Electromagnetic waves in a topological insulator thin film stack: helicon-like wave mode and photonic band structure

Jun-ichi Inoue

doi:10.1364/OE.21.021317

1. Introduction

Helicon waves are one of the canonical elements that characterize electromagnetic (EM) properties of bulk semiconductors in the low energy region [1–3]. Bulk semiconductors generally fail to propagate transverse EM waves whose frequencies are lower than the plasma frequency, because carriers in the sample can instantaneously follow a slow modulation induced by the EM wave, resulting in total reflection. However, once an external magnetic field is applied, for instance along the z-axis, a branch of the EM mode that is active for a specific circular polarization appears below the plasma frequency. This low energy mode, or helicon wave, propagates parallel along the direction of the applied magnetic field. The features of this mode are that in the long wavelength limit k_z → 0, the dispersion relation is gapless and quadratic, $ω \propto k_{z}^{2}$ ; also, the proportionality coefficient depends on the amplitude of the magnetic field.

The helicon wave can also propagate in a semiconductor system with a long period structure, as shown by Tselis et al. [4]. They considered a semiconductor superlattice in which each interface is assumed to be an infinitely thin two dimensional electron layer (2DEL) with conductivity tensor σ. Then they modeled the superlattice as stacked 2DELs along the z direction under an external magnetic field. This one dimensional periodic system, schematically shown in Fig. 1(a), is the simplest in the sense that each unit “cell” contains a single “site.” The authors in [4] had a keen interest in the long wavelength limit; they found a low energy EM mode whose nature was identical to that of the helicon wave in isotropic bulk semiconductors. They further applied their theory to a novel system called a quantum Hall layer stack (QHLS); in this stack each 2DEL was assumed to be in the quantum Hall state. In this case, since the diagonal conductivity is null, each layer is characterized solely by its own Hall conductivity. Although the helicon wave dispersion expected in the QHLS was also found to be gapless and quadratic, the curvature of the dispersion relation no longer depends on the magnetic field.

Fig. 1 (a) Schematic of a stack of quantum Hall layers, QHLS, which was considered in [4]. This system shows distinct EM properties between the two circular polarizations E_± and in the long wavelength limit, it has a gapless quadratic dispersion relation that is active only for E₊. (b) Schematic of our system, AQHLS, which is a stack of quantum Hall layers whose conductivities change signs layer by layer. The EM properties for the two polarizations are identical. (c) Topological insulator superlattice, which can be model by the system in (b). Gray (white) regions indicate topological (conventional) insulator thin films. (d) Asymmetric AQHLS (0 < α < 2).

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A natural extension of the QHLS shown in Fig. 1(a) is a one dimensional periodic system that has two kinds of sites in a unit cell. The two sites correspond to a pair of quantum Hall layers in which the sign of the Hall conductivity in one layer is opposite to that in the other layer, as shown schematically in Fig. 1(b). We refer to this system as an alternating QHLS (AQHLS). In the context of the previous study [4], the assessment of a low energy helicon like mode in AQHLS is a major concern.

One might think an AQHLS to be artificial, but it connects to a realistic system encountered in modern condensed matter physics; i.e., an AQHLS mimics a topological insulator superlattice. A topological insulator, whose typical examples are Bi₂Te₃ and Bi₂Se₃, has bulk properties similar to those of conventional band insulators, but has a novel property in that its surface state is metallic [5–7]. The surface state is expected to exhibit the quantum charge/spin Hall effect without an external magnetic field. The following discussion emphasizes the point that the two interfaces of a topological insulator in slab geometry have quantized Hall conductivities with opposite signs (shown later) [8]. Note that a similar system is expected by considering bianisotropic photonic metamaterials [9]. Exploiting these facts, we can use the system shown in Fig. 1(b) to model the topological insulator superlattice shown in Fig. 1(c).

The system considered here is also intriguing from the viewpoint of photonic band structures [10]. The fundamental physics of photonic bands has been extensively studied, but we find a nontrivial aspect to our system: Since the AQHLS is periodic, a finite band gap is naturally expected to open at the zone boundary. However, it might also be considered as a “vacuum” under an appropriate condition, because of the two interfaces having opposite signs in their Hall conductivities.

