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An elastic electrode model for wave propagation analysis in piezoelectric layered structures of film bulk acoustic resonators
Accepted Manuscript
An elastic electrode model for wave propagation analysis in piezoelectric layered structures of film bulk acoustic resonators Feng Zhu , Yuxing Zhang , Bin Wang , Zhenghua Qian PII: DOI: Reference:
S0894-9166(17)30006-X 10.1016/j.camss.2017.04.001 CAMSS 25
To appear in:
Acta Mechanica Solida Sinica
Received date: Revised date: Accepted date:
5 January 2017 15 April 2017 18 April 2017
Please cite this article as: Feng Zhu , Yuxing Zhang , Bin Wang , Zhenghua Qian , An elastic electrode model for wave propagation analysis in piezoelectric layered structures of film bulk acoustic resonators, Acta Mechanica Solida Sinica (2017), doi: 10.1016/j.camss.2017.04.001
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An elastic electrode model for wave propagation analysis in piezoelectric layered structures of film bulk acoustic resonators Bin Wang, Zhenghua Qian *
Feng Zhu, Yuxing Zhang,
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State Key Laboratory of Mechanics and Control of Mechanical Structures/College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China * Corresponding author, E-mail: [email protected], Tel: 86-25-84892696.
Abstract-- Wave propagation in a piezoelectric layered structure of a film bulk acoustic resonator (FBAR) is studied. The accurate results of dispersion relation are calculated using the proposed elastic electrode model for both electroded and
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unelectroded layered plates. The differences of calculated cut-off frequencies between the current elastic electrode model and the simplified inertial electrode
model (often used in the quartz resonator analysis) are illustrated in detail, which shows that an elastic electrode model is indeed needed for the accurate analysis of FBAR. These results can be used as an accurate criterion to calibrate the 2-D
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theoretical model for a real finite-size structure of FBAR.
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Keywords: FBARs; Elastic electrode; Waves propagation; Dispersion curves
1. INTRODUCTION
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The thin film bulk acoustic resonator (FBAR) consists of a thin film of AlN or ZnO with a
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proper electrode layout on a silicon substrate and has the working frequency ranging from one to dozens of GHz [1-2]. Compared with the dielectric or surface acoustic wave (SAW)
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resonators, FBARs present much more advantages in size, working frequency, Q-factor and fabrication, and thus have broad applications and great potentials in telecommunication, signal processing, control, guidance, and sensing [3]. Typical FBARs operate in thickness-extensional modes of thin films, giving rise to three different types of structures for FBARs, i.e. the back etching type, air gap type and solidly mounted type. For the sake of analysis and design of FBARs, accurate predictions of their vibration frequencies and mode shapes are not only useful and necessary but also increasingly demanded [4], which have drawn much attention from researchers to theoretical and
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numerical modeling of FBARs. Qin et al. [5] theoretically analyzed thickness-longitudinal and thickness-shear dual-mode FBARs based on the c-axis-tilted ZnO and AlN films. Wang et al. [6] calculated the quality factor of FBARs with the consideration of viscosity. Wang et al. [7] also calculated the electrical parameters of FBARs by analyzing the vibration modes of layered structures of FBARs. Huang et al. [8] analyzed the high-frequency vibrations of layered anisotropic plates for the applications of FBARs. Du et al. [9] performed a theoretical
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study on thickness vibrations in AlN plates and found that for certain material orientations the thickness electric field was electrically coupled to thickness-shear only or thickness-stretch only. Recently, Huang et al. [10] investigated the effect of temperature on resonance properties of FBARs in thickness-stretch mode. The theoretical models used for FBARs in the
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above-mentioned analyses are all one-dimensional, which cannot describe the in-plane mode variations and mode couplings in practical devices of finite size.
