General solution of 2D and 3D superconducting quasiclassical systems: coalescing vortices and nanoisland geometries

An extension of quasiclassical Keldysh-Usadel theory to higher spatial dimensions than one is crucial in order to describe physical phenomena like charge/spin Hall effects and topological excitations like vortices and skyrmions, none of which are captured in one-dimensional models. We here present a numerical finite element method which solves the non-linearized 2D and 3D quasiclassical Usadel equation relevant for the diffusive regime. We show the application of this on three model systems with non-trivial geometries: (i) a bottlenecked Josephson junction with external flux, (ii) a nanodisk ferromagnet deposited on top of a superconductor and (iii) superconducting islands in contact with a ferromagnet. In case (i), we demonstrate that one may control externally not only the geometrical array in which superconducting vortices arrange themselves, but also to cause coalescence and tune the number of vortices. In case (iii), we show that the supercurrent path can be tailored by incorporating magnetic elements in planar Josephson junctions which also lead to a strong modulation of the density of states. The finite element method presented herein paves the way for gaining insight in physical phenomena which have remained largely unexplored due to the complexity of solving the full quasiclassical equations in higher dimensions.Comment: 16 pages, 8 figures. Added several new result

Field-free nucleation of antivortices and giant vortices in non-superconducting materials

Giant vortices with higher phase-winding than $2\pi$ are usually energetically unfavorable, but geometric symmetry constraints on a superconductor in a magnetic field are known to stabilize such objects. Here, we show via microscopic calculations that giant vortices can appear in intrinsically non-superconducting materials, even without any applied magnetic field. The enabling mechanism is the proximity effect to a host superconductor where a current flows, and we also demonstrate that antivortices can appear in this setup. Our results open the possibility to study electrically controllable topological defects in unusual environments, which do not have to be exposed to magnetic fields or intrinsically superconducting, but instead display other types of order.Comment: Revised version; 4 pages manuscript, 4 pages supplemental, 6 figure

Spin accumulation induced by a singlet supercurrent

We show that a supercurrent carried by spinless singlet Cooper pairs can induce a spin accumulation in the normal metal interlayer of a Josephson junction. This phenomenon occurs when a nonequilibrium spin-energy mode is excited in the normal metal, for instance by an applied temperature gradient between ferromagnetic electrodes. Without supercurrent, the spin accumulation vanishes in the Josephson junction. With supercurrent, a spatially antisymmetric spin accumulation is generated that can be measured by tunneling to a polarized detector electrode. We explain the physical origin of the induced spin accumulation by the combined effect of a Doppler shift induced by a flow of singlet Cooper pairs, and the spin-energy mode excited in the normal metal. This effect shows that spin control is possible even with singlet Cooper pairs in conventional superconductors, a finding which could open new perspectives in superconducting spintronics.Comment: 4 pages, 4 figure

Quasiclassical boundary conditions for spin-orbit coupled interfaces with spin-charge conversion

The quasiclassical theory of superconductivity provides a methodology to study emergent phenomena in hybrid structures comprised of superconductors interfaced with other materials. A key component in this theory is the boundary condition that the Green functions describing the materials must satisfy. Recently, progress has been made toward formulating such a boundary condition for interfaces with spin-orbit coupling, the latter playing an important role for several phenomena in spintronics. Here, we derive a boundary condition for spin-orbit coupled interfaces that includes gradient terms which enables the description of spin-Hall like effects with superconductors due to such interfaces. As an example, we show that the boundary conditions predict that a supercurrent flowing through a superconductor that is coupled to a normal metal via a spin-orbit interface can induce a non-local magnetization in the normal metal.Comment: 9 pages, 2 figures. Corrected here a few minor typos present in published versio

Grain-boundary topological superconductor

We show that a grain boundary (GB) defect, constituted by a one-dimensional dislocation superlattice, can host a topological superconductor (SC) with a pair of co-habitating Majorana zero modes (MZMs) at its end when immersed in a parent two-dimensional topological SC. For the realization of our proposal, it is essential that the single-dislocation defects feature MZMs, implying that the parent topological SC should be gapped with the Fermi surface away from the $\Gamma-$point. As we demonstrate, both numerically and analytically, the GB topological SC with two localized MZMs then emerge in a finite range of both the angle and the magnitude of the applied exchange magnetic field. Finally, we argue that the proposed GB SC can be realized in various superconducting materials where the GB defects can be readily manipulated.Comment: 6 pages, 4 figure

Superconducting phase diagram and spin diode effect via spin accumulation

Spin-split superconductors offer new functionality compared to conventional superconductors such as diode-effects and efficient thermoelectricity. The superconducting state can nevertheless only withstand a small amount of spin-splitting $m$. Here, we self-consistently determine the spin transport properties and the phase diagram of a spin-split superconductor in the presence of an injected spin accumulation. Energy and spin relaxation are accounted for in the relaxation time approximation via a single effective inelastic scattering parameter. We find that the spin-splitting field in the superconductor enables a spin diode effect in the presence of such spin relaxation. Moreover, we consider the superconducting phase diagram of such a system when it is in contact with a spin accumulation, and find that the inclusion of energy and spin relaxation alters the phase diagram qualitatively. In particular, these mechanisms turn out to induce a superconducting state in large parts of the phase diagram where a normal state would otherwise be the ground-state. We identify an FFLO-like state even in the presence of impurity scattering which can be controllably tuned on and off via the electrically induced spin accumulation. We explain the underlying physics from how the superconducting order parameter depends on the non-equilibrium modes in the system as well as the behavior of these modes in the presence of energy and spin relaxation when a spin-splitting field is present.Comment: 8 pages, 5 figure

