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Wave propagation in piezoelectric layered structures of film bulk acoustic resonators
Ultrasonics 67 (2016) 105–111
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Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Wave propagation in piezoelectric layered structures of film bulk acoustic resonators Feng Zhu, Zheng-hua Qian ⇑, Bin Wang State Key Laboratory of Mechanics and Control of Mechanical Structures/College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
a r t i c l e
i n f o
Article history: Received 23 October 2015 Received in revised form 5 January 2016 Accepted 7 January 2016 Available online 13 January 2016 Keywords: Piezoelectric layered structure FBARs Dispersion curve Vibration
a b s t r a c t In this paper, we studied the wave propagation in a piezoelectric layered plate consisting of a piezoelectric thin film on an electroded elastic substrate with or without a driving electrode. Both plane-strain and anti-plane waves were taken into account for the sake of completeness. Numerical results on dispersion relations, cut-off frequencies and vibration distributions of selected modes were given. The effects of mass ratio of driving electrode layer to film layer on the dispersion curve patterns and cut-off frequencies of the plane-strain waves were discussed in detail. Results show that the mass ratio does not change the trend of dispersion curves but larger mass ratio lowers corresponding frequency at a fixed wave number and may extend the frequency range for energy trapping. Those results are of fundamental importance and can be used as a reference to develop effective two-dimensional plate equations for structural analysis and design of film bulk acoustic resonators. Ó 2016 Elsevier B.V. All rights reserved.
1. Introduction Acoustic wave resonators made from piezoelectric crystals are key components of circuits for alternating currents and have been widely used for time keeping, frequency operation, signal generation and processing as frequency standards. During the last one to two decades, researchers succeeded in depositing with good enough quality a thin piezoelectric film of AlN or ZnO with proper electrode configuration on a silicon layer to form thin film bulk acoustic wave resonators (FBARs) [1–3]. Compared with conventional acoustic wave resonators, FBARs have a series of advantages such as much smaller size and compatible fabrication with other on-chip and integrated circuit (IC) technologies and have become research hot recently. There are several structural types of FBARs operating with thickness modes [4–8] or solidly mounted on an elastic substrate [9,10]. Fig. 1 shows the structure of a typical, basic, and widely used type of FBARs which is the one that is going to be studied in the current paper. Structurally, the FBAR in Fig. 1 is a multilayered plate with metal electrodes, a piezoelectric film, and an elastic layer. The c-axis of the piezoelectric film is in the x3 direction. To aid the analysis and design of FBARs, it is necessary to obtain accurate predictions of their frequency and mode shapes. Starting from the three-dimensional linear piezoelectric equations, however, direct solution to such a multilayered structure is still ⇑ Corresponding author. Tel.: +86 25 84892696; fax: +86 25 84895759. E-mail address: [email protected] (Z.-h. Qian). http://dx.doi.org/10.1016/j.ultras.2016.01.004 0041-624X/Ó 2016 Elsevier B.V. All rights reserved.
mathematically changing, no matter theoretically or numerically [11–13]. Up to now, most theoretical analyses on FBARs are based on one-dimensional models which can describe the most basic vibration characteristics of FBARs [7,14–18]. However, for real devices of finite plates, one-dimensional models are inadequate. They cannot describe the in-plane mode variations associated with finite plates. They are also incapable of describing mode couplings induced by wave reflections at the plate edges in finite devices. Recently, Qian’s group [19] has studied the free vibrations of FBAR multilayered structures by using the Steven–Tiersten’s twodimensional scalar differential equations [20] which are mathematically simple and accurate, and can describe the in-plane variation of the operating thickness-extensional mode and the related energy trapping in FBARs. The main limitation of the scalar equation is that it is for the single operating mode of the FBAR only and therefore cannot describe mode coupling, although it has inplane mode variation in the model [20]. At present, the design and operation of FBARs are adversely affected by various couplings between the operating mode and other unwanted modes. The understanding of the mechanism of these mode couplings is limited. Therefore, it is extremely needed to develop effective twodimensional plate equations which can simultaneously describe both the in-plane mode couplings and the energy trapping vibration phenomenon. One way is to follow Mindlin’s approach in treating mode couplings and in-plane mode variations in conventional quartz resonators [21–25]. The structures of FBARs are multilayered and hence are more complicated than single-layered
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Driving electrode Ground electrode
T 11 ¼ c11 u1;1 þ c13 u3;3 þ e31 u;3 ;
x3
h
T 33 ¼ c13 u1;1 þ c33 u3;3 þ e33 u;3 ; hf
hs Silicon
h
T 31 ¼ T 13 ¼ c44 ðu3;1 þ u1;3 Þ þ e15 u;1 ;
x1
Film
ð2Þ
and
D1 ¼ e15 ðu3;1 þ u1;3 Þ e11 u;1 ;
hˆ s
ð3Þ
D3 ¼ e31 u1;1 þ e33 u3;3 e33 u;3 :
Fig. 1. Cross section of a typical thin AlN or ZnO film on a silicon layer as an FBAR.
The substitution of Eqs. (2) and (3) into Eq. (1) gives
€1 ; c11 u1;11 þ c44 u1;33 þ ðc13 þ c44 Þu3;13 þ ðe31 þ e15 Þu;13 ¼ qu quartz resonators, for which we first need to obtain accurate results of dispersion curves and cut-off frequencies in the wave number range of interest for the operation of FBARs. This motivates the current work in this paper. Since the structures of FBARs shown in Fig. 1 have two different regions, with or without a top driving electrode, we consider wave propagation in plate models for FBARs with a top electrode layer and without a top electrode layer, respectively. Both plane and aniplane waves were taken into account. From the three-dimensional piezoelectro-elastic equations, an exact procedure was established to calculate dispersion relations, cut-off frequencies and vibration distributions of selected modes. Those results are of fundamental importance and can be used as a reference to improve the accuracy of the two-dimensional plate equations by requiring the cutoff frequencies and the curvatures of the dispersion curves at cutoff frequencies predicted by the two-dimensional plate equations and the three-dimensional equations to be the same.
€3 ; c44 u3;11 þ c33 u3;33 þ ðc44 þ c13 Þu1;31 þ e15 u;11 þ e33 u;33 ¼ qu
ð4Þ
ðe15 þ e31 Þu1;13 þ e15 u3;11 þ e33 u3;33 e11 u;11 e33 u;33 ¼ 0:
Similarly, for silicon layer which is a cubic crystal, from the equations of anisotropic elasticity we have
€1 ; cs11 u1;11 þ cs44 u1;33 þ cs13 þ cs44 u3;13 ¼ qs u s s s s s€ c44 u3;11 þ c33 u3;33 þ c44 þ c13 u1;31 ¼ q u3 ;
ð5Þ
where we have used a superscript ‘s’ to indicate the material constants of silicon layer. At the top of the composite plate where x3 = hf, mechanically the surface is free. Electrically the surface may be electroded and grounded, or unelectroded. For an electroded surface, the boundary conditions are 0
f
0
f
€ 1 ðh Þ; T 31 ðh Þ ¼ q0 h u f
€ 3 ðh Þ; T 33 ðh Þ ¼ q0 h u f
ð6Þ
uðh Þ ¼ 0; f
2. Theoretical derivation Consider the composite plates in Fig. 2. We study straightcrested waves without x2 dependence, i.e., o/ox2 = 0. In this case, the equations of linear piezoelectricity for ZnO or other crystals of class 6 mm with the c axis along x3 decouples into two groups. One gives the displacement components u1 and u3 as well as the electric potential u, which corresponds to plane-strain wave case. The other is for u2 alone which corresponds to anti-plane wave case. The two cases will be dealt with below, respectively.
