京都大学 / 0048 / 新制・課程博士 / 博士(工学) / 甲第19720号 / 工博第4175号 / 新制||工||1644(附属図書館) / 32756 / 京都大学大学院工学研究科電子工学専攻 / (主査)教授 川上 養一, 教授 田中 勝久, 准教授 掛谷 一弘 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(工学) / 甲第21370号 / 工博第4529号 / 新制||工||1705(附属図書館) / 京都大学大学院工学研究科電子工学専攻 / (主査)教授 竹内 繁樹, 教授 雨宮 尚之, 准教授 掛谷 一弘 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DFAM

The experimental pieces of evidence for the existence of Majorana states in topo- logical superconductors have so far been inconclusive despite intense research in the past two decades [Zha+20; Kay+20]. Combined with promising applications in quantum computing [Nay+08; Ali+11] and the resulting technological development of society, the elusiveness of Majorana states keeps motivating theoretical and ex- perimental research to this day. Our analytical findings and numerical explorations in new topological superconducting platforms suggest several tools and solutions for their future realisation in condensed matter systems. The planar Josephson junction (pJJ) introduced in 2017 by F. Pientka et al. [Pie+17] and M. Hell et al. [HLF17] is a versatile platform for topological superconductivity. It harnesses the tunability of the superconducting phase difference across the Josephson junction as an external control parameter that switches the pJJ between the trivial and topological phases of matter. The junction between the (trivial) superconductors is quasi-one-dimensional and hosts one new Majorana zero mode at each of its ends following each topological phase transition. However, the creation of a second Majorana zero mode on one end triggers a return to the trivial regime as both zero modes hybridize into a regular non-topological fermion. It is then crucial to identify the model parameters that lead to topological phases with a single Majorana state per end. Our main result on the pJJ establishes the general constraint on its microscopic parameters—including the phase difference and a magnetic field—to cross the topo- logical phase transitions. The identification of sectors in parameter space leading to a single Majorana mode becomes then straightforward. In some limits the pJJ develops a topological sector at small magnetic field for a phase difference close to the value p while it remains trivial at the same field near zero phase difference. Since the phase is sufficient to turn on and off the topology, we call this feature “switchable topology”. Looking for switchable topology is experimentally relevant as it makes the topology easily tunable while keeping intact the proximitized su- perconductivity otherwise jeopardized by the applied field. Concretely, we found switchable topology in three configurations: in wide junctions with a transparent interface with the superconducting regions, in fine-tuned narrow junctions weakly coupled to the superconducting regions, and in junctions with a strong Zeeman energy when they are ultranarrow and transparent. Thanks to our exact analytical results, setups interpolating between these limits can adjust the desired properties at will. The other important finding about the pJJ concerns the stability of its topological phases, by which we mean the presence of a sizable spectral gap in the topological sector. We observed that the Rashba spin-orbit coupling is responsible for strongly decreasing the gap in the relevant topological sector at low Zeeman field, but sym- metry arguments justify that wide, transparent junctions are generically immune to this effect for large enough Rashba coupling. After 2017, other platforms started to use the Josephson superconducting phase difference as a knob to trigger topological superconductivity [Liu+19; JY21]. We introduce here the stacked Josephson junction (sJJ) as a new platform for topological superconductivity, which is made of two non-centrosymmetric superconductors sandwiching a two-dimensional magnet around which chiral Majorana edge modes propagate. Unlike the Majorana zero modes in the pJJ, chiral Majorana modes can add to each other if they propagate in the same direction, as indicated by the integer Chern number of their topological phase. The bulk-edge correspondence, however, only constrains the net number of topological edge states and allows room for other non-topological states to coexist with the chiral Majorana states without interacting with them. We found that the presence of trivial chiral edge modes in the