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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.
Exploring the transport properties of the three-dimensional topological insulator material HgTe
(2015)
In the present thesis the transport properties of strained bulk HgTe devices are investigated. Strained HgTe forms a 3D TI and is of special interest for studying topological surface states, since it can be grown by MBE in high crystal quality. The low defect density leads to considerable mobility values, well above the mobilities of other TI materials. However, strained HgTe has a small band gap of ca. 20 meV. With respect to possible applications the question is important, under which conditions the surface transport occurs. To answer this question, the HgTe devices are investigated at dilution refrigerator temperatures (T<100 mK) in high magnetic fields of different orientation. The influence of top and back gate electrodes as well as surface protecting layers is discussed.
On the basis of an analysis of the quantum Hall behaviour it is shown that transport is dominated by the topological surface states in a surprisingly large parameter range. A dependence on the applied top gate voltage is presented for the topological surface states. It enables the first demonstration of an odd integer QHE sequence from the surfaces perpendicular to the magnetic field. Furthermore, the p-type QHE from the surface states is observed for the first time in any 3D TI. This is achieved in samples of high surface quality. It is concluded from the gate response that the screening behaviour in 3D TI devices is non-trivial. The transport data are qualitatively analysed by means of intuitive theoretical models.
The quantum Hall (QH) effect, which can be induced in a two-dimensional (2D) electron gas by an external magnetic field, paved the way for topological concepts in condensed matter physics. While the QH effect can for that reason not exist without Landau levels, there is a plethora of topological phases of matter that can exist even in the absence of a magnetic field. For instance, the quantum spin Hall (QSH), the quantum anomalous Hall (QAH), and the three-dimensional (3D) topological insulator (TI) phase are insulating phases of matter that owe their nontrivial topology to an inverted band structure. The latter results from a strong spin-orbit interaction or, generally, from strong relativistic corrections. The main objective of this thesis is to explore the fate of these preexisting topological states of matter, when they are subjected to an external magnetic field, and analyze their connection to quantum anomalies. In particular, the realization of the parity anomaly in solid state systems is discussed. Furthermore, band structure engineering, i.e., changing the quantum well thickness, the strain, and the material composition, is employed to manipulate and investigate various topological properties of the prototype TI HgTe.
Like the QH phase, the QAH phase exhibits unidirectionally propagating metallic edge channels. But in contrast to the QH phase, it can exist without Landau levels. As such, the QAH phase is a condensed matter analog of the parity anomaly. We demonstrate that this connection facilitates a distinction between QH and QAH states in the presence of a magnetic field. We debunk therefore the widespread belief that these two topological phases of matter cannot be distinguished, since they are both described by a $\mathbb{Z}$ topological invariant. To be more precise, we demonstrate that the QAH topology remains encoded in a peculiar topological quantity, the spectral asymmetry, which quantifies the differences in the number of states between the conduction and valence band. Deriving the effective action of QAH insulators in magnetic fields, we show that the spectral asymmetry is thereby linked to a unique Chern-Simons term which contains the information about the QAH edge states. As a consequence, we reveal that counterpropagating QH and QAH edge states can emerge when a QAH insulator is subjected to an external magnetic field. These helical-like states exhibit exotic properties which make it possible to disentangle QH and QAH phases. Our findings are of particular importance for paramagnetic TIs in which an external magnetic field is required to induce the QAH phase.
A byproduct of the band inversion is the formation of additional extrema in the valence band dispersion at large momenta (the `camelback'). We develop a numerical implementation of the $8 \times 8$ Kane model to investigate signatures of the camelback in (Hg,Mn)Te quantum wells. Varying the quantum well thickness, as well as the Mn-concentration, we show that the class of topologically nontrivial quantum wells can be subdivided into direct gap and indirect gap TIs. In direct gap TIs, we show that, in the bulk $p$-regime, pinning of the chemical potential to the camelback can cause an onset to QH plateaus at exceptionally low magnetic fields (tens of mT). In contrast, in indirect gap TIs, the camelback prevents the observation of QH plateaus in the bulk $p$-regime up to large magnetic fields (a few tesla). These findings allowed us to attribute recent experimental observations in (Hg,Mn)Te quantum wells to the camelback. Although our discussion focuses on (Hg,Mn)Te, our model should likewise apply to other topological materials which exhibit a camelback feature in their valence band dispersion.
