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Band Structure of Three-Dimensional Phononic Crystals

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Band Structure of Three-Dimensional Phononic Crystals CHINESE JOURNAL OF CHEMICAL PHYSICS VOLUME 19, NUMBER 2 APRIL 27, 2006 ARTICLE Band Structure of Three-dimensional Phononic Crystals Lin Yana,b; He-ping Zhaoa,b¤; Xiao-yun Wanga; Guo-sheng Huanga; Xiu-yan Penga a. College of Physics Science and Information Engineering, Jishou University, Jishou 416000, China; b. Faculty of Material Science and Photoelectronic Physics, Xiangtan University, Xiangtan 411105, China (Dated: Received on January 2, 2005; Accepted on June 24, 2005) By using the plane-wave-expansion method, the band structure of three-dimension phononic crystals was calculated, in which the cuboid scatterers were arranged in a host with a face-centered-cubic (FCC) structure. The influences of a few factors such as the component materials, the ¯lling fraction of scatterers and the ratio (RHL) of the scatterer's height to its length on the band-gaps of phononic crystals were investigated. It is found that in the three-dimension solid phononic crystals with FCC structure, the optimum case to obtain band-gaps is to embed high-velocity and high-density scatterers in a low-velocity and low-density host. The maximum value of band-gap can be obtained when the ¯lling fraction is in the middle value. It is also found that the symmetry of the scatterers strongly influences the band-gaps. For RHL¸1, the width of the band-gap decreases as RHL increases. On the contrary, the width of the band-gap increases with the increase of RHL when RHL is smaller than 1. Key words: Phononic crystals, Band-gap, Face-centered-cubic (FCC), Plane wave expansion method I. INTRODUCTION The research on the band structure of periodic com- posites and the propagation of elastic (acoustic) waves The periodic composites have attracted a great deal in di®erent periodic composites is very important. It of attentions in recent years [1-10]. It is well known will help us realize the influences of the di®erent com- that theoretic basis of the semiconductor is the energy ponents and the structures of phononic crystals on the band theory. When the electrons propagate in the peri- band structure and will help us understand the theory odical potential ¯eld, the band structure (i.e. the con- of the phononic crystals. These theories will guide us duction band and the forbidden band) will form, and to design new function materials to satisfy our needs. the electrons can only move freely in the conduction At present, several methods such as the plane-wave- band. People can design and modulate the band-gaps expansion method [1-7], ¯nite-di®erence-time-domain by adjusting the physical parameter in a semiconductor method, multiple scattering theory, transfer matrix and super lattice, and thus promote the development of the variational approach are used to calculate the band semiconductor science and technology. Earlier in 1960s, structure of phononic crystals. It has been found that it was found that the propagation of light-waves in the band-gaps can be influenced by the following factors [1- periodic dielectric structure is similar to that of electron 7,9]: (i) the ratio of mass density, sound velocity and in the semiconductor. If the wavelength of electromag- impedance of the scatterers to those of the host(ii) the netic wave has the same order of magnitude as the pe- geometrical size and the ¯lling fraction of the scatterers riodicity of the dielectric structure, the band structure, and (iii) the topology of the scatterers in the crystal. In called photon band, will appear. The gap between pho- this work, the band structure of binary three-dimension ton bands is called photon band-gap, in which the light (3-D) phononic crystals is calculated by using the plane- wave can not propagate, and the corresponding mate- wave-expansion method. The band-gaps of 3-D com- rials are called photonic crystals. Recently, the simi- posites which consist of cuboid scatterers (A) arranged lar research has been extended to the propagation of in a face-centered-cubic (FCC) structure in host (B) elastic/acoustic waves in the periodic composites called are calculated. Comparisons are made among the bad- phononic crystals. Because elastic/acoustic wave is a gaps of 3-D phononic crystals which are composed of full vector and each component includes three inde- di®erent component materials. The influences of some pendent parameters, namely, the mass of density, the factors such as the ¯lling fraction of scatterer and the transverse and longitudinal speed of