We theoretically explore the electromagnetic modes specific to a topological insulator superlattice in which topological and conventional insulator thin films are stacked periodically. In particular, we obtain analytic formulas for low energy mode that corresponds to a helicon wave, as well as those for photonic bands. We illustrate that the system can be modeled as a stack of quantum Hall layers whose conductivity tensors alternately change signs, and then we analyze the photonic band structures. This subject is a natural extension of a previous study by Tselis et al., which took into consideration a stack of identical quantum Hall layers but their discussion was limited into a low energy mode. Thus we provide analytic formulas for photonic bands and compare their features between the two systems. Our central findings in the topological insulator superlattice are that a low energy mode corresponding to a helicon wave has linear dispersion instead of the conventional quadratic form, and that a robust gapless photonic band appears although the system considered has spacial periodicity. In addition, we demonstrate that the photonic bands agree with the numerically calculated transmission spectra.
© 2013 OSA
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 kz → 0, the dispersion relation is gapless and quadratic, ; 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. . 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  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.
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 , 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) . Note that a similar system is expected by considering bianisotropic photonic metamaterials . 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 . 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 , 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 . 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. . 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) byEq. (1) is 11]. The current in turn is related to the electric field by Ohm’s law, J±(z) = σ±(z)E±(z), where σ±(z) ≡ σ̂xx(z) ∓ iσ̂xy(z), and the conductivity tensor is defined by Eq. (2) can be written in terms of E± as 4] 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 , 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 . In our case, for instance, transparent ferromagnetic material, TiO2 codoped with Fe and Nb , 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 ℒax = 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 ℒax is essentially a term given by the total derivative of another quantity . 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 − atopB and H = (1/μ0)B + btopE [12, 13]. The second terms in the latter two equations are referred to as cross correlated terms .
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 as12, 13], substitution of the constitutive relations into the Maxwell equation for z ≥ 0 provides us with Eq. (11) is equivalent to that given by the quantized Hall current on the surface at z = 0 . 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 ; this can be easily seen from the fact that the cross correlated term in H has the form btop[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 byEq. (7), and Eq. (14): Eqs. (15) and (17) has nontrivial solutions, we obtain an equation to determine a dispersion relation Eq. (8) is obtained:
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
From a careful classification, we find that the nth branch of the photonic band for E+ polarization is , , and that for E− polarization is , . These are combined into a single equationFig. 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 ≡ 2πσ̂xy/(ε1/2c) 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 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 .
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, , which satisfies the equation . Expanding the left and right hand sides up to the second and fourth orders, respectively, one finds the relation , 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, 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),Eq. (23) are Fig. 4(a). This band structure has identical energy gaps ΔΩ1± = 2θ and bandwidth 2(π−θ), except for the lowest photonic band whose width is π−θ.
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 Ω1± in the vicinity of the origin. Indeed, we reproduce the relation, Ω1± ≈ 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, Ω2± = 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 Ω2± does not involve the lower energy mode Eq. (19), although Ω2± 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 Ω2± does not. Thus, only Ω1± 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 transmitted through the system against an electric field incident on the surface from the normal direction. These two electric fields are related by , 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 Ω1±. As in the case of QHLS, the transmission spectrum shows good agreement with the photonic band structure Ω1±. 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 Ω1± ∼ K, and the presence of a gapless photonic band Ω2± = 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 beEq. (26) reduces to (cosΩ − cosK)2 = 0, by intentionally setting α = 0 and 2. Consequently, we obtain doubly degenerated gapless bands, Ω1± = Ω2± = K. Note that when the cross correlated terms, or equivalently σ±, are removed from AQHLS, a conventional dispersion relation of EM waves, ω = (c/ε1/2)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 Ω1± = 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 Ω1± = 0 and K = 0 is still its solution, Ω1± = 2nπ ceases to be a solution at K = 0. Instead, Ω1± opens finite band gaps at these points. Thus, the zero gap at (K, Ω1±) = (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.
The author was supported in part by Grant-in-Aid for Scientific Research (C) 22540340 from MEXT, Japan.
References and links
1. E. D. Palik and J. K. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Rep. Prog. Phys. 33, 1193–1322 (1970). [CrossRef]
2. P. M. Platzman and P. A. Wolff, Waves and Interactions in Solids State Plasma, Solid State Phys. 13 (Academic Press, 1972).
3. J. J. Quinn and K-s. Yi, Solid State Physics: Principles and Modern Applications (Springer, 2009).
4. A. C. Tselis and J. J. Quinn, “Retardation effects and transverse collective excitations in semiconductor superlattices,” Phys. Rev. B 29, 2021–2027 (1984). [CrossRef]
5. M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010). [CrossRef]
6. X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011). [CrossRef]
7. M. Z. Hasan and J. E. Moore, “Three-dimensional topological insulators,” Ann. Rev. Condens. Matter Phys. 2, 55–78 (2011). [CrossRef]
8. A. M. Essin, J. E. Moore, and D. Vanderbilt, “Magnetoelectric polarizability and axion electrodynamics in crystalline insulators,” Phys. Rev. Lett. 102, 146805-1–146805-4 (2009). [CrossRef]
9. C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,”IEEE J. Sel. Top. Quant. Electron. 16, 367–375 (2010).
10. T. Ochiai, “Theory of Light Scattering in Axion Electrodynamics,” J. Phys. Soc. Jpn. 81, 094401–094408 (2012). [CrossRef]
11. K. W. Chiu and J. J. Quinn, Phys. Rev. B 9, “Plasma oscillations of a two-dimensional electron gas in a strong magnetic field,” 4724–4732 (1974). [CrossRef]
12. X. L. Qi, T. L. Hughes, and S. C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B 78, 195424-1–195424-43 (2008). [CrossRef]
14. J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803-1–166803-4 (2010). [CrossRef]
15. W. K. Tse and A. H. MacDonald, “Giant magneto-optical Kerr effect and universal Faraday effect in thin-film topological insulators,” Phys. Rev. Lett. 105, 057401-1–057401-4 (2010). [CrossRef]
16. M. C. Chang and M. F. Yang, “Optical signature of topological insulators,” Phys. Rev. B 80, 113304-1–113304-4 (2009). [CrossRef]
18. E. Sakai, A. Chikamatsu, Y. Hirose, T. Shimada, and T. Hasegawa, “Magnetic and transport properties of anatase TiO2 codoped with Fe and Nb,” Appl. Phys. Express 3, 043001-1–043001-3 (2010).
20. J. Inoue and A. Tanaka, “Photoinduced spin Chern number change in a two-dimensional quantum spin Hall insulator with broken spin rotational symmetry,” Phys. Rev. B 85, 125425-1–125425-7 (2012). [CrossRef]
21. W. Dittrich and M. Reuter, Selected Topics in Gauge Theories (Springer, 1986).
22. M. Fiebig, “Revival of the magnetoelectric effect,” J. Phys. D: Appl. Phys. 38, R123–R152 (2005). [CrossRef]