Selective spatial localization of the atom displacements in one-dimensional hybrid quasi-regular (Thue–Morse and Rudin–Shapiro)/periodic structures

Abstract

We have studied the vibrational frequencies and atom displacements of one-dimensional systems formed by combinations of Thue–Morse and Rudin–Shapiro quasi-regular stackings with periodic ones. The materials are described by nearest-neighbor force constants and the corresponding atom masses. These systems exhibit differences in the frequency spectrum as compared to the original simple quasi-regular generations and periodic structures. The most important feature is the presence of separate confinement of the atom displacements in one of the parts forming the total composite structure for different frequency ranges, thus acting as a kind of phononic cavity.

Introduction

After the discovery of the quasicrystals [1], [2], [3] quasi-regular structures have been intensively studied. These structures do not appear in the nature, but can be produced in the laboratory as multilayer systems [4], [5], [6], [7] by molecular beam epitaxy (MBE) techniques. The interest on these systems was increased after the predictions that they would exhibit peculiar electron and phonon critical states and highly fragmented fractal energy spectra [8], [9], [10], [11], [12], [13], [14], [15]. Many works have been devoted to the study of the basic properties of quasi-regular structures, as it can be seen in [16], [17]. These systems can be characterized by the presence of two different orders at different length scales. The periodic order of the crystalline arrangement of atoms in each layer is present at the atomic level, whereas the quasi-regular order due to the disposition of the different atomic layers following a given building sequence is the main feature at the long scale. This order is artificially produced during the growth process and is carefully controlled. Because the relevant physical scales influence different physical phenomena it is possible, in principle, to exploit the quasi-regular order introduced in the system by tuning the corresponding length scales, thus opening new possible applications. The peculiar characteristics of these systems come from the interplay of these two different orders. Aspects of the role of this quasi-regular or aperiodic order in science and technology can be found in a recent review [18].

These systems present some additional physical characteristics or better performances than the periodic structures for specific applications. This has been realized in the optical capabilities of quasi-regular systems concerning second- [19] and third-harmonic generation [20], as well as the localization of light in these systems [21], [22]. Hybrid-order devices formed by periodic and Fibonacci quasi-regular blocks have been found to exhibit complementary optical responses [23]. Perfect optical transmission has been found in symmetric Fibonacci-class multilayers [24], [25]. Broad omnidirectional reflection bands have been predicted when combining Fibonacci sequences and periodic 1D photonic crystals [26].

The vibrational spectrum of quasi-regular structures presents a highly fragmented character [12], [15], [27]. By using different materials as the starting ones we can have different realizations (ABAAB…, BABBA…, etc.), and thus we can have systems with primary and secondary gaps in different frequency ranges. By combining these systems with finite periodic structures it could be possible to further modify the frequency spectrum, and perhaps the vibrational properties of the resulting system as compared to those of the constituent quasi-regular and periodic systems. These are the structures to be studied here. They can be considered as a more complicated variation of the systems used in phonon cavities [28], [29], [30], where a spacer layer is sandwiched between two finite superlattices. It remains to be seen if these hybrid systems possess equivalent properties. In order to describe the properties of real quasi-regular systems it is necessary to describe these structures with enough physical realism in spite of the simplicity of the models.

We shall maintain in our study the basic simplicity employed in the big majority of calculations [16], thus employing 1D linear chains while keeping all the basic physical ingredients in the model. The Fibonacci systems are the most studied ones because they can serve as 1D realizations of the quasicrystals [1], [2], [3]. Instead we shall consider here those systems obtained by means of the Thue–Morse [31] and Rudin–Shapiro [32] sequences, that are less frequently considered. In particular Rudin–Shapiro sequences exhibit peculiar characteristics and are not covered by the theorems valid for other quasi-regular sequences [13].

The theoretical model and method of calculation are presented in Section 2. Section 3 deals with the results for the Thue–Morse and Section 4 with those of the Rudin–Shapiro systems. Conclusions are presented in Section 5.