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SH waves in multilayered piezoelectric semiconductor plates with imperfect interfaces
Journal Pre-proof SH waves in multilayered piezoelectric semiconductor plates with imperfect interfaces Ru Tian, Jinxi Liu, Ernian Pan, Yuesheng Wang PII:
S0997-7538(19)30996-9
DOI:
https://doi.org/10.1016/j.euromechsol.2020.103961
Reference:
EJMSOL 103961
To appear in:
European Journal of Mechanics / A Solids
Received Date: 10 December 2019 Revised Date:
24 January 2020
Accepted Date: 27 January 2020
Please cite this article as: Tian, R., Liu, J., Pan, E., Wang, Y., SH waves in multilayered piezoelectric semiconductor plates with imperfect interfaces, European Journal of Mechanics / A Solids (2020), doi: https://doi.org/10.1016/j.euromechsol.2020.103961. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Masson SAS.
SH waves in multilayered piezoelectric semiconductor plates with imperfect interfaces Ru Tian1,4, Jinxi Liu2,3∗ Ernian Pan4, Yuesheng Wang1 1.
Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, P. R. China
2.
Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, P. R. China
3.
Hebei Key Laboratory of Mechanics of Intelligent Materials and Structures, Shijiazhuang Tiedao University, Shijiazhuang 050043, P. R. China
4.
Department of Civil Engineering, University of Akron, Akron, OH 44325-3905, USA
Abstract: In this paper, analytical solutions for SH waves in transversely isotropic multilayered piezoelectric semiconductor (PSC) plates with imperfect interfaces are obtained. The extended displacements and stresses are expressed in terms of the eigenvalues and eigenvectors by introducing the extended Stroh formalism. Making use of the dual variable and position (DVP) method and the imperfect interface conditions, the transfer matrix which relates the extended displacement and traction on the lower and upper interfaces of the multilayered plates is derived. Then the dispersion relation is obtained by using the boundary conditions on the top and bottom surfaces of the multilayered plates. Effect of the steady-state carrier density in single-layer ZnO plate, and effect of stacking sequences and imperfect interfaces in sandwich plates are discussed via numerical examples. Particularly, the critical elastic (E), piezoelectric (PE), and PSC wave domains are identified for the given carrier density, plate thickness and frequency, which could be very helpful as theoretical guidance for the design of PSC devices. Keywords: SH wave; piezoelectric semiconductor plate; dispersion relation; ∗
Corresponding author, E−mail: [email protected](J.-X. Liu) 1
multilayered plates; imperfect interface
1. Introduction
Recently, researchers obtained various novel PSC nanostructures such as nanobelts, nanofibers, nanowires and thin film (Wang, 2003; Heo et al., 2004; Olson et al., 2006; Duraisamy et al., 2014). The PSC materials with electromechanical coupling and semiconductor (SC) characteristics have attracted intensive attention. During the past two decades, various studies on PSC have been done, which focused on two aspects. One is the prediction of the properties of PSC materials and devices (Pearton et al., 2005; Liu et al., 2011; Song et al., 2017; Mishra and Pak, 2017; Latiff et al., 2018; Morales-Mendoza et al., 2019), and the other is the investigation of the multi-field coupling mechanics among elasticity, electricity and semiconductor such as the dynamic bending and extension of fibers (Araneo et al., 2012; Zhang et al., 2016; Wang et al., 2018), elastic waves propagation (Yang et al., 2004; Gu and Jin, 2016; Cao et al., 2019; Jiao et al., 2019a) and fracture mechanics (Yang, 2005; Sladek et al., 2014; Zhao et al., 2018; Qin et al., 2019). Due to the simultaneous possession of piezoelectricity and SC properties, the PSC materials can be used as sensors (Kalantar-Zadeh et al., 2001; Lin et al., 2011; Rahman et al., 2018), transistors (Fernandes and Mulato, 2014; Rasheed et al., 2017; Li et al., 2018; Yang et al., 2018;), resonators (Takeuchi et al., 2003; Luo et al., 2013; Bian et al., 2015) as well as other PSC devices. These applications call for a better understanding of the dynamic behaviors of PSC composites, especially the propagation characteristics of the elastic waves in the multilayered PSC structures. Since the potential created in the PSC layer has a strong effect on the carrier transport at the interface (Wang, 2012), the interface plays an important role in the PSC multilayered structures. At the same time, due to various causes such as microinhomogeneities, microdefects, and microdebonding (Nie et al., 2016), the interfaces may not be well-bonded which we call imperfect in layered structures. The interfacial imperfection can affect the wave propagation behavior, so we need to 2
consider the effect of imperfect interfaces. It is known that the wave in the PSC is both dispersive and diffusive due to the interaction between PE and SC (Hutson and White, 1962; White, 1962; White, 1967). As a result, wave propagation in PSC structures is of great interest for researchers, particularly when it propagates in layered structures. For a single layer PSC plate, and based on the power series expansions method, Yang and Zhou (2005) obtained the two-dimensional equation of thin PSC plate and further numerically calculated the effect of SC characteristic and biasing electric field on the propagation characteristics of thickness-shear waves; Tian et al. (2019) systematically studied the effect of the surface boundary condition, steady-state carrier density, plate thickness and biasing electric field on the wave speed and attenuation of both SH and Lamb waves in the transversely isotropic ZnO PSC plate by using the extended Stroh formalism. For semi-infinite PSC, propagation of shear horizontal (SH) waves (Gu and Jin, 2015) and generalized Rayleigh surface waves (Cao et al., 2019) as well as coupled elastic waves (Jiao et al., 2019b) were all studied. Jiao et al. (2019a) investigated the reflection and transmission of elastic waves in a sandwich PSC slab, and found that the steady-state carrier density and the biasing electric field have a great influence on the reflection and transmission coefficients. Yang et al. (2004) discussed the dispersion curves in the PSC multilayered plates by using the power series expansions method. To the authors' best knowledge, however, no work on wave propagation in multilayered plates with imperfect interfaces has been reported, which motivates the present study. This paper focus on the propagation characteristic of SH waves in the PSC multilayered plates with imperfect interfaces. This paper is organized as follows. In section 2, we describe the problem to be solved via the basic equations, interface and boundary conditions. In section 3, the dispersion equation in multilayered PSC plates with imperfect interfaces is derived based on the Stroh formalism and DVP method. In section 4.1, 3D dispersion curves of SH wave in a single-layer PSC plate with varying steady-state carrier density and the corresponding mode shapes are presented. The effect of stacking sequences and imperfect interfaces on the dispersion and 3
attenuation curves of sandwich plates are investigated in Section 4.2. Finally, conclusions are drawn in Section 5.
2. Problem statements with basic equations
As shown in Fig. 1, we consider the propagation of SH waves in the multilayered transverse isotropic PSC plates which are horizontally infinite but vertically finite. The SH wave propagates along positive x1 direction. The jth PSC plate bonded by its and upper interface xj2 with thickness hj. The layered structure has lower interface xj−1 2 n layers and the total thickness of the multilayered PSC plates is h=h1+h2+…+hn.
Fig. 1. SH waves propagation in x1-drection in the PSC multilayered plates.
