Bogoliubov-de Gennes Equations Research Articles

Impact of Correlated Disorder on Surface Superconductivity: Revealing the Robustness of the Surface Ordering Effect.

Surface superconductivity, wherein electron pairing occurs at material surfaces or interfaces, has attracted a remarkable amount of attention since its discovery. Recent theoretical predictions have unveiled increased critical temperatures, especially at the surfaces of certain compounds and/or structures. The notion of "surface ordering" has been advanced to elucidate this phenomenon. Employing the framework of self-consistent Bogoliubov-de Gennes equations and a model incorporating correlated disorder, our study demonstrates the persistence of the surface ordering effect in the presence of weak to moderate bulk disorder. Intriguingly, our findings indicate that under moderate disorder conditions the surface critical temperature can be further increased, depending on the intensity and correlation of the disorder.

Fully Microscopic Treatment of Magnetic Field Using Bogoliubov–De Gennes Approach

This work introduces an algorithm designed to solve the Bogoliubov–de Gennes equations of superconductivity theory. What sets this algorithm apart is its remarkable ability to precisely and consistently consider the impact of an external magnetic field, all within the microscopic approach. The computation scheme’s convergence is guaranteed by addressing the Biot–Savart equation for the field where the vector potential appears on both of its sides. To showcase the capabilities of this approach, we provide several key examples: the Abrikosov lattice, vortex core states, and the vortex structure in the intermediate mixed state of a superconductor. This method promises to offer valuable insights into the microscopic physics of intertype superconductivity.

Superconductivity in correlated carbon nanotubes under pressure: A Bogoliubov-de Gennes study

In contrast to most microscopic theories of superconductivity based on the reciprocal space, the Bogoliubov-de Gennes (BdG) formalism provides a real-space alternative for addressing inhomogeneous systems. In this article, we study the superconducting states in correlated single-walled carbon nanotubes (SWNTs) with curvature and spin-orbit corrections, as well as the inter-tube interaction through a connecting molecule using an attractive Hubbard model. The results reveal a close relationship between the on-site superconducting gap and the single-electron local density of states. For the limiting case of independent large-diameter nanotubes, the BdG equations can be reduced to the standard Bardeen-Cooper-Schrieffer one with analytical solutions. Moreover, an optimal separation between nanotubes is found, which leads to a maximal superconducting critical temperature. This finding has a remarkable accordance with the experimental data obtained from Buckypapers built of boron doped SWNTs under external pressure.

Surface superconductor-insulator transition: Reduction of the critical electric field by Hartree-Fock potential

Recently, a surface superconductor-insulator transition has been predicted for a bulk superconductor in an electric field applied perpendicular to its surface. The related calculations were performed within a one-dimensional Hubbard model by numerically solving the Bogoliubov-de Gennes (BdG) equations without the Hartree-Fock (HF) interaction potential. The phase diagram of the surface superconducting, metallic, and insulating states was obtained as dependent on the electric field and temperature. This diagram was found to be in agreement with experimental results reported previously for (Li,Fe)OHFeSe thin flakes. In the present work, by taking into account the HF potential, we find that the latter acts as a kind of an extra electrostatic potential that enhances the electric-field effects on the surface states. The qualitative features of the phase diagram remain the same but the surface superconductor-insulator transition occurs at significantly lower electric fields, which supports prospects of its experimental observation in bulk samples.

Bogoliubov-de Gennes equation on graphs: A model for tree-branched Majorana wire network

We consider Bogoliubov-de Gennes equation on a metric tree graph. Formulation of the problem for arbitrary graph topology is provided. Self-adjoint vertex boundary conditions are derived. Exact solutions of the problem is obtained for quantum tree graph. A quantum graph based model for tree-branched Majorana wire network is proposed.