In this regard, an analytic formula for photonic bands would be helpful. An interest in photonic bands did not appear in the previous study of QHLS [4], and thus a comparison of photonic bands for QHLS and AQHLS is central issue addressed in this work. Furthermore, the photonic band structure is relevant when applications are considered. Since a transmission spectrum is a direct measure of a photonic band structure, calculation of the spectrum is essential not only to confirm the photonic band structure, but also to examine the possible applications for the system at hand.

In this paper, we explore properties of EM waves in a topological insulator superlattice. In the long wavelength limit, we find a gapless EM mode, which is a known feature of conventional helicon waves. However, in marked contrast, the dispersion relation is linear instead of quadratic, and the mode is active for both circular polarizations. We then analytically obtain explicit equations for the photonic band structures of the geometries in both Figs.1(a) and 1(b). The former gives a distinct photonic band for each circular polarization, one of which is gap-less and thus continuously connects to the quadratic helicon mode found in [4]. The latter, the model of the topological insulator superlattice, has two kinds of photonic bands, since the unit “cell” has two “sites.” Both photonic bands do not depend on the polarization direction. What should be emphasized is that one of the photonic bands has nowhere energy gaps, in spite of the presence of periodicity. This fact indicates that the AQHLS can be interpreted as a “vacuum” due to the contributions from pairs of layers having opposite signs for the conductivities. The resulting photonic band structure is compared with a numerically calculated transmission spectrum. Finally, by introducing asymmetry in AQHLS, as Fig. 1(d), the two features, the gapless linear low energy mode and a presence of the gapless band, are demonstrated not to be limited to the case of Fig. 1(b).

2. Helicon waves in stacked quantum Hall layers

The helicon wave in the QHLS shown in Fig. 1(a) was explored by Tselis et al. [4]. We begin by rewriting their theory in a form suitable for the below discussion, and we rederive the central results. Consider such an EM plane wave with monochromatic frequency ω that propagates along the z-axis in a medium uniform in the xy direction. From the Maxwell equation in a circularly polarized basis, the electric field E_±(z) is related to a current J_±(z) by

(\frac{\partial^{2}}{\partial z^{2}} - β^{2}) E_{\pm} (z) = \frac{4 π i ω}{c^{2}} J_{\pm} (z),

where X_±(z) = X_x(z) ± iX_y(z) (X = E, J), β² ≡ −εω²/c², ε is a dielectric constant of the medium and c is the speed of light in vacuum. The solution of Eq. (1) is

E_{\pm} (z) = \frac{1}{ξ} \int_{- \infty}^{\infty} d z^{'} exp (- β | z - z^{'} |) J_{\pm} (z^{'}),

where ξ = −βc²/(2πiω) = −ε^1/2c/(2π) [11]. The current in turn is related to the electric field by Ohm’s law, J_±(z) = σ_±(z)E_±(z), where σ_±(z) ≡ σ̂_xx(z) ∓ iσ̂_xy(z), and the conductivity tensor is defined by

σ (z) = (\begin{matrix} {\hat{σ}}_{x x} (z) & {\hat{σ}}_{x y} (z) \\ - {\hat{σ}}_{x y} (z) & {\hat{σ}}_{x x} (z) \end{matrix}) .

Then the solution Eq. (2) can be written in terms of E_± as

E_{\pm} (z) = \frac{1}{ξ} \int_{- \infty}^{\infty} d z^{'} exp (- β | z - z^{'} |) σ_{\pm} (z^{'}) E_{\pm} (z^{'}) .

The system under study has infinitely thin 2DELs regularly distributed along the z-axis with period 2d. As such, the conductivity tensor has a finite value within the layers, and σ_±(z) for the whole system can be represented using an integer m,

σ_{\pm} (z) = σ_{\pm} \sum_{m} δ (z - 2 d m) .

Here we consider the case in which each 2DEL is in the quantum Hall state. Thus the diagonal component σ̂_xx is set to zero, leading to σ_± = ∓iσ̂_xy. Using the plane wave ansatz E_±(z) = E_±0 exp(ikz), we have a simple algebraic equation

ξ = σ_{\pm} S,

which determines a dispersion relation. In this equation,

S \equiv \frac{i sin Ω}{cos Ω - cos K},

iΩ ≡ 2βd = iωε^1/2(2d/c) and K ≡ 2kd are defined. Considering the low energy mode Ω ≪ 1 in the long wave length limit K → 0 and expanding the trigonometric functions up to lowest order, we reproduce the result [4]

Ω_{+} ~ K^{2} .