2.1 Governing equations
Using the extended notation introduced by Barnett and Lothe (1975) for PE material and generalized to magneto-electro-elastic coupling by Pan (2002), the equilibrium equations and the constitutive relations of each layer can be written as
σ iJ ,i = FJ 4
(1)
σ iJ = ciJMlγ Ml
(2)
where repeated lowercase subscripts take the summation from 1 to 3, whilst those of the uppercase subscripts take the summation from 1 to 5; and the extended displacements, stresses, strains and body forces are defined as
ui uI = ϕ n
( I = i = 1, 2,3) , ( I = 4) ( I = 5)
γ ij = 0.5 ( ui , j + u j ,i ) γ Ij = − E j = ϕ, j N = n ,j j
σ ij σ iJ = Di J i
( I = i = 1, 2,3) , ( I = 4) ( I = 5)
( J = j = 1, 2,3) , ( J = 4) ( J = 5)
ρ u&&j ( J = j = 1, 2,3) FJ = qn ( J = 4) −qn& ( J = 5)
(3)
where uk, φ and n are the displacements, electric potential, and carrier density, respectively; σij, Di and Ji are the stresses, electric displacements and electric currents, respectively; γij, Ej and Nj are strains, electric field, and carrier density gradient, respectively; ρ is the mass density, q=1.602×10−19C is the carrier charge constant, a dot over a quantity denotes the differentiation with respect to time, a subscript comma denotes the partial differentiation with respect to the coordinates. Also in Eq. (2), the extended material coefficients are defined as
ciJMl
cijml ( J = M = j, m = 1, 2,3) ( J = j = 1, 2,3; M = 4 ) elij 0 ( J = 1, 2,3, 4; M = 5) ( J = 4; M = m = 1, 2,3) e = iml ( J = M = 4) −ε il 0 ( J = 5; M = 1, 2,3) − qn0 µil ( J = 5; M = 4 ) − qd ( J = M = 5) il
(4)
where cijkl, ekij and εij are the elastic, piezoelectric and dielectric coefficients, respectively; µij and dij are the carrier mobility and diffusion coefficients, respectively; and n0 is the steady-state carrier density. For the propagation of SH wave in a transverse isotropic PSC plate, the extended displacements, stresses, strains, body forces and material coefficients that we need to consider are the components in Eqs. (3) and (4) with lowercase subscripts from 1 to 2, 5
and the uppercase subscripts from 3 to 5. Then by substituting the nonzero components into Eqs. (1) and (2), the equilibrium equations and the constitutive relations are obtained.
2.2 boundary conditions
In the multilayered PSC plates, we usually assume that the extended displacements and tractions (defined below) are perfectly bonded on the top and bottom interfaces of the layer. However, if the interface is imperfect, the interface conditions say at x2=xj2 need to be generalized as (Wang and Pan, 2007; Pan, 2019) T ( x2j + ) =T ( x2j − ) ,
U ( x2j + ) − U ( x2j − ) = α j T ( x2j )
(5)
with
α11j 0 0 α j = 0 α 22j 0 0 0 α 33j
(6)
where U=[u3, φ, n]T is the extended displacement vector, and T=[σ23, D2, J2]T the extended traction vector; T denotes vector or matrix transpose; j=0,1,2,…,n, and α0=αn=0 (i.e., on the top and bottom surfaces of the layered structure). It should be noted that if αii=0, the general interface is reduced to the perfect one. Furthermore, if αii are very large, the interface corresponds to a completely debonded and separate one. On the surfaces of the multilayered PSC plates, we assume that the extended traction is zero, namely T ( x20 ) = 0, T ( x2n ) = 0
3. Dispersion equation
6
(7)
For the wave propagating in the positive x1 direction, the time-harmonic solutions of the extended displacements can be assumed as
u I =aI eikx1 −iωt eiskx2
(8)
where k is the complex wavenumber. The real part of k, i.e. Re(k), indicates the wave propagation; the imaginary part of k, i.e. Im(k), denotes the wave attenuation. For the problem we considered in this paper we require Re(k) >0 for wave which propagates along the positive x1 direction and Im(k)>0 for wave which decays; ω is the angular frequency; i is the imaginary unit; s is the eigenvalue, and aI (I=3-5;) the unknown coefficients, both to be determined. Substituting Eq. (8) into Eq. (2), the generalized traction on the plane perpendicular to the x2-axis is found to be
σ 2 J =ikbJ eikx −iωt eiskx 1
2
(9)
where bJ (J=3-5) are the unknown coefficients. Inserting Eqs. (8) and (9) into Eq. (2), yields b = s[M ]a
(10)
where a=[ a3 a4 a5]T is related to the generalized displacement vector, b=[b3 b4 b5]T to the generalized traction vector; and the elements of matrices [M] are
M ( J − 2)( L − 2) = c2 JL 2
(11)
where J, L=3−5. Substituting Eq. (8)-(9) into Eq. (1) yields
[Q+s2 M ]a = 0
(12)
Q( J − 2)( L − 2) = c1JL1 +X ( J −2)( L −2) ( j = 1 − 3),
(13)
where
X ( J −2)( L−2)
− ρω 2 / k 2 2 q / k = 2 −iqω / k 0
( J = L = 3) ( J = 4; L = 5) ( J = 5; L = 5) ( J = 3; L = 4 − 5 or J = 4 − 5; L = 3) 7
(14)
Then combining Eqs. (10) and (12), the following linear eigenvalue system can be obtained
a a = s b b
[N]
(15)
where 0 [ N ]= −Q
M −1 0
(16)
From Eq.(15), we obtain six eigenvalues and the corresponding six eigenvectors a and b. Based on them, the general solutions of the extended displacement and traction vectors can be written as (omitting the proportional coefficient e ikx1 − iω t )
ikU ( x2 ) E11 = x T ( ) 2 E 21
ik ( x2 − x2j ) s− e E12 E 22 0
0 e
(
)
ik x2 − x2j −1 s+
c− c +
(17)
where <> denotes the diagonal matrix of 3×3. Notice that the six eigenvalues are ordered as such that Im(s+)Re(k)+ Im(k)Re(s+) ≥ Im(s−)Re(k)+Im(k)Re(s−), and [E] is the eigenmatrix made of the corresponding eigenvectors. The coefficient vectors c− and c+ can be determined for the given boundary and interface conditions discussed below. In this paper, instead of applying the traditional transfer matrix method and stiffness matrix method, we use the DVP method which was recently developed (see, e.g., Pan, 2019). Compared to the transfer and stiffness matrix methods, the DVP method is much stable at large/small k and thin/thick layer and is computationally more efficient, especially when the structures involve many layers and there is an evanescent wave in the layered structures. Then, by following Liu and Pan (2018a, 2018b), the layer matrix relation for layer j from xj−1+ to xj−2 can be derived as 2
