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We present a comprehensive theoretical study of the static spin response in HgTe quantum wells, revealing distinctive behavior for the topologically nontrivial inverted structure. Most strikingly, the q=0 (long-wavelength) spin susceptibility of the undoped topological-insulator system is constant and equal to the value found for the gapless Dirac-like structure, whereas the same quantity shows the typical decrease with increasing band gap in the normal-insulator regime. We discuss ramifications for the ordering of localized magnetic moments present in the quantum well, both in the insulating and electron-doped situations. The spin response of edge states is also considered, and we extract effective Landé g factors for the bulk and edge electrons. The variety of counterintuitive spin-response properties revealed in our study arises from the system’s versatility in accessing situations where the charge-carrier dynamics can be governed by ordinary Schrödinger-type physics; it mimics the behavior of chiral Dirac fermions or reflects the material’s symmetry-protected topological order.
We report magnetotransport studies on a gated strained HgTe device. This material is a three-dimensional topological insulator and exclusively shows surface-state transport. Remarkably, the Landau-level dispersion and the accuracy of the Hall quantization remain unchanged over a wide density range (3×1011 cm−2<n<2×1012 cm−2). These observations imply that even at large carrier densities, the transport is surface-state dominated, where bulk transport would have been expected to coexist already. Moreover, the density dependence of the Dirac-type quantum Hall effect allows us to identify the contributions from the individual surfaces. A k⋅p model can describe the experiments but only when assuming a steep band bending across the regions where the topological surface states are contained. This steep potential originates from the specific screening properties of Dirac systems and causes the gate voltage to influence the position of the Dirac points rather than that of the Fermi level.
Topological insulators are electronic phases that insulate in the bulk and accommodate a peculiar, metallic edge liquid with a spin-dependent dispersion.
They are regarded to be of considerable future use in spintronics and for quantum computation.
Besides determining the intrinsic properties of this rather novel electronic phase, considering its combination with well-known physical systems can generate genuinely new physics.
In this thesis, we report on such combinations including topological insulators. Specifically, we analyze an attached Rashba impurity, a Kondo dot in the two channel setup, magnetic impurities on the surface of a strong three-dimensional topological insulator, the proximity coupling of the latter system to a superconductor, and hybrid systems consisting of a topological insulator and a semimetal.
Let us summarize our primary results.
Firstly, we determine an analytical formula for the Kondo cloud and describe its possible detection in current correlations far away from the Kondo region.
We thereby rely on and extend the method of refermionizable points.
Furthermore, we find a class of gapless topological superconductors and semimetals, which accommodate edge states that behave similarly to the ones of globally gapped topological phases. Unexpectedly, we also find edge states that change their chirality when affected by sufficiently strong disorder.
We regard the presented research helpful in future classifications and applications of systems containing topological insulators, of which we propose some examples.
Topological insulators interacting with magnetic impurities have been reported to host several unconventional effects. These phenomena are described within the framework of gapping Dirac quasiparticles due to broken time-reversal symmetry. However, the overwhelming majority of studies demonstrate the presence of a finite density of states near the Dirac point even once topological insulators become magnetic. Here, we map the response of topological states to magnetic impurities at the atomic scale. We demonstrate that magnetic order and gapless states can coexist. We show how this is the result of the delicate balance between two opposite trends, that is, gap opening and emergence of a Dirac node impurity band, both induced by the magnetic dopants. Our results evidence a more intricate and rich scenario with respect to the once generally assumed, showing how different electronic and magnetic states may be generated and controlled in this fascinating class of materials.
Effective lifting of the topological protection of quantum spin Hall edge states by edge coupling
(2022)
The scientific interest in two-dimensional topological insulators (2D TIs) is currently shifting from a more fundamental perspective to the exploration and design of novel functionalities. Key concepts for the use of 2D TIs in spintronics are based on the topological protection and spin-momentum locking of their helical edge states. In this study we present experimental evidence that topological protection can be (partially) lifted by pairwise coupling of 2D TI edges in close proximity. Using direct wave function mapping via scanning tunneling microscopy/spectroscopy (STM/STS) we compare isolated and coupled topological edges in the 2D TI bismuthene. The latter situation is realized by natural lattice line defects and reveals distinct quasi-particle interference (QPI) patterns, identified as electronic Fabry-Pérot resonator modes. In contrast, free edges show no sign of any single-particle backscattering. These results pave the way for novel device concepts based on active control of topological protection through inter-edge hybridization for, e.g., electronic Fabry-Pérot interferometry.