Furthermore, we employ the numerical implementation of the $8\times 8$ Kane model to explore the crossover from a 2D QSH to a 3D TI phase in strained HgTe quantum wells. The latter exhibit 2D topological surface states at their interfaces which, as we demonstrate, are very sensitive to the local symmetry of the crystal lattice and electrostatic gating. We determine the classical cyclotron frequency of surface electrons and compare our findings with experiments on strained HgTe.
This thesis treats the thermopower and other thermal effects in single quantum dots (QD) and quantum dot systems. It contributes new experimental results to the broad and active field of research on thermoelectrics in low dimensional systems. The thermopower experiments discussed in this work focus on QDs which exhibit a net spin and on tunnel-coupled double QDs (DQD). Furthermore, experiments are presented which address the realization of a QD device which extracts thermal energy from a heat reservoir and converts it into a directed charge current in a novel way.
The samples used for these investigations have been fabricated from GaAs/AlGaAs heterostructures which contain a two dimensional electron gas. Using optical and electron beam lithography, the devices have been realized by means of the top-gate technology. All experiments have been performed at low temperature. In order to create a controllable temperature difference in the samples the current heating technique has been used. These experimental basics as well as fundamentals of electric and thermoelectric transport are introduced in Part I of this thesis.
The experiments on the thermopower of a single QD are described in Part II. Essentially, they deal with the problem of how a single spin situated on a QD influences the thermoelectric properties of the system. In this context, the Kondo-effect plays a crucial role. Generally, the Kondo effect is the result of a many-body state which arises from an antiferromagnetic coupling of a magnetic impurity with the surrounding conduction electrons. Here, the magnetic impurity is represented by a QD which is occupied with an odd number of electrons so that it exhibits a net spin. For the first time the thermopower of a Kondo-QD has been studied systematically as a function of two parameters, namely the QD coupling energy and the sample temperature. Both parameters are crucial quantities for Kondo-physics to be observed. Based on these data, it is shown that the thermopower line shape as a function of QD energy is mainly determined by two competing contributions: On the one hand by the enhanced density of states around the Fermi level due to Kondo-correlations and on the other hand by thermopower contributions from the Coulomb resonances. Furthermore, the experiments confirm theoretical predictions which claim that the spectral DOS arising from Kondo-correlations shifts away from the Fermi level for those QD level configurations which are not electron-hole symmetric. Comparison with model calculations by T. Costi and V. Zlatic [Phys. Rev. B 81, 235127 (2010)] shows qualitative and partly even quantitative agreement. A finite thermovoltage at the center of the Kondo-region, which occurred in previous investigations, is also observed in the experiments presented here. It is not covered by the current theory of the Kondo effect. The dependence of this signal on temperature, coupling energy and magnetic field, which differ from non-Kondo regions, is analyzed. In order to clarify the physics behind this phenomenon further studies are desirable.
Furthermore, it is shown by variation of the QD coupling energy over a wide range that Kondo-correlations can be detected in the thermopower even in the regime of very weak coupling. In contrast, no Kondo signatures are visible in the conductance in this energy range. It is found that in the limit of weak coupling the Kondo effect causes the thermopower to exhibit a diminished amplitude in close vicinity of a conductance resonance. Subsequent filling of spin-degenerate states then leads to a thermopower amplitude modulation (odd-even-effect). Although this effect had been observed in previous studies, no connection to Kondo physics had been established in order to explain the observations.
Hence, the experiments on a single QD presented in this thesis provide unique insight into the complex interplay of different transport mechanisms in a spin-correlated QD. Moreover, the results confirm the potential of thermopower measurements as a highly sensitive tool to probe Kondo-correlations.
In Part III thermal effects are investigated in systems which contain two coupled QDs.
Such QD-systems are particularly interesting with respect to thermoelectric applications: Many proposals utilize the extremely sharp energy filtering properties of such coupled QDs and also different kinds of inter dot coupling to construct novel and highly efficient thermoelectric devices. In the present work, thermopower characterizations are performed on a tunnel-coupled DQD for the first time. The key result of these investigations is the thermopower stability diagram. Here it is found, that in such a system maximal thermopower is generated in the vicinity of the so-called triple points (TP) at which three charge states of the DQD are degenerate. Along the axis of total energy, which connects two adjacent TP, a typical thermopower line shape is observed. It is explained and modeled within an intuitive picture that assumes two transport channels across the DQD, representing the TP. For those regions which are far away from the TP, the thermopower turns out to be very sensitive to the relative configuration of the QD energies. The conductance and thermopower data are well reproduced within a model that assumes transport via molecular states. Integration of both models into one then allows model calculations for a complete stability cell in conductance and thermopower to be done.
