NMR spectroscopy of paramagnetic solids provides detailed information about the local configuration and the chemical environment of the NMR observed center, as well as about the structural, magnetic and electronic properties of the coordianted paramagnetic centres. In the case of complex paramagnetic solids such as cathode materials for (rechargeable) batteries, NMR represents an invaluable tool to provide insight into the structural and electronic properties of the systems, which are at the base of the electrochemical performance of these materials. However, the paramagnetism makes the interpretation of the NMR data very challenging. This is primarily due to the interactions of the unpaired electrons with the NMR observed nucleus, and the interpretation of the NMR spectra often requires the aid of reliable theoretical and computational methods. Often the dominant interaction contributing to the measured isotropic shifts is the hyperfine interaction between the unpaired electrons and the observed nucleus, which results from the transfer of unpaired electrons from the paramagnetic centre(s) to the NMR observed site. In systems such as the ones studied here, in which the paramagnetic ions are a major constituent of the lattice, the multitide of different local environments results in a complex distribution of resonances. As in the case of the Li$_x$V$_6$O$_{13}$ cathode material, a methodical investigation of the configurational stability from first principles gives insight into the preferred site configurations. The combination of experimental $^7$Li NMR spectra and hyperfine shift DFT calculations of the so-found stable Li environments allows to unravel the complex lithiation mechanism of this material. In the other case of the LiTi$_x$Mn$_{2-x}$O$_4$ cathode materials, the $^7$Li hyperfine shifts calculated from first principles for a variety of Li environments are combined in a lattice model which allows to assign the isotropic regions of the experimental $^7$Li NMR spectra, helping to resolve the complex cation ordering as a function of Mn/Ti content in the series. For paramagnetic centres with an unquenched orbital component of the electron magnetic moment(s), the spin-orbit coupling effects also contribute to the paramagnetic NMR shift and shift anisotropy. A first principles model is derived, which describes how spin-orbit coupling and the single-ion $g$-tensor are defined and calculated in periodic paramagnetic solids, and how they can be coupled with the hyperfine interaction to model their effects on the NMR spectrum. The method is applied to a series of olivine-type LiTMPO$_4$ cathode materials (with TM = Mn, Fe, Co, and Ni) and the respective $^7$Li and $^{31}$P NMR spectra are simulated and compared with the experiments. The other paramagnetic effect considered in this thesis involves the bulk magnetic susceptibility (BMS), which is particularly important for paramagnetic single crystals and solids of complex shape. The BMS effect results from the discontinuity of the bulk susceptibility at the surface of the crystal, inducing a demagnetizing field throughout the sample which changes the measured NMR shift and shift anisotropy. A method to analytically calculate the demagnetising field and the BMS shift in crystals of different shapes is derived, and it is applied to a series of LiFePO$_4$ single crystals for which the $^7$Li NMR spectra are also measured experimentally. The study confirms that, particularly for $^7$Li NMR, the macroscopic shape-dependent BMS shift can indeed be a significant contribution to the measured resonances, determining the large variation in shift measured for the crystals of different shapes.

