of controlling microwave electromagnetic properties of topological insulators by changing their geometric parameters and permittivity of the constituents. From a practical point of view, topological insulators can be used as components of microwave transmission lines and devices featuring very small propagation loss. Fig. 5. Ref.: 17 items.

Structures called topological insulators stand out in solid state physics. Th eir main distinguishing feature is that a topological insulator is insulating inside the bulk like a normal insulator, while on its surface it shows conductivity like a metal. Since their advent, the structures of the type have been of great interest among researchers and engineers [1,2]. Also, with advances in telecommunication technologies, physicists got a new opportunity to study the nature of unusual conductivity for spin [3] and electromagnetic waves, too [4][5][6]. Th e signifi cance of "topological" properties is highlighted in lots of theoretical and experimental publi-cations in the fi eld of topological photonics, displaying that with topological insulators a low-loss electromagnetic energy transmission is possible in both the optical [7,8] and microwave ranges [9][10][11][12][13][14]. Topological insulators of the type act as spin and electromagnetic analogues of "classical" electronic topological insulators, demonstrating similar properties for spin and electromagnetic waves.
Topological insulators can be constructed of either metal [9,10] or dielectric elements [11][12][13]. Notice that dielectric ones have a signifi cant advantage over other materials. Th ey can be used at the optical frequencies due to their low loss. Topo-L.I. Ivzhenko, S.Yu. Polevoy, E.N. Odarenko, S.I. Tarapov logical properties can also arise in structures based on magnetized ferrites [14]. However, these topological insulators require a fairly strong magnetic fi eld. Another thing is with topological insulators that are formed of ordinary dielectrics and do not require an external magnetic fi eld application. If made of ferrite, these topological insulators can work without external magnetic fi elds. Yet a weak magnetic fi eld provides extra tuning options, enabling magnetically tunable topological insulators.
Th e aim of the work is to determine the infl uence of geometrical and constitutive parameters of topological insulator elements on spectral and dispersion properties of a combined artifi cial topological insulator based on a two-dimensional and double-periodic array of dielectric elements. Numerical results demonstrating electromagnetic properties of two-dimensional (2D) dielectric topological insulators of various types are presented.
1. Dispersion properties of artifi cial topological insulators based on a double-periodic array of elliptical quartz elements. Firstly, in this section, the features of electromagnetic properties of all-dielectric electromagnetic analogues of topological insulators [11][12][13] are studied as applied to a double-periodic array of elliptical elements [16,17] made of quartz. Th e choice of this topological insulator type is reasoned by a possibility to transfer the microwave range results to the optical region, considering that metal elements in the optical range have higher losses than their dielectric analogs.
Th e use of a numerical simulation soft ware package MPB [15] made it possible to compute dispersion diagrams of the above-mentioned 2D topological insulator calculations over a band of frequencies.
For a 2D topological insulator, a double-periodic structure with a hexagonal unit cell and a cell period p (Fig. 1) is taken. Th e unit cell consists of six dielectric z-infi nite elliptical cylinders in a vacuum. Each cylinder axis and the unit cell axis are l spaced. Th e lines connecting the centers of two neighboring ellipses and the unit cell center make the angle   60 [11,16,17].
Each ellipse with half-axes a and b is rotated around its center through an angle C measured between the main half-axis a of the ellipse and the line going from the unit cell center to the ellipse center (see Fig. 1).
Initially, the structure made of elliptical quartz cylinders is studied for the infl uence of geometric and constitutive parameters of the structure elements on the electromagnetic properties. Th e geometric parameters of the unit cell and the structure period p are selected subject to certain conditions. Namely, the topological insulator operating frequency is about 10 GHz, and the band gap should be wide enough to overlap with the band gap of another topological insulator whose unit cell has somewhat diff erent parameters.
As known, to demonstrate the existence of surface electromagnetic oscillations and fi eld concentration, a structure is oft en used that is made up of two adjoining topological insulators diff ering in shapes of their unit cells [11][12][13]. In this case, the necessary condition for the existence of surface oscillations is overlap of band gaps of these topological insulators. Th e further search for parameter values at the interface between diff erent topological insulators shows that surface electromagnetic oscillations are possible. Th ese oscillations depend on the type of wave polarization (right circular or left circular, RCP or LCP) and exist in this case at any frequency within the overlapped band gap. At some frequencies, these oscillations have the same wavelength in the dispersion diagram. By numerical simulation, geometrical parameters of the unit cells were selected for two topological insulators and operating frequency area around 10 GHz. For elliptical quartz cylinders, the unit cell period is p  20.7 mm. Th e distance from the unit cell center to the center of each ellipse is l = p/3. Th e semi-axes of the ellipses making elliptical cylinders are a  0.155 p and b  0.095 p. Th e rotation angles of ellipses in two diff erent unit cells were obtained given a suffi ciently wide overlap of the band gaps of both topological insulators and measure С 1  0 and С 2  90 for topological properties to be achieved.