Superconductivity of metallic graphene

It has been found that the states at the level of the Fermi energy of undoped graphene sheet reduces to zero, making it difficult for the Cooper pairing of such a system to exist in its superconductive state. Instead, T-graphene, having physical properties close to graphene, exhibits a metallic behavior which means the existence of the available states for electrons near to Fermi level. For this two-dimensional metallic system, the gap equation of the s-wave superconductive state is derived based on the Bogoliubov de Gennes equation within the attractive Hubbard model. Using the gap equation, a nonzero critical temperature, Tc, is found for different amounts of electron–electron interaction, U. Tc has higher values when the system possesses electronic half band-filling, whereas the reverse is true when the system does not have half band-filling. However, close to band edges, Tc vanishes when U becomes small enough.

Tunability of Andreev levels in a spin-active Ising superconductor/half-metal Josephson junction

We study the Andreev levels, supercurrent and tunnelling conductance in a clean Ising superconductor (ISC)/half-metal (HM)/ISC Josephson junction (JJ) with spin-active interfaces using Bogoliubov–de Gennes equations. We theoretically demonstrate the effect of spin mixing, spin flipping processes and spin–orbit coupling (SOC) of the ISC on Andreev bound states spectra, the current phase relation and tunnelling conductance in a transparent and opaque barrier limit. We witness an additional splitting of the Andreev levels due to the SOC of the ISC and transition for different barrier magnetic moments. Also, different φ-junctions can be achieved by tuning the strength of the barrier magnetic moment and spin mismatch angle. Moreover, a possible transition can also be achieved for a SOC stronger than the chemical potential of the ISC using suitable control parameters. The tunnelling conductance is found to be dependent on the spin mismatch angle. A finite subgap conductance is also observed, indicating different probabilities of Andreev reflected electrons and holes in the presence of SOC. Furthermore, anomalous Andreev levels are observed for different HM length scales, signifying their role in the tunability of the φ-phase JJ.

Light-modulated graphene-based φ 0 Josephson junction and −φ 0 to φ 0 transition

We investigate the chiral edge states-induced Josephson current–phase relation in a graphene-based Josephson junction modulated by the off-resonant circularly polarized light and the staggered sublattice potential. By solving the Bogoliubov–de Gennes equation, a φ 0 Josephson junction is induced in the coaction of the off-resonant circularly polarized light and the staggered sublattice potential, which arises from the fact that the center of-mass wave vector of Cooper pair becomes finite and the opposite center of-mass wave vector to compensate is lacking in the nonsuperconducting region. Interestingly, when the direction of polarization of light is changed, −φ 0 to φ 0 transition generates, which generalizes the concept of traditional 0–π transition. Our findings provide a purely optical way to manipulate a phase-controllable Josephson device and guidelines for future experiments to confirm the presence of graphene-based φ 0 Josephson junction.

Formation, propagation, and excitation of matter solitons and rogue waves in chiral BECs with a current nonlinearity trapped in external potentials.

In this paper, we investigate formation and propagation of matter solitons and rogue waves (RWs) in chiral Bose-Einstein condensates modulated by different external potentials, modeled by the chiral Gross-Pitaevskii (GP) equation with the current nonlinearity and external potentials. On the one hand, the introduction of two potentials (Pöschl-Teller and harmonic-Gaussian potentials) enables the discovery of exact soliton solutions in both focusing and defocusing cases. We analyze the interplay effects of current nonlinearity and potential on soliton stability via associated Bogoliubov-de Gennes equations. Moreover, multiple families of numerical solitons (ground-state and dipole modes) trapped in potentials are found, exhibiting distinctive structures. The interactions between solitons trapped in potentials are studied, which exhibit the inelastic trajectories and repulsive interactions. On the other hand, we introduce the time-dependent potentials such that the controlled RWs are found in both focusing and defocusing GP equations with current nonlinearity. Furthermore, through the interaction between potentials and current nonlinearity, it is possible to enlarge the region of modulational instability, leading to the generation of RWs and chiral solitons. High-order RWs are generated from several Gaussian perturbations on a continuous wave. The presence of current nonlinearity disrupts the structures of these high-order RWs, causing them to undergo a transform into chiral lower-amplitude solitons. Finally, various types of soliton excitations are investigated by varying the strengths of potential and current nonlinearity, showing the abundant dynamic transforms of chrital matter solitons.

