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

Optics Express, Vol. 21, Issue 18, pp. 21317-21328 (2013)

http://dx.doi.org/10.1364/OE.21.021317

Acrobat PDF (825 KB)

### Abstract

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

## 1. Introduction

1. E. D. Palik and J. K. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Rep. Prog. Phys. **33**, 1193–1322 (1970). [CrossRef]

*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*→ 0, the dispersion relation is gapless and quadratic,

_{z}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]

_{2}Te

_{3}and Bi

_{2}Se

_{3}, has bulk properties similar to those of conventional band insulators, but has a novel property in that its surface state is metallic [5

5. M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys. **82**, 3045–3067 (2010). [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]

10. T. Ochiai, “Theory of Light Scattering in Axion Electrodynamics,” J. Phys. Soc. Jpn. **81**, 094401–094408 (2012). [CrossRef]

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]

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]

## 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

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]

16. M. C. Chang and M. F. Yang, “Optical signature of topological insulators,” Phys. Rev. B **80**, 113304-1–113304-4 (2009). [CrossRef]

17. F. Wilczek, “Two applications of axion electrodynamics,” Phys. Rev. Lett. **58**, 1799–1802 (1987). [CrossRef] [PubMed]

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]

_{2}codoped with Fe and Nb [18], is suitable [19

19. J. Inoue, “An optical test for identifying topological insulator thin films,” Opt. Express **21**, 8564–8569 (2013). [CrossRef] [PubMed]

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]

*ε*

_{2}/

*ε*

_{1}= 1 throughout this work.

### 3.2. Gapless linear EM mode

*m*. Consequently, the electric field is written in the form where the symbol ∑

_{z′:±}indicates the sum with respect to the position

*z′*of a layer with ±

*σ*,

*E*

^{(+)}(

*z*= 2

*dm*) and

*E*

^{(−)}(

*z*= 2

*dm*+

*d*) are electric fields at an interface with +

*σ*and −

*σ*, respectively.

*S*is given in Eq. (7), and is newly introduced. Similarly, we find the other secular equation by considering

*K*→ 0 and confining ourselves to a low energy region, as in Sec. 2, a dispersion relation corresponding to Eq. (8) is obtained: which is once again gapless and

*A*≡ −(

*σ̂*

_{±}/

*ξ*)

^{2}= (2

*πσ̂*/

_{xy}*ε*

^{1/2}

*c*)

^{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

*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 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

*ξ*

^{2}= (

*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*.

*n*th branch of the photonic band for

*E*

_{+}polarization is

*E*

_{−}polarization is

_{1}, for instance, in the vicinity of

*K*= 0 and Ω

_{1}= 2

*π*, whose curves are shown in Fig. 2. Rewrite the equation for

*E*

_{+}(

*E*

_{−}) polarization,

*ξ*=

*σ*

_{+}

*S*(

*ξ*=

*σ*

_{−}

*S*), in the form cosΩ

_{1}− cos

*K*= −

*a*sinΩ

_{1}(cos Ω

_{1}− cos

*K*= +

*a*sinΩ

_{1}), where

*a*≡ 2

*πσ̂*/(

_{xy}*ε*

^{1/2}

*c*) is assumed to be positive. For

*K*= 0, the left-hand side of the equation is cosΩ

_{1}− 1 ≤ 0, and thus the right-hand side should be negative. This indicates that Ω

_{1}≥ 2

*π*(Ω

_{1}< 2

*π*) in the vicinity of Ω

_{1}= 2

*π*. 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

*θ*= cos

^{−1}[(1 −

*A*)/(1 +

*A*)]. Thus the bandwidth

*π*−

*θ*is identical for all branches in

_{+}in the long wavelength limit. Consider the lowest branch of the photonic band,

*E*

_{+}polarization. On the other hand,

*K*→ 0 limit, and thus there is no low energy EM mode active for

*E*

_{−}polarization.

*E*

_{±}, in contrast to the case of QHLS. The two solutions of Eq. (23) are 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±}= 2

*θ*and bandwidth 2(

*π*−

*θ*), except for the lowest photonic band whose width is

*π*−

*θ*.

*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.

## 5. Transmission spectra

*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

_{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.

*α*≠ 1, we find an aspect that is limited in the symmetric case; the EM modes of Ω

_{1±}= 2

*nπ*(

*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±}= 2

*nπ*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, 2

*nπ*) is limited in the case of

*α*= 1.

