Bogoliubov-de Gennes Research Articles

Josephson junctions with chiral p-wave ferromagnetic superconductor reservoirs in presence of spin-orbit coupling interaction

The spin transfer torque (STT), is studied in new cases of Josephson junctions containing triplet p-wave chiral ferromagnetic superconductor (FS) reservoirs. To be more precise, FS1|spin orbit coupling layer (SOC1)|Normal metal (N)|SOC2|FS2 materials have been investigated. Specifically, FS1|N|FS2 structure with one layer of SOC at FS1|N interface and also two layers of SOC at both FS1|N and N|FS2 interfaces are investigated. For the structures including two SOC layers, both symmetric and asymmetric cases with respectively equal and different potential strengths are studied. The ballistic Josephson junctions are studied solving the Bogoliubov–de-Gennes equation. In particular, the cases with short normal metal contacts with thickness much smaller than the superconducting coherence length, ψ, are studied. It is obtained for the case with only one SOC layer, while the absolute value of the out-of-plane STT remains almost unchanged, its direction reverses for lower values of α, the angle between exchange fields of the two FSs. Also, the nanosystem in presence of two similar SOCs, exhibits lower out-of-plane STT. This is justified as the unit vector along the interface normal for the SOC1 is in the opposite direction of the second SOC2 layer. Hence, one can expect that these potentials fade the effect of each other. When the SOCs are not the same and have different values of potential strength, the out-of-plane STT reduces less which is attributed to slighter cancellation due to unequal potential strength. Moreover, the barrier strength is varied. It has been shown that the barrier strength increment changes the behavior of the STT. Particularly, it is found that it results in sign and value change of the STT. Further, the emergence of in-plane STTs in presence of SOC are confirmed and investigated for various systems.

Shiba states in systems with density of states singularities

Magnetic impurities placed in the superconductor can lead to emergence of the Yu-Shiba-Rusinov bound states. Coupling between the impurity and the substrate depends on density of states (DOS) at the Fermi level and can be tuned by DOS singularities. In this paper, we study the role of DOS singularities using the real space Bogoliubov-de Gennes equations for chosen lattice models. To uncover the role of these singularities (Dirac point, van Hove singularity, or the flat band), we study honeycomb, kagome, and Lieb lattices. We show that the properties of the Shiba state strongly depends on the type of lattice. Nevertheless some behaviors are generic, e.g. dependence of the critical magnetic coupling on the DOS at the Fermi level. However, the Shiba states realized in the Lieb lattice exhibit extraordinary properties, which can be explained by the presence of a few nonequivalent sublattices. Depending on the location of the magnetic impurity in the chosen sublattice, the value of critical magnetic coupling $J_\text{c}$ can be reduced or enhanced when the flat band is located at the Fermi level. In this context, we also present differences in the local DOS and coherence lengths for different sublattices in the Lieb lattice.

Density functional Bogoliubov-de Gennes analysis of superconducting Nb and Nb(110) surfaces

We report on the implementation of the Bogoliubov-de Gennes method into the JuKKR code, an implementation of the relativistic all-electron, full-potential Korringa-Kohn-Rostoker Green function method, which allows a material-specific description of inhomogeneous superconductors and heterostructures on the basis of density functional theory. We describe the formalism and report on calculations for the $s$-wave superconductor Nb, a potential component of the materials platform enabling the realization of the Majorana zero modes in the field of topological quantum computing. We compare the properties of the superconducting state both in the bulk and for (110) surfaces. We compare slab calculations for different thicknesses, the effect of surface relaxations, and the influence of a softening of phonon modes on the surface for the resulting superconducting gap.

