Spectrum of Andreev bound states in Josephson junctions with a ferromagnetic insulator

Ferromagnetic-insulator (FI) based Josephson junctions are promising candidates for a coherent superconducting quantum bit as well as a classical superconducting logic circuit. Recently the appearance of an intriguing atomic-scale 0–π0–π transition has been theoretically predicted. In order to uncover the mechanism of this phenomena, we numerically calculate the spectrum of Andreev bound states in a FI barrier by diagonalizing the Bogoliubov–de Gennes equation. We show that Andreev spectrum drastically depends on the parity of the FI-layer number L   and accordingly the π(0)π(0) state is always more stable than the 0 (ππ) state if L is odd (even).