Physica A 362 (2006) 289–294
A multifractal analysis of optical phonon excitations
in quasicrystals
D.H.A.L. Anselmo
a,b, A.L. Dantas
a,c, E.L. Albuquerque
c,aDepartamento de Fı´sica, Universidade do Estado do Rio Grande do Norte, 59600-900, Mossoro´-RN, Brazil bDepartamento de Fı´sica, Universidade Federal do Ceara´, Campus do Pici, Fortaleza-CE, 60455-760, Brazil
cDepartamento de Fı´sica, Universidade Federal do Rio Grande do Norte, Natal-RN, 59072-970, Brazil
Received in revised form 21 April 2005 Available online 8 September 2005
Abstract
It is shown that the allowed frequencies of optical phonon modes propagating in quasiperiodic AlN–GaN structures display a multifractal behavior. This feature is confirmed through a scaling-exponent analysis of a parametric spectrum of singularities f ðaÞ. This is done by using an alternative counting method, avoiding the use of a Legendre transform, which could lead to distortions. The Cantor-like spectrum is also investigated, and scaling properties are shown for Fibonacci, Thue–Morse and double-period quasiperiodic superlattices.
r2005 Elsevier B.V. All rights reserved.
Keywords: Optical phonons; Multifractals; Superlattices
1. Introduction
The use of multifractal and monofractal analysis has proved to be an important tool to simulate (in many cases with striking agreement with experimental data) real systems, providing their characterization and describing the dynamics of the events occurring on them[1]. Their main concepts were introduced in the early 80s, focusing on the study of complex structures ranging from problems of aggregation to the behavior of chaotic dynamical systems[2,3].
On the other hand, the discovery of quasicrystals by Shechtman et al.[4]has attracted a lot of attention, both theoretically and experimentally (for an up to date account of this subject see Ref.[5]). One reason for that is because quasiperiodic systems have a unique feature, namely, they can be considered as intermediate systems between a periodic crystal and the random amorphous solids, thus defining a novel description of disorder. While periodic potentials lead to continuous spectra and extended eigenstates, random potentials lead to pure point spectra and exponentially localized states, and this last feature is strongly observed in the present study. A practical way to simulate these crystals consists of considering artificial quasiperiodic
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multilayers, with well defined rules of growth. The quasi-periodicity of such structures is a consequence of the substitutional aspect of these rules.
In this paper, we study the multifractal profile of optical phonons’ spectra in quasiperiodic multilayers, which follow the Fibonacci (FB), Thue–Morse (TM) and double-period (DP) mathematical sequences. These quasiperiodic structures can be generated by their inflation rules (in what follows, A and B are building blocks characterized by the geometric thicknesses da and db, respectively): A ! AB, B ! A (Fibonacci sequence);
A ! AB, B ! AA (double-period sequence); and A ! AB, B ! BA (Thue–Morse sequence). Although several theoretical techniques have been used to study the propagation of collective excitations in these structures, in the present work we make use of the transfer matrix approach to analyze the superlattice modes, simplifying the algebra which would be otherwise quite involved.
Fractality is in general a common property of strange attractors in nonlinear systems[6]. Multifractals differ mainly from monofractals in the sense that they need an infinite set of exponents (rather than a single exponent that suffices for a fractal) to characterize their spectra. In this sense, one must introduce the Re´yni dimensions[7](actually, there is a large number of different definitions of dimension[8,9]), which are a family of dimensions of order q (where q is any real number). Also, they arise mainly from multiplicative processes, and while fractal analysis looks at the geometry of a pattern, multifractal analysis concentrates on the arrangement of measurable physical as well as biological, chemical, etc., quantities on a particular system. In simple words, a multifractal can be defined as an object with two or more scaling regions of different fractal dimensions.
It has been proved for different kinds of collective excitation’s propagating modes in quasiperiodic systems, that their energy spectra’s bandwidth for the bulk solutions obey a linear scale law, which is a typical signature of a monofractal system (for details see Ref.[10]). The aim of our paper is to show that, besides the already much fragmented spectrum of the confined optical phonon modes, the set of solutions for these modes propagating in quasiperiodic structures is in fact a multifractal object, characterized by their f ðaÞ spectrum. In this context, the study of the f ðaÞ function is very important: it describes the distribution of different fractal dimensions of the object upon variation of the singularities of strength a[11].
