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

Abstract

We study Andreev bound states and $π$-junction transition in a superconductor / quantum-dot / superconductor (S-QD-S) system by Green function method. We derive an equation to describe the Andreev bound states in S-QD-S system, and provide a unified understanding of the $π$-junction transition caused by three different mechanisms: (1) {\it Zeeman splitting.} For QD with two spin levels $E_{\uparrow}$ and $E_{\downarrow}$, we find that the surface of the Josephson current $I(ϕ=\frac π2)$ vs the configuration of $(E_{\uparrow},E_{\downarrow})$ exhibits interesting profile: a sharp peak around $E_{\uparrow}=E_{\downarrow}=0$; a positive ridge in the region of $E_{\uparrow}\cdot E_{\downarrow}>0$; and a {\em % negative}, flat, shallow plain in the region of $E_{\uparrow}\cdot E_{\downarrow}<0$. (2){\it \ Intra-dot interaction.} We deal with the intra-dot Coulomb interaction by Hartree-Fock approximation, and find that the system behaves as a $π$-junction when QD becomes a magnetic dot due to the interaction. The conditions for $π$-junction transition are also discussed. (3) {\it \ Non-equilibrium distribution.} We replace the Fermi distribution $f(ω)$ by a non-equilibrium one $\frac 12[ f(ω-V_c)+f(ω+V_c)] $, and allow Zeeman splitting in QD where $% E_{\uparrow}=-E_{\downarrow}=h.$ The curves of $I(ϕ=\frac π2)$ vs $% V_c$ show the novel effect of interplay of non-equilibrium distribution with magnetization in QD.