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Phonon–polariton in two-dimensional piezoelectric phononic crystals
Physics Letters A 372 (2008) 4730–4735
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Phonon–polariton in two-dimensional piezoelectric phononic crystals Ming-Yi Yang a , Liang-Chieh Wu a , Jiun-Yi Tseng b,∗ a b
Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China Materials and Chemical Research Laboratories, Industrial Technology Research Institute, Hsinchu, Taiwan 31040, Republic of China
a r t i c l e
i n f o
Article history: Received 14 March 2008 Received in revised form 15 April 2008 Accepted 5 May 2008 Available online 15 May 2008 Communicated by R. Wu PACS: 43.20.+g 43.40.+s 73.50.Rb
a b s t r a c t The phonon–polariton behaviors of two-dimensional piezoelectric phononic crystals (PPCs) were studied using the plane wave expansion method. The governing equations combine Maxwell’s equations and Newton’s equations of motion. A mode-repulsion can be formed by strong coupling between electromagnetic (EM) waves and elastic waves in the vicinity of the center of the first Brillouin zone for PPC that comprises piezoelectric material and with opposite polarization in different periodically organized areas. Take a 2D ZnO PPC as a numerical example, it was decoupled into two independent groups. One refers to the mixed mode of the in-plane elastic waves and the transverse-magnetic (TM) mode EM waves. The other group refers to the mixed mode of the out-of-plane elastic waves and the transverse-electric (TE) mode EM waves. Coupling repulsion is also observed in these two groups. © 2008 Elsevier B.V. All rights reserved.
Keywords: Phonon–polariton Piezoelectric phononic crystals Plane-wave-expansion method Repulsion
1. Introduction As micro-fabrication technology advances, artificial microstructures, such as periodic structures, with desired properties can be manufactured more easily. In solid state physics, the band gaps that prohibiting the electronic wave propagation are generated by a periodic potential [1]. A similar phenomenon is also observed in such dielectric-modulated materials as photonic crystals [2–5]. The applications of photonic crystals include spontaneous emission suppression, light path manipulation and lasers. Recently, spacemodulated elastic structures, known as the phononic crystals, have been investigated [6–9]. Their band gap characteristics are more complex than those of photonic crystals and depend on the elastic constants, piezoelectric coefficients, density and other factors. Phononic crystals have been developed for vibration suppression, elastic wave guides and localized waves. In photonic/phononic crystals, Bragg scattering is responsible for most band gaps. However, the band gap has also been demonstrated in locally resonant structures [10–15], in which the order of magnitude of the central frequency of the spectral gaps is lower than that of structures with the same lattice constant. Outside the spectral gaps, the characteristics of the flat dispersion curves and
*
Corresponding author. Tel.: +886 3 5919281; fax: +886 3 5820386. E-mail address: [email protected] (J.-Y. Tseng).
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the modes with zero group velocity, which correspond to a localized mode, have been found [11,13]. Huang [16] investigated ionic crystals, which are also regarded as locally resonant structures. He found that exciting lattice vibration by applying an external radiation field causes a polaritonlike band gap at infrared frequencies. Zhu and Chen [17] study the polariton band gap at microwave frequencies in one-dimensional superlattice structures of LiNbO3 . This structure is also called ionic-type phononic crystals or piezoelectric phononic crystals. The periodic space modulation of the piezoelectric coefficients causes the elastic fields to be coupled with the electromagnetic (EM) fields. Consequently, external EM fields can induce the crossed-field or in-line field acoustic excitation and the resonant frequency depends on the periodicity of the reversed domain. The methods of plane wave expansion and transfer matrix are utilized to investigate piezoelectric phononic crystals (PPCs) that are composed of various materials to verify the dependence of the band-gap frequency on the periodicity of the structure [17–24]. According to these references, it is worthwhile to do more studies on this subject. This work explores the hybridization of photons and phonons (also called as phonon–polaritons) in the generalized two-dimensional PPCs. The coupled equations are derived from the combination of Maxwell’s equations with Newton’s equations of motion. The plane-wave expansion method yielded the band structures of the PPCs. The equations of PPCs that were composed of class 6mm materials could be decoupled into two groups: the first comprised
M.-Y. Yang et al. / Physics Letters A 372 (2008) 4730–4735
in-plane elastic waves and the TM-mode EM waves which the polarizations of magnetic fields are parallel to x1 –x2 plane, and the other comprised out-of-plane elastic waves and the TE-mode EM waves which the polarizations of electric fields are parallel to x1 –x2 plane. A two-dimensional PPC composed of ZnO was demonstrated as an example. The simulation showed that the difference between the wavelengths of photons and phonons was responsible for the fact that the band structures in the first Brillouin zone were similar to those associated with pure phonons. Also, the periodicity of the piezoelectric coupling effect reveals that photons and phonons interacted with each other within the normalized wave vector region of order 10−4 . This coupling effect was responsible for repulsion. If the mode-repulsion effect is large enough, it will induce a gap in the band structures. This investigation demonstrates that the coupling effect of two-dimensional PPCs was more complex than that of traditional one-dimensional PPCs, indicating that the interval between two arising frequencies of repulsion or the gaps is not a multiple of fundamental frequency. 2. Governing equations in two-dimensional PPCs Fig. 1 depicts a two-dimensional PPC structure, comprising two materials, A and B, with periodicity a and embedded circular cylinders of radius r0 . The x1 -axis and x2 -axis are in-plane, and the x3 -axis lies in the thickness direction; the thickness is assumed to be infinite. The structure consists of triclinic material, which has the lowest symmetry of all crystal systems. The interaction in this structure between the EM fields and the elastic fields, which does not occur in photonic crystals or phononic crystals, was considered. Based on the assumption that the media were lossless without a source, the coupled equations of the EM fields and the elastic fields can be expressed as [25]
−∇ × ∇ × E = με s · ∇
∂2E ∂t2
2 · C : (∇s · u) = ρ ∂∂ t 2u
E
+ μe : ∇ s ·
∂2u ∂t2
,
(1)
+ ∇ · (e · E), T
where
T
(∇×) = (−∇×) = ∇· = (∇s ·)T =
0
−∂/∂ x3
∂/∂ x3 −∂/∂ x2
∂/∂ x1
∂/∂ x1
0
0 0
∂/∂ x2 0
0 0 0
∂/∂ x2 −∂/∂ x1 , 0 0
∂/∂ x3 ∂/∂ x3 ∂/∂ x2
4731
For plane waves that are traveling through PPC structures and propagating in the x1 –x2 plane, the partial derivatives of the electric and elastic fields with respect to x3 can be assumed to be zero. Also, according to the Bloch theorem, the displacement and electric fields are given by,
u i (x, t ) = e i (k·x−ωt ) G U¯ i (G)e iG·x , E i (x, t ) = e i (k·x−ωt ) G E¯ i (G)e iG·x ,
(4)
where k = (k1 , k2 ) is the wave vector in the first Brillouin zone; U¯ i (G) and E¯ i (G) are the coefficients of the plane waves, and G = (G 1 , G 2 ) = (2π n1 /a, 2π n2 /a) is the reciprocal lattice vector in which n1 , n2 ∈ I . In the unit-cell of PPC, the elastic constants, piezoelectric constants, permittivity and density can all be expanded as Fourier series:
⎧ pq iG·x ⎪ ⎪ C pq = G C G e , ⎪ ip iG·x ⎨e = , ip G eG e i j iG·x ⎪ ε = ε e , ⎪ ij ⎪ G G ⎩ ρ = G ρG e iG·x ,
( p , q = 1–6; i , j = 1–3).