This quadratic form of the dispersion relation is one of the hallmarks of the conventional helicon wave in semiconductors under a magnetic field. This mode is only active for E₊, as demonstrated in Sec. 4, although this is not explicitly stated in the theory in [4].

3. Low energy EM waves in a stack of alternating quantum Hall layers

3.1. Equivalence of a topological insulator superlattice and an AQHLS

We verify our claim that the AQHLS is a model for a topological insulator superlattice. This fact relies on a novel EM property inherent to topological insulators [12–16]. The topological EM response is described within a theoretical framework termed axion electrodynamics [17], originally introduced in the context of high energy physics. As a prerequisite, time reversal symmetry should be broken. One of the methods proposed for that is to make a contact with a magnetic material [12]. In our case, for instance, transparent ferromagnetic material, TiO₂ codoped with Fe and Nb [18], is suitable [19,20], and this kind of material having a finite energy gap is assumed as a conventional insulator in the following discussion, although fabrication of AQHLS on target, in particular alignment of magnetization direction, is challenging.

Peculiar EM responses from topological insulators are captured by considering an axion term whose Lagrangian is ℒ_ax = θ ∫ drE · B. Here, the coupling coefficient θ is a topological number intrinsic to a given material. However, the addition of this term to the conventional Lagrangian for the Maxwell theory does not produce a substantial difference in the corresponding Euler–Lagrange equation, since ℒ_ax is essentially a term given by the total derivative of another quantity [21]. The net effect of the term θ is incorporated into the Maxwell theory by an extension of conventional constitutive relations from D = εE and H = (1/μ₀)B to D = εE − a_topB and H = (1/μ₀)B + b_topE [12, 13]. The second terms in the latter two equations are referred to as cross correlated terms [22].

At the interface of a topological insulator, the effect of the cross correlated terms manifests itself as quantized Hall conductivity on the interface. Consider the interface at z = 0 between topological and conventional insulators; the former (latter) fills the entire region of z ≥ 0(z < 0). In this geometry, the constitutive relations can be written as

D = ε (z) E - a_{top} Θ (z) B,

H = \frac{1}{μ_{0}} B + b_{top} Θ (z) E,

where Θ(z) is the Heaviside step function and the dielectric function is assumed to be ε(z) = ε₁Θ(−z)+ε₂Θ(z). With use of the fact that a_top = b_top ∝ θ, which can be proved from a field theory for topological insulators [12, 13], substitution of the constitutive relations into the Maxwell equation for z ≥ 0 provides us with

\frac{1}{μ_{0}} \frac{\partial}{\partial z} (\begin{matrix} - B_{y} \\ B_{x} \end{matrix}) = - i ε_{2} ω (\begin{matrix} E_{x} \\ E_{y} \end{matrix}) + b_{top} δ (z) (\begin{matrix} E_{y} \\ - E_{x} \end{matrix}),

where, once again, a monochromatic plane EM wave is assumed. Next starting from the Maxwell equation with a current term ∇ × H = (∂D/∂t) + J, the quantized Hall current J_x(y) = δ(z)σ̂_xy(yx)E_y(x) and Onsager’s relation σ̂_yx = −σ̂_xy, we arrive at

\frac{1}{μ_{0}} \frac{\partial}{\partial z} (\begin{matrix} - B_{y} \\ B_{x} \end{matrix}) = - i ε_{2} ω (\begin{matrix} E_{x} \\ E_{y} \end{matrix}) + {\hat{σ}}_{x y} δ (z) (\begin{matrix} E_{y} \\ - E_{x} \end{matrix}) .

We can immediately observe that the second term on the right hand side of Eq. (11) is equivalent to that given by the quantized Hall current on the surface at z = 0 [19]. Accordingly, we now have two ways to describe EM properties in topological materials: the use of the cross correlated terms, and the use of surface quantized Hall conductivity.

