## Periodic Reporting for period 4 - 4-TOPS (Four experiments in Topological Superconductivity.)

Reporting period: 2021-12-01 to 2022-05-31

The ERC-AG 4-TOPS project focused at advancing our understanding of induced superconductivity in topological materials. Few technologically viable topological superconductor materials are known today. The concept of inducing superconductivity into the surface or edge states of topological insulators via the proximity effect is thus the most promising route for studying the properties of topological superconductivity at present. In this project, we aimed at laying the groundwork for future applications such as topological qubits for quantum computation in topological materials.

Our research is grouped around four central topics (“experiments”), each addressing a defining aspect of induced topological superconductivity in HgTe 2-dimensional and 3-dimensional topological insulators (2DTI, 3DTI) and other topological materials.

The first experiment concerned the study of Andreev bound states in topological Josephson junctions and their topological properties. Andreev bound states are energy levels in a metallic weak link between two superconducting electrodes that make up the (topological) Josephson device. They carry the supercurrent and depend on the relative superconducting phase. We performed measurements of the supercurrent in HgTe 2DTI Josephson devices as a function of relative phase to determine the current-phase relation (CPR) in the asymmetric DC SQUID configuration, i.e. another topological Josephson junction serves as a phase-reference device. The CPR was found to be negatively skewed as expected for high transmission, quasi-ballistic, topological Josephson devices. A faster measurement on the time-scale of the quasiparticle life-time in the test device showed that the switching events could not be clearly attributed to either the test and reference junctions. To overcome this obstacle, we embedded the test device in an RF SQUID circuit, which allowed phase-biasing of a single junction with the applied magnetic field. The current in the device was read out by coupling it inductively to a co-planar microwave resonator with large quality factor (Q>100 000) using the “flip-chip” method to bond the RF SQUID sample to the read-out resonator circuit. This arrangement allowed us to apply large magnetic fields and modulate the superconducting phase along the electrodes of the test junction laterally as well as phase-bias the device. Similar to inverting the ordinary Fraunhofer diffraction pattern, we mapped out the CPR and the losses by transitions between Andreev bound states as a function of spatial location in the junction to gain understanding about the distribution of Andreev bound states in the device. This constitutes an important step toward engineering topological Josephson devices as part of more complex quantum circuits.

Developing the tunneling spectroscopy method for probing the energy and spatial distribution of Andreev states was center objective of the second experiment. We pushed the boundaries of state-of-the-art microfabrication technology and developed a process to wet-etch a constriction in the HgTe 2DTI while minimizing the damage to the material that causes scattering in electronic transport. Using this breakthrough technology, we observed for the first time quantized conductance in a Quantum Point Contact (QPC) of the HgTe quantum well and investigated interaction effects in the few channel limit. This demonstrates the feasibility of using QPCs as tunable interaction centers as envisioned in topological quantum computing proposals. By further narrowing the constrictions in the TI, we realized tunneling contacts, suitable for tunnel spectroscopy measurements. As a proof of concept for this technology, we measured the induced superconducting gap energy in the 2DTI next to a Nb electrode and inside the weak link of a topological Josephson device to find it matches the scale given by the product of the critical current and the normal state resistance as conjectured in our previous analysis.

The third experiment approached the outstanding problem of demonstrating the induced proximity effect in the quantum Hall edge channel, e.g. by measuring a supercurrent flow between superconducting electrodes. This is a difficult feat due to contradicting requirements of relatively high magnetic fields for the quantum Hall regime while preserving the superconducting state in the electrode. The problem could be circumvented by using topological materials that show the quantum anomalous Hall effect, which show edge channel transport for filling factor ν=1 without an applied magnetic field such as the previously studied vanadium-doped (Bi1-xSbx)2Te3 (ERC project 3TOP). We fabricated SNS and NSN devices to test for an induced proximity effect and for chiral Majoranas in such devices. While supercurrent flow was not detected, we contributed to the clarification of the conditions for the observation of chiral Majoranas. A second candidate material is Mn-doped HgTe quantum wells which exhibits edge channel transport for small applied magnetic fields as low as 50 mT. In the course of the project, we clarified the physical mechanism of the edge channel transport and demonstrated for the first time that superconductivity can be induced in such a dilute magnetically-doped topological insulator.

With strained bulk HgTe, we can access a 3DTI or a Dirac/Weyl semimetal phase. In both materials, we obtained a robust Josephson effect. The direction of the fourth experiment was to clarify the effect of charge carrier distribution and search for band structure effects (such as finite-momentum-pairing in an applied magnetic field) in the supercurrent transport in bulk HgTe. By improving the microfabrication technology and careful device geometry choices, we were able to study supercurrent interference from one or both surface states by top-gating bulk HgTe Josephson devices of (as-grown) low carrier density. In parallel, we investigated the interplay of gating and the formation of topological surface bands in quantum transport which helped us understand the role carrier distribution in the evolution of the interference patterns. Due to its low carrier density, the resolution of chiral Weyl physics in the supercurrent transport remains a topic of on-going research.