sJJ restricts access to the Majorana states themselves. The symmetry protection of the trivial modes, fortunately, disappears with an in-plane magnetic field applied through the magnet or with superconducting leads different on the top and at the bottom of the stacked junction. The theoretical investigations of topological platforms have currently outnum- bered the experiments with convincing signatures of Majorana edge states. This imbalance calls for new ways to probe the agreement between topological models and laboratory setups. The critical current of a Josephson junction acts as a link between the microscopic description and macroscopic observables. Thermoelectric measurements, which distinguish between supercurrent and quasiparticle current, modify this model-dependent connection, and would provide an electrical probe to estimate the validity of a model like that of the pJJ. We computed the contribution to the thermoelectric coefficient of the bulk states of a uniform superconductor, that has a similar environment to that of the pJJ (i.e., Rashba coupling and in-plane Zeeman field). The results were not conclusive and motivated us to suggest new analytical and numerical approaches to obtain the thermoelectric response of the pJJ, in particular by including the contribution of the Andreev bound states and non-linear effects.:Foreword — how to read this thesis 1 Preamble A popular short story: pencils and lightbulbs 5 Basics and concepts 1 Introduction to Majorana physics 13 1.1 The electrons & their properties 13 1.1.1 Hamiltonian for the planar Josephson junction 17 1.2 The scattering matrix for bound states 19 1.3 Andreev bound states for topology 24 1.4 Topological superconductivity & Majorana edge states 28 1.5 Induced topological superconductivity 34 1.6 Summary 36 Appendices 37 1.A Microscopic dynamics 37 1.A.1 Origin of spin–orbit coupling 37 1.A.2 Bogoliubov-deGennes symmetrization 37 1.A.3 Andreev reflection below the coherence length 38 1.A.4 Proximity-induced superconductivity 40 1.A.5 From s- to p-wave superconductivity 41 1.B Scattering theory for bound states 44 1.B.1 Bound states as trapped waves 44 1.B.2 Scattering theory for an open region 45 1.B.3 Scattering theory for two open regions 46 1.B.4 Bound states recovered from an open region 47 1.B.5 Numerical scattering theory for bound states 48 2 Perspectives on electronic transport 53 2.1 Electric current in a metal 53 2.2 Quantum-mechanical current 54 2.2.1 Expression for the microscopic current 55 2.3 Thermoelectric current 57 2.3.1 The Boltzmann transport equation 61 2.4 Supercurrents and the superconducting coherence phase 64 2.4.1 Josephson currents 67 Appendices 71 2.A Electric current from a potential difference 71 2.B Scattering and current 71 2.C Hole-based current in metals 73 Introduction Introduction to the Research Projects 77 i Topological properties of Josephson junctions 3 Switchable topology in the planar Josephson junction 85 Motivation & Overview of the Study 85 3.1 The planar Josephson junction and the nanowire setup 87 3.1.1 Comparison with the nanowire setup. 89 3.2 Model 92 3.3 General formula for the phase transitions 94 3.3.1 Spin decoupling for the phase transitions 96 3.3.2 Exact reflection coefficients 97 3.3.3 Exact scattering formula and Andreev reflectivity 98 3.3.4 Andreev approximation 100 3.3.5 Dimensionless formulation 101 3.3.6 Numerical and analytical checks 103 3.4 Three regimes for switchable topology 105 3.4.1 Diamond-shape regime 108 3.4.2 V-shape regime 110 3.4.3 Nanowire regime 111 3.4.4 Summary: extent of the topological transitions 114 3.5 Avoiding regimes with a small topological gap 117 3.5.1 Gapless lines as BDI phase transitions 119 3.5.2 Opening the gap in f = p 120 3.5.3 Role of the Rashba coupling 121 3.6 Conclusion 125 Appendices 129 3.A Limiting cases of the pJJ 129 3.A.1 Andreev approximation 129 3.A.2 Small field limit 131 3.A.3 Delta-barrier junction 131 3.A.4 Semiconductor nanowire 132 3.B Normal reflection via surface impurity and surface refraction 134 3.C Symmetry-constrained gap closings 136 3.D Linear deviation of the gapless line near f = p 138 3.E Calculations for the scattering formula 141 3.E.1 Boundary conditions 141 3.E.2 Combinations of scattering coefficients 142 3.E.3 Andreev coefficients for the phase transitions 143 3.E.4 Formula for B > μ 145 4 Topological and trivial chiral states in the stacked Josephson