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

Abstract

We study Andreev bound states and $\pi$-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 $\pi$-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 $I(\varphi =\frac{\pi }{2})$ 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 $\pi$-junction when QD becomes a magnetic dot due to the interaction. The conditions for $\pi$-junction transition are also discussed. (3) {\it \ Non-equilibrium distribution.} We replace the Fermi distribution $f(\omega )$ by a non-equilibrium one $\frac{1}{2}[f(\omega -V_c)+f(\omega +V_c)]$, and allow Zeeman splitting in QD where The curves of $I(\varphi =\frac{\pi }{2})$ vs show the novel effect of interplay of non-equilibrium distribution with magnetization in QD.

18 pages, 8 figures, LateX

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