Vibrations in one-dimensional hybrid Fibonacci/periodic structures

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1 REVISTA MEXICANA DE FÍSICA S 53 (7) DICIEMBRE 2007 Vibrations in one-dimensional hybrid Fibonacci/periodic structures A. Montalbán Departamento de Ciencia y Tecnología de Materiales, División de Óptica, Universidad Miguel Hernández, Elche, Spain. V.R. Velasco Instituto de Ciencia de Materiales de Madrid, CSIC, Sor Juana Inés de la Cruz 3, Madrid, Spain. J. Tutor Advanced Products España, S.L. Av. de la Industria, Alcobendas, Madrid, Spain. F.J. Fernández-Velicia Departamento de Física de Materiales, Universidad Nacional de Educación a Distancia, Senda del Rey 9, Madrid, Spain. Recibido el 30 de noviembre de 2006; aceptado el 8 de octubre de 2007 We have studied the vibrational frequencies and atom displacements of one-dimensional systems formed by combinations of Fibonacci quasiperiodic 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 Fibonacci generations. 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. Keywords: Multilayer systems; quasi-periodic structures, phonons. Se han estudiado las frecuencias de vibración y los desplazamientos atómicos de sistemas unidimensionales formados por combinaciones de apilamientos cuasi-periódicos de Fibonacci y bloques periódicos. Los materiales se describen mediante modelos de constantes de fuerza a primeros vecinos y sus correspondientes masas atómicas. Estos sistemas presentan diferencias en sus espectros de frecuencia con respecto a los de las generaciones de Fibonacci. El rasgo más importante es la presencia de confinamiento de las vibraciones en uno de los bloques que forman la estructura compuesta, para diferentes rangos de frecuencia. Descriptores: Multicapas; estructuras cuasi-periódicas; fonones. PACS: m; k 1. Introduction Quasi-regular structures have been intensively studied after the discovery of the quasicrystals [1 3]. These structures do not appear in nature, but can be produced in the laboratory [4 7] by molecular beam epitaxy (MBE) techniques. The interest of these systems was increased after the predictions that they would exhibit peculiar electron and phonon critical states and highly fragmented fractal energy spectra [8 15]. Many works have been devoted to study the 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 quasiregular or aperiodic order in science and technology can be found in a recent review [18]. It is also important to find out if these materials can exhibit some additional physical characteristics or better performances than the periodic structures for specific applications. This has been found 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 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

2 VIBRATIONS IN ONE-DIMENSIONAL HYBRID FIBONACCI/PERIODIC STRUCTURES 13 have systems with primary and secondary gaps in different frequency ranges. By combining them, it could be possible to modify the frequency spectrum, and thus the vibrational properties of the resulting system as compared to those of the constituent quasi-regular systems. These are the structures to be studied here. 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 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 3]. The theoretical model and method of calculation are presented in Sec. 2. Section 3 deals with the results for the hybrid systems. Conclusions are presented in Sec Theoretical model and method of calculation We shall consider systems formed by combining a Fibonacci sequence, let us say ABAAB, and a periodic one ABABAB. 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 [28] and finite stage Si-Ge Fibonacci superlattices [29], longitudinal phonons of alkali-metal graphite intercalation compounds [30] compared with the corresponding neutron scattering data [31]. 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 are the force constants k A, k B and the atom masses m A, m B, corresponding to the bulk materials A and B, respectively. The interactions between both materials will be represented by a force constant k I, which we shall take as the mathematical average of both force constants k A and k B, k I = (k A + k B )/2, without loss of generality. Different choices of k I would modify numerical values in the frequency spectrum but not the overall physical picture. 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 [32] and superlattices [33]. The force constants for the bulk materials calculated from their elastic constants together with the atom masses are given in Table I. TABLE I. Force constants (k) and atom masses (m) of Al and Ag for the dynamic model discussed in the text. Material k (dyne/cm) m (g) Al Ag Our building blocks will be: (i) a finite periodic repetition of blocks A-B formed by N(N A + N B ) atoms, where N is the number of A-B periods and N A (N B ) is the number of atoms of material A (B) included; (ii) a finite Fibonacci generation grown by recursive stacking with generator blocks A and B, mapping the mathematical rule in the Fibonacci sequence. We consider our structures as the period of a polytype superlattice, with N A =4 and N B =3, and obtain the eigenvalues by means of a direct diagonalization. The eigenvectors (atom displacements) are obtained by using the method described in Refs. 34 to 36 for block-tridiagonal matrices. Other different boundary conditions, as the case with the extreme free atoms, give essentially the same frequency spectrum with the addition of some frequencies in the gaps corresponding to localized modes, as it could be expected. 3. Hybrid Fibonacci systems The Fibonacci systems are produced by stacking recursively with two generator blocks A and B, mapping the mathematical rule in the Fibonacci sequence S 1 = {A}, S 2 = {AB} S 3 = {ABA},, S n = S n 1 S n 2, (1) In Fig. 1a we present the frequency spectrum versus the order number for a periodic system formed by the repetition of the AB blocks 49 times. Figure 1b presents this information for a tenth order Fibonacci generation (composed of 220 A atoms and 102 B atoms). Figure 1c presents the same information for a hybrid structure having the former Fibonacci generation sandwiched between two blocks, each one formed by the periodic repetition of the AB blocks 49 times. Figure 1d gives the same information for a hybrid structure formed by the former periodic block sandwiched between two tenth order Fibonacci generations. One can see how the addition of the periodic block modifies the frequency spectrum of the Fibonacci generation. The fragmentation of the spectrum is kept, but the gaps at middle and intermediate frequencies are modified, in such a way that we can hope that some effects on the vibration patterns will also be affected. Only from the spectra presented in the above figures, it is not possible to ascertain if some new features will be present in these structures. The atom displacements are important in the analysis of spectroscopic experiments [37]. To see if

