collective excitations in a three-dimensional diatomic penrose lattice

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14 March 1994

Physics Letters A 186 (1994) 250-254

PHYSICS LETTERS A

Collective excitations in a three-dimensional diatomic Penrose lattice

Ying H u a, Decheng Tian a,b, Zhengyou Liu a

a Department of Physics, Wuhan University, Wuhan 430072, China b International Center for Material Physics, Academia Sinica, Shengyang 110015, China

Received 16 November 1993; revised manuscript received 12 January 1994; accepted for publication 17 January 1994 Communicated by J. Flouquet

Abstract

We present detailed investigations of the vibrational modes in a three-dimensional diatomic Penrose lattice. The vibrational densities of states are calculated using a recursion technique. These spectra show an apparently branched structure, we classify the branches by means of the same convention as used for crystals. The acoustic branches appear to be smooth and continuous and exhibit a linear feature near zero frequency, while the optical-like branches display a rich structure. The results demonstrate that the density of states in the acoustic branches can be attributed to the phonon and the fracton-like excitations and that in the optical-like branches to some localized vibrational modes.

Since the discovery of quasicrystals by Schecht- man et al. [ 1 ], the invest igat ion o f the character of the electronic and v ibra t ional eigenstates of the quasi lat t ices has received much a t tent ion [ 2 -4 ]. For one-dimensional quasicrystals, such as the Fibonacci chain, which seems to be the sole one analyt ical ly solvable at present, it is found that the v ibra t ional spect rum consists o f a Cantor set and three kinds o f v ibra t ional states, namely the extended, the localized and the in te rmedia te (or cr i t ical) ones [ 4 ], coexist. The numerica l results for the two- and three-dimensional quasicrystals indicate that the second conclu- sion above also holds [2 ,5 -8 ] . So far the investiga- t ions on the v ibra t ional proper t ies o f quas iper iodic lattices are based on a s imple mona tomic model. Analogous to the complex lat t ice of a c o m m o n crys- tal, we consider the quas iper iodic lat t ice composed o f mul t ip le a toms in order to extend the unders tand- ing to the dynamical behavior of quasicrystals. In this paper we would like to repor t the results on the vibra-

t ional densi ty of states on a three-dimensional di- a tomic Penrose lattice.

The perfect three-dimensional Penrose lat t ice is constructed by the project ion method [9,10]. Ini- tially, all vertices of the rhombohedra are covered with one sort o f a toms marked A. Modif ica t ions are made in a Penrose t i l ing by inserting subsets of a toms (B) in the middle o f the original bonds. Only the in- teract ion between nearest neighbors being considered, the potent ia l energy of the latt ice [11 ] is ex- pressed as

N N N N

V =½ (o~- f l ) ~ [ (u i -u j ) . r i j ]2+½f l ~ lu i - -u j l 2, o ij

where ui is a small d isplacement of the ith site about its equi l ibr ium posi t ion ri, r 0 is the unit vector from site i to site j , a is the bond-stre tching force constant and fl is the bond-bend ing force constant. The vector nature o f the elastic force is included natural ly in this

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Y. Hu et al. / Physics Letters A 186 (1994) 250--254 251

equation. When ot is chosen to equal fl, the system considered becomes isotropic.

We utilize a recursion method [ 12,13 ] which is especially convenient for a disorder system and has been checked to be valid for quasicrystals. A numerical computat ion is performed on such a Penrose lattice with 6623 A atoms and 19428 B atoms with free boundary conditions. In the calculation, both a and flare unity and the masses o f A and B atoms take two sets o f values: ( 1 ) rnA = 1, rnB = 4 and (2) mA = 4, mB---- l.

Fig. 1 shows the global integrated density o f states for the 3D diatomic Penrose lattice, (a) is for case 1, mA= 1, mB=4; and (b) is for case 2, mA=4, mB= 1. It deserves to be noticed that on the whole the two spectra are alike in structure and the only difference between them is the spectral width. This implies that the contour o f the vibrational spectrum for the diatomic Penrose lattice is determined by the structure o f the lattice, more or less independently o f the mass o f the atoms in it, thus we only consider case 1. The spectrum apparently consists o f two parts, which are separated by a rather large gap. The low-frequency

.g.

.~o

4 t t t t t

! I .-,. 2 4

o)

0 5

(h)

/ f - -

8 16 co

Fig. 1. Global integrated DOS for the diatomic Pearose lattice with force constants a = p = 1. (a) mA= 1, roB=4, (b) mB=4, roB---- 1.

part is smooth and continuous, with a sharp step at the edge; and the high-frequency part exhibits a rich structure. It is noticed that the global IDOS curve has an infinite slope at the left edge of the gap (mode o~ ), which implies that there is a strong degeneracy effect near the upper edge o f the low-frequency part for a general diatomic Penrose lattice (Fig. 1 ). With no in- consistency in principle, we regard the low-frequency part as acoustic branches and the high-frequency part as optical-like branches as in the convention adopted for crystals.

Fig. 2 is the global DOS corresponding to the integrated DOS mentioned above. We use a logarithmic coordinate system here in order to bring out the feature o f acoustic branches. We note that near zero frequency, the spectrum appears to be linear with the frequency. There exist two crossover frequencies cos and coc. Without doubt, below coc the VDOS are to be attributed to phonon excitations. A scaling law pGOcO)d-1 holds in this regime. Below cos d equals 2, which corresponds to surface phonon excitations;

(a)

] c G .

-1

-3 j -2 2

loglo~o

0

-2

- 4 _ 2 1 ~ I 91~

log~0w

Fig. 2. Global densities for the diatomic Penrose lattice with force constants c~ = fl= 1. (a) mA = 1, mB = 4, ( b ) mA = 4, mB = 1.