ikU ( x2j −1+ ) S j = 11 j− T ( x2 ) S 21j 8
j− S12j ikU ( x2 ) S 22j T ( x2j −1+ )
(18)
where
S S
j 11 j 21
- ikh j s− S E11 e = S E 21 j 12 j 22
E12 E 22 e
ikh j s+
E11 E e- ikh j s− 21
E12 e
ikh j s+
E 22
−1
(19)
to xj+2 (passing Combined Eqs. (5) and (18), the propagate matrix of layer j from xj−1+ 2 also the imperfect interface) can be expressed as ikU ( x2j −1+ ) S j = 11int j+ T ( x2 ) S 21j int
j+ S12j int ikU ( x2 ) S22j int T ( x2j −1+ )
(20)
where −1
S11j int = S11j − ikS11j α j I + ikS21j α j S21j , −1
S12j int = S12j − ikS11j α j I + ikS21j α j S22j , S
j 21int
−1
= I + ikS α j S , j 21
(21)
j 21
−1
S22j int = I + ikS21j α j S22j Similarly, for layer j+1 the transfer matrix from xj+2 on its bottom to xj+1+ on its top 2 can be obtained. Then the transfer matrix from layer j to layer j+1 can be found as ikU ( x2j −1+ ) S j: j +1 = 11 j +1+ T ( x2 ) S 21j: j +1
j +1+ S12j: j +1 ikU ( x2 ) S 22j: j +1 T ( x2j −1+ )
(22)
where −1
S11j: j +1 = S11j int S12j +int1 I − S21j int S12j +int1 S21j int S11j +int1 + S11j int S11j +int1 , −1
S12j: j +1 = S11j int S12j +int1 I − S21j int S12j +int1 S22j int + S12j int , −1
1 j j +1 j j +1 j +1 S21j: j +1 = S22j +int I − S 21int S12 int S 21int S11int + S 21int ,
(23)
−1
1 I − S 21j int S12j +int1 S 22j int S22j: j +1 = S22j +int
Making use of the recursive relation (22) repeatedly, the following important relation between the top and bottom surfaces of the layered structure can be derived ikU ( x20 − ) S 0 −:n + = 11 0 −:n + T ( x2n + ) S 21
n+ S120 −:n + ikU ( x2 ) 0 −:n + 0− S 22 T ( x2 )
9
(24)
Using the boundary conditions on the surfaces (i.e. Eq. (7)), the dispersion relation can be finally obtained as 0 −:n + det[ S21 ]=0
(25)
In other words, for a given frequency ω, the wavenumber k can be obtained by solving the above equation.
4. Numerical results and discussions
In this section, we study the propagation characteristics of the SH wave in the transversely isotropic single ZnO PSC plate of thickness h, and in the sandwich plates composed of PSC ZnO and PSC GaN. For the sandwich plates, each layer has the same thickness (e.g. the thickness of each layer is h/3). The material parameters of ZnO (Qin et al., 2010; Qin et al., 2011) and GaN (Pan and Chen, 2015; Zhao et al., 2018) are listed in Table 1. In the following numerical analysis, we will use the dimensionless wavenumber kh, dimensionless frequency Ω, and dimensionless interface moduli r1, r2 and r3, as defined below
Ω = ω h / c11ZnO / ρ ZnO , r1 = c11ZnOα11 / h,
(26)
r2 = −ε11ZnOα 22 / h, r3 = − qd11ZnOα 33 / h Table 1 Material properties properties
ZnO
GaN
properties
ZnO
GaN
c11(GPa)
210
379
µ11(m2/V)
1
6.53×10−2
c44(GPa)
43
98
µ22(m2/V)
1
6.53×10−2
e15(C/m2)
−0.48
−0.33
d11(m2/s)
0.026
16.99×10−4
ε11(10−11C/Vm)
7.61
8.41
d22(m2/s)
0.026
16.99×10−4
ε22(10−11C/Vm)
7.61
8.41
ρ(kg/m3)
5700
6095
4.1 SH wave in single ZnO plate 10
The 3D dispersion curves of SH wave in the single ZnO plate of fixed h=3mm and n0=1×1015m-3 are shown in Fig. 2a, and the corresponding 2D front-view and side-view are plotted in Figs. 2b and 2c. From Fig. 2. we can see that: 1) The first three modes all start from Re(kh)=0; 2) There exist the imaginary part of kh which is positive, showing that the wave is attenuating; 3) With decreasing Ω, the imaginary of kh increases; 4) The attenuation of the 0th mode is much smaller than that of the 1st and 2nd modes at low frequency (Ω<1.5).
Fig. 2 The dispersion and attenuation curves of SH wave in the single ZnO plate with fixed h=3mm and n0=1×1015m-3. (a) The 3D curves of the first three modes, (b) the dispersion curves of the first three modes, and (c) the attenuation curves of the first three modes.
Figure 3 shows the variation of dimensionless frequency with the dimensionless wavenumber for different values of n0 with fixed h=3mm. The following interesting 11
features can be observed from Fig. 3: 1) When n0=1×105m-3, the dispersion relation of the SH wave in the single PSC plate is the same as that in the corresponding PE plate; 2) When n0=1×1015m−3, the 1st and 2nd modes of the SH wave begin at Re(kh)=0, which is different from the corresponding pure elastic and PE plate; 3) When n0=1×1025m−3, the SH wave propagates with the wave speed of SH wave in the corresponding elastic plate. As such, for given h and Ω, there should exist a critical n0 corresponding to the wave speed of SH wave in the ZnO plate from PE material to PSC material and then to elastic (E) material. Figure 4 shows at which combined values n0, h and Ω, one could achieve the purely PE, PSC and E response for the 0th mode. We observe from Fig. 4a that: 1) The combination of n0, h and Ω constitutes two planes; 2) Above the upper plane, the propagation of SH wave in the PSC plate is the same as that in the corresponding E plate; 3) Between the upper and lower planes, the SH wave is dispersive and diffusive; 4) Below the lower plane, the SH wave propagates with the wave speed in the corresponding PE plate.
12
Fig. 3. The 3D dispersion relations and the front-view of the 3D dispersion curves of the first three modes. (a) The 0th mode, (b) the 1st mode, and (c) the 2nd mode.
13
Fig. 4. Phase diagram of the critical values of 0th mode. (a) The 3D phase diagram with n0, h and Ω, and (b) the 2D phase diagram with fixed Ω=1.
The following interesting features can be observed from Fig. 4b: for the critical values from PE to PSC and PSC to E: 1) The relation between log(h) and log(n0) is linear for small value of log(h) (log(h)<−6.2m for PE to PSC and log(h)<−4.9m for PSC to E) and large value of log(h) (−3.3m
Fig. 5. Normalized displacements and stresses on the wave with Re(kh)=0.1. (a) The 0th mode, (b) the 1st mode, and (c) the 2nd mode.
14
Fig. 6. Normalized displacements and stresses on the wave with Re(kh)=4. (a) The 0th mode, (b) the 1st mode, and (c) the 2nd mode.
