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Fractal nanostructures with the Hilbert curve geometry as a SERS substrate

We suggest a new type of substrates for the Surface Enhanced Raman Scattering measurements with the geometry based on self-similar fractal space filling curves. As an example, we have studied theoretically the dielectric response properties of doped semiconductor nanostructures, where the conducting electrons are trapped in the effective potential having the geometry of the Hilbert curve. We have found that the system may exhibit the induced charge distribution specific for either two dimensional or one dimensional systems, depending on the frequency of the external applied field. We have demonstrated that with the increasing of the depth of the trapping potential the resonance of the system counterintuitively shifts to lower frequencies.