For the material of the elliptical cylinders, uniaxial-anisotropy quartz has been chosen. Let us begin with the infl uence of the anisotropy axis direction on the TI dispersion properties. Th e permittivity tensor of anisotropic quartz is with any two components on the main diagonal being the same, the third one is diff erent (uniaxial anisotropy). For the elliptical -quartz cylinders, the x-directed uniaxial anisotropy is initially taken so that  xx  4.847,  yy  4.643,  zz  4.643, which corresponds to the real values of the permittivity tensor components.
Th e dispersion diagrams calculated for the two topological insulators with the ellipse rotation angles С 1  0 and С 2  90 (Fig. 2) are plotted in Fig. 3. In it, the ordinate is the normalized frequency fp / c, where c is the speed of light in a vacuum and p is the structure period. Th e abscissa is the wave vector. Th e points Μ, Γ, Κ are special points inside the Brillouin zone in the case of a 2D photonic crystal (PC) with a hexagonal lattice [11].
Notice that for the topological insulator with the angle С 1  0 (Fig. 3, a), the band gap is in the frequency range fp / c  0.689…0.702, and its width is 1.89%. For the topological insulator with С 2  90 (Fig. 3, b), the band gap is in the frequency range fp / c  0.688…0.702, and its width is 2.03%.
Consider the action of the quartz anisotropy direction on the width and shift of the band gap. When uniaxial anisotropy of -quarts is along the y-axis, the permittivity tensor compo-nents are  xx  4.643,  yy  4.847, and  zz  4.643. For the topological insulator with the angle С 1  0 (Fig. 3, а), the band gap is in the frequency range fp / c  0.688…0.702, and it is 2.03% wide. For the topological insulator with С 2  90 (Fig. 3, b), the band gap is in the frequency range fp / c  0.689…0.702, and it is 1.89% wide.
When uniaxial anisotropy of -quartz is along the z-axis, the permittivity tensor components are  xx  4.643,  yy  4.643, and  zz  4.847. For the topological insulator with the angle С 1  0 (Fig. 3, а), the band gap is in the frequency range fp / c  0.675…0.691, and its width is 2.27%. For the topological insulator with the angle С 2  90 (Fig. 3, b), the band gap is within fp / c   0.676…0.691, and its width is 2.10%.
Th us, for all the three directions of the quartz uniaxial anisotropy axis, we have signifi cant over- laps of the band gaps of the topological insulators with the ellipse rotation angles С 1  0 and С 2  90. Th is makes it possible to design combined structures of two topological insulators, with the interface between them expected to support electromagnetic surface oscillations [11][12][13]. When the uniaxial anisotropy is z-directed, the band gap expands up to 10% against the other two cases (the x-and y-directed anisotropy). So, it has been just demonstrated that by choosing the uniaxial anisotropy direction of the material, the topological insulator electromagnetic properties can be controlled without any changes in other parameters of the structure.
In addition, the electromagnetic properties of the topological insulator are tunable by varying its geometric parameters. As the structure period p increases, with the relationship between elements and period fi xed, the band gap moves to the lower frequencies. Another way to fi ne-tune the band gap parameters is by varying distance l from the unit cell center to any ellipse center.
Th us, a possibility of controlling the electromagnetic properties of the topological insulator by changing its geometric parameters and by permittivity of its constituents has been numerically demonstrated.

Dispersion properties of combined artificial topological insulators based on a doubleperiodic array of elliptical quartz elements.
It is known that electromagnetic analogues of topological insulators can support electromagnetic surface oscillations on the surface area and behave as insulators in the bulk interior [4][5][6]. Th is effect can arise in a certain frequency range -within the band gaps of topological insulators. As soon as conditions for the maintenance of electromagnetic surface oscillations arise on the topological insulator surface, a low-loss energy transfer takes place owing to the waves travelling along the topological insulator boundary. In the previous section, these frequency ranges were determined for a certain topological insulator type. Th e role of the topological insulator boundary can be played by a metal wall [14] or by a topological insulator with diff erent unit cell parameters [11][12][13]. Th is section is devoted to a combined topological insulator in which two topological insulators with diff erent unit cells are interfaced with each other. Th e unit cell scheme (Fig. 1) for both topological insulators is generally the same. Yet some geometric parameters diff er. Th e virtue is that this combined structure is all-dielectric.