Furthermore, experiments on two capacitively coupled QDs are presented. In these studies the focus lies on testing the feasibility of such systems for the manipulation and generation of charge currents from thermal energy. In a series of experiments it is shown that such a system of QDs can be utilized to increase or decrease a current flowing between two electron reservoirs by varying the temperature in a third reservoir. This effect is based on the cross-correlation of occupation fluctuations of the individual QDs. These are positive for certain QD energy level configurations and negative for others, which increases or decreases the charge current in the experiments, respectively. In the stability diagram this is manifested in a characteristic clover leaf shaped structure of positive and negative current changes in vicinity of the TP. All main experimental results are reproduced qualitatively in simple model calculations. Due to the close analogy between electrical and thermal conductance of a QD, this effect of thermal switching can, in principle, also be used to built a thermal transistor.
Finally, it is shown that a system consisting of two Coulomb-coupled QDs, which couple a hot electron reservoir electrostatically to two cold electron reservoirs, can be utilized as a novel device which extracts heat from its environment and converts it into a directed charge current. The idea of this heat-to-current converter (HCC) was first proposed by R. Sánchez and M. Büttiker [Phys. Rev. B 83, 085428 (2011)]. It is not only characterized by the novelty of its working principle but also by the fact, that it decouples the directions of charge current and energy flow. In the experiments presented here, such HCC-currents are identified unambiguously: For certain QD-level configurations an electric current between the two cold reservoirs is observed if the temperature in the third reservoir is increased. The direction of this current is shown to be independent of an external voltage. In contrast, the direction of the current exhibits a characteristic dependence on the tunneling coefficients of the QDs, as predicted by theory: By adjusting the thickness and the shape of the respective tunnel junctions, a charge current can be generated between two cold reservoirs, and it can even be inverted. The experimental observations are quantitatively reproduced by model calculations by R. Sánchez and B. Sothmann. Thus, the results represent direct evidence for the existence of HCC-currents. Due to the novelty of the working principle of the HCC and its relevance from a fundamental scientific point of view, the results presented here are an important step towards energy harvesting devices at the nano scale.
Herstellung und Charakterisierung von Halbleiterbauelementen für die integrierte Quantenphotonik
(2014)
Der Schwerpunkt dieser Arbeit liegt auf der Entwicklung quantenphotonischer Komponenten, welche für eine monolithische Integration auf einem Halbleiter-Chip geeignet sind. Das GaAs-Materialsystem stellt für solch einen optischen Schaltkreis die ideale Plattform dar, weil es flexible Einzelphotonenquellen bereithält und mittels ausgereifter Technologien auf vielfältige Weise prozessiert werden kann.
Als Photonenemitter werden Quantenpunkte genutzt. Man kann sie mit komplexen Bauelementen kombinieren, um ihre optischen Eigenschaften weiter zu verbessern.
Im Rahmen dieser Arbeit konnte eine erhöhte Effizienz der Photonenemission beobachtet werden, wenn Quantenpunkte in Wellenleiter eingebaut werden, die durch photonische Kristalle gebildet werden. Die reduzierte Gruppengeschwindigkeit die diesem Effekt zugrunde liegt konnte anhand des Modenspektrums von kurzen Wellenleitern nachgewiesen werden. Durch zeitaufgelöste Messungen konnte ermittelt werden, dass die Zerfallszeit der spontanen Emission um einen Faktor von 1,7 erhöht wird, wenn die Emitter zur Mode spektrale Resonanz aufweisen. Damit verbunden ist eine sehr hohe Modeneinkopplungseffizienz von 80%.
Das Experiment wurde erweitert, indem die zuvor undotierte Membran des Wellenleiters durch eine Diodenstruktur ersetzt und elektrische Kontakte ergänzt wurden. Durch Anlegen von elektrischen Feldern konnte die Emissionsenergie der Quantenpunkte über einen weiten spektralen Bereich von etwa 7meV abgestimmt werden. Das Verfahren kann genutzt werden, um die exzitonischen Quantenpunktzustände in einen spektralen Bereich der Wellenleitermode mit besonders stark reduzierter Gruppengeschwindigkeit zu verschieben. Hierbei konnten für Purcell-Faktor und Kopplungseffizienz Bestwerte von 2,3 und 90% ermittelt werden. Mithilfe einer Autokorrelationsmessung wurde außerdem nachgewiesen, dass die Bauelemente als Emitter für einzelne Photonen geeignet sind.
