Model For Layered Superconductors Research Articles

Exact solutions of the Lawrence-Doniach model for layered superconductors

We solve the problem of exact minimization of the Lawrence-Doniach (LD) free-energy functional in parallel magnetic fields. We consider both the infinite in the layering direction case (the infinite LD model) and the finite one (the finite LD model). We prove that, contrary to a prevailing view, the infinite LD model does not admit solutions in the form of isolated Josephson vortices. For the infinite LD model, we derive a closed, self-consistent system of mean-field equations involving only two variables. Exact solutions to these equations prove simultaneous penetration of Josephson vortices into all the barriers, accompanied by oscillations and jumps of the magnetization, and yield a completely new expression for the lower critical field. Moreover, the obtained equations allow us to make self-consistent refinements on such well-known results as the Meissner state, Fraunhofer oscillations of the critical Josephson current, the upper critical field, and the vortex solution of Theodorakis [S. Theodorakis, Phys. Rev. B 42, 10 172 (1990)]. Our consideration of the finite LD model illuminates the role of the boundary effect. In contrast to the infinite case, an explicit analytical solution to the Maxwell equations of the finite case does not preclude the existence of localized Josephson vortex configurations. By the use of this solution, we obtain a self-consistent description of the Meissner state. Finally, we discuss some theoretical and experimental implications.

Two-square-well model for layered superconductors: Effect of intralayer and interlayer Coulomb interactions on thed-wave gap

On the basis of Fermi liquid theory, we study the effect of intralayer and interlayer Coulomb interactions on the energy gap of layered superconductors assuming the BCS d-wave pairing model. The gap equation is derived for the two-square-well model, assuming a boson-mediated attraction and a long-range Coulomb repulsion that leads to intralayer and interlayer Coulomb scattering of Cooper pairs. The d-wave gap amplitude ${\ensuremath{\Delta}}_{1}$ at $T=0\mathrm{K}$ is evaluated as a function of the attractive pairing strength and of the screened Coulomb interactions. For optimally doped ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ we find that intralyer Coulomb scattering reduces ${\ensuremath{\Delta}}_{1}$ by a factor between 1.5 and 2. The additional effect of interlayer Coulomb scattering depends in a crucial manner on the relative phases of the gap functions in adjacent layers. If Josephson and boson-assisted interlayer pair tunneling (ILT) are responsible, the three-dimensional phase coherence, that is, the phase difference between adjacent layers $\ensuremath{\Delta}\ensuremath{\varphi}=0,$ interlayer Coulomb scattering leads to a further reduction of the gap that can be comparable with or larger than the effect of intralyer scattering. If, however, the phase of the gap function alternates, that is $\ensuremath{\Delta}\ensuremath{\varphi}=\ensuremath{\pi},$ because the interlayer Coulomb scattering dominates the ILT effect, the interlayer Coulomb scattering of Cooper pairs enhances ${\ensuremath{\Delta}}_{1}.$ This effect is studied as a function of the strength of the Coulomb interaction and it is shown that, with increasing strength, the pseudo-Coulomb parameter ${\mathrm{\ensuremath{\mu}}}^{\mathrm{*}}$ decreases and may change sign when the Coulomb interaction is sufficiently strong. The possible role of the interlayer Coulomb effect is discussed in relation to scanning tunneling microscopy experiments.

On the Lawrence–Doniach model for layered superconductors

The Lawrence–Doniach model for superconducting layered compounds in which the Ginzburg–Landau order parameters in adjacent layers are coupled by Josephson tunneling is considered. The main purpose of this paper is to demonstrate that the time-dependent Ginzburg–Landau model is the limit of the Lawrence–Doniach model as the layer spacing goes to zero.

Finite element analysis on the lawrence-doniach model for layered superconductors

A simple fully discrete finite element method is proposed to solve the time-dependent nonlinear Lawrence-Doniach model for layered superconductors. Numerical stability and optimal energy-norm error estimates are established for the proposed numerical method.

Coupled two-dimensional Fermi liquids as a model for layered superconductors: Basic equations and elementary results.