More recently, Zhao et al. used the Tiersten’s 2-D scalar equations [11] to investigate the free vibrations of multilayered structures of film bulk acoustic wave resonators [12] and film
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bulk acoustic wave filters [13], and successfully obtained the in-plane variational thickness-extensional mode and the phenomenon of energy trapping. However, the scalar
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equation still has limitations, e.g. it is for the thickness-extensional mode only and cannot describe its coupling with other modes. At present, knowledge of the mechanism of mode
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couplings existing in real finite-size FBARs is still limited. There is an increasing demand for developing effective theoretical or numerical tools to describe and predict mode couplings in
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FBARs as well as to guide their design. To deal with this problem, Mindlin’s 2-D theory for quartz resonators can be an alternative reference [14-18]. Before doing so, it is natural to
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make use of the detailed analysis of acoustic waves propagating in the FBAR structures, from which an accurate criterion can be obtained for the calibration of dispersion relation within the operating frequency range of FBARs. In this respect, Zhu et al. [19] proposed an inertia electrode model to investigate the waves propagating in the layered structures of FBARs based on the 3-D elastic and piezoelectric equations. For the multilayered structure of FBARs, however, the thicknesses of electrodes are no longer far less than that of the piezoelectric film, which is different from the case in quartz resonators. Therefore, both the mass inertia and the elastic deformation of electrodes are needed to be taken into account when analyzing the 2
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wave propagation problem in such a multilayered structure, which motivates the current work.
Driving electrode Ground
x3
ht
hf
electrode
hb
hˆ s
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hs Silicon
x1
Film
Fig. 1. Side view of the schematic drawing of a typical FBAR structure
To show the error caused by ignoring the elastic deformation of electrodes, the same
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model with elastic electrode layers shown in Fig. 1 is considered. From this elastic electrode model, we calculate dispersion equations, cut-off frequencies and vibration modes. By comparing the results with those obtained from the previous inertia model [19], difference of thousands of ppm in the operating frequency is observed. The obtained results are both
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fundamental and important, and can be used to accurately calibrate an effective theoretical or numerical model describing and predicting mode couplings in FBARs as well as guiding their
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design.
2. THEORETICAL DERIVATION
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2.1 Model settings
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The structure of FBAR shown in Fig. 1 is made up of electrodes, an elastic supporting layer, and a piezoelectric film polarized along the x3 direction. It should be noticed that the top
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driving electrode only covers part of the piezoelectric film, which means that two theoretical models exist in the FBAR structure, i.e., one plate model with a driving electrode, as shown in Fig. 2(a), representing the electroded part of FBAR, and the other plate model without driving electrode, as shown in Fig. 2(b), representing the unelectroded part of FBAR. Therefore, the waves propagating in these two models are investigated respectively.
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x3 Electrode
hb
x3
ht′ hf
Film
hs
Silicon
x1
Electrode
hb
(a)
hf
Film
hs
Silicon
x1
(b)
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Fig. 2. The two schematic drawings of the layered models: (a) one representing the unelectroded part of FBAR; (b) the other representing the electroded part of FBAR.
2.2 Equations and derivation
Considering the composite plates in Fig. 2, we study the straight-crested waves without x2 dependence, i.e. / x2 0 . In this case, the equations of linear piezoelectricity for ZnO or
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other crystals of class 6mm with the c axis along x3 decouple into two groups. One gives the displacement components u1 and u3 as well as the electric potential ; the other is for u2 alone, which is not of interest here. The relevant equations of motion and the charge equation of electrostatics are
T11,1 T31,3 u1
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T13,1 T33,3 u3
(1)
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D1,1 D3,3 0
where the stress components Tij and the electric displacement components Di are related to
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and
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the displacements and the electric potential gradients through the constitutive relations T11 c11u1,1 c13u3,3 e31,3 T33 c13u1,1 c33u3,3 e33,3
(2)
T31 T13 c44 (u3,1 u1,3 ) e15,1
D1 e15 (u3,1 u1,3 ) 11,1 D3 e31u1,1 e33u3,3 33,3
(3)
The substitution of Eq. (2) and Eq. (3) into Eq. (1) gives c11u1,11 c44u1,33 (c13 c44 )u3,13 (e31 e15 ),13 u1 c44 u3,11 c33u3,33 (c44 c13 )u1,31 e15,11 e33,33 u3
(4)
(e15 e31 )u1,13 e15u3,11 e33u3,33 11,11 33,33 0
Similarly, for silicon which is a cubic crystal, from the equations of anisotropic elasticity,
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we have s s c11s u1,11 c44 u1,33 (c13s c44 )u3,13 s u1 s s c44 u3,11 c33s u3,33 (c44 c13s )u1,31 s u3
(5)
where the superscript ‘s’ is used to indicate the material constants of silicon. In the ZnO film, consider the possibility of the following waves: u1 A exp( x3 ) cos( x1 ) exp(it )
(6)
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u3 B exp( x3 ) sin( x1 ) exp(it )
C exp( x3 ) sin( x1 ) exp(it )
where A, B and C are undetermined constants. Substituting Eq. (6) into Eq. (4), we obtain three homogeneous linear equations for A, B and C. For the existence of nontrivial solutions, the determinant of the coefficient matrix of the equations has to vanish, which leads to a
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polynomial equation of degree six for . The six roots of the equation are denoted by ( m) (, ) , 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
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ratios, i.e., A:B:C= 1:α(m):β(m). Then the general solution to Eq. (4) in the form of Eq. (6) is 6
u1 A( m ) exp(
( m)
m 1
x3 ) cos( x1 ) exp(it )
6
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u3 A( m ) ( m ) exp(
( m)
m 1
x3 ) sin( x1 ) exp(it )
(7)
6
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A( m ) ( m ) exp( ( m ) x3 ) sin( x1 ) exp(it ) m 1
where A(m) represents six undetermined constants.