Considering N plane waves in Eq. (4) and substituting Eqs. (4) and (5) into Eq. (1), yields M · U = 0,
(6)
where M is a matrix of dimensions 6N × 6N, whose explicit form is given in Appendix A, and U is a vector of dimension 6N as
U = U¯ 1 (G )
U¯ 2 (G )
U¯ 3 (G )
E¯ 1 (G )
E¯ 2 (G )
∂/∂ x3 ∂/∂ x2 0 ∂/∂ x1 , ∂/∂ x1 0 (3)
are matrices of space-differential operators; E is the electric field; ε s is the permittivity matrix in a constant strain; e is the piezoelectric-stress matrix; u is the elastic field; C E is the elastic matrix in a constant electric field, and ρ is the density.
E¯ 3 (G )
T
.
(7)
The existence of a nontrivial solution U indicates that the determinant of the matrix M must be zero, such that |M| = 0. Solving this eigenvalues/eigenvectors problem yields the dispersion relations between the frequency ω and the wave vector k. A 6mm-symmetric PPC structure, in which materials A and B are identical but with opposite polarization directions along the x3 -axis, is considered. The piezoelectric constants of materials A and B differ only in sign. The elastic constants, piezoelectric constants and permittivity of material A can be expressed as follows;
⎤
⎡
(2)
(5)
C 11 C 12 C 13 0 0 0 0 0 ⎥ ⎢ C 12 C 11 C 13 0 ⎥ ⎢ C 13 C 33 0 0 0 ⎥ ⎢C C = ⎢ 13 ⎥, 0 0 C 44 0 0 ⎥ ⎢ 0 ⎦ ⎣ 0 0 0 0 C 44 0 1 0 0 0 0 0 (C 11 − C 12 ) 2 0 0 0 0 e 15 0 e= 0 0 0 e 15 0 0 , e 31 e 31 e 33 0 0 0 ε11 0 0 ε = 0 ε11 0 . 0 0 ε33
(8)
(9)
(10)
Substituting Eqs. (8), (9) and (10) into Eq. (6), yield a set of equations,
⎡
(11)
MG,G
⎢ (21) ⎢ MG,G ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎣ 0 (61) MG,G
Fig. 1. Schematic diagram of a 2D PPC.
(12)
0
0
0
MG,G
MG,G
(22)
0
0
0
MG,G ⎥
0
MG,G
(33)
MG,G
(34)
MG,G
(35)
0
0
MG,G
(43)
MG,G
(44)
MG,G
(45)
0
0
MG,G
(53)
MG,G
(54)
MG,G
0
0
(66) MG,G
(62) MG,G
0
0
(55)
(16)
⎤
MG,G
(26) ⎥
⎥ ⎥ ⎥ ⎥ · U = 0, ⎥ ⎥ ⎥ ⎦
(11)
where matrix elements are as listed in Appendix B. Eq. (11) reveals that the set of equations can be decoupled into two groups. One group represents a mixed mode of the in-plane elastic wave and the TM-mode EM wave, which is composed of u 1 , u 2 and E 3 , where the E 3 component reveals that there will be H 1 and H 2
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M.-Y. Yang et al. / Physics Letters A 372 (2008) 4730–4735
Table 1 Material properties of ZnO Stiffness (1010 N/m2 )
ZnO
Piezoelectric constant (C/m2 )
Permittivity (1/ε0 )
Density (kg/m3 )
C 11
C 33
C 44
C 12
C 13
e 15
e 31
e 33
ε11
ε33
ρ
20.97
21.09
4.247
12.11
10.51
−0.48
−0.573
1.32
8.55
10.2
5680
Fig. 2. Band structure of mixed mode of in-plane elastic and TM electromagnetic waves.
