Discrete aperiodically ordered structures in low dimensions

Abstract: This thesis deals with discrete models for one-dimensional (1D) and two-dimensional (2D) aperiodic systems such as quasicrystals and aperiodic superlattices. Various model systems are investigated using different forms of the tight-binding (TB) approximation. The electronic energy-spectra and corresponding wavefunctions are studied, as well as the dynamical behaviour of wave packets, obtained from, for example, an added nonlinearity.The aperiodicity enters in most of the models either by the order of the on-site potentials or the hopping matrix elements (the transfer model). The order is in the studied deterministic systems in most cases determined by different binary sequences such as the Thue-Morse sequence, the Rudin-Shapiro sequence, the Fibonacci sequence, etc. In the only nondeterministic structure treated the order is determined according to a sequence of correlated random dimers.First a numerical study of the dynamical properties of solitons in a non-linear lattice is performed. The aperiodicity enters through an on-site potential chosen according to the Thue-Morse sequence. The influence of the speed of the soliton on the dispersion, radiation and trapping in the lattice is studied. Examples of soliton collisions are shown as well. In this study the used model is equivalent with the discrete non-linear Schrodinger equation (DNLS).The DNLS is also studied when the on-site potential is obtained from the asymmetric multifractal third fourth Cantor set (a variation of the middle third Cantor set). In the linear limit the eigenvalue spectrum and eigenvectors are successfully compared to those of a set of isolated wells of different lengths. When non-linear, the lowest and highest modes in the largest well show solitonlike character. We show examples of soliton dynamics with and without an additional external linear potential corresponding to a constant applied external field.The nature of the eigenstates is studied using the transfer model with variable off-diagonal matrix elements, assuming the two values t or -t. It is proved that if the distribution of the matrix elements constitutes a deterministic aperiodic sequence, the eigenstate corresponding to the middle eigenvalue is periodic for e.g. the Thue-Morse sequence, the Rudin-Shapiro sequence and many of the generalised Thue-Morse sequences, but not for instance the wellknown Fibonacci sequence. Somewhat remarkable is that even a distribution as correlated random dimers not only has all eigenstates extended but on top of that gives rise to a periodic middle eigenstate.When aperiodically ordered the entities having the specific order are all binary valued. In one study however the on-site potentials are obtained by adding Fourier components with different phases and amplitudes. The resulting potential barrier is used to discriminate between wave packets depending on their velocities. With the given examples as guidelines it is possible to design filters for plane waves and wave packets, with known spatial form, with almost any location of the passband(s) in the transmission coefficient.The structures above are all 1D. But the effects on the Hofstadter butterfly of different distributions of on-site potentials are numerically studied in a 2D model where the influences of an external magnetic field are included through the so-called Peierls substitution.