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In this thesis, the Josephson effect in mercury telluride based superconducting quantum point contacts (SQPCs) is studied. Implementing such confined structures into topological superconductors has been proposed as a means to detect and braid Majorana fermions. For the successful realization of such experiments though, coherent transport across the constriction is essential. By demonstrating the Josephson effect in a confined topological system, the presented experiments lay the foundation for future quantum devices that can be used for quantum computation. In addition, the experiments also provide valuable insights into the behavior of the Josephson effect in the low-channel limit (N<20). Due to the confinement of the weak link, we can also study the Josephson effect in a topological insulator, where the edge modes interact.
In conclusion, this thesis discusses the fabrication of, and low-temperature measurements on mercury telluride quantum point contacts embedded within Josephson junctions. We find that the merging of the currently used fabrication methods for mercury telluride quantum point contacts and Josephson junctions does not yield a good enough device quality to resolve subbands of the quantum point contact as quantization effects in the transport properties. As we attribute this to the long dry etching time that is necessary for a top-contact, the fabrication process was adapted to reduce the defect density at the superconductor-semiconductor interface. Employing a technique that involves side contacting the mercury telluride quantum well and reducing the size of the mercury telluride mesa to sub-micrometer dimensions yields a quantized supercurrent across the junction. The observed supercurrent per mode is in good agreement with theoretical predictions for ballistic, one-dimensional modes that are longer than the Josephson penetration depth. Moreover, we find that oscillatory features superimpose the plateaus of the supercurrent and the conductance. The strength of these oscillatory features are sample-dependent and complicate the identification of plateaus. We suggest that the oscillatory features originate mainly from local defects and the short gate electrode. Additionally, resonances are promoted within the weak link if the transparency of the superconductor-HgTe interface differs from one.
Furthermore, the research explores the regimes of the quantum spin Hall effect and the 0.5 anomaly. Notably, a small yet finite supercurrent is detected in the QSH regime. In samples fabricated from thick mercury telluride quantum wells, the supercurrent appears to vanish when the quantum point contact is tuned into the regime of the 0.5 anomaly. For samples fabricated from thin mercury telluride quantum wells, the conductance as well as the supercurrent vanish for strong depopulation. In these samples though, the supercurrent remains detectable even for conductance values significantly below 2 e²/h.
Numerical calculation reproduce the transport behavior of the superconducting quantum point contacts.
Additionally, the topological nature of the weak link is thoroughly investigated using the supercurrent diffraction pattern and the absorption of radio frequency photons. The diffraction pattern reveals a gate independent, monotonous decay of $I_\text{sw}(B)$, which is associated with the quantum interference of Andreev bound states funneled through the quantum point contact. Interestingly, the current distribution in the weak link appears unaffected as the quantum point contact is depleted. In the RF measurements, indications of a 4π periodic supercurrent are observed as a suppression of odd Shapiro steps. The ratio of the 4π periodic current to the 2π periodic current appears to decrease for smaller supercurrents, as odd Shapiro steps are exclusively suppressed for large supercurrents. Additionally, considering the observation that the supercurrent is small when the bulk modes in the quantum point contact are fully depleted, we suggest that the re-emerging of odd Shapiro steps is a consequence of the group velocity of the edge modes being significantly suppressed when the bulk modes are absent. Consequently, the topological nature of the superconducting quantum point contact is only noticeable in the transport properties when bulk modes are transmitted through the superconducting quantum point contact.
The shown experiments are the first demonstration of mercury telluride superconducting quantum point contacts that exhibit signatures of quantization effects in the conductance as well as the supercurrent. Moreover, the experiments suggest that the regime of interacting topological edge channels is also accessible in mercury telluride superconducting quantum point contacts. This is potentially relevant for the realization of Majorana fermions and their application in the field of quantum computation.
Over the last two decades, accompanied by their prediction and ensuing realization, topological non-trivial materials like topological insulators, Dirac semimetals, and Weyl semimetals have been in the focus of mesoscopic condensed matter research. While hosting a plethora of intriguing physical phenomena all on their own, even more fascinating features emerge when superconducting order is included. Their intrinsically pronounced spin-orbit coupling leads to peculiar, time-reversal symmetry protected surface states, unconventional superconductivity, and even to the emergence of exotic bound states in appropriate setups.
