Electromagnetic proximity effect in superconducting spin valves heterostrucutres in the clean limit*

We theoretically study the magnetic field effect produced by the exchange coupling in ferromagnet-superconductor junctions. The inverse proximity effect induces magnetism inside the superconductor which then generates a spontaneous current. The spontaneous current in turn affects the vector potential via Maxwell's equations, thereby adjusting the Bogoliubov-de Gennes (BdG) Hamiltonian. To self-consistently determine physical quantities such as the currents and superconducting gap, we simultaneously find solutions to the BdG equations and Maxwell's equations iteratively. An alternative approach to solve this problem is the quasi-classical Green's function technique, but it requires a strong boundary condition on the vector potential to fix the gauge freedom in order to obtain a unique solution. In addition, this technique is not adequate for a strong ferromagnet. In the literature, the dirty limit is usually assumed to reduce the Eilenberger equation into a simpler Usadel equation. Still, it is difficult to self-consistently solve the Usadel equation along with Maxwell's equation. In contrast, we demonstrate that our BdG approach is natural to achieve self-consistency from a step-by-step analysis with Hamiltonain, symmetry and free energy. Finally, our work also offers a direction to find the relation between the odd-frequency triplet pairing and electromagnetism which can be controlled in the spin valve structure.