## An optical test for identifying topological insulator thin films |

Optics Express, Vol. 21, Issue 7, pp. 8564-8569 (2013)

http://dx.doi.org/10.1364/OE.21.008564

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### Abstract

As the search for new compounds of a topological insulator (TI) becomes more extensive, it is increasingly important to develop an experimental technique that can identify TIs. In this work, we theoretically propose a simple optical method for distinguishing between topological and conventional insulator thin films. An electromagnetic interference wave consisting of waves transmitted through and reflected by the TI thin film is sensitive to the circular polarization direction of the incident electromagnetic wave. Based on this fact, we can identify a TI by observing the interference wave. This method is straightforward, and thus should propel TI research.

© 2013 OSA

## 1. Introduction

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

3. M. Z. Hasan and J. E. Moore, “Three-dimensional topological insulators,” Ann. Rev. Condens. Matter Phys. **2**, 55–78 (2011) [CrossRef] .

*I*of two EM interference waves as a function of frequency

*ω*. The absence of a signal in Fig. 1(a) indicates that the insulator thin film measured is a conventional one, whereas the finite signal in Fig. 1(b), regardless of the shape, indicates that it is a TI thin film. These TI test results are obtained as described below. Take a conventional/topological thin film sandwiched by other thin films or substrates, as shown in Fig. 2(a). By injecting EM waves with electric polarization

*E*(+ : +)|

^{2}, where

*E*(+ : +) is the sum of the transmitted wave of

*E*(+ : −)|

^{2}. See Figs. 1(e) and 1(f). Then, subtract the two records Δ

*I*(+ : ±) ≡ |

*E*(+ : −)|

^{2}− |

*E*(+ : +)|

^{2}, resulting in Figs. 1(a) and 1(b). As mentioned earlier, the total cancellation of the two plots |

*E*(+ : ±)|

^{2}indicates that it is a conventional insulator thin film, whereas a finite signal indicates a TI, regardless of the signal form. Note that the test results provide us with a

*qualitative*difference, and thus, no further analysis of the signal shape is required.

## 2. Topological electromagnetic effects

## 3. Optical topological insulator test

*qualitative*difference between the two kinds of insulators is devised by exploiting the interference phenomenon of EM waves. To show this, we consider the other three remaining geometries. The relations of each transmittance and reflectance with

*z*< 0 and

*z*>

*d*) denote the polarization direction of the incident waves and the incident surface, respectively. The latter relation, Eq. (10), shows that

*r*depends on the sign of

*σ*̃

*. This fact seems to be valuable in devising an optical scheme that enables us to distinguish the insulators. However, the reflectance alone does not meet the current aim, because the quantity that is straightforwardly measured, namely the reflection coefficient*

_{H}*R*= |

*r*|

^{2}, is indeed insensitive to the sign of

*σ*̃

*. Therefore, in order to deduce the phase information inherent in*

_{H}*r*, we use an interference signal of the transmitted and reflected waves. The quantity suitable for our purpose is the subtraction of the two interference waves, defined as which is plotted in Figs. 1(a) and 1(b) for a conventional and topological insulator, respectively. In Fig. 1(b), the fact that

*r*in the TI depends on both the incident wave polarization and the incident surface is observed through the interplay with the transmitted wave, such that Δ

*I*remains finite. However, since both of

*t*and

*r*in conventional insulators are totally symmetric with respect to

*E*, Δ

_{±}*I*(+ : ±) becomes zero. The results obtained in this feasible scheme no longer require further analysis of the shape of Δ

*I*and the two kinds of insulators can be identified without any ambiguity. Although the theoretical foundation of the TI test has been fully presented in the above analytical discussion of a free-standing case, the results shown in Fig. 1 are numerically obtained in a rather realistic sandwiched geometry with using the transfer matrix method, as seen in Fig.2(a), where

*n*

_{1}/

*n*

_{0}= 1.5 and

*n*

_{2}/

*n*

_{0}= 2.0 are used. Note that the scheme is also effective for an asymmetric geometry.