This Thesis explores various junctions built from - or incorporating - topological materials in contact with superconducting order, placing particular emphasis on the transport properties and the proximity effect.
We begin with the analysis of Josephson junctions where planar samples of mercury telluride are sandwiched between conventional superconducting contacts. The surprising observation of pronounced excess currents in experiments, which can be well described by the Blonder-Tinkham-Klapwijk theory, has long been an ambiguous issue in this field, since the necessary presumptions are seemingly not met. We propose a resolution to this predicament by demonstrating that the interface properties in hybrid nanostructures of distinctly different materials yet corroborate these assumptions and explain the outcome. An experimental realization is feasible by gating the contacts. We then proceed with NSN junctions based on time-reversal symmetry broken Weyl semimetals and including superconducting order. Due to the anisotropy of the electron band structure, both the transport properties as well as the proximity effect depend substantially on the orientation of the interfaces between the materials. Moreover, an imbalance can be induced in the electron population between Weyl nodes of opposite chirality, resulting in a non-vanishing spin polarization of the Cooper pairs leaking into the normal contacts. We show that such a system features a tunable dipole character with possible applications in spintronics. Finally, we consider partially superconducting surface states of three-dimensional topological insulators. Tuning such a system into the so-called bipolar setup, this results in the formation of equal-spin Cooper pairs inside the superconductor, while simultaneously acting as a filter for non-local singlet pairing. The creation and manipulation of these spin-polarized Cooper pairs can be achieved by mere electronic switching processes and in the absence of any magnetic order, rendering such a nanostructure an interesting system for superconducting spintronics. The inherent spin-orbit coupling of the surface state is crucial for this observation, as is the bipolar setup which strongly promotes non-local Andreev processes.
In this thesis we discuss the potential of nanodevices based on topological insulators. This novel class of matter is characterized by an insulating bulk with simultaneously conducting boundaries. To lowest order, the states that are evoking the conducting behavior in TIs are typically described by a Dirac theory. In the two-dimensional case, together with time- reversal symmetry, this implies a helical nature of respective states. Then, interesting physics appears when two such helical edge state pairs are brought close together in a two-dimensional topological insulator quantum constriction. This has several advantages. Inside the constriction, the system obeys essentially the same number of fermionic fields as a conventional quantum wire, however, it possesses more symmetries. Moreover, such a constriction can be naturally contacted by helical probes, which eventually allows spin- resolved transport measurements.
We use these intriguing properties of such devices to predict the formation and detection of several profound physical effects. We demonstrate that narrow trenches in quantum spin Hall materials – a structure we coin anti-wire – are able to show a topological super- conducting phase, hosting isolated non-Abelian Majorana modes. They can be detected by means of a simple conductance experiment using a weak coupling to passing by helical edge states. The presence of Majorana modes implies the formation of unconventional odd-frequency superconductivity. Interestingly, however, we find that regardless of the presence or absence of Majoranas, related (superconducting) devices possess an uncon- ventional odd-frequency superconducting pairing component, which can be associated to a particular transport channel. Eventually, this enables us to prove the existence of odd- frequency pairing in superconducting quantum spin Hall quantum constrictions. The symmetries that are present in quantum spin Hall quantum constrictions play an essen- tial role for many physical effects. As distinguished from quantum wires, quantum spin Hall quantum constrictions additionally possess an inbuilt charge-conjugation symmetry. This can be used to form a non-equilibrium Floquet topological phase in the presence of a time-periodic electro-magnetic field. This non-equilibrium phase is accompanied by topological bound states that are detectable in transport characteristics of the system. Despite single-particle effects, symmetries are particularly important when electronic in- teractions are considered. As such, charge-conjugation symmetry implies the presence of a Dirac point, which in turn enables the formation of interaction induced gaps. Unlike single-particle gaps, interaction induced gaps can lead to large ground state manifolds. In combination with ordinary superconductivity, this eventually evokes exotic non-Abelian anyons beyond the Majorana. In the present case, these interactions gaps can even form in the weakly interacting regime (which is rather untypical), so that the coexistence with superconductivity is no longer contradictory. Eventually this leads to the simultaneous presence of a Z4 parafermion and a Majorana mode bound at interfaces between quantum constrictions and superconducting regions.
