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The fractional AC Josephson effect is a discerning property of topological superconductivity in hybrid Josephson junctions. Recent experimental observations of missing odd Shapiro steps and half Josephson frequency emission in various materials have sparked significant debate regarding their potential origin in the effect. In this study, we present microwave emission measurements on a resistively shunted Josephson junction based on a HgTe quantum well. We demonstrate that, with significant spurious inductance in the shunt wiring, the experiment operates in a nonlinear dynamic regime characterized by period-doubling. This leads to additional microwave emission peaks at half of the Josephson frequency, $f_J/2$, which can mimic the $4\pi$-periodicity of topological Andreev states. The observed current-voltage characteristics and emission spectra are well-described by a simple RCLSJ model. Furthermore, we show that the nonlinear dynamics of the junction can be controlled using gate voltage, magnetic field, and temperature, with our model accurately reproducing these effects without incorporating any topological attributes. Our observations urge caution in interpreting emission at $f_J/2$ as evidence for gapless Andreev bound states in topological junctions and suggest the appropriate parameter range for future experiments., Comment: 13 pages, 8 figures

The software project kdotpy provides a Python application for simulating electronic band structures of semiconductor devices with $\mathbf{k}\cdot\mathbf{p}$ theory on a lattice. The application implements the widely used Kane model, capable of reliable predictions of transport and optical properties for a large variety of topological and non-topological materials with a zincblende crystal structure. The application automates the tedious steps of simulating band structures. The user inputs the relevant physical parameters on the command line, for example materials and dimensions of the device, magnetic field, and temperature. The program constructs the appropriate matrix Hamiltonian on a discretized lattice of spatial coordinates and diagonalizes it. The physical observables are extracted from the eigenvalues and eigenvectors and saved as output. The program is highly customizable with a large set of configuration options and material parameters., Comment: 143 pages, submission to SciPost Physics Codebases

2D topological insulators promise novel approaches towards electronic, spintronic, and quantum device applications. This is owing to unique features of their electronic band structure, in which bulk-boundary correspondences enforces the existence of 1D spin-momentum locked metallic edge states - both helical and chiral - surrounding an electrically insulating bulk. Forty years since the first discoveries of topological phases in condensed matter, the abstract concept of band topology has sprung into realization with several materials now available in which sizable bulk energy gaps - up to a few hundred meV - promise to enable topology for applications even at room-temperature. Further, the possibility of combining 2D TIs in heterostructures with functional materials such as multiferroics, ferromagnets, and superconductors, vastly extends the range of applicability beyond their intrinsic properties. While 2D TIs remain a unique testbed for questions of fundamental condensed matter physics, proposals seek to control the topologically protected bulk or boundary states electrically, or even induce topological phase transitions to engender switching functionality. Induction of superconducting pairing in 2D TIs strives to realize non-Abelian quasiparticles, promising avenues towards fault-tolerant topological quantum computing. This roadmap aims to present a status update of the field, reviewing recent advances and remaining challenges in theoretical understanding, materials synthesis, physical characterization and, ultimately, device perspectives.

We present overlapping top gate electrodes for the formation of gate defined lateral junctions in semiconducting layers as an alternative to the back gate/top gate combination and to the split gate configuration. The optical lithography microfabrication of the overlapping top gates is based on multiple layers of low-temperature atomic layer deposited hafnium oxide, which acts as a gate dielectric and as a robust insulating layer between two overlapping gate electrodes exhibiting a large dielectric breakdown field of > 1E9 V/m. The advantage of overlapping gates over the split gate approach is confirmed in model calculations of the electrostatics of the gate stack. The overlapping gate process is applied to Hall bar devices of mercury telluride in order to study the interaction of different quantum Hall states in the nn', np, pn and pp' regime., Comment: 7 pages, 3 figures

The band inversion of topological materials in three spatial dimensions is intimately connected to the parity anomaly of two-dimensional massless Dirac fermions. At finite magnetic fields, the parity anomaly reveals itself as a non-zero spectral asymmetry, i.e., a non-zero difference between the number of conduction and valence band Landau levels, due to the unpaired zero Landau level. Here, we realize this two-dimensional Dirac physics at a single surface of the three-dimensional topological insulator (Hg,Mn)Te. We observe an unconventional re-entrant quantum Hall effect that can be directly related to the occurrence of spectral asymmetry in a single topological surface state. The effect should be observable in any topological insulator where the transport is dominated by a single Dirac surface state., Comment: 13 pages, 10 figures, originally submitted version, includes supplementary material

