Pinning of flux lines by planar defects

The influence of randomly distributed point impurities and planar defects on order and transport in type-II superconductors and related systems is studied. It is shown that the Bragg glass phase is unstable with respect to planar defects. Even a single weak defect plane oriented parallel to the magnetic field as well as to one of the main axes of the Abrikosov flux-line lattice is a relevant perturbation in the Bragg glass. A defect that is aligned with the magnetic field restores the flux density oscillations, which decay algebraically with the distance from the defect. The theory exhibits striking similarities to the physics of a Luttinger liquid with a frozen impurity. The exponent for the flux-line creep in the direction perpendicular to a relevant defect is derived. We find that the flux-line lattice exhibits in the presence of many randomly distributed parallel planar defects aligned to the magnetic field a glassy phase which we call planar glass. The planar glass is characterized by diverging shear and tilt moduli, a transverse Meissner effect, and resistance against shear deformations. We also obtain sample-to-sample fluctuations of the longitudinal magnetic susceptibility and an exponential decay of translational long-range order in the direction perpendicular to the defects. The flux creep perpendicular to the defects leads to a nonlinear resistivity $\rho \left(J\to 0\right)\sim \text{exp}\left[-{\left(J_D/J\right)}^{3/2}\right]$. Strong planar defects enforce arrays of dislocations that are located at the defects with a Burgers vector parallel to the defects in order to relax shear strain.

Figure 1

Schematic phase diagram of disordered flux-line lattices resulting from impurities, columnar and planar defects of concentration $n_{imp}$, $n_{cd}$, and $n_{pd}$, respectively. The stability of the phases with respect to different kinds of disorder is indicated by arrows. $\mu$ denotes the creep exponent.

Figure 2

(Color online) Vectors of the triangular flux-line lattice $\left({\mathbf{a}}_1,{\mathbf{a}}_2\right)$ and of its reciprocal lattice $\left({\mathbf{b}}_1,{\mathbf{b}}_2\right)$, and the angles $\alpha$ and $\beta$ that define the orientation of the defect plane.

Figure 3

Two possible orientations of defect planes relative to the flux-line lattice, corresponding to $g=\eta /2$ (dotted lines) and $3\eta /2$ (dashed lines).

Figure 4

Schematic illustration of a droplet with radius $R$ and width $w$ at a defect plane.

Figure 5

Schematic representation of a cylindrically shaped droplet of radius $R$ and length $L_x$ that extends across more than one defect plane. It drives the system into the new metastable state $u^{\left(n\right)}+\ell$. Note that domain walls parallel to the defects are wide (not shown here), while the cylindrically shaped domain wall is narrow (of width $w$).

Figure 6

(Color online) Array of edge dislocations that is located at the defects (thick blue lines) in order to relax shear strain. The Burgers vector is parallel to the defects.