Fig. 3. Band structure of mixed mode of out-of-plane elastic and TE electromagnetic waves.
magnetic fields according to the Maxwell’s equations. Here, we defined that the TM mode is the polarization of magnetic fields parallel to x1 –x2 plane. The other group represents a mixed mode of the out-of-plane elastic wave and the TE-mode EM wave, which is composed of u 3 , E 1 and E 2 , where the E 1 and E 2 components parallel to x1 –x2 plane so we defined it as the TE mode. The determinants of the associated matrix of the two groups yield the corresponding dispersion relations (also called band structures). 3. Numerical results and discussion A PPC structure that consists of ZnO piezoelectric material was used as an example. Table 1 lists the material properties of ip ZnO [25]. In this case, the Fourier coefficients of e G in Eq. (5) are
ip
eG =
A B e ip f + e ip (1 − f ) G = 0, A B (e ip − e ip ) F (G)
f = π r02 /a2 ,
G = 0,
1 F (G) = A − C
(12)
exp(−iG · x) d A ,
(13)
A
where f is the filling fraction; F (G) is the structure function, and A C is the cross-section area of the unit-cell. The structure function can be further expressed as 2 f J 1 (Gr0 )/(Gr0 ) where J 1 is the first order Bessel function. The dispersion relations are calculated using the method of plane-wave expansion. The superposition of 121 plane waves is selected, which means that n1 , n2 ∈ [−5, 5]. Figs. 2 and 3 show the band structures with r0 /a = 0.25 in the mixed modes of u 1 , u 2 and E 3 and u 3 , E 1 and E 2 , respectively. The frequency is normalized with C L , which C L = (C 11 /ρ )1/2 . For PPC that comprises ZnO but with different polarization, the band gaps due to Bragg scattering were almost absent in the overall band structure. In Figs. 2 and 3 the number of dispersion curves start from Γ point, which energy component is almost the elastic energy, is two and one respectively according to Eq. (11). In this region in which the reduced wave vector was large, the behaviors of the mixed modes were approximately quasi-static, and phonon–photon interaction was absent.
Fig. 4. Dispersion relation of mixed mode of in-plane elastic and TM electromagnetic waves near point Γ .
However, in the region where the reduced wave vector was smaller, about ka/(2π ) ≈ 10−4 , phonon–polariton was observed. Figs. 4 and 5 plot the phonon–polariton dispersion relations for different mixed modes near point Γ , where polaritons dominated. The electromagnetic dominated dispersion curve starts from Γ point and has strong coupling with the folded-phonon dispersion curve near the center of the Brillouin zone. It generated the moderepulsion. This phenomenon has also been investigated in locally resonant materials [19]. However, in this study, elastic waves that were excited by external EM fields and repulsion at particular frequencies were observed, for example the points a and b in Fig. 4. These frequencies were not integer multiples of the fundamental frequency. The same situation is also found in one-dimensional quasi-periodic PPC [22], which is never happened in traditional one-dimensional PPC [21,23]. At the same frequency, the order of the period of this two-dimensional PPC structure is 10−4 less than
M.-Y. Yang et al. / Physics Letters A 372 (2008) 4730–4735
4733
Fig. 5. Dispersion relation of mixed mode of out-of-plane elastic and TE electromagnetic waves near point Γ .
Fig. 7. Contour plot of mode shape at point b, where (k1 a/(2π ), k2 a/(2π )) = (1.06 × 10−4 , 0) and ωaC L /(2π ) = 1.409.
at x2 /a = 0, ±1, ±2, . . ., etc., and the S 2 field in materials A and B have opposite phase along x2 direction at x1 /a = 0, ±1, ±2, . . ., etc. Also, Fig. 7 reveals that the S 1 and S 2 fields in materials A and B have opposite phase along x1 direction at x2 /a = 0, ±1, ±2, . . ., etc., and along x2 direction at x1 /a = 0, ±1, ±2, . . ., etc. The first equation in Eq. (1) indicates that the charges induced by the time-varying strain fields reradiate the EM field. The periodic modulation of the piezoelectric coefficient cooperated with the periodic variation of the stain fields enable the reradiated EM fields, whose wavelengths significantly exceeds the elastic wavelengths, to interact with the external EM fields and finally induce a repulsion gap. Other than at points a and b, the simulation shows high-order modes that exhibit similar behaviors. 4. Conclusions
Fig. 6. Contour plot of mode shape at point a, where (k1 a/(2π ), k2 a/(2π )) = (7.2 × 10−5 , 0) and ωaC L /(2π ) = 0.9893.
that of the two-dimensional photonic crystals, because the velocity of photons markedly exceeds that of phonons. The dimensions of the two-dimensional PPC support potential applications of miniaturized EM wave filters or transducers. Figs. 6 and 7 show the mode shape contours plot of points a and b in Fig. 4, where the reduced wave vectors are (k1 a/(2π ), k2 a/(2π )) = (7.2 × 10−5 , 0) and (k1 a/(2π ), k2 a/(2π )) = (1.06 × 10−4 , 0) respectively, the reduced frequencies are 0.9893 and 1.409, respectively, and the region of plot is within one unit cell. Fig. 6 reveals that the S 1 field in material A has the same phase while those in material B has opposite phase along x1 direction
This investigation derived the phonon–polariton dispersion relation in the generalized two-dimensional PPCs. For a twodimensional PPC that consists of 6mm material, the dispersion relation can be decoupled into two independent groups: one group represented a mixed mode of the in-plane elastic waves and the TM-mode EM waves; the other represented a mixed mode of the out-of-plane elastic waves and the TE-mode EM waves. Both systems exhibited phonon–polariton behaviors near the Γ point of the first Brillouin zone. The constitutive equations of piezoelectric material and the periodic structures reveal interaction between the EM waves and the elastic waves and strong coupling between phonons and photons. Numerous bands involved in the strong coupling of the elastic waves and the EM waves were formed; these could be applied to multi-band filters whose dimensions can be significantly smaller than those of filters that use two-dimensional photonic crystals.