where q0 and h0 are the mass density and thickness of the top electrode. In Eq. (6), we have assumed that the electrodes are very thin and neglected their stiffness. For an unelectroded top surface, we have f
T 31 ðh Þ ¼ 0;
f
T 33 ðh Þ ¼ 0;
f
D3 ðh Þ ¼ 0:
ð7Þ
At the interface between the ZnO film and the silicon layer where x3 = 0, the continuity conditions are
u1 ð0þ Þ ¼ u1 ð0 Þ;
u3 ð0þ Þ ¼ u3 ð0 Þ;
uð0þ Þ ¼ 0;
00 00 €
T 31 ð0 Þ T 31 ð0 Þ ¼ q h u1 ð0Þ; þ
2.1. Plane-strain waves
ð8Þ
00 00 €
T 33 ð0 Þ T 33 ð0 Þ ¼ q h u3 ð0Þ; þ
The relevant equations of motion and the charge equation of electrostatics are
€1 ; T 11;1 þ T 31;3 ¼ qu €3 ; T 13;1 þ T 33;3 ¼ qu
ð1Þ
D1;1 þ D3;3 ¼ 0; where the stress components Tij and the electric displacement components Di are related to the displacement and electric displacement gradients through the constitutive relations
where q00 and h00 are the mass density and thickness of the interface electrode. At the bottom of the composite plate where x3 = hs the traction-free boundary conditions are s
T 31 ðh Þ ¼ 0;
s
T 33 ðh Þ ¼ 0:
We look for time-harmonic wave solutions that may exist in the composite plate. The ZnO film and the silicon layer need to be treated separately initially, and then boundary and continuity conditions will be applied.
x3 Electrode
h (a)
h' hf
Film
hs
Silicon
x1
ð9Þ
x3
Electrode
h
hf
Film
hs
Silicon
(b)
Fig. 2. Plate models for FBARs: (a) without a driving electrode; (b) with a driving electrode.
x1
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F. Zhu et al. / Ultrasonics 67 (2016) 105–111
In the ZnO film, consider the possibility of the following waves:
u1 ¼ A exp fx3 cos nx1 expðixtÞ; u3 ¼ B exp fx3 sin nx1 expðixtÞ; u ¼ C exp fx3 sin nx1 expðixtÞ;
ð10Þ
where A, B, and C are undetermined constants. Substituting Eq. (10) into Eq. (4), we obtain three homogeneous linear equations for A, B, and C. For nontrivial solutions the determinant of the coefficient matrix of the equations has to vanish, which leads to a polynomial equation of degree six for f. We denote the six roots of the equation by fðmÞ ðx; nÞ, with m = 1, 2, . . . , 6. Corresponding to each m, the three homogeneous linear equations for A, B, and C have nontrivial solutions which determine the so-called amplitude ratios, i.e., A:B: C = 1:a(m):b(m). Then the general solution to Eq. (4) in the form of Eq. (10) is
u1 ¼
AðmÞ exp fðmÞ x3 cos nx1 expðixtÞ;
6 X AðmÞ aðmÞ exp fðmÞ x3 sin nx1 expðixtÞ;
ð11Þ
6 X
AðmÞ bðmÞ exp fðmÞ x3 sin nx1 expðixtÞ;
where A(m) represent six undetermined constants. Similarly, for the silicon layer, we let
u3 ¼ G exp fx3 sin nx1 expðixtÞ;
ð12Þ
where F and G are undetermined constants. Substituting Eq. (12) into Eq. (5) gives a system of two homogeneous linear equations for F and G. For nontrivial solutions, we can obtain four roots of fðnÞ with the corresponding amplitude ratios, i.e., F:G = 1:c(n), with n = 1, 2, . . . , 4. Then the general solution to Eq. (5) in the form of Eq. (12) can be written as
u1 ¼
4 X F ðnÞ exp fðnÞ x3 cos nx1 expðixtÞ; n¼1
4 X u3 ¼ F ðnÞ cðnÞ exp fðnÞ x3 sin nx1 expðixtÞ;
ð13Þ
where F(n) are four undetermined constants. Finally, we substitute Eqs. (11) and (13) into Eqs. (6), (8) and (9) for an electroded plate, or into Eqs. (7)–(9) for an unelectroded plate. In each case, this results in ten homogeneous linear equations for the ten undetermined constants of A(m) and F(n). For nontrivial solutions, the determinant of the coefficient matrix of these equations has to vanish, which determines the dispersion relations of the waves. The nontrivial solutions of A(m) and F(n) gives the displacement and potential fields of the waves. These can be done on a computer. 2.2. Anti-plane waves In this part, we will deal with anti-plane waves with displacement u2 only, i.e.u1 = u3 = 0, u2 = u2 (x1, x3, t), u = u (x1, x3, t). The relevant equations of motion and the charge equation of electrostatics are
T 11;1 þ T 21;2 þ T 31;3 ¼ 0; T 13;1 þ T 23;2 þ T 33;3 ¼ 0; D1;1 þ D2;2 þ D3;3 ¼ 0 ;
T 31 ¼ e15 u;1 ;
T 33 ¼ e33 u;3 ;
T 12 ¼ c66 u2;1 ;
ð15Þ
and
D1 ¼ e11 u;1 ; ð16Þ
D2 ¼ e15 u2;3 ;
D3 ¼ e33 u;3 ; The substitution of Eqs. (15) and (16) into Eq. (14) gives
ðe31 þ e15 Þu;13 ¼ 0; ð17Þ
e11 u;11 e33 u;33 ¼ 0;
Similarly, for silicon which is a cubic crystal, from the equations of anisotropic elasticity we have
ð18Þ
where we have used a superscript ‘s’ to indicate the material constants of silicon. At the top of the composite plate where x3 = hf, mechanically the surface is free. Electrically the surface may be electroded and grounded, or unelectroded. For an electroded surface, the boundary conditions are 0
€ 2 ðh Þ; T 33 ðh Þ ¼ 0; uðh Þ ¼ 0; T 31 ðh Þ ¼ 0; T 32 ðh Þ ¼ q0 h u f
f
f
f
f
ð19Þ 0
where q and h are the mass density and thickness of the top electrode. In Eq. (19), we have assumed that the electrodes are very thin and neglected their stiffness. For an unelectroded top surface, we have 0
f
f
T 31 ðh Þ ¼ 0;
T 32 ðh Þ ¼ 0;
f
T 33 ðh Þ ¼ 0;
f
D3 ðh Þ ¼ 0:
ð20Þ
At the interface between the ZnO film and the silicon layer where x3 = 0, the continuity conditions are
n¼1
€2 ; T 12;1 þ T 22;2 þ T 32;3 ¼ qu
T 22 ¼ e31 u;3 ;
€2 ; cs66 u2;11 þ cs44 u2;33 ¼ qs u
m¼1
u1 ¼ F exp fx3 cos nx1 expðixtÞ;
T 23 ¼ c44 u2;3 ;
e15 u;11 þ e33 u;33 ¼ 0;
m¼1
u¼
T 11 ¼ e31 u;3 ;
€2 ; c66 u2;11 þ c44 u2;33 ¼ qu
6 X
m¼1
u3 ¼
where the stress components Tij and the electric displacement components Di are related to the displacement and electric displacement gradients through the constitutive relations
ð14Þ
u2 ð0þ Þ ¼ u2 ð0 Þ; þ
þ
uð0þ Þ ¼ 0; 00 € 2 ð0Þ; T 32 ð0þ Þ T 32 ð0 Þ ¼ q00 h u
T 31 ð0 Þ T 31 ð0 Þ ¼ 0;
ð21Þ
T 33 ð0 Þ T 33 ð0 Þ ¼ 0; 00
where q00 and h are the mass density and thickness of the interface electrode. At the bottom of the composite plate where x3 = hs the traction-free boundary conditions are s
T 31 ðh Þ ¼ 0;
s
T 32 ðh Þ ¼ 0;
s
T 33 ðh Þ ¼ 0:
ð22Þ
For Eq. (17), we make the following observation. (e31 + e15) u,31 = 0 in Eq. (17)1 gives u = F(x1) + G(x3), whilst from e15u,11 + e33u,33 = 0 in Eq. (17)3 and e11u,11 e33u,33 = 0 in Eq. (17)4, we can obtain u,11 = u,33 = 0, giving rise to a general solution u = ax1 + bx3 + c. For an electroded plate, substitution of u = ax1 + bx3 + c into u(x3 = hf) = 0 in Eq. (19) and u(x3 = 0+) = 0 in Eq. (21) gives u = 0. Similarly, for an unelectroded plate, substitution of u = ax1 + bx3 + c into D3 = e33u,3 in Eq. (16), D3(hf) = 0 in Eq. (20), and u(x3 = 0+) = 0 in Eq. (21) gives u = 0 too. Therefore, for both electroded and unelectroded plate, u = 0 holds on in the case of anti-plane wave motion. The wave motion Eq. (17) will be simplified as
€2 : c66 u2;11 þ c44 u2;33 ¼ qu
ð23Þ
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F. Zhu et al. / Ultrasonics 67 (2016) 105–111
For the boundary and continuity conditions, the Eqs. (19)–(22) are separately simplified as 0
€ 2 ðh Þ; T 32 ðh Þ ¼ q0 h u f
f
f
T 32 ðh Þ ¼ 0; þ
u2 ð0 Þ ¼ u2 ð0 Þ;
00 00 €
T 32 ð0 Þ T 32 ð0 Þ ¼ q h u2 ð0Þ; þ
s
T 32 ðh Þ ¼ 0:
ð25Þ
^eij ¼ eij =ðc44 e33 Þ2 ; ^eij ¼ eij =e33 ; F^ ¼ F;
ð26Þ
ð28Þ
Substituting Eq. (28) into Eq. (23), we obtain
fð2Þ ¼
qx2 c66 n2
!12 i;
c44
qx2 c66 n2
ð29Þ
!12
The general solution for u2 in the ZnO film is given as