junction 147 Motivation & Overview of the Study 147 4.1 The basics of the stacked Josephson junction 149 4.2 Continuous and lattice models 151 4.3 Topological index 155 4.3.1 Methodology for the Chern number 155 4.3.2 Interpretation of the results 156 4.4 Topological and trivial edge states 162 4.5 BDI phase transitions 167 4.5.1 Dimensional reduction 168 4.5.2 Link between topological invariants 170 4.5.3 Explaining the low-energy sector 171 4.6 Conclusion 174 Appendices 177 4.A Symmetries of the Hamiltonian 177 4.A.1 Class D 177 4.A.2 Class BDI 177 4.A.3 Gapless line in f = p 178 4.A.4 Symmetry around f = p 179 4.B The parity index in 2D TSC 180 ii Transport properties of the planar Josephson junction 5 An approach to thermoelectric effects in the planar Josephson junction 183 Motivation & Overview of the Study 183 5.1 From the Josephson junction to a homogeneous superconductor 185 5.2 Model and Phenomenology 187 5.2.1 Homogeneous superconductor 187 5.2.2 Analytical spectrum and two-surface approximation 188 5.2.3 Magnetoelectric supercurrent: phenomenology 191 5.3 Electric current in a spin–orbit coupled superconductor 194 5.3.1 Formula for the current 196 5.3.2 Zero-temperature current 198 5.3.3 Small perturbations at finite temperature 200 5.4 Thermoelectric current in a spin–orbit coupled superconductor 206 5.4.1 Distribution imbalance under temperature bias 208 5.4.2 Explicit formula for the thermoelectric current 209 5.5 Discussion and Outlook 213 Appendices 219 5.A The Boltzmann equation in temperature-biased superconductors 219 5.A.1 The linear approximation 220 5.A.2 The low-temperature approximation 220 5.A.3 Integral solution of the Boltzmann equation 223 5.B Diagonalisation of the planar superconductor 225 5.B.1 Eigenstates of spin–orbit coupled superconductor 225 5.B.2 Eigenstates with a small Zeeman field 227 Conclusion Majorana quasiparticles in Josephson junctions 233 Extra Material 6 Mathematical details of Scattering theory 241 6.1 Asymmetric quantum well 241 6.2 Scattering theory for an open region 243 6.2.1 Change in potential over a small region 243 6.2.2 Change in spin-orbit coupling over a small region 245 6.2.3 Change in mass over a small region 245 7 Numerical codes for chapter 4 247 7.1 BDI index 247 7.2 Chern number 255 7.3 Spectral gap 257 7.4 Localized edge states 258 8 Short courses 261 8.1 Two formulations of superconductivity 261 8.1.1 The BCS Hamiltonian 261 8.1.2 The Bogoliubov transformation 263 8.1.3 Bogoliubov-de Gennes symmetrization 264 8.1.4 Building the semiconductor representation 266 8.2 Topological band theory 270 8.3 Majorana physics in 1D 274 8.3.1 The SSH chain 275 8.3.2 The Kitaev chain 277 Bibliography 283

Introduction. Defect-induced superconductivity is an important phenomenon manifested in triggering the superconducting state due to defects and disorder in the material lattice. Promising materials for this investigation are carbon-based. Josephson behavior has been reported in 1974 for a disordered graphite powder, which is considered to be the first hint of a room temperature graphite-based superconductor. Theoretical and experimental studies support the idea that certain two-dimensional stacking faults (SFs) in the semiconducting matrix contribute to the granular superconducting-like behavior of graphene-based materials. Hints for the existence of high-temperature superconductivity at certain SFs in graphite were demonstrated. This phenomenon is considered to be caused by flat band regions at the SF. Especially the SFs between Bernal and rhombohedral stacking orders (without any twist angle around the common c-axis) have the largest probability to show robust superconductivity due to an extended and robust flat band behavior. In this work, a permanent current path in graphite, after the application of a magnetic field, is investigated to show clear evidence for the existence of room temperature superconductivity (RTS). Preliminary results for the existence of such permanent current path were obtained with magnetic force microscopy (MFM) and published a few years ago. Thus, the objectives of this work are to investigate trapped magnetic flux with magnetic force microscopy, to reveal the reasons for the difficulties of finding such permanent current path in the remanent state of the sample and to give an additional hints to the semiconducting behavior and energy gaps of an ideal graphite using