Graphene fractals: Energy gap and spin polarization

Graphene is a natural material platform for experimental realization of triangular fractals. Hence, forming triangular holes in the hexagonal lattice following a self-similar recipe leads to fractal Sierpinski triangles. Two subclasses characterized by armchair and zigzag edges are investigated using a mean-field Hubbard approach. We find that zigzag-edged fractals support a significant spin polarization whereas the armchair subclass is spin balanced. Despite this difference, the energy gap between like-spin states is large for both types. A self-similar distribution of energy levels is found in late-generation fractals. In an external magnetic field, the energy gaps tend to decrease, and an intricate dispersion of individual levels is found. This leads to optical Hall conductivities with a seemingly chaotic field dependence in late-generation fractals.