u2 ¼ Að1Þ exp fð1Þ x3 þ Að2Þ exp fð2Þ x3 expðinx1 Þ expðixtÞ
ð30Þ
Similarly, for the silicon layer, we let
u2 ¼ B exp fx3 exp inx1 expðixtÞ:
ð31Þ
Substituting Eq. (31) into Eq. (18), we obtain
fð1Þ ¼ f
ð2Þ
¼
qs x2 cs66 n2
!12 i;
cs44
qs x2 cs66 n2 cs44
1
q^ ¼ q=qZnO ;
^f ¼ f ðh f þ hs Þ; ^ ¼ h=ðh f þ hs Þ; ^ ¼ G; h G
^ ¼ C ðe33 =c44 Þ12 ; C
ð32Þ
!12 i:
The general solution for u2 in the silicon layer is given as
u2 ¼ Bð1Þ exp fð1Þ x3 þ Bð2Þ exp fð2Þ x3 expðinx1 Þ expðixtÞ: ð33Þ Finally, we substitute Eqs. (30) and (33) into Eqs. (24), (26) and (27) for an electroded plate, or into Eqs. (25)–(27) for an unelectroded plate. In each case, this results in four homogeneous linear equations for the four undetermined constants of A(1), A(2), B(1) and B(2). For nontrivial solutions, the determinant of the coefficient matrix of these equations has to vanish, which determines the dispersion relations of the anti-plane waves. The nontrivial solutions of A(1), A(2), B(1) and B(2) gives the displacement of the waves. These are done on a computer as well.
^ ¼ x ðqZnO =c44 Þ2 ðh f þ hs Þ; x 1
where we have used a superscript ‘^’ to indicate the dimensionless parameters. If the imaginary part of the frequency x doesn’t vanish, the displacements in the form of Eq. (11) will decay exponentially as time goes on and the corresponding waves are meaningless. Hence, the frequency x is required to be real while the wave number n can be complex. The waves with imaginary wave number decay along the x1 direction, thus they are restricted in some small regions, and cannot propagate.
The dispersion curves are shown in Fig. 3 for electroded plate and Fig. 4 for unelectroded plate, where the left-hand side is for imaginary wave number and the right-hand side is for the real wave number. All dots in the figures are independent, and each dot represents a solution to the above-mentioned mathematical ^ in Figs. 3 and 4 is dimensionless as derivation. The frequency x the definition above, while the dimensional frequency is at the level of GHz, and much higher than that of traditional resonators. For the curves in Figs. 3 and 4, they can be divided into two groups. One group goes to the zero frequency as the wave numbers change from real to pure imaginary and the other one is shaped like ‘U’. These U-shaped curves extend to the complex wave number space and tend to the zero frequency. It can be seen from the comparison of Figs. 3 and 4 that the dispersion curves of electroded plate have similar patterns to those of unelectroded plate in real wave number range, but different patterns in imaginary wave number range. Further, the distributions of displacement for different branches of the real part of the curves are calculated to ascertain the vibration modes. Starting from zero frequency, different branches of the real part of the curves are named as first order curve, second order curve, and so on. For a fixed dot in the curves, the displacements in the forms of Eqs. (11) and (13) are complex values. The real part and the imaginary part of the displacements have the same distributions when they are normalized by the modulus of the complex displacements, thus the real part and the imaginary part are indiscriminate in the ^ ¼ 0:1 in the first order curve results. For a fixed wave number n
3. Numerical results and discussion For real applications, thickness of the FBARs is in decade microns range. Hence, we used the following parameter values in our calculation. For the structure with a top driving electrode, the top electrode is treated as an inertial layer and its stiffness is neglected here. In that case, we don’t need to know how much the density and the thickness of the top electrode are. Their product is enough, thus the mass ratio of the piezoelectric layer to the top electrode layer is given here. 00
h ¼ 15 106 m; h ¼ 5 106 m; h ¼ 2 107 m; f
s
ð34Þ
3.1. Plane-strain waves
i:
c44
^n ¼ n ðh f þ hs Þ; ^ ¼ B; ^f ¼ f ðh f þ hs Þ; B
^cij ¼ cij =c44 ;
In the same way as the plane-strain case, we look for timeharmonic wave solutions that may exist in the composite plate. The ZnO film and the silicon layer need to be treated separately initially, and then boundary and continuity conditions will be applied. In the ZnO film, consider the possibility of the following waves:
fð1Þ ¼
^ ¼ A; A
ð24Þ
ð27Þ
u2 ¼ A exp fx3 exp inx1 expðixtÞ
The dimensionless quantities used in the calculation are defined as follows,
q0 h0 =ðqZnO hf Þ ¼ 0:01; q00 ¼ 19; 300 kg=m3 :
Fig. 3. Dispersion curves of plane-strain waves in electroded plate.
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F. Zhu et al. / Ultrasonics 67 (2016) 105–111
Fig. 6. The distribution of u3 on the first order curve of plane-strain waves.
Fig. 4. Dispersion curves of plane-strain waves in unelectroded plate.
Table 1 ^ at the zero wave number. Results of dimensionless frequency x Model
in Fig. 3, the distributions of u1 and u3 are separately shown in Figs. 5 and 6. ^ ¼ 0:1, one cycle for x1 is at the range of As for wave number n 10p to 10p as shown in Figs. 5 and 6. It can be seen from Fig. 6 that the vibration mode of the first order curve is flexure mode (as indicated by ‘‘F” in Figs. 3 and 4), whilst Fig. 5 shows the corresponding displacement u1 along x1 caused by bending. Similarly, as shown in Figs. 3 and 4, the second order curve is the extensional mode (E), the third order curve is the first order thickness-shear mode (TS1), the fourth order curve is the second order thicknessshear mode (TS2), the fifth order is the thickness-extensional mode (TE), and the sixth order curve is the third order thickness-shear mode (TS3). The vibration modes of electroded plate and unelectroded plate are corresponding with each other from the first order to the sixth order. The FBARs shown in Fig. 1 normally operate at the thickness-extensional mode, thus the high order modes are out of interest for the operation of FBARs and are neglected here, but can be obtained in the same way. 3.1.1. Zero wave number Cut-off frequency, i.e. a frequency when wave number takes zero value, is an important consideration when one is to develop effective two-dimensional plate equations. This section will show the cut-off frequency of the first a few modes within the scope of interest for the operation of FBARs. From the mentioned mathematical derivation shown in Section 2, theoretically, a ^ can be obtained complex transcendental equation about x directly from the determinant of the coefficient matrix of the ten ^ ¼ 0. The solutions can homogeneous linear equations by setting n be obtained by solving the equation numerically, which gives the cut-off frequencies of the first nine order curves listed in Table 1. ^ range from 0 to 20. In fact, an Table 1 shows the results of x ^ at the zero wave number can be obtained infinite number of x from the complex transcendental equation. The results of electroded plate and unelectroded plate are correspondingly given. The latter values are slightly higher than the former due to the fact that electroded plate has larger mass inertia than the unelectroded
Fig. 5. The distribution of u1 on the first order curve of plane-strain waves.