a new PF-TUNA method. Summary. The experimental pre-characterization of graphite samples was conducted using XRD and Raman spectroscopy. The spectra show well-ordered structure of the samples with a sufficient content of the rhombohedral phase. The grounded samples were examined with PF-TUNA mode at bias voltages applied between the conductive tip and the sample surface. The samples with Bernal phase and with mixed phases showed semiconductor-like behavior. Using the semiconductor model, the obtained simulations of registered I-V curves could estimate the energy gap in a range from 12 to 37 meV. This is in a good agreement with the values of energy gaps, observed in transport measurements. Additionally, the shift in the position of the minimum of the tunneling conductance was explained by the tip-induced band bending. The results of this thesis confirm the existence of the peak in the density of states, that is correlated to the flat band in a sufficiently thick multigraphene flake with a 3R stacking order (thickness should be much greater than 3 nm to observe it) at room temperature and the existence of the trapped magnetic flux, expulsed by the weakly coupled superconducting patches in the natural graphite sample. The trapped flux was identified and examined by MFM measurements at the surface of natural graphite sample in the remanent state. Therefore, we successfully reproduced the results reported in and performed field and time dependent measurements, that prove the superconducting origin of this phenomena. The modeling of the MFM signal was done according to the monopole tip approximation. The value of the permanent current was estimated in the range of 0.2 μA to 6 μA, which is consistent with literature. An accidental scratch on the sample surface allowed us to estimate the depth of the aforementioned superconducting patches, ≲ 10 nm, and gave additional evidence to its origin by changing the route due to the superconducting patches nearby. This investigation provides hints for room temperature superconductivity at certain SFs in graphite and clarifies the reasons for the difficulties of the trapped flux identification in graphite. Further research should be focus on the identification of the permanent currents by MFM at lower temperatures. Moreover, it would be helpful to understand, how to artificially produce extended SFs. Finally, it should be noted, that additional measurements should be performed in order to clarify the field dependence of trapped magnetic flux in graphite and the role of Pearl vortices. Collaboration and External Contributions. This work was conducted under the supervision of Prof. Dr. Pablo Esquinazi, Felix-Bloch-Institute for solid state physics, Division of Superconductivity and Magnetism, University of Leipzig. STEM images were made by Dr. W. Bölmann, University of Leipzig. X-ray diffraction was made by Mr. O. Baehre and Mr. T. Muenster at Institute of Mineralogy, Crystallography and Materials Science at the University of Leipzig. The Raman spectra were recorded by Mr. Tom Venus and Dr. Irina Estrela-Lopis, Institute of Medical Physics and Biophysics, University of Leipzig. The natural graphite samples from Brazil were provided by Prof. Dr. Ana Melva Champi Farfan from Universidade Federal do ABC in Santo Andre, Sao Paulo, Brazil. The natural graphite from Sri-Lanka by Mr. Henning Beth from Golden Bowerbird Pty Ltd. in Mullumbimby, Australia. The magnetoresistance measurement of a natural graphite sample from Sri-Lanka was performed by Dr. Christian E. Precker, AIMEN Technology Centre, Smart Systems and Smart Manufacturing, Artificial Intelligence and Data Analytics Laboratory, PI. Cataboi, Pontevedra, Spain. The calculations, related to modeling of the tunneling current based on the tip-induced band bending, were performed by Dr. Michael Schnedler, Peter Gruenberg Institut, Forschungszentrum Juelich.

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Transport properties of a few hundreds of nanometers thick (in the graphene plane direction) lamellae of highly oriented pyrolytic graphite (HOPG) have been investigated. Current–voltage characteristics as well as the temperature dependence of the voltage at different fixed input currents provide evidence for Josephson-coupled superconducting regions embedded in the internal two-dimensional interfaces of HOPG, reaching zero resistance at low enough temperatures.