The quantum anomalous Hall effect in magnetic topological insulators has been recognized as a promising platform for applications in quantum metrology. The primary reason for this is the electronic conductance quantization at zero external magnetic field, which allows to combine it with the quantum standard of voltage. Here we demonstrate a measurement scheme that increases the robustness of the zero magnetic field quantum anomalous Hall resistor, allowing for higher operational currents. This is achieved by simultaneous current injection into the two disconnected perimeters of a multi-terminal Corbino device to balance the electrochemical potential between the edges, screening the electric field that drives back-scattering through the bulk, and thus improving the stability of the quantization at increased currents. This approach is not only applicable to devices based on the quantum anomalous Hall effect, but more generally can also be applied to existing quantum resistance standards that rely on the integer quantum Hall effect.

Nonlinear planar magnetotransport is ubiquitous in topological HgTe structures, both in tensile (topological insulator) or compressively strained layers (Weyl semimetal phase). We show that the common reason for the nonlinear planar magnetotransport is the presence of tilted Dirac cones combined with the formation of charge puddles. The origin of the tilted Dirac cones is the mix of the Zeeman term due to the in-plane magnetic field and quadratic contributions to the dispersion relation. We develop a network model that mimics transport of tilted Dirac fermions in the landscape of charge puddles. The model captures the essential features of the experimental data. It should be relevant for nonlinear planar magnetotransport in a variety of topological and small band gap materials., Comment: 12 pages, 11 figures, 1 table

The unambiguous identification of Majorana zero modes (MZMs) is one of the most outstanding problems of condensed matter physics. Thermal transport provides a detection tool that is sensitive to these chargeless quasiparticles. We study thermoelectric transport between metallic leads transverse to a Josephson junction. The central double quantum dot hosts conventional or topological Andreev states that depend on the phase difference $\phi$. We show that the presence of MZMs can be identified by a significant amplification of both the electrical and thermal conductance at $\phi \approx \pi$ as well as the Seebeck coefficient at $\phi \approx 0$. In addition, we show that the Wiedemann-Franz law is strongly violated in the presence of MZMs around $\phi \approx \pi$ when compared to the conventional case. We further investigate the robustness of our results against Cooper pair splitting processes., Comment: 13 pages, 8 figures, content corresponds to the published version

High-frequency transport in the edge states of the quantum spin Hall (QSH) effect has to date rarely been explored, though it could cast light on the scattering mechanisms taking place therein. We here report on the measurement of the plasmon velocity in topological HgTe quantum wells both in the QSH and quantum Hall (QH) regimes, using harmonic GHz excitations and phase-resolved detection. We observe low plasmon velocities corresponding to large transverse widths, which we ascribe to the prominent influence of charge puddles forming in the vicinity of edge channels. Together with other recent works, it suggests that puddles play an essential role in the edge state physics and probably constitute a main hurdle on the way to clean and robust edge transport., Comment: 8 pages, 6 figures, + supplementary material

HgTe is a versatile topological material and has enabled the realization of a variety of topological states, including two- and three-dimensional (3D) topological insulators and topological semimetals. Nevertheless, a quantitative understanding of its electronic structure remains challenging, in particular due to coupling of the Te 5p-derived valence electrons to Hg 5d core states at shallow binding energy. We present a joint experimental and theoretical study of the electronic structure in strained HgTe(001) films in the 3D topological-insulator regime, based on angle-resolved photoelectron spectroscopy and density functional theory. The results establish detailed agreement in terms of (i) electronic band dispersions and orbital symmetries, (ii) surface and bulk contributions to the electronic structure, and (iii) the importance of Hg 5d states in the valence-band formation. Supported by theory, our experiments directly image the paradigmatic band inversion in HgTe, underlying its non-trivial band topology.