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M.-Y. Yang et al. / Physics Letters A 372 (2008) 4730–4735
(55)
M G, G =
Acknowledgements The authors would like to thank the MOEA, Taiwan, for financially supporting this research under Contract No. 7354DB3220. Prof. Chou of National Taiwan University is appreciated for his valuable discussions. Appendix A The explicit expressions of submatrices in Eq. (6) are ρG−G ω2 − C G11−G (k1 + G 1 )(k1 + G 1 ) − C G16−G (k2 + G 2 )(k1 + G 1 ) (11) , M G, G = −C G16−G (k1 + G 1 )(k2 + G 2 ) − C G66−G (k2 + G 2 )(k2 + G 2 )
(12)
M G, G =
(13)
M G, G =
−C G16−G (k1 + G 1 )(k1 + G 1 ) − C G12−G (k2 + G 2 )(k1 + G 1 ) , −C G66−G (k1 + G 1 )(k2 + G 2 ) − C G26−G (k2 + G 2 )(k2 + G 2 )
(15) (16)
M G, G (21)
M G, G
−C G15−G (k1 + G 1 )(k1 + G 1 ) − C G14−G (k2 + G 2 )(k1 + G 1 ) , −C G56−G (k1 + G 1 )(k2 + G 2 ) − C G46−G (k2 + G 2 )(k2 + G 2 )
(A.3) (A.4)
26 = −i e 21 (A.5) G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 31 = −i e G−G (k1 + G 1 ) + e 36 (A.6) G−G (k2 + G 2 ) , 16 66 −C G−G (k1 + G 1 )(k1 + G 1 ) − C G−G (k2 + G 2 )(k1 + G 1 ) = , 26 12 −C G−G (k1 + G 1 )(k2 + G 2 ) − C G−G (k2 + G 2 )(k2 + G 2 )
(A.7) ρG−G ω2 − C G66−G (k1 + G 1 )(k1 + G 1 ) − C G26−G (k2 + G 2 )(k1 + G 1 ) (22) , M G, G = 26 22 −C G−G (k1 + G 1 )(k2 + G 2 ) − C G−G (k2 + G 2 )(k2 + G 2 ) (A.8)
(23) M G, G
(A.27)
= μ0 ω2 εG23−G , 36 = i μ0 ω2 e 31 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 32 = i μ0 ω2 e 36 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 34 = i μ0 ω2 e 35 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , = μ0 ω2 εG13−G , = μ0 ω2 εG23−G , = μ0 ω2 εG33−G − (k1 + G 1 )2 δG,G − (k2 + G 2 )2 δG,G ,
(A.28) (A.29) (A.30) (A.31) (A.32) (A.33) (A.34)
Appendix B The explicit expressions of submatrices in Eq. (11) are
(A.2)
16 M G,G = −i e 11 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) ,
M G, G
μ0 ω2 εG22−G − (k1 + G 1 )2 δG,G ,
(A.1)
(14)
(56) M G, G (61) M G, G (62) M G, G (63) M G, G (64) M G, G (65) M G, G (66) M G, G
−C G56−G (k1 + G 1 )(k1 + G 1 ) − C G46−G (k2 + G 2 )(k1 + G 1 ) = , −C G25−G (k1 + G 1 )(k2 + G 2 ) − C G24−G (k2 + G 2 )(k2 + G 2 )
(24)
(25)
(26)
12 M G,G = −i e 16 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 22 M G,G = −i e 26 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 32 M G,G = −i e 36 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) ,
(A.9) (A.10) (A.11) (A.12)
(32) M G, G
−C G56−G (k1 + G 1 )(k1 + G 1 ) − C G25−G (k2 + G 2 )(k1 + G 1 ) = , −C G46−G (k1 + G 1 )(k2 + G 2 ) − C G24−G (k2 + G 2 )(k2 + G 2 )
(34) M G, G (35) M G, G (36) M G, G (41) M G, G (42) M G, G (43) M G, G (44) M G, G (45) M G, G (46) M G, G (51) M G, G (52) M G, G (53) M G, G (54) M G, G
= −i + + G2) , 24 = −i e 25 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 34 = −i e 35 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 16 = i μ0 ω2 e 11 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 12 = i μ0 ω2 e 16 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 14 = i μ0 ω2 e 15 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , = μ0 ω2 εG11−G − (k2 + G 2 )2 δG,G , = μ0 ω2 εG12−G + (k1 + G 1 )(k2 + G 2 )δG,G , = μ0 ω2 εG13−G , 26 = i μ0 ω2 e 21 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 22 = i μ0 ω2 e 26 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 24 = i μ0 ω2 e 25 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , = μ0 ω2 εG12−G + (k1 + G 1 )(k2 + G 2 )δG,G ,
e 15 G−G (k1
G 1 ) + e 14 G−G (k2
(A.13) (A.14) (A.15)
(11)
M G, G = (12) M G, G (16) M G, G (21) M G, G (22) M G, G (26) M G, G (33) M G, G (34) M G, G (35) M G, G (43) M G, G (44) M G, G (45) M G, G (53) M G, G (54) M G, G (55) M G, G (61) M G, G (62) M G, G (66) M G, G
ρω2 − C 11 (k1 + G 1 )2 − C 66 (k2 + G 2 )2 δG,G ,
= −(C 12 + C 66 )(k1 + G 1 )(k2 + G 2 ) δG,G , = −i e 31 G−G (k1 + G 1 ) , = −(C 12 + C 66 )(k1 + G 1 )(k2 + G 2 ) δG,G , = ρω2 − C 66 (k1 + G 1 )2 − C 11 (k2 + G 2 )2 δG,G , = −i e 31 G−G (k2 + G 2 ) , = ρω2 − C 44 (k1 + G 1 )2 − C 44 (k2 + G 2 )2 δG,G , = −i e 15 G−G (k1 + G 1 ) , = −i e 15 G−G (k2 + G 2 ) , = i μ0 ω2 e 15 G−G (k1 + G 1 ) , = μ0 ω2 ε11 δG,G − (k2 + G 2 )2 δG,G ,
(B.1) (B.2) (B.3) (B.4) (B.5) (B.6) (B.7) (B.8) (B.9) (B.10) (B.11)
= (k1 + G 1 )(k2 + G 2 )δG,G , = i μ0 ω2 e 15 G−G (k2 + G 2 ) ,
(B.12)
= (k1 + G 1 )(k2 + G 2 )δG,G , = μ0 ω2 ε11 δG,G − (k1 + G 1 )2 δG,G , = i μ0 ω2 e 31 G−G (k1 + G 1 ) , = i μ0 ω2 e 31 G−G (k2 + G 2 ) , = μ0 ω2 ε33 δG,G − (k1 + G 1 )2 δG,G − (k2 + G 2 )2 δG,G .
