## Periodic Reporting for period 3 - MiTopMat (Microstructured Topological Materials: A novel route towards topological electronics)

Reporting period: 2020-12-01 to 2022-05-31

Why are some materials great electric conductors, like copper, and others not, like silicon? Like everything, both contain vast numbers of electrons, yet one conducts and the other does not. Solving this puzzle was one of the major successes of quantum mechanics. In essence, it shows how the classical picture of electrons as charged, billiard-ball-like particles breaks down and one needs to consider electrons as waves. Anybody who ever played a guitar knows well that the tone is set by the length of the guitar string. Pressed against a fret, the string vibrates only at one specific tone, and its overtones. Similarly, quantum mechanics tells us that a material only supports some specific electron waves. Some happen to be mobile waves, and we have a metal, and some are immobile, static patterns, as in insulators. This fundamental principle, called band theory, is the foundation of our understanding of electrons in matter. Like changing the fret on the guitar gives a different note, changing the electronic waves can turn a conductor into an insulator – a process that happens now billions of times per second in computers or smart phones. The transistor undoubtably was one of the most transformative technological developments in history that changed virtually any aspect of life. A dramatic example are self-driving cars which must possess the ability to process sensor information, assess highly complex traffic situations, and react within milliseconds – while lives are at stake. As the computational power is growing, these and many ideas currently locked in the realm of sci-fi suddenly become possible. However, silicon technology is approaching saturation. Radical new ideas that go beyond incremental improvements are necessary to achieve giant leaps in computational power.

One exciting recent development, leading to the Nobel prize in physics, is the discovery of a loophole in band theory that may allow entirely new ways of encoding and processing information in the future. The electron waves may vibrate at a certain frequency, but their patterns inside the metal are of importance too. This property, called band structure topology, is related to the mathematics of knots. Electron waves can be twisted in space, distinguishing them well from untwisted ones. In the analogy of the guitar, the tone will certainly change if one ties a knot into the string. Such topological properties are very robust. One could move the knot up and down on the string or change its shape, but to remove it, one has to remove at least one end from the guitar. Similarly, topological aspects of electronic waves are very robust, an ideal aspect if one wants to encode information.

This field of research is in its very beginning. In our project MiTopMat, we develop new processes to build prototype devices from conductors that host these topological electron waves, so called Dirac/Weyl semi-metals. These new materials exist only in small particles so far, and new concepts to fabricate them into functional devices are critical. MiTopMat develops a new approach based on focused ion beam milling. Here, an ion beam is focused onto a nanometric spot on the particle, and the impacting ions very locally cut the target material. One could imaging this as a very precise blade that can carve small metallic crystals. This way, we remove most of the target particle, until only the microdevice in the desired shape remains which can be used to prototype its performance.

The main goal we address here is to test these ideas of using topological materials in applications. Can we realize a microdevice in which electrons, due to their topological nature, do something radically different compared to copper or silicon, which are normal, non-topological metal? If this can be robustly demonstrated, naturally new technologies can emerge that exploit these differences. To this end, we screen multiple materials in which topological electrons have been predicted and compare their behavior in microdevices. A key distinction is a principle called bulk-surface correspondence that inherently links electron waves on the surface and deep inside the bulk. We could show that magnetic fields can be used to shift electron charge from the bulk into the surface and back, which is a first signature of an electron process that could not happen in silicon.

One exciting recent development, leading to the Nobel prize in physics, is the discovery of a loophole in band theory that may allow entirely new ways of encoding and processing information in the future. The electron waves may vibrate at a certain frequency, but their patterns inside the metal are of importance too. This property, called band structure topology, is related to the mathematics of knots. Electron waves can be twisted in space, distinguishing them well from untwisted ones. In the analogy of the guitar, the tone will certainly change if one ties a knot into the string. Such topological properties are very robust. One could move the knot up and down on the string or change its shape, but to remove it, one has to remove at least one end from the guitar. Similarly, topological aspects of electronic waves are very robust, an ideal aspect if one wants to encode information.

This field of research is in its very beginning. In our project MiTopMat, we develop new processes to build prototype devices from conductors that host these topological electron waves, so called Dirac/Weyl semi-metals. These new materials exist only in small particles so far, and new concepts to fabricate them into functional devices are critical. MiTopMat develops a new approach based on focused ion beam milling. Here, an ion beam is focused onto a nanometric spot on the particle, and the impacting ions very locally cut the target material. One could imaging this as a very precise blade that can carve small metallic crystals. This way, we remove most of the target particle, until only the microdevice in the desired shape remains which can be used to prototype its performance.

