Universal rotation gauge via quantum anomalous Hall effect

Quantized resistivity in the quantum Hall effect¹ (QHE) provides a unique possibility to measure the fine-structure constant $α≈1/137$ that quantifies the strength of electromagnetic interaction. Extending the static QHE measurement to the optical range suggests that the dimensionless α will correspond to an angle, which can be measured in a metrological calibration experiment. Indeed, in a simple static-dynamic correspondence, the optical polarization rotation θ_F in perpendicular magnetic field is expected to be just an integer times fine-structure constant. The relation is especially simple for a two-dimensional quantum well and simply reads $θF≈Nα$ with N being an integer. Therefore, optical analogy of the QHE can provide contact-free and dimensionless quantum of rotation. However, optical experiments on the quantized Faraday rotation on GaAs-based quantum wells turned out to be difficult due to the smearing out of the quantum plateaus at elevated frequencies.^2,3 This additional uncertainty is not well understood⁴ even taking for a moment the experimental challenges out of discussion.

One possible solution of the quantum Faraday metrology problem could be provided by three-dimensional topological insulators (TIs).^5,6 Due to the robustness of chiral surface bands, the available frequency range of quantized optical experiments can be extended to the terahertz frequency region. In such experiments, static magnetic fields of several tesla still have to be applied leading to further experimental errors and demanding quantum Hall systems in low or even zero magnetic fields.

A possibility of QHE in zero magnetic field has been suggested already about 30 years ago.⁷ A system of interest requires a time reversal symmetry breaking that can be realized experimentally, e.g., in a magnetic material. In such case, spontaneous magnetization can lead to a combination of anomalous Hall effect⁸ (AHE) and of the QHE, resulting in quantum anomalous Hall effect (QAHE) (see Refs. 9 and 10 for recent reviews). Indeed, after the first observation¹¹ of the QAHE in 2013, a vivid field of research has emerged that now includes such new phenomena like Majorana modes^12,13 or axion insulators.^14,15 In a nutshell, introduction of magnetic exchange interaction into the system opens a gap in linear dispersion of a topological insulator and leads to an appearance of a chiral edge state with quantized Hall conductivity $σxy=±e2/h$⁠. An important difference between static and contact-free optical QHE experiments should be mentioned at this point: in the former, the current is carried by one-dimensional edge channels whereby in the latter the edges are avoided as the light beam is restricted to the center of the sample.

In order to calibrate a rotation angle equal to the fine-structure constant, several difficulties have to be overcome. Most important one is the sample quality suppressing the effect,¹⁷ as in the optical experiment, the quantum coherence has to be obtained across the area of several millimeters. In addition, the effect of the substrate has to be avoided as it reduces the effect of Faraday rotation by roughly a factor $1/(1+ns)$⁠.¹⁸ Here, n_s is the refractive index of the substrate. Therefore, in the experiments, either a correction has to be included directly or an additional experimental parameter, like Kerr angle, has to be measured.^19,20 As a typical example, in topological insulators, the quantized value of the Faraday rotation can be obtained in high external magnetic fields.^21,22

In this work, we demonstrate experimentally that in similar conditions a universal Faraday rotation due to QAHE can be reached. In addition, we utilize the continuous wave technique at the maxima of Fabry–Pérot resonances. In this case, the universal rotation angle is directly obtained in the experiment without any calculations and additional parameters. Finally, as the optical experiment does not rely on one-dimensional conducting channels at the sample edges, a universal quantized rotation is seen although the static resistivity is still in the classical regime and is far below the resistance quantum $h/e2≈26$ kΩ.

The samples of the present work are single-crystalline (Cr_0.12Bi_0.26Sb_0.62)₂Te₃ films^23,24 on insulating (111) GaAs substrates grown by molecular beam epitaxy.^24–26 The Cr doping level (12%) and the (Bi/Sb) ratio (0.3/0.7) were optimized so that the Fermi level positions of the as-grown samples were close to the charge neutrality point. The growth was monitored by reflection high-energy electron diffraction and the films with a thickness of six quintuple layers (⁠$∼6$ nm) were obtained. To measure the static resistivity, indium contacts were made at the corners of the hexagon-like samples of typical size $∼10$ mm. The static Hall resistance has been measured simultaneously during the polarization rotation experiments in the frequency range of 80–600 GHz [Fig. 1(a)]. The contacts are not affected by the radiation that is focused to a spot with the diameter $ϕ<5$ mm.

Terahertz polarization rotation experiments at frequencies 80 $< ν<$ 600 GHz were carried out in transmittance geometry^16,27 which allows measurements of the amplitude and phase shift of the electromagnetic radiation with controlled polarization. The spectrometer utilizes linearly polarized monochromatic radiation which is provided by backward-wave oscillators, and a He-cooled bolometer is used as a detector. The amplitude $|t|$ and the phase shift $φ$ of the radiation transmitted through the sample are measured by using the Mach–Zehnder interferometer setup. Static magnetic field up to ±1 T is applied to the sample using a split-coil superconducting magnet with mylar windows. The polarization state of the transmitted beam is determined by measuring the amplitude and phase shift of the radiation both with parallel and crossed polarizer and analyzer. This procedure provides the complex values of t_xx and t_xy, respectively. We note that, however, t_xx in the maxima of the Fabry–Pérot interferences is close to unity for both samples.