Josephson junctions (JJs) in the presence of a magnetic field exhibit qualitatively different interference patterns depending on the spatial distribution of the supercurrent through the junction. In JJs based on two-dimensional topological insulators (2DTIs), the electrons/holes forming a Cooper pair (CP) can either propagate along the same edge or be split into the two edges. The former leads to a SQUID-like interference pattern, with the superconducting flux quantum ϕ\(_0\) (where ϕ\(_0\)=h/2e) as a fundamental period. If CPs’ splitting is additionally included, the resultant periodicity doubles. Since the edge states are typically considered to be strongly localized, the critical current does not decay as a function of the magnetic field. The present paper goes beyond this approach and inspects a topological JJ in the tunneling regime featuring extended edge states. It is here considered the possibility that the two electrons of a CP propagate and explore the junction independently over length scales comparable to the superconducting coherence length. As a consequence of the spatial extension, a decaying pattern with different possible periods is obtained. In particular, it is shown that, if crossed Andreev reflections (CARs) are dominant and the edge states overlap, the resulting interference pattern features oscillations whose periodicity approaches 2ϕ\(_0\).
This thesis describes the studies of topological superconductivity, which is predicted to
emerge when pair correlations are induced into the surface states of 2D and 3D topolog-
ical insulators (TIs). In this regard, experiments have been designed to investigate the
theoretical ideas first pioneered by Fu and Kane that in such system Majorana bound
states occur at vortices or edges of the system [Phys. Rev. Lett. 100, 096407 (2008), Phys.
Rev. B 79, 161408 (2009)]. These states are of great interest as they constitute a new
quasiparticle which is its own antiparticle and can be used as building blocks for fault
tolerant topological quantum computing.
After an introduction in chapter 1, chapter 2 of the thesis lays the foundation for the
understanding of the field of topology in the context of condensed matter physics with a
focus on topological band insulators and topological superconductors. Starting from a
Chern insulator, the concepts of topological band theory and the bulk boundary corre-
spondence are explained. It is then shown that the low energy Hamiltonian of mercury
telluride (HgTe) quantum wells of an appropriate thickness can be written as two time
reversal symmetric copies of a Chern insulator. This leads to the quantum spin Hall effect.
In such a system, spin-polarized one dimensional conducting states form at the edges
of the material, while the bulk is insulating. This concept is extended to 3D topological
insulators with conducting 2D surface states. As a preliminary step to treating topological
superconductivity, a short review of the microscopic theory of superconductivity, i.e. the
theory of Bardeen, Cooper, and Shrieffer (BCS theory) is presented. The presence of
Majorana end modes in a one dimensional superconducting chain is explained using the
Kitaev model. Finally, topological band insulators and conventional superconductivity
are combined to effectively engineer p-wave superconductivity. One way to investigate
these states is by measuring the periodicity of the phase of the Josephson supercurrent
in a topological Josephson junction. The signature is a 4π-periodicity compared to the
2π-periodicity in conventional Josephson junctions. The proof of the presence of this
effect in HgTe based Josephson junction is the main goal of this thesis and is discussed in
chapters 3 to 6.
Chapter 3 describes in detail the transport of a 3D topological insulator based weak
link under radio-frequency radiation. The chapter starts with a review of the state of
research of (i) strained HgTe as 3D topological insulator and (ii) the progress of induc-
ing superconducting correlations into the topological surface states and the theoretical
predictions of 3D TI based Josephson junctions. Josephson junctions based on strained
HgTe are successfully fabricated. Before studying the ac driven Josephson junctions, the
dc transport of the devices is analysed. The critical current as a function of temperature
is measured and it is possible to determine the induced superconducting gap. Under
rf illumination Shapiro steps form in the current voltage characteristic. A missing first
step at low frequencies and low powers is found in our devices. This is a signature of
a 4π-periodic supercurrent. By studying the device in a wide parameter range - as a
147148 SUMMARY
function of frequency, power, device geometry and magnetic field - it is shown that the
results are in agreement with the presence of a single gapless Andreev doublet and several
conventional modes.
Chapter 4 gives results of the numerical modelling of the I −V dynamics in a Josephson
junction where both a 2π- and a 4π-periodic supercurrents are present. This is done in
the framework of an equivalent circuit representation, namely the resistively shunted
Josephson junction model (RSJ-model). The numerical modelling is in agreement with
the experimental results in chapter 3. First, the missing of odd Shapiro steps can be
understood by a small 4π-periodic supercurrent contribution and a large number of
modes which have a conventional 2π-periodicity. Second, the missing of odd Shapiro
steps occurs at low frequency and low rf power. Third, it is shown that stochastic processes
like Landau Zener tunnelling are most probably not responsible for the 4π contribution.