Electroded Unelectroded
Order 1&2
3
4
5
6
7
8
9
0 0
3.53 3.56
7.13 7.19
7.83 8.20
10.78 10.87
14.34 14.46
15.86 15.97
17.65 17.78
plate and the electrode inertia lowers resonant frequencies. This subtle difference of the cut-off frequency is useful for the design of FBARs. In real application, as shown in Fig. 1, the film layer of FBARs is partly covered by a driving electrode. When an operating frequency of FBARs is chosen between the cut-off frequencies of the TE mode in a plate with and without electrode, as shown in Table 1 and Figs. 3 and 4, the wave number for electroded part is real while the wave number for unelectroded part is pure imaginary due to the decreasing pattern of frequency as wave number goes from real to pure imaginary. This leads to an energy trapping phenomenon because the real wave number represents a cyclical wave while the imaginary wave number represents a decayed wave. For the other mode curves, if the frequency decreases as wave number goes from real to pure imaginary, when an operating frequency is chosen between the corresponding cut-off frequencies, the energy trapping phenomenon appears as well. 3.1.2. The influence of the mass ratio The mass ratio of driving electrode layer to piezoelectric layer plays an important role in the design of resonators since it directly relates to resonant frequency and vibration distribution. Especially in the case of FBARs, structurally, the thickness of its driving electrode is comparable to that of its film layer. In this subsection, we present the influence of the mass ratio on dispersion curves of plain waves in the electroded plate. Three different values of the 0 mass ratio R = q0 h /(qZnOhf) are chosen to find out how it influences the dispersion curves, as shown in Fig. 7. It can be seen from Fig. 7 that the mass ratio does not change the pattern of dispersion curves of plain waves. But larger mass ratio leads to larger structural inertia, which lowers the corresponding frequency at a fixed wave number. This phenomenon becomes more obvious for higher order dispersion curves, e.g., the thickness-extensional mode of FBARs. According to the results above, the mass ratio may be applied to extend the frequency range for energy trapping by fixing the parameters of film layer and changing that of driving electrode layer, which can give a theoretical guidance to the design of FBARs. The FBARs normally operate at the thickness-extensional mode. Thus, the influence of mass ratio on the thickness-extensional mode of FBARs is discussed in detail as shown in Table 2. It can be seen from Table 2 that the mass ratio has a great effect on the resonance frequency of the thickness-extensional mode of FBARs. That is to say, the operating frequency of FBARs can be adjusted by setting different mass ratios. It is useful for the design and optimization of FBARs.
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4. Conclusion In this paper, we theoretically studied the wave propagation in plate models for FBARs with and without a top electrode layer, respectively. Both plane-strain and anti-plane waves were taken into account. Numerical results on dispersion relations, cut-off frequencies and vibration distributions of selected modes were given and discussed in detail. Further discussion on how the mass ratio of piezoelectric layer to driving electrode layer influences the dispersion curves was made, which results in that the mass ratio does not change the trend of dispersion curves but larger mass ratio lowers corresponding frequency at a fixed wave number and may extend the frequency range for energy trapping. Those results are of fundamental importance and can be used as a reference to develop effective two-dimensional plate equations for structural analysis and design of FBARs. Fig. 7. Dispersion curves of plane-strain waves in electroded plate under different mass ratios.
Table 2 ^ on TE mode with different mass ratios. Dimensionless cut-off frequencies x Mass ratio Frequency
0.002 7.88909
0.005 7.86863
0.008 7.84829
0.01 7.83481
0.02 7.76820
0.05 7.57698
Acknowledgements This work was supported by the Program for New Century Excellent Talents in Universities (No. NCET-12-0625), the Natural Science Foundation for Distinguished Young Scholars of Jiangsu Province (No. SBK2014010134), the Fundamental Research Funds for Central Universities (No. NE2013101), the National Natural Science Foundation of China (Nos. 11502108, 11232007), and a project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). References
Fig. 8. Dispersion curves of anti-plane wave in electroded and unelectroded plates.
3.2. Anti-plane waves For the sake of completeness, we put the discussions on the dispersion curves of anti-plane waves here. In fact, from the theoretical observation made in Section 2.2, it is clear that the plate model in the anti-plane case is degenerated into an elastic layered plate due to u = 0, i.e., a pure elastic wave propagation problem. Fig. 8 shows the dispersion curves of the anti-plane wave in the plate with and without a top electrode, where the left-hand side is for pure imaginary wave number case and the right-hand side for real wave number case. Different from the case of plane-strain waves, there is only one group of dispersion curves, i.e., the one going to the zero frequency as wave number changes from real to pure imaginary. Furthermore, the comparison between electroded and unelectroded plates shows that the dispersion curves of the anti-plane waves in the plate model with and without a top electrode are qualitatively similar. The only difference is that the latter has a higher frequency value at any fixed wave number point, which is due to that electroded plate has larger mass ratio than unelectroded plate, i.e., larger inertia lowers vibration frequency.
[1] K.M. Lakin, Thin film resonators and filters, in: Proc. IEEE Ultrasonics Symp., 1999, pp. 895–906. [2] G.F. Iriarte, F. Engelmark, I.V. Katardjiev, Reactive sputter deposition of highly oriented AlN films at room temperature, J. Mater. Res. 17 (2002) 1469–1475. [3] F. Martin, M.-E. Jan, S. Rey-Mermet, B. Belgacem, D. Su, M. Cantoni, P. Muralt, Shear mode coupling and tilted grain growth of AlN thin films in BAW resonators, IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. 53 (2006) 1339– 1343. [4] T. Yanagitani, N. Mishima, M. Matsukawa, Y. Watanabe, Electromechanical coupling coefficient k15 of polycrystalline ZnO films with the c-axis lie in the substrate plane, IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. 54 (2007) 701– 704. [5] T. Yanagitani, M. Kiuchi, M. Matsukawa, Y. Watanabe, Characteristics of pure textured ZnO films, IEEE shear mode BAW resonators consisting of (1120) Trans. Ultrason. Ferroelectr. Freq. Contr. 54 (2007) 1680–1686. [6] J.K. Du, K. Xian, J. Wang, J.S. Yang, Thickness vibration of piezoelectric plates of 6mm crystals with tilted six-fold axis and two-layered thick electrodes, Ultrasonics 49 (2009) 149–152. [7] L.F. Qin, Q.M. Chen, H.B. Cheng, Q.-M. Wang, Analytical study of dual-mode thin film bulk acoustic resonators (FBARs) based on ZnO and AlN films with tilted c-axis orientation, IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. 57 (2010) 1840–1853. [8] T. Yanagitani, N. Morisato, S. Takayanagi, M. Matsukawa, Y. Watanabe, C-axis zig–zag ZnO film ultrasonic transducers for designing longitudinal and shear wave resonant frequencies and modes, IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. 58 (2011) 1062–1068. [9] S.-H. Lee, K.H. Yoon, J.-K. Lee, Influence of electrode configurations on the quality factor and piezoelectric coupling constant of solidly mounted bulk acoustic wave resonators, J. Appl. Phys. 92 (2002) 4062–4069. [10] Y.F. Zhang, D. Chen, Multilayer Integrated Film Bulk Acoustic Resonators, Shanghai Jiaotong University Press, Shanghai, 2013. [11] J.H. Jung, Y.H. Lee, J.H. Lee, H.C. Choi, Vibration-mode analysis of an RF filmbulk-acoustic-wave resonator by using the finite element method, J. Korean Phys. Soc. 43 (2003) 1648–1650. [12] C. Buccela, V. De Santis, M. Feliziani, P. Tognolatti, Finite element modeling of a thin-film bulk acoustic wave resonator (FBAR), Int. J. Comput. Math. Electr. Electron. Eng. 27 (2008) 1296–1306. [13] P. Katus, L.M. Reindl, Reduced order finite element models of bulk acoustic wave resonators, in: Proc. IEEE International Ultrasonics Symposium, 2009, pp. 2123–2128. [14] J. Wang, J. Liu, J. Du, et al., The analysis of film acoustic wave resonators with the consideration of film piezoelectric properties, Proc. SPIE – Int. Soc. Opt. Eng. (2009). [15] J. Wang, J. Liu, J. Du, et al., The calculation of electrical parameters of film bulk acoustic wave resonators from vibrations of layered piezoelectric structures, in: Ultrasonics Symposium (IUS), 2009 IEEE International, 2009, pp. 2027– 2030.