*E*(+ : ±)|

^{2}. One might think that |

*E*(+ : +)|

^{2}or |

*E*(+ : −)|

^{2}alone is sufficient for identifying a TI, and hence, there is no need to subtract the terms |

*E*(+ : ±)|

^{2}, since a finite deferential coefficient

*d*|

*E*(+ : ±)|

^{2}/

*dω*|

_{ω}_{=0}would become a subtle indicator of the insulator, as observed in Fig. 1. However, the measurement in D.C. limit using EM waves is far from realistic, and it is practical to rely on A.C. measurement in a frequency range

*ω*∈ [

*ω*,

_{a}*ω*].

_{b}*ε*

_{2}and

*μ*

_{0}are real and insensitive to

*ω*. Therefore, we should avoid the region with a specific spectroscopic structure, e.g., a resonance. To obtain a clear signal, it is also favorable to avoid the region with

*n*

_{2}/

*σ*̃

*≫ 1. The frequency range that meets this condition depends on the individual material being studied. Although this careful identification of the frequencies*

_{H}*ω*and

_{a}*ω*is required, it is feasible. The frequency selection also depends on a property of magnetic thin films that break time reversal symmetry. In our discussion, the EM properties are assumed to be described by real refractive index

_{b}*n*

_{1}. This ideal condition would be satisfied by using, for instance, a ferromagnetic thin film that is transparent up to the visible frequency range. Recently, the synthesis of such a material, TiO

_{2}codopoed with Fe and Nb, has been reported [13

13. E. Sakai, A. Chikamatsu, Y. Hirose, T. Shimada, and T. Hasegawa, “Magnetic and transport properties of anatase TiO_{2} codoped with Fe and Nb,” Appl. Phys. Express **3**, 043001 (2010) [CrossRef] .

14. C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,” IEEE J. Sel. Top. Quantum Electron. **16**, 367–375 (2010) [CrossRef] .

*I*≠ 0, since constitutive relations in this metamaterial are identical in form with those in topological insulators. In this sense, the metamaterial would fall into a class of topological materials.

## 4. Conclusion

## Acknowledgments

## References and links

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

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

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

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

5. | W. K. Tse and A. H. MacDonald, “Giant magneto-optical Kerr effect and universal Faraday effect in thin-film topological insulators,” Phys. Rev. Lett. |

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

7. | J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. |

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

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

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

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

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

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

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

**OCIS Codes**

(310.3840) Thin films : Materials and process characterization

(310.6860) Thin films : Thin films, optical properties

(310.6805) Thin films : Theory and design

**ToC Category:**

Thin Films

**History**

Original Manuscript: December 27, 2012

Revised Manuscript: March 12, 2013

Manuscript Accepted: March 14, 2013

Published: April 1, 2013

**Citation**

Jun-ichi Inoue, "An optical test for identifying topological insulator thin films," Opt. Express **21**, 8564-8569 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8564

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### References

- 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]
- X. L. Qi, T. L. Hughes, and S. C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B78, 195424-1 (2008). [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 (2010). [CrossRef] [PubMed]
- M. C. Chang and M. F. Yang, “Optical signature of topological insulators,” Phys. Rev. B80, 113304 (2009). [CrossRef]
- 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]
- F. Wilczek, “Two applications of axion electrodynamics,” Phys. Rev. Lett.58, 1799–1802 (1987). [CrossRef] [PubMed]
- W. Dittrich and M. Reuter, Selected Topics in Gauge Theories (Springer, 1986).
- X. L. Qi, J. Zang, and S. C. Zhang, “Inducing a magnetic monopole with topological surface states,” Science323, 1184–1187 (2009). [CrossRef] [PubMed]
- M. Fiebig, “Revival of the magnetoelectric effect,” J. Phys. D: Appl. Phys.38, R123–R152 (2005). [CrossRef]
- A. M. Essin, J. E. Moore, and D. Vanderbilt, “Magnetoelectric polarizability and axion electrodynamics in crystalline insulators,” Phys. Rev. Lett.102, 146805 (2009). [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 (2010). [CrossRef]
- C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,” IEEE J. Sel. Top. Quantum Electron.16, 367–375 (2010). [CrossRef]

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