A striking feature of 3 dimensional (3D) topological insulators (TIs) is the theoretically expected topological magneto-electric (TME) effect, which gives rise to additional terms in Maxwell's laws of electromagnetism with an universal quantized coefficient proportional to half-integer multiples of the fine structure constant $\alpha$. In an ideal scenario one therefore expects also quantized contributions in the magneto-optical response of TIs. We review this premise by taking into account the trivial dielectric background of the TI bulk and potential host substrates, and the often present contribution of itinerant bulk carriers. We show that (i) one obtains a non-universal magneto-optical response whenever there is impedance mismatch between different layers and (ii) that the detectable signals due to the TME rapidly approach vanishingly small values as the impedance mismatch is detuned from zero. We demonstrate that it is methodologically impossible to deduce the existence of a TME exclusively from an optical experiment in the thin film limit of 3D TIs at high magnetic fields.

The search for topological superconductivity has motivated investigations into materials that combine topological and superconducting properties. The half-Heusler compound YPtBi appears to be such a material, however experiments have thus far been limited to bulk single crystals, drastically limiting the scope of available experiments. This has made it impossible to investigate the potential topological nature of the superconductivity in this material. Experiments to access details about the superconducting state require sophisticated lithographic structures, typically based on thin films. Here we report on the establishment of high crystalline quality epitaxial thin films of YPtBi(111), grown using molecular beam epitaxy on Al2O3(0001) substrates. A robust superconducting state is observed, with both critical temperature and critical field consistent with that previously reported for bulk crystals. Moreover we find that AlOx capping sufficiently protects the sample surface from degradation to allow for proper lithography. Our results pave a path towards the development of advanced lithographic structures, that will allow the exploration of the potentially topological nature of superconductivity in YPtBi., Comment: 5 pages, 4 figures

Achieving efficient, high-power harmonic generation in the terahertz spectral domain has technological applications, for example in sixth generation (6G) communication networks. Massless Dirac fermions possess extremely large terahertz nonlinear susceptibilities and harmonic conversion efficiencies. However, the observed maximum generated harmonic power is limited, because of saturation effects at increasing incident powers, as shown recently for graphene. Here, we demonstrate room-temperature terahertz harmonic generation in a Bi$_2$Se$_3$ topological insulator and topological-insulator-grating metamaterial structures with surface-selective terahertz field enhancement. We obtain a third-harmonic power approaching the milliwatt range for an incident power of 75 mW - an improvement by two orders of magnitude compared to a benchmarked graphene sample. We establish a framework in which this exceptional performance is the result of thermodynamic harmonic generation by the massless topological surface states, benefiting from ultrafast dissipation of electronic heat via surface-bulk Coulomb interactions. These results are an important step towards on-chip terahertz (opto)electronic applications.

We observe a localized cnoidal (LCn) state in an electric circuit network. Its formation derives from the interplay of non-linearity and the topology inherent to a Su-Schrieffer-Heeger (SSH) chain of inductors. Varicap diodes act as voltage-dependent capacitors, and create a non-linear on-site potential. For a sinusoidal voltage excitation around midgap frequency, we show that the voltage response in the non-linear SSH circuit follows the Korteweg-de Vries equation. The topological SSH boundary state which relates to a midgap impedance peak in the linearized limit is distorted into the LCn state in the non-linear regime, where the cnoidal eccentricity decreases from edge to bulk.

The recent advent of topological states of matter spawned many significant discoveries. The quantum anomalous Hall effect[1-3] is a prime example due to its potential for applications in quantum metrology[4, 5] as well as its influence on fundamental research into the underlying topological and magnetic states[6-11] and axion electrodynamics[2, 12-14]. Here, we perform electronic transport studies on a (V,Bi,Sb)2Te3 ferromagnetic topological insulator nanostructure in the quantum anomalous Hall regime. This allows us access to the dynamics of an individual ferromagnetic domain. The volume of the domain is estimated to be about 85 000 nm3, containing some 50 000 vanadium atoms, spread over a macroscopic distance of 115 nm. Telegraph noise resulting from the magnetization fluctuations of this domain is observed in the Hall signal. Careful analysis of the influence of temperature and external magnetic field on the domain switching statistics provides evidence for quantum tunneling of magnetization[15-22] in a macrospin state. This ferromagnetic macrospin is not only the largest magnetic object in which quantum tunneling has been observed, but also the first observation of the effect in a topological state of matter.

