Recursion method for nonhomogeneous superconductors: Proximity effect in superconductor-ferromagnet nanostructures

We present a theoretical method to study the electronic local spectral density of hybrid nanostructures consisting of a normal (N) or ferromagnetic (F) region deposited on top of a superconductor $\mathrm{}\left(S\right).$ Our approach is based on a lattice Hamiltonian model which allows to describe the spatial variation of the superconducting order parameter in nanostructures of arbitrary geometry. In order to obtain the local density of states we develop a generalization of the recursion method valid for systems containing superconducting and ferromagnetic regions. As a first step we analyze the proximity effect and the detailed behavior of Andreev states in one-dimensional (1D) $N-S$ and $F-S$ structures. We study the transition from the 1D case to the limit of infinite lateral dimensions in the ballistic regime. Finally we analyze the spatial variation of the proximity effect as a function of the exchange field in $F-S$ nanostructures. It is found that the oscillations in the induced pairing amplitude in the scale of the ferromagnetic coherence length can be correlated to the crossing of Andreev states through the Fermi energy as a function of the ferromagnetic region size.