Ein weiteres zentrales Thema dieser Arbeit war die Entwicklung spektraler Filterelemente. Aufgrund des selbstorganisierten Wachstums und der großen räumlichen Oberflächendichte von Quantenpunkten werden von typischen Anregungsmechanismen Photonen mit einer Vielzahl unterschiedlicher Energien erzeugt. Um die Emission eines einzelnen Quantenpunktes zu selektieren, muss der Transmissionsbereich des Filters kleiner sein als der Abstand zwischen benachbarten Spektrallinien. Ein Filter konnte durch die Variation des effektiven Brechungsindex entlang von indexgeführten Wellenleitern realisiert werden. Es wurde untersucht wie sich die optischen Eigenschaften durch strukturelle Anpassungen verbessern lassen. Ein weiterer Ansatz wurde mithilfe photonischer Kristalle umgesetzt. Es wurde gezeigt, dass der Filter hierbei eine hohe Güte von 1700 erreicht und gleichzeitig die Emission des Quantenpunkt-Ensembles abgetrennt werden kann. Die Bauelemente wurden so konzipiert, dass die im photonischen Kristall geführten Moden effizient in indexgeführte Stegwellenleiter einkoppeln können.
Ein Teil dieser Arbeit beschäftigte sich zudem mit den Auswirkungen von anisotropen Verspannungen auf die exzitonischen Zustände der Quantenpunkte. Besonders starke Verspannungsfelder konnten induziert werden, wenn der aktive Teil der Bauelemente vom Halbleitersubstrat abgetrennt wurde. Dies wurde durch ein neu entwickeltes Fabrikationsverfahren ermöglicht.
Infolgedessen konnten die Emissionsenergien reversibel um mehr als 5meV abgestimmt werden, ohne dass die optischen Eigenschaften signifikant beeinträchtigt wurden.
Die auf den aktiven Teil der Probe wirkende Verspannung wurde durch die Anwendung verschiedener Modelle abgeschätzt. Darüberhinaus wurde gezeigt, dass durch Verspannungen der spektrale Abstand zwischen den Emissionen von Exziton und Biexziton gezielt beeinflusst werden kann. Die Kontrolle dieser exzitonischen Bindungsenergie kann für die Erzeugung quantenmechanisch verschränkter Photonen genutzt werden. Dieses Ziel kann auch durch die Reduzierung der Feinstrukturaufspaltung des Exzitons erreicht werden. Die experimentell untersuchten Quantenpunkte weisen Feinstrukturaufspaltungen in der Größenordnung von 100meV auf. Durch genau angepasste Verspannungsfelder konnte der Wert erheblich auf 5,1meV verringert werden. Beim Durchfahren des Energieminimums der Feinstrukturaufspaltung wurde eine Drehung der Polarisationsrichtung um nahezu 90° beobachtet. Desweiteren wurde ein Zusammenhang des Polarisationsgrades mit der Feinstrukturaufspaltung nachgewiesen.
Es wurde ein weiterer Prozessablauf entworfen, um komplexe Halbleiterstrukturen auf piezoelektrische Elemente übertragen zu können. Damit war es möglich den Einfluss der Verspannungsfelder auf Systeme aus Quantenpunkten und Mikroresonatoren zu untersuchen. Zunächst wurde demonstriert, dass die Modenaufspaltung von Mikrosäulenresonatoren reversibel angepasst werden kann. Dies ist ebenfalls von Interesse für die Erzeugung polarisationsverschränkter Photonen. An Resonatoren aus photonischen Kristallen konnte schließlich gezeigt werden, dass das Verhältnis der spektralen Abstimmbarkeiten von exzitonischen Emissionslinien und Resonatormode etwa fünf beträgt, sodass beide Linien in Resonanz gebracht werden können. Dieses Verhalten konnte zur Beeinflussung der Licht-Materie-Wechselwirkung genutzt werden.
Recently a new state of matter was discovered in which the bulk insulating state in a material is accompanied by conducting surface or edge states. This new state of matter can be distinguished from a conventional insulator phase by the topological properties of its band structure which led to the name "topological insulators". Experimentally, topological insulator states are mostly found in systems characterized by a band inversion compared to conventional systems. In most topological insulator systems, this is caused by a combination of energetically close bands and spin orbit coupling. Such properties are found in systems with heavy elements like Hg and Bi. And indeed, the first experimental discovery of a topological insulator succeeded in HgTe quantum wells and later also in BiSb bulk systems.