A finite element toolbox for the Bogoliubov-de Gennes stability analysis of Bose-Einstein condensates

We present a finite element toolbox for the computation of Bogoliubov-de Gennes modes used to assess the linear stability of stationary solutions of the Gross-Pitaevskii (GP) equation. Applications concern one (single GP equation) or two-component (a system of coupled GP equations) Bose-Einstein condensates in one, two and three dimensions of space. An implementation using the free software ▪is distributed with this paper. For the computation of the GP stationary (complex or real) solutions we use a Newton algorithm coupled with a continuation method exploring the parameter space (the chemical potential or the interaction constant). Bogoliubov-de Gennes equations are then solved using dedicated libraries for the associated eigenvalue problem. Mesh adaptivity is proved to considerably reduce the computational time for cases implying complex vortex states. Programs are validated through comparisons with known theoretical results for simple cases and numerical results reported in the literature. Program summaryProgram Title: FFEM_BdG_toolbox.zipCPC Library link to program files:https://doi.org/10.17632/dgypyc34gb.1Licensing provisions: Apache 2.0Programming language:▪(v 4.12) free software (www.freefem.org)Nature of problem: The software computes Bogoliubov-de Gennes (BdG) complex modes of Bose-Einstein condensates described by the Gross-Pitaevskii (GP) equation. BdG equations are obtained by linearizing the GP equation (or the system of coupled GP equations) around a stationary solution. Obtained BdG modes are used to assess on the stability of stationary states.Solution method: Stationary states of the GP equation are obtained by a Newton algorithm. Parameter space is explored using a continuation on the chemical potential. Once the stationary (complex or real) state is captured accurately, BdG modes are computed by solving the associated eigenvalue problem with the ARPACK library. Complex eigenvalues and eigenvectors are computed and stored. The wave function is discretized by P2 (piece-wise quadratic) Galerkin triangular (in 2D) or tetrahedral (in 3D) finite elements. Mesh adaptation is implemented to reduce the computational time. Examples are given for stationary states in one- and two-component Bose-Einstein condensates.

Nambu-goldstone Modes in Two Segregated Bose-einstein Condensates Limited by Two Hard Walls

The Nambu-Goldstone (NG) modes in the system of two segregated Bose-Einstein condensates (BECs) limited by two hard walls are studied by means of the Gross-Pitaevskii (GP) theory. Based on the double-parabola approximation (DPA) combining with the Bogoliubov-de Gennes (BdG) equations we found four NG modes that proves the failure of the Watanabe-Brauner counting rule and, furthermore, their dispersion relations depend explicitly on the geometrical structure.
 
 

Engineering low-temperature proximity effect in clean metals by spectral singularities

The present study investigates the behavior of the Cooper pair wave function in a normal metal (NM) near superconductor-NM-junctions, specifically focusing on the ballistic regime at zero temperature. It is widely assumed that the wave function follows a power-law decay, with the decay exponents dependent on the system’s dimensionality. Our work reveals that the multiband nature of a compound significantly influences the damping degree of pair amplitudes in an NM, rendering it sensitive to the position of the Fermi level. To explore this phenomenon, we employ the numerical method of self-consistent Bogoliubov–de Gennes equations, utilizing a nanowire as a model for an electronic multiband system. By analyzing the obtained pair amplitudes, we extract relevant lengths and exponents that characterize the leakage of superconducting correlations. We further examine this phenomenon by varying the sample’s cross-sectional size and the superconducting coupling constant. Consequently, our findings demonstrate that the properties of a superconducting/NM junction’s proximity effect can be manipulated not only through temperature, total impurity and defect density, but also by controlling the position of the Fermi level. This tunability enables the transition from a long-range regime to a short-range one, providing valuable insights for designing and understanding such junctions in practical applications.