3 14 A. MONTALBA N, V.R. VELASCO, J. TUTOR, AND F.J. FERNA NDEZ-VELICIA some special feature of the vibrational spectrum of the hybrid structures exists, we shall look now to the atom displacements of some frequencies in different ranges of the spectrum. Figure 2 gives information on the system formed by combining a periodic repetition of 49 AB blocks sandwiched between two tenth order Fibonacci generations. In Fig. 2a we present the frequency spectrum, whereas in the other panels F IGURE 1. Frequency eigenvalues versus their ordering number for: (a) a periodic system formed by the repetition of the AB blocks 49 times; (b) a tenth order Fibonacci generation; (c) a hybrid structure having the former Fibonacci generation sandwiched between two blocks, each one formed by the periodic repetition of the AB blocks 49 times; (d) a hybrid structure formed by the former periodic block sandwiched between two tenth order Fibonacci generations. F IGURE 2. (a) Frequency eigenvalues versus their ordering number for the hybrid structure formed by the periodic repetition 49 times of the AB blocks sandwiched between two tenth order Fibonacci generations (solid horizontal lines indicate the frequencies whose atom displacements are represented in the figure). Normalized atom displacement u(n) of the former multi-layer structure versus the atom order number n for the frequencies: (b) ω= Hz; (c) ω= Hz; (d) ω= Hz. we present the atom displacement versus the atom number. Figure 2b corresponds to ω= Hz. This frequency belongs to the Fibonacci frequency spectrum and not to that corresponding to the periodic structure. It is seen now that only the atoms in the left block corresponding to the tenth Fibonacci generation exhibit non-negligible atom displacements. Figure 2c corresponds to ω= Hz. This frequency does not belong to the Fibonacci frequency spectrum, but to that corresponding to the periodic structure. It is seen now that only the atoms in the periodic block at the center exhibit non-negligible atom displacements and a notable regularity. Figure 2d corresponds to ω= Hz. This frequency belongs to the Fibonacci frequency spectrum and not to that corresponding to the periodic structure. It is seen now that only the atoms in the right block corresponding to the tenth Fibonacci generation exhibit non-negligible atom displacements. The regularity of the atom displacements in the periodic block can be noted. Figure 3 presents information for the structure formed by combining a tenth Fibonacci generation sandwiched between two periodic repetitions of 49 AB blocks. Figure 3a gives the frequency spectrum, and the remaining panels give the atom displacement versus the atom number. Figure 3b corresponds to ω= Hz. This frequency belongs to the periodic structure frequency spectrum, and we can see how the displacement is essentially confined to the left periodic block. Figure 3c corresponds to ω= Hz. This frequency corresponds to the Fibonacci frequency spectrum. It is seen now that only the atoms in the central block corresponding to the Fibonacci structure exhibit nonnegligible atom displacements. Figure 3d corresponds to F IGURE 3. (a) Frequency eigenvalues versus their ordering number for the hybrid structure having the tenth order Fibonacci generation sandwiched between two blocks, each one formed by the periodic repetition of the AB blocks 49 times (solid horizontal lines indicate the frequencies whose atom displacements are represented in the figure). Normalized atom displacement u(n) of the above multi-layer structure versus the atom order number n for the frequencies: (b) ω= Hz; (c) ω= Hz; (d) ω= Hz. Rev. Mex. Fı s. S 53 (7) (2007) 12 16