The main goal we address here is to test these ideas of using topological materials in applications. Can we realize a microdevice in which electrons, due to their topological nature, do something radically different compared to copper or silicon, which are normal, non-topological metal? If this can be robustly demonstrated, naturally new technologies can emerge that exploit these differences. To this end, we screen multiple materials in which topological electrons have been predicted and compare their behavior in microdevices. A key distinction is a principle called bulk-surface correspondence that inherently links electron waves on the surface and deep inside the bulk. We could show that magnetic fields can be used to shift electron charge from the bulk into the surface and back, which is a first signature of an electron process that could not happen in silicon.

The project MiTopMat is structured into three research thrusts.

In the first, “Hallmark of Fermi Arc transport”, we search for direct signatures of topologically protected surface states in topological semi-metals, the so called “Fermi-arc” state. Here, strong magnetic fields are used to further twist the electronic wave functions, and by studying their response to further twisting, we can make statements about their intrinsic topological state. First, have set up a high field magnet capable of reaching 20T for this project. Using this new setup, we have characterized crystals of the Dirac semi-metal Cd3As2 and screened them to investigate the number of free electrons each of these crystals contained. This allows us to pick a desired electron density for further FIB fabrication, which has now been achieved. In the coming month, we will prepare devices to investigate how the topological charge transport scales with the electron density.

In the second, “Fermi Arcs in Weyl materials”, we mostly screened candidate materials for topological transport, casting the net a bit wider than only into Weyl semi-metals. NbAs is a prototypical Weyl material, and the microstructures of it are very interesting. Our fabrication method enriches Nb at the surface, and because Nb is an elemental superconductor, these structures become superconducting and open a new route towards topological superconductivity. In the search for new materials, we found a surprising phenomenon in the metal PdCoO2. We discovered that the magnetoresistance of this material, when cut into microscopic bars, oscillates as a function of the applied magnetic field. The oscillation period is set by fundamental constants of nature, the electron charge and the Planck constant, and hence appears to be directly related to the wave-like nature of electrons. Surprisingly, this phenomenon persists up to temperatures of T~60K, which is very high compared to the ultra-low temperatures such wave phenomena are usually observed. Currently we investigate the role of orbital topology in this long-ranged coherence, a major discovery by MiTopMat.

The third thrust aims to build a “Topological Voltage Inverter”. Here we pursue a theoretical prediction that the unconventional twist of the wave functions in topological semi-metals should lead to an counterintuitive electron flow. Ususally, electrons follow locally the applied electric field and flow along the field lines. But as these topological electron waves live both on the surface and in the bulk, inducing a surface current inherently induces a bulk current, and in particular, also a current on the opposite side of a crystal. Essentially, the predicted pattern is a vortex in which the current on the other side flows into the opposite direction. This should lead to a non-local voltage signal on the far side of a crystal opposite to the voltage applied to the other side, hence a voltage inverter. We have successfully built and characterized such non-local microdevices, and indeed observed strong non-local voltage signals. However, the results are quantitatively at odds with the prediction, and we discovered that the existing theory needs to be refined to include the effects of magnetoresistance anisotropy. Essentially, magnetic fields themselves create a type of non-local signature. We built a quantitative model to capture this pheonomenon, which is a critical step towards a unified model of charge transport in topological matter.

In the first, “Hallmark of Fermi Arc transport”, we search for direct signatures of topologically protected surface states in topological semi-metals, the so called “Fermi-arc” state. Here, strong magnetic fields are used to further twist the electronic wave functions, and by studying their response to further twisting, we can make statements about their intrinsic topological state. First, have set up a high field magnet capable of reaching 20T for this project. Using this new setup, we have characterized crystals of the Dirac semi-metal Cd3As2 and screened them to investigate the number of free electrons each of these crystals contained. This allows us to pick a desired electron density for further FIB fabrication, which has now been achieved. In the coming month, we will prepare devices to investigate how the topological charge transport scales with the electron density.

In the second, “Fermi Arcs in Weyl materials”, we mostly screened candidate materials for topological transport, casting the net a bit wider than only into Weyl semi-metals. NbAs is a prototypical Weyl material, and the microstructures of it are very interesting. Our fabrication method enriches Nb at the surface, and because Nb is an elemental superconductor, these structures become superconducting and open a new route towards topological superconductivity. In the search for new materials, we found a surprising phenomenon in the metal PdCoO2. We discovered that the magnetoresistance of this material, when cut into microscopic bars, oscillates as a function of the applied magnetic field. The oscillation period is set by fundamental constants of nature, the electron charge and the Planck constant, and hence appears to be directly related to the wave-like nature of electrons. Surprisingly, this phenomenon persists up to temperatures of T~60K, which is very high compared to the ultra-low temperatures such wave phenomena are usually observed. Currently we investigate the role of orbital topology in this long-ranged coherence, a major discovery by MiTopMat.