Figure 1(b) shows the transmittance spectra of the (Cr_0.12Bi_0.26Sb_0.62)₂Te₃ film no. 1 in the frequency range of the present experiment. Due to Fabry–Pérot resonances within the substrate, a clear periodic modulation is seen in the spectra. The frequency positions of the maxima correspond to a resonance relation $2nsdsν=m$⁠, where m is an integer, $ds=0.447$ mm is the sample thickness, and $ns=3.5$ is the refractive index of the substrate. Doing magnetic field-dependent experiments at the maxima of the resonances²² leads to the Faraday rotation angle that equals to that of the free-standing film, thus strongly simplifying the handling of the data. To increase the accuracy, exact expressions^16,22 can be used to calculate the angle of the polarization rotation. The film transmission in the maxima of the substrate interferences is close to unity, indicating that the 2D films are transparent within the accuracy of our experiment. Indeed, in these conditions, the transmission in parallel polarizers t_xx is given by¹⁷ $txx≈1−σxxZ0/2≈1$ with $σxxZ0≪1$⁠. Here, $Z0=μ0/ε0$ is the impedance of the free space and σ_xx is the diagonal film conductivity that can be estimated as $σxx∼10−5Ω−1$⁠. As expected, no magnetic field dependence is observed in $txx(B)$⁠. On the contrary, in the experiment with crossed polarizers that is sensitive to the off diagonal transmission t_xy, a step in the amplitude and in the phase shift is clearly observed around zero field and at T < 40 K [see Figs. 1(d) and 1(e)]. Simultaneously, a step in the static Hall resistivity is observed and is shown in Fig. 1(c). We note that even at the lowest temperature of 1.8 K the step in $Rxy∼6$ kΩ is substantially smaller than $h/e2≈26$ kΩ, indicating that dissipation-free edge channels are still not established in this sample.

Faraday rotation step θ_F can be calculated either using the exact procedure^16,21,28 or within the approximations of the present experiment via¹⁷ 

With the quantized value of the optical Hall conductivity $σxy=e2/h$⁠, the rotation angle is obtained as

We estimate the error of the simplification procedure $Δθ/θ∼1 %$⁠. Alternatively, the Faraday rotation can be obtained in the phase-free experiment by measuring the transmission amplitude with polarizer and analyzer rotated by $±45°$⁠. These data as well as similar results on sample no. 2 are given in the supplementary material.

Figure 2 shows the Faraday rotation angle measured at different frequencies and temperatures. Panel (a) demonstrates that $θF≈α$ at liquid helium temperatures and it reveals a clear magnetic hysteresis. Rotation experiments have been carried out on two samples and for different Fabry–Pérot maxima. Panels (b) and (c) show that the data are concentrated around the value of the fine-structure constant within uncertainties of $Δθ∼1$ mrad independently on the frequency and that they saturate around 1.8 K.

At the same time, the step in the static Hall resistance R_xy for both samples is substantially smaller than $h/e2$ [see Fig. 1(c)]. In order to compare static and dynamic results, we transfer the resistance into the rotation angle using $θ(0)∼α·ΔRxye2/h$ and show these points as zero-frequency data in Fig. 2(b). Substantial difference between optical and terahertz data reflects differences in conduction mechanisms in static and dynamic experiments. As mentioned above, in the former case, the dissipation-free one-dimensional conducting channels have to be formed on the sample edges, and they are easily susceptible to distortions. On the contrary, in the optical experiments, the electromagnetic field is focused in the finite area of the sample center providing direct response by 2D conductivity.

In conclusion, in this work we investigated the Faraday rotation due to quantum anomalous Hall effect in (Cr_0.12Bi_0.26Sb_0.62)₂Te₃ thin films. We utilize the continuous wave technique in the millimeter wave range at the maxima of the Fabry–Pérot resonances. This technique allows to directly observe the universal rotation angle equal to the fine structure constant without any calculations and additional parameters. Finally, as the optical experiment does not rely on one-dimensional conducting channels at the sample edges, a universal quantized rotation is seen although the static resistivity is far below the resistance quantum $h/e2≈26$ kΩ.

See the supplementary material for additional results supporting the main text: Faraday rotation on sample no. 1 measured in $±45°$ geometry, providing an alternative way to determine the rotation angle, and the data on sample no. 2 with similar results.

This work was supported by the Austrian Science Funds (Nos. I 3456-N27 and I 5539-N). The authors acknowledge financial support from TU Wien library through the Open Access Funding Programme.