We further study the mode shapes of the SH wave in the ZnO plate. Six sampling points in Fig. 2a are chosen for the calculation of mode shapes: the points of first three modes with Re(kh) equal to 0.1 and 4. Since all the extended displacements (or stresses) have similar mode shapes except for different numerical values, we only present the mode shapes of the elastic displacements and stresses (tractions). Points of 1st and 2nd modes with Re(kh)=0.1 have a very large Im(kh) compared to Re(kh), so the normalized displacement and stress are all calculated in 0.1 wavelength equals to 0.1×2π/Re(k)=2πh along the wave propagation direction; points of 0th mode with Re(kh) equal to 0.1 and 4 and 1st and 2nd modes with Re(kh)=4 have a very large Re(kh) compared to Im(kh). The normalized displacement and stress are calculated in one wavelength equals to 20πh and 2π/ Re(k)=0.5πh along the wave propagation direction, respectively. Figures 5 and 6 show the variation of first three normalized u3 and σ23 for fixed h=3mm and n0=1×1015m-3 along the thickness and wave propagation directions. The following features can be observed from Figs. 5 and 6: 1) For the 0th mode, the normalized displacements are all constant along the thickness direction of the plate; the normalized stresses are all zero (Figs. 5a and 6a); 2) The 1st and 2nd modes are 15
antisymmetric and symmetric, respectively; 3) For the points with large ratio of Im(kh)/Re(kh), the attenuation of SH wave is obvious along the wave propagation direction (Figs. 5b and 5c); for the points with large ratio of Re(kh)/Im(kh), the attenuation is not obvious (Figs. 5a, 6b and 6c).
4.2 SH waves in sandwich plates
We now apply our solution to the sandwich plates composed of materials ZnO and GaN to investigate the influence of the stacking sequences on dispersion curves. Four sandwich plates are considered: Z/Z/Z, Z/G/Z, G/Z/G, and G/G/G. The letter “Z” denotes ZnO and “G” denotes GaN, with the first and last being homogeneous plates. Again, these three layers are assumed to have equal thickness h/3 with h=3mm. Figure 7 shows the 3D dispersion curves of SH waves for the first three modes in different sandwich plates, and Fig. 8 is the comparison of each mode for the four sandwich plates with fixed n0=1×1015m−3. The interfaces are assumed to be perfect. From Figs. 7 and 8, we observe that: 1) For 0th and 2nd modes, with increasing material GaN, the real part of kh decreases. 2) For SH waves in Z/Z/Z and G/G/G plates, the imaginary part of kh is equal when Ω=0; 3) The imaginary part of kh decreases with increasing material GaN for the 0th mode.
16
Fig. 7. The 3D dispersion curves of SH waves in the sandwich plates. (a) Z/Z/Z, (b) Z/G/Z, (c) G/Z/G, and (d) G/G/G.
17
Fig. 8. The first three modes of SH waves in sadwich plates.
18
19
Fig. 9. The effect of interface modulus r1 on the first three dispersion curves of Z/G/Z plates (r2=r3=0).
20
Fig. 10. The effect of interface modulus r1 on the first three dispersion curves of G/Z/G plates (r2=r3=0). 21
We further study the effect of imperfect interfaces on the dispersion curves of the same sandwich plates. Figures 9 and 10 show the dispersion curves of the first three modes with varying interface parameter r1 (r2=r3=0) for two sandwich plates with fixed n0=1×1015m−3. The two interfaces in the sandwich plates are assumend to have the same imperfection. Figure 9 for the Z/G/Z plate and Fig. 10 for the G/Z/G plate. It is observed that, in general, at a fixed frequency, with increasing interface parameter, the real part of the wavenumber for both Z/G/Z and G/Z/G plates increases; the imaginary part of the wavenumber increases first and then decreases for the 0th mode; the imaginary part of the wavenumber for the 1st and 2nd modes decreases. We further observed that the wavenumber on the 0th dispersive curves are nearly independent of the interface parameter when frequency is low. For the 1st and 2nd dispersion curves, the dimensionless frequency Ω of the critical point (a point which has a larger imaginary part and a smaller real part of kh) decreases with increasing mechianical interface modulus r1. To study the effect of interface parameters r2 and r3 on the dispersion curves of the sandwich plates, the wavenumber of the first three modes of Z/G/Z plates for Ω=3, and n0=1×1015m-3 are listed in Tables 2 (r1=r3=0) and 3 (r1=r2=0). The two interfaces in the sandwich plates are assumed to have the same imperfection. From Tables 2 and 3, we notice that with increasing interface parameters , the dimensionless wavenumber on the same mode changes only slightly.
Table 2 Dimensionless wavenumber kh at Ω=3 in the sandwich Z/G/Z plates with varying interface parameter r2 (r1=r3=0). 0
1
2
Mode r2
Re(kh)
Im(kh)
Re(kh)
Im(kh)
Re(kh)
Im(kh)
0
5.7485
0.0429
5.1244
0.0652
0.2773
0.9320
1
5.7489
0.0450
5.1251
0.0668
0.2779
0.9309
10
5.7490
0.0456
5.1253
0.0672
0.0313
0.9155
100
5.7491
0.0457
5.1254
0.0672
0.2681
0.9369
22
Table 3 Demensionless wavenumber kh at Ω=3 in the sandwich Z/G/Z plates with varying interface parameter r3 (r1=r2=0). 0
1
2
Mode r3
Re(kh)
Im(kh)
Re(kh)
Im(kh)
Re(kh)
Im(kh)
0
5.7485
0.0429
5.1244
0.0652
0.2773
0.9320
1
5.7485
0.0427
5.1245
0.0645
0.2730
0.9285
10
5.7485
0.0427
5.1245
0.0644
0.2730
0.9280
100
5.7485
0.0427
5.1245
0.0644
0.2730
0.9279
5. Conclusions The dispersion and attenuation curves of the SH wave in multilayered PSC plates with imperfect interfaces are derived via the extended Stroh formalism and the DVP method. The effects of the steady-state carrier density on the single-layer ZnO PSC plate, and the effect of stacking sequences and imperfect interfaces on the sandwich plates composed of the PSC ZnO and PSC GaN are studied in detail. Numerical results demonstrate the following features: (1) Due to the semiconductor characteristic of PSC, the SH wave in PSC plate is diffusive, the first three modes of SH wave all start from Re(kh)=0 which is quite different from the SH wave in the corresponding elastic and PE plate. (2) The critical points with combined values of n0, h and Ω are identified where the SH wave propagates in the corresponding PE, PSC, and E plates. These points form two planes to separate the space into three domains: above the upper plane, between the upper and lower planes, and below the lower plane. The SH wave propagates, respectively, with the corresponding wave speed in the E plate, PSC plate and PE plate. (3) For the critical value from PE to PSC and PSC to E, n0h2 is a constant for small value of log(h) (e.g. log(h)<−6.2m for PE to PSC and log(h)<−4.9m for PSC to E for the 0th mode), and n0h is a constant for large value of log(h) (e.g. 23
−3.3m
Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 11472182 and 11272222). The first author is grateful to the support of China Scholarship Council (CSC Grant No. 201907090051).
References
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Highlights The imperfect interfaces characterized by discontinuous mechanical displacement, electric potential and carrier density are considered to study the behavior of SH waves in transversely isotropic multilayered piezoelectric semiconductor (PSC) plates For single-layer ZnO plate, the critical elastic (E), piezoelectric (PE), and PSC wave domains are identified for the given carrier density, plate thickness and frequency. The effect of stacking sequences and imperfect interfaces in sandwich plates are shown. The mechanical imperfect interface would increase the real part of the wavenumber and decrease the imaginary part of wavenumber for the 1st and 2nd mode, while the electric or electric current imperfect interface nearly has no influence on the dispersion and attenuation curves.
