Non-local Andreev reflection in superconducting quantum dots

With the aid of the Keldysh technique we develop a microscopic theory of non-local electron transport in three-terminal NSN structures consisting of a chaotic superconducting quantum dot attached to one superconducting and two normal electrodes. Our theory fully accounts for non-equilibrium effects and disorder in a superconducting terminal. We go beyond perturbation theory in tunneling and derive a general expression for the system conductance matrix which remains valid in both weak and strong tunneling limits. We demonstrate that the proximity effect yields a decrease of crossed Andreev reflection (CAR). Beyond weak tunneling limit the contribution of CAR to the non-local conductance does not cancel that of direct electron transfer between two normal terminals. We argue that temperature dependence of the non-local resistance of NSN devices is determined by the two competing processes -- Andreev reflection and charge imbalance -- and it has a pronounced peak occurring at the crossover between these two processes. This behavior is in a good agreement with recent experimental observations.