The outline of this paper is as follows: in the next section, we present the main equations related with the theory of multifractals. The numerical results and the conclusions of the paper are presented in Section 3. 2. General theory
A traditional method employed to determine the multifractal spectra, consists in defining a set of generalized dimensions Dq, whose associated spectrum of singularities f ðaÞ is obtained through a Legendre
transform [12]. Both of these quantities can be used to determine the whole spectra. The process starts by constructing a partition function wq¼Pip
q
i. The curve Dq versus q is then defined by the expression
Dq¼ 1 q 1N!1lim ln wq ln N , (1) where D1¼ lim N!1 P ipiln pi ln N . (2)
Here pi¼Rboxdm denotes the integrated measure on the ith (i ¼ 1; 2;. . . N) cube of the support for the individual measure m. Such cube belongs to a grid that covers the set and has a linear size ¼ 1=N.
The singularity strength of the measure a is defined by aðxÞ ¼ lim N!1 ln pðxÞ ln N , (3)
where pðxÞ is the same integral defined above, but evaluated over a box with center in x. The f ðaÞ function is then determined by the relation
with ! 0. In this way, Nða; Þ is the number of boxes of linear size , with singularity strength lying between a and a þ da. To evaluate the multifractal spectra, one should first obtain the measures from real or computer experiments, and then use Legendre transform. The difficulty of this method lies on the Legendre transform itself, depending on the system considered, and on eventual discontinuities that can arise on the f ðaÞ curves [13].
A different approach for this problem was introduced by Chhabra and Jensen[14], where the authors link the Hausdorff dimension of the support to the generalized dimensions, by following some relationships between thermodynamic and multifractal formalisms [15]. This approach is used in the present work to analyze the multifractal spectra of optical phonons in quasiperiodic superlattices.
The first suggestion that the spectra of the optical phonons in these quasiperiodic structures is a fractal comes from their scaling behaviors. In analogy with the behavior observed in spin waves[16], such modes have a linear scale behavior when one consider the logarithm of the sum of the allowed bandwidth against the logarithm of the growth rate of the desired sequence [17]. By defining D ¼Pidoi (the Lebesgue measure of
the energy spectrum), we will show that this quantity obey a power law, defined by a basis and a scaling exponent. For FB superlattice, this basis is equal to FN, the corresponding Fibonacci number. For the TM
and DP sequences, the basis is equal to 2N, where N is the generation index of the sequences. Using the substitutional rules of the quasiperiodic structures considered here, it can be easily proved that the scale law for FB superlattice is DðFNÞd, while for Thue–Morse and double-period sequence it is Dð2NÞd, where d,
the scale exponent, is the diffusion constant of the spectra [10]. Furthermore, we observed that the linear scaling behavior has another feature: it seems to increase linearly with the reduced in-plane wavevector K ¼ kxda, da being the thickness of the building block A, for a limited region of K. The above results and the
details of the numerical calculations will be considered in the next section. 3. Numerical results and conclusions
To set up our quasiperiodic structure, we consider two different building blocks A and B, which are arranged following a Fibonacci, Thue–Morse or double-period sequence, having thicknesses da and db,
respectively. We consider also that the optical-phonon propagation is along the xy-plane, with the optical axis c coinciding with the cartesian axis z. We use physical parameters modelling the nitride semiconductors AlN (building block A) and GaN (building block B). Their hexagonal wurtzite structure allows the propagation of three bulk polar optical-phonons, among the nine optical-phonon modes, which are Raman and infrared active in the irreducible representation of A1ðzÞ (z-axis) and E1ðxyÞ (xy-plane) at the G point. Two of them are
extraordinary waves associated with z- and xy-polarized vibrations. The z-polarized mode has A1ðzÞ
symmetry, while the xy-polarized one has E1ðxyÞ symmetry. The other one is an ordinary wave, which is
always transverse and polarized in the xy-plane, with E1ðxyÞ symmetry[18]. The artificial structures (governed
by their respective mathematical sequences) have a strong influence on the localization of the optical-phonon propagating modes, defining a Cantor-like feature. Fig. 1(a) describes the reduced allowed bandwidths, defined by O ¼ o=o0 (with o0¼oTO;A1ðGaNÞ) as a function of the generation index N, for optical phonon modes propagating on the quasiperiodic FB structure, taking a fixed value of the reduced in-plane wavevector (namely kxda¼1:0). The dashed horizontal lines correspond to the resonant frequencies (in units of o0) given
by
o1¼oTO;E1ðGaNÞ; o2¼oTO;A1ðAlNÞ; o3¼oTO;E1ðAlNÞ, (5)
o4¼oLO;A1ðGaNÞ; o5¼oLO;E1ðGaNÞ; o6¼oLO;A1ðAlNÞ, (6)
o7¼oLO;E1ðAlNÞ. (7)
Here oTO;X (oLO;X), with X ¼ A1ðzÞ, E1ðxyÞ, is the transverse optical (longitudinal optical) phonon angular
frequency for the mode X. The physical parameters are given elsewhere [17]. It can be seen clearly the localization of the modes, when one goes to higher values of the sequence generation. InFig. 1b we show the allowed bandwidth of the confined spectrum for the double-period sequence, considering the same physical parameters as before, except that here we take kxda¼0:6. As we can see, there exist forbidden regions in both
spectra, limited by the o3 and o4 reduced resonant frequencies, which separate their lower and higher
frequency regions. If one analyzes both higher and lower allowed regions separately, one can observe (for both FB and DP superlattices) that the Cantor-like feature is preserved. This remind us the concept of self-similarity, an universal behavior of a fractal. We do not show the spectrum for the TM sequence, because it is qualitatively similar to the DP sequence.