Electronic structure of a Josephson vortex in a SIS junction

The Josephson vortex formed in a superconductor-insulator-superconductor (SIS) junction can affect the quantum mechanics of quasiparticles by creating an effective adiabatic potential determined by the inhomogeneous distribution of the phase difference of the order parameter along the junction. Starting from the quasiclassical version of the Bogoliubov-de Gennes (BdG) theory, we found the quasiparticle spectrum and the local density of states (DOS) both for the isolated Josephson vortex and the vortex chain. The spatially resolved DOS reveals a peculiar two-peak structure for each Josephson vortex, which can be detected experimentally using the scanning tunneling microscopy/spectroscopy techniques.

Probing two Higgs oscillations in a one-dimensional Fermi superfluid with Raman-type spin—orbit coupling

We theoretically investigate the Higgs oscillation in a one-dimensional Raman-type spin-orbit-coupled Fermi superfluid with the time-dependent Bogoliubov-de Gennes equations. By linearly ramping or abruptly changing the effective Zeeman field in both the Bardeen-Cooper-Schrieffer state and the topological superfluid state, we find the amplitude of the order parameter exhibits an oscillating behaviour over time with two different frequencies (i.e., two Higgs oscillations) in contrast to the single one in a conventional Fermi superfluid. The observed period of oscillations has a great agreement with the one calculated using the previous prediction [Volkov and Kogan, J. Exp. Theor. Phys. 38, 1018 (1974)], where the oscillating periods are now determined by the minimums of two quasi-particle spectrum in this system. We further verify the existence of two Higgs oscillations using a periodic ramp strategy with theoretically calculated driving frequency. Our predictions would be useful for further theoretical and experimental studies of these Higgs oscillations in spin-orbit-coupled systems.

In-gap states of vortices at the surface of a topological insulator

By solving the Bogoliubov–de Gennes equations we study fermion bound states in the core of a vortex of a two-dimensional superconductor. The metallic surface states of a strong topological insulator become superconducting via proximity effect with an s-wave superconductor. The strong spin-orbit coupling locks the spin perpendicular to the momentum (Rashba interaction) giving rise to a zero-energy Majorana state. We investigate the case of a large first excitation energy of the order of the superconducting gap (Δ∞), i.e. for the Fermi level close to the apex of the Dirac cone. We obtain the local density of states (LDOS) for the bulk and the bound states. The particle-hole symmetry is broken in the LDOS for up-spin as a consequence of the spin-orbit coupling and the chirality of the vortex. On the other hand, the down-spin LDOS is particle-hole symmetric. The spin-polarization of the bound state peaks is studied in detail. We present a waterfall-like spectrum that shows that for a distance less than 3 times the coherence length ξ from the core, the bound states can be observed.

Inherited topological superconductivity in two-dimensional Dirac semimetals

Under what conditions does a superconductor inherit topologically protected nodes from its parent normal state? In the context of inter-Fermi-surface pairing in three-dimensional Weyl semimetals with broken time-reversal symmetry, the pairing order parameter is classified by monopole harmonics and is necessarily nodal [Li and Haldane, Phys. Rev. Lett. 120, 067003 (2018)]. Here, we show that a similar conclusion could also be drawn for 2D Dirac semimetals, although the conditions for the existence of nodes are more complex, depending on the pairing matrix structure in the valley and sublattice space. We analytically and numerically analyze the Bogoliubov-de Gennes quasiparticle spectra for Dirac systems based on the monolayer as well as twisted bilayer graphene. We find that in the cases of intravalley intrasublattice and intervalley intersublattice pairings, the point nodes in the BdG spectra (which are inherited from the Dirac cone in the normal state) are protected by a 1D winding number. The nodal structure of the superconductivity is confirmed numerically using tight-binding models of monolayer and twisted bilayer graphene. Notably, the BdG spectrum is nodal even with a momentum-independent ``bare'' pairing, which, however, acquires a momentum dependence and point nodes upon projection to the Bloch states on the topologically nontrivial Fermi surface, similar in spirit to the Li-Haldane monopole superconductor and the Fu-Kane proximity-induced superconductor on the surface of a topological insulator.