u1 F exp( x3 ) cos( x1 ) exp(it ) u3 G exp( x3 )sin( x1 ) exp(it )
(8)
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Similarly, for the silicon layer, we let
where F and G are undetermined constants. Substituting Eq. (8) into Eq. (5) gives a system of two homogeneous linear equations for F and G. For the existence of nontrivial solutions, we can obtain four roots of ( n ) with the corresponding amplitude ratios, i.e. F:G=1:γ(n). Then the general solution to Eq. (5) in the form of Eq. (8) can be written as 4
u1 F ( n ) exp( ( n ) x3 ) cos( x1 ) exp(it ) n 1
(9)
4
u3 F (n)
(n)
exp(
n 1
5
(n)
x3 ) sin( x1 ) exp(it )
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u1 H ( n ) exp( ' x3 ) cos( x1 ) exp(it ) (n)
n 1 4
(10)
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u3 H ( n ) ( n ) exp( '( n ) x3 ) sin( x1 ) exp(it ) n 1
where H(n) represents four undetermined constants, and ( n ) denotes amplitude ratios.
For the bottom electrode, the general solution to Eq. (5) in the form of Eq. (8) can be written as 4
n 1 4
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u1 J ( n ) exp( '' x3 ) cos( x1 ) exp(it ) (n)
u3 J ( n ) ( n ) exp( ''( n ) x3 ) sin( x1 ) exp(it ) n 1
(11)
where J(n) represents four undetermined constants, and ( n ) denotes amplitude ratios.
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For boundary and continuity conditions, since the electrodes are treated as elastic layers, the mechanical boundary conditions are free traction and the continuity conditions are
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continuous stresses and displacements. For different models, the electric boundary conditions are different. The model without top driving electrode represents open circuit; while the
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model with a top driving electrode represents short circuit. The electric boundary conditions correspond to different circuits. As a conclusion, boundary and continuity conditions are as
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follows.
For the model with a top driving electrode, at the top of the composite plate, where
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x3 h f ht hb / 2 , the boundary conditions are
T31 (h f ht hb / 2) 0, T33 (h f ht hb / 2) 0
(12)
At the interface between the top electrode and the film layer, where x3 h f hb / 2 , the continuity conditions are u1[(hb / 2 h f ) ] u1[(hb / 2 h f ) ], u3 [(hb / 2 h f ) ] u3 [(hb / 2 h f ) ] T31[(hb / 2 h f ) ] T31[(hb / 2 h f ) ], T33 [(hb / 2 h f ) ] T33 [(hb / 2 h f ) ]
(h / 2 h ) 0 b
f
6
(13)
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At the interface between the bottom electrode and the film layer, where x3 hb / 2 , the continuity conditions are u1[(hb / 2) ] u1[(hb / 2) ], u3 [( hb / 2) ] u3 [( hb / 2) ] T31[(hb / 2) ] T31[(hb / 2) ], T33 [(hb / 2) ] T33 [(hb / 2) ]
(14)
(h / 2) 0 b
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At the interface between the bottom electrode and the silicon layer, where x3 hb / 2 , the continuity conditions are
u1[(hb / 2) ] u1[(hb / 2) ], u3 [(hb / 2) ] u3 [(hb / 2) ]
T31[(hb / 2) ] T31[(hb / 2) ], T33 [(hb / 2) ] T33 [(hb / 2) ]
(15)
At the bottom of the composite plate, where x3 hs hb / 2 , the traction-free boundary
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conditions are
T31 (hs hb / 2) 0, T33 (hs hb / 2) 0
(16)
For the model without top driving electrode, Eqs. (12) and (13) are replaced by T31 (h f hb / 2) 0, T33 (h f hb / 2) 0
(17)
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D3 (h f hb / 2) 0
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The other equations are the same as those for electroded plates. Finally, we substitute Eq. (7) and Eqs. (9)-(11) into Eqs. (12)-(16) for an electroded plate,
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or into Eqs. (14)-(17) for an unelectroded plate. This results in eighteen homogeneous linear equations for the undetermined constants of A( m) , F ( n ) , H(n) and J(n) in the electroded plates
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and fourteen homogeneous linear equations for the undetermined constants of A( m) , F ( n ) and J(n) in the unelectroded plates. For the existence of nontrivial solutions, the determinant of the
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coefficient matrix of these equations has to vanish, which determines the dispersion relations of the waves. The nontrivial solutions of A( m) , F ( n ) , H(n) and J(n) give the displacement and potential fields of the waves. In the previous work [19], since the electrodes were simplified as inertia layers, the number of homogeneous linear equations was ten for both electroded and unelectroded plates, which was much simpler than the problem under question here.