The third thrust aims to build a “Topological Voltage Inverter”. Here we pursue a theoretical prediction that the unconventional twist of the wave functions in topological semi-metals should lead to an counterintuitive electron flow. Ususally, electrons follow locally the applied electric field and flow along the field lines. But as these topological electron waves live both on the surface and in the bulk, inducing a surface current inherently induces a bulk current, and in particular, also a current on the opposite side of a crystal. Essentially, the predicted pattern is a vortex in which the current on the other side flows into the opposite direction. This should lead to a non-local voltage signal on the far side of a crystal opposite to the voltage applied to the other side, hence a voltage inverter. We have successfully built and characterized such non-local microdevices, and indeed observed strong non-local voltage signals. However, the results are quantitatively at odds with the prediction, and we discovered that the existing theory needs to be refined to include the effects of magnetoresistance anisotropy. Essentially, magnetic fields themselves create a type of non-local signature. We built a quantitative model to capture this pheonomenon, which is a critical step towards a unified model of charge transport in topological matter.

The crystalline microdevices developed in MiTopMat push the technical boundaries of how we can shape electronic devices in complex materials. We have developed workflows and procedures that standardize this time-consuming prototyping process and greatly enhance the efficiency. While initially every microstructure was a unique project, using the methods of MiTopMat now turns microstructure characterization into a simple and standard approach allowing us to quickly screen multiple materials.

A major technical development involves free standing structures. While traditional thin-film processing methods are mostly happening in direct contact with a substrate, the carved crystalline structure of MiTopMat can easily be pivoted over an edge, leading to free-standing cantilevers. We have developed an apparatus to bend these microstructures to extreme angles, leading to significant structural distortions within the crystal. New ideas currently emerge around the general concepts of pseudo-magnetic fields. In essence, a mathematical correspondence exists between the influence of strain gradients and magnetic fields on the electrons. The momentum transfer due to the magnetic field, the Lorentz force, is highly similar in topological materials to the momentum transfer as the electron moves in a strain gradient landscape. By bending microcrystalline circuits, we can generate, control and tune such gradients and study the effect of these pseudo-magnetic fields in a new way. There are two main improvements over existing bending techniques. Due to the sub-micrometer thickness of the devices, extreme bending strain gradiends exceeding 2%/micrometer can be generated. Also, as there is no substrate involved, the neutral strain plane goes through the sample, allowing for both compressive and tensile strain to be generated simultaneously.

By the end of the project we aim to elucidate new transport phenomena in topological semi-metals that are inherently linked to the topology in these materials. The goal is to provide unambiguous experimental signatures in transport that demonstrate this non-trivial topology. We will then measure the size of these responses, which is poorly understood theoretical at present yet makes a key difference for its use in technology. MiTopMat may be successful at detecting the expected transport anomalies using laboratory-grade testing equipment and thereby confirm these new physical concepts, yet if their signatures are so faint it will be difficult to exploit them in practical applications. We will establish this signal size, and work towards new geometries and device designs to amplify them, should they turn out to be weak.

A major technical development involves free standing structures. While traditional thin-film processing methods are mostly happening in direct contact with a substrate, the carved crystalline structure of MiTopMat can easily be pivoted over an edge, leading to free-standing cantilevers. We have developed an apparatus to bend these microstructures to extreme angles, leading to significant structural distortions within the crystal. New ideas currently emerge around the general concepts of pseudo-magnetic fields. In essence, a mathematical correspondence exists between the influence of strain gradients and magnetic fields on the electrons. The momentum transfer due to the magnetic field, the Lorentz force, is highly similar in topological materials to the momentum transfer as the electron moves in a strain gradient landscape. By bending microcrystalline circuits, we can generate, control and tune such gradients and study the effect of these pseudo-magnetic fields in a new way. There are two main improvements over existing bending techniques. Due to the sub-micrometer thickness of the devices, extreme bending strain gradiends exceeding 2%/micrometer can be generated. Also, as there is no substrate involved, the neutral strain plane goes through the sample, allowing for both compressive and tensile strain to be generated simultaneously.

By the end of the project we aim to elucidate new transport phenomena in topological semi-metals that are inherently linked to the topology in these materials. The goal is to provide unambiguous experimental signatures in transport that demonstrate this non-trivial topology. We will then measure the size of these responses, which is poorly understood theoretical at present yet makes a key difference for its use in technology. MiTopMat may be successful at detecting the expected transport anomalies using laboratory-grade testing equipment and thereby confirm these new physical concepts, yet if their signatures are so faint it will be difficult to exploit them in practical applications. We will establish this signal size, and work towards new geometries and device designs to amplify them, should they turn out to be weak.