Improving the Correctness of Automated Program Repair

Developers spend much of their time fixing bugs in software programs. Automated program repair (APR) techniques aim to alleviate the burden of bug fixing from developers by generating patches at the source-code level. Recently, Generate-and-Validate (G&V) APR techniques show great potential to repair general bugs in real-world applications. Recent evaluations show that G&V techniques repair 8–17.7% of the collected bugs from mature Java or C open-source projects. Despite the promising results, G&V techniques may generate many incorrect patches and are not able to repair every single bug. This thesis makes contributions to improve the correctness of APR by improving the quality assurance of the automatically-generated patches and generating more correct patches by leveraging human knowledge. First, this thesis investigates whether improving the test-suite-based validation can precisely identify incorrect patches that are generated by G&V, and whether it can help G&V generate more correct patches. The result of this investigation, Opad, which combines new fuzz-generated test cases and additional oracles (i.e., memory oracles), is proposed to identify incorrect patches and help G&V repair more bugs correctly. The evaluation of Opad shows that the improved test-suite-based validation identifies 75.2% incorrect patches from G&V techniques. With the integration of Opad, SPR, one of the most promising G&V techniques, repairs one additional bug. Second, this thesis proposes novel APR techniques to repair more bugs correctly, by leveraging human knowledge. Thus, APR techniques can repair new types of bugs that are not currently targeted by G&V APR techniques. Human knowledge in bug-fixing activities is noted in the forms such as commits of bug fixes, developers’ expertise, and documentation pages. Two techniques (APARE and Priv) are proposed to target two types of defects respectively: project-specific recurring bugs and vulnerability warnings by static analysis. APARE automatically learns fix patterns from historical bug fixes (i.e., originally crafted by developers), utilizes spectrum-based fault-localization technique to identify highly-likely faulty methods, and applies the learned fix patterns to generate patches for developers to review. The key innovation of APARE is to utilize a percentage semantic-aware matching algorithm between fix patterns and faulty locations. For the 20 recurring bugs, APARE generates 34 method fixes, 24 of which (70.6%) are correct; 83.3% (20 out of 24) are identical to the fixes generated by developers. In addition, APARE complements current repair systems by generating 20 high-quality method fixes that RSRepair and PAR cannot generate. Priv is a multi-stage remediation system specifically designed for static-analysis security-testing (SAST) techniques. The prototype is built and evaluated on a commercial SAST product. The first stage of Priv is to prioritize workloads of fixing vulnerability warnings based on shared fix locations. The likely fix locations are suggested based on a set of rules. The rules are concluded and developed through the collaboration with two security experts. The second stage of Priv provides additional essential information for improving the efficiency of diagnosis and fixing. Priv offers two types of additional information: identifying true database/attribute-related warnings, and providing customized fix suggestions per warning. The evaluation shows that Priv suggests identical fix locations to the ones suggested by developers for 50–100% of the evaluated vulnerability findings. Priv identifies up to 2170 actionable vulnerability findings for the evaluated six projects. The manual examination confirms that Priv can generate patches of high-quality for many of the evaluated vulnerability warnings

Inverse proximity effect in ss-wave and dd-wave superconductors coupled to topological insulators

We study the inverse proximity effect in a bilayer consisting of a thin $s$- or $d$-wave superconductor (S) and a topological insulator (TI). Integrating out the topological fermions of the TI, we find that spin-orbit coupling is induced in the S, which leads to spin-triplet $p$-wave ($f$-wave) correlations in the anomalous Green's function for an $s$-wave ($d$-wave) superconductor. Solving the self-consistency equation for the superconducting order parameter, we find that the inverse proximity effect can be strong for parameters for which the Fermi momenta of the S and TI coincide. The suppression of the gap is approximately proportional to $e^{-1/\lambda}$, where $\lambda$ is the dimensionless superconducting coupling constant. This is consistent with the fact that a higher $\lambda$ gives a more robust superconducting state. For an $s$-wave S, the interval of TI chemical potentials for which the suppression of the gap is strong is centered at $\mu_{TI} = \pm\sqrt{2mv_F^2\mu}$, and increases quadratically with the hopping parameter $t$. Since the S chemical potential $\mu$ typically is high for conventional superconductors, the inverse proximity effect is negligible except for $t$ above a critical value. For sufficiently low $t$, however, the inverse proximity effect is negligible, in agreement with what has thus far been assumed in most works studying the proximity effect in S-TI structures. In superconductors with low Fermi energies, such as high-$T_c$ cuprates with $d$-wave symmetry, we again find a suppression of the order parameter. However, since $\mu$ is much smaller in this case, a strong inverse proximity effect can occur at $\mu_{TI}=0$ for much lower values of $t$. Moreover, the onset of a strong inverse proximity effect is preceded by an increase in the order parameter, allowing the gap to be tuned by several orders of magnitude by small variations in $\mu_{TI}$.Comment: 11 pages, 4 figures, updated versio

High magnetic field superconductivity in a two-band superconductor

When applying an external magnetic field to a superconductor, orbital and Pauli paramagnetic pairbreaking effects govern the limit of the upper critical magnetic field that can be supported before superconductivity breaks down. Experimental studies have shown that many multiband superconductors exhibit values of the upper critical magnetic field that violate the theoretically predicted limit, giving rise to many studies treating the underlying mechanisms that allow this. In this work we consider spin-splitting induced by an external magnetic field in a superconductor with two relevant bands close to the Fermi level, and show that the presence of interband superconducting pairing produces high-field reentrant superconductivity violating the Pauli-Chandrasekhar-Clogston limit for the value of the upper critical magnetic field