Neste trabalho apresentamos a determinação autoconsistente da constante de interação spin-órbita em heteroestruturas com duas sub-bandas. Como recentemente proposto, ao obter o hamiltoneano de um sistema com duas sub-bandas na aproximação de massa efetiva, constata-se a presença de um acoplamento inter-subbanda que não se anula mesmo em heteroestruturas simétricas. Apresentamos aqui as deduções teóricas que levaram à proposição desse novo acoplamento e mostramos o cálculo autoconsistente da intensidade do acoplamento e a comparamos com a intensidade do acoplamento Rashba, já amplamente estudado. Discutimos o método k.p e a Aproximação da Função Envelope e mostramos a obtenção do modelo de Kane 8x8 para semicondutores com estrutura zincblende. Aplicamos o método do \"folding down\'\' ao hamiltoneano de Kane isolando o setor correspondente à banda de condução. Escrevemos dessa forma um hamiltoneano efetivo para a banda de condução no contexto de um poço quântico com uma barreira. Através da projeção desse hamiltoneano nos dois primeiros estados da parte orbital verifica-se o surgimento de um acoplamento inter-subbanda. Finalmente escrevemos o hamiltoneano efetivo 4x4 que descreve as duas primeiras subbandas de um poço quântico e obtivemos seus autoestados e autoenergias. Finalmente fizemos o cálculo autoconsistente das funções de onda e energias de um gás de elétrons em poços quânticos simples e duplos através da aproximação de Hartree e a partir desses resultados determinamos o valor da constante de acoplamento Rashba e da nova constante inter-subbanda. Entre os resultados obtidos destacam-se o controle elétrico da constante de acoplamento inter-subbanda através de um eletrodo externo e um efeito de renormalização da massa efetiva que pode chegar até 5% em algumas estruturas. / In this work we present the self-consistent determination of the spin-orbit coupling constant in heterostructure with two subbands.As recently proposed, the effective hamiltonian for the conduction band in the effective mass approximation contains an inter-subband spin-orbit coupling which is non-zero even for symmetric heterostructures. We present the theoretical derivation which leads to this proposal and show a selfconsistent determination of the coupling constant. We also compare the magnitude of the new coupling constant with the usual Rashba coupling. Starting with a discussion of the k.p method and the Envelope Function Approximation (EFA) we show the derivation of the 8x8 Kane model for semiconductors with zincblende structure. We then apply the \"folding down\'\' method, isolating the conduction band sector of the EFA hamiltonian. By projecting this hamiltonian in the first two states of the orbital part, we find an effective 4x4 hamiltonian that contains an inter-subband spin orbit coupling. The eingenvalues and eigenvectors of this hamiltonian are shown and, specializing the model for single and double quantum wells, we self-consistently determine the inter-subband and Rashba coupling constants in the Hartree approximation. The results indicate the possibility of electrical control of the coupling constant and show an effective mass renormalization effect that can be up to 5% in some cases.

Apresento neste trabalho uma derivação alternativa da hamiltoniana efetiva para um elétron na banda de condução de uma heteroestrutura semicondutora de rede Zincblende. Partindo do modelo de Kane 8 × 8 e da aproximação das funções envelope, esta hamiltoniana efetiva foi obtida com a linearização dos denominadores (dependentes das autoenergias) presentes na equação para a banda de condução, sob a hipótese de que o gap de energia seja muito maior que todas as demais diferenças de energia envolvidas (verdade para a maioria das estruturas Zincblende). A partir de um procedimento introduzido previamente1,3, desenvolvi um procedimento mais geral que implementa sistematicamente esta linearização até segunda ordem no inverso do gap de energia e que corrige a normalização do spinor da banda de condução usando as bandas de valência. Este procedimento é idêntico à expansão em série de potência no inverso da velocidade da luz utilizada para se obter aproximações relativísticas da equação de Dirac. Uma vantagem deste procedimento é a arbitrariedade na forma dos potenciais, o que implica na validade da hamiltoniana resultante para poços, fios e pontos quânticos. Evidencio também as consequências de cada termo desta hamiltoniana efetiva para os autoestados eletrônicos em poços retangulares, incluindo termos independentes de spin inéditos (Darwin e interação momento linearcampo elétrico). Estes resultados estão de acordo com os estudos anteriores4. A fim de estudar transições ópticas dentro da banda de condução, mostro que o acoplamento mínimo pode ser realizado diretamente na hamiltoniana de Kane se os campos externos variam tão lentamente quanto as funções envelope. Repetindo a linearização dos denominadores de energia, derivo uma hamiltoniana efetiva para a banda de condução com acoplamentos elétron-fótons. Um destes acoplamentos, induzido exclusivamente pela banda de valência, origina transições mediadas pelo spin eletrônico. Estas transições assistidas por spin possibilitam mudanças, opticamente induzidas, na orientação do spin eletrônico, um fato que talvez possa ser útil na construção de dispositivos spintrônicos. Por fim, as taxas de transição deste acoplamento apresentam saturação e linhas de máximos e mínimos no espaço recíproco. Espero que estas acoplamentos ópticos possam auxiliar na observação dos efeitos dos acoplamentos spin-órbita intra (Rashba) e intersubbandas. / In this work, I present an alternative derivation of the conduction band effective hamiltonian for Zincblende semiconductor heterostructures. Starting from the 8×8 Kane model and envelope function approximation, this effective hamiltonian was obtained by means of a linearization in the eigenenergy-dependent denominators present the conduction band equation, under the hypothesis that the energy gap is bigger than any other energy difference in the system (true for mostly every Zincblende semiconductor bulk or heterostructure). Based on a previous procedure1,3, I have developed a more general procedure that implements sistematicaly (i) this linearization and (ii) renormalizes the conduction band spinor using the valence bands, both (i) and (ii) up to second order in the inverse of the energy gap. This procedure is identical to the expansion in power series of the inverse of the light speed used to derive non-relativistic approximations of the Dirac equation. One advantage of this procedure is the generality of the potentials adopted in our derivation: the same results hold for quantum wells, wires and dots. I show the consequences of each term of this hamiltonian for the electron eigenstates in retangular wells, including novel spin-independent terms (Darwin and linear momentumelectric field couplings). These resulties agree with previous works4. In order to study conduction band optical transitions, I show that the minimal substitution can be performed directly in the Kane hamiltonian if the external fields vary slowly (at least, as slowly as the envelope functions). Repeating the linearization of the energy denominators, I derive a new effective hamiltonian, up to second order in the inverse of the energy gap, with electron-photons couplings. One of these couplings, induced exclusively by the valence bands, gives rise to optical transitions mediated by the electron spin. This spin-assisted coupling enable optically-induced spin flipps in conduction subband transitions, which can be useful in the construction of spintronic devices. Finaly, the spin-assisted transitions rates show saturation and lines of maxima and minima in the reciprocal lattice. I hope that these optical couplings can be of any help in the observation of interesting effects induced by the intra and intersubband spin-orbit coupling.

