The quasi-regular systems have been intensively studied in the last years [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35] and [36]. The interest was triggered from the theoretical side by the prediction that these systems should manifest non-conventional electron and phonon states [9], [11], [23] and [25], and exhibit energy spectra with a high fragmentation and fractal character [7], [17] and [24]. The experimental growth of Fibonacci [2] and [3] and Thue–Morse [4] multilayer structures has provided the practical realization of these systems. The Fibonacci system, a linear lattice constructed recursively, is the one-dimensional (1D) version of the quasi-crystals [5], [6], [37] and [38], and has been the subject of many theoretical studies. The electronic structure of the Fibonacci system has been investigated mainly in the single-band tight-binding limit. In these studies, it was found that the energy spectrum is self-similar, in the sense that the energy bands divide into three subbands, each of which further subdivides into three, and so on [13], [14], [15] and [16], thus producing a singular continuous spectrum [21], which in the infinite limit reduces to a Cantor-like spectrum with dense energy gaps everywhere [7], [8], [9] and [10]. More realistic studies, using an empirical tight-binding (ETB) sp3s* Hamiltonian [39], were presented in [28], [29], [30], [31], [32], [33] and [34]. In these works, the Fibonacci spectrum was only estimated for certain energy ranges and for wave vectors in the vicinity of the superlattice Γ point.
We delve here into the properties arising from the simple models, by considering possible variations in the basic structure of these models, as a parallel to the more sophisticated and realistic sp3s* Hamiltonians.
In Section 2, a brief description of the theoretical models is given and conclusions are provided in Section 3.