252 Y. Hu et aL / Physics Letters A 186 (1994) 250-254

(a)

9

f ,,-.""T I I I g,

(~ 2 4-

0.5

(hi

J S

r

8

I I I :, 2 4 oJ

5" 0.5

(c)

f . . . . . .--'11 I I ,,

2 4

3" -Z 05

(d)

/ , I I Z co

4

i I (e)

-~_0.5 5

I 0 2

-5"- 0.5

I - 4 " 0

ff)

y

,/

Z 4

~ o 5

(g)

y... . . . .

, 'J~ l 2 4

~ 0 . 5

(h?

. _ _

/ 2

I I -,. 4

co Fig. 3. Local i n t e g r a t e d d e n s i t i e s o f s t a t e s w i t h f o r c e c o n s t a n t s a = f l = 1 f o r a t o m s a t t h e v e r t e x (a) V12, (b ) Vjo, (c) V9, (d ) Vs, (e) VT, ( f ) V6, (g) V~, (h ) V4.

above cos d equals 3, which corresponds to bulk phonons. Analysis of local DOS indicates that all states below o)c are extended, which is a typical feature of

acoustic waves. The effect of free boundaries might unavoidably introduce some features near the spectral edge that are not genuine bulk ones. So there is

Y. Hu et al. / Physics Letters A 186 (1994) 250-254 253

0.5

r

r / ' l

4

0.5 ,ff

tb)

i l /

/

J

2 4 co

Fig. 4. Local integrated densities of states with force constants a = p = 1 for atoms in the middle of bonds but with different next nearest neighbor.

the first crossover frequency ogs from two-dimensional surface phonons to three-dimensional bulk phonons. Another linear region on the spectrum above o9¢ can be attributed to the fracton-like excitations introduced by Liu and Tian [ I 1 ]. This is due to the fractal feature (self-similarity) of quasicrystals. It is in the sense of self-similarity we call quasicrystals fractal structures, thus are possible fracton- like excitations, ogc is attributed to the crossover frequency from phonon excitations to fracton-like excitations. The spectrum above ogc satisfies a scaling law Pc (o9) oc ogd,-1 with d, = 3, where d, is the spectral di- mension. The states in this regime are neither extended nor localized but critical. In addition, from Fig. 1 we see that the derivative of this integrated DOS, i.e. the DOS, experiences an increase and a de- crease with frequency on the low-frequency part. This kind of spectral shape cannot in general be produced by only one acoustic branch, it should be a combination of two or more acoustic branches, including TA modes and LA modes.

We investigate the local DOS either for eight vertex stars that can occur in subset A (Fig. 3) labelled

as V~ with I denoting its coordination number, or for the sites in subset B (Fig. 4). Since the local DOS for a site is proportional to the square of the vibrational amplitude of the site, the information about the vibrational state is naturally included in local DOS. Our knowledge of vibrational states indicates that modes lower than o9¢ are all extended states. Critical vibrational modes are found in the frequency regime above ogc and localized vibrational modes are found at the upper edge of the spectrum labeled as modes 1-9. It is implied that mode 1 is completely contributed by the vibrational modes of B atoms. Modes 2-9 are all centered at A atoms. Comparing Fig. 1 with Fig. 3, we find that the different local modes exhibited in the global density of states are favored by the special local configurations. Furthermore, we can establish a simple relation between the local configuration and some of the localized modes: the larger the coordination number the higher the frequency of the mode it favors.

The physical significance of the optical-like branches at present is far from being as clear as that in crystals. The notion of wave vector, for the vibrational modes, especially for localized modes in a disorder system, being somewhat obscure except for the extremely long acoustic wave for which any system can be considered as being homogeneous, continuous and elastic, it is difficult to obtain a dispersion relation as that for crystals. A simple analogy to the usual complex lattice seems that the DOS in the high- frequency part should also be a combination of a few optical-like branches, but unfortunately direct evi- dence and effective analysis cannot be given yet.

To review our model from the viewpoint of projec- tion may be helpful to understand such spectral structures. This diatomic Penrose lattice also can be obtained by projecting a six-dimensional regular diatomic lattice on a three-dimensional space. The six- dimensional complex lattice is constructed in such a way that atom A is at the vertex and atom B in the middle of the edge. The lattice is of course a complex lattice with the same periodicity as the simple lattice, each basic unit contains seven atoms: one A atom and six B atoms. The vibrational spectrum for this six- dimensional diatomic lattice may consist of six acoustic branches and more optical-like branches in the 6D superspace. In principle, the vibrational spectrum for the real three-dimensional diatomic lattice

254 Y. Hu et al. / Physics Letters A 186 (1994) 250-254

should be a mapping of this spectrum in a suitable way, thus the vibrat ional spectrum should exhibit two-band structures, regrettably this is not such a straightforward course.

To summarize: we have presented the first inves- t igation of the vibrat ional properties of a 3D diatomic Penrose lattice. The vibrat ional DOS of the lattice consists of acoustic branches and optical-like branches, quite like usual crystals. The DOS of the acoustic branches is contr ibuted by phonons and fracton-like excitations, and that of the optical-like branches is contr ibuted by a few localized vibrat ional modes. A possible explanation for such a spectral

structure is presented.

Support from the Nat ional Natural Science Foun- dat ion of China and the Founda t ion for Ph.D. Grad- uates of the Nat ional Commit tee of Education is

acknowledged.

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collective excitations in a three-dimensional diatomic penrose lattice

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