In a next step the periodicity of Josephson junctions based on quantum spin Hall
insulators using are investigated in chapter 5. A fabrication process of Josephson junctions
based on inverted HgTe quantum wells was successfully developed. In order to achieve a
good proximity effect the barrier material was removed and the superconductor deposited
without exposing the structure to air. In a next step a gate electrode was fabricated which
allows the chemical potential of the quantum well to be tuned. The measurement of the
diffraction pattern of the critical current Ic due to a magnetic field applied perpendicular
to the sample plane was conducted. In the vicinity to the expected quantum spin Hall
phase, the pattern resembles that of a superconducting quantum interference device
(SQUID). This shows that the current flows predominantly on the edges of the mesa.
This observation is taken as a proof of the presence of edge currents. By irradiating the
sample with rf, missing odd Shapiro steps up to step index n = 9 have been observed. This
evidences the presence of a 4π-periodic contribution to the supercurrent. The experiment
is repeated using a weak link based on a non-inverted HgTe quantum well. This material
is expected to be a normal band insulator without helical edge channels. In this device,
all the expected Shapiro steps are observed even at low frequencies and over the whole
gate voltage range. This shows that the observed phenomena are directly connected
to the topological band structure. Both features, namely the missing of odd Shapiro
steps and the SQUID like diffraction pattern, appear strongest towards the quantum spin
Hall regime, and thus provide evidence for induced topological superconductivity in the
helical edge states.
A more direct way to probe the periodicity of the Josephson supercurrent than using
Shapiro steps is the measurement of the emitted radiation of a weak link. This experiment
is presented in chapter 6. A conventional Josephson junction converts a dc bias V to
an ac current with a characteristic Josephson frequency fJ
= eV /h. In a topological
Josephson junction a frequency at half the Josephson frequency fJ /2 is expected. A
new measurement setup was developed in order to measure the emitted spectrum of a
single Josephson junction. With this setup the spectrum of a HgTe quantum well based
Josephson junction was measured and the emission at half the Josephson frequency fJ /2
was detected. In addition, fJ emission is also detected depending on the gate voltage and
detection frequency. The spectrum is again dominated by half the Josephson emission at
low voltages while the conventional emission is determines the spectrum at high voltages.
A non-inverted quantum well shows only conventional emission over the whole gateSUMMARY 149
voltage and frequency range. The linewidth of the detected frequencies gives a measure
on the lifetime of the bound states: From there, a coherence time of 0.3–4ns for the fJ /2
line has been deduced. This is generally shorter than for the fJ line (3–4ns).
The last part of the thesis, chapter 7, reports on the induced superconducting state
in a strained HgTe layer investigated by point-contact Andreev reflection spectroscopy.
For the experiment, a HgTe mesa was fabricated with a small constriction. The diameter
of the orifice was chosen to be smaller than the mean free path estimated from magne-
totransport measurements. Thus one gets a ballistic point-contact which allows energy
resolved spectroscopy. One part of the mesa is covered with a superconductor which
induces superconducting correlations into the surface states of the topological insulator.
This experiment therefore probes a single superconductor normal interface. In contrast to
the Josephson junctions studied previously, the geometry allows the acquisition of energy
resolved information of the induced superconducting state through the measurement
of the differential conductance dI/dV as a function of applied dc bias for various gate
voltages, temperatures and magnetic fields. An induced superconducting order parame-
ter of about 70µeV was extracted but also signatures of the niobium gap at the expected
value around Δ Nb
≈ 1.1meV have been found. Simulations using the theory developed by
Blonder, Tinkham and Klapwijk and an extended model taking the topological surface
states into account were used to fit the data. The simulations are in agreement with a
small barrier at the topological insulator-induced topological superconductor interface
and a high barrier at the Nb to topological insulator interface. To understand the full con-
ductance curve as a function of applied voltage, a non-equilibrium driven transformation
is suggested. The induced superconductivity is suppressed at a certain bias value due to
local electron population. In accordance with this suppression, the relevant scattering
regions change spatially as a function of applied bias.