Magnesium diboride MgB2 is an interesting material that was discovered to be a superconductor in 2001. It has a remarkably high critical temperature of 39 K which is much greater than was previously thought possible for a phonon-mediated superconductor. MgB2 was also the first material found to exhibit multiple gap superconductivity. It has two energy gaps, the pi gap with a value of 2.3 meV, and the sigma gap with a value of 7.1 meV. Both the high critical temperature and the multiple large energy gaps make MgB2 an attractive candidate for superconducting devices. While the initial discovery of MgB2 was accompanied by much excitement, the enthusiasm has mostly disappeared due to the lack of progress made in implementing MgB2 in practical devices. The aim of this thesis is to attempt to reinvigorate interest in this remarkable material through a study of a variety of practical superconducting devices made with MgB2 thin films grown by hybrid physical-chemical vapor deposition (HPCVD). Two different methods of fabricating MgB2 Josephson junctions are explored. The first is a sandwich type trilayer configuration with a barrier made by magnetron sputtered MgO. Junctions of this sort have been previously studied and implemented in a variety of devices. While they do show some attractive properties, the on-chip spread in critical current due to barrier non-uniformity was too high to be considered a viable option for use in many-junction devices. By developing a fabrication scheme which utilizes electron beam lithography, modest improvements were made in the on-chip parameter spread, and miniaturization of junction size yielded some insight into the non-uniform barriers. The second approach of creating MgB2 Josephson junctions utilized a planar geometry with a normal metal barrier created by irradiating nano-sized strips of the material with a focused helium ion beam. The properties of these junctions are investigated for different irradiation doses. This new technique is capable of producing high quality junctions and furthermore the parameter spread is greatly reduced as compared to the sandwich type junctions. While more research is necessary in order to increase the IcRn products, these junctions show promise for use in many-junction devices such as RSFQ circuits. Prior to this work, the largest substrates that could be coated with HPCVD grown MgB2 were 2" in diameter. A new chamber was designed and constructed which demonstrated the ability to coat substrates as large as 4". This scaled-up system was used to grow MgB2 films on 1 x 10 cm flexible substrates. A method of fabrication was developed which could pattern these 10 cm long samples into ribbon cables consisting of many high frequency transmission lines. This technology can be utilized to increase the cooling efficiency of cryogenic systems used for RSFQ systems which require many connections between low temperature and room temperature electronics. Finally, a method of producing MgB2 films with thicknesses as low as 8 nm was developed. This is achieved by first growing thicker films and using a low angle ion milling step to gradually reduce the film thickness while still maintaining well connected high quality films. A procedure was developed for fabricating meandering nanowires in these films with widths as low as 100 nm for use as superconducting nanowire single photon detectors (SNSPDs). A study of the transport properties of these devices is first presented. Measurements show low values of kinetic inductance which is ideal for high count rates in SNSPDs. The kinetic inductance measurements also yielded the first measurements of the penetration depth of MgB2 films in the ultra-thin regime. Devices made from these ultra-thin films were found to be photon sensitive by measurements made by our collaborators. / Physics

Nous avons étudié la dynamique des sauts quantiques de phase (quantum phase-slips) dans différents types de chaînes de jonctions Josephson. Les sauts de phase sont contrôlés par le rapport entre l'énergie Josephson et l'énergie de charge de chaque jonction. Nous avons mesuré l'effet des sauts de phase sur l'état fondamental de la chaîne et nous avons observé l'interférence quantique de sauts de phase (effet Aharonov-Casher). Les résultats de nos mesures sont en très bon accord avec les prédictions théoriques. Nous avons montré qu'une chaîne de jonctions Josephson polarisée en phase, présente un comportement collectif, similaire à un objet macroscopique. Les résultats de cette thèse ouvrent la voie pour la conception de nouveaux circuits Josephson, comme par exemple un qubit topologiquement protégé ou un dispositif quantique pour la conversion courant-fréquence. / In this thesis we presented detailed measurements of quantum phase-slips in Josephson junction chains. The measured phase-slips are the result of fluctuations induced by the finite charging energy of each junction. Our experimental results can be fitted in very good agreement by considering a simple tight-binding model for QPS. We have shown that under phase-bias, a chain of Josephson junctions or rhombi can behave in a collective way very similar to a single macroscopic quantum object. These results open the way for possible use of quantum phase-slips for the design of novel Josephson junction circuits, such as topologically protected rhombi qubits or current-to-frequency conversion devices.