Declaration of interests ☐ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
S0997-7538(19)30996-9
DOI:
https://doi.org/10.1016/j.euromechsol.2020.103961
Reference:
EJMSOL 103961
To appear in:
European Journal of Mechanics / A Solids
Received Date: 10 December 2019 Revised Date:
24 January 2020
Accepted Date: 27 January 2020
Please cite this article as: Tian, R., Liu, J., Pan, E., Wang, Y., SH waves in multilayered piezoelectric semiconductor plates with imperfect interfaces, European Journal of Mechanics / A Solids (2020), doi: https://doi.org/10.1016/j.euromechsol.2020.103961. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Masson SAS.
SH waves in multilayered piezoelectric semiconductor plates with imperfect interfaces Ru Tian1,4, Jinxi Liu2,3∗ Ernian Pan4, Yuesheng Wang1 1.
Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, P. R. China
2.
Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, P. R. China
3.
Hebei Key Laboratory of Mechanics of Intelligent Materials and Structures, Shijiazhuang Tiedao University, Shijiazhuang 050043, P. R. China
4.
Department of Civil Engineering, University of Akron, Akron, OH 44325-3905, USA
Abstract: In this paper, analytical solutions for SH waves in transversely isotropic multilayered piezoelectric semiconductor (PSC) plates with imperfect interfaces are obtained. The extended displacements and stresses are expressed in terms of the eigenvalues and eigenvectors by introducing the extended Stroh formalism. Making use of the dual variable and position (DVP) method and the imperfect interface conditions, the transfer matrix which relates the extended displacement and traction on the lower and upper interfaces of the multilayered plates is derived. Then the dispersion relation is obtained by using the boundary conditions on the top and bottom surfaces of the multilayered plates. Effect of the steady-state carrier density in single-layer ZnO plate, and effect of stacking sequences and imperfect interfaces in sandwich plates are discussed via numerical examples. Particularly, the critical elastic (E), piezoelectric (PE), and PSC wave domains are identified for the given carrier density, plate thickness and frequency, which could be very helpful as theoretical guidance for the design of PSC devices. Keywords: SH wave; piezoelectric semiconductor plate; dispersion relation; ∗
Corresponding author, E−mail: [email protected](J.-X. Liu) 1
multilayered plates; imperfect interface
1. Introduction
Recently, researchers obtained various novel PSC nanostructures such as nanobelts, nanofibers, nanowires and thin film (Wang, 2003; Heo et al., 2004; Olson et al., 2006; Duraisamy et al., 2014). The PSC materials with electromechanical coupling and semiconductor (SC) characteristics have attracted intensive attention. During the past two decades, various studies on PSC have been done, which focused on two aspects. One is the prediction of the properties of PSC materials and devices (Pearton et al., 2005; Liu et al., 2011; Song et al., 2017; Mishra and Pak, 2017; Latiff et al., 2018; Morales-Mendoza et al., 2019), and the other is the investigation of the multi-field coupling mechanics among elasticity, electricity and semiconductor such as the dynamic bending and extension of fibers (Araneo et al., 2012; Zhang et al., 2016; Wang et al., 2018), elastic waves propagation (Yang et al., 2004; Gu and Jin, 2016; Cao et al., 2019; Jiao et al., 2019a) and fracture mechanics (Yang, 2005; Sladek et al., 2014; Zhao et al., 2018; Qin et al., 2019). Due to the simultaneous possession of piezoelectricity and SC properties, the PSC materials can be used as sensors (Kalantar-Zadeh et al., 2001; Lin et al., 2011; Rahman et al., 2018), transistors (Fernandes and Mulato, 2014; Rasheed et al., 2017; Li et al., 2018; Yang et al., 2018;), resonators (Takeuchi et al., 2003; Luo et al., 2013; Bian et al., 2015) as well as other PSC devices. These applications call for a better understanding of the dynamic behaviors of PSC composites, especially the propagation characteristics of the elastic waves in the multilayered PSC structures. Since the potential created in the PSC layer has a strong effect on the carrier transport at the interface (Wang, 2012), the interface plays an important role in the PSC multilayered structures. At the same time, due to various causes such as microinhomogeneities, microdefects, and microdebonding (Nie et al., 2016), the interfaces may not be well-bonded which we call imperfect in layered structures. The interfacial imperfection can affect the wave propagation behavior, so we need to 2
consider the effect of imperfect interfaces. It is known that the wave in the PSC is both dispersive and diffusive due to the interaction between PE and SC (Hutson and White, 1962; White, 1962; White, 1967). As a result, wave propagation in PSC structures is of great interest for researchers, particularly when it propagates in layered structures. For a single layer PSC plate, and based on the power series expansions method, Yang and Zhou (2005) obtained the two-dimensional equation of thin PSC plate and further numerically calculated the effect of SC characteristic and biasing electric field on the propagation characteristics of thickness-shear waves; Tian et al. (2019) systematically studied the effect of the surface boundary condition, steady-state carrier density, plate thickness and biasing electric field on the wave speed and attenuation of both SH and Lamb waves in the transversely isotropic ZnO PSC plate by using the extended Stroh formalism. For semi-infinite PSC, propagation of shear horizontal (SH) waves (Gu and Jin, 2015) and generalized Rayleigh surface waves (Cao et al., 2019) as well as coupled elastic waves (Jiao et al., 2019b) were all studied. Jiao et al. (2019a) investigated the reflection and transmission of elastic waves in a sandwich PSC slab, and found that the steady-state carrier density and the biasing electric field have a great influence on the reflection and transmission coefficients. Yang et al. (2004) discussed the dispersion curves in the PSC multilayered plates by using the power series expansions method. To the authors' best knowledge, however, no work on wave propagation in multilayered plates with imperfect interfaces has been reported, which motivates the present study. This paper focus on the propagation characteristic of SH waves in the PSC multilayered plates with imperfect interfaces. This paper is organized as follows. In section 2, we describe the problem to be solved via the basic equations, interface and boundary conditions. In section 3, the dispersion equation in multilayered PSC plates with imperfect interfaces is derived based on the Stroh formalism and DVP method. In section 4.1, 3D dispersion curves of SH wave in a single-layer PSC plate with varying steady-state carrier density and the corresponding mode shapes are presented. The effect of stacking sequences and imperfect interfaces on the dispersion and 3
attenuation curves of sandwich plates are investigated in Section 4.2. Finally, conclusions are drawn in Section 5.
2. Problem statements with basic equations
As shown in Fig. 1, we consider the propagation of SH waves in the multilayered transverse isotropic PSC plates which are horizontally infinite but vertically finite. The SH wave propagates along positive x1 direction. The jth PSC plate bonded by its and upper interface xj2 with thickness hj. The layered structure has lower interface xj−1 2 n layers and the total thickness of the multilayered PSC plates is h=h1+h2+…+hn.
Fig. 1. SH waves propagation in x1-drection in the PSC multilayered plates.