## 7. Summary and conclusions

## Acknowledgments

## References and links

1. | E. D. Palik and J. K. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Rep. Prog. Phys. |

2. | P. M. Platzman and P. A. Wolff, |

3. | J. J. Quinn and K-s. Yi, |

4. | A. C. Tselis and J. J. Quinn, “Retardation effects and transverse collective excitations in semiconductor superlattices,” Phys. Rev. B |

5. | M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys. |

6. | X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. |

7. | M. Z. Hasan and J. E. Moore, “Three-dimensional topological insulators,” Ann. Rev. Condens. Matter Phys. |

8. | A. M. Essin, J. E. Moore, and D. Vanderbilt, “Magnetoelectric polarizability and axion electrodynamics in crystalline insulators,” Phys. Rev. Lett. |

9. | C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,”IEEE J. Sel. Top. Quant. Electron. |

10. | T. Ochiai, “Theory of Light Scattering in Axion Electrodynamics,” J. Phys. Soc. Jpn. |

11. | K. W. Chiu and J. J. Quinn, Phys. Rev. B |

12. | X. L. Qi, T. L. Hughes, and S. C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B |

13. | X. L. Qi, J. Zang, and S. C. Zhang, “Inducing a magnetic monopole with topological surface states,” Science |

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. |

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. |

16. | M. C. Chang and M. F. Yang, “Optical signature of topological insulators,” Phys. Rev. B |

17. | F. Wilczek, “Two applications of axion electrodynamics,” Phys. Rev. Lett. |

18. | E. Sakai, A. Chikamatsu, Y. Hirose, T. Shimada, and T. Hasegawa, “Magnetic and transport properties of anatase TiO |

19. | J. Inoue, “An optical test for identifying topological insulator thin films,” Opt. Express |

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 |

21. | W. Dittrich and M. Reuter, |

22. | M. Fiebig, “Revival of the magnetoelectric effect,” J. Phys. D: Appl. Phys. |

**OCIS Codes**

(310.6860) Thin films : Thin films, optical properties

(160.5293) Materials : Photonic bandgap materials

(310.6805) Thin films : Theory and design

**ToC Category:**

Thin Films

**History**

Original Manuscript: July 10, 2013

Revised Manuscript: August 26, 2013

Manuscript Accepted: August 26, 2013

Published: September 4, 2013

**Citation**

Jun-ichi Inoue, "Electromagnetic waves in a topological insulator thin film stack: helicon-like wave mode and photonic band structure," Opt. Express **21**, 21317-21328 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-18-21317

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### References

- E. D. Palik and J. K. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Rep. Prog. Phys.33, 1193–1322 (1970). [CrossRef]
- P. M. Platzman and P. A. Wolff, Waves and Interactions in Solids State Plasma, Solid State Phys. 13 (Academic Press, 1972).
- J. J. Quinn and K-s. Yi, Solid State Physics: Principles and Modern Applications (Springer, 2009).
- A. C. Tselis and J. J. Quinn, “Retardation effects and transverse collective excitations in semiconductor superlattices,” Phys. Rev. B29, 2021–2027 (1984). [CrossRef]
- M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys.82, 3045–3067 (2010). [CrossRef]
- X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys.83, 1057–1110 (2011). [CrossRef]
- M. Z. Hasan and J. E. Moore, “Three-dimensional topological insulators,” Ann. Rev. Condens. Matter Phys.2, 55–78 (2011). [CrossRef]
- 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]
- C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,”IEEE J. Sel. Top. Quant. Electron.16, 367–375 (2010).
- T. Ochiai, “Theory of Light Scattering in Axion Electrodynamics,” J. Phys. Soc. Jpn.81, 094401–094408 (2012). [CrossRef]
- K. W. Chiu and J. J. Quinn, Phys. Rev. B9, “Plasma oscillations of a two-dimensional electron gas in a strong magnetic field,” 4724–4732 (1974). [CrossRef]
- X. L. Qi, T. L. Hughes, and S. C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B78, 195424-1–195424-43 (2008). [CrossRef]
- X. L. Qi, J. Zang, and S. C. Zhang, “Inducing a magnetic monopole with topological surface states,” Science323, 1184–1187 (2009). [CrossRef] [PubMed]
- 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]
- 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]
- M. C. Chang and M. F. Yang, “Optical signature of topological insulators,” Phys. Rev. B80, 113304-1–113304-4 (2009). [CrossRef]
- F. Wilczek, “Two applications of axion electrodynamics,” Phys. Rev. Lett.58, 1799–1802 (1987). [CrossRef] [PubMed]
- 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. Express3, 043001-1–043001-3 (2010).
- J. Inoue, “An optical test for identifying topological insulator thin films,” Opt. Express21, 8564–8569 (2013). [CrossRef] [PubMed]
- 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. B85, 125425-1–125425-7 (2012). [CrossRef]
- W. Dittrich and M. Reuter, Selected Topics in Gauge Theories (Springer, 1986).
- M. Fiebig, “Revival of the magnetoelectric effect,” J. Phys. D: Appl. Phys.38, R123–R152 (2005). [CrossRef]

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