Superconducting-magnatic proximity systems and mathematical analogies to classical mechanics

Abstract

We present a model of a magnetic thin film that accurately replicates the features of exchange springs and use it to study the superconducting proximity effects when placed between two superconductors. The exchange spring is found to possess a mathematical analogy to the frictionless spherical pendulum at constant azimuthal frequency, also known as the bead on a hoop, which is occasionally used in introductory classical mechanics courses as an example of Least Action Principles. We provide the exact closed form, analytic solution of the bead and hoop through the use of Jacobi elliptic functions to this nearly 200 year old problem. The general solution strategy used to solve the mechanics problem is used to obtain the order parameter of a wide, dirty superconductor-ferromagnet-superconductor (SFS) trilayer to find the Green's functions analytically in the case of a uniform exchange field. The exchange spring is then substituted for the homogeneous ferromagnet and used to numerically investigate the emergence of long range triplet pairing as a function of the twisting magnetization profile.