Тысячи-1 / Selective confinement of vibrations in composite systems

Selective confinement of vibrations in composite systems

with alternate quasi-regular sequences

A. Montalba. na, V.R. Velascob,_, J. Tutorc, F.J. Ferna. ndez-Veliciad

Abstract

We have studied the atom displacements and the vibrational frequencies of 1D systems formed by combinations of Fibonacci,

Thue–Morse and Rudin–Shapiro quasi-regular stacks and their alternate ones. The materials are described by nearest-neighbor force

constants and the corresponding atom masses, particularized to the Al, Ag systems. These structures exhibit differences in the frequency

spectrum as compared to the original simple quasi-regular generations but the most important feature is the presence of separate

confinement of the atom displacements in one of the sequences forming the total composite structure for different frequency ranges.

1. Introduction

Quasi-regular structures have been intensively studied from the moment of the discovery of the quasi-crystals [1–3]. 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 [4–11]. Real samples of these systems have been obtained in the laboratory as multilayer systems [12–15] by molecular beam epitaxy (MBE) techniques. More properties and features of these systems in different fields of the science and technology can be found in a recent revision paper [16]. Many works have been devoted to the study of the basic properties of quasi-regular systems, as it can be seen in Refs. [17,18]. 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. The peculiar characteristics of these systems come from the interplay of these two different orders. Many of these studies on quasi-regular systems have been devoted to the character of the spectrum. In order to do this the limit N !1 must be considered, N being the number of the atomic layers forming the structure. It must be also noted that real systems are finite and that the properties of these systems must be studied with more realistic models than those simple models usually employed in the study of the nature of the spectrum [17,18]. The characterization of the kind of spectrum of the different structures is an important aspect, but more important is to find if these structures can present some additional physical characteristics or better performances than the periodic structures for specific applications. This has been realized in the optical capabilities of quasiregular 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 quasi-regular blocks have been found to exhibit complementary optical responses [23] and separate confinement of the atom displacements in the different parts of the hybrid structure for given frequency ranges [24]. Perfect optical transmission has been found in symmetric Fibonacci- class multilayers [25,26]. The vibrational spectrum of quasi-regular structures presents a highly fragmented character [8,11,24]. 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 them 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 systems. By combining alternate sequences we are further modifying the different orders in the total structure as compared with those present in each constituent sequence. These are the structures to be studied here. We must consider that in the theoretical and experimental studies one does not consider a full infinite quasi-regular sequence S1, but reaches only up to some high, but finite generation SN, after repeated application of the generating rule. It is then a reasonable assumption to expect that the systems for sufficiently high N will display the essential features of the ideal infinite sequence. This poses some problems in order to study these systems with realistic models, because of the size of the systems to be studied. Thus, it can prove convenient to make a first study with simple physical 1D models to see if some additional characteristic can be found. It would then be reasonable to make a more realistic study later. We shall maintain in our study the basic simplicity employed in the big majority of calculations [17], thus employing 1D linear chains while keeping all the basic physical ingredients in the model. We shall center on the Fibonacci systems, because they are the most studied ones and they can serve as 1D realizations of the quasi-crystals [1–3], but we shall consider also those systems obtained by means of the Thue–Morse [27] and Rudin–Shapiro [28] sequences. The theoretical model and method of calculation are presented in Section 2. Section 3 deals with the Fibonacci systems. The results for the Thue–Morse and Rudin– Shapiro systems are presented in Section 4. Conclusions are presented in Section 5. 2. Theoretical model and method of calculation We shall study systems formed by combining a quasiregular sequence, let us say ABAAB, and its alternate one BABBA. The total system would then be ABAAB— BABBA. We shall use the simplest model enabling us to get the essential physical data. Thus, we shall consider 1D linear chains with nearest-neighbor interactions. In spite of its simplicity this kind of model has been applied to the study of the properties of real materials with good results: theoretical analysis of Raman spectra of ultrathin Si–Ge superlattices [29] and finite stage Si–Ge Fibonacci superlattices [30], longitudinal phonons of alkali–metal graphite intercalation compounds [31] compared with the corresponding neutron scattering data [32]. This model can describe in a simple, but reasonably realistic way, the longitudinal phonons of a system in which two different materials A and B form the generating blocks of the different structures. The basic requirements will be the force constants kA, kB and the atom masses mA, mB, corresponding to the bulk materials A and B, respectively. The interactions between both materials will be represented by a force constant kI, which we shall take as the mathematical average of both force constants kA and kB, kI . .kA . kB.=2, without loss of generality. The minimum of requirements for the analysis of the problem is thus satisfied. We shall specialize our model to the case of metals Al (medium A) and Ag (medium B). They have a very good lattice-parameter matching (within 0.3%) and they can be grown forming good quality interfaces [33] and superlattices [34]. The force constants for the bulk materials calculated from their elastic constants together with the atom masses are given in Table 1. We consider our structures as the period of a polytype superlattice, with different values of NA and NB (Ni being the number of atoms of material i . A; B), and obtain the eigenvalues by means of a direct diagonalization. The eigenvectors (atom displacements) are obtained by using the method described in Refs. [35–37] for block-tridiagonal matrices.