Modelling the propagation of a signal through a layered nanostructure: connections between the statistical properties of waves and random walks

It is possible to discuss the propagation of an electronic current through certain layered nanostructures modelling them as a collection of random one-dimensional interfaces, through which a coherent signal can be transmitted or reflected while being scattered at each interface. We present a simple model in which a persistent random walk (the 't-r' model in one dimension) is used as a representation of the propagation of a signal in a medium with such random interfaces. In this model all the possible paths through the system leading to transmission or reflection can be enumerated in an expansion in the number of loops described by the path. This expansion allows us to conduct a statistical analysis of the length of the paths for different geometries and boundary conditions and to understand their scaling with the size of the system. By tuning the parameters of the model it is possible to interpolate smoothly between the ballistic and the diffusive regimes of propagation. An extension of this model to higher dimensions is presented. We show Monte Carlo simulations that support the theoretical results obtained.