Measuring angular momentum of $p_x+i{p}_y$ topological superfluids: A proposal

Topological superconductors are among the most actively studied materials these days. They are a promising candidate for hosting Majorana fermions either on their boundaries or in vortex cores. Detecting one-dimensional (1D) edge current around the periphery of a 2D $p_x+i{p}_y$ superconductor would be a hallmark signature of topological superconductivity, but Majorana fermions are not amenable to electronic current measurements due to their charge-neutral nature. Thermal conductivity measurements, such as the thermal Hall effect, are alternatively proposed, but material synthesis must come first. Superfluid ${}^3\mathrm{He}-A$, on the other hand, is a known $p_x+i{p}_y$ superfluid whose angular momentum generated by the edge current can be measured with a gyroscopic technique. Here, we propose a microelectromechanical system based gyroscope that will not only have enough signal sensitivity to measure the angular momentum but also be used to observe dimensionality induced phase transitions between different topological superfluids.

Figure 1

Proposed gyroscopic design. The central plate of dimension $l_x\times l_y\times t_g\approx 100\mu \text{m}\times 100\mu \text{m}\times 100\text{nm}$ is held by four identical springs at the four corners of the gyroscope. Then the gyroscope can have two almost degenerate torsional modes orthogonal to each other by making $l_x\approx l_y$. Our analysis in the text is carried out with such an assumption. Grey and yellow lines on the gyroscope represent electrical leads for drive and detection, respectively.

Figure 2

Estimated signal strength for three different values of $Q=Q_x=Q_y$ in comparison to the noise floor. The noise floor is dominated by the input noise of the cryogenic low noise amplifier $(\approx 300\mathrm{pV}/\sqrt{\mathrm{Hz}})$, where the measurement bandwidth is set to 1 Hz. $B=2000$ G and $t_f=100$ nm are used in Eq. (4). For a moderate quality factor of ${10}^3$ or less, the two orthogonal resonance frequencies are well-matched and the product $Q_x{Q}_y={10}^6$ enters into Eq. (4). When the quality factor is as high as ${10}^4$, due to the narrow resonances, matching two frequencies becomes more difficult and only $Q_y={10}^4$ is relevant. This is why the signal for $Q={10}^4$ is estimated to be smaller than that for $Q={10}^3$.

Figure 3

The Gyroscope will be placed above a ${}^3\mathrm{He}$ liquid bath as shown above. At temperatures above 300 mK, evaporated ${}^3\mathrm{He}$ from the bath will be deposited onto the gyroscope surface. With $\sim$ 100 nm film, the device and the film have strong enough thermal link to the bath to follow the bath temperature. Once superfluidity is obtained, one can control the thickness of the film by adjusting the bath level. For thicker films, IDCs patterned on the gyroscope surface can be used to draw films as thick as a few $\mu \mathrm{m}$.