3. NUMERICAL RESULTS AND DISCUSSION
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In our calculation, the size parameters for the models are set as,
h f 15 106 m, hs 5 106 m, hb 2 107 m, ht 3.1614 107 m The thicknesses of the film and the silicon substrate layer are adopted to correspond with the real size of FBAR, which is in the range of decade microns. The material of the film is zinc oxide; the material of the substrate layer is silicon; the top electrode is crystal aluminum; and
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the bottom electrode is crystal gold. The thickness of the top electrode is adopted to ensure that the mass ratio between the top electrode and the ZnO film is 0.01. Detailed material parameters can be found in the book [20]; and the dimensionless quantities are unique in our calculation, which are all listed as follows, 1
c 'ij cij / c44 , e 'ij eij / c44 33 2 , 'ij ij / 33 1
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' (h f h s ), ' （ ZnO∕c44）2 (h f h s ) ' ∕ ZnO , F ' F , G ' G，J ' J
(18)
h '=h /(h h ), H ' H , I ' I , K ' K f
s
1
A ' A, B ' B, C ' C 33 / c44 2
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where the superscript ‘′’ is used to indicate the dimensionless parameters. 3.1. Dispersion curves
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In this part, we compare the calculated dispersion curves with the results obtained by the simplified inertial electrode model in [19]. The two sets of results are plotted in the same
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figure in order to observe the slight difference between each other.
Fig. 3. Dispersion curves of electroded plate in Fig. 2 (b)
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Fig. 4.
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Dispersion curves of unelectroded plate in Fig. 2 (a)
Figures 3 and 4 show the dispersion curves of electroded plates and unelectroded plates, respectively. Each of them has two kinds of deriving models: one is the accurate elastic
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electrode model which is proposed here; the other is the simplified inertial electrode model which has been analyzed in [19]. From Figs. 3 and 4, it can be seen that the dispersion curves of these two kinds of deriving models are consistent with each other in both electroded and unelectroded plates, with only a subtle difference.