Topological insulators are of large interest due to their unique properties: In 2-dimensional topological insulators one dimensional edge states form without the need of an external magnetic field (in contrast to the quantum Hall effect). These edge states feature a linear band dispersion, a so called Dirac dispersion. The quantum spin Hall states are helical edge states, which means they consist of counterpropagating oppositely spin polarized edge channels. They are therefore of great potential for spintronic applications as well as building blocks for new more exotic states like Majorana Fermions. 3-dimensional topological insulators feature 2-dimensional surface states with only one Dirac band (also called Dirac cone) on each surface and an interesting spin texture where spin and momentum are locked perpendicular to each other in the surface plane. This unique surface band structure is predicted to be able to host several exotic states like e.g. Majorana Fermions (in combination with superconductors) and magnetic monopole like excitations.
This PhD thesis will summarize the discovery of topological insulators and highlights the developments on their experimental observations. The work focuses on HgTe which is up to now the only topological insulator material where the expected properties are unambiguously demonstrated in transport experiments. In HgTe, the topological insulator properties arise from the inversion of the Gamma_6 and Gamma_8 bands. The band inversion in HgTe is due to a combination of a high spin orbit splitting in Te and large energy corrections (due to the mass-velocity term) to the energy levels in Hg. Bulk HgTe, however, is a semimetal, which means for the conversion into a topological insulator a band gap has to be opened. In two dimensions (HgTe quantum well structures) this is achieved via quantum confinement, which opens a band gap between the quantum well subbands. In three dimensions, strain is used to lift the degeneracy of the semimetallic Gamma_8 bands opening up a band gap.
The thesis is structured as follows:
- The first chapter of this thesis will give a brief overview on discoveries in the field of topological insulators. It focuses on works relevant to experimental results presented in the following chapters. This includes a short outline of the early predictions and a summary of important results concerning 2-dimensional topological insulators while the final section discusses observations concerning 3-dimensional topological insulators.
- The discovery of the quantum spin Hall effect in HgTe marked the first experimental observation of a topological insulator. Chapter 2 will focus on HgTe quantum wells and the quantum spin Hall effect.
Above a critical thickness, HgTe quantum wells are predicted to host the quantum spin Hall state, the signature of a 2-dimensional topological insulator. HgTe quantum wells exhibiting low carrier concentrations and at the same time high carrier mobilities are required to be able to measure the quantum spin Hall effect. The growth of such high quality HgTe quantum wells was one of the major goals for this work. Continuous optimization of the substrate preparation and growth conditions resulted in controlled carrier densities down to a few 10^10 cm^-2. At the same time, carrier mobilities exceeding 1 x 10^6 cm^2/Vs have been achieved, which provides mean free paths of several micrometers in the material. Thus the first experimental evidence for the existence of the quantum spin Hall edge states succeeded in transport experiments on microstructures: When the Fermi energy was located in the bulk band gap a residual quantized resistance of 2e^2/h was found. Further experiments focused on investigating the nature of transport in this regime. By non-local measurements the edge state character could be established. The measured non-local resistances corresponded well with predictions from the Landauer-Büttiker theory applied to transport in helical edge channels.
In a final set of experiments the spin polarization of the edge channels was investigated. Here, we could make use of the advantage that HgTe quantum well structures exhibit a large Rashba spin orbit splitting. In systems with a large Rashba spin orbit splitting a spin accumulation is expected to occur at the edge of the sample perpendicular to a current flow. This so-called spin Hall effect was then used as a spin injector and detector. Using split gate devices it was possible to bring spin Hall and quantum spin Hall state into direct contact, which enabled an all electrical detection of the spin polarization of the quantum spin Hall edge channels.
- HgTe as a 3-dimensional topological insulator will be presented in chapter 3. Straining the HgTe layer enables the observation of topological insulator behavior. It was found that strain can be easily implemented during growth by using CdTe substrates. CdTe has a slightly larger lattice constant than HgTe and therefore leads to tensile strain in the HgTe layer as long as the growth is pseudomorphic. Magnetotransport studies showed the emergence of quantum Hall transport with characteristic signatures of a Dirac type bandstructure. Thus, this result marks the first observation of the quantum Hall effect in the surface states of a 3-dimensional topological insulator.
