Microwave Measurements of Chiral Edge States of the Quantum Anomalous Hall Effect

Chiral 1D edge states of the quantum Hall effect can be used as “optical fibers” for electrons. Thanks to their long phase-coherence time, it has been possible to demonstrate interference of coherent single electrons – giving rise to the field of electron quantum optics [1].

While the formation of edge states in the quantum Hall effect requires the presence of an external magnetic field, some materials exhibit chiral edge states without an external magnetic field, a phenomenon known as the quantum anomalous Hall effect. Working at zero magnetic field paves the route to combining edge states with superconducting materials to exploit new properties.

To gain a deeper understanding of the transport in chiral edge states of the quantum anomalous Hall effect, the dispersion relation and the attenuation of edge excitations in the microwave regime can be studied. Recently, we visited the group of Prof. Erwann Bocquillon at the University of Cologne and performed measurements on a magnetically doped (Bi_xSb_1-x)₂Te₃ sample using our SHFLI 8.5 GHz Lock-in Amplifier.

This blog post will give you an overview of the insights that you can get about the sample properties by two types of measurements: 1) The transmission of sinusoidal microwave signals as a function of magnetic field and microwave frequency. 2) The magnetic-field-dependant transmission of different harmonics when a square wave microwave signal is applied.

Experimental Setup

An optical micrograph of the sample is shown in Figure 1. It consists of the V-doped topological insulator (Bi_xSb_1-x)₂Te₃ in a Hall bar geometry. The sample is mounted inside a dilution refrigerator where a magnetic field can be applied perpendicular to the sample. Two RF lines were connected to two of the contacts to measure the transmission through the sample. The measurements were performed with the SHFLI Lock-in Amplifier which allows to perform lock-in measurements of microwave signals up to 8.5 GHz using fully integrated double super-heterodyne frequency up- and down-conversion.

Frequency- and Magnetic-field-dependent Measurements

For the first experiment, we generated a sinusoidal microwave signal using the signal output channel of the SHFLI. We then swept the microwave frequency from 300 MHz to 8 GHz and demodulated the transmitted signal. The frequency sweep was repeated for different magnetic field strengths between 2 T and -2 T (and back) using a Python script.

Figure 2(a) shows the demodulated amplitude as a function of frequency at -2 T. The change in amplitude is related to several effects: transport through the edge channel, stray coupling between the contacts, and the impedance mismatch between the 50 Ω waveguide and the sample which leads to large oscillations as a function of frequency.

To eliminate the oscillations due to the impedance mismatch we used the measurement at -2 T as a baseline, as in this configuration the transmitted signal due to edge state transport is very low. Then we subtracted it from all other measurement results to get the difference ΔX and ΔY of the in-phase component X and the out-of-phase component Y. Afterwards the amplitude difference ΔR was calculated from ΔX and ΔY, and finally ΔR was divided by the amplitude R of the baseline measurement. One should note that this normalization eliminates the oscillations due to the impedance mismatch while removing the effects of the stray coupling would require an additional calibration process.

Figure 3 shows the normalized amplitude ΔR/R when sweeping the field from -2 T to 2 T in the left panel and the backward sweep from 2 T to -2 T in the right panel. A step in the normalized amplitude can be observed when increasing the magnetic field above approximately 0.8 T for the up sweep, and at approximately -0.8 T for the down sweep. This change is related to the single chiral edge channel circulating in a direction set by the magnetization in an adequately magnetically doped topological insulator. In a conductance measurement, it would be visible as a change from one resistance quantum +R_K = h/e² ≃ 25.9 kΩ to -R_K [2,3]. In the microwave measurement performed here, the amplitude of the step is more difficult to interpret, because it depends on the dissipation of the plasmons in the edge channel which increases with the distance between the two contacts: at 2 T we observe a large transmitted amplitude because the edge channel transport is in the clockwise direction where there is a short distance of 35 µm between the output and input contacts, while for anti-clockwise transport at -2 T the distance between the ports is around 2 mm and most of the signal is dissipated.

Harmonics Measurements

In a second experiment, the transmission of a square wave signal was measured for different magnetic field strengths. For this, a square wave signal with a frequency of 230 MHz was generated using an external arbitrary waveform generator and applied to contact 1. The transmitted signal at contact 2 was then measured with the SHFLI. Since a square wave contains information both in the fundamental frequency as well as its harmonics, we used all 8 demodulators simultaneously to capture the information of the harmonics.

In the SHFLI, each signal input channel provides a 1 GHz wide analysis window [4] around a specified center frequency. To measure the information of the harmonics of 230 MHz, we split the signal between the two signal input channels, and used a different center frequency for each channel. To increase the number of harmonics beyond 8, we performed the measurement twice for every magnetic field strength with a different center frequency to reach a measurement of 13 frequencies in total.

Figure 4 shows the amplitude of the odd harmonics as a function of magnetic field strength when sweeping the magnetic field from 2 T to -2 T and back. Several effects can be observed:

• There is hysteresis between the up and down sweep, similar to what was observed in the first measurement.
• The amplitude of the odd harmonics changes as a function of the magnetic field. The change of amplitude shows a different behavior for the different harmonics: while some harmonics increase in amplitude when increasing the magnetic field (e.g. 1 and 3), others decrease in amplitude (e.g. 7 and 9).

Comparing these observations to a suitable theoretical model would allow us to isolate the effect of the edge channel transport from the stray coupling of the contacts and thereby provide valuable insights into the experiment and the properties of the sample. However, until the launch of the SHFLI Lock-in Amplifier, measuring multiple harmonics simultaneously was not possible out of the box and appropriate theoretical models are still lacking. With the advance of state-of-the-art lock-in detection to the microwave regime, new analysis methods can now be explored.

Conclusion

In this blog post, we gave an overview of how to measure the microwave transport in chiral edge channels of the quantum anomalous Hall effect in a magnetically doped (Bi_xSb_1-x)₂Te₃ sample. In addition to measuring magnetic-field and frequency-dependent transport, the SHFLI 8.5 GHz Lock-in Amplifier also allows us to measure multiple harmonics simultaneously - thereby paving the road towards a deeper understanding of the experiment and the sample properties.

Acknowledgments

We thank Prof. Erwann Bocquillon and Torsten Röper for their time and the possibility of conducting these measurements with their samples in their lab!

The measurements were done together with my colleague Avishek Chowdhury.

References

[1] E. Bocquillon et al. Electron quantum optics in ballistic chiral conductors. Ann. Phys. (Berlin) 526, 1–30 (2014)

[2] C.-Z. Chang et al. Quantum anomalous Hall effect in time-reversal-symmetry breaking topological insulators. J. Phys.: Condens. Matter 28, 123002 (2016)

[3] Y. Deng et al. Quantum anomalous Hall effect in intrinsic magnetic topological insulator MnBi2Te4. Science 367, 895 (202)