(B.14)
(B.13)
(B.15) (B.16) (B.17) (B.18)
References
(A.16) (A.17) (A.18) (A.19) (A.20)
[1] [2] [3] [4] [5] [6]
(A.21)
[7] [8] [9]
(A.22)
[10]
(A.23)
[11]
(A.24)
[12] [13] [14] [15] [16]
(A.25) (A.26)
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Phonon–polariton in two-dimensional piezoelectric phononic crystals Ming-Yi Yang a , Liang-Chieh Wu a , Jiun-Yi Tseng b,∗ a b
Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China Materials and Chemical Research Laboratories, Industrial Technology Research Institute, Hsinchu, Taiwan 31040, Republic of China
a r t i c l e
i n f o
Article history: Received 14 March 2008 Received in revised form 15 April 2008 Accepted 5 May 2008 Available online 15 May 2008 Communicated by R. Wu PACS: 43.20.+g 43.40.+s 73.50.Rb
a b s t r a c t The phonon–polariton behaviors of two-dimensional piezoelectric phononic crystals (PPCs) were studied using the plane wave expansion method. The governing equations combine Maxwell’s equations and Newton’s equations of motion. A mode-repulsion can be formed by strong coupling between electromagnetic (EM) waves and elastic waves in the vicinity of the center of the first Brillouin zone for PPC that comprises piezoelectric material and with opposite polarization in different periodically organized areas. Take a 2D ZnO PPC as a numerical example, it was decoupled into two independent groups. One refers to the mixed mode of the in-plane elastic waves and the transverse-magnetic (TM) mode EM waves. The other group refers to the mixed mode of the out-of-plane elastic waves and the transverse-electric (TE) mode EM waves. Coupling repulsion is also observed in these two groups. © 2008 Elsevier B.V. All rights reserved.
Keywords: Phonon–polariton Piezoelectric phononic crystals Plane-wave-expansion method Repulsion
1. Introduction As micro-fabrication technology advances, artificial microstructures, such as periodic structures, with desired properties can be manufactured more easily. In solid state physics, the band gaps that prohibiting the electronic wave propagation are generated by a periodic potential [1]. A similar phenomenon is also observed in such dielectric-modulated materials as photonic crystals [2–5]. The applications of photonic crystals include spontaneous emission suppression, light path manipulation and lasers. Recently, spacemodulated elastic structures, known as the phononic crystals, have been investigated [6–9]. Their band gap characteristics are more complex than those of photonic crystals and depend on the elastic constants, piezoelectric coefficients, density and other factors. Phononic crystals have been developed for vibration suppression, elastic wave guides and localized waves. In photonic/phononic crystals, Bragg scattering is responsible for most band gaps. However, the band gap has also been demonstrated in locally resonant structures [10–15], in which the order of magnitude of the central frequency of the spectral gaps is lower than that of structures with the same lattice constant. Outside the spectral gaps, the characteristics of the flat dispersion curves and
*
Corresponding author. Tel.: +886 3 5919281; fax: +886 3 5820386. E-mail address: [email protected] (J.-Y. Tseng).
0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.05.012
the modes with zero group velocity, which correspond to a localized mode, have been found [11,13]. Huang [16] investigated ionic crystals, which are also regarded as locally resonant structures. He found that exciting lattice vibration by applying an external radiation field causes a polaritonlike band gap at infrared frequencies. Zhu and Chen [17] study the polariton band gap at microwave frequencies in one-dimensional superlattice structures of LiNbO3 . This structure is also called ionic-type phononic crystals or piezoelectric phononic crystals. The periodic space modulation of the piezoelectric coefficients causes the elastic fields to be coupled with the electromagnetic (EM) fields. Consequently, external EM fields can induce the crossed-field or in-line field acoustic excitation and the resonant frequency depends on the periodicity of the reversed domain. The methods of plane wave expansion and transfer matrix are utilized to investigate piezoelectric phononic crystals (PPCs) that are composed of various materials to verify the dependence of the band-gap frequency on the periodicity of the structure [17–24]. According to these references, it is worthwhile to do more studies on this subject. This work explores the hybridization of photons and phonons (also called as phonon–polaritons) in the generalized two-dimensional PPCs. The coupled equations are derived from the combination of Maxwell’s equations with Newton’s equations of motion. The plane-wave expansion method yielded the band structures of the PPCs. The equations of PPCs that were composed of class 6mm materials could be decoupled into two groups: the first comprised
M.-Y. Yang et al. / Physics Letters A 372 (2008) 4730–4735
in-plane elastic waves and the TM-mode EM waves which the polarizations of magnetic fields are parallel to x1 –x2 plane, and the other comprised out-of-plane elastic waves and the TE-mode EM waves which the polarizations of electric fields are parallel to x1 –x2 plane. A two-dimensional PPC composed of ZnO was demonstrated as an example. The simulation showed that the difference between the wavelengths of photons and phonons was responsible for the fact that the band structures in the first Brillouin zone were similar to those associated with pure phonons. Also, the periodicity of the piezoelectric coupling effect reveals that photons and phonons interacted with each other within the normalized wave vector region of order 10−4 . This coupling effect was responsible for repulsion. If the mode-repulsion effect is large enough, it will induce a gap in the band structures. This investigation demonstrates that the coupling effect of two-dimensional PPCs was more complex than that of traditional one-dimensional PPCs, indicating that the interval between two arising frequencies of repulsion or the gaps is not a multiple of fundamental frequency. 2. Governing equations in two-dimensional PPCs Fig. 1 depicts a two-dimensional PPC structure, comprising two materials, A and B, with periodicity a and embedded circular cylinders of radius r0 . The x1 -axis and x2 -axis are in-plane, and the x3 -axis lies in the thickness direction; the thickness is assumed to be infinite. The structure consists of triclinic material, which has the lowest symmetry of all crystal systems. The interaction in this structure between the EM fields and the elastic fields, which does not occur in photonic crystals or phononic crystals, was considered. Based on the assumption that the media were lossless without a source, the coupled equations of the EM fields and the elastic fields can be expressed as [25]
−∇ × ∇ × E = με s · ∇
∂2E ∂t2
2 · C : (∇s · u) = ρ ∂∂ t 2u
E
+ μe : ∇ s ·
∂2u ∂t2
,
(1)
+ ∇ · (e · E), T
where
T
(∇×) = (−∇×) = ∇· = (∇s ·)T =
0
−∂/∂ x3
∂/∂ x3 −∂/∂ x2
∂/∂ x1
∂/∂ x1
0
0 0
∂/∂ x2 0
0 0 0
∂/∂ x2 −∂/∂ x1 , 0 0
∂/∂ x3 ∂/∂ x3 ∂/∂ x2
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For plane waves that are traveling through PPC structures and propagating in the x1 –x2 plane, the partial derivatives of the electric and elastic fields with respect to x3 can be assumed to be zero. Also, according to the Bloch theorem, the displacement and electric fields are given by,
u i (x, t ) = e i (k·x−ωt ) G U¯ i (G)e iG·x , E i (x, t ) = e i (k·x−ωt ) G E¯ i (G)e iG·x ,
(4)
where k = (k1 , k2 ) is the wave vector in the first Brillouin zone; U¯ i (G) and E¯ i (G) are the coefficients of the plane waves, and G = (G 1 , G 2 ) = (2π n1 /a, 2π n2 /a) is the reciprocal lattice vector in which n1 , n2 ∈ I . In the unit-cell of PPC, the elastic constants, piezoelectric constants, permittivity and density can all be expanded as Fourier series:
⎧ pq iG·x ⎪ ⎪ C pq = G C G e , ⎪ ip iG·x ⎨e = , ip G eG e i j iG·x ⎪ ε = ε e , ⎪ ij ⎪ G G ⎩ ρ = G ρG e iG·x ,
( p , q = 1–6; i , j = 1–3).