We can generalize the above discussion to a case with two interfaces. A topological insulator thin film sandwiched by conventional insulators has surface quantized Hall conductivities on both interfaces. Note that the signs of the Hall conductivities are opposite to one another [8]; this can be easily seen from the fact that the cross correlated term in H has the form b_top[Θ(z) − Θ(z − d)]E for a film with thickness d. Thus, the topological insulator superlattice in Fig. 1(c) can be modeled by the system shown in Fig. 1(b).

Before moving to a discussion about low energy EM modes, we emphasize that the scheme of using the surface quantized Hall conductivity is advantageous over the alternatives. One reason is that the scheme fully utilizes the novel nature of topological insulators. In addition, compared to other approaches, the method reduces the complexity of the problem and enables a simple analysis. The latter advantage is essential in deriving analytic formulas for photonic band structures in Sec. 4. To highlight the role of quantum Hall layers and to emphasize differences between AQHLS and QHLS, we set ε₂/ε₁ = 1 throughout this work.

3.2. Gapless linear EM mode

Having demonstrated the equivalence between the AQHLS shown in Fig. 1(b) and the topological insulator superlattice, we now characterize the expected low energy EM modes. The analysis can be performed by generalizing the procedure in Sec. 2. The Hall conductivity in each layer changes sign layer by layer, and the conductivity for the system is denoted by

σ_{\pm} (z) = σ_{\pm} [\sum_{m} δ (z - 2 d m) - \sum_{m} δ (z - 2 d m - d)],

with an integer m. Consequently, the electric field is written in the form

E_{\pm} (z) = \frac{σ_{\pm}}{ξ} [\sum_{z^{'} : +} exp (- β | z - z^{'} |) E_{\pm}^{(+)} (z^{'}) - \sum_{z^{'} : -} exp (- β | z - z^{'} |) E_{\pm}^{(-)} (z^{'})],

where the symbol ∑_z′:± indicates the sum with respect to the position z′ of a layer with ±σ, E⁽⁺⁾(z = 2dm) and E⁽⁻⁾(z = 2dm + d) are electric fields at an interface with +σ and −σ, respectively.

Now consider an electric field $E_{\pm}^{(+)} (z = 2 d m)$ on the left hand side of Eq. (14). Substituting the ansatz $E_{\pm}^{(+)} (2 d m) = E_{\pm 0}^{(+)} exp (i k 2 d m)$ and $E_{\pm}^{(-)} (2 d m + d) = E_{\pm 0}^{(-)} exp (i k (2 d m + d))$ in Eq. (14), we are lead to one of the secular equations,

ξ E_{\pm 0}^{(+)} = σ_{\pm} S E_{\pm 0}^{(+)} - σ_{\pm} S^{'} E_{\pm 0}^{(-)},

where S is given in Eq. (7), and

S^{'} = \frac{2 i sin \frac{Ω}{2} cos \frac{K}{2}}{cos Ω - cos K},

is newly introduced. Similarly, we find the other secular equation by considering

E_{\pm}^{(-)}

on the left hand side of Eq. (14):

ξ E_{\pm 0}^{(-)} = σ_{\pm} S^{'} E_{\pm 0}^{(+)} - σ_{\pm} S E_{\pm 0}^{(-)} .

From the condition that the set of Eqs. (15) and (17) has nontrivial solutions, we obtain an equation to determine a dispersion relation

ξ^{2} - σ_{\pm}^{2} (S^{2} - {S^{'}}^{2}) = 0 .

Taking the long wavelength limit K → 0 and confining ourselves to a low energy region, as in Sec. 2, a dispersion relation corresponding to Eq. (8) is obtained:

Ω_{1 \pm} \approx \frac{K}{{(1 + A)}^{1 / 2}},

which is once again gapless and A≡ −(σ̂_±/ξ)² = (2πσ̂_xy/ε^1/2c)² is defined. However, we can observe two sharp contrasts: the mode is active for both circular polarizations, and the wavenumber dependence is linear instead of quadratic.

4. Electromagnetic dispersion relation: analytic form of one dimensional photonic bands

Since the present systems contain a spatial periodicity, the EM dispersion relation should reflect this structure and thus construct a one dimensional photonic band. To show this, Eqs. (6) and (18), which were used to explore the low-energy EM modes, are also applied over the entire energy region, as long as we assumed that material parameters are insensitive to the frequency under consideration.

