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Scaling behavior in the transmission coefficient for a self-affine multi-barrier system using graphene

By means of a deposited or epitaxial graphene model, we study the transmission coefficient as a function of the incident electron's energy, for a multi-barrier system which is finitely self-affine (i.e., it is self-similar but with different scaling ratios in the x and energy axis) and has mirror symmetry with respect to the center of the structure. The main result is the scaling behavior in the transmission coefficient (which in fact resembles the form of the multi-barrier structure) and the appearance of a scaling relation between curves of different parameter values. This system is finitely self-affine as the number of scaled pieces is finite and the scaling is only made in the energy axis. In order to study the transmission properties of the proposed structure, we consider first different “generations” of its construction, we compute their transmission coefficient curves and then search for some resemblance of the geometric properties of the multi-barrier structure in the form of scaling relations between transmission curves. We find that not only such scaling relations exist, but they are surprisingly simple. In fact they are simple enough to write down a closed algebraic expression that describe them. We thought that this is due to the finite self-affinity property and that it could be used as a basic model to analyze more complicated multi-barrier profiles.