Considering N plane waves in Eq. (4) and substituting Eqs. (4) and (5) into Eq. (1), yields M · U = 0,
(6)
where M is a matrix of dimensions 6N × 6N, whose explicit form is given in Appendix A, and U is a vector of dimension 6N as
U = U¯ 1 (G )
U¯ 2 (G )
U¯ 3 (G )
E¯ 1 (G )
E¯ 2 (G )
∂/∂ x3 ∂/∂ x2 0 ∂/∂ x1 , ∂/∂ x1 0 (3)
are matrices of space-differential operators; E is the electric field; ε s is the permittivity matrix in a constant strain; e is the piezoelectric-stress matrix; u is the elastic field; C E is the elastic matrix in a constant electric field, and ρ is the density.
E¯ 3 (G )
T
.
(7)
The existence of a nontrivial solution U indicates that the determinant of the matrix M must be zero, such that |M| = 0. Solving this eigenvalues/eigenvectors problem yields the dispersion relations between the frequency ω and the wave vector k. A 6mm-symmetric PPC structure, in which materials A and B are identical but with opposite polarization directions along the x3 -axis, is considered. The piezoelectric constants of materials A and B differ only in sign. The elastic constants, piezoelectric constants and permittivity of material A can be expressed as follows;
⎤
⎡
(2)
(5)
C 11 C 12 C 13 0 0 0 0 0 ⎥ ⎢ C 12 C 11 C 13 0 ⎥ ⎢ C 13 C 33 0 0 0 ⎥ ⎢C C = ⎢ 13 ⎥, 0 0 C 44 0 0 ⎥ ⎢ 0 ⎦ ⎣ 0 0 0 0 C 44 0 1 0 0 0 0 0 (C 11 − C 12 ) 2 0 0 0 0 e 15 0 e= 0 0 0 e 15 0 0 , e 31 e 31 e 33 0 0 0 ε11 0 0 ε = 0 ε11 0 . 0 0 ε33
(8)
(9)
(10)
Substituting Eqs. (8), (9) and (10) into Eq. (6), yield a set of equations,
⎡
(11)
MG,G
⎢ (21) ⎢ MG,G ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎣ 0 (61) MG,G
Fig. 1. Schematic diagram of a 2D PPC.
(12)
0
0
0
MG,G
MG,G
(22)
0
0
0
MG,G ⎥
0
MG,G
(33)
MG,G
(34)
MG,G
(35)
0
0
MG,G
(43)
MG,G
(44)
MG,G
(45)
0
0
MG,G
(53)
MG,G
(54)
MG,G
0
0
(66) MG,G
(62) MG,G
0
0
(55)
(16)
⎤
MG,G
(26) ⎥
⎥ ⎥ ⎥ ⎥ · U = 0, ⎥ ⎥ ⎥ ⎦
(11)
where matrix elements are as listed in Appendix B. Eq. (11) reveals that the set of equations can be decoupled into two groups. One group represents a mixed mode of the in-plane elastic wave and the TM-mode EM wave, which is composed of u 1 , u 2 and E 3 , where the E 3 component reveals that there will be H 1 and H 2
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Table 1 Material properties of ZnO Stiffness (1010 N/m2 )
ZnO
Piezoelectric constant (C/m2 )
Permittivity (1/ε0 )
Density (kg/m3 )
C 11
C 33
C 44
C 12
C 13
e 15
e 31
e 33
ε11
ε33
ρ
20.97
21.09
4.247
12.11
10.51
−0.48
−0.573
1.32
8.55
10.2
5680
Fig. 2. Band structure of mixed mode of in-plane elastic and TM electromagnetic waves.
Fig. 3. Band structure of mixed mode of out-of-plane elastic and TE electromagnetic waves.
