Vortices in layered superconductors with columnar pins: A density-functional study

Abstract

We use numerical minimization of a model free energy functional to study the effects of columnar pinning centers on the structure and thermodynamics of a system of pancake vortices in the mixed phase of highly anisotropic layered superconductors. The magnetic field and the columnar pins are assumed to be perpendicular to the layers. Our methods allow us to study in detail the density distribution of vortices in real space. We present results for the dependence of the average number of vortices trapped at a pinning center on temperature and pinning strength, and for the effective interaction between nearby pinned vortices arising from short-range correlations in the vortex liquid. For a commensurate, periodic array of pinning centers, we find a line of first order vortex lattice melting transitions in the temperature T vs. pin concentration c plane, which terminates at an experimentally accessible critical point as c is increased. Beyond this point, the transition is replaced by a crossover. Our results should also apply, with little change, to thin-film superconductors with strong point pinning.

Phase diagram of vortex matter in layered high-temperature superconductors with random point pinning

“…As discussed in detail in Ref. 17, many theoretical and experimental studies indicate that this model is appropriate for describing the properties of vortex matter in highly anisotropic materials such as BSCCO (Bi 2 Sr 2 CaCu 2 O 8+x ) in a large region of the field-temperature plane, although this conclusion has been questioned in a recent theoretical analysis 23 . The free energy functional of such a system of pancake vortices lying on the superconducting layers and interacting with a time-independent pinning potential can be written in the form:…”

“…In this limit, vortices in the same layer repel each other with a potential logarithmic in their separation, while there is a much weaker attractive interaction between vortices on different layers which falls off exponentially with layer separation and varies logarithmically with the separation in the layer plane. This model for the vortex system was used in our earlier studies 17,18 of the effects of columnar pinning. As discussed in detail in Ref.…”

“…If the time averaged vortex density is the same on all layers (as would be the case for columnar pins perpendicular to the layers), the free energy could be written in an effectively two-dimensional form 17,18 with the quantityC(r) ≡ n C mn (r) playing the role of the two-dimensional direct correlation function. But in the present case, with a pinning potential that varies from one layer to another, the full three dimensional problem has to be considered.…”

“…On the sites of this lattice we define discretized density variables ρ n,i ≡ ρ n (r i )A 0 , where ρ n (r i ) is the density at mesh point i on layer n and A 0 the area of the unit cell of the in-plane computational lattice. A numerical procedure developed earlier 17,26 is used to find local minima of the free energy written as a function of the variables {ρ n,i }.…”

“…In this highly anisotropic case, the Josephson coupling between layers is vanishingly small, and pancake vortices on different layers are coupled only through the electromagnetic interaction. Under these circumstances, one can use (as in our recent studies 17,18 of vortex systems in the presence of columnar pinning) a spatially discretized version of the Ramakrishnan-Yussouff free energy functional 19 for a system of pancake vortices. In this formalism, the free energy of the system is expressed as a functional of the time-averaged local density of vortices.…”