Magnetic systems underlie the physics of quantum mechanics when reaching the limit of few or even single atoms. This behavior limits the minimum size of magnetic bits in data storage devices as spontaneous switching of the magnetization leads to the loss of information. On the other hand, exactly these quantum mechanic properties allow to use such systems in quantum computers. Proposals to realize qubits involve the spin states of single atoms as well as topologically protected Majorana zero modes, that emerge in coupled systems of magnetic atoms in proximity to a superconductor. In order to implement and control the proposed applications, a detailed understanding of atomic spins and their interaction with the environment is required.
In this thesis, two different systems of magnetic adatoms coupled to metallic and superconducting surfaces are studied by means of scanning tunneling microscopy (STM) and spectroscopy: Co atoms on the clean Cu(111) were among the first systems exhibiting signatures of the Kondo effect in an individual atom. Yet, a recent theoretical work proposed an alternative interpretation of these early experimental results, involving a newly described many-body state. Spin-averaged and -polarized experiments in high magnetic fields presented in this thesis confirm effects beyond the Kondo effect that determine the physics in these Co atoms and suggest a potentially even richer phenomenology than proposed by theory.
The second studied system are single and coupled Fe atoms on the superconducting Nb(110) surface. Magnetic impurities on superconducting surfaces locally induce Yu-Shiba-Rusinov (YSR) states inside the superconducting gap due to their pair breaking potential. Coupled systems of such impurities exhibit YSR bands and, if the bands cross the Fermi level such that the band structure is inverted, host Majorana zero modes. Using the example of Fe atoms on Nb(110), the YSR states’ dependence on the adatom–substrate interaction as well as the interatomic YSR state coupling is investigated. In the presence of oxygen on the Nb surface, the adatom–substrate interaction is shown to be heavily modified and the YSR states are found to undergo a quantum phase transition, which can be directly linked to a modified Kondo screening.
STM tips functionalized with CO molecules allow to resolve self-assembled one-dimensional chains of Fe atoms on the clean Nb(110) surface to study the YSR states’ coupling. Mapping out the states’ wave functions reveals their symmetry, which is shown to alter as a function of the states’ energy and number of atoms in the chain. These experimental results are reproduced in a simple tight-binding model, demonstrating a straightforward possibility to describe also more complex YSR systems toward engineered, potentially topologically non-trivial states.
This thesis contains two major parts: The first part introduces the reader into three independent concepts of treating strongly correlated many body physics. These are, on the analytical side the SO(5)-theory (Chap.3), which poses the general frame. On the numerical side these are the Stochastic Series Expansion (SSE) (Chap.1) and the Contractor Renormalization Group (CORE) approach (Chap. 2}). The central idea of this thesis was to combine these above concepts, in order to achieve a better understanding of the high-T_c superconductors (HTSC). The results obtained by this combination can be found in the second major part of this thesis (chapters 4 and 5). The main idea of this thesis, i.e., to combine the SO(5)-theory with the capabilities of bosonic Quantum-Monte Carlo simulations and those of the CORE approach, has been proven to be a very successful Ansatz. Two different approaches, one based on symmetry and one on renormalization-group arguments, motivate an effective bosonic Hamiltonian. In a subsequent step the effective Hamiltonian has been simulated efficiently using the SSE. The results reproduce salient experiments on high-T_c superconductors. In addition, it has been shown that the model can be extended to capture also charge ordering. These results also form a profound basis for further studies, for example one could address the open question of SO(5)-symmetry restoration at a multicritical point in the extended pSO(5) model, where longer ranged interactions are included.