2.1 Governing equations
Using the extended notation introduced by Barnett and Lothe (1975) for PE material and generalized to magneto-electro-elastic coupling by Pan (2002), the equilibrium equations and the constitutive relations of each layer can be written as
σ iJ ,i = FJ 4
(1)
σ iJ = ciJMlγ Ml
(2)
where repeated lowercase subscripts take the summation from 1 to 3, whilst those of the uppercase subscripts take the summation from 1 to 5; and the extended displacements, stresses, strains and body forces are defined as
ui uI = ϕ n
( I = i = 1, 2,3) , ( I = 4) ( I = 5)
γ ij = 0.5 ( ui , j + u j ,i ) γ Ij = − E j = ϕ, j N = n ,j j
σ ij σ iJ = Di J i
( I = i = 1, 2,3) , ( I = 4) ( I = 5)
( J = j = 1, 2,3) , ( J = 4) ( J = 5)
ρ u&&j ( J = j = 1, 2,3) FJ = qn ( J = 4) −qn& ( J = 5)
(3)
where uk, φ and n are the displacements, electric potential, and carrier density, respectively; σij, Di and Ji are the stresses, electric displacements and electric currents, respectively; γij, Ej and Nj are strains, electric field, and carrier density gradient, respectively; ρ is the mass density, q=1.602×10−19C is the carrier charge constant, a dot over a quantity denotes the differentiation with respect to time, a subscript comma denotes the partial differentiation with respect to the coordinates. Also in Eq. (2), the extended material coefficients are defined as
ciJMl
cijml ( J = M = j, m = 1, 2,3) ( J = j = 1, 2,3; M = 4 ) elij 0 ( J = 1, 2,3, 4; M = 5) ( J = 4; M = m = 1, 2,3) e = iml ( J = M = 4) −ε il 0 ( J = 5; M = 1, 2,3) − qn0 µil ( J = 5; M = 4 ) − qd ( J = M = 5) il
(4)
where cijkl, ekij and εij are the elastic, piezoelectric and dielectric coefficients, respectively; µij and dij are the carrier mobility and diffusion coefficients, respectively; and n0 is the steady-state carrier density. For the propagation of SH wave in a transverse isotropic PSC plate, the extended displacements, stresses, strains, body forces and material coefficients that we need to consider are the components in Eqs. (3) and (4) with lowercase subscripts from 1 to 2, 5
and the uppercase subscripts from 3 to 5. Then by substituting the nonzero components into Eqs. (1) and (2), the equilibrium equations and the constitutive relations are obtained.
2.2 boundary conditions
In the multilayered PSC plates, we usually assume that the extended displacements and tractions (defined below) are perfectly bonded on the top and bottom interfaces of the layer. However, if the interface is imperfect, the interface conditions say at x2=xj2 need to be generalized as (Wang and Pan, 2007; Pan, 2019) T ( x2j + ) =T ( x2j − ) ,
U ( x2j + ) − U ( x2j − ) = α j T ( x2j )
(5)
with
α11j 0 0 α j = 0 α 22j 0 0 0 α 33j
(6)
where U=[u3, φ, n]T is the extended displacement vector, and T=[σ23, D2, J2]T the extended traction vector; T denotes vector or matrix transpose; j=0,1,2,…,n, and α0=αn=0 (i.e., on the top and bottom surfaces of the layered structure). It should be noted that if αii=0, the general interface is reduced to the perfect one. Furthermore, if αii are very large, the interface corresponds to a completely debonded and separate one. On the surfaces of the multilayered PSC plates, we assume that the extended traction is zero, namely T ( x20 ) = 0, T ( x2n ) = 0
3. Dispersion equation
6
(7)
For the wave propagating in the positive x1 direction, the time-harmonic solutions of the extended displacements can be assumed as
u I =aI eikx1 −iωt eiskx2
(8)
where k is the complex wavenumber. The real part of k, i.e. Re(k), indicates the wave propagation; the imaginary part of k, i.e. Im(k), denotes the wave attenuation. For the problem we considered in this paper we require Re(k) >0 for wave which propagates along the positive x1 direction and Im(k)>0 for wave which decays; ω is the angular frequency; i is the imaginary unit; s is the eigenvalue, and aI (I=3-5;) the unknown coefficients, both to be determined. Substituting Eq. (8) into Eq. (2), the generalized traction on the plane perpendicular to the x2-axis is found to be
σ 2 J =ikbJ eikx −iωt eiskx 1
2
(9)
where bJ (J=3-5) are the unknown coefficients. Inserting Eqs. (8) and (9) into Eq. (2), yields b = s[M ]a
(10)
where a=[ a3 a4 a5]T is related to the generalized displacement vector, b=[b3 b4 b5]T to the generalized traction vector; and the elements of matrices [M] are
M ( J − 2)( L − 2) = c2 JL 2
(11)
where J, L=3−5. Substituting Eq. (8)-(9) into Eq. (1) yields
[Q+s2 M ]a = 0
(12)
Q( J − 2)( L − 2) = c1JL1 +X ( J −2)( L −2) ( j = 1 − 3),
(13)
where
X ( J −2)( L−2)
− ρω 2 / k 2 2 q / k = 2 −iqω / k 0
( J = L = 3) ( J = 4; L = 5) ( J = 5; L = 5) ( J = 3; L = 4 − 5 or J = 4 − 5; L = 3) 7
(14)
Then combining Eqs. (10) and (12), the following linear eigenvalue system can be obtained
a a = s b b
[N]
(15)
where 0 [ N ]= −Q
M −1 0
(16)
From Eq.(15), we obtain six eigenvalues and the corresponding six eigenvectors a and b. Based on them, the general solutions of the extended displacement and traction vectors can be written as (omitting the proportional coefficient e ikx1 − iω t )
ikU ( x2 ) E11 = x T ( ) 2 E 21
ik ( x2 − x2j ) s− e E12 E 22 0
0 e
(
)
ik x2 − x2j −1 s+
c− c +
(17)
where <> denotes the diagonal matrix of 3×3. Notice that the six eigenvalues are ordered as such that Im(s+)Re(k)+ Im(k)Re(s+) ≥ Im(s−)Re(k)+Im(k)Re(s−), and [E] is the eigenmatrix made of the corresponding eigenvectors. The coefficient vectors c− and c+ can be determined for the given boundary and interface conditions discussed below. In this paper, instead of applying the traditional transfer matrix method and stiffness matrix method, we use the DVP method which was recently developed (see, e.g., Pan, 2019). Compared to the transfer and stiffness matrix methods, the DVP method is much stable at large/small k and thin/thick layer and is computationally more efficient, especially when the structures involve many layers and there is an evanescent wave in the layered structures. Then, by following Liu and Pan (2018a, 2018b), the layer matrix relation for layer j from xj−1+ to xj−2 can be derived as 2
ikU ( x2j −1+ ) S j = 11 j− T ( x2 ) S 21j 8
j− S12j ikU ( x2 ) S 22j T ( x2j −1+ )
(18)
where
S S
j 11 j 21
- ikh j s− S E11 e = S E 21 j 12 j 22
E12 E 22 e
ikh j s+
E11 E e- ikh j s− 21
E12 e
ikh j s+
E 22
−1
(19)
to xj+2 (passing Combined Eqs. (5) and (18), the propagate matrix of layer j from xj−1+ 2 also the imperfect interface) can be expressed as ikU ( x2j −1+ ) S j = 11int j+ T ( x2 ) S 21j int
j+ S12j int ikU ( x2 ) S22j int T ( x2j −1+ )
(20)
where −1
S11j int = S11j − ikS11j α j I + ikS21j α j S21j , −1
S12j int = S12j − ikS11j α j I + ikS21j α j S22j , S
j 21int
−1
= I + ikS α j S , j 21
(21)
j 21
−1
S22j int = I + ikS21j α j S22j Similarly, for layer j+1 the transfer matrix from xj+2 on its bottom to xj+1+ on its top 2 can be obtained. Then the transfer matrix from layer j to layer j+1 can be found as ikU ( x2j −1+ ) S j: j +1 = 11 j +1+ T ( x2 ) S 21j: j +1
j +1+ S12j: j +1 ikU ( x2 ) S 22j: j +1 T ( x2j −1+ )
(22)
where −1
S11j: j +1 = S11j int S12j +int1 I − S21j int S12j +int1 S21j int S11j +int1 + S11j int S11j +int1 , −1
S12j: j +1 = S11j int S12j +int1 I − S21j int S12j +int1 S22j int + S12j int , −1
1 j j +1 j j +1 j +1 S21j: j +1 = S22j +int I − S 21int S12 int S 21int S11int + S 21int ,
(23)
−1
1 I − S 21j int S12j +int1 S 22j int S22j: j +1 = S22j +int
Making use of the recursive relation (22) repeatedly, the following important relation between the top and bottom surfaces of the layered structure can be derived ikU ( x20 − ) S 0 −:n + = 11 0 −:n + T ( x2n + ) S 21
n+ S120 −:n + ikU ( x2 ) 0 −:n + 0− S 22 T ( x2 )
9
(24)
Using the boundary conditions on the surfaces (i.e. Eq. (7)), the dispersion relation can be finally obtained as 0 −:n + det[ S21 ]=0
(25)
In other words, for a given frequency ω, the wavenumber k can be obtained by solving the above equation.