Quantum Squeezing of Matter-Wave Solitons in Bose-Einstein Condensates

We investigate the quantum squeezing of matter-wave solitons in atomic Bose–Einstein condensates. By calculating quantum fluctuations of the solitons via solving the Bogoliubov–de Gennes equations, we show that significant quantum squeezing can be realized for both bright and dark solitons. We also show that the squeezing efficiency of the solitons can be enhanced and manipulated by atom–atom interaction and soliton blackness. The results reported here are beneficial not only for understanding quantum property of matter-wave solitons, but also for promising applications of Bose-condensed quantum gases.

Aplicación de las teorías de Klein-Gordon y de Bogoliubov-deGennes a Niquelatos

In the present work we show the generalities of the classical field theory (CFT), we study its extension to the quantum field theory (QFT), where as an example of numerical analysis and combination with the field theory technique, we solve a system Klein-Gordon type (KGS) in two space-time dimensions (1+1) studying its stability through the spectral parameter λ(k), principle of convergence due to the parameters of the numerical network and the solution for the field ф (x;t), obtaining novel results. Also, we briefly study the technique of creation and destruction ladder operators from the perspective of the quantum harmonic oscillator, to define some properties and extensions to the problem in canonical quantization. Finally, we apply the topics studied to a problem of unconventional superconductivity in Nickelates compounds by solving the system of Bogoliubov-deGennes (BdG) Equations in the mean expansion of the field, obtaining the superconducting energy band.

Subgap states and quantum phase transitions in one-dimensional superconductor–ferromagnetic insulator heterostructures

We theoretically study the spectral properties of a one dimensional semiconductor-superconductor-ferromagnetic insulator (SE-SU-FMI) hybrid nanostructure, motivated by recent experiments where such devices have been fabricated using epitaxial growing techniques. We model the hybrid structure as a one-dimensional single-channel semiconductor nanowire under the simultaneous effect of two proximity-induced interactions: superconducting pairing and a (spatially inhomogeneous) Zeeman exchange field. The coexistence of these competing mechanisms generates a rich quantum phase diagram and a complex subgap Andreev bound state (ABS) spectrum. By exploiting the symmetries of the problem, we classify the solutions of the Bogoliubov-de Gennes equations into even and odd ABS with respect to the spatial inversion symmetry $x\ensuremath{\rightarrow}\ensuremath{-}x$. We find the ABS spectrum of the device as a function of the different parameters of the model: the length $L$ of the coexisting SU-FMI region, the induced Zeeman exchange field ${h}_{0}$, and the induced superconducting coherence length $\ensuremath{\xi}$. In particular we analyze the evolution of the subgap spectrum as a function of the length $L$. Interestingly, we generically find spin-polarized ABS emerging in the subgap region, which, depending on the ratio ${h}_{0}/\mathrm{\ensuremath{\Delta}}$, can eventually cross below the Fermi energy at certain critical values ${L}_{c}$, and induce spin- and fermion parity-changing quantum phase transitions. We argue that this type of device constitute a promising highly-tunable platform to engineer subgap ABS.

Anisotropic vortex core in the nematic state in electron-doped iron-pnictide superconductors

We investigated the vortex states in the electron-doped BaFe1.8Co0.2As2 using Bogoliubov-de Gennes equations, based on a phenomenological model with competing nematic order and extended s-wave superconductivity. Specifically, we considered the spin-driven nematic order for the optimally doped compound. Our findings suggest that there are no Andreev bound states at the Fermi energy in the nematic phase, and the local density of states exhibits a double-peak structure, consistent with scanning tunneling microscopy experiments. Moreover, we propose an elliptic vortex structure in the local density of states map, which we attribute to the presence of the nematic order.