F. Zhu et al. / Ultrasonics 67 (2016) 105–111 [16] D. Huang, J. Wang, J. Du, The analysis of high frequency vibrations of layered anisotropic plates for FBAR applications, in: Frequency Control Symposium, 2008 IEEE International IEEE, 2008, pp. 204–208. [17] J. Du, X. Kai, W. Ji, et al., Thickness vibration of piezoelectric plates of 6mm crystals with tilted six-fold axis and two-layered thick electrodes, Ultrasonics 49 (2) (2009) 149–152. [18] D. Huang, J. Wang, Effects of temperature on frequency properties of FBAR, in: Symposium on Piezoelectricity, Acoustic Waves, and Device Applications (SPAWDA), 2014, IEEE, 2014, pp. 379–382. [19] Z.N. Zhao, Z.H. Qian, B. Wang, et al., Energy trapping of thickness-extensional modes in thin film bulk acoustic wave resonators, J. Mech. Sci. Tech. 29 (7) (2015) 2767–2773. [20] H.F. Tiersten, D.S. Stevens, An analysis of thickness-extensional trapped energy resonant device structures with rectangular electrodes in the
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Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Wave propagation in piezoelectric layered structures of film bulk acoustic resonators Feng Zhu, Zheng-hua Qian ⇑, Bin Wang State Key Laboratory of Mechanics and Control of Mechanical Structures/College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
a r t i c l e
i n f o
Article history: Received 23 October 2015 Received in revised form 5 January 2016 Accepted 7 January 2016 Available online 13 January 2016 Keywords: Piezoelectric layered structure FBARs Dispersion curve Vibration
a b s t r a c t In this paper, we studied the wave propagation in a piezoelectric layered plate consisting of a piezoelectric thin film on an electroded elastic substrate with or without a driving electrode. Both plane-strain and anti-plane waves were taken into account for the sake of completeness. Numerical results on dispersion relations, cut-off frequencies and vibration distributions of selected modes were given. The effects of mass ratio of driving electrode layer to film layer on the dispersion curve patterns and cut-off frequencies of the plane-strain waves were discussed in detail. Results show that the mass ratio does not change the trend of dispersion curves but larger mass ratio lowers corresponding frequency at a fixed wave number and may extend the frequency range for energy trapping. Those results are of fundamental importance and can be used as a reference to develop effective two-dimensional plate equations for structural analysis and design of film bulk acoustic resonators. Ó 2016 Elsevier B.V. All rights reserved.
1. Introduction Acoustic wave resonators made from piezoelectric crystals are key components of circuits for alternating currents and have been widely used for time keeping, frequency operation, signal generation and processing as frequency standards. During the last one to two decades, researchers succeeded in depositing with good enough quality a thin piezoelectric film of AlN or ZnO with proper electrode configuration on a silicon layer to form thin film bulk acoustic wave resonators (FBARs) [1–3]. Compared with conventional acoustic wave resonators, FBARs have a series of advantages such as much smaller size and compatible fabrication with other on-chip and integrated circuit (IC) technologies and have become research hot recently. There are several structural types of FBARs operating with thickness modes [4–8] or solidly mounted on an elastic substrate [9,10]. Fig. 1 shows the structure of a typical, basic, and widely used type of FBARs which is the one that is going to be studied in the current paper. Structurally, the FBAR in Fig. 1 is a multilayered plate with metal electrodes, a piezoelectric film, and an elastic layer. The c-axis of the piezoelectric film is in the x3 direction. To aid the analysis and design of FBARs, it is necessary to obtain accurate predictions of their frequency and mode shapes. Starting from the three-dimensional linear piezoelectric equations, however, direct solution to such a multilayered structure is still ⇑ Corresponding author. Tel.: +86 25 84892696; fax: +86 25 84895759. E-mail address: [email protected] (Z.-h. Qian). http://dx.doi.org/10.1016/j.ultras.2016.01.004 0041-624X/Ó 2016 Elsevier B.V. All rights reserved.
mathematically changing, no matter theoretically or numerically [11–13]. Up to now, most theoretical analyses on FBARs are based on one-dimensional models which can describe the most basic vibration characteristics of FBARs [7,14–18]. However, for real devices of finite plates, one-dimensional models are inadequate. They cannot describe the in-plane mode variations associated with finite plates. They are also incapable of describing mode couplings induced by wave reflections at the plate edges in finite devices. Recently, Qian’s group [19] has studied the free vibrations of FBAR multilayered structures by using the Steven–Tiersten’s twodimensional scalar differential equations [20] which are mathematically simple and accurate, and can describe the in-plane variation of the operating thickness-extensional mode and the related energy trapping in FBARs. The main limitation of the scalar equation is that it is for the single operating mode of the FBAR only and therefore cannot describe mode coupling, although it has inplane mode variation in the model [20]. At present, the design and operation of FBARs are adversely affected by various couplings between the operating mode and other unwanted modes. The understanding of the mechanism of these mode couplings is limited. Therefore, it is extremely needed to develop effective twodimensional plate equations which can simultaneously describe both the in-plane mode couplings and the energy trapping vibration phenomenon. One way is to follow Mindlin’s approach in treating mode couplings and in-plane mode variations in conventional quartz resonators [21–25]. The structures of FBARs are multilayered and hence are more complicated than single-layered
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Driving electrode Ground electrode
T 11 ¼ c11 u1;1 þ c13 u3;3 þ e31 u;3 ;
x3
h
T 33 ¼ c13 u1;1 þ c33 u3;3 þ e33 u;3 ; hf
hs Silicon
h
T 31 ¼ T 13 ¼ c44 ðu3;1 þ u1;3 Þ þ e15 u;1 ;
x1
Film
ð2Þ
and
D1 ¼ e15 ðu3;1 þ u1;3 Þ e11 u;1 ;
hˆ s
ð3Þ
D3 ¼ e31 u1;1 þ e33 u3;3 e33 u;3 :
Fig. 1. Cross section of a typical thin AlN or ZnO film on a silicon layer as an FBAR.
The substitution of Eqs. (2) and (3) into Eq. (1) gives
€1 ; c11 u1;11 þ c44 u1;33 þ ðc13 þ c44 Þu3;13 þ ðe31 þ e15 Þu;13 ¼ qu quartz resonators, for which we first need to obtain accurate results of dispersion curves and cut-off frequencies in the wave number range of interest for the operation of FBARs. This motivates the current work in this paper. Since the structures of FBARs shown in Fig. 1 have two different regions, with or without a top driving electrode, we consider wave propagation in plate models for FBARs with a top electrode layer and without a top electrode layer, respectively. Both plane and aniplane waves were taken into account. From the three-dimensional piezoelectro-elastic equations, an exact procedure was established to calculate dispersion relations, cut-off frequencies and vibration distributions of selected modes. Those results are of fundamental importance and can be used as a reference to improve the accuracy of the two-dimensional plate equations by requiring the cutoff frequencies and the curvatures of the dispersion curves at cutoff frequencies predicted by the two-dimensional plate equations and the three-dimensional equations to be the same.
