Different methods of calculating the pinning energy of plane vortices in a 3D Josephson medium: A comparative study

The pinning energy of plane (laminar) vortices in a 3D Josephson medium is calculated within a continuous vortex model considering functions of two types: V=1−cosϕ and V= 2/π⁴ϕ²(2π−ϕ)². The shape and energy of the stable and unstable vortices are found with an algorithm for the exact numerical solution of a set of difference equations. The vortex magnetic and Josephson energies diverge. The magnetic and Josephson components of the pinning energy are close in magnitude but differ in sign; as a result, the total pinning energy is smaller than its components by one order of magnitude. This result is confirmed analytically. An analytical computing method within the continuous vortex model is suggested. This method preserves the difference terms in the energy expression. The magnetic energy found by this method differs from the Josephson energy in magnitude, and the magnetic component of the pinning energy is opposite in sign to the Josephson component. Comparative analysis of the approximate approaches to energy calculation within the continuous vortex model when the difference terms are retained and when they are replaced by derivatives is performed. It is shown that the continuous vortex model gives incorrect values of the Josephson and magnetic components of the pinning energy. The actual values are several tens or several hundreds of times higher than those obtained with the continuous vortex model. Yet, since the Josephson and magnetic components of the pinning energy have different signs, the exact value of the total pinning energy and the approximate value obtained within the continuous vortex model differ insignificantly.