magnetic fields according to the Maxwell’s equations. Here, we defined that the TM mode is the polarization of magnetic fields parallel to x1 –x2 plane. The other group represents a mixed mode of the out-of-plane elastic wave and the TE-mode EM wave, which is composed of u 3 , E 1 and E 2 , where the E 1 and E 2 components parallel to x1 –x2 plane so we defined it as the TE mode. The determinants of the associated matrix of the two groups yield the corresponding dispersion relations (also called band structures). 3. Numerical results and discussion A PPC structure that consists of ZnO piezoelectric material was used as an example. Table 1 lists the material properties of ip ZnO [25]. In this case, the Fourier coefficients of e G in Eq. (5) are
ip
eG =
A B e ip f + e ip (1 − f ) G = 0, A B (e ip − e ip ) F (G)
f = π r02 /a2 ,
G = 0,
1 F (G) = A − C
(12)
exp(−iG · x) d A ,
(13)
A
where f is the filling fraction; F (G) is the structure function, and A C is the cross-section area of the unit-cell. The structure function can be further expressed as 2 f J 1 (Gr0 )/(Gr0 ) where J 1 is the first order Bessel function. The dispersion relations are calculated using the method of plane-wave expansion. The superposition of 121 plane waves is selected, which means that n1 , n2 ∈ [−5, 5]. Figs. 2 and 3 show the band structures with r0 /a = 0.25 in the mixed modes of u 1 , u 2 and E 3 and u 3 , E 1 and E 2 , respectively. The frequency is normalized with C L , which C L = (C 11 /ρ )1/2 . For PPC that comprises ZnO but with different polarization, the band gaps due to Bragg scattering were almost absent in the overall band structure. In Figs. 2 and 3 the number of dispersion curves start from Γ point, which energy component is almost the elastic energy, is two and one respectively according to Eq. (11). In this region in which the reduced wave vector was large, the behaviors of the mixed modes were approximately quasi-static, and phonon–photon interaction was absent.
Fig. 4. Dispersion relation of mixed mode of in-plane elastic and TM electromagnetic waves near point Γ .
However, in the region where the reduced wave vector was smaller, about ka/(2π ) ≈ 10−4 , phonon–polariton was observed. Figs. 4 and 5 plot the phonon–polariton dispersion relations for different mixed modes near point Γ , where polaritons dominated. The electromagnetic dominated dispersion curve starts from Γ point and has strong coupling with the folded-phonon dispersion curve near the center of the Brillouin zone. It generated the moderepulsion. This phenomenon has also been investigated in locally resonant materials [19]. However, in this study, elastic waves that were excited by external EM fields and repulsion at particular frequencies were observed, for example the points a and b in Fig. 4. These frequencies were not integer multiples of the fundamental frequency. The same situation is also found in one-dimensional quasi-periodic PPC [22], which is never happened in traditional one-dimensional PPC [21,23]. At the same frequency, the order of the period of this two-dimensional PPC structure is 10−4 less than
M.-Y. Yang et al. / Physics Letters A 372 (2008) 4730–4735
4733
Fig. 5. Dispersion relation of mixed mode of out-of-plane elastic and TE electromagnetic waves near point Γ .
Fig. 7. Contour plot of mode shape at point b, where (k1 a/(2π ), k2 a/(2π )) = (1.06 × 10−4 , 0) and ωaC L /(2π ) = 1.409.
at x2 /a = 0, ±1, ±2, . . ., etc., and the S 2 field in materials A and B have opposite phase along x2 direction at x1 /a = 0, ±1, ±2, . . ., etc. Also, Fig. 7 reveals that the S 1 and S 2 fields in materials A and B have opposite phase along x1 direction at x2 /a = 0, ±1, ±2, . . ., etc., and along x2 direction at x1 /a = 0, ±1, ±2, . . ., etc. The first equation in Eq. (1) indicates that the charges induced by the time-varying strain fields reradiate the EM field. The periodic modulation of the piezoelectric coefficient cooperated with the periodic variation of the stain fields enable the reradiated EM fields, whose wavelengths significantly exceeds the elastic wavelengths, to interact with the external EM fields and finally induce a repulsion gap. Other than at points a and b, the simulation shows high-order modes that exhibit similar behaviors. 4. Conclusions
Fig. 6. Contour plot of mode shape at point a, where (k1 a/(2π ), k2 a/(2π )) = (7.2 × 10−5 , 0) and ωaC L /(2π ) = 0.9893.
that of the two-dimensional photonic crystals, because the velocity of photons markedly exceeds that of phonons. The dimensions of the two-dimensional PPC support potential applications of miniaturized EM wave filters or transducers. Figs. 6 and 7 show the mode shape contours plot of points a and b in Fig. 4, where the reduced wave vectors are (k1 a/(2π ), k2 a/(2π )) = (7.2 × 10−5 , 0) and (k1 a/(2π ), k2 a/(2π )) = (1.06 × 10−4 , 0) respectively, the reduced frequencies are 0.9893 and 1.409, respectively, and the region of plot is within one unit cell. Fig. 6 reveals that the S 1 field in material A has the same phase while those in material B has opposite phase along x1 direction
This investigation derived the phonon–polariton dispersion relation in the generalized two-dimensional PPCs. For a twodimensional PPC that consists of 6mm material, the dispersion relation can be decoupled into two independent groups: one group represented a mixed mode of the in-plane elastic waves and the TM-mode EM waves; the other represented a mixed mode of the out-of-plane elastic waves and the TE-mode EM waves. Both systems exhibited phonon–polariton behaviors near the Γ point of the first Brillouin zone. The constitutive equations of piezoelectric material and the periodic structures reveal interaction between the EM waves and the elastic waves and strong coupling between phonons and photons. Numerous bands involved in the strong coupling of the elastic waves and the EM waves were formed; these could be applied to multi-band filters whose dimensions can be significantly smaller than those of filters that use two-dimensional photonic crystals.