A general theory for all classes of unconventional superconductors is still one of the unsolved key issues in condensed-matter physics. Actually, it is not yet fully settled if there is a common underlying pairing mechanism. Instead, it might be possible that several distinct sources for unconventional (not phonon-mediated) superconductivity have to be considered, or an electron-phonon interaction is not negligible. The focus of this thesis is on the most probable mechanism for the formation of Cooper pairs in unconventional superconductors, namely a strictly electronic one where spin fluctuations are the mediators. Studying different superconductors in this thesis, the emphasis is put on material-independent features of the pairing mechanism. In addition, the investigation of the phase diagrams enables a view on the vicinity of superconductivity. Thus, it is possible to clarify which competing quantum fluctuations enhance or weaken the propensity for a superconducting state. The broad range of superconducting materials requires the use of more than one numerical technique to study an appropriate microscopic description. This is not a problem but a big advantage because this facilitates the approach-independent description of common underlying physics. For this evaluation, the strongly correlated cuprates are simulated with the variational cluster approach. Especially the question of a pairing glue is taken into consideration. Furthermore, it is possible to distinguish between retarded and non-retarded contributions to the gap function. The cuprates are confronted with the cobaltate NaCoO and graphene. These weakly correlated materials are investigated with the functional renormalization group (fRG) and reveal a comprehensive phase diagram, including a d+id-wave superconductivity, which breaks time-reversal symmetry. The corresponding gap function is nodeless, but for NaCoO, it features a doping-dependent anisotropy. In addition, some general considerations on the kagome lattice are completing the discussion, where a sublattice interference dramatically affects the Fermi-surface instabilities, suppressing the usual spin-density wave and d+id-wave superconductivity. Thereby, some different fascinating charge and bond orders as well as a nematic are observable. In short, this thesis provides an insight to distinct classes of unconventional superconductors with appropriate simulation techniques. This facilitates to separate the material specific properties from the universal ones.
This Thesis explores hybrid structures on the basis of quantum spin Hall insulators, and in particular the interplay of their edge states and superconducting and magnetic order. Quantum spin Hall insulators are one example of topological condensed matter systems, where the topology of the bulk bands is the key for the understanding of their physical properties. A remarkable consequence is the appearance of states at the boundary of the system, a phenomenon coined bulk-boundary correspondence. In the case of the two-dimensional quantum spin Hall insulator, this is manifested by so-called helical edge states of counter-propagating electrons with opposite spins. They hold great promise, \emph{e.g.}, for applications in spintronics -- a paradigm for the transmission and manipulation of information based on spin instead of charge -- and as a basis for quantum computers. The beginning of the Thesis consists of an introduction to one-dimensional topological superconductors, which illustrates basic concepts and ideas. In particular, this includes the topological distinction of phases and the accompanying appearance of Majorana modes at their ends. Owing to their topological origin, Majorana modes potentially are essential building-blocks for topological quantum computation, since they can be exploited for protected operations on quantum bits. The helical edge states of quantum spin Hall insulators in conjunction with $s$-wave superconductivity and magnetism are a suitable candidate for the realization of a one-dimensional topological superconductor. Consequently, this Thesis investigates the conditions in which Majorana modes can appear. Typically, this happens between regions subjected to either only superconductivity, or to both superconductivity and magnetism. If more than one superconductor is present, the phase difference is of paramount importance, and can even be used to manipulate and move Majorana modes. Furthermore, the Thesis addresses the effects of the helical edge states on the anomalous correlation functions characterizing proximity-induced superconductivity. It is found that helicity and magnetism profoundly enrich their physical structure and lead to unconventional, exotic pairing amplitudes. Strikingly, the nonlocal correlation functions can be connected to the Majorana bound states within the system. Finally, a possible thermoelectric device on the basis of hybrid systems at the quantum spin Hall edge is discussed. It utilizes the peculiar properties of the proximity-induced superconductivity in order to create spin-polarized Cooper pairs from a temperature bias. Cooper pairs with finite net spin are the cornerstone of superconducting spintronics and offer tremendous potential for efficient information technologies.
The combination of a topological insulator (TI) and a superconductor (S), which together
form a TI/S interface, is expected to influence the possible surface states in the
TI. It is of special interest, if the theoretical prediction of zero energy Majorana states
in this system is verifiable. This thesis presents the experimental realization of such
an interface between the TI strained bulk HgTe and the S Nb and studies if the afore
mentioned expectations are met.