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The vibration modes of the first six order curves are also shown in Figs. 3 and 4. From the first order to the sixth order, the vibration modes are respectively the flexural mode, the
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extensional mode, the first order thickness-shear mode, the second order thickness-shear mode, the thickness-extensional mode and the third order thickness-shear mode. The results
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are also consistent with the ones obtained in the simplified inertial electrode model [19]. Such
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a calculation comparison on general dispersion curves shows the validity of the current elastic electrode model to some extent. The calculation of these modes will be discussed later in
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detail. Here, it is noted that if the ZnO film is replaced by AlN, dispersion curves similar to Figs. 3 and 4 can be obtained, but the cut-off frequency of the second order thickness-shear mode is above that of the thickness-extensional mode. 3.2. Vibration modes For the research of vibration modes, the distributions of displacements u1 and u3 along the boundaries of the plates are both calculated by arbitrarily picking a point on the dispersion curve of each order. For example, the point of 0.1 on the first order to the sixth order is chosen to calculate the displacements u1 and u3. 9
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(b)
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(a)
(d)
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(c)
(e)
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(f) Fig. 5. The vibration distributions of the first six modes: (a) Flexural mode, (b) Extensional mode, (c) First order thickness-shear mode, (d) Second order thickness-shear mode, (e) Thickness-extensional mode, (f) Third order thickness-shear mode
Fig. 6. The distribution of u1 for the first order curve on the boundary
x1 10π
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Figure 5 shows the vibration distributions of the first six order modes along the
boundaries. As the dimensionless quantities in Eq. (18), h '=h / (h f hs ) , the thickness of the model, is in the range of -0.255~0.7708 after dimensionless process. The thickness ranging from -0.005 to 0.005 represents the bottom electrode; the thickness ranging from -0.005 to -0.255 represents the silicon substrate; the thickness ranging from 0.005 to 0.755 represents the piezoelectric film; and the thickness ranging from 0.755 to 0.7708 represents the top electrode. The length of x1 is a cycle of -10π~10π due to the selected wavenumber 0.1. 11
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In Fig. 5a, the distributions of displacements are bent on the top and bottom boundaries. On the left and right boundaries, the displacements seem to vanish. Actually, the thickness of the composite layer is much smaller than its size in the direction of x1, thus the ratio between coordinates x1 and x3 is very big in Fig. 5, which makes it difficult to observe the tiny displacements in the direction of x1 on the left and right boundaries. The normalized displacements on the right boundary x1 10π in Fig. 5a are individually shown in Fig. 6. It
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can be seen that the distributions of displacements are in an oblique straight line, which represents the slant of cross section due to the bend and satisfies the plane cross-section assumption. It is reasonable for the composite layer to bend as a Bernoulli-Euler beam due to the big slenderness ratio.
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In Fig. 5b, it is obvious that the composite layer is pulled along x1 within the range of 0~10π and compressed along x1 within the range of -10π~0 . Meanwhile, the composite layer
shrinks and extends along x3 in the corresponding ranges due to Poisson effect. It is obviously the extensional mode. While in Fig. 5e, the composite layer is pulled and compressed along the
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thickness direction. Thus, it is clearly the thickness-extensional mode.
In Fig. 5c, the top boundary shifts in the negative direction of x1, while the bottom
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boundary shifts in the positive direction of x1. They have opposite directions and the distributions along thickness are curves with one null point. It is the so-called first order
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thickness-shear mode. In Fig. 5d, the direction of the composite layer’s shifting changes twice from the top boundary to the bottom boundary, and the distribution curves along thickness
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have two null points. It is the so-called second order thickness-shear mode. Similarly, the one shown in Fig. 5f is the so-called third order thickness-shear mode.
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Compared with the results in the simplified inertial electrode model [19], it can be
concluded that the two kinds of deriving models lead to the same modes except for a subtle difference in the frequency. 3.3. Quantitative discussion on cut-off frequency As mentioned above, the simplified inertial electrode model is simple but inaccurate to deal with the layered structure of FBAR, while the elastic electrode model is accurate but complicated. If the difference between the frequencies calculated using the two models are sufficiently small, the simplified inertial electrode model may replace the complicated elastic 12
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electrode model completely. Therefore, it is necessary to calculate the cut-off frequencies using both models and make a detailed comparison. Table 1 and Table 2 show the dimensionless cut-off frequencies of some selected modes calculated using both the elastic electrode model and the inertial electrode model for electroded plate (Fig. 2(b)) and unelectroded plate (Fig. 2(a)), respectively. It can be seen from Table 1 and Table 2 that the cut-off frequency difference changes a lot for different
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modes. For the thickness-shear modes, the cut-off frequency difference increases quickly from the first order to the third order. Compared with the results of the corresponding modes between both electroded plate and unelectroded plate, the cut-off frequency differences in thickness-shear modes do not change much, while the cut-off frequency difference in the
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thickness-extensional (TE) mode changes greatly. Besides, FBARs normally operate in the TE mode. Thus, it is necessary to study if the cut-off frequency difference in the TE mode between the two models would change further when the mass ratio of the top electrode to the film changes.