€3 ; c44 u3;11 þ c33 u3;33 þ ðc44 þ c13 Þu1;31 þ e15 u;11 þ e33 u;33 ¼ qu
ð4Þ
ðe15 þ e31 Þu1;13 þ e15 u3;11 þ e33 u3;33 e11 u;11 e33 u;33 ¼ 0:
Similarly, for silicon layer which is a cubic crystal, from the equations of anisotropic elasticity we have
€1 ; cs11 u1;11 þ cs44 u1;33 þ cs13 þ cs44 u3;13 ¼ qs u s s s s s€ c44 u3;11 þ c33 u3;33 þ c44 þ c13 u1;31 ¼ q u3 ;
ð5Þ
where we have used a superscript ‘s’ to indicate the material constants of silicon layer. At the top of the composite plate where x3 = hf, mechanically the surface is free. Electrically the surface may be electroded and grounded, or unelectroded. For an electroded surface, the boundary conditions are 0
f
0
f
€ 1 ðh Þ; T 31 ðh Þ ¼ q0 h u f
€ 3 ðh Þ; T 33 ðh Þ ¼ q0 h u f
ð6Þ
uðh Þ ¼ 0; f
2. Theoretical derivation Consider the composite plates in Fig. 2. We study straightcrested waves without x2 dependence, i.e., o/ox2 = 0. In this case, the equations of linear piezoelectricity for ZnO or other crystals of class 6 mm with the c axis along x3 decouples into two groups. One gives the displacement components u1 and u3 as well as the electric potential u, which corresponds to plane-strain wave case. The other is for u2 alone which corresponds to anti-plane wave case. The two cases will be dealt with below, respectively.
where q0 and h0 are the mass density and thickness of the top electrode. In Eq. (6), we have assumed that the electrodes are very thin and neglected their stiffness. For an unelectroded top surface, we have f
T 31 ðh Þ ¼ 0;
f
T 33 ðh Þ ¼ 0;
f
D3 ðh Þ ¼ 0:
ð7Þ
At the interface between the ZnO film and the silicon layer where x3 = 0, the continuity conditions are
u1 ð0þ Þ ¼ u1 ð0 Þ;
u3 ð0þ Þ ¼ u3 ð0 Þ;
uð0þ Þ ¼ 0;
00 00 €
T 31 ð0 Þ T 31 ð0 Þ ¼ q h u1 ð0Þ; þ
2.1. Plane-strain waves
ð8Þ
00 00 €
T 33 ð0 Þ T 33 ð0 Þ ¼ q h u3 ð0Þ; þ
The relevant equations of motion and the charge equation of electrostatics are
€1 ; T 11;1 þ T 31;3 ¼ qu €3 ; T 13;1 þ T 33;3 ¼ qu
ð1Þ
D1;1 þ D3;3 ¼ 0; where the stress components Tij and the electric displacement components Di are related to the displacement and electric displacement gradients through the constitutive relations
where q00 and h00 are the mass density and thickness of the interface electrode. At the bottom of the composite plate where x3 = hs the traction-free boundary conditions are s
T 31 ðh Þ ¼ 0;
s
T 33 ðh Þ ¼ 0:
We look for time-harmonic wave solutions that may exist in the composite plate. The ZnO film and the silicon layer need to be treated separately initially, and then boundary and continuity conditions will be applied.
x3 Electrode
h (a)
h' hf
Film
hs
Silicon
x1
ð9Þ
x3
Electrode
h
hf
Film
hs
Silicon
(b)
Fig. 2. Plate models for FBARs: (a) without a driving electrode; (b) with a driving electrode.
x1
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F. Zhu et al. / Ultrasonics 67 (2016) 105–111
In the ZnO film, consider the possibility of the following waves:
u1 ¼ A exp fx3 cos nx1 expðixtÞ; u3 ¼ B exp fx3 sin nx1 expðixtÞ; u ¼ C exp fx3 sin nx1 expðixtÞ;
ð10Þ
where A, B, and C are undetermined constants. Substituting Eq. (10) into Eq. (4), we obtain three homogeneous linear equations for A, B, and C. For nontrivial solutions the determinant of the coefficient matrix of the equations has to vanish, which leads to a polynomial equation of degree six for f. We denote the six roots of the equation by fðmÞ ðx; nÞ, with m = 1, 2, . . . , 6. Corresponding to each m, the three homogeneous linear equations for A, B, and C have nontrivial solutions which determine the so-called amplitude ratios, i.e., A:B: C = 1:a(m):b(m). Then the general solution to Eq. (4) in the form of Eq. (10) is
u1 ¼
AðmÞ exp fðmÞ x3 cos nx1 expðixtÞ;
6 X AðmÞ aðmÞ exp fðmÞ x3 sin nx1 expðixtÞ;
ð11Þ
6 X
AðmÞ bðmÞ exp fðmÞ x3 sin nx1 expðixtÞ;
where A(m) represent six undetermined constants. Similarly, for the silicon layer, we let
u3 ¼ G exp fx3 sin nx1 expðixtÞ;
ð12Þ
where F and G are undetermined constants. Substituting Eq. (12) into Eq. (5) gives a system of two homogeneous linear equations for F and G. For nontrivial solutions, we can obtain four roots of fðnÞ with the corresponding amplitude ratios, i.e., F:G = 1:c(n), with n = 1, 2, . . . , 4. Then the general solution to Eq. (5) in the form of Eq. (12) can be written as
u1 ¼
4 X F ðnÞ exp fðnÞ x3 cos nx1 expðixtÞ; n¼1
4 X u3 ¼ F ðnÞ cðnÞ exp fðnÞ x3 sin nx1 expðixtÞ;
ð13Þ
where F(n) are four undetermined constants. Finally, we substitute Eqs. (11) and (13) into Eqs. (6), (8) and (9) for an electroded plate, or into Eqs. (7)–(9) for an unelectroded plate. In each case, this results in ten homogeneous linear equations for the ten undetermined constants of A(m) and F(n). For nontrivial solutions, the determinant of the coefficient matrix of these equations has to vanish, which determines the dispersion relations of the waves. The nontrivial solutions of A(m) and F(n) gives the displacement and potential fields of the waves. These can be done on a computer. 2.2. Anti-plane waves In this part, we will deal with anti-plane waves with displacement u2 only, i.e.u1 = u3 = 0, u2 = u2 (x1, x3, t), u = u (x1, x3, t). The relevant equations of motion and the charge equation of electrostatics are
T 11;1 þ T 21;2 þ T 31;3 ¼ 0; T 13;1 þ T 23;2 þ T 33;3 ¼ 0; D1;1 þ D2;2 þ D3;3 ¼ 0 ;
T 31 ¼ e15 u;1 ;
T 33 ¼ e33 u;3 ;
T 12 ¼ c66 u2;1 ;
ð15Þ
and
D1 ¼ e11 u;1 ; ð16Þ
D2 ¼ e15 u2;3 ;
D3 ¼ e33 u;3 ; The substitution of Eqs. (15) and (16) into Eq. (14) gives
ðe31 þ e15 Þu;13 ¼ 0; ð17Þ
e11 u;11 e33 u;33 ¼ 0;
Similarly, for silicon which is a cubic crystal, from the equations of anisotropic elasticity we have
ð18Þ
where we have used a superscript ‘s’ to indicate the material constants of silicon. At the top of the composite plate where x3 = hf, mechanically the surface is free. Electrically the surface may be electroded and grounded, or unelectroded. For an electroded surface, the boundary conditions are 0
€ 2 ðh Þ; T 33 ðh Þ ¼ 0; uðh Þ ¼ 0; T 31 ðh Þ ¼ 0; T 32 ðh Þ ¼ q0 h u f
f
f
f
f
ð19Þ 0
where q and h are the mass density and thickness of the top electrode. In Eq. (19), we have assumed that the electrodes are very thin and neglected their stiffness. For an unelectroded top surface, we have 0
f
f
T 31 ðh Þ ¼ 0;
T 32 ðh Þ ¼ 0;
f
T 33 ðh Þ ¼ 0;
f
D3 ðh Þ ¼ 0:
ð20Þ
At the interface between the ZnO film and the silicon layer where x3 = 0, the continuity conditions are
n¼1
€2 ; T 12;1 þ T 22;2 þ T 32;3 ¼ qu
T 22 ¼ e31 u;3 ;
€2 ; cs66 u2;11 þ cs44 u2;33 ¼ qs u
m¼1
u1 ¼ F exp fx3 cos nx1 expðixtÞ;
