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

Abstract

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.

Self-similar conductance patterns in graphene Cantor-like structures

“…As we already mentioned, in previous works we have shown that graphene under appropriate nanostructuring can present self-similar characteristics 19 – 21 . To be specific, the transmission probability or transmittance sustains self-similar patterns with well defined rules.…”

“…The main goal of this work is to investigate the existence of scaling rules, and hence self-similarity, in the conductance patterns. For this, we follow the guidelines reported in our previous works 19 – 21 . Specifically, we explore the three scaling rules reported for the transmittance: 1) between generations, 2) between different heights of the main barrier, and 3) between different lengths of the structure.…”

“…Several works claim that the transmission probability sustains self-similar characteristics in Fibonacci and Thue-Morse superlattices 15 – 18 , but once again the scale factors that connect the transmission patterns are not derived. Recently, we have shown that the transmission properties of graphene subjected to self-similar and self-affine multi-barrier structures present self-similar characteristics 19 – 21 . Even more, we have determined the scale factors that connect the transmission patterns.…”

Self-similar transport, spin polarization and thermoelectricity in complex silicene structures

“…Here, it is important to mention that substrates with different degrees of interaction with the graphene sheet (different barrier heights) are needed in order to obtain the self-similar graphene structures. It is also reported that simpler complex graphene structures present self-similar physical properties 16 – 19 . In particular, self-similar graphene structures, in which only the length of the barriers is scaled according to the Cantor set rules, display self-similar characteristics in the transmission probability or transmittance.…”

“…In particular, self-similar graphene structures, in which only the length of the barriers is scaled according to the Cantor set rules, display self-similar characteristics in the transmission probability or transmittance. These self-similar structures can be generated by nanostructured substrates 16 – 18 and inhomogeneous magnetic fields 19 . As we have documented graphene is the only material that manifests a fractal complex behavior in the electron transport.…”