Influence of Rashba spin-orbit coupling on Josephson effect in triplet superconductor/two-dimensional semiconductor/triplet superconductor junctions

We study theoretically Josephson effect in a planar ballistic junction between two triplet superconductors with p-wave orbital symmetries and separated by a two-dimensional (2D) semiconductor channel with strong Rashba spin–orbit coupling. In triplet superconductors, three types of orbital symmetries are considered. We use Bogoliubov–de Gennes formalism to describe quasiparticle propagations through the junction and the supercurrents are calculated in terms of Andreev reflection coefficients. The features of the variation of the supercurrents with the change of the strength of Rashba spin–orbit coupling are investigated in some detail. It is found that for the three types of orbital symmetries considered, both the magnitudes of supercurrent and the current-phase relations can be manipulated effectively by tuning the strength of Rashba spin–orbit coupling. The interplay of Rashba spin–orbit coupling and Zeeman magnetic field on supercurrent is also investigated in some detail.

Supercurrent decay in ballistic magnetic Josephson junctions

We investigate transport properties of ballistic magnetic Josephson junctions and establish that suppression of supercurrent is an intrinsic property of the junctions, even in absence of disorder. By studying the role of ferromagnet thickness, magnetization, and crystal orientation we show how the supercurrent decays exponentially with thickness and identify two mechanisms responsible for the effect: (i) large exchange splitting may gap out minority or majority carriers leading to the suppression of Andreev reflection in the junction, (ii) loss of synchronization between different modes due to the significant dispersion of the quasiparticle velocity with the transverse momentum. Our results for Nb/Ni/Nb junctions are in good agreement with recent experimental studies. Our approach combines density functional theory and the Bogoliubov-de Gennes model and opens a path for material composition optimization in magnetic Josephson junctions and superconducting magnetic spin valves.

Full orbital decomposition of Yu-Shiba-Rusinov states based on first principles

We have implemented the Bogoliubov-de Gennes (BdG) equation in a screened Korringa-Kohn- Rostoker (KKR) method for solving, self-consistently, the superconducting state for 3d crystals including substitutional impurities. In this report we extend this theoretical framework to allow for collinear magnetism and apply it to fcc Pb with 3d magnetic impurities. In the presence of magnetic impurities, there is a pair-breaking effect that results in sup-gap Yu-Shiba-Rusinov (YSR) states which we decompose into contributions from the individual orbital character. We determine the spatial extent of these impurity states, showing how the different orbital character affects the details of the YSR states within the superconducting gap. Our work highlights the importance of the first principles based description which captures the quantitative details making direct comparisons possible with experimental findings.

Modified Bogoliubov-de Gennes treatment for Majorana conductances in three-terminal transports

We consider a two-lead (three-terminal) setup of nonlocal transport through Majorana zero modes (MZMs) and construct a Majorana master equation (which is also valid for small bias voltage). We first carry out representative results of current and then show that a modified Bogoliubov-de Gennes (BdG) treatment can consistently recover the same results. Based on the interplay of the two approaches, we reveal the existence of nonvanishing channels of teleportation and crossed Andreev reflections even at the limit $\epsilon_M\to 0$ (zero coupling energy of the MZMs), which leads to new predictions for the height of the zero-bias-peak of the local conductance and the $\epsilon_M$-scaling behavior of the teleportation conductance, for verification by experiments.

Two-Dimensional Topological Superconductors with Spin-Orbit Coupling and Zeeman Term

Topological superconductors have attracted much attention for their potential applications in realizing topological quantum computing. In this paper, we show a system with 8 × 8 Bogoliubov-de Gennes Hamiltonian. The system has particle-hole symmetry. By adding a Zeeman term to the model, we discuss this system from two situations. In the first case of breaking the inversion symmetry while preserving the mirror symmetry, it is obtained from the topological phase diagram of the system that it is a topological superconductor. In another case where the mirror symmetry is broken while preserving the inversion symmetry, the system has two nodes connected by a flat band of zero-energy Andreev edge states and the topological number is non-zero . It can be concluded that the system is a topological nodal superconductor.

