Andreev bound states and the π-junction transition in a superconductor/quantum-dot/superconductor system

We study Andreev bound states and the pi -junction transition in a superconductor/quantum-dot/superconductor (S-QD-S) system by a Green function method. We derive an equation for describing the Andreev bound states in the S-QD-S system, and provide a unified understanding of the pi -junction transition caused by three different mechanisms. (1) Zeeman splitting. For a QD with two spin levels E-up arrow and E-down arrow we find that the surface of the Josephson current I(phi = pi /2) versus the (E-up arrow, E-down arrow configuration exhibits an interesting profile: a sharp peak around E-up arrow = E-down arrow = 0; a positive ridge in the region of E-up arrow E-down arrow > 0; and a negative, flat, shallow plane in the region of Eup arrowEdown arrow < 0. (2) Intra-dot interaction. We deal with the intra-dot Coulomb interaction by using the Hartree-Fock approximation, and find that the system behaves as a pi -junction when the QD becomes a magnetic dot due to the interaction. The conditions for pi -junction transition are also discussed. (3) Non-equilibrium distribution. We replace the Fermi distribution, f (co), by a non-equilibrium one, (1)/(2) [f (omega - V-c) + f (omega + V-c)], and allow Zeeman splitting in the QD where E = -E4 = h. The curves of 1 (q5 =.7/2) versus V, show the novel effect of the interplay of the non-equilibrium distribution with magnetization in the QD.