Chiral Josephson effect in double layers: The role of particle–hole duality

AbstractThe Josephson effect of inter‐layer s‐wave pairing in a double layer of two chiral metals with Josephson junctions along the direction is considered. Starting from the Bogoliubov de Gennes equations of electron–electron and electron–hole double layers, we employ the duality transformation between the two systems to determine the relation of the zero‐energy quasiparticle modes at a Josephson junction in both systems. The appearance of an exceptional point at zero energy is observed, where the four‐dimensional eigenspace coalesces to a two‐dimensional eigenspace. In the second part of the article, the coupling of the quasiparticle currents and the supercurrent is studied. The coupling is based on the quasiparticle charge conservation in the form of a continuity equation. Although the quasiparticle modes differ between the electron–electron and the electron–hole double layers, their corresponding currents are the same.

Anomalous Josephson effect between d-wave superconductors through a two-dimensional electron gas with both Rashba spin–orbit coupling and Zeeman splitting

Based on the Bogoliubov–de Gennes equation and the extended McMillan’s Green’s function formalism, we study theoretically the Josephson effect between two d-wave superconductors bridged by a ballistic two-dimensional electron gas with both Rashba spin–orbit coupling and Zeeman splitting. We show that due to the interplay of Rashba spin–orbit coupling and Zeeman splitting and d-wave pairing, the current–phase relation in such a heterostructure may exhibit a series of novel features and can change significantly as some relevant parameters are tuned. In particular, anomalous Josephson current may occur at zero phase bias under various different situations if both time reversal symmetry and inversion symmetry of the system are simultaneously broken, which can be realized by tuning some relevant parameters of the system, including the relative orientations and the strengths of the Zeeman field and the spin–orbit field in the bridge region, the relative orientations of the a axes in two superconductor leads, or the relative orientations between the Zeeman field in the bridge region and the a axes in the superconductor leads. We show that both the magnitude and the direction of the anomalous Josephson current may depend sensitively on these relevant parameters.

Effects of strong spin-orbit coupling on Shiba states from magnetic adatoms using first-principles theory

Magnetic impurities at surfaces of superconductors can induce bound states referred to as Yu–Shiba–Rusinov states (i.e. Shiba states) within superconducting (SC) gaps. For superconductors with strong spin–orbit coupling (SOC), Shiba states arising from even single magnetic adatoms are too complex to be fully understood using effective models alone because SOC cannot be treated perturbatively and multiple orbitals are strongly mixed with spin projections. Here we investigate Shiba states of single magnetic adatoms at the surface of strongly spin-orbit coupled SC Pb, by solving the fully relativistic Dirac–Bogoliubov–de Gennes equations using multiple scattering Green’s function methods. For Fe and Co adatoms on Pb(110), we show that the Shiba states are better characterized by total angular momentum, J, and its projections on the z axis, m J . As a hallmark of the SOC effect, the Shiba states show a strong dependence of the orientation of the adatom moment. As the orientation of the Fe/Co moment changes, the deepest Shiba states merge at zero energy. This zero-energy state disappears with an additional non-magnetic adatom next to the magnetic adatom, although the other Shiba states unchange. For a Mn adatom on Pb, our Shiba states overall agree with experiments. The characteristics of our Shiba states are also observed with the similar energies and characters in the experiments. The deepest Shiba states that we compute, however, do not appear as close to the Fermi level as the experimental data. It would be interesting to compute the Shiba states with continuously varying vertical distances of the Mn adatom from the surface or with varying the charge state of the adatom, and to calculate the spatial dependence of the spectral density. Our findings will be also useful for understanding of Shiba states for dimers and longer spin chains on the Pb surface considering noncollinear magnetic structures in them.

The magnetic field driven superconductor–metal transition in disordered hole-overdoped cuprates

By solving the Bogoliubov–de Gennes equations for a d-wave superconductor, we explore how the interplay between disorder and the orbital depairing of an external magnetic field influences the superconductor–metal transition of the hole-overdoped cuprates. For highly disordered systems, we find granular Cooper paring to persist above the critical field where the superfluid stiffness goes to zero. We also show that because the vortices are attracted to regions where the superconducting pairing is already weak, the Caroli–de Gennes–Matricon zero-bias peak in the local density of states at the vortex cores disappears already at moderate disorder.