Transport experiments on samples fitted with a top gate enabled the identification of contributions from individual surfaces. Furthermore, the surface state quantum Hall effect was found to be surprisingly stable, perturbations due to additional bulk transport could not be found, even at high carrier densities of the system.
- Chapters 4 - 6 serve as in depth overviews of selected works: Chapter 4 presents a detailed overview on the all electrical detection of the spin Hall effect in HgTe quantum wells. The detection of the spin polarization of the quantum spin Hall effect is shown in chapter 5 and chapter 6 gives a detailed overview on the quantum Hall effect originating from the topological surface state in strained bulk HgTe.
The investigations discussed in this thesis pioneered the experimental work on the transport properties of topological insulator systems. The understanding of the fundamental properties of topological insulators enables new experiments in which e.g. the inclusion of magnetic dopants or the interplay between topological insulator and superconductors can be investigated in detail.
A novel growth method has been developed, allowing for the growth of strained HgTe shells on CdTe nanowires (NWs). The growth of CdTe-HgTe core-shell NWs required high attention in controlling basic parameters like substrate temperature and the intensity of supplied material fluxes. The difficulties in finding optimized growth conditions have been successfully overcome in this work.
We found the lateral redistribution of liquid growth seeds with a ZnTe growth start to be crucial to trigger vertical CdTe NW growth. Single crystalline zinc blende CdTe NWs grew, oriented along [111]B. The substrate temperature was the most critical parameter to achieve straight and long wires. In order to adjust it, the growth was monitored by reflection high-energy electron diffraction, which was used for fine tuning of the temperature over time in each growth run individually. For optimized growth conditions, a periodic diffraction pattern allowed for the detailed analysis of atomic arrangement on the surfaces and in the bulk. The ability to do so reflected the high crystal quality and ensemble uniformity of our CdTe NWs. The NW sides were formed by twelve stable, low-index crystalline facets. We observed two types stepped and polar sides, separated by in total six flat and non-polar facets.
The high crystalline quality of the cores allowed to grow epitaxial HgTe shells around. We reported on two different heterostructure geometries. In the first one, the CdTe NWs exhibit a closed HgTe shell, while for the second one, the CdTe NWs are overgrown mainly on one side. Scanning electron microscopy and scanning transmission electron microscopy confirmed, that many of the core-shell NWs are single crystalline zinc blende and have a high uniformity. The symmetry of the zinc blende unit cell was reduced by residual lattice strain. We used high-resolution X-ray diffraction to reveal the strain level caused by the small lattice mismatch in the heterostructures. Shear strain has been induced by the stepped hetero-interface, thereby stretching the lattice of the HgTe shell by 0.06 % along a direction oriented with an angle of 35 ° to the interface.
The different heterostructures obtained, were the base for further investigation of quasi-one-dimensional crystallites of HgTe. We therefore developed methods to reliably manipulate, align, localize and contact individual NWs, in order to characterize the charge transport in our samples. Bare CdTe cores were insulating, while the HgTe shells were conducting. At low temperature we found the mean free path of charge carriers to be smaller, but the phase coherence length to be larger than the sample size of several hundred nanometers. We observed universal conductance fluctuations and therefore drew the conclusion, that the trajectories of charge carriers are defined by elastic backscattering at randomly distributed scattering sites. When contacted with superconducting leads, we saw induced superconductivity, multiple Andreev reflections and the associated excess current. Thus, we achieved HgTe/superconductor interfaces with high interfacial transparency.
In addition, we reported on the appearance of peaks in differential resistance at Delta/e for HgTe-NW/superconductor and 2*Delta/e for superconductor/HgTe-NW/superconductor junctions, which is possibly related to unconventional pairing at the HgTe/superconductor interface. We noticed that the great advantage of our self-organized growth is the possibility to employ the metallic droplet, formerly seeding the NW growth, as a superconducting contact. The insulating wire cores with a metallic droplet at the tip have been overgrown with HgTe in a fully in-situ process. A very high interface quality was achieved in this case.
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.
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.
Novel appraches to the molecular beam epitaxy of core-shell nanowires in the group II telluride material system were explored in this work. Significant advances in growth spurred the development of a flexible and reliable platform for a charge transport characterization of the topological insulator HgTe in a tubular nanowire geometry. The transport results presented provide an important basis for the design of future studies that strive for the experimental realization of topological charge transport in the quantum wire limit.