T 23 ¼ c44 u2;3 ;
e15 u;11 þ e33 u;33 ¼ 0;
m¼1
u¼
T 11 ¼ e31 u;3 ;
€2 ; c66 u2;11 þ c44 u2;33 ¼ qu
6 X
m¼1
u3 ¼
where the stress components Tij and the electric displacement components Di are related to the displacement and electric displacement gradients through the constitutive relations
ð14Þ
u2 ð0þ Þ ¼ u2 ð0 Þ; þ
þ
uð0þ Þ ¼ 0; 00 € 2 ð0Þ; T 32 ð0þ Þ T 32 ð0 Þ ¼ q00 h u
T 31 ð0 Þ T 31 ð0 Þ ¼ 0;
ð21Þ
T 33 ð0 Þ T 33 ð0 Þ ¼ 0; 00
where q00 and h are the mass density and thickness of the interface electrode. At the bottom of the composite plate where x3 = hs the traction-free boundary conditions are s
T 31 ðh Þ ¼ 0;
s
T 32 ðh Þ ¼ 0;
s
T 33 ðh Þ ¼ 0:
ð22Þ
For Eq. (17), we make the following observation. (e31 + e15) u,31 = 0 in Eq. (17)1 gives u = F(x1) + G(x3), whilst from e15u,11 + e33u,33 = 0 in Eq. (17)3 and e11u,11 e33u,33 = 0 in Eq. (17)4, we can obtain u,11 = u,33 = 0, giving rise to a general solution u = ax1 + bx3 + c. For an electroded plate, substitution of u = ax1 + bx3 + c into u(x3 = hf) = 0 in Eq. (19) and u(x3 = 0+) = 0 in Eq. (21) gives u = 0. Similarly, for an unelectroded plate, substitution of u = ax1 + bx3 + c into D3 = e33u,3 in Eq. (16), D3(hf) = 0 in Eq. (20), and u(x3 = 0+) = 0 in Eq. (21) gives u = 0 too. Therefore, for both electroded and unelectroded plate, u = 0 holds on in the case of anti-plane wave motion. The wave motion Eq. (17) will be simplified as
€2 : c66 u2;11 þ c44 u2;33 ¼ qu
ð23Þ
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F. Zhu et al. / Ultrasonics 67 (2016) 105–111
For the boundary and continuity conditions, the Eqs. (19)–(22) are separately simplified as 0
€ 2 ðh Þ; T 32 ðh Þ ¼ q0 h u f
f
f
T 32 ðh Þ ¼ 0; þ
u2 ð0 Þ ¼ u2 ð0 Þ;
00 00 €
T 32 ð0 Þ T 32 ð0 Þ ¼ q h u2 ð0Þ; þ
s
T 32 ðh Þ ¼ 0:
ð25Þ
^eij ¼ eij =ðc44 e33 Þ2 ; ^eij ¼ eij =e33 ; F^ ¼ F;
ð26Þ
ð28Þ
Substituting Eq. (28) into Eq. (23), we obtain
fð2Þ ¼
qx2 c66 n2
!12 i;
c44
qx2 c66 n2
ð29Þ
!12
The general solution for u2 in the ZnO film is given as
u2 ¼ Að1Þ exp fð1Þ x3 þ Að2Þ exp fð2Þ x3 expðinx1 Þ expðixtÞ
ð30Þ
Similarly, for the silicon layer, we let
u2 ¼ B exp fx3 exp inx1 expðixtÞ:
ð31Þ
Substituting Eq. (31) into Eq. (18), we obtain
fð1Þ ¼ f
ð2Þ
¼
qs x2 cs66 n2
!12 i;
cs44
qs x2 cs66 n2 cs44
1
q^ ¼ q=qZnO ;
^f ¼ f ðh f þ hs Þ; ^ ¼ h=ðh f þ hs Þ; ^ ¼ G; h G
^ ¼ C ðe33 =c44 Þ12 ; C
ð32Þ
!12 i:
The general solution for u2 in the silicon layer is given as
u2 ¼ Bð1Þ exp fð1Þ x3 þ Bð2Þ exp fð2Þ x3 expðinx1 Þ expðixtÞ: ð33Þ Finally, we substitute Eqs. (30) and (33) into Eqs. (24), (26) and (27) for an electroded plate, or into Eqs. (25)–(27) for an unelectroded plate. In each case, this results in four homogeneous linear equations for the four undetermined constants of A(1), A(2), B(1) and B(2). For nontrivial solutions, the determinant of the coefficient matrix of these equations has to vanish, which determines the dispersion relations of the anti-plane waves. The nontrivial solutions of A(1), A(2), B(1) and B(2) gives the displacement of the waves. These are done on a computer as well.
^ ¼ x ðqZnO =c44 Þ2 ðh f þ hs Þ; x 1
where we have used a superscript ‘^’ to indicate the dimensionless parameters. If the imaginary part of the frequency x doesn’t vanish, the displacements in the form of Eq. (11) will decay exponentially as time goes on and the corresponding waves are meaningless. Hence, the frequency x is required to be real while the wave number n can be complex. The waves with imaginary wave number decay along the x1 direction, thus they are restricted in some small regions, and cannot propagate.
The dispersion curves are shown in Fig. 3 for electroded plate and Fig. 4 for unelectroded plate, where the left-hand side is for imaginary wave number and the right-hand side is for the real wave number. All dots in the figures are independent, and each dot represents a solution to the above-mentioned mathematical ^ in Figs. 3 and 4 is dimensionless as derivation. The frequency x the definition above, while the dimensional frequency is at the level of GHz, and much higher than that of traditional resonators. For the curves in Figs. 3 and 4, they can be divided into two groups. One group goes to the zero frequency as the wave numbers change from real to pure imaginary and the other one is shaped like ‘U’. These U-shaped curves extend to the complex wave number space and tend to the zero frequency. It can be seen from the comparison of Figs. 3 and 4 that the dispersion curves of electroded plate have similar patterns to those of unelectroded plate in real wave number range, but different patterns in imaginary wave number range. Further, the distributions of displacement for different branches of the real part of the curves are calculated to ascertain the vibration modes. Starting from zero frequency, different branches of the real part of the curves are named as first order curve, second order curve, and so on. For a fixed dot in the curves, the displacements in the forms of Eqs. (11) and (13) are complex values. The real part and the imaginary part of the displacements have the same distributions when they are normalized by the modulus of the complex displacements, thus the real part and the imaginary part are indiscriminate in the ^ ¼ 0:1 in the first order curve results. For a fixed wave number n
3. Numerical results and discussion For real applications, thickness of the FBARs is in decade microns range. Hence, we used the following parameter values in our calculation. For the structure with a top driving electrode, the top electrode is treated as an inertial layer and its stiffness is neglected here. In that case, we don’t need to know how much the density and the thickness of the top electrode are. Their product is enough, thus the mass ratio of the piezoelectric layer to the top electrode layer is given here. 00
h ¼ 15 106 m; h ¼ 5 106 m; h ¼ 2 107 m; f
s
ð34Þ
3.1. Plane-strain waves
i:
c44
^n ¼ n ðh f þ hs Þ; ^ ¼ B; ^f ¼ f ðh f þ hs Þ; B
^cij ¼ cij =c44 ;
In the same way as the plane-strain case, we look for timeharmonic wave solutions that may exist in the composite plate. The ZnO film and the silicon layer need to be treated separately initially, and then boundary and continuity conditions will be applied. In the ZnO film, consider the possibility of the following waves:
fð1Þ ¼
^ ¼ A; A
ð24Þ
ð27Þ
u2 ¼ A exp fx3 exp inx1 expðixtÞ
The dimensionless quantities used in the calculation are defined as follows,
q0 h0 =ðqZnO hf Þ ¼ 0:01; q00 ¼ 19; 300 kg=m3 :
Fig. 3. Dispersion curves of plane-strain waves in electroded plate.
