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Faraday Rotation Due to Quantum Anomalous Hall Effect in Cr-Doped (Bi,Sb)_{2}Te_{3}

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## Abstract

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## 1. Introduction

## 2. Results and Discussion

## 3. Conclusions

## 4. Materials and Methods

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Scheme of the optical experiment. Linearly polarized incident light is transformed into the elliptical polarization after the sample, and is characterized by Faraday rotation angle $\theta $ and ellipticity $\eta $. The analyzer in front of the detector projects the ellipse either into the same direction as the incident beam leading to parallel transmittance ${t}_{xx}$ or to perpendicular direction leading to the crossed transmittance ${t}_{xy}$. An external magnetic field is applied parallel to the propagation direction (Faraday geometry).

**Figure 2.**Magnetic field scans of the transmittance in (Cr${}_{0.12}$Bi${}_{0.26}$Sb${}_{0.62}$)${}_{2}$Te${}_{3}$ film and in crossed polarizers geometry ${t}_{xy}=\left|{t}_{xy}\right|{e}^{i\phi}$. The external field is applied parallel to the propagation direction (Faraday geometry, see Figure 1). The parameters of the experiment are given in the plot. Bottom panel: amplitude of the crossed signal. Top panel: relative optical thickness (phase shift) of the sample.

**Figure 3.**Complex polarization rotation angle $\theta +i\eta $ in (Cr${}_{0.12}$Bi${}_{0.26}$Sb${}_{0.62}$)${}_{2}$Te${}_{3}$ at $T=1.85$ K and at different frequencies. The frequencies were selected at the maxima of the Fabry-Pérot interferences to suppress the effect of the substrate in the spectra. Bottom panel: Faraday rotation angle $\theta $, top panel: ellipticity $\eta $. The inset shows the absolute values of the rotation angle due to quantum anomalous Hall effect (QAHE) in the units of the fine structure constant at zero magnetic field and as a function of frequency. Straight dashed line is to guide the eye.

**Figure 4.**Complex polarization rotation angle $\theta +i\eta $ in (Cr${}_{0.12}$Bi${}_{0.26}$Sb${}_{0.62}$)${}_{2}$Te${}_{3}$ at $\nu =203$ GHz and at different temperatures. Bottom panel: Faraday rotation angle $\theta $, top panel: ellipticity $\eta $. The inset shows the absolute values of the step of the rotation angle in the units of the fine structure constant at zero magnetic field and as a function of temperature.

**Figure 5.**Magnetoresistance data in (Cr${}_{0.12}$Bi${}_{0.26}$Sb${}_{0.62}$)${}_{2}$Te${}_{3}$ film at various temperatures. Top: diagonal resistivity; bottom: Hall resistivity. The inset shows the absolute values of the step in the Hall resistance at zero magnetic field and as a function of temperature.

**Figure 6.**Transmittance spectrum of the sample used in this work in zero magnetic field and at temperature $T=1.85$ K. This spectrum was measured in the parallel polarizers geometry, ${t}_{xx}$, and shows a series of Fabry-Pérot resonances due to reflections on the substrate surfaces.

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**MDPI and ACS Style**

Shuvaev, A.; Pan, L.; Zhang, P.; Wang, K.L.; Pimenov, A.
Faraday Rotation Due to Quantum Anomalous Hall Effect in Cr-Doped (Bi,Sb)_{2}Te_{3}. *Crystals* **2021**, *11*, 154.
https://doi.org/10.3390/cryst11020154

**AMA Style**

Shuvaev A, Pan L, Zhang P, Wang KL, Pimenov A.
Faraday Rotation Due to Quantum Anomalous Hall Effect in Cr-Doped (Bi,Sb)_{2}Te_{3}. *Crystals*. 2021; 11(2):154.
https://doi.org/10.3390/cryst11020154

**Chicago/Turabian Style**

Shuvaev, Alexey, Lei Pan, Peng Zhang, Kang L. Wang, and Andrei Pimenov.
2021. "Faraday Rotation Due to Quantum Anomalous Hall Effect in Cr-Doped (Bi,Sb)_{2}Te_{3}" *Crystals* 11, no. 2: 154.
https://doi.org/10.3390/cryst11020154