Modes
elastic electrode model (R=0.01)
inertial electrode model (R=0.01)
Difference (ppm)
3.5240487 7.0911815 7.8170531 10.6752217
3.5305297 7.1325251 7.8348066 10.7790716
1839.08 5830.28 2271.12 9728.13
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Table 1 Dimensionless cut-off frequencies by different models for electroded plate
TS2 TE
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TS3
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Table 2 Dimensionless cut-off frequencies by different models for unelectroded plate Modes
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TS1 TS2 TE
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elastic electrode model (R=0)
inertial electrode model (R=0)
Difference (ppm)
3.5541280 7.1525560 8.1798774 10.7672421
3.5608000 7.1949549 8.2005978 10.8731797
1877.25 5927.80 2533.09 9838.88
Table 3 shows the dimensionless cut-off frequencies of the TE mode calculated using both the elastic electrode model and the inertial electrode model for different mass ratios of the top electrode to the film. It can be seen from Table 3 that the cut-off frequency difference (ppm) between the two models decreases more and more slowly when the mass ratio changes
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from 0.001 to 0.015, while the difference starts to increase more and more quickly when the mass ratio is over 0.015. In fact, the difference of the TE mode does not decrease significantly when the mass ratio changes from 0.001 to 0.015, and the minimum value of the difference is about 2261. In general, the single-digit ppm is acceptable for the design of resonators. Therefore, it is obvious that the simplified inertial electrode model leads to an unacceptable error, and is not suitable for calculating the working frequencies of FBARs. The simplified
modes.
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inertial electrode model can only be used for qualitative analysis, such as the determination of
Table 3 Dimensionless cut-off frequencies of TE mode by different models elastic electrode model
inertial electrode model
Difference (ppm)
0.001 0.002 0.005 0.010 0.012 0.015 0.016 0.018 0.020 0.025 0.030 0.050 0.080 0.100
7.87769437 7.87090978 7.85062925 7.81705309 7.80369399 7.78372433 7.77708501 7.76383052 7.75060639 7.71766480 7.68486701 7.55452106 7.35873647 7.22607038
7.89593449 7.88908811 7.86863010 7.83480656 7.82137376 7.80132903 7.79467546 7.78141053 7.76820210 7.73542972 7.70301518 7.57698067 7.39887978 7.28736896
2315.41 2309.56 2292.92 2271.12 2265.56 2261.73 2261.83 2264.35 2270.24 2301.85 2361.55 2973.00 5455.19 8482.98
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Mass ratios R
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4. CONCLUSION
In this paper, the accurate dispersion curves in a piezoelectric layered structure of FBARs are
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obtained using the elastic electrode model for both electroded and unelectroded layered plates. The vibration modes are calculated and verified as the flexural mode, the extensional mode, the first order thickness-shear mode, the second order thickness-shear mode, the thickness-extensional mode and the third order thickness-shear mode for dispersion curves of the first six orders. The differences of calculated cut-off frequencies between the current elastic electrode model and the simplified inertial electrode model are analyzed in detail. As a result, the vibration modes obtained by the two models correspond with each other, but the simplified inertial electrode model leads to an unacceptable error in the cut-off frequencies, 14
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and cannot be used for the accurate analysis of FBARs. ACKNOWLEDGEMENTS This work was supported by the National Natural Science Foundation of China (Nos. 11502108, 11232007, 51405225), the Natural Science Foundation for Distinguished Young Scholars of Jiangsu Province (Nos. BK20140037, BK20140808), the Fundamental Research
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Funds for Central Universities (No. NE2013101), the Program for New Century Excellent Talents in Universities (No. NCET-12-0625), and a project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). REFERENCES
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[9] J.K. Du, X. Kai, 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. [10] D. Huang, J. Wang, Effects of temperature on frequency properties of FBAR，IEEE Symposium on Piezoelectricity Acoustic Waves and Device Applications (SPAWDA) (2014) 379-382. [11] H.F. Tiersten, D.S. Stevens, An analysis of thickness-extensional trapped energy resonant
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device structures with rectangular electrodes in the piezoelectric thin film on silicon configuration, Journal of Applied Physics 54 (1983) 5893-5910.
[12] Z.N. Zhao, Z.H. Qian, et al., Energy trapping of thickness-extensional modes in thin film bulk acoustic wave resonators, Journal of Mechanical Science and Technology 29(7)
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(2015) 2767-2773.
[13] Z.N. Zhao, Z.H. Qian, et al., Energy trapping of thickness-extensional modes in thin film bulk acoustic wave filters, AIP Advances 6 (2016) 015002.
[14] R.D. Mindlin, An Introduction to the Mathematical Theory of Vibrations of Elastic Plates,
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Yang J.S. ed., World Scientific, Beijing, 2006.
