Comparison of various techniques for calculating vortex pinning energies in a long periodically modulated Josephson contact

The pinning energy of a vortex in a long periodically modulated Josephson c ontact is calculated. An algorithm is proposed for the exact numerical solu tion of a system of difference equations describing the shape and energy of the vortex in the stable and unstable states. The magnetic and Josephson e nergies of a vortex are different. The magnetic and Josephson components of the pinning energy are close in magnitude and have opposite signs, so that the total pinning energy is an order of magnitude lower than its component s. A theoretical analysis is carried out, confirming this behaviour. A tech nique is proposed for analytical calculation in terms of the continuous vor tex model, retaining the difference terms in the expression for energy. The magnetic energy E-H calculated by this method differs from the Josephson e nergy E-J, and the magnetic component of the pinning energy has a sign oppo site to that of the Josephson component. A comparative analysis of the poss ibility of calculating the energies by approximate methods in the framework of the continuous vortex model with difference terms retained or replaced by derivatives is made. The analysis shows that the continuous vortex model fails to provide correct values of the Josephson and magnetic components o f the pinning energy. The exact values of these parameters are tens and hun dreds of times greater than those obtained using the continuous vortex mode l. This notwithstanding, the exact values of the total pinning energy diffe r insignificantly from those obtained by applying the continuous vortex mod el because of the opposite signs of the Josephson and magnetic components o f the pinning energy.