sound. Thus, the ratio of the scatterer's height to its length on the band research on phononic crystals is of greater physical sig- structure are investigated. ni¯cance [1-4]. II. MODEL AND FORMULA ¤Author to whom correspondence should be addressed. E-mail: The basic idea of the plane-wave-expansion method [email protected], Tel.: 0743-8563623 is to expand the density and elastic coe±cient of mate- ISSN 1003-7713/DOI:10.1360/cjcp2006.19(2).155.4 155 °c 2006 Chinese Physical Society 156 Chin. J. Chem. Phys., Vol. 19, No. 2 Lin Yan et al. ¡! rial to the superposition of plane wave in reciprocal- where K is a 3-D Bloch vector which is restricted within lattice vectors space. Then the acoustic wave equa- the ¯rst Brillouin zone (BZ) (Fig.1), and ¡!u (¡!r ) is a ¡! K ¡! ¡! tion is translated into an eigen equation. By solving periodic function of r just like ¸, ¹ and ½. u K ( r ) this equation, we can get the dispersion relations of the can also be expanded into Fourier series according to spreading phonon. Eq.(4). Thus, we have, The three-dimension phononic crystal is composed of X ¡! ¡! ¡! various shapes of scatterers embedded in a host. Re- ¡! ¡! ¡!¡! ¡! i(K+G)¢ r u ( r ) = u K+G e (7) gardless of the factor of time, the three-dimension elas- ¡! G tic wave equation in a medium with local isotropy can be written as the following: Substituting Eq.(4)(with f = ¸, ¹, ½¡1) and Eq.(7) into Eq.(2), we can obtain: ¡! ¡! 2¡! (¸ + 2¹)r(r ¢ u ) ¡ ¹r £ r £ u + ½! u = 0 (1) X n X h 2 i ¡1 ¡! ¡!0 ! u¡! ¡! = ½¡! ¡! ¸¡! ¡! (K + G )l K+G G¡G 00 G 00¡G 0 In a rectangular coordinate system, equation (1) can be ¡! ¡! rewritten as follows [1,4,7]: G 0 l;G 00 ³¡! ¡! ¡! ¡! ¡! ¡! i K + G 00) + ¹¡! ¡! (K + G 0) (K + G 00) 2 i ½ µ l ¶ · µ i l ¶¸¾ i G 00¡G 0 i l @ u 1 @ @u @ @u @u X h 2 = ¸ + ¹ + (2) l ¡1 @t ½ @xi @xl @xl @xl @xl £ u¡! ¡! + ½¡! ¡! ¹¡! ¡! K+G 0 G¡G 00 G 00¡G 0 ¡! G 00 where ui(i = x; y; z) are the Cartesian components of X i o ¡! ¡!0 ¡! ¡!00 i ¡! ¡! £ (K + G ) (K + G ) u¡! ¡! (8) the displacement vector u ( r ), xl(l = x; y; z) are the j j K+G 0 Cartesian components of the position vectors ¡!r , ¸(¡!r ) j and u(¡!r ) are Lam¶ecoe±cients, ½(¡!r ) is mass density. In this work, we have investigated the binary 3-D ¸, ¹ and ½(¡!r ) are periodic functions of ¡!r : phononic crystals, in which every unit cell is composed ¡! ¡! ¡! of two materials. One is the scatterer and the other f( r + R) = f( r ) (3) is the host. They are labeled with A and B respec- ¡! tively. Thus material A(B) can be characterized by the In Eq.(3), R = n ¡!a + n ¡!a + n ¡!a , in which n , n , ¡!1 1¡! 2¡!2 3 3 1 2 parameters ½A(½B), ¹A(¹B) and ¸A(¸B). We speci¯ed n3 are integers, a 1, a 2, a 3 are crystals lattice basis the occupancy ration of the volume of each material by ¡! ¡! ¡! ¡1 ¡! vector. f( r ) stands for ¸( r ), ¹( r ) or ½ ( r ). Owing F (for A) and 1¡F (for B) respectively. The Fourier co- to its periodicity, it can be expanded in a 3-D Fourier e±cients in Eq.(4) now take a particularly simple form series: and can be written as, X ¡! ¡! iG¢¡!r Z f( r ) = f¡!e (4) 1 ¡! G ¡! ¡! ¡! 3¡! ¡! fG = f( r )exp(¡iG ¢ r )d r (9) G V The summation extends over all the reciprocal vec- The integration covers the unit cell, V is the volume of ¡! ¡! tors G, which is de¯ned by the following relation: the unit cell. We de¯ne a structure function P (G), and ¡! ¡! ¡! ¡! Z G=m1 b 1+m2 b 2+m3 b 3. in which m1, m2 and m3 ¡! 1 ¡! ¡! ¡! ¡! P (G) = exp(¡iG ¢ ¡!r )d3¡!r (10) are integers, b 1, b 2 and b 3 are reciprocal lattice ba- V sis vectors. They can be de¯ned as follows: where the integration only covers the volume of scatter- ¡! ¡! ¡! 2¼( a 2 £ a 3) ers A. It is found that the form of the structure function b 1 = ¡! ¡! ¡! ¡! a 1 ¢ ( a 2 £ a 3) P (G) does not depend on the type of the Bravais lat- ¡! ¡! tice, but on the geometry of scatterers A only. Now ¡! 2¼( a 3 £ a 1) b 2 = ¡! ¡! ¡! (5) Eq.(9) can be written as: a 1 ¢ ( a 2 £ a 3) ¡! ¡! ¡! ¡! ¡! 2¼( a 1 £ a 2) f(G) = fAF + fB(1 ¡ F ) ´ f (G = 0) (11) b 3 = ¡! ¡! ¡! a 1 ¢ ( a 2 £ a 3) ¡! ¡! ¡! ¡! f(G) = (fA ¡ fB)P (G) = ¢fP (G)(G 6= 0)(12) ¡! Reciprocal vector G is a periodic function with the same periodicity as the phononic crystals. Owing to If the scatterers are cuboids with l1 in length, l2 in the common periodicity of all the coe±cients in Eq.(3) width and l in height, we can write the structure func- ¡! 3 and (4), its eigenvalues, according to the Bloch's theo- tion P (G) as the following, rem, can be written as the follows: ¡! sin(Gxl1=2) sin(Gyl2=2) sin(Gzl3=2) ¡! ¡! P (G) = F (13) ¡! ¡! iK¢ r ¡!¡! ¡! u ( r ) = e u K ( r ) (6) Gxl1=2 Gyl2=2 Gzl3=2 Chin.
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