As these types of interfaces were produced for the first time the initial step was
to develop a new lithographic process. Optimization of the S deposition technique as
well as the application of cleaning processes allowed for reproducible fabrication of
structures. In parallel the measurement setup was upgraded to be able to execute the
sensitive measurements at low energy. Furthermore several filters have been implemented
into the system to reduce high frequency noise and the magnetic field control
unit was additionally replaced to achieve the needed resolution in the μT range.
Two kinds of basic geometries have been studied: Josephson junctions (JJs) and
superconducting quantum interference devices (SQUIDs). A JJ consists of two Nb contacts
with a small separation on a HgTe layer. These S/TI/S junctions are one of the
most basic structures possible and are studied via transport measurements. The transport
through this geometry is strongly influenced by the behavior at the two S/TI
interfaces. In voltage dependent differential resistance measurements it was possible
to detect multiple Andreev reflections in the JJ, indicating that electrons and holes are
able to traverse the HgTe gap between both interfaces multiple times while keeping
phase coherence. Additionally using BTK theory it was possible to extract the interface
transparency of several junctions. This allowed iterative optimization for the highest
transparency via lithographic improvements at these interfaces. The increased transparency
and thus the increased coupling of the Nb’s superconductivity to the HgTe
results in a deeper penetration of the induced superconductivity into the HgTe. Due
to this strong coupling it was possible to enter the regime, where a supercurrent is
carried through the complete HgTe layer. For the first time the passing of an induced
supercurrent through strained bulk HgTe was achieved and thus opened the area for
detailed studies. The magnetic dependence of the supercurrent in the JJ was recorded,
which is also known as a Fraunhofer pattern. The periodicity of this pattern in magnetic
field compared to the JJ geometry allowed to conclude how the junction depends
on the phase difference between both superconducting contacts. Theoretical calculations
predicted a phase periodicity of 4p instead of 2p, if a TI is used as weak link
material between the contacts, due to the presence of Majorana modes. It could clearly
be shown that despite the usage of a TI the phase still was 2p periodic. By varying
further influencing factors, like number of modes and phase coherence length in the
junction, it might still be possible to reach the 4p regime with bound Majorana states
in the future. A good candidate for further experiments was found in capped HgTe
samples, but here the fabrication process still has to be developed to the same quality
as for the uncapped HgTe samples.
The second type of geometry studied in this thesis was a DC-SQUID, which consists
of two parallel JJs and can also be described as an interference device between two JJs.
The DC-SQUID devices were produced in two configurations: The symmetric SQUID,
where both JJs were identical, and the asymmetric SQUID, where one JJ was not linear,
but instead has a 90° bent. These configurations allow to test, if the predicted
uniformity of the superconducting band gap for induced superconductivity in a TI
is valid. While the phase of the symmetric SQUID is not influenced by the shape of
the band gap, the asymmetric SQUID would be in phase with the symmetric SQUID
in case of an uniform band gap and out of phase if p- or d-wave superconductivity
is dominating the transport, due to the 90° junction. As both devices are measured
one after another, the problem of drift in the coil used to create the magnetic field has
to be overcome in order to decide if the oscillations of both types of SQUIDs are in
phase. With an oscillation period of 0.5 mT and a drift rate in the range of 5.5 μT/h
the measurements on both configurations have to be conducted in a few hours. Only
then the total shift is small enough to compare them with each other. For this to be
possible a novel measurement system based on a real time micro controller was programmed,
which allows a much faster extraction of the critical current of a device. The
measurement times were reduced from days to hours, circumventing the drift problems
and enabling the wanted comparison. After the final system optimizations it has
been shown that the comparison should now be possible. Initial measurements with
the old system hinted that both types of SQUIDs are in phase and thus the expected
uniform band gap is more likely. With all needed optimizations in place it is now up
to the successors of this project to conclusively prove this last point.