The first experiment concerned the study of Andreev bound states in topological Josephson junctions and their topological properties. Andreev bound states are energy levels in a metallic weak link between two superconducting electrodes that make up the (topological) Josephson device. They carry the supercurrent and depend on the relative superconducting phase. We performed measurements of the supercurrent in HgTe 2DTI Josephson devices as a function of relative phase to determine the current-phase relation (CPR) in the asymmetric DC SQUID configuration, i.e. another topological Josephson junction serves as a phase-reference device. The CPR was found to be negatively skewed as expected for high transmission, quasi-ballistic, topological Josephson devices. A faster measurement on the time-scale of the quasiparticle life-time in the test device showed that the switching events could not be clearly attributed to either the test and reference junctions. To overcome this obstacle, we embedded the test device in an RF SQUID circuit, which allowed phase-biasing of a single junction with the applied magnetic field. The current in the device was read out by coupling it inductively to a co-planar microwave resonator with large quality factor (Q>100 000) using the “flip-chip” method to bond the RF SQUID sample to the read-out resonator circuit. This arrangement allowed us to apply large magnetic fields and modulate the superconducting phase along the electrodes of the test junction laterally as well as phase-bias the device. Similar to inverting the ordinary Fraunhofer diffraction pattern, we mapped out the CPR and the losses by transitions between Andreev bound states as a function of spatial location in the junction to gain understanding about the distribution of Andreev bound states in the device. This constitutes an important step toward engineering topological Josephson devices as part of more complex quantum circuits.

Developing the tunneling spectroscopy method for probing the energy and spatial distribution of Andreev states was center objective of the second experiment. We pushed the boundaries of state-of-the-art microfabrication technology and developed a process to wet-etch a constriction in the HgTe 2DTI while minimizing the damage to the material that causes scattering in electronic transport. Using this breakthrough technology, we observed for the first time quantized conductance in a Quantum Point Contact (QPC) of the HgTe quantum well and investigated interaction effects in the few channel limit. This demonstrates the feasibility of using QPCs as tunable interaction centers as envisioned in topological quantum computing proposals. By further narrowing the constrictions in the TI, we realized tunneling contacts, suitable for tunnel spectroscopy measurements. As a proof of concept for this technology, we measured the induced superconducting gap energy in the 2DTI next to a Nb electrode and inside the weak link of a topological Josephson device to find it matches the scale given by the product of the critical current and the normal state resistance as conjectured in our previous analysis.

The third experiment approached the outstanding problem of demonstrating the induced proximity effect in the quantum Hall edge channel, e.g. by measuring a supercurrent flow between superconducting electrodes. This is a difficult feat due to contradicting requirements of relatively high magnetic fields for the quantum Hall regime while preserving the superconducting state in the electrode. The problem could be circumvented by using topological materials that show the quantum anomalous Hall effect, which show edge channel transport for filling factor ν=1 without an applied magnetic field such as the previously studied vanadium-doped (Bi1-xSbx)2Te3 (ERC project 3TOP). We fabricated SNS and NSN devices to test for an induced proximity effect and for chiral Majoranas in such devices. While supercurrent flow was not detected, we contributed to the clarification of the conditions for the observation of chiral Majoranas. A second candidate material is Mn-doped HgTe quantum wells which exhibits edge channel transport for small applied magnetic fields as low as 50 mT. In the course of the project, we clarified the physical mechanism of the edge channel transport and demonstrated for the first time that superconductivity can be induced in such a dilute magnetically-doped topological insulator.

With strained bulk HgTe, we can access a 3DTI or a Dirac/Weyl semimetal phase. In both materials, we obtained a robust Josephson effect. The direction of the fourth experiment was to clarify the effect of charge carrier distribution and search for band structure effects (such as finite-momentum-pairing in an applied magnetic field) in the supercurrent transport in bulk HgTe. By improving the microfabrication technology and careful device geometry choices, we were able to study supercurrent interference from one or both surface states by top-gating bulk HgTe Josephson devices of (as-grown) low carrier density. In parallel, we investigated the interplay of gating and the formation of topological surface bands in quantum transport which helped us understand the role carrier distribution in the evolution of the interference patterns. Due to its low carrier density, the resolution of chiral Weyl physics in the supercurrent transport remains a topic of on-going research.

ERC AdG 4TOPS laid the technological groundwork to push from the original observation of supercurrent in HgTe topological insulator Josephson devices to engineering superconducting quantum devices that can be integrated in RF circuits or used for the spectroscopy of topological quantum states. In particular, the development of QPCs and constriction tunnel contacts are important milestones as they serve as key functional elements in topological quantum circuit proposals. The research of this project contributed to the start of a local research initiative that includes the founding of a new Institute for Topological Insulator research, as joint endeavor of the PI’s research group, and local state and federal governments. The work of this project will be continued and expanded on in the framework of this new institute.