We develop a microscopic model of layered superconductors, designed specifically for the high-${\mathit{T}}_{\mathit{c}}$ cuprates, which is valid at all temperatures below the degeneracy temperature and for all relevant length scales. The model is based on the original idea of Bulaevskii, and consists of a stack of two-dimensional conducting planes coupled via interlayer diffusion of charge carriers. The microscopic model presented here reduces to the phenomenological Ginzburg-Landau, Lawrence-Doniach, and anisotropic London models in the appropriate limits. We derive the main equations of the interlayer diffusion model, then use the model to examine the role of dimensionality and interlayer scattering on the magnetic properties of layered superconductors.

Effect of d-wave energy-gap symmetry on Raman shifts.

We have calculated the shift and change in width, due to superconductivity, for a q=0 Raman-active phonon in a two-dimensional, tight-binding band structure with a d-wave pairing interaction. We find that additional structure appears in the phonon self-energy due to the van Hove singularity, which occurs in two dimensions. A more general model of a superconductor with nodes in the gap function is also examined and related to both the d-wave model and another model for layered superconductors. In contrast to the usual isotropic s-wave case where a phonon with frequency less than twice the gap value is softened and narrowed, we find softening with broadening in these models. Comparison of the anisotropic models is made with experiment.

Dimensional crossover in the upper critical field of layered superconductors.

Using the recently introduced ${\mathit{g}}_{3}$ model for layered superconductors, we calculate the angular and temperature dependence of the upper critical field ${\mathit{H}}_{\mathit{c}2}$. By lowering the temperature we find a crossover from three-dimensional bulk behavior to decoupled two-dimensional layers in both the temperature and angular dependence of ${\mathit{H}}_{\mathit{c}2}$. This behavior is remarkably consistent with recent experiments and the comparison with the measured angular dependence provides an estimate for the interlayer coupling ${\mathit{g}}_{3}$.

From low to high-temperature superconductivity: A dimensional crossover phenomenon? A finite size effect?

We explore the Ginzburg-Landau version of a model for layered superconductors, in which superconducting material consisting ofM unit cells is separated by insulating layers that areN unit cells thick. Invoking the recently proposed interlayer coupling—which enhances pairing within the superconducting sheets—the order parameter coupling turns out to be of Heisenberg type, analogous to layered magnets. The mean field result forTc (M, N) is found to be in remarkable agreement with experiment for ultrathin NbSe2 crystals and Sn−SiO, Al−AlO2, YBCO/PrBCO superlattices. The fact thatTc (M, N) increases with increasingM is traced back to a finite size effect, corresponding to a crossover from nearly two to three-dimensional superconductivity. This change from low to high-temperature superconductivity is found to be independent of the sign of the interlayer interaction. An electrostatic model for the interaction on YBCO/PrBCO superlattices is in reasonable agreement with experiment. The orbital upper critical fieldHc2‖ turns out to be unlimited ited whileHc2⊥ is proportional toTc (M, N).

A GINZBURG-LANDAU THEORY FOR CRITICAL TEMPERATURE OF LAYERED HIGH-Tc SUPERCONDUCTORS

Using a phenomenological model for layered superconductors, we derive a formalism for the critical temperature T c in terms of the layer numbers. We show that T c of a system of n identical layers is the same as that of a single layer for all values of n. In a periodic layered structure composed of two kinds of layers, T c increases monotonically with the number of higher-T c layers. When the carrier number depends on structure, T c shows a behavior consistent with the experimental data on some high-T c superconductors such as Tl and Bi families. We make a suggestion to achieve a higher T c .

Critical temperature of layered highT c superconductors

We present and investigate a model for layered superconductors with single-electron interlayer tunneling and a unit cell containing an arbitrary number of sheets. For an effective pairing of two-dimensional origin,T c is found to be independent of the layer spacings. However, pairing of three-dimensional nature, allowing for spacing dependent coupling strength, fits remarkably well the observed climb inT c for the Bi−O and Tl−O layered systems, as the number of CuO2 planes in the unit cell is increased from two to four and six. This agreement points to a weak coupling pairing mechanism of three-dimensional origin.

A Model for Layered Superconductors and First Order Transition in the Presence of Magnetic Field Parallel to Layers

A model for layered superconductors is proposed and the possibility of the first order transition caused by the transformation of flux structure in the presence of the magnetic field parallel to the layers is pointed out by means of the boson theory. Flux structure is calculated for a simple case.