A photonic band structure in QHLS for E₊ and E₋ polarizations should be separately determined from the equation ξ = σ₊S and ξ = σ₋S, respectively. The solutions are straightforwardly found to be

Ω_{1} = {cos}^{- 1} [\frac{cos K + {(A^{2} + A {sin}^{2} K)}^{1 / 2}}{1 + A}],

Ω_{2} = {cos}^{- 1} [\frac{cos K - {(A^{2} + A {sin}^{2} K)}^{1 / 2}}{1 + A}] .

Here there is a subtlety; in the course of solving those equations, we square both sides of the equations. This procedure, however, provides the common equation ξ² = (σ₊S)² = (σ₋S)² for the two polarizations, and information with respect to the polarization has been lost. Consequently, there is no one to one correspondence between the solutions of Ω_1,2 and the EM modes Ω_± for E_±. To recover the correspondence, we need to return to the original equations ξ = σ_±S.

From a careful classification, we find that the nth branch of the photonic band for E₊ polarization is $Ω_{+}^{(n = 2 m - 1)} = Ω_{1}$ , $Ω_{+}^{(n = 2 m)} = Ω_{2}$ , and that for E₋ polarization is $Ω_{-}^{(n = 2 m - 1)} = Ω_{2}$ , $Ω_{-}^{(n = 2 m)} = Ω_{1}$ . These are combined into a single equation

Ω_{\pm}^{(n)} = {cos}^{- 1} [\frac{cos K \mp {(- 1)}^{n} {(A^{2} + A {sin}^{2} K)}^{1 / 2}}{1 + A}] .

To illustrate this formula, consider Ω₁, for instance, in the vicinity of K = 0 and Ω₁ = 2π, whose curves are shown in Fig. 2. Rewrite the equation for E₊ (E₋) polarization, ξ = σ₊S (ξ = σ₋S), in the form cosΩ₁ − cosK = −asinΩ₁ (cos Ω₁ − cosK = +asinΩ₁), where a ≡ 2πσ̂_xy/(ε^1/2c) is assumed to be positive. For K = 0, the left-hand side of the equation is cosΩ₁ − 1 ≤ 0, and thus the right-hand side should be negative. This indicates that Ω₁ ≥ 2π (Ω₁ < 2π) in the vicinity of Ω₁ = 2π. Consequently, the upper (lower) branch in Fig. 2 is assigned to the mode for E₊ (E₋). Repeating this sort of discussion, we finally arrive at Eq. (22). The photonic band structures

Ω_{\pm}^{(n)}

with corresponding transmission spectra (discussed in Sec. 5) are illustrated in Figs. 3(a) and 3(b), respectively. The band gaps appear between the branches, all with equal magnitude given by θ = cos⁻¹[(1 − A)/(1 + A)]. Thus the bandwidth π−θ is identical for all branches in

Ω_{\pm}^{(n)}

.

Fig. 2 Two branches of Ω₁, from Eq. (20), in the vicinity of K = 0 and Ω₁ = 2π.

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Fig. 3 Photonic band structures (a) Ω₊ and (b) Ω₋ for E₊ and E₋, respectively, in a QHLS as functions of K. The right panels in (a) and (b) are corresponding transmission spectra T as functions of Ω.

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The helicon wave discussed in Sec. 2 should appear in the gapless photonic band structure Ω₊ in the long wavelength limit. Consider the lowest branch of the photonic band, $Ω_{+}^{(1)}$ , which satisfies the equation $cos Ω_{+}^{(1)} = {(1 + A)}^{- 1} (cos K + {(A^{2} + A {sin}^{2} K)}^{1 / 2})$ . Expanding the left and right hand sides up to the second and fourth orders, respectively, one finds the relation ${(Ω_{+}^{(1)})}^{2} ~ K^{4}$ , which reproduces the gapless quadratic dispersion relation. This derivation also proves that the helicon mode obtained in Sec. 2 is active only for E₊ polarization. On the other hand, $Ω_{-}^{(1)}$ has a finite energy gap in the K → 0 limit, and thus there is no low energy EM mode active for E₋ polarization.