Hybrid samples combining superconductors with magnetic topological insulators are a promising platform for exploring exotic new transport physics. We examine a Josephson junction of such a system, based on the dilute magnetic topological insulator (Hg,Mn)Te and the type II superconductor MoRe. In the zero and very low field limit, to the best of our knowledge, the device shows, for the first time, induced supercurrent through a magnetically doped semiconductor, in this case a topological insulator. At higher fields, a rich and hysteretic magnetoresistance is revealed. Careful analysis shows that the explanation of this behaviour can be found in magnetic flux focusing stemming from the Meissner effect in the superconductor, without invoking any role of proximity induced superconductivity. The phenomena is important, as it will ubiquitously co-exist with any exotic new physics that may be present in this class of devices., Comment: This document is the unedited Author's version of a Submitted Work that was subsequently accepted for publication in Nano Letters, copyright \c{opyright} 2022 American Chemical Society after peer review. To access the final edited and published work see doi.org/10.1021/acs.nanolett.1c04903

The survival of the quantum spin Hall edge channels in presence of an external magnetic field has been a subject of experimental and theoretical research. The inversion of Landau levels that accommodates the quantum spin Hall effect is destroyed at a critical magnetic field, and a trivial insulating gap appears in the spectrum for stronger fields. In this work, we report the absence of this transport gap in disordered two dimensional topological insulators in perpendicular magnetic fields of up to 16 T. Instead, we observe that a topological edge channel (from band inversion) coexists with a counterpropagating quantum Hall edge channel for magnetic fields at which the transition to the insulating regime is expected. For larger fields, we observe only the quantum Hall edge channel with transverse resistance close to $h/e^2$. By tuning the disorder using different fabrication processes, we find evidence that this unexpected $\nu=1$ plateau originates from extended quantum Hall edge channels along a continuous network of charge puddles at the edges of the device., Comment: 8+3 pages, 5+2 figures

Magneto-transport measurements on gated high mobility heterostructures containing a 60 nm layer of tensile strained HgTe, a three-dimensional topological insulator, show well-developed Hall quantization from surface states both in the n- as well as in the p-type regime. While the n-type behavior is due to transport in the topological surface state of the material, we find from 8-orbital k.p calculations that the p-type transport results from massive Volkov-Pankratov states. Their formation prevents the Dirac point and thus the p-conducting topological surface state from being accessible in transport experiments. This interpretation is supported by low-field magneto-transport experiments demonstrating the coexistence of n-conducting topological surface states and p-conducting Volkov-Pankratov states at the relevant gate voltages., Comment: 15+5 pages, 5 figures; initially submitted version

Achieving metrological precision of quantum anomalous Hall resistance quantization at zero magnetic field so far remains limited to temperatures of the order of 20 mK, while the Curie temperature in the involved material is as high as 20 K. The reason for this discrepancy remains one of the biggest open questions surrounding the effect, and is the focus of this article. Here we show, through a careful analysis of the non-local voltages on a multi-terminal Corbino geometry, that the chiral edge channels continue to exist without applied magnetic field up to the Curie temperature of bulk ferromagnetism of the magnetic topological insulator, and that thermally activated bulk conductance is responsible for this quantization breakdown. Our results offer important insights on the nature of the topological protection of these edge channels, provide an encouraging sign for potential applications, and establish the multi-terminal Corbino geometry as a powerful tool for the study of edge channel transport in topological materials.

We experimentally investigate the effect of electron temperature on transport in the two-dimensional Dirac surface states of the three-dimensional topological insulator HgTe. We find that around the minimal conductivity point, where both electrons and holes are present, heating the carriers with a DC current results in a non-monotonic differential resistance of narrow channels. We show that the observed initial increase in resistance can be attributed to electron-hole scattering, while the decrease follows naturally from the change in Fermi energy of the charge carriers. Both effects are governed dominantly by a van Hove singularity in the bulk valence band. The results demonstrate the importance of interband electron-hole scattering in the transport properties of topological insulators.

