Self-consistent Bogoliubov-de Gennes Research Articles

Yu-Shiba-Rusinov States in s-Wave Kagome Superconductors: Self-Consistent Bogoliubov-de Gennes Calculations.

Significant research has recently been conducted into the Yu-Shiba-Rusinov (YSR) states in kagome superconductors through theoretical modeling and experimental investigations. However, additional efforts are still needed to further understand the local superconductivity near magnetic impurities in the kagome lattice and clarify how relevant quantities depend on the interaction strength, J, between such impurities and electrons. In this study, we explore a self-consistent numerical solution of the Bogoliubov-de Gennes equations for an s-wave superconducting Kagome model with a single classical magnetic impurity. Our study reveals that with increasing J, the local pair potential is systematically depressed in the vicinity of the impurity, similar to previous results obtained for the square and triangular lattices. Moreover, when J is further increased, the system undergoes a first-order phase transition with the appearance of stable and metastable states, reflecting the presence of the hysteresis loop in the pertinent quantities. As a consequence of this transition, the minimal energy of the stable YSR state is nonzero at any J, contrary to the expectations based on the assumption of a constant pair potential. A distinctive feature of the kagome lattice is that characteristics of the first-order transition are very sensitive to the position of the chemical potential within the kagome energy spectrum.

Manipulating chiral Majorana mode with additional potential in superconductor-Chern insulator heterostructures

We employed the self-consistent Bogoliubov–de Gennes equations to explore the states of chiral Majorana mode in quantum anomalous Hall insulators in proximity to a superconductor, leading to the development of an extensive topological phase diagram. Our investigation focused on how an additional potential affects the separation of chiral Majorana modes across different phase conditions. We substantiated our findings by examining the zero-energy Local Density of States spectrum and the probability distribution of the chiral Majorana modes. We established the universality of chiral Majorana mode separation by applying an additional potential. This finding serves as a vital resource for future endeavors aimed at controlling and detecting these particles, thereby contributing to the advancement of quantum computing and condensed matter physics.

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.

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.

Simulating superconducting properties of overdoped cuprates: The role of inhomogeneity

Theoretical studies of disordered $d$-wave superconductors have focused, with a few exceptions, on optimally doped models with strong scatterers. Addressing recent controversies about the nature of the overdoped cuprates, however, requires studies of the weaker scattering associated with dopant atoms. Here we study simple models of such systems in the self-consistent Bogoliubov-de Gennes (BdG) framework, and compare to disorder-averaged results using the self-consistent-T-matrix-approximation (SCTMA). Despite surprisingly linear in energy behavior of the low-energy density of states even for quite disordered systems, the superfluid density in such cases retains a quadratic low-temperature variation of the penetration depth, unlike other BdG results reported recently. We trace the discrepancy to smaller effective system size employed in that work. Overall, the SCTMA performs remarkably well, with the exception of highly disordered systems with strongly suppressed superfluid density. We explore this interesting region where gap inhomogeneity dominates measured superconducting properties, and compare with overdoped cuprates.

Topological superconductivity driven by correlations and linear defects in multiband superconductors

There have been several proposals for platforms sustaining topological superconductivity in high temperature superconductors, in order to make use of the larger superconducting gap and the expected robustness of Majorana zero modes towards perturbations. In particular, the iron-based materials offer relatively large $T_c$ and nodeless energy gaps. In addition, atomically flat surfaces enable the engineering of defect structures and the subsequent measurement of spectroscopic properties to reveal topological aspects. From a theory perspective, a materials-specific description is challenging due to the correlated nature of the materials and complications arising from the multiband nature of the electronic structure. Here we include both aspects in realistic interacting models, and find that the correlations themselves can lead to local magnetic order close to linear potential scattering defects at the surface of the superconductor. Using a self-consistent Bogoliubov-de Gennes framework in a real-space setup using a prototype electronic structure, we allow for arbitrary magnetic orders and show how a topological superconducting state emerges. The calculation of the topological invariant and the topological gap allows us to map out the phase diagram for the case of a linear chain of potential scatterers. While intrinsic spin-orbit coupling is not needed to enter the topological state in presence of spin-spiral states, it enlarges the topological phase. We discuss the interplay of a triplet component of the superconducting order parameter and the spin spiral leading effectively to extended spin orbit coupling terms, and connect our results to experimental efforts on the Fe(Se,Te) system.