The AQHLS provides photonic bands qualitatively different from those in QHLS. The photonic bands are obtained from Eq. (18),

A (S^{2} - {S^{'}}^{2}) = - 1 .

This equation commonly holds for both polarizations E_±, in contrast to the case of QHLS. The two solutions of Eq. (23) are

Ω_{1 \pm} = {cos}^{- 1} [\frac{A + cos K}{1 + A}],

Ω_{2 \pm} = K .

The former photonic band Ω_1± with corresponding transmission spectrum is shown in Fig. 4(a). This band structure has identical energy gaps ΔΩ_1± = 2θ and bandwidth 2(π−θ), except for the lowest photonic band whose width is π−θ.

Fig. 4 (a) Photonic band structure Ω_1± and transmission spectrum in AQHLS. (b) “Vacuum” like photonic band Ω_2± found in AQHLS.

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In the long wavelength limit K → 0 the dispersion is gapless and hence should coincide with the low energy mode in Eq. (19). This fact is confirmed by expanding Ω_1± in the vicinity of the origin. Indeed, we reproduce the relation, Ω_1± ≈ K/(1 + A)^1/2, from Eq. (24). As aforementioned, the mode is linear, gapless, and active for both polarizations E_±. Hence, this mode is a generalization of the conventional helicon wave found in bulk semiconductors under magnetic fields.

The other mode, Ω_2± = K, is quite characteristic in that it is identical to the mode in vacuum. The dispersion relation in a reduced zone scheme is shown in Fig. 4(b). The interesting point is that there are no band gaps, although it is well known that even an empty lattice would be sufficient to open a finite energy gap at the zone boundaries. Our finding may allow us to have a simple view: the present system contains an aspect of an effective “vacuum”. The photonic band Ω_2± does not involve the lower energy mode Eq. (19), although Ω_2± is also linear and gapless. This is because the “velocity” of the low energy helicon related mode Eq. (19) depends on material details A, while Ω_2± does not. Thus, only Ω_1± contains the low energy mode.

In the present study of photonic bands, the scheme using the surface Hall conductivity is essential; this scheme enables us to obtain analytic formulas of photonic bands, Eqs. (22), (24) and (25). Furthermore, the interpretation that one aspect of the AQHLS is effectively “vacuum” in the context of EM response would not be available unless this scheme is employed.

5. Transmission spectra

The analytic formulas for photonic bands are also supported by their comparison with independently obtained transmission spectra. Photonic band structures directly connect to transmission spectra; the spectra have finite values in an energy range that contains photonic bands, but they are zero in photonic gap regions. The latter indicates total reflection. Consider the geometries in Figs. 1(a) and 1(b) as before, but now let them have finite lengths N × 2d in the z direction (we set N = 30). We numerically calculate transmission spectra for this geometry. For this purpose, we employ the transfer matrix method and calculate electric field amplitudes $E_{\pm}^{out}$ transmitted through the system against an electric field $E_{\pm}^{in}$ incident on the surface from the normal direction. These two electric fields are related by $E_{\pm}^{out} = t_{\pm} (ω) E_{\pm}^{in}$ , and the transmission spectrum is defined as T_±(ω) = |t_±(ω)|² for each polarization.

The transmission spectra for QHLS are shown in the right panels in Figs. 3(a) and 3(b) for E₊ and E₋ polarizations, respectively. The fine fringes in the spectra are due to finite sized effects. In both polarizations, the transmission spectra have two values: T = 1 and T = 0 (note that real dielectric constants are assumed). The former appears in energy regions where the photonic bands are constructed, whereas the latter corresponds to photonic band gaps. Similarly, the transmission spectrum for AQHLS is shown in the right panel in Fig. 4(a). This spectrum corresponds to Ω_1±. As in the case of QHLS, the transmission spectrum shows good agreement with the photonic band structure Ω_1±. We confirm that the spectra are common to both polarizations E_±.

6. Asymmetric AQHLS

So far, we have emphasized the two features in AQHLS: the linear gapless low energy mode Ω_1± ∼ K, and the presence of a gapless photonic band Ω_2± = K. Both of these are common for the two circular polarizations. Since these results are derived in a limited case, we now relax the conditions imposed on AQHLS and introduce asymmetry in the model. We show that these two features are still valid.