An example of the 2D combined topological insulator can be provided by the two previously considered topological insulator structures with diff erent angles С 1  0 (Fig. 2, a) and С 2  90 (Fig. 2, b) of the element rotation [11]. Th ese angles were selected by analysis of the dispersion diagrams (Fig. 3) of these structures in view of that the band gaps of the structures should overlap in some frequency interval, and the circular polarization directions for the allowed states at the ends of the band gap at the point Γ (k x  k y  0) in the dispersion diagram should be opposite. Th e so obtained combined structure of two interfaced topological insulators based on an array of elliptical quartz cylinders is schematized in Fig. 4, a.   5 presents dispersion diagrams calculated in the Γ-Κ direction for the combined structure (Fig. 4, a) composed of two adjoining topological insulators with the angles С 1  0 and С 2  90 of the ellipse rotation in the unit cells. In Fig. 5, the ordinate is the normalized frequency fp / c, the abscissa is the wave vector k x . Th e calculation results for the combined structures are presented for the quartz anisotropy axis along the z-axis (Fig. 5, a) and along the x-axis (Fig. 5, b).
As seen from Fig. 5, in the band gap of the combined structure of adjoining topological insulators, there exist two degenerate surface-state modes of p-type and d-type. Th ey are indicated by bold x Wavevector, k x Quartz anisotropy axis z lines and correspond to circular polarization waves (RCP and LCP). Th ese surface states meet at the point Γ (at k x  0) in the dispersion diagram [11]. It has been numerically shown that by choosing the quartz uniaxial anisotropy direction, the frequencies of topological insulator surface states can be controlled. Th us, for the structure of quartz cylinders with z-directed uniaxial anisotropy, the surface-state resonant modes meet near the point fp / c  0.685 (Fig. 5, a). For the structure with xdirected uniaxial anisotropy, they meet near the point fp / c  0.697 (Fig. 5, b). One more way of surface-state frequency control is by varying the periods of the structures. As the periods of the structures increase with the relationship between elements and period fi xed, the surface state modes move to the lower frequencies.
To observe surface oscillations in terms of electromagnetic fi eld concentration around the interface of the two topological insulators, numerical calculations were performed for the spatial distribution of the normal (z) component of the electric fi eld of the combined structure at the normalized frequency fp / c  0.69029 (the wave vectors are k x  0.02174 and k y  0.32609) (Fig. 4, b). Th is frequency falls within the band gap of each individual topological insulator. Th e quartz anisotropy axis is along the z-axis. Fig. 4, b shows (by darkening) that certain fi eld concentration exists at the interface between the two adjoining topological insulators and rapidly dies away from it along the x-axis.
Conclusions. Th e features of electromagnetic properties of electromagnetic analogues of topological insulators have been considered on the example of a two-dimensional double-periodic array of elliptical quartz cylinders.
A numerical method evaluating the spectral and dispersion properties and patterns of the topologi-cal insulator electromagnetic fi eld spatial distribution in the microwave range has been developed.
It has been shown that by choosing the uniaxial anisotropy direction of the topological insulator material, the electromagnetic properties of topological insulators are tuned without any changes in other parameters of the structure. Th e infl uence of the quartz anisotropy axis direction on the width and position of the band gap of a topological insulator based on a 2D double-periodic array of elliptical quartz cylinders has been studied.
Th e dispersion diagrams and patterns of electromagnetic fi eld spatial distribution have been evaluated for a combined structure consisting of two adjoining topological insulators whose unit cells differ in the angles, 0 and 90, of ellipse rotation. It has been numerically shown that the frequencies of the surface states in the dispersion diagram of the combined structure of adjoining topological insulators can be controlled by choosing the quartz uniaxial anisotropy direction. It has been demonstrated that the fi eld concentration occurs at the interface between the two topological insulators and rapidly decreases away from it.
As for practical applications, artifi cial combined topological insulators as those discussed in the paper can be part of microwave transmission lines and devices off ering very low-loss wave propagation.
Th e work is funded by the Ukrainian-Polish project "Properties and propagation of microwaves in nonreciprocal topological insulators based on magnetically tunable resonances in metamaterials" (Agreement M/104-2020 on 21.10.2020 between the Ministry of Education and Science of Ukraine and the O.Ya. Usikov Institute for Radiophysics and Electronics of the National Academy of Sciences of Ukraine).