Superfluidity of fermionic pairs in a harmonic trap. Comparative studies: Local Density Approximation and Bogoliubov-de Gennes solutions

Experiments with ultracold gases on the lattice give the opportunity to realize superfluid fermionic mixtures in a trapping potential. The external trap modifies the chemical potential locally. Moreover, this trap also introduces non-homogeneity in the superconducting order parameter. There are, among other approaches, two methods which can be used to describe the system of two-component mixtures loaded into an optical lattice: the Local Density Approximation (LDA) and the self-consistent Bogoliubov–de Gennes equations. Here, we compare results obtained within these two methods. We conclude that the results can be distinguishable only in the case of a small value of the pairing interaction.

Charge density wave and superconductivity in transition metal dichalcogenides

Competing orders in condensed matter give rise to the emergence of fascinating, new phenomena. Here, we investigate the competition between superconductivity and charge density wave in the context of layered-metallic compounds, transition metal dichalcogenides, in which the superconducting state is usually suppressed by the charge density wave. We show, using real-space self-consistent Bogoliubov-de Gennes calculations and momentum-space calculations involving density-functional theory and dynamical mean-field theory, that there is a surprising reappearance of superconductivity in the presence of non-magnetic disorder fluctuations, as observed in recent experiments.

Phase separation and proximity effects in itinerant ferromagnet/superconductor heterostructures

Heterostructures made of itinerant ferromagnets and superconductors are studied. In contrast to most previous models, ferromagnetism is not enforced as an external Zeeman field but induced in a correlated single-band model (CSBM) that displays itinerant ferromagnetism as a mean-field ground state. This allows us to investigate the influence of an adjacent superconducting layer on the properties of the ferromagnet in a self-consistent Bogoliubov-de Gennes approach. The CSBM displays a variety features not present in the Zeeman exchange model that influence the behavior of order parameters close to the interface, as e.g. phase separation and the competition between magnetism and superconducting orders.

Topological features of the thermal Hall conductivity in a chiral p-wave superconductor under strain

Motivated by experimental studies on the effect of tensile and compressive strain or uniaxial pressure in the superconducting phase of the transition metal oxide Sr2RuO4, the pairing symmetry assuming the chiral p-wave superconducting state and the transport property are investigated theoretically by means of the self-consistent Bogoliubov-de Gennes approach. The uniaxial strain leads to the Fermi surface deformation. The topological properties are sensitive to the Fermi surface geometry induced by the strain and switch around the Lifshitz transition in the γ band. The temperature dependence of the thermal Hall conductivity is connected with the amplitude of the strain. While the topological number is reflected in the temperature linear contribution in the low-temperature limit of the thermal Hall conductivity, the deviation from this behavior depends on the pairing channel. Our result may enable us to provide information on the gap function of the chiral p-wave phase.

The nematicity induced d-symmetry charge density wave in electron-doped iron-pnictide superconductors

The interplay among the nematicity, the stripe spin-density-wave (SDW) order and superconductivity in iron-pnictides is studied in a self-consistent Bogoliubov-de Gennes equations. Our calculations have shown that the nematic-order breaks the degeneracy of dxz and dyz orbitals and causes the elliptic Fermi surface near the Γ point in the normal state. In addition, the appearance of the orthorhombic magnetic fluctuations generates two uneven pairs of peaks at ( ± π, 0) and (0, ± π) in its Fourier transformation. All these are comparing favorably with experimental measurements. In the nematic phase, our results indicate that the charge density and its spatial image in the local density of states exhibit a dx2−y2-like symmetry. Finally, the complete phase diagram is obtained and the nematic phase is found to be in a narrow region close to the SDW transition in the electron-doped iron-pnictide superconductors.