[15] R.D. Mindlin, High frequency vibrations of piezoelectric crystal plates, International
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Journal of Solids and Structures 8 (1972) 895-906. [16] J. Wang, J.S. Yang, Higher-order theories of piezoelectric plates and applications,
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Applied Mechanics Reviews 53 (2000) 87-99. [17] H.F. Tiersten, On the thickness expansion of the electric potential in the determination of
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two-dimensional equations for the vibration of electroded piezoelectric plates, Journal of Applied Physics 91 (2002) 2277-2283.
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[18] J.S. Yang, The Mechanics of Piezoelectric Structures, World Scientific, Beijing, 2006. [19] F. Zhu, Z.H. Qian, et al., Wave propagation in piezoelectric layered structures of film bulk acoustic resonators, Ultrasonics 67 (2016) 105-111.
[20] B.A. Auld, Acoustic fields and waves in solids. Wiley-Interscience Publication, New York, 1973.
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An elastic electrode model for wave propagation analysis in piezoelectric layered structures of film bulk acoustic resonators Feng Zhu , Yuxing Zhang , Bin Wang , Zhenghua Qian PII: DOI: Reference:
S0894-9166(17)30006-X 10.1016/j.camss.2017.04.001 CAMSS 25
To appear in:
Acta Mechanica Solida Sinica
Received date: Revised date: Accepted date:
5 January 2017 15 April 2017 18 April 2017
Please cite this article as: Feng Zhu , Yuxing Zhang , Bin Wang , Zhenghua Qian , An elastic electrode model for wave propagation analysis in piezoelectric layered structures of film bulk acoustic resonators, Acta Mechanica Solida Sinica (2017), doi: 10.1016/j.camss.2017.04.001
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An elastic electrode model for wave propagation analysis in piezoelectric layered structures of film bulk acoustic resonators Bin Wang, Zhenghua Qian *
Feng Zhu, Yuxing Zhang,
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State Key Laboratory of Mechanics and Control of Mechanical Structures/College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China * Corresponding author, E-mail: [email protected], Tel: 86-25-84892696.
Abstract-- Wave propagation in a piezoelectric layered structure of a film bulk acoustic resonator (FBAR) is studied. The accurate results of dispersion relation are calculated using the proposed elastic electrode model for both electroded and
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unelectroded layered plates. The differences of calculated cut-off frequencies between the current elastic electrode model and the simplified inertial electrode
model (often used in the quartz resonator analysis) are illustrated in detail, which shows that an elastic electrode model is indeed needed for the accurate analysis of FBAR. These results can be used as an accurate criterion to calibrate the 2-D
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theoretical model for a real finite-size structure of FBAR.
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Keywords: FBARs; Elastic electrode; Waves propagation; Dispersion curves
1. INTRODUCTION
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The thin film bulk acoustic resonator (FBAR) consists of a thin film of AlN or ZnO with a
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proper electrode layout on a silicon substrate and has the working frequency ranging from one to dozens of GHz [1-2]. Compared with the dielectric or surface acoustic wave (SAW)
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resonators, FBARs present much more advantages in size, working frequency, Q-factor and fabrication, and thus have broad applications and great potentials in telecommunication, signal processing, control, guidance, and sensing [3]. Typical FBARs operate in thickness-extensional modes of thin films, giving rise to three different types of structures for FBARs, i.e. the back etching type, air gap type and solidly mounted type. For the sake of analysis and design of FBARs, accurate predictions of their vibration frequencies and mode shapes are not only useful and necessary but also increasingly demanded [4], which have drawn much attention from researchers to theoretical and
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numerical modeling of FBARs. Qin et al. [5] theoretically analyzed thickness-longitudinal and thickness-shear dual-mode FBARs based on the c-axis-tilted ZnO and AlN films. Wang et al. [6] calculated the quality factor of FBARs with the consideration of viscosity. Wang et al. [7] also calculated the electrical parameters of FBARs by analyzing the vibration modes of layered structures of FBARs. Huang et al. [8] analyzed the high-frequency vibrations of layered anisotropic plates for the applications of FBARs. Du et al. [9] performed a theoretical
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study on thickness vibrations in AlN plates and found that for certain material orientations the thickness electric field was electrically coupled to thickness-shear only or thickness-stretch only. Recently, Huang et al. [10] investigated the effect of temperature on resonance properties of FBARs in thickness-stretch mode. The theoretical models used for FBARs in the
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above-mentioned analyses are all one-dimensional, which cannot describe the in-plane mode variations and mode couplings in practical devices of finite size.