Les amplificateurs paramétriques Josephson (JPA) se sont révélés être un outil indispensablepour l’étude expérimentale de dispositifs quantiques dans le régime micro-onde ; car ilsrajoutent uniquement le minimum de bruit imposé par la mécanique quantique[1]. Cependant,ces amplificateurs sont beaucoup plus difficiles à utiliser et optimiser que leurs homologuesclassiques. Récemment, plusieurs expériences réalisées avec des circuits supraconducteurs,composés d’une jonction Josephson polarisée en tension en série avec un résonateur microonde,ont montré qu’une paire de Cooper peut traverser la barrière de la jonction par effettunnel en émettant un ou plusieurs photons avec une énergie totale de 2e fois la tensionappliquée. Dans cette thèse, nous montrerons qu’un tel circuit permet de mettre en place unamplificateur micro-onde préservant la phase que nous appelons « Amplificateur basé sur letunneling inélastique de paires de Cooper » (ICTA). Il est alimenté par une tension continueet peut fonctionner avec un bruit très proche de la limite quantique.Nous commencerons en présentant le principe du fonctionnement de l’ICTA. Par analogieavec la théorie quantique des JPAs[2], nous avons étudié les performances de cet amplificateurcomme le gain, la bande passante et le bruit. Ensuite, nous présenterons la premièrepreuve expérimentale d’une amplification proche de la limite quantique sans utilisation d’unepompe micro-onde externe, mais simplement d’une tension continue dans une configurationextrêmement simple. Ces mesures ont été faites sur des échantillons avec des jonctionsen aluminium, dénommés ICTA de première génération. Selon nos résultats théoriques etexpérimentaux, nous avons conçu des circuits hyperfréquences où l’impédance présentéeà la jonction dépend de fréquences spécifiques afin d’optimiser les performances de notreamplificateur. Ces échantillons, dénommés ICTA de seconde génération, ont été fabriquésavec du nitrure de niobium. Une amélioration significative du gain et du bruit a été prouvée.Un tel amplificateur, alimenté par une simple tension continue, pourrait rendre la mesurede signaux micro-ondes au niveau du photon unique beaucoup plus faciles et permettred’intégrer plusieurs amplificateurs sur une seule puce. Il pourrait donc être un élémentimportant pour la lecture de qubit dans les processeurs quantiques à grande échelle. / Josephson parametric amplifiers (JPA), have proven to be an indispensable tool for awide range of experiments on quantum devices in the microwave frequency regime, becausethey provide the lowest possible noise. However, JPAs remain much more difficult to use andoptimize than conventional microwave amplifiers. Recent experiments with superconductingcircuits consisting of a DC voltage-biased Josephson junction in series with a resonator,showed that a tunneling Cooper pair can emit one or several photons with a total energyof 2e times the applied voltage. In this thesis we show that such q circuit can be used toimplement a new type of phase preserving microwave amplifier, which we call InelasticCooper pair Tunneling Amplifier (ICTA). It is powered by a simple DC bias and offers nearquantum-limited noise performance.We start this work by presenting a brief and simple picture of the basic ICTA operatingprinciples. In analogy with the quantum theory of JPAs, we calculate the performances ofthis amplifier such as the gain, bandwidth and noise. Then, we present the first experimentalproof that amplification close to the quantum limit is possible without microwave drive inan extremely simple setup. These measurements are made on a first generation of samplesbased on aluminium junctions. According to our theoretical and experimental results, wehave designed microwave circuits presenting specific frequency-dependent impedances tothe junction in order to optimize the performances of our amplifier. This second generationof ICTA samples is fabricated from niobium nitride and provide significantly lower noiseand higher gain.We expect that once fully optimized, such an amplifier, powered by simple DC voltagescould then make measuring microwave signals at the single photon level much easier andallow to deploy many amplifiers on a chip. It could therefore be an important ingredient forqubit readout in large-scale quantum processors.