4. Numerical results and discussions
In this section, we study the propagation characteristics of the SH wave in the transversely isotropic single ZnO PSC plate of thickness h, and in the sandwich plates composed of PSC ZnO and PSC GaN. For the sandwich plates, each layer has the same thickness (e.g. the thickness of each layer is h/3). The material parameters of ZnO (Qin et al., 2010; Qin et al., 2011) and GaN (Pan and Chen, 2015; Zhao et al., 2018) are listed in Table 1. In the following numerical analysis, we will use the dimensionless wavenumber kh, dimensionless frequency Ω, and dimensionless interface moduli r1, r2 and r3, as defined below
Ω = ω h / c11ZnO / ρ ZnO , r1 = c11ZnOα11 / h,
(26)
r2 = −ε11ZnOα 22 / h, r3 = − qd11ZnOα 33 / h Table 1 Material properties properties
ZnO
GaN
properties
ZnO
GaN
c11(GPa)
210
379
µ11(m2/V)
1
6.53×10−2
c44(GPa)
43
98
µ22(m2/V)
1
6.53×10−2
e15(C/m2)
−0.48
−0.33
d11(m2/s)
0.026
16.99×10−4
ε11(10−11C/Vm)
7.61
8.41
d22(m2/s)
0.026
16.99×10−4
ε22(10−11C/Vm)
7.61
8.41
ρ(kg/m3)
5700
6095
4.1 SH wave in single ZnO plate 10
The 3D dispersion curves of SH wave in the single ZnO plate of fixed h=3mm and n0=1×1015m-3 are shown in Fig. 2a, and the corresponding 2D front-view and side-view are plotted in Figs. 2b and 2c. From Fig. 2. we can see that: 1) The first three modes all start from Re(kh)=0; 2) There exist the imaginary part of kh which is positive, showing that the wave is attenuating; 3) With decreasing Ω, the imaginary of kh increases; 4) The attenuation of the 0th mode is much smaller than that of the 1st and 2nd modes at low frequency (Ω<1.5).
Fig. 2 The dispersion and attenuation curves of SH wave in the single ZnO plate with fixed h=3mm and n0=1×1015m-3. (a) The 3D curves of the first three modes, (b) the dispersion curves of the first three modes, and (c) the attenuation curves of the first three modes.
Figure 3 shows the variation of dimensionless frequency with the dimensionless wavenumber for different values of n0 with fixed h=3mm. The following interesting 11
features can be observed from Fig. 3: 1) When n0=1×105m-3, the dispersion relation of the SH wave in the single PSC plate is the same as that in the corresponding PE plate; 2) When n0=1×1015m−3, the 1st and 2nd modes of the SH wave begin at Re(kh)=0, which is different from the corresponding pure elastic and PE plate; 3) When n0=1×1025m−3, the SH wave propagates with the wave speed of SH wave in the corresponding elastic plate. As such, for given h and Ω, there should exist a critical n0 corresponding to the wave speed of SH wave in the ZnO plate from PE material to PSC material and then to elastic (E) material. Figure 4 shows at which combined values n0, h and Ω, one could achieve the purely PE, PSC and E response for the 0th mode. We observe from Fig. 4a that: 1) The combination of n0, h and Ω constitutes two planes; 2) Above the upper plane, the propagation of SH wave in the PSC plate is the same as that in the corresponding E plate; 3) Between the upper and lower planes, the SH wave is dispersive and diffusive; 4) Below the lower plane, the SH wave propagates with the wave speed in the corresponding PE plate.
12
Fig. 3. The 3D dispersion relations and the front-view of the 3D dispersion curves of the first three modes. (a) The 0th mode, (b) the 1st mode, and (c) the 2nd mode.
13
Fig. 4. Phase diagram of the critical values of 0th mode. (a) The 3D phase diagram with n0, h and Ω, and (b) the 2D phase diagram with fixed Ω=1.
The following interesting features can be observed from Fig. 4b: for the critical values from PE to PSC and PSC to E: 1) The relation between log(h) and log(n0) is linear for small value of log(h) (log(h)<−6.2m for PE to PSC and log(h)<−4.9m for PSC to E) and large value of log(h) (−3.3m
Fig. 5. Normalized displacements and stresses on the wave with Re(kh)=0.1. (a) The 0th mode, (b) the 1st mode, and (c) the 2nd mode.
14
Fig. 6. Normalized displacements and stresses on the wave with Re(kh)=4. (a) The 0th mode, (b) the 1st mode, and (c) the 2nd mode.