To conclude, it is emphasized that the experiments conducted in this thesis found
clear signatures of induced topological superconductivity in HgTe based quantum well
and bulk devices and opens up the avenue to many experiments. It would be interesting
to apply the developed concepts to other topological matter-superconductor hybrid
systems. The direct spectroscopy and manipulation of the Andreev bound states using
circuit quantum electrodynamic techniques should be the next steps for HgTe based
samples. This was already achieved in superconducting atomic break junctions by the
group in Saclay [Science 2015, 349, 1199-1202 (2015)]. Another possible development
would be the on-chip detection of the emitted spectrum as a function of the phase φ
through the junction. In this connection, the topological junction needs to be shunted
by a parallel ancillary junction. Such a setup would allow the current phase relation
I(φ) directly and the lifetime of the bound states to be measured directly. By coupling
this system to a spectrometer, which can be another Josephson junction, the energy
dependence of the Andreev bound states E(φ) could be obtained. The experiments on
the Andreev reflection spectroscopy described in this thesis could easily be extended to
two dimensional topological insulators and to more complex geometries, like a phase
bias loop or a tunable barrier at the point-contact. This work might also be useful for
answering the question how and why Majorana bound states can be localized in quantum
spin Hall systems.
Strained bulk HgTe is a three-dimensional topological insulator, whose surface electrons have a high mobility (~ 30 000 cm\(^2\)=Vs), while its bulk is effectively free of mobile charge carriers. These properties enable a study of transport through its unconventional surface states without being hindered by a parallel bulk conductance. Here, we show transport experiments on HgTe-based Josephson junctions to investigate the appearance of the predicted Majorana states at the interface between a topological insulator and a superconductor. Interestingly, we observe a dissipationless supercurrent flow through the topological surface states of HgTe. The current-voltage characteristics are hysteretic at temperatures below 1 K, with critical supercurrents of several microamperes. Moreover, we observe a magnetic-field-induced Fraunhofer pattern of the critical supercurrent, indicating a dominant \(2\pi\)-periodic Josephson effect in the unconventional surface states. Our results show that strained bulk HgTe is a promising material system to get a better understanding of the Josephson effect in topological surface states, and to search for the manifestation of zero-energy Majorana states in transport experiments.
Since the early days of Dirac flux quantization, magnetic monopoles have been sought after as a potential corollary of quantized electric charge. As opposed to magnetic monopoles embedded into the theory of electromagnetism, Weyl semimetals (WSM) exhibit Berry flux monopoles in reciprocal parameter space. As a function of crystal momentum, such monopoles locate at the crossing point of spin-polarized bands forming the Weyl cone. Here, we report momentum-resolved spectroscopic signatures of Berry flux monopoles in TaAs as a paradigmatic WSM. We carried out angle-resolved photoelectron spectroscopy at bulk-sensitive soft X-ray energies (SX-ARPES) combined with photoelectron spin detection and circular dichroism. The experiments reveal large spin- and orbital-angular-momentum (SAM and OAM) polarizations of the Weyl-fermion states, resulting from the broken crystalline inversion symmetry in TaAs. Supported by first-principles calculations, our measurements image signatures of a topologically non-trivial winding of the OAM at the Weyl nodes and unveil a chirality-dependent SAM of the Weyl bands. Our results provide directly bulk-sensitive spectroscopic support for the non-trivial band topology in the WSM TaAs, promising to have profound implications for the study of quantum-geometric effects in solids. Weyl semimetals exhibit Berry flux monopoles in momentum-space, but direct experimental evidence has remained elusive. Here, the authors reveal topologically non-trivial winding of the orbital-angular-momentum at the Weyl nodes and a chirality-dependent spin-angular-momentum of the Weyl bands, as a direct signature of the Berry flux monopoles in TaAs.
The topological classification of electronic band structures is based on symmetry properties of Bloch eigenstates of single-particle Hamiltonians. In parallel, topological field theory has opened the doors to the formulation and characterization of non-trivial phases of matter driven by strong electron-electron interaction. Even though important examples of topological Mott insulators have been constructed, the relevance of the underlying non-interacting band topology to the physics of the Mott phase has remained unexplored. Here, we show that the momentum structure of the Green’s function zeros defining the “Luttinger surface" provides a topological characterization of the Mott phase related, in the simplest description, to the one of the single-particle electronic dispersion. Considerations on the zeros lead to the prediction of new phenomena: a topological Mott insulator with an inverted gap for the bulk zeros must possess gapless zeros at the boundary, which behave as a form of “topological antimatter” annihilating conventional edge states. Placing band and Mott topological insulators in contact produces distinctive observable signatures at the interface, revealing the otherwise spectroscopically elusive Green’s function zeros.