Magnetic oscillations of Dirac surface states of topological insulators are expected to be associated with the formation of Landau levels or the Aharonov-Bohm effect. We instead study the conductance of Dirac surface states subjected to an in-plane magnetic field in presence of a barrier potential. Strikingly, we find that, in the case of large barrier potentials, the surface states exhibit pronounced oscillations in the conductance when varying the magnetic field, in the \textit{absence} of Landau levels or the Aharonov-Bohm effect. These novel magnetic oscillations are attributed to the emergence of \textit{super-resonant regimes} by tuning the magnetic field, in which almost all propagating electrons cross the barrier with perfect transmission. In the case of small and moderate barrier potentials, we also identify a positive magnetoconductance which is due to the increase of the Fermi surface by tilting the surface Dirac cone. Moreover, we show that for weak magnetic fields, the conductance displays a shifted sinusoidal dependence on the field direction with period $\pi$ and phase shift determined by the tilting direction with respect to the field direction. Our predictions can be applied to many topological insulators, such as HgTe and Bi$_{2}$Se$_{3}$, and provide important insights into exploring and understanding exotic magnetotransport properties of topological surface states., Comment: 5+2 pages, 4 figures

The kinetic equation used for the description of Dirac systems does not fully take into account two features that play an important role in the vicinity of the Dirac point: (i) the spin degree of freedom, in particular if the spin-flip energy $2 vp$ is not large anymore; and (ii) the failure of the semiclassical approximation due to the large Fermi wavelength. In our work, we propose a novel quantum kinetic equation, which does not have these two drawbacks. Exploiting it in the presence of short range disorder, we demonstrate how it predicts the correct minimal conductivity in 2D Dirac system, a result that has so far been obtained only by other methods like the Kubo formula. The nature of the presented kinetic equation opens up the possibility for the kinetic description of deeply quantum and even strongly correlated Dirac systems., Comment: 5 pages

We report growth of CuMnSb thin films by molecular beam epitaxy on InAs(001) substrates. The CuMnSb layers are compressively strained ($0.6~\text{%}$) due to lattice mismatch. The thin films have a $\omega$ full width half max of $7.7^{''}$ according to high resolution X-ray diffraction, and a root mean square roughness of $0.14~\text{nm}$ as determined by atomic force microscopy. Magnetic and electrical properties are found to be consistent with reported values from bulk samples. We find a N\'eel temperature of $62~\text{K}$, a Curie-Weiss temperature of $-65~\text{K}$ and an effective moment of $5.9~\mu_{\text{B}}/\text{f.u.}$. Transport measurements confirm the antiferromagetic transition and show a residual resistivity at $4~\text{K}$ of $35~\mu\Omega\cdot \text{cm}$., Comment: 6 pages, 5 figures, accepted in PRM

The realization of the quantum spin Hall effect in HgTe quantum wells has led to the development of topological materials which, in combination with magnetism and superconductivity, are predicted to host chiral Majorana fermions. However, the large magnetization ($\sim$ a few tesla) in conventional quantum anomalous Hall system, makes it challenging to induce superconductivity. Here, we report two different emergent quantum Hall effects in HgTe quantum wells dilutely alloyed with Mn. Firstly, a novel quantum Hall state emerges from the quantum spin Hall state at an exceptionally low magnetic field of $\sim 50$ mT. Secondly, tuning towards the bulk $p$-regime, we resolve multiple quantum Hall plateaus at fields as low as $20 - 30$ mT, where transport is dominated by a van Hove singularity in the valence band. These emergent quantum Hall phenomena rely critically on the topological band structure of HgTe and their occurrence at very low fields make them an ideal candidate for interfacing with superconductors to realize chiral Majorana fermions.

Since the discovery of the quantum anomalous Hall effect in the magnetically doped topological insulators (MTI) Cr:(Bi,Sb)$_2$Te$_3$ and V:(Bi,Sb)$_2$Te$_3$, the search for the exchange coupling mechanisms underlying the onset of ferromagnetism has been a central issue, and a variety of different scenarios have been put forward. By combining resonant photoemission, X-ray magnetic dichroism and multiplet ligand field theory, we determine the local electronic and magnetic configurations of V and Cr impurities in (Bi,Sb)$_2$Te$_3$. While strong pd hybridisation is found for both dopant types, their 3d densities of states show pronounced differences. State-of-the-art first-principles calculations show how these impurity states mediate characteristic short-range pd exchange interactions, whose strength sensitively varies with the position of the 3d states relative to the Fermi level. Measurements on films with varying host stoichiometry support this trend. Our results establish the essential role of impurity-state mediated exchange interactions in the magnetism of MTI.