This thesis has proven that it is possible to induce superconductivity in strained
bulk HgTe. It has thus realized the most basic sample geometry proposed by Fu and
Kane in 2008 for the appearance of Majorana bound states. Based on this work it is
now possible to further explore induced superconductivity in strained bulk HgTe to
finally reach a regime, where the Majorana states are both stable and detectable.
The superconducting properties of complex materials like the recently discovered iron-pnictides or strontium-ruthenate are often governed by multi-orbital effects. In order to unravel the superconductivity of those materials, we develop a multi-orbital implementation of the functional renormalization group and study the pairing states of several characteristic material systems. Starting with the iron-pnictides, we find competing spin-fluctuation channels that become attractive if the superconducting gap changes sign between the nested portions of the Fermi surface. Depending on material details like doping or pnictogen height, these spin fluctuations then give rise to $s_{\pm}$-wave pairing with or without gap nodes and, in some cases, also change the symmetry to $d$-wave. Near the transition from nodal $s_{\pm}$-wave to $d$-wave pairing, we predict the occurrence of a time-reversal symmetry-broken $(s+id)$-pairing state which avoids gap nodes and is therefore energetically favored. We further study the electronic instabilities of doped graphene, another fascinating material which has recently become accessible and which can effectively be regarded as multi-orbital system. Here, the hexagonal lattice structure assures the degeneracy of two $d$-wave pairing channels, and the system then realizes a chiral $(d+id)$-pairing state in a wide doping range around van-Hove filling. In addition, we also find spin-triplet pairing as well as an exotic spin-density wave phase which both become leading if the long-ranged hopping or interaction parameters are slightly modified, for example, by choosing different substrate materials. Finally, we consider the superconducting state of strontium-ruthenate, a possible candidate for chiral spin-triplet pairing with fascinating properties like the existence of half-quantum vortices obeying non-Abelian statistics. Using a microscopic three orbital description including spin-orbit coupling, we demonstrate that ferromagnetic fluctuations are still sufficient to induce this $\bs{\hat{z}}(p_x\pm ip_y)$-pairing state. The resulting superconducting gap reveals strong anisotropies on the $d_{xy}$-dominated Fermi-surface pocket and nearly vanishes on the other remaining two pockets.
Relativistic effects crucially influence the fundamental properties of many quantum materials. In the accelerated reference frame of an electron, the electric field of the nuclei is transformed into a magnetic field that couples to the electron spin. The resulting interaction between an electron spin and its orbital angular momentum, known as spin-orbit coupling (SOC), is hence fundamental to the physics of many condensed matter phenomena. It is particularly important quantitatively in low-dimensional quantum systems, where its coexistence with inversion symmetry breaking can lead to a splitting of spin degeneracy and spin momentum locking. Using the paradigm of Landau Fermi liquid theory, the physics of SOC can be adequately incorporated in an effective single particle picture. In a weak coupling approach, electronic correlation effects beyond single particle propagator renormalization can trigger Fermi surface instabilities such as itinerant magnetism, electron nematic phases, superconductivity, or other symmetry broken states of matter.
In this thesis, we use a weak coupling-based approach to study the effect of SOC on Fermi surface instabilities and, in particular, superconductivity. This encompasses a weak coupling renormalization group formulation of unconventional superconductivity as well as the random phase approximation. We propose a unified formulation for both of these two-particle Green’s function approaches based on the notion of a generalized susceptibility.
In the half-Heusler semimetal and superconductor LuPtBi, both SOC and electronic correlation
effects are prominent, and thus indispensable for any concise theoretical description. The metallic and weakly dispersive surface states of this material feature spin momentum locked Fermi surfaces, which we propose as a possible domain for the onset of unconventional surface superconductivity. Using our framework for the analysis of Fermi surface instability and combining it with ab-initio density functional theory calculations, we analyse the surface band structure of LuPtBi, and particularly its propensity towards Cooper pair formation. We study how the presence of strong SOC modifies the classification of two-electron wave functions as well as the screening of electron-electron interactions. Assuming an electronic mechanism, we identify a chiral superconducting condensate featuring Majorana edge modes to be energetically favoured over a wide range of model parameters.