Quantum transport and microwave scattering on fractal lattices

Studying the wave-particle nature of electrons in different ways has lead to many fundamental discoveries. Particularly, the dimensionality dependent electronic behavior in the Luttinger Liquid (1D), Quantum Hall (2D) and non-interacting Fermi Liquid (3D) regimes have already revolutionized our understanding of the mechanisms behind quantum electronics. In this work, the theoretical and experimental studies focus on the non-integer dimension represented by an sp(2)-carbon-based Sierpinski triangular structure with a 1.58D space occupancy. In the tight-binding approach, the spectral distribution of electronic states of such a structure exhibits distinct peak patterns, which are well-separated by gaps. Through quantum transport simulation, the conductance of electrons in 1.58D was studied. Both delocalized, conducting and localized, non-conducting states identified, which differ from the established features of both the fully 2D graphene sheet and 1D carbon nanotubes. In microwave scattering measurements on an adequate experimental setting and the respective simulations on the Sierpinski triangle, the obtained diffraction patterns showed interesting peculiarities such as a reduced number of minima and magic angle, next to diffraction regions of high and low intensity, as well as forbidden regions. The fractal geometry of the structure affects the propagation of waves by manipulating the way they interact with each other which results in structural metamaterial-like interference characteristics, decreasing or amplifying the transmitted or reflected signals, or blocking the transport completely.