Self-consistent Bogoliubov–de Gennes theory of the vortex lattice state in a two-dimensional strongly type-II superconductor at high magnetic fields

A self-consistent Bogoliubov deGennes theory of the vortex lattice state in a 2D strong type-II superconductor at high magnetic fields reveals a novel quantum mixed state around the semiclassical Hc2, characterized by a well-defined Landau--Bloch band structure in the quasi-particle spectrum and suppressed order-parameter amplitude, which sharply crossover into the well-known semiclassical (Helfand-Werthamer) results upon decreasing magnetic field. Application to the 2D superconducting state observed recently on the surface of the topological insulator Sb2Te3, accounts well for the experimental data, revealing a strong type-II superconductor, with unusually low carrier density and very small cyclotron mass, which can be realized only in the strong coupling superconductor limit.

Bogoliubov-de Gennes study of nanoscale Hubbard superconductors

The effects of quantum confinement on the superconducting ground state are studied within the Bogoliubov–de Gennes (BdG) formalism and an attractive Hubbard model. We consider a periodic arrangement of two-dimensional superconducting grains composed by N × N atoms surrounded by insulating, metallic or superconducting stripes with a thickness of s atoms, leading to 2(N + s)2 coupled self-consistent BdG equations for a supercell of (N + s) × (N + s) atoms. These equations determine the spatial variation of superconducting gap as functions of temperature, electron–electron interaction, and hopping integrals analyzing three types of boundary stripes. The results show a clear enhancement of the superconducting gap and critical temperature induced by the electron confinement in the grain, being larger for the insulating boundary case. Finally, the numerical solutions of BdG equations are compared with those obtained by applying the BCS theory to each grain site.

Effects of single- and multi-substituted Zn ions in doped 122-type iron-based superconductors

Recent experiments on Zn-substituted 122-type iron-based superconductors (FeSCs) at electron- and hole- doped region provide us with a testing ground for understanding the effect of Zn impurities in these systems. Our first-principle calculations of the electronic structure reveal that the Zn 3d orbitals are far below the Fermi level and chemically inactive, while the Zn 4s-orbital is partially occupied and its wave function overlapping with those 3d-orbitals of neighboring Fe-ions. This suggests that the impurity effect is originating in the Zn 4s-orbital, not its 3d-orbitals. Employing a phenomenological two-orbital lattice model for 122-FeSCs and the self-consistent Bogoliubov-de Gennes equations, we study how the Zn-impurities suppress the superconductivity in electron- and hole- doped compounds. Our obtained results qualitatively agree with the experimental measurements.

Multiband theory of superconductivity at theLaAlO3/SrTiO3interface

We present a multi-band model for superconductivity at the metallic interface between insulating oxides LaAlO$_3$ and SrTiO$_3$ (001). Using a self-consistent Bogoliubov-de Gennes theory, formulated with the realistic bands at the interface, we investigate the spin-singlet and spin-triplet pairings in intra-band and inter-band channels. We find that the Rashba and atomic spin-orbit interactions at the interface induce singlet pairing in the inter-band channel and triplet pairing in both the intra-band and inter-band channels when the pairing amplitude in the singlet intra-band channel is finite. The gate-voltage variation of superconductivity is resolved in different pairing channels, compared with experimental results and found to match quite well. Interestingly, an enhancement of the superconducting transition temperature by external in-plane magnetic field is found revealing the existence of a hidden superconducting state above the observed one. As the interface is known to possess high level of inhomogeneity, we explore the role of non-magnetic disorder incorporating thermal phase fluctuations by using a Monte-Carlo method. We show that even after the transition to the non-superconducting phase, driven by temperature or magnetic field, the interface possesses localized Cooper pairs whose signature was observed in previous experiments.

Effect of single interstitial impurity in iron-based superconductors with sign-changed s-wave pairing symmetry

We employ the self-consistent Bogoliubov-de Gennes (BdG) formulation to investigate the effect of single interstitial nonmagnetic/magnetic impurity in iron-based superconductors with s ± -wave pairing symmetry. We find that both the nonmagnetic and magnetic impurities can induce bound states within the superconducting (SC) gap and a π phase shift of SC order parameter at the impurity site. However, different from the interstitial-nonmagnetic-impurity case characterized by two symmetric peaks with respect to zero energy, the interstitial magnetic one only induces single bound-state peak. In the strong scattering regime this peak can appear at the Fermi level, which has been observed in the recent scanning tunneling microscope (STM) experiment of Fe(Te,Se) superconductor with interstitial Fe impurities (Yin et al. 2015 [44]). This novel single in-gap peak feature also distinguishes the interstitial case from the substitutional one with two peaks. These results provide important information for comparing the different impurity effects in the iron-based superconductors.