We further study the mode shapes of the SH wave in the ZnO plate. Six sampling points in Fig. 2a are chosen for the calculation of mode shapes: the points of first three modes with Re(kh) equal to 0.1 and 4. Since all the extended displacements (or stresses) have similar mode shapes except for different numerical values, we only present the mode shapes of the elastic displacements and stresses (tractions). Points of 1st and 2nd modes with Re(kh)=0.1 have a very large Im(kh) compared to Re(kh), so the normalized displacement and stress are all calculated in 0.1 wavelength equals to 0.1×2π/Re(k)=2πh along the wave propagation direction; points of 0th mode with Re(kh) equal to 0.1 and 4 and 1st and 2nd modes with Re(kh)=4 have a very large Re(kh) compared to Im(kh). The normalized displacement and stress are calculated in one wavelength equals to 20πh and 2π/ Re(k)=0.5πh along the wave propagation direction, respectively. Figures 5 and 6 show the variation of first three normalized u3 and σ23 for fixed h=3mm and n0=1×1015m-3 along the thickness and wave propagation directions. The following features can be observed from Figs. 5 and 6: 1) For the 0th mode, the normalized displacements are all constant along the thickness direction of the plate; the normalized stresses are all zero (Figs. 5a and 6a); 2) The 1st and 2nd modes are 15
antisymmetric and symmetric, respectively; 3) For the points with large ratio of Im(kh)/Re(kh), the attenuation of SH wave is obvious along the wave propagation direction (Figs. 5b and 5c); for the points with large ratio of Re(kh)/Im(kh), the attenuation is not obvious (Figs. 5a, 6b and 6c).
4.2 SH waves in sandwich plates
We now apply our solution to the sandwich plates composed of materials ZnO and GaN to investigate the influence of the stacking sequences on dispersion curves. Four sandwich plates are considered: Z/Z/Z, Z/G/Z, G/Z/G, and G/G/G. The letter “Z” denotes ZnO and “G” denotes GaN, with the first and last being homogeneous plates. Again, these three layers are assumed to have equal thickness h/3 with h=3mm. Figure 7 shows the 3D dispersion curves of SH waves for the first three modes in different sandwich plates, and Fig. 8 is the comparison of each mode for the four sandwich plates with fixed n0=1×1015m−3. The interfaces are assumed to be perfect. From Figs. 7 and 8, we observe that: 1) For 0th and 2nd modes, with increasing material GaN, the real part of kh decreases. 2) For SH waves in Z/Z/Z and G/G/G plates, the imaginary part of kh is equal when Ω=0; 3) The imaginary part of kh decreases with increasing material GaN for the 0th mode.
16
Fig. 7. The 3D dispersion curves of SH waves in the sandwich plates. (a) Z/Z/Z, (b) Z/G/Z, (c) G/Z/G, and (d) G/G/G.
17
Fig. 8. The first three modes of SH waves in sadwich plates.
18
19
Fig. 9. The effect of interface modulus r1 on the first three dispersion curves of Z/G/Z plates (r2=r3=0).
20
Fig. 10. The effect of interface modulus r1 on the first three dispersion curves of G/Z/G plates (r2=r3=0). 21
We further study the effect of imperfect interfaces on the dispersion curves of the same sandwich plates. Figures 9 and 10 show the dispersion curves of the first three modes with varying interface parameter r1 (r2=r3=0) for two sandwich plates with fixed n0=1×1015m−3. The two interfaces in the sandwich plates are assumend to have the same imperfection. Figure 9 for the Z/G/Z plate and Fig. 10 for the G/Z/G plate. It is observed that, in general, at a fixed frequency, with increasing interface parameter, the real part of the wavenumber for both Z/G/Z and G/Z/G plates increases; the imaginary part of the wavenumber increases first and then decreases for the 0th mode; the imaginary part of the wavenumber for the 1st and 2nd modes decreases. We further observed that the wavenumber on the 0th dispersive curves are nearly independent of the interface parameter when frequency is low. For the 1st and 2nd dispersion curves, the dimensionless frequency Ω of the critical point (a point which has a larger imaginary part and a smaller real part of kh) decreases with increasing mechianical interface modulus r1. To study the effect of interface parameters r2 and r3 on the dispersion curves of the sandwich plates, the wavenumber of the first three modes of Z/G/Z plates for Ω=3, and n0=1×1015m-3 are listed in Tables 2 (r1=r3=0) and 3 (r1=r2=0). The two interfaces in the sandwich plates are assumed to have the same imperfection. From Tables 2 and 3, we notice that with increasing interface parameters , the dimensionless wavenumber on the same mode changes only slightly.
Table 2 Dimensionless wavenumber kh at Ω=3 in the sandwich Z/G/Z plates with varying interface parameter r2 (r1=r3=0). 0
1
2
Mode r2
Re(kh)
Im(kh)
Re(kh)
Im(kh)
Re(kh)
Im(kh)
0
5.7485
0.0429
5.1244
0.0652
0.2773
0.9320
1
5.7489
0.0450
5.1251
0.0668
0.2779
0.9309
10
5.7490
0.0456
5.1253
0.0672
0.0313
0.9155
100
5.7491
0.0457
5.1254
0.0672
0.2681
0.9369
22
Table 3 Demensionless wavenumber kh at Ω=3 in the sandwich Z/G/Z plates with varying interface parameter r3 (r1=r2=0). 0
1
2
Mode r3
Re(kh)
Im(kh)
Re(kh)
Im(kh)
Re(kh)
Im(kh)
0
5.7485
0.0429
5.1244
0.0652
0.2773
0.9320
1
5.7485
0.0427
5.1245
0.0645
0.2730
0.9285
10
5.7485
0.0427
5.1245
0.0644
0.2730
0.9280
100
5.7485
0.0427
5.1245
0.0644
0.2730
0.9279
5. Conclusions The dispersion and attenuation curves of the SH wave in multilayered PSC plates with imperfect interfaces are derived via the extended Stroh formalism and the DVP method. The effects of the steady-state carrier density on the single-layer ZnO PSC plate, and the effect of stacking sequences and imperfect interfaces on the sandwich plates composed of the PSC ZnO and PSC GaN are studied in detail. Numerical results demonstrate the following features: (1) Due to the semiconductor characteristic of PSC, the SH wave in PSC plate is diffusive, the first three modes of SH wave all start from Re(kh)=0 which is quite different from the SH wave in the corresponding elastic and PE plate. (2) The critical points with combined values of n0, h and Ω are identified where the SH wave propagates in the corresponding PE, PSC, and E plates. These points form two planes to separate the space into three domains: above the upper plane, between the upper and lower planes, and below the lower plane. The SH wave propagates, respectively, with the corresponding wave speed in the E plate, PSC plate and PE plate. (3) For the critical value from PE to PSC and PSC to E, n0h2 is a constant for small value of log(h) (e.g. log(h)<−6.2m for PE to PSC and log(h)<−4.9m for PSC to E for the 0th mode), and n0h is a constant for large value of log(h) (e.g. 23
−3.3m
Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 11472182 and 11272222). The first author is grateful to the support of China Scholarship Council (CSC Grant No. 201907090051).
References
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Highlights The imperfect interfaces characterized by discontinuous mechanical displacement, electric potential and carrier density are considered to study the behavior of SH waves in transversely isotropic multilayered piezoelectric semiconductor (PSC) plates For single-layer ZnO plate, the critical elastic (E), piezoelectric (PE), and PSC wave domains are identified for the given carrier density, plate thickness and frequency. The effect of stacking sequences and imperfect interfaces in sandwich plates are shown. The mechanical imperfect interface would increase the real part of the wavenumber and decrease the imaginary part of wavenumber for the 1st and 2nd mode, while the electric or electric current imperfect interface nearly has no influence on the dispersion and attenuation curves.
Declaration of interests ☐ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: