Spectrum of Andreev bound states in Josepshon 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-pi 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 pi (0) state is always more stable than the 0 (pi) state if L is odd (even). 
Shiro Kawabata, Yukio Tanaka, Alexander A. Golubov, Andrey S. Vasenko,   Yasuhiro Asano
cond-mat.mes-hall cond-mat.supr-con