Rolf Riklund

THE THUE-MORSE APERIODIC CRYSTAL, A LINK BETWEEN THE FIBONACCI QUASICRYSTAL AND THE PERIODIC CRYSTAL

The electronic spectrum and eigenstates of a one-dimensional aperiodic Thue-Morse crystal isstudied with an on-site tight-binding model. The relation between the constructing elements andthe hierarchical splitting of the bands into subbands is analysed. The eigenstates are shown to be much more similar to those of a periodic crystal than those of a Fibonacci quasicrystal. We thus claim that the Thue-Morse aperiodic crystal is a link between the Fibonacci quasicrystal and theperiodic crystal, and that the study of non-Fibonaccian aperiodic crystals is a promising steptowards the desired unified theory of disordered, aperiodic and periodic systems. Since the experimentally studied MBE-grown aperiodic crystals typically has 5% fluctuation in layer thickness, we also investigate the density of states and eigenstates for a model system withfluctuating site-energies.

Optical properties of perfect and non-perfect quasi-periodic multilayers: a comparison with periodic and disordered multilayers

A detailed study is presented of the optical transmission through quasi-periodically layered media using a transfer matrix technique. The overall behaviour in multilayers arranged according to the Thue-Morse and Fibonacci sequences is found to be very similar. This is in strong contrast to the marked difference recently found in the electronic wavefunctions for the two corresponding aperiodic crystals in a tight-binding model. Other types of quasi-periodic multilayers are shown to have rather different transmission properties compared to the Thue-Morse and the Fibonacci layering. A thorough comparison with both periodic and disordered multilayers shows clearly that the quasi-periodic multilayers have quite different optical behaviour. When the number of layers is allowed to approach infinity the transmission coefficient is claimed to show a multifractal behaviour. Inspired by the recent interest in the effect of fluctuations in the layer thickness on the physical properties of quasi-periodic superlattices prepared with molecular beam epitaxy, the authors have also studied the sensitivity of the optical transmission to a few per cent variation in the thickness of the layers. The optical transmission is found to be less sensitive to fluctuations in the layer size in the quasi-periodic case than in the periodic case. Compared to randomly layered media on the other hand the transmission coefficient in the quasi-periodic multilayer case is more sensitive to small variations in the length of the layers.

Periodic and quasiperiodic wavefunctions in a class of one-dimensional quasicrystals: an analytical treatment

The one-dimensional, discrete Schrodinger equation is studied analytically for a class of quasiperiodic Hamiltonians known as the copper mean. The lattice is described by a recursion formula SL+1=SL-1SL-1SL, L=2,3, . . ., given the two initial sequences S0 and S1. Extended states are shown to exist for energies satisfying TrTL=0 and Tr((T0)-2T1)=2 where TL is the transfer matrix for the Lth generation of the quasicrystal. Also, periodic states are shown to exist quite generally in a subclass of the copper mean. A specific one-dimensional quasicrystal is given as an example of this, and is shown to have exclusively periodic states.

Renormalization of aperiodic model lattices: spectral properties

Many of the published results for one-dimensional deterministic aperiodic systems treat rather simplified electron models with either a constant site energy or a constant hopping integral. Here we ...

Solitary wave propagation in periodic and aperiodic diatomic Toda lattices

We investigate numerically how a solitary wave propagates in some one-dimensional diatomic periodic and aperiodic Toda lattices. It is found that a nearly stable wave of rather high amplitude can propagate in the periodic lattice. For several of the deterministic aperiodic sequences considered, the damping of the wave in the corresponding lattice is considerable less than for a random lattice. The short range correlation between the atoms in the aperiodic lattices seems to be of main importance for how much the wave is damped. We suggest therefore that the entropy according to Shannon might be a relevant measure for the properties of the lattices in this case. It is shown that this measure yields at least an approximative agreement with what is actually achieved by our numerical experiments. It is also shown that the earlier proposed idea of viewing the process as multiple scattering cannot be applied to other cases than random sequences with small mass-differences.

Using the Fourier spectrum to classify families of generalised extensions of the Fibonaccian lattice

The interest in non-Fibonaccian aperiodic crystals has increased during the last few years, since it appears that the physical properties of the Fibonaccian quasicrystal may in some fundamental aspects not be generic. As an example the property that all the electronic eigenstates in a tight-binding model for a Fibonaccian lattice are chaotic and correspond to a singular continuous spectrum is not shared by all aperiodic crystals. Gumbs and Ali (1988) have recently studied the trace map of the two families given by the inflational rule A to ABB. . .B, B to with n Bs and A to AAA. . .AB, B to A with n As. They conjectured that these two families, which for n greater than one are both extensions of the Fibonaccian sequence, should have rather different physical properties. The authors confirm their conjecture by showing that the structure of the Fourier spectra of the two families are quite different when n is greater than one. The Fibonaccian sequence is found to be much more like the members of the second family than the first one. This classification is then related to the definition of a quasicrystal using the diffraction criterion of Bombieri and Taylor (1986), and the member with n equal to two in the first family is shown to be a marginal case.

Extended spin waves in a one-dimensional ferromagnetic aperiodic crystal

Since the character of the low-lying spin excitations in a one-dimensional disordered system is still to some extent under debate, and the quasi-crystals can, in some sense, be said to fill the space between the disordered and periodic systems, the authors consider it of fundamental interest to study the character of the excited states in a one-dimensional deterministic aperiodic spin system. In contrast to the disordered case they find that the Thue-Morse aperiodic chain also has extended spin waves in the high-energy region.

Deterministic Aperiodic One-Dimensional Systems with All States Extended,One of Which is Periodic

A one-dimensional discrete tight-binding model with nearest-neighbour interaction is studied. We use the transfer model with variable hopping matrix elements, here assuming the two values t or - t , and constant on-site potential. Under this conditions all the eigenstates are known to be extended. It is shown that if the distribution of the off-diagonal matrix elements constitutes a deterministic aperiodic sequence, the eigenstate corresponding to the middle eigenvalue is periodic for some choices of the sequence, but not for all. The studied sequences that turn out to have a periodic middle state are the Thue-Morse sequence, the Rudin-Shapiro sequence and many of the generalised Thue-Morse sequences but not for instance the well known Fibonacci sequence.

Extension of the hopping range in incommensurate systems can generate localised states

The effect of hopping beyond nearest neighbours on the electronic properties of incommensurate systems is discussed very little in the literature even though for example structural phase transitions from commensurate to incommensurate systems are impossible without interactions beyond nearest neighbours. The authors show here that extending the range of hopping in the well known Aubry model to include next-nearest-neighbour hopping gives rise to localised states in regions where the original model with only nearest-neighbour hopping gives all states extended. A short discussion of the relation of this result to the Aubry duality and the Soukoulis-Economou model is given.

Metal-insulator transition and self-similarity in the Aubry model

The Aubry model near the critical region is analysed with the numerical renormalisation group method. The transition is shown to be energy dependent and the self-similarity is demonstrated.