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 filling 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 filling 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 composites and the propagation of elastic (acoustic) waves The periodic composites have attracted a great deal in different 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 different 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 field, 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], finite-difference-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 filling 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- different component materials. The influences of some pendent parameters, namely, the mass of density, the factors such as the filling 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. nificance [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 coeﬃcient 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 first 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 λ, µ and ρ. 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 = λ, µ, ρ−1) and Eq.(7) into Eq.(2), we can obtain: −→ −→ 2−→ (λ + 2µ)∇(∇ · u ) − µ∇ × ∇ × u + ρω u = 0 (1) X n X h 2 i −1 −→ −→0 ω u−→ −→ = ρ−→ −→ λ−→ −→ (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) + µ−→ −→ (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 = λ + µ + (2) l −1 ∂t ρ ∂xi ∂xl ∂xl ∂xl ∂xl × u−→ −→ + ρ−→ −→ µ−→ −→ 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 , λ(−→r ) j and u(−→r ) are Lamécoefficients, ρ(−→r ) is mass density. In this work, we have investigated the binary 3-D λ, µ and ρ(−→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 ρA(ρB), µA(µB) and λA(λB). We specified 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 λ( r ), µ( r ) or ρ ( 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 efficients 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 defined by the following relation: the unit cell. We define 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 defined as follows: where the integration only covers the volume of scatter- −→ −→ −→ 2π( 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 −→ 2π( a 3 × a 1) b 2 = −→ −→ −→ (5) Eq.(9) can be written as: a 1 · ( a 2 × a 3) −→ −→ −→ −→ −→ 2π( 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 coefficients 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. J. Chem. Phys., Vol. 19, No. 2, Band Structure of Three-dimensional Phononic Crystals 157

By using Eqs.(11) and (12), it is not difficult to frequency is ∆Ω/Ωg = 0.4565, and the value of the nor- rewrite Eq.(8) in the form of a standard eigenvalue malized gap width is ∆Ω = 0.6437. equation. If the infinite series in Eqs.(4) and (7) are approximated by a sum of M reciprocal vectors, Eq.(8) can be reduced to a 3M × 3M matrix eigenvalue equa- TABLE I The main physical parameters of the materials i tion about 3M unknown coefficients u−→ −→(i = x, y, z). 3 3 K+G Material ρ/(10 kg/m ) Cl/(m/s) Ct/(m/s) This equation can be solved by a numerical method and Epoxy 1.85 2830 1160 the desired precision can be achieved by increasing the −→ Si 2.33 8950 5359 value of M. To a given wave vector K in the first BZ, Au 19.5 3360 1240 an infinite number of eigenvalues can be obtained, ev- Pb 11.4 2158 860 ery of which is characterized by a natural number n. −→ The corresponding eigen frequency is ωK,n. Let K scan the irreducible BZ, we can get the band structure by −→ solving the eigenvalue equation. The value of K is usu- ally focused on the middle and edge of the first BZ. We illustrate the first BZ−→ of 3-D face-centered-cube lattice in Fig.1, in which K is varied along straight segments UL-LΓ-ΓX-XW -WK.

FIG. 1 Frist Brillouin-zone of three-dimensional face-center- cube lattic(FCC)

III. NUMERICAL RESULTS

Firstly, we have calculated the band gaps of a 3-D periodic system which is composed of an Au cuboid scatterer embedded in Epoxy with FCC structure. The FIG. 2 Elastic wave band structure of three-dimension sys- corresponding physical parameters are listed in Table I. tem with FCC lattice. (a) The case of Au cuboid embeded In our calculation, m , m and m are permitted to take in epoxy host. (b) The case of Pb cuboid embedded in epoxy 1 2 3 host. The ratio of scatterer’s height to length is 1.1/1, the the integer values between −3 and +3. Then Eq.(8) be- filling fraction of scatterer is 0.3, the lattice constants is come a 1029×1029 matrix elements, thus, a very good −→ 4.0 cm. convergence can be obtained. When K varies along the high symmetric straight segments of the first BZ, we −→ We changed the material of the scatterers by sub- can obtain a series curves of K vs. ωK,n. The band stituting Pb for Au (namely Pb/epoxy system). Pb structure for a=4.0 cm, F =0.3 and l1=l2, l3=1.1l1 is cuboids are arranged periodically in an FCC lattice shown in Fig.2(a), where the abscissa stands for the di- within an Epoxy host and the other parameters are mensionless Bloch wave vector of high-symmetry points fixed. The calculated results of the band structure in the first BZ and the ordinate stands for normalized are shown in Fig.2(b). It is found that a band- frequency Ω = ωa/(2πCt), in which a is the lattice con- gap also appears between the sixth and the seventh stant, ω is the angular frequency and Ct is the trans- band, and the parameters of the band-gap are Ωg=1.13, verse velocities in the host. In the lowest 25 bands, we ∆Ω/Ωg=0.148 and ∆Ω=0.168. If we substitute silicon found a relative band-gap which located between the for Epoxy (Au/Si system) and keep the other param- sixth and the seventh band. The normalized frequency eters the same as Au/Epoxy system mentioned above, of the mid-gap is Ωg ≈ 1.41, the value of gap/mid-gap no band-gap appears. 158 Chin. J. Chem. Phys., Vol. 19, No. 2 Lin Yan et al.

on the condition of RHL<0.4 and RHL>2.5. When 0.4

IV. CONCLUSIONS

By using the plane-wave-expansion method, we calculated the band structure of the binary three-dimensional solid phononic crystals consisting of cuboid scatterers embedded in a host. The scatterers are arranged in face-centered-cubic structure. Several factors, which FIG. 3 The relative width of the gap, ∆Ω/Ωg versus the help to obtain band-gaps and influence on the prop- filling fraction of the scatterer. B and C curves correspond erties of band-gaps are investigated. The results show to Au/epoxy and Pb/epoxy system. (i) Scatterers with High-velocity and high-density embedded in a host with low-velocity, and low-density pro- vide an optimum situation to obtain large band-gaps. (ii) The symmetry of scatterers has a great influence on the band-gaps. For example, cube scatterers are more favorable than cuboid scatterers for band-gaps. The higher the symmetry of scatterers, the easier the ap- pearance of band-gaps. (iii) The widest gap can be obtained when the filling fraction is in the middle value. If we choose appropriate physical parameters of the composites, we can obtain wider band-gaps.

V. ACKNOWLEDGMENT

This work was supported by the Natural Science Foundation of Hu’nan Province (Grant No. 00JJY2072) FIG. 4 The relative width of the gap, ∆Ω/Ωg, versus the ratio of scatterer’s height to length for Au/epoxy system. and the Foundation of Educational Committee of Hu’nan Province (Grant No. 01B019).

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