We now define our measure, namely the normalized local allowed bandwidth ðDiÞ, i.e.,
xi¼
Di
P
iDi
. (8)
The next step is to construct a parameterized family of these normalized measures miðqÞ ¼ xqi P ix q i , (9)
which are generalizations of the original measures xi. The multifractal spectrum f ðaÞ is then obtained by
varying the parameter q in Eq. (9) and calculating f ðaqÞ ¼ lim N!1 P imiln mi ln N , (10) aq¼ lim N!1 P imiln xi ln N . (11)
InFig. 2a we show the f ðaÞ functions for the 8th generation of the FB sequence, considering three different values of the reduced wavevector kxda. One can note that, differently from the multifractal Fibonacci spectra
of magnetostatic modes[19], the curves in the present case are qualitatively insensitive to the kxdavalues. The
spectra for the double-period sequence is given inFig. 2b, for its 4th generation, while inFig. 2c we show the f ðaÞ spectra for the 4th generation of the Thue–Morse sequence. The narrowing or the broadening of the curves are quantified by the multifractal strength of the system, which in turn is determined by the slope of each of the curves, given by the q exponents[12].
Fig. 1. The distribution of the bandwidths as a function of the quasiperiodic generation numbers. (a) Fibonacci sequence, with kxda¼1:0,
The linear behavior of the scale exponent d against the reduced wavevector K ¼ kxdafor the FB superlattice
is depicted inFig. 3a. The better linear fit for the Fibonacci case, compared with the TM and DP structures (Fig. 3b), is due to its relatively small growth rate, and consequent smaller localization. As it can be seen, there is a very strong dependence of the d exponent with the wavevector K for the double period case, when compared to the FB and TM structures. This is a completely different behavior from the one found for magnetostatic modes propagating in these structures[16], where the linear coefficient is virtually the same for the three sequences.
In the center of the Brillouin zone, we have the imposition of the retardation effect (finite speed of light). Further, when one gets close to the first Brillouin zone boundary, the localization of the modes for higher values of the wavevector is very strong, giving rise to a loss of precision in the numerical results. It can, however, be seen that there is a monotonically increasing behavior of the scaling exponent, in a fair range of wavevectors, and we do expect that this linearization occurs at the whole Brillouin zone.
In summary, we have described the scaling, localization and multifractal behavior for optical phonons propagating in quasiperiodic semiconducting superlattices. Multifractal analysis is a suitable statistical description for the study of long term dynamical behavior of a physical system. For all cases the extremes amin
and amaxof the abscissa of the f ðaÞ curves represent the minimum and maximum of the singularity exponent a,
which acts as an appropriate weight in the reciprocal space. In fact, amin ¼limN0
!þ1Dq and amax¼
limN0
!1Dq characterize the scaling properties of the most concentrated and most rarified region of the
intensity measure, respectively. The value of Da amaxamin may be used as a parameter reflecting the
randomness of the intensity measure. The above multifractal analysis revealed a smooth f ðaÞ function
distributed in a finite range [amin, amax] for all the quasiperiodic structures, with a summit at f ða0Þ ¼1. Our
investigations demonstrated that the optical-phonon spectra discussed here correspond to highly nonuniform intensity distributions, and therefore they possess the scaling properties of a multifractal.
Acknowledgements
The authors would like to acknowledge the financial support provided by the Brazilian Research Agencies CAPES-Procad, CNPq, Finep-CTEnerg and Finep-CTInfra.
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Fig. 3. (a) The linear scaling behavior of the d exponent for the Fibonacci superlattice, against the reduced wavevector K ¼ kxda, (b)
linear scaling for Thue–Morse (open circles) and double-period (triangles) cases. The linear fitting equations are d ¼ ð0:52 0:02ÞK 0:01 for Fibonacci, d ¼ ð1:1 0:04ÞK 0:14 for Thue–Morse and d ¼ ð3:14 0:03ÞK 0:41 for double-period sequences, respectively.
