Electronic properties of a tight-binding and a Kronig-Penney model of the Thue-Morse chain

We study electronic properties of the energy spectrum in a one-dimensional tight-binding model with site energies arranged in the Thue-Morse sequence. Using the trace map, we obtain the branching rules of the spectrum and perform a scaling analysis of bandwidths. It is shown that the spectrum consists of absolutely continuous parts and singular continuous parts. We consider a Kronig-Penney model to study the properties of the spectrum and eigenstates in a Thue-Morse superlattice. For this purpose, we calculate the density of states and the resistance using the symmetry of the Thue-Morse lattice, and wave functions via the Poincaré map. The calculations reproduce the results obtained in the tight-binding model. We also discuss transport properties. Our results for the two models clearly show that the Thue-Morse lattice is intermediate between periodic and quasiperiodic lattices.