4 VIBRATIONS IN ONE-DIMENSIONAL HYBRID FIBONACCI/PERIODIC STRUCTURES 15 present in the hybrid structures, including only periodic systems. 4. Conclusions FIGURE 4. Normalized atom displacement u(n) of the hybrid structure having the tenth order Fibonacci generation sandwiched between two blocks, each one formed by the periodic repetition of the AB blocks 49 times for the frequencies: (a) ω= Hz; (b) ω= Hz. ω= Hz. This frequency belongs to the periodic structure frequency spectrum, and we can see how the displacement is essentially confined to the right block corresponding to the periodic structure. The regularity of the atom displacements in the periodic blocks is easily seen. Other modes can exhibit different displacement patterns as those shown in Fig. 4 for the modes of the structure formed by combining a tenth Fibonacci generation sandwiched between two periodic repetitions of 49 AB blocks with frequencies ω= Hz and ω= Hz. We can see here how these modes can propagate along the whole structure. Other patterns seen in aperiodic systems, such as the critical localization, are also present in the frequency spectrum of the hybrid structure. The separate confinement of the displacements seen here is also possible in hybrid structures combining different periodic superlattices having different gaps. Hybrid structures including aperiodic sequences open new possibilities due to the spectrum fragmentation not We have studied the vibrational frequencies and atom displacements of structures formed by sandwiched periodic- Fibonacci-periodic, or Fibonacci-periodic-Fibonacci 1D heterostructures. In all the cases, we have seen modifications in the frequency spectrum specially in the primary and secondary gaps. The most notable feature found in these structures is the existence of modes in different frequency ranges exhibiting atom vibrations confined to only one of the sequences forming the total structure, as in the case of normal cavities. This selective confinement of the atom vibrations is achieved with structures formed by two different materials only, but with the interplay of different orders at different length scales. These structures could be useful, at least in principle, in filtering and guiding systems. This effect found at the center of the Brillouin zone presents similarities with the surface, avoiding waves recently found in periodic superlattices on a substrate [38]. There are also similarities with the properties seen in the light propagation through Fibonacci quasicrystals [39, 40]. Acknowledgments This work was partially supported by the Spanish Ministry of Science and Technology through Grant MAT D. Shechtman, I. Blech, D. Gratias, and J.W. Cahn, Phys. Rev. Lett. 53 (1984) D. Shechtman and I. Blech, Metall. Trans. A 16 (1985) A.I. Goldman and R.F. Kelton, Rev. Mod. Phys. 65 (1993) R. Merlin, K. Bajema, R. Clarke, F.-Y. Juang, and P.K. Bhattacharya, Phys. Rev. Lett. 55 (1985) J. Todd, R. Merlin, R. Clarke, K.M. Mohanty, and J.D. Ax, Phys. Rev. Lett. 57 (1986) R. Merlin, K. Bajema, J. Nagle, and K. Ploog, J. de Physique 48 (1987) C A.A. Yamaguchi et al., Solid State Commun. 75 (1990) S. Ostlund and R. Pandit, Phys. Rev. B 29 (1984) M. Kohmoto, B. Sutherland, and C. Tang, Phys. Rev. B 35 (1987) E. Maciá and F. Domínguez-Adame, Phys. Rev. Lett. 76 (1996) E. Maciá, Phys. Rev. B 60 (1999) M. Kohmoto, L.P. Kadanoff, and C. Tang, Phys. Rev. Lett. 50 (1983) A. Süttő, J. Stat. Phys. 56 (1989) J. Bellissard, B. Iochum, E. Scoppola, and D. Testard, Commun. Math. Phys. 125 (1986) E. Maciá and F. Domínguez-Adame, Semicond. Sci.Technol. 11 (1996) E. Maciá and F. Domínguez-Adame, Electrons, Phonons and Excitons in Low Dimensional Aperiodic Systems (Madrid, Editorial Complutense, 2000). 17. E.L. Albuquerque and M.G. Cottam, Phys. Rep. 376 (2003) 225; Polaritons in Periodic and Quasiperiodic Structures (Elsevier, Amsterdam, 2004). 18. E. Maciá, Rep. Prog. Phys. 69 (2006) S.N. Zhu et al., Phys. Rev. Lett. 78 (1997) Y.B. Chen, C. Zhang, Y.Y. Zhu, H.T. Wang, and N.B. Ming, Appl. Phys. Lett. 78 (2001) W. Gellermann, M. Kohmoto, B. Sutherland, and P.C. Taylor, Phys. Rev. Lett. 72 (1994) T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, Phys. Rev. B 50 (1994) E. Maciá, Phys. Rev. B 63 (2001)

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