Cette thèse est consacrée à l'étude des propriétés de transport statique et dynamique du gaz d'électrons bidimensionnel supraconducteur à l'interface LaAlO3/SrTiO3. Dans un premier temps, nous étudions l'effet du désordre microscopique induit par le dopage en Chrome, sur la supraconductivité et le couplage spin-orbite en fonction de la densité de porteur modulée par effet de champ. Dans une géométrie de grille locale au-dessus du gaz, nous montrons le contrôle électrostatique de la transition supraconducteur-isolant. De même, nous analysons l'ajustement du couplage spin-orbite contrôlé par effet de champ. A l'aide de méthodes de nanofabrication par lithographie électronique, nous démontrons la première réalisation d'un point critique quantique dans LaAlO3/SrTiO3. En changeant le confinement latéral et le niveau de Fermi par effet de champ, nous sommes capables de régler le nombre de canaux conducteurs dans l'état normal et de mesurer la quantification de la conductance. Enfin, nous présentons des mesures radio-fréquence qui donnent accès aux propriétés dynamiques du gaz supraconducteur. L'évolution de la conductivité en fonction de la densité de porteurs et de la température est comparée avec la théorie standard BCS/Mattis-Bardeen d'une part, et avec la théorie BKT d'autre part. / This thesis is devoted to the study of static and dynamical transport properties of the superconducting two-dimensional electron gas at the LaAlO3/SrTiO3 interface. Under strong 2D confinement, the degeneracy of the t$_{2g}$ bands of SrTiO$_3$ is lifted at the interface, generating a rich and complex band structure. Starting from a free electron model, we derive numerically a self-consistent calculation of the potential well and the band structure (chapter 1). These simulations highlight the presence of two types of bands d$_{xy}$ and d$_{xz/yz}$ with very different transport properties. We investigate first the effect of microscopic disorder introduced by Cr doping, on superconductivity and spin-orbit coupling over a wide range of back-gate doping (chapter 3). We also describe the first implementation of a field-effect device where the superconductor-insulator transition could be continuously tuned with a top-gate. The presence of a strong spin-orbit coupling that could be controlled with the top-gate voltage is also demonstrated by analyzing the magneto-transport measurements. The gate dependence of the spin-splitting energy, of the order of a few meV, is found to be consistent with Rashba spin-orbit coupling. Going one step further in nanofabrication, we report on the first realization of a quantum point contact in LaAlO$_3$/SrTiO$_3$ using split gates (chapter 6). To go further in the understanding of the LaAlO$_3$/SrTiO$_3$ interface, we present high frequency measurements of the conductivity $\sigma$ (chapter 5). This measurement gives us access to the superfluid stiffness and to the gap energy via the BCS theory. We show that the competition between these two energy scales controls the superconducting Tc in the phase diagram.

