Depinning of a Discrete Elastic String from a Random Array of Weak Pinning Points with Finite Dimensions

Abstract

In possible connection with dislocation pinning by foreign atoms in alloys and vortex pinning in type II superconductors, we compute the external force required to drag an elastic string along a discrete two-dimensional random array with finite dimensions. The obstacles, with a maximum pinning force fm are distributed randomly on a rectangular lattice with square symmetry. The system dimensions are fixed by the total course of the elastic string Lx and the string length Ly. Our study shows that Larkin’s length is larger than Ly when fm is less than a certain bound depending on the system size as well as on the obstacle density cs. Below such a bound an analytical theory is developed to compute the depinning threshold. Some numerical simulations allow us to demonstrate the accuracy of the theory for an obstacle density ranging from 1 to 50% and for different geometries.