Higher-order topological superconductivity from repulsive interactions in kagome and honeycomb systems

We discuss a pairing mechanism in interacting two-dimensional multipartite lattices that intrinsically leads to a second order topological superconducting state with a spatially modulated gap. When the chemical potential is close to Dirac points, oppositely moving electrons on the Fermi surface undergo an interference phenomenon in which the Berry phase converts a repulsive electron–electron interaction into an effective attraction. The topology of the superconducting phase manifests as gapped edge modes in the quasiparticle spectrum and Majorana Kramers pairs at the corners. We present symmetry arguments which constrain the possible form of the electron–electron interactions in these systems and classify the possible superconducting phases which result. Exact diagonalization of the Bogoliubov-de Gennes Hamiltonian confirms the existence of gapped edge states and Majorana corner states, which strongly depend on the spatial structure of the gap. Possible applications to vanadium-based superconducting kagome metals AV3Sb5 (A = K, Rb, Cs) are discussed.

Shape resonances and the T c dependence on film thickness of Ni/Bi systems

We report on the experimentally observed variation of the superconducting critical temperature (T c ) of Ni/Bi systems as a function of the total deposited film thickness and on its explanation using a theoretical model. Two series of Ni/Bi systems have been analyzed which were obtained by depositions of Ni onto Bi in the proportions Ni3Bix (3 nm of Ni onto x nm of Bi) and NiyBi6y (y nm Ni onto 6 y nm of Bi). As shown recently, the formation of the superconducting compound NiBi3 at Ni/Bi interfaces in the resulting NiBi3-Bi films is thermodynamically favored by a volume contraction. Here we corroborate this result and estimate the thickness of the resulting NiBi3 and of the remaining Bi layers for the Ni3Bix and NiyBi6y series using the laws of mass and conservation of number of atoms. We consider the resulting film as being made up of two homogeneous and uniform layers of NiBi3 and Bi, respectively, and study this idealizing model using the Bogoliubov de Gennes (BdG) equations. It is assumed that superconductivity originates in the NiBi3 layer and penetrates the Bi layers via a potential barrier. Our theoretical calculations predict the dependence of T c with respect to the thicknesses of the NiBi3 and Bi layers, and also with the strength of the potential barrier that blocks the migration of electrons from the NiBi3 to the Bi layer. The calculations show that the superconducting gap also exists in Bi, although much weaker than in the NiBi3 layer. We compare the predicted T c values with the experimental data and find sufficient agreement to suggest that our model can explain the experimentally observed variation of T c with thickness. We interpret this dependence as shape resonance oscillations which are derived from the BdG theory applied to thin superconducting films.

Effect of Rashba spin-orbit and Rabi couplings on the excitation spectrum of binary Bose-Einstein condensates

We present the collective excitation spectrum analysis of binary Bose-Einstein condensates (BECs) with spin-orbit (SO) and Rabi couplings in a quasi-two-dimensional system. In particular, we investigate the role of SO and Rabi coupling strengths in determining the dynamical stability of the coupled BECs using Bogoliubov-de Gennes (BdG) theory. Using the eigenergy of BdG spectrum, we confirm the existence of phonon, roton, and maxon modes with weak repulsive intra- and inter-species contact interactions. The depth of the minimum corresponding to the roton mode depends strongly on the coupling strength. We find that the increase of the SO coupling leads to instability, while the increase in the Rabi coupling stabilizes the system. Also the eigenvectors of BdG spectrum indicates the presence of density like mode in the stable regime and spin like modes in unstable regimes. A phase diagram demonstrating the stability regime in the plane of SO and Rabi coupling strengths is obtained. Finally, we complement the observation of the excitation spectrum with the direct numerical simulation results of coupled Gross-Pitaevskii equations.