In the recent years, interest in TM oxides with 5d valence electrons has grown immensely due to the realization of novel spin-orbit coupled ground states. In these compounds, e.g., iridates and osmates, the intriguing situation arises where the spin-orbit and electron-electron interactions meet on the same energy scale. This has created a new window of interest in these compounds since the interplay of crystal field effects, local multiplet physics, spin-orbit couplings, and intersite hopping can offer novel types of correlated ground states and excitations. In 5d5 iridates, a spin-orbit entangled j = 1/2 Mott insulating state has been realized recently. A remarkable feature of such a ground state is that it gives rise to anisotropic magnetic interactions. The 2D honeycomb-lattice 213 iridium oxides, A2IrO3 (A=Li,Na), have been put forward to host highly anisotropic bond-dependent spin-spin interactions that resemble the Kitaev spin model, which supports various types of topological phases relevant in quantum computing. The 2D square-lattice 214 iridates Sr2IrO4 and Ba2IrO4 are, on the other hand, appealing because of their perceived structural and magnetic simi- larity to La2CuO4, the mother compound of the cuprate high-Tc superconductors. This has promoted the latter iridium oxide compounds as novel platforms for the search of high-Tc superconductivity. To put such considerations on a firm footing, it is essential to quantify the different coupling strengths and energy scales, as they for instance appear in effective Hamiltonian descriptions of these correlated systems. Moreover, it is important to correctly describe their effects. In this thesis, the electronic structure and magnetic properties of 5d5 (mainly 214 and 213) iridates are studied using wave-function-based quantum chemistry methods. These methods are fully ab initio and are capable of accurately treating the electron-electron interactions without using any ad hoc parameters. The spin-orbit entangled j = 1/2 ground state in 214, 213 and other lower symmetry Sr3CuIrO6 and Na4Ir3O8 iridates is first analyzed in detail, by studying the local electronic structure of the 5d5 Ir4+ ion. We establish that the longer-range crystal anisotropy, i.e., low-symmetry fields related to ionic sites beyond the nearest neighbor oxygen cage, strongly influence the energies of Ir d levels. The ground state in all the compounds studied is j = 1/2 like with admixture from j ≃ 3/2 states ranging from 1 – 15 %. Further, the average j ≃ 1/2 → j ≃ 3/2 excitation energy we find is around 0.6 eV. The NN magnetic exchange interactions we computed for 214 iridates are predominantly isotropic Heisenberg-like with J ~ 60 meV, 3 – 4 times smaller than found in isostructural copper oxides. However, the anisotropic interactions are an order of magnitude larger than those in cuprates. Our estimates are in excellent agreement with those extracted from experiments, e.g., resonant inelastic x-ray scattering measurements. For the 213 honeycomb-lattice Na2IrO3 our calculations show that the relevant spin Hamiltonian contains further anisotropic terms beyond the Kitaev-Heisenberg model. Nevertheless, we predict that the largest energy scale is the Kitaev interaction, 10 to 20 meV, while the Heisenberg superexchange and off-diagonal symmetric anisotropic couplings are significantly weaker. In the sister compound Li2IrO3, we find that the structural inequivalence between the two types of Ir-Ir links has a striking influence on the effective spin Hamiltonian, leading in particular to two very different NN superexchange pathways, one weakly AF (~ 1 meV) and another strongly FM (−19 meV). The latter gives rise to rigid spin-1 triplets on a triangular lattice.

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I derive and apply quantum propagator techniques to atomic and condensed matter systems. I observe many interesting features by following the evolution of a wavepacket. In atomic systems, I revisit the Stern-Gerlach effect and study the spin dynamics inside an inhomogeneous magnetic field. The results I obtained are not exactly the same as the textbook description of the effect which is usually a manifestation of a perfect space and spin entanglement. This discovery can provide insight on more reliable quantum computation device designs. In condensed matter systems, the doping concentration inhomogeneity leads to the Rashba spin-orbit interaction. This makes it possible to control the spin without the external magnetic field. By propagating the wave packet in systems exhibiting Rashba spin-orbit interactions, I discover several features such as spin separation, spin accumulation, persistent spin-helix, and ripple formation.

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

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