Consider an asymmetric AQHLS shown in Fig. 1(d), where the layers with σ₋ are located at z = 2dm +αd (0 < α < 2), while those with σ₊ are at z = 2dm. From a cumbersome but parallel calculation with that for symmetric case (i.e., α = 1), the secular equation for general α is found to be

2 A (cos Ω - cos K) sin (\frac{α}{2} Ω) sin (\frac{2 - α}{2} Ω) = {(cos Ω - cos K)}^{2} .

Thus, we can immediately find relations that the two modes, Ω_1± and Ω_2±, should satisfy:

2 A sin (\frac{α}{2} Ω_{1 \pm}) sin (\frac{2 - α}{2} Ω_{1 \pm}) = cos Ω_{1 \pm} - cos K,

cos Ω_{2 \pm} - cos K = 0 .

From the former, when Ω, K ≪ 1, the gapless linear low energy mode

Ω_{1 \pm} \approx \frac{K}{{1 + A α (2 - α)}^{1 / 2}},

and from the latter, the gapless photonic band Ω_2± = K, are obtained, respectively. Thus those two features found in the symmetric case are not accidental, but robust against the introduction of the asymmetry. Interestingly, we can observe that Eq. (26) reduces to (cosΩ − cosK)² = 0, by intentionally setting α = 0 and 2. Consequently, we obtain doubly degenerated gapless bands, Ω_1± = Ω_2± = K. Note that when the cross correlated terms, or equivalently σ_±, are removed from AQHLS, a conventional dispersion relation of EM waves, ω = (c/ε^1/2)k, or Ω = K, should be obtained, since the AQHLS now loses the spatial periodicity and then turns into a uniform system.

In contrast to those features robust against α ≠ 1, we find an aspect that is limited in the symmetric case; the EM modes of Ω_1± = 2nπ (n = 1,2,···) at K = 0 in symmetric AQHLS disappear once the asymmetry is introduced. Indeed, we can see in Eq. (27) that when α ≠ 1, although Ω_1± = 0 and K = 0 is still its solution, Ω_1± = 2nπ ceases to be a solution at K = 0. Instead, Ω_1± opens finite band gaps at these points. Thus, the zero gap at (K, Ω_1±) = (0, 2nπ) is limited in the case of α = 1.

7. Summary and conclusions

We have discussed EM mode properties in a topological insulator superlattice. We demonstrated that the system is equivalent to a stack of quantum Hall layers whose conductivity tensors alternately change sign; thus, the topological insulator superlattice is a natural extension of the quantum Hall layers stack that was investigated previously. Particular interest was paid to both the low energy EM modes and the overall dispersion relations that construct the photonic bands due to the periodicity of the spatial structure. The former is motivated by previous work in the context of helicon waves. After rewriting the previous theory in a form useful for this work, we found that the low energy EM mode in our system is gapless, similar to that in the previous theory, but it exhibits linear dependence on the wave number instead of quadratic dependence. We also found that the mode is commonly active for both circular polarizations. Generalizing the procedure for helicon wave analysis, we derived analytic formulas for the photonic band structures of AQHLS as well as QHLS. A specific feature found in AQHLS is a presence of a gapless band in spite of the periodicity that the system maintains. Also, we numerically calculated the transmission spectra in both systems and confirmed that the spectra agree with photonic band structures obtained analytically. This fact demonstrates the effectiveness of the approach taken in this work. Finally, the two features found in AQHLS, gapless linear low energy mode and a presence of a gapless band, still hold when the system contains asymmetry.

Acknowledgments

The author was supported in part by Grant-in-Aid for Scientific Research (C) 22540340 from MEXT, Japan.

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Electromagnetic waves in a topological insulator thin film stack: helicon-like wave mode and photonic band structure

Abstract

1. Introduction

2. Helicon waves in stacked quantum Hall layers

3. Low energy EM waves in a stack of alternating quantum Hall layers

3.1. Equivalence of a topological insulator superlattice and an AQHLS

3.2. Gapless linear EM mode

4. Electromagnetic dispersion relation: analytic form of one dimensional photonic bands

5. Transmission spectra

6. Asymmetric AQHLS

7. Summary and conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Equations (29)

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