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M.-Y. Yang et al. / Physics Letters A 372 (2008) 4730–4735
(55)
M G, G =
Acknowledgements The authors would like to thank the MOEA, Taiwan, for financially supporting this research under Contract No. 7354DB3220. Prof. Chou of National Taiwan University is appreciated for his valuable discussions. Appendix A The explicit expressions of submatrices in Eq. (6) are ρG−G ω2 − C G11−G (k1 + G 1 )(k1 + G 1 ) − C G16−G (k2 + G 2 )(k1 + G 1 ) (11) , M G, G = −C G16−G (k1 + G 1 )(k2 + G 2 ) − C G66−G (k2 + G 2 )(k2 + G 2 )
(12)
M G, G =
(13)
M G, G =
−C G16−G (k1 + G 1 )(k1 + G 1 ) − C G12−G (k2 + G 2 )(k1 + G 1 ) , −C G66−G (k1 + G 1 )(k2 + G 2 ) − C G26−G (k2 + G 2 )(k2 + G 2 )
(15) (16)
M G, G (21)
M G, G
−C G15−G (k1 + G 1 )(k1 + G 1 ) − C G14−G (k2 + G 2 )(k1 + G 1 ) , −C G56−G (k1 + G 1 )(k2 + G 2 ) − C G46−G (k2 + G 2 )(k2 + G 2 )
(A.3) (A.4)
26 = −i e 21 (A.5) G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 31 = −i e G−G (k1 + G 1 ) + e 36 (A.6) G−G (k2 + G 2 ) , 16 66 −C G−G (k1 + G 1 )(k1 + G 1 ) − C G−G (k2 + G 2 )(k1 + G 1 ) = , 26 12 −C G−G (k1 + G 1 )(k2 + G 2 ) − C G−G (k2 + G 2 )(k2 + G 2 )
(A.7) ρG−G ω2 − C G66−G (k1 + G 1 )(k1 + G 1 ) − C G26−G (k2 + G 2 )(k1 + G 1 ) (22) , M G, G = 26 22 −C G−G (k1 + G 1 )(k2 + G 2 ) − C G−G (k2 + G 2 )(k2 + G 2 ) (A.8)
(23) M G, G
(A.27)
= μ0 ω2 εG23−G , 36 = i μ0 ω2 e 31 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 32 = i μ0 ω2 e 36 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 34 = i μ0 ω2 e 35 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , = μ0 ω2 εG13−G , = μ0 ω2 εG23−G , = μ0 ω2 εG33−G − (k1 + G 1 )2 δG,G − (k2 + G 2 )2 δG,G ,
(A.28) (A.29) (A.30) (A.31) (A.32) (A.33) (A.34)
Appendix B The explicit expressions of submatrices in Eq. (11) are
(A.2)
16 M G,G = −i e 11 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) ,
M G, G
μ0 ω2 εG22−G − (k1 + G 1 )2 δG,G ,
(A.1)
(14)
(56) M G, G (61) M G, G (62) M G, G (63) M G, G (64) M G, G (65) M G, G (66) M G, G
−C G56−G (k1 + G 1 )(k1 + G 1 ) − C G46−G (k2 + G 2 )(k1 + G 1 ) = , −C G25−G (k1 + G 1 )(k2 + G 2 ) − C G24−G (k2 + G 2 )(k2 + G 2 )
(24)
(25)
(26)
12 M G,G = −i e 16 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 22 M G,G = −i e 26 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 32 M G,G = −i e 36 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) ,
(A.9) (A.10) (A.11) (A.12)
(32) M G, G
−C G56−G (k1 + G 1 )(k1 + G 1 ) − C G25−G (k2 + G 2 )(k1 + G 1 ) = , −C G46−G (k1 + G 1 )(k2 + G 2 ) − C G24−G (k2 + G 2 )(k2 + G 2 )
(34) M G, G (35) M G, G (36) M G, G (41) M G, G (42) M G, G (43) M G, G (44) M G, G (45) M G, G (46) M G, G (51) M G, G (52) M G, G (53) M G, G (54) M G, G
= −i + + G2) , 24 = −i e 25 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 34 = −i e 35 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 16 = i μ0 ω2 e 11 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 12 = i μ0 ω2 e 16 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 14 = i μ0 ω2 e 15 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , = μ0 ω2 εG11−G − (k2 + G 2 )2 δG,G , = μ0 ω2 εG12−G + (k1 + G 1 )(k2 + G 2 )δG,G , = μ0 ω2 εG13−G , 26 = i μ0 ω2 e 21 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 22 = i μ0 ω2 e 26 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , 24 = i μ0 ω2 e 25 G−G (k1 + G 1 ) + e G−G (k2 + G 2 ) , = μ0 ω2 εG12−G + (k1 + G 1 )(k2 + G 2 )δG,G ,
e 15 G−G (k1
G 1 ) + e 14 G−G (k2
(A.13) (A.14) (A.15)
(11)
M G, G = (12) M G, G (16) M G, G (21) M G, G (22) M G, G (26) M G, G (33) M G, G (34) M G, G (35) M G, G (43) M G, G (44) M G, G (45) M G, G (53) M G, G (54) M G, G (55) M G, G (61) M G, G (62) M G, G (66) M G, G
ρω2 − C 11 (k1 + G 1 )2 − C 66 (k2 + G 2 )2 δG,G ,
= −(C 12 + C 66 )(k1 + G 1 )(k2 + G 2 ) δG,G , = −i e 31 G−G (k1 + G 1 ) , = −(C 12 + C 66 )(k1 + G 1 )(k2 + G 2 ) δG,G , = ρω2 − C 66 (k1 + G 1 )2 − C 11 (k2 + G 2 )2 δG,G , = −i e 31 G−G (k2 + G 2 ) , = ρω2 − C 44 (k1 + G 1 )2 − C 44 (k2 + G 2 )2 δG,G , = −i e 15 G−G (k1 + G 1 ) , = −i e 15 G−G (k2 + G 2 ) , = i μ0 ω2 e 15 G−G (k1 + G 1 ) , = μ0 ω2 ε11 δG,G − (k2 + G 2 )2 δG,G ,
(B.1) (B.2) (B.3) (B.4) (B.5) (B.6) (B.7) (B.8) (B.9) (B.10) (B.11)
= (k1 + G 1 )(k2 + G 2 )δG,G , = i μ0 ω2 e 15 G−G (k2 + G 2 ) ,
(B.12)
= (k1 + G 1 )(k2 + G 2 )δG,G , = μ0 ω2 ε11 δG,G − (k1 + G 1 )2 δG,G , = i μ0 ω2 e 31 G−G (k1 + G 1 ) , = i μ0 ω2 e 31 G−G (k2 + G 2 ) , = μ0 ω2 ε33 δG,G − (k1 + G 1 )2 δG,G − (k2 + G 2 )2 δG,G .
(B.14)
(B.13)
(B.15) (B.16) (B.17) (B.18)
References
(A.16) (A.17) (A.18) (A.19) (A.20)
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(A.21)
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(A.22)
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(A.24)
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(A.25) (A.26)
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