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F. Zhu et al. / Ultrasonics 67 (2016) 105–111
Fig. 6. The distribution of u3 on the first order curve of plane-strain waves.
Fig. 4. Dispersion curves of plane-strain waves in unelectroded plate.
Table 1 ^ at the zero wave number. Results of dimensionless frequency x Model
in Fig. 3, the distributions of u1 and u3 are separately shown in Figs. 5 and 6. ^ ¼ 0:1, one cycle for x1 is at the range of As for wave number n 10p to 10p as shown in Figs. 5 and 6. It can be seen from Fig. 6 that the vibration mode of the first order curve is flexure mode (as indicated by ‘‘F” in Figs. 3 and 4), whilst Fig. 5 shows the corresponding displacement u1 along x1 caused by bending. Similarly, as shown in Figs. 3 and 4, the second order curve is the extensional mode (E), the third order curve is the first order thickness-shear mode (TS1), the fourth order curve is the second order thicknessshear mode (TS2), the fifth order is the thickness-extensional mode (TE), and the sixth order curve is the third order thickness-shear mode (TS3). The vibration modes of electroded plate and unelectroded plate are corresponding with each other from the first order to the sixth order. The FBARs shown in Fig. 1 normally operate at the thickness-extensional mode, thus the high order modes are out of interest for the operation of FBARs and are neglected here, but can be obtained in the same way. 3.1.1. Zero wave number Cut-off frequency, i.e. a frequency when wave number takes zero value, is an important consideration when one is to develop effective two-dimensional plate equations. This section will show the cut-off frequency of the first a few modes within the scope of interest for the operation of FBARs. From the mentioned mathematical derivation shown in Section 2, theoretically, a ^ can be obtained complex transcendental equation about x directly from the determinant of the coefficient matrix of the ten ^ ¼ 0. The solutions can homogeneous linear equations by setting n be obtained by solving the equation numerically, which gives the cut-off frequencies of the first nine order curves listed in Table 1. ^ range from 0 to 20. In fact, an Table 1 shows the results of x ^ at the zero wave number can be obtained infinite number of x from the complex transcendental equation. The results of electroded plate and unelectroded plate are correspondingly given. The latter values are slightly higher than the former due to the fact that electroded plate has larger mass inertia than the unelectroded
Fig. 5. The distribution of u1 on the first order curve of plane-strain waves.
Electroded Unelectroded
Order 1&2
3
4
5
6
7
8
9
0 0
3.53 3.56
7.13 7.19
7.83 8.20
10.78 10.87
14.34 14.46
15.86 15.97
17.65 17.78
plate and the electrode inertia lowers resonant frequencies. This subtle difference of the cut-off frequency is useful for the design of FBARs. In real application, as shown in Fig. 1, the film layer of FBARs is partly covered by a driving electrode. When an operating frequency of FBARs is chosen between the cut-off frequencies of the TE mode in a plate with and without electrode, as shown in Table 1 and Figs. 3 and 4, the wave number for electroded part is real while the wave number for unelectroded part is pure imaginary due to the decreasing pattern of frequency as wave number goes from real to pure imaginary. This leads to an energy trapping phenomenon because the real wave number represents a cyclical wave while the imaginary wave number represents a decayed wave. For the other mode curves, if the frequency decreases as wave number goes from real to pure imaginary, when an operating frequency is chosen between the corresponding cut-off frequencies, the energy trapping phenomenon appears as well. 3.1.2. The influence of the mass ratio The mass ratio of driving electrode layer to piezoelectric layer plays an important role in the design of resonators since it directly relates to resonant frequency and vibration distribution. Especially in the case of FBARs, structurally, the thickness of its driving electrode is comparable to that of its film layer. In this subsection, we present the influence of the mass ratio on dispersion curves of plain waves in the electroded plate. Three different values of the 0 mass ratio R = q0 h /(qZnOhf) are chosen to find out how it influences the dispersion curves, as shown in Fig. 7. It can be seen from Fig. 7 that the mass ratio does not change the pattern of dispersion curves of plain waves. But larger mass ratio leads to larger structural inertia, which lowers the corresponding frequency at a fixed wave number. This phenomenon becomes more obvious for higher order dispersion curves, e.g., the thickness-extensional mode of FBARs. According to the results above, the mass ratio may be applied to extend the frequency range for energy trapping by fixing the parameters of film layer and changing that of driving electrode layer, which can give a theoretical guidance to the design of FBARs. The FBARs normally operate at the thickness-extensional mode. Thus, the influence of mass ratio on the thickness-extensional mode of FBARs is discussed in detail as shown in Table 2. It can be seen from Table 2 that the mass ratio has a great effect on the resonance frequency of the thickness-extensional mode of FBARs. That is to say, the operating frequency of FBARs can be adjusted by setting different mass ratios. It is useful for the design and optimization of FBARs.
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4. Conclusion In this paper, we theoretically studied the wave propagation in plate models for FBARs with and without a top electrode layer, respectively. Both plane-strain and anti-plane waves were taken into account. Numerical results on dispersion relations, cut-off frequencies and vibration distributions of selected modes were given and discussed in detail. Further discussion on how the mass ratio of piezoelectric layer to driving electrode layer influences the dispersion curves was made, which results in that the mass ratio does not change the trend of dispersion curves but larger mass ratio lowers corresponding frequency at a fixed wave number and may extend the frequency range for energy trapping. Those results are of fundamental importance and can be used as a reference to develop effective two-dimensional plate equations for structural analysis and design of FBARs. Fig. 7. Dispersion curves of plane-strain waves in electroded plate under different mass ratios.
Table 2 ^ on TE mode with different mass ratios. Dimensionless cut-off frequencies x Mass ratio Frequency
0.002 7.88909
0.005 7.86863
0.008 7.84829
0.01 7.83481
0.02 7.76820
0.05 7.57698
Acknowledgements This work was supported by the Program for New Century Excellent Talents in Universities (No. NCET-12-0625), the Natural Science Foundation for Distinguished Young Scholars of Jiangsu Province (No. SBK2014010134), the Fundamental Research Funds for Central Universities (No. NE2013101), the National Natural Science Foundation of China (Nos. 11502108, 11232007), and a project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). References
Fig. 8. Dispersion curves of anti-plane wave in electroded and unelectroded plates.
3.2. Anti-plane waves For the sake of completeness, we put the discussions on the dispersion curves of anti-plane waves here. In fact, from the theoretical observation made in Section 2.2, it is clear that the plate model in the anti-plane case is degenerated into an elastic layered plate due to u = 0, i.e., a pure elastic wave propagation problem. Fig. 8 shows the dispersion curves of the anti-plane wave in the plate with and without a top electrode, where the left-hand side is for pure imaginary wave number case and the right-hand side for real wave number case. Different from the case of plane-strain waves, there is only one group of dispersion curves, i.e., the one going to the zero frequency as wave number changes from real to pure imaginary. Furthermore, the comparison between electroded and unelectroded plates shows that the dispersion curves of the anti-plane waves in the plate model with and without a top electrode are qualitatively similar. The only difference is that the latter has a higher frequency value at any fixed wave number point, which is due to that electroded plate has larger mass ratio than unelectroded plate, i.e., larger inertia lowers vibration frequency.
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