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Bodge is a free and open-source Python package for constructing large-scale real-space tight-binding models for calculations in condensed matter physics. "Large-scale" means that it should remain performant even for lattices with millions of atoms, and "real-space" means that the model is formulated in terms of individual lattice sites and not in momentum space, for example. Although general tight-binding models can be constructed with this package, the main focus is on the Bogoliubov-De Gennes ("BoDGe") Hamiltonian used to model superconductivity in the clean limit. The package is designed to be easy to use, flexible, and extensible - and very few lines of code are required to model heterostructures containing, e.g., conventional and unconventional superconductors, ferromagnets and antiferromagnets, altermagnetism, and spin-orbit coupling. In other words: If you want a lattice model for superconducting nanostructures, and want something that is computationally efficient yet easy to use, Bodge should be a good choice., Comment: Software documentation (4 pages + references)

Interacting impurity spins adsorbed on surfaces have been suggested as basic components for applications in quantum computation and spintronics. Such spins usually prefer a parallel or antiparallel configuration, but weakly non-collinear alignments are possible due to the Dzyaloshinskii-Moriya interaction (DMI) that arises in the presence of relativistic spin-orbit coupling. Here, we show that an effective DMI can emerge purely from superconducting correlations without any spin-orbit interaction. We give an analytical proof and provide a numerical study which shows that DMI arises in mixed-parity superconductors solely from the superconducting pairing. These results enable a way to engineer spin textures using the superconducting phase transition., Comment: 4 pages, 3 figures

The Josephson effect is a hallmark signature of the superconducting state, which, however, has been sparsely explored in non-crystalline superconducting materials. Motivated by this, we consider a Josephson junction consisting of two superconductors with a fractal metallic interlayer, which is patterned as a Sierpi\'nski carpet by removing atomic sites in a self-similar and scale-invariant manner. We here show that the fractal geometry has direct observable consequences on the Josephson effect. In particular, we demonstrate that the form of the supercurrent-magnetic field relation as the fractal generation number increases can be directly related to the self-similar fractal geometry of the normal metallic layer. Furthermore, the maxima of the corresponding diffraction pattern directly encode the self-repeating fractal structure in the course of fractal generation, implying that the corresponding magnetic length directly probes the shortest length scale in the given fractal generation. Our results should motivate future experimental efforts to verify these predictions in designer quantum materials and motivate future pursuits regarding fractal-based SQUID devices., Comment: 5 pages, 3 figures. This article may be downloaded for personal use only. Any other use requires prior permission of the authors and AIP Publishing

The Ruderman--Kittel--Kasuya--Yosida (RKKY) interaction governs the coupling between localized spins and is strongly affected by the environment in which these spins reside. In superconductors, this interaction becomes long-ranged and provides information about the orbital symmetry of the superconducting order parameter. In this work, we consider the RKKY interaction between localized spins mediated by $p$-wave triplet superconductors. In contrast to the well-studied RKKY interaction in $d$-wave superconductors, we find that the spin of the Cooper pair in a triplet state also modulates the spin--spin coupling. We consider several different types of $p$-wave triplet states, and find that the form of the RKKY interaction changes significantly with the symmetries of the order parameter. For non-unitary superconducting states, two new terms appear in the RKKY interaction: a background spin magnetization coupling to the individual spins and, more interestingly, an effective Dzyaloshinskii--Moriya term. The latter term oscillates with the separation distance between the impurity spins. Finally, we find that the finite spin expectation value in non-unitary superconductors in concert with the conventional RKKY interaction can lead to non-collinear magnetic ground states even when the Dzyaloshinskii--Moriya term is negligible. The RKKY interaction in $p$-wave triplet superconductors thus offers a way to achieve new ground state spin configurations of impurity spins and simultaneously provides information about the underlying superconducting state., Comment: 21 pages (including 10 figures, 4 appendices, and references)

Adding a Rashba term to the Hubbard Hamiltonian produces a model which can be used to learn how spin-orbit interactions impact correlated electrons on a lattice. Previous works have studied such a model using a variety of theoretical frameworks, mainly close to half-filling. In this work, we determine the magnetic phase-diagram for the Rashba-Hubbard model for arbitrary hole doping using a sine square deformed lattice mean-field model with an unrestricted ansatz, thus suppressing finite size effects and allowing for inhomogeneous order. We find that the introduction of Rashba spin-orbit coupling significantly alters the ground state properties of the Hubbard model and we observe an increasing complexity of the ground state phase composition for increasing spin-orbit strength. We also introduce a gradual deformed envelope (GDE) technique building on the sine square methodology to facilitate convergence towards ordered and defect-free ground state configurations which is a challenge with the unrestricted ansatz at high interaction strengths. We observe that the use of the GDE technique significantly lowers the free energy of the obtained configurations. Moreover, we consider transient dynamics in the Rashba-Hubbard model by quenching the interaction strength. We find that the quench dynamics within a sine-square methodology allows for the simulation of quasi-open systems by using the zero-energy edge states as a particle reservoir. Interaction quenches at half-filling show a tendency towards quench-induced spatial spin-magnitude inhomogeneity and a non-equilibrium system magnetization lower than equilibrium predictions, possibly related to a build-up of non-local correlations on the lattice., Comment: 19 pages, 15 figures

The ability of magnetic materials to modify superconducting systems is an active research area for possible applications in thermoelectricity, quantum sensing, and spintronics. We consider the fundamental properties of the Josephson effect in a third class of magnetic materials beyond ferromagnets and antiferromagnets: altermagnets. We show that despite having no net magnetization, altermagnets induce $0$-$\pi$ oscillations. The decay length and oscillation period of the Josephson coupling are qualitatively different from ferromagnetic junctions and depend on the crystallographic orientation of the altermagnet. The Josephson effect in altermagnets thus serves a dual purpose: it acts as a signature that distinguishes altermagnetism from conventional (anti)ferromagnetism and offers a way to tune the supercurrent via flow direction anisotropy., Comment: 4 pages, 4 figures

Superconductor-ferromagnet tunnel junctions demonstrate giant thermoelectric effects which are being exploited to engineer ultra-sensitive terahertz radiation detectors. Here, we experimentally observe the recently predicted complete magnetic control over thermoelectric effects in a superconducting spin valve, including the dependence of its sign on the magnetic state of the spin valve. The description of the experimental results is improved by the introduction of an interfacial domain wall in the spin filter layer interfacing the superconductor. Surprisingly, the application of high in-plane magnetic fields induces a double sign inversion of the thermoelectric effect, which exhibits large values even at applied fields twice the superconducting critical field.

Giant thermoelectric effects are known to arise at the interface between superconductors and strongly polarized ferromagnets, enabling the construction of efficient thermoelectric generators. We predict that the thermopower of such a generator can be completely controlled by a magnetic input signal: Not only can the thermopower be toggled on and off by rotating a magnet, but it can even be entirely reversed. This in situ control diverges from conventional thermoelectrics, where the thermopower is usually fixed by the device design., Comment: Significantly expanded manuscript: 8 pages, 6 figures, 29 subfigures

Conventional energy production based on fossil fuels causes emissions which contribute to global warming. Accurate energy system models are required for a cost-optimal transition to a zero-emission energy system, an endeavor that requires an accurate modeling of cost reductions due to technological learning effects. In this review, we summarize common methodologies for modeling technological learning and associated cost reductions. The focus is on learning effects in hydrogen production technologies due to their importance in a low-carbon energy system, as well as the application of endogenous learning in energy system models. Finally, we present an overview of the learning rates of relevant low-carbon technologies required to model future energy systems., Comment: review paper: main article (11 pages), appendices (8 pages), references (4 pages)

Transport of liquefied natural gas (LNG) by ship occurs globally on a massive scale. The large temperature difference between LNG and water means LNG will boil violently if spilled onto water. This may cause a physical explosion known as rapid phase transition (RPT). Since RPT results from a complex interplay between physical phenomena on several scales, the risk of its occurrence is difficult to estimate. In this work, we present a combined fluid-dynamic and thermodynamic model to predict the onset of delayed RPT. On the basis of the full coupled model, we derive analytical solutions for the location and time of delayed RPT in an axisymmetric steady-state spill of LNG onto water. These equations are shown to be accurate when compared to simulation results for a range of relevant parameters. The relative discrepancy between the analytic solutions and predictions from the full coupled model is within 2% for the RPT position and within 8% for the time of RPT. This provides a simple procedure to quantify the risk of occurrence for delayed RPT for LNG on water. Due to its modular formulation, the full coupled model can straightforwardly be extended to study RPT in other systems., Comment: 22 pages, 11 figures

Conventional superconductors respond to external magnetic fields by generating diamagnetic screening currents. However, theoretical work has shown that one can engineer systems where the screening current is paramagnetic, causing them to attract magnetic flux -- a prediction that has recently been experimentally verified. In contrast to previous studies, we show that this effect can be realized in simple superconductor/normal-metal structures with no special properties, using only a simple voltage bias to drive the system out of equilibrium. This is of fundamental interest, since it opens up a new avenue of research, and at the same time highlights how one can realize paramagnetic Meissner effects without having odd-frequency states at the Fermi level. Moreover, a voltage-tunable electromagnetic response in such a simple system may be interesting for future device design., Comment: Article: 4 pages, 3 figures; Supplemental: 3 pages, no figures

We propose a mechanism whereby spin supercurrents can be manipulated in superconductor/ferromagnet proximity systems via nonequilibrium spin injection. We find that if a spin supercurrent exists in equilibrium, a nonequilibrium spin accumulation will exert a torque on the spins transported by this current. This interaction causes a new spin supercurrent contribution to manifest out of equilibrium, which is proportional to and polarized perpendicularly to both the injected spins and equilibrium spin current. This is interesting for several reasons: as a fundamental physical effect; due to possible applications as a way to control spin supercurrents; and timeliness in light of recent experiments on spin injection in proximitized superconductors., Comment: Revised version; Manuscript: 5 pages, 3 figures; Supplemental: 7 pages, 0 figures

Exerting control of the magnetic exchange interaction in heterostructures is of both basic interest and has potential for use in spin-based applications relying on quantum effects. We here show that the sign of the exchange interaction in a spin-valve, determining whether the ferro- or antiferromagnetic configuration is favored, can be controlled via an electric voltage. This occurs due to an interplay between a nonequilibrium quasiparticle distribution and the presence of spin-polarized Cooper pairs. Additionally, we show that a voltage-induced distribution controls the anomalous supercurrent that occurs in magnetic Josephson junctions, obviating the challenging task to manipulate the magnetic texture of the system. This demonstrates that two key phenomena in superconducting spintronics, the magnetic exchange interaction and the phase shift generating the anomalous Josephson effect, can be controlled electrically. Our findings are of relevance for spin-based superconducting devices which in practice most likely have to be operated precisely by nonequilibrium effects., Comment: 4 pages, 3 figures, 10 subfigures, revised edition

We predict that superconductivity in thin films can be stabilized in high magnetic fields if the superconductor is driven out of equilibrium by a DC voltage bias. For realistic material parameters and temperatures, we show that superconductivity is restored in fields many times larger than the Chandrasekhar-Clogston limit. After motivating the effect analytically, we perform rigorous numerical calculations to corroborate the findings, and present concrete experimental signatures. On the technical side, we also introduce a new form for the nonequilibrium kinetic equations, which generalizes and simplifies previous formulations of the problem., Comment: Revised version; 4 pages, 4 figures, supplemental information

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 figures

We demonstrate that spin supercurrents are conserved upon transmission through a conventional superconductor, even in the presence of spin-dependent scattering by impurities with magnetic moments or spin-orbit coupling. This is fundamentally different from conventional spin currents, which decay in the presence of such scattering, and this has important implications for the usage of superconducting materials in spintronic hybrid structures., Comment: 6 pages, 4 figures, 13 subfigures

We study non-equilibrium quantum transport of spin, heat, and charge in diffusive heterostructures including both superconductors and materials with spin-dependent fields, such as textured ferromagnets and spin-orbit coupled materials. Using the quasiclassical theory of superconductivity, we derive a general analytical formula for the zero-temperature charge conductance valid in the non-linear regime. The analytical results are in excellent agreement with previous numerical calculations at small but finite temperatures. We predict a strong deviation between the conductance spectra and local density of states in certain junctions, which is important to correctly identify spectroscopic fingerprints of unconventional superconductivity., Comment: Manuscript contains incorrect assumption, analytical method not valid for general spin-textured systems

Spin-triplet Cooper pairs induced in ferromagnets form the centrepiece of the emerging field of superconducting spintronics [1,2]. Usually the focus of research is on the spin polarization of the triplets, potentially enabling low-dissipation magnetization switching and domain wall motion. However, the fundamental mechanism for generating triplet pairs [3,4] also permits control over a parameter which has not been addressed before, namely the spatial distribution of the supercurrent. Here we demonstrate this control by tailoring distinct supercurrent pathways in the ferromagnetic weak link of a Josephson junction. Combining micromagnetic simulations with three-dimensional critical current calculations, based on the Usadel description of mesoscopic superconductivity [5], we designed a disk-shaped structure with a magnetic vortex, which induces two distinct supercurrent channels across the junction. The design was successfully tested with superconducting quantum interferometry (SQI). Moreover, we show how the position of the pathways can be controlled by moving the vortex with a magnetic field. This novel approach allows adaptable supercurrent paths to be dynamically reconfigured to switch between different functionalities in the same device., Comment: 27 pages, 10 figures

Spin filter superconducting S/I/N tunnel junctions (NbN/GdN/TiN) show a robust and pronounced zero bias conductance peak at low temperatures, the magnitude of which is several times the normal state conductance of the junction. Such a conductance anomaly is representative of unconventional superconductivity and is interpreted as a direct signature of an odd frequency superconducting order., Comment: 5 pages, 3 figures + supplementary information

With a combination of simple analytical arguments and extensive numerical simulations, we theoretically propose a Josephson junction with $n+1$ superconductors where the current-phase relation can be toggled in situ between a $\sin(\delta\varphi)$ and $\sin(\delta\varphi/n)$ shape using an applied magnetic field. Focusing in particular on the case $n=2$, we show that by using realistic system parameters such as unequal interface transparencies, the $\sin(\delta\varphi/2)$-shaped solution retains its $2\pi$-periodicity due to discontinuities at $\delta\varphi = \pm \pi$. Moreover, we demonstrate that as one toggles between the $\sin(\delta\varphi)$- and $\sin(\delta\varphi/2)$-shaped solutions, the system acts as an on--off switch, and can acheive more than two orders of magnitude difference between the supercurrent in the on and off states. Finally, we argue that the same approach can be generalized to switchable $\sin(\delta\varphi/n)$ junctions for arbitrary integers~$n$, which we motivate by analytically solving the Josephson equations for double- and triple-barrier junctions., Comment: 7 pages, 7 figures, revised version

Multiterminal Josephson junctions have recently been proposed as a route to artificially mimic topological matter with the distinct advantage that its properties can be controlled via the superconducting phase difference, giving rise to Weyl points in 4-terminal geometries. A key goal is to accurately determine when the system makes a transition from a gapped to non-gapped state as a function of the phase differences in the system, the latter effectively playing the role of quasiparticle momenta in conventional topological matter. We here determine the proximity gap phase diagram of diffusive n-terminal Josephson junctions, both numerically and analytically, by identifying a class of solutions to the Usadel equation at zero energy in the full proximity effect regime. We present an analytical equation which provides the phase diagram for an arbitrary number of terminals n. After briefly demonstrating the validity of the analytical approach in the previously studied 2- and 3-terminal cases, we focus on the 4-terminal case and map out the regimes where the electronic excitations in the system are gapped and non-gapped, respectively, demonstrating also in this case full agreement between the analytical and numerical approach., Comment: 10 pages, 6 figures

Interfacing superconductors with strongly spin-polarized magnetic materials opens the possibility to discover new spintronic devices in which spin-triplet Cooper pairs play a key role. Motivated by the recent derivation of spin-polarized quasiclassical boundary conditions capable of describing such a scenario in the diffusive limit, we consider the emergent physics in hybrid structures comprised of a conventional s-wave superconductor (e.g. Nb, Al) and either strongly spin-polarized ferromagnetic insulators (e.g. EuO, GdN) or halfmetallic ferromagnets (e.g. CrO2, LCMO). In contrast to most previous works, we focus on how the superconductor itself is influenced by the proximity effect, and how the generated triplet Cooper pairs manifest themselves in the self-consistently computed density of states (DOS) and the superconducting critical temperature Tc. We provide a comprehensive treatment of how the superconductor and its properties are affected by the triplet pairs, demonstrating that our theory can reproduce the recent observation of an unusually large zero-energy peak in a superconductor interfaced with a half-metal, which even exceeds the normal-state DOS. We also discuss the recent observation of a large superconducting spin-valve effect with a Tc change ~1K in superconductor/half-metal structures, in which case our results indicate that the experiment cannot be explained fully by a long-ranged triplet proximity effect., Comment: 16 pages, 11 figures, published version

We demonstrate theoretically that shining light on superconducting proximity structures controls the minigap and triplet proximity effect in normal and magnetic metals, respectively. Considering both a bilayer and Josephson junction geometry, we show that microwave irradiation with frequency $\omega$ qualitatively alters the spectral properties of the system, controlling the minigap size $E_\text{mg}$ and even replacing the minigap with a strong peak of quasiparticle accumulation at zero energy when $\omega=E_\text{mg}$. The interaction between light and Cooper pairs may thus open a route to active control of quantum coherent phenomena in superconducting proximity structures., Comment: 9 pages, 4 figures

We demonstrate theoretically all-electric control of the superconducting transition temperature using a device comprised of a conventional superconductor, a ferromagnetic insulator, and semiconducting layers with intrinsic spin-orbit coupling. By using analytical calculations and numerical simulations, we show that the transition temperature of such a device can be controlled by electric gating which alters the ratio of Rashba to Dresselhaus spin-orbit coupling. The results offer a new pathway to control superconductivity in spintronic devices., Comment: 9 pages, 5 figures; added a new Methods section, and added references

We investigate theoretically how the proximity effect in superconductor/ferromagnet hybrid structures with intrinsic spin-orbit coupling manifests in the density of states and critical temperature. To describe a general scenario, we allow for both Rashba and Dresselhaus type spin-orbit coupling. Our results are obtained via the quasiclassical theory of superconductivity, extended to include spin-orbit coupling in the Usadel equation and Kupriyanov--Lukichev boundary conditions. Unlike previous works, we have derived a Riccati parametrization of the Usadel equation with spin-orbit coupling which allows us to address the full proximity regime. First, we consider the density of states in both SF bilayers and SFS trilayers, where the spectroscopic features in the latter case are sensitive to the phase difference between the two superconductors. We find that the presence of spin-orbit coupling leaves clear spectroscopic fingerprints in the density of states due to its role in creating spin-triplet Cooper pairs. Unlike SF and SFS structures without spin-orbit coupling, the density of states in the present case depends strongly on the direction of magnetization. We show that the spin-orbit coupling can stabilize singlet superconductivity even in the presence of a strong exchange field $h \gg \Delta$. This leads to the possibility of a magnetically tunable minigap: changing the direction of the exchange field opens and closes the minigap. We also determine how the critical temperature $T_c$ of an SF bilayer is affected by spin-orbit coupling and demonstrate that one can achieve a spin-valve effect with a single ferromagnet. We find that $T_c$ displays highly non-monotonic behavior both as a function of the magnetization direction and the type and direction of the spin-orbit coupling, offering a new way to exert control over the superconductivity of proximity structures., Comment: 25 pages, 21 figures. Accepted for publication in Phys. Rev. B