Energy spectrum of the graphene-based Fibonacci superlattice

Energy spectra of the graphene-based Fibonacci superlattice (SL) in the presence of the band gap in graphene have been investigated. The lattice consists of rectangular barriers, which are arranged along axis $Ox$. The quasi-periodic modulation is performed due to the difference in the values of the mass term of the Hamiltonian in various SL elements. It is shown that effective splitting of allowed bands (and thereby the formation of a series of gaps) under the effect of the quasi-periodic factor is implemented with both oblique and normal incidence of the electron wave on the SL surface. The energy spectra have a clearly pronounced periodic character over the entire energy scale. The bands split in separate fragments of the spectrum (conventionally periods) according to the Fibonacci inflation rule in each new generation. The band gap associated with a new Dirac point is formed in all Fibonacci generations similarly to the periodic graphene-based SLs. The location of the Dirac point is independent of the SL period; it is very sensitive to the potential barrier height and to the width ratio between the quantum well and the barrier and depends weakly on the mass term in the Hamiltonian. The dependence of the spectra on the incidence angle of the electron wave is insignificant.