In this thesis, we study the unconventional superconductivity in Sr2RuO4 (SRO) and related systems. The superconducting state in SRO remains a puzzle after more than 28 years of study. Early experiments had pointed toward a topological non-trivial time-reversal symmetry breaking (TRSB) chiral p-wave order. This pairing candidate has attracted a large amount of attention, partly in relation to the possibility of topological quantum computation, and has stimulated studies on higher chirality superconducting systems. In the first part of this thesis, we study the spontaneous edge current in chiral d- and f-wave superconductors. We show that these currents, which vanish in the continuum limit at zero temperature, are generally non-vanishing but tiny, compared to the simplest chiral p-wave case. In the presence of strong surface roughness, the direction of the edge current in the chiral d-wave case can be reversed, compared with that of a specular ideal surface with specular scattering. However, it is shown that this current reversal is non-universal beyond the continuum limit. The chiral p-wave scenario in SRO is overturned by recent Knight shift measurements, highlighting the importance of exploring different pairing symmetries for SRO. Recently, $d_{x^2-y^2} \pm ig_{(x^2-y^2)xy}$, $s' \pm id_{xy}$ and mixed helical p-wave pairings have been proposed as order parameter candidates. However, the stability of these states, especially of the $d_{x^2-y^2} \pm ig$ pairing, remains unclear. In the second part of the thesis, we study the leading superconducting instabilities in SRO in the presence of sizable atomic spin-orbit coupling (SOC), non-local SOC, and non-local interactions. We find that it is difficult to stabilize chiral p-wave pairing in SRO models; this is because, among the triplet p-wave states, the atomic SOC favors helical states over the chiral state. The presence of both d- and g-wave pairings, including a $d_{x^2-y^2} \pm ig$ state, is found when the second nearest neighbor (in-plane) repulsions, together with orbital-anisotropy of the non-local interactions and/or the B2g channel non-local SOC are included. We further analyze the properties, such as nodal structures, in-plane field spin-susceptibility, and spontaneous edge current, of the realized $d_{x^2-y^2} \pm ig$ pairing and find that this state is more compatible with existing experimental measurements than the $s' \pm id_{xy}$ and the mixed helical p-wave proposals. / Dissertation / Doctor of Philosophy (PhD)

京都大学 / 新制・課程博士 / 博士(理学) / 甲第24176号 / 理博第4867号 / 新制||理||1696(附属図書館) / 京都大学大学院理学研究科化学専攻 / (主査)准教授 倉重 佑輝, 教授 谷村 吉隆, 教授 吉村 一良 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

The following treatise is a collection of three experimental reports, detailing measurements made over the last several years on the magnetic properties of specific correlated electron systems. Each of these systems is an unconventional superconductor at low temperatures, but in each the metallic state from which the superconductivity condenses is poorly understood. The experiments presented will focus on temperatures greater than the superconducting transition temperature, and in particular on magnetic properties of the normal state, which are thought to be important. Original work is contained in Chapters 3, 4 and 5. Chapter 3 describes our search for the presence of time-reversal symmetry breaking in the pseudo-gap state of La2-xSrxCuO4 with zero-field μSR, and is largely based on previously published data. Additional data on the related systems La(1.875)Ba(0.125)CuO(4) and HgBa(2)CuO(4+δ) are also presented. Based on this data, we put strict upper limits on any time-reversal symmetry breaking field which can be associated with the pseudo-gap, and show that the current interpretation of recent neutron scattering results in the literature cannot be correct. Chapter 4 summarizes our explorations of overdoped La(2)-(x)Sr(x)CuO(4) in applied magnetic field with transverse-field μSR. We see an unconventional broadening of the local magnetic field distribution in response to applied field, and discuss possible interpretations. This chapter has also been prepared for publication. Chapter 5 describes measurements of the non-linear magnetic susceptibility of URu(2)Si(2) as a function of temperature and hydrostatic pressure. By examining the temperature dependence, we draw conclusions about the existence of the anti-ferromagnetism and 'hidden order' at each pressure, and construct a preliminary pressure-temperature phase diagram. / Thesis / Doctor of Philosophy (PhD)

<p> The effects of hydrostatic pressure on the superconducting transition temperatures of the simple metals aluminum, lead, mercury, tin and indium are investigated within the strong-coupling theory of superconductivity. The experimental variation of the transition temperature with volume change can be understood within a simple scaling model. Strong-coupling effects in the pressure dependence of superconductivity in lead and mercury are investigated.</p> <p> A formalism which includes effects of force constant disorder on superconductivity in a binary alloy of simple metals is established within pseudopotential theory.</p> / Thesis / Doctor of Philosophy (PhD)

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.

BSCCO-2212 is a high-temperature cuprate superconductor whose microscopic behavior is currently poorly understood. In particular, it is unclear whether its order parameter is consistent with s-wave or d-wave symmetry. It has been suggested that its order parameter might take one of several forms that are consistent with d-wave behavior. We present some calculations using the many-body theory approach to superconductivity that suggest that such order parameters would lead to a suppression of the critical temperature in the presence of impurities. Because some experiments have suggested the critical temperature of BSCCO-2212 is relatively independent of the concentration of impurities, this lends support to the hypothesis that its order parameter has s-wave symmetry.

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