Nanostructures with the Hilbert curve geometry as surface enhanced Raman scattering substrates

In this work, we consider fractal substrates for Surface Enhanced Raman Scattering measurements. The shape of the substrates is based on self-similar space filling Hilbert curves, which possess properties of both one dimensional and two dimensional geometries. The dielectric response of a doped semiconductor nanostructure, where conducting electrons are trapped in an effective potential having the geometry of the Hilbert curve is calculated and analysed. It is found that the system may exhibit electronic collective excitations specific for either a two dimensional or one dimensional system, depending on the excitation frequency.