Anderson localization of excitations in disordered Gross-Pitaevskii lattices

We examine the one-dimensional Gross-Pitaevskii lattice at zero temperature in the presence of uncorrelated disorder. We obtain analytical expressions for the thermodynamic properties of the ground state field and compare them with numerical simulations both in the weak and strong interaction regimes. We analyze weak excitations above the ground state and compute the localization properties of Bogoliubov-de Gennes modes. In the long-wavelength limit, these modes delocalize in accordance with the extended nature of the ground state. For strong interactions, we observe and derive a divergence of their localization length at finite energy due to an effective correlated disorder induced by the weak ground state field fluctuations. We derive effective strong interaction field equations for the excitations and generalize to higher dimensions.

Distinct properties of vortex bound states driven by temperature

We investigate the behavior of vortex bound states in the quantum limit by self-consistently solving the Bogoliubov-de Gennes equation. We find that the energies of the vortex bound states deviates from the analytical result with the half-integer angular momentum μ at very low temperature. Specifically, the energy ratio for the first three orders is closer to 1: 2: 3 than to 1: 3: 5 in the extreme quantum limit . The local density of states reveals a Friedel-like behavior associated with that of the pair potential, which will be smoothed out by the thermal effect above the quantum limit. Our studies show that the vortex bound states can exhibit very distinct features in different temperature regimes, which provides a comprehensive understanding and should stimulate more experimental efforts for verification.

Vortex images influenced by superconducting gap and Fermi surface

A theoretical study on the vortex image is carried out near a vortex core in the presence of the anisotropic superconducting gap or the anisotropic Fermi surface with fourfold symmetry by solving Bogoliubov-de Gennes (BdG) equations. Both results show the anisotropic vortex core with fourfold symmetry, but detailed structures are rather different. At zero energy, the local density of states (LDOS) is always larger along curvature minima than along curvature maxima of the Fermi surface at the same radial distances from the vortex center in the presence of the anisotropic Fermi surface. In contrast, in the presence of the anisotropic superconducting gap, the LDOS is larger along gap maxima than along gap minima at short distances, but with an inversion at larger distances. With increasing energy, the evolutions of vortex images in these two situations also behave quite differently. Our results provide an effective method to distinguish these two contributions from the anisotropic superconducting gap and Fermi surface to the vortex image.

Band superconductivity in a periodic constricted nanoribbon structure

Periodic geometrically constricted 2D NbSe2 nanowire is investigated with the Anderson approximation of Bogoliubov-de Gennes (BdG) equations. The superconducting order parameter is related to the single particle DOS around the Fermi level, but the introduction of an inter-cell phase difference changes this relation from average-like (accounting for the miniband-proximity effect) to strictly local. A miniband crossing related “shape resonance” is identified as a function of the periodic constriction. The phase difference leads to modulation or suppression of the superconducting gap which is shown to typically be strictly coupled with the vanishing of the order parameter. However, an exception to the general rule is discussed, where significant Δ(r⃗) exists with a closed gap as a consequence of the mixing of different transverse symmetries. Possibility to redefine the properties of the system via manipulation of its geometry or the electron concentration is presented.

Stability of a Bose-condensed mixture on a bubble trap

Stability and dynamical behavior of binary Bose-Einstein condensed mixtures trapped on the surface of a rigid spherical shell are investigated in the mean-field level, exploring the miscibility with and without vortex charges, considering repulsive and attractive interactions. In order to compute the critical points for the stability, we follow the Bogoliubov-de Gennes method for the analysis of perturbed solutions, with the constraint that initially the stationary states are in a complete miscible configuration. For the perturbed equal density mixture, of a homogeneous uniform gas and when hidden vorticity is verified, with the species having opposite azimuthal circulation, we consider small perturbation analysis for each unstable mode, providing a complete diagram with the intra- and inter-species interaction role on the stability of the miscible system. Finally, beyond small perturbation analysis, we explore the dynamics of some repulsive and attractive inter-species states by full numerical solutions of the time-dependent Gross-Pitaevskii equation.