Zero-energy peak and triplet correlations in nanoscale superconductor/ferromagnet/ferromagnet spin valves

Using a self-consistent Bogoliubov-de Gennes approach, we theoretically study the proximity-induced density of states (DOS) in clean SFF spin-valves with noncollinear exchange fields. Our results clearly demonstrate a direct correlation between the presence of a zero energy peak (ZEP) in the DOS spectrum and the persistence of spin-1 triplet pair correlations. By systematically varying the geometrical and material parameters governing the spin-valve, we point out to experimentally optimal system configurations where the ZEPs are most pronounced, and which can be effectively probed via scanning tunneling microscopy. We complement these findings in the ballistic regime by employing the Usadel formalism in the full proximity limit to investigate their diffusive SFF counterparts. We determine the optimal normalized ferromagnetic layer thicknesses which result in the largest ZEPs. Our results can serve as guidelines in designing samples for future experiments.

Spin-Orbit Coupled s-Wave Superconductor in One-Dimensional Optical Lattice* *Supported by National Program for Basic Research of MOST (973 grant), and by National Natural Science Foundation of China under Grant Nos. 11121063, 11174360, 11374354, 11274195, 2011CB606405 and 2013CB922000

We study the topological properties of spin-orbit coupled s-wave superconductor in one-dimensional optical lattice. Compared to its corresponding continuum model, the single particle spectrum is modified by the optical lattice and the topological phase which is characterized by the Majorana edge modes can survive in two regions of the single-particle spectrum. With the help of the self-consistent Bogoliubov-de Gennes calculation in the harmonic trap, we find that the existence of an upper critical magnetic field removes the topological superconductor phase to the trap wings. We also study the effects of nonmagnetic and magnetic impurity on the topological properties, and find the universal behavior of the mid-gap state induced by impurity in the topological superconductor phase in strong scattering limit.

Topological aspect and transport property in multi-band spin-triplet chiral p-wave superconductor Sr2RuO4

Considering the superconductor Sr2RuO4, we analyze a three-band tight-binding model with one hole-like and two electron-like Fermi surfaces corresponding to the α, β and γ bands of Sr2RuO4 by means of a self-consistent Bogoliubov-de Gennes approach for ribbonshaped system to investigate topological properties and edge states. In the superconducting phase two types of gapless edge states can be identified, one of which displays an almost flat dispersion at zero energy, while the other, originating from the γ band, has a linear dispersion and constitutes a genuine chiral edge states. Not only a charge current appears at the edges but also a spin current due to the multi-band effect in the superconducting phase. In particular, the chiral edge state from the γ band is closely tied to topological properties, and the chiral p-wave superconducting states are characterized by an integer topological number, the so-called Chern number. We show that the γ band is close to a Lifshitz transition. Since the sign of the Chern number may be very sensitive to the surface condition, we consider the effect of the surface reconstruction observed in Sr2RuO4 on the topological property and show the possibility of the hole-like Fermi surface at the surface.

Probe-type of superconductivity by impurity in materials with short coherence length: the s-wave and η-wave phases study

The effects of a single non-magnetic impurity on superconducting states in the Penson–Kolb–Hubbard model have been analyzed. The investigations have been performed within the Hartree–Fock mean field approximation in two steps: (i) the homogeneous system is analysed using the Bogoliubov transformation, whereas (ii) the inhomogeneous system is investigated by self-consistent Bogoliubov-de Gennes equations (with the exact diagonalization and the kernel polynomial method). We analysed both signs of the pair hopping, which correspond to s-wave and η-wave superconductivity. Our results show that an enhancement of the local superconducting gap at the impurity-site occurs for both cases. We obtained that Cooper pairs are scattered (at the impurity site) into the states which are from the neighborhoods of the states, which are commensurate ones with the crystal lattice. Additionally, in the η-phase there are peaks in the local-energy gap (in momentum space), which are connected with long-range oscillations in the spatial distribution of the energy gap, superconducting order parameter (SOP), as well as effective pairing potential. Our results can be contrasted with the experiment and predicts how to experimentally differentiate these two different symmetries of SOP by the scanning tunneling microscopy technique.