More recently, Zhao et al. used the Tiersten’s 2-D scalar equations [11] to investigate the free vibrations of multilayered structures of film bulk acoustic wave resonators [12] and film
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bulk acoustic wave filters [13], and successfully obtained the in-plane variational thickness-extensional mode and the phenomenon of energy trapping. However, the scalar
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equation still has limitations, e.g. it is for the thickness-extensional mode only and cannot describe its coupling with other modes. At present, knowledge of the mechanism of mode
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couplings existing in real finite-size FBARs is still limited. There is an increasing demand for developing effective theoretical or numerical tools to describe and predict mode couplings in
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FBARs as well as to guide their design. To deal with this problem, Mindlin’s 2-D theory for quartz resonators can be an alternative reference [14-18]. Before doing so, it is natural to
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make use of the detailed analysis of acoustic waves propagating in the FBAR structures, from which an accurate criterion can be obtained for the calibration of dispersion relation within the operating frequency range of FBARs. In this respect, Zhu et al. [19] proposed an inertia electrode model to investigate the waves propagating in the layered structures of FBARs based on the 3-D elastic and piezoelectric equations. For the multilayered structure of FBARs, however, the thicknesses of electrodes are no longer far less than that of the piezoelectric film, which is different from the case in quartz resonators. Therefore, both the mass inertia and the elastic deformation of electrodes are needed to be taken into account when analyzing the 2
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wave propagation problem in such a multilayered structure, which motivates the current work.
Driving electrode Ground
x3
ht
hf
electrode
hb
hˆ s
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hs Silicon
x1
Film
Fig. 1. Side view of the schematic drawing of a typical FBAR structure
To show the error caused by ignoring the elastic deformation of electrodes, the same
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model with elastic electrode layers shown in Fig. 1 is considered. From this elastic electrode model, we calculate dispersion equations, cut-off frequencies and vibration modes. By comparing the results with those obtained from the previous inertia model [19], difference of thousands of ppm in the operating frequency is observed. The obtained results are both
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fundamental and important, and can be used to accurately calibrate an effective theoretical or numerical model describing and predicting mode couplings in FBARs as well as guiding their
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design.
2. THEORETICAL DERIVATION
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2.1 Model settings
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The structure of FBAR shown in Fig. 1 is made up of electrodes, an elastic supporting layer, and a piezoelectric film polarized along the x3 direction. It should be noticed that the top
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driving electrode only covers part of the piezoelectric film, which means that two theoretical models exist in the FBAR structure, i.e., one plate model with a driving electrode, as shown in Fig. 2(a), representing the electroded part of FBAR, and the other plate model without driving electrode, as shown in Fig. 2(b), representing the unelectroded part of FBAR. Therefore, the waves propagating in these two models are investigated respectively.
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x3 Electrode
hb
x3
ht′ hf
Film
hs
Silicon
x1
Electrode
hb
(a)
hf
Film
hs
Silicon
x1
(b)
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Fig. 2. The two schematic drawings of the layered models: (a) one representing the unelectroded part of FBAR; (b) the other representing the electroded part of FBAR.
2.2 Equations and derivation
Considering the composite plates in Fig. 2, we study the straight-crested waves without x2 dependence, i.e. / x2 0 . In this case, the equations of linear piezoelectricity for ZnO or
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other crystals of class 6mm with the c axis along x3 decouple into two groups. One gives the displacement components u1 and u3 as well as the electric potential ; the other is for u2 alone, which is not of interest here. The relevant equations of motion and the charge equation of electrostatics are
T11,1 T31,3 u1
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T13,1 T33,3 u3
(1)
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D1,1 D3,3 0
where the stress components Tij and the electric displacement components Di are related to
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and
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the displacements and the electric potential gradients through the constitutive relations T11 c11u1,1 c13u3,3 e31,3 T33 c13u1,1 c33u3,3 e33,3
(2)
T31 T13 c44 (u3,1 u1,3 ) e15,1
D1 e15 (u3,1 u1,3 ) 11,1 D3 e31u1,1 e33u3,3 33,3
(3)
The substitution of Eq. (2) and Eq. (3) into Eq. (1) gives c11u1,11 c44u1,33 (c13 c44 )u3,13 (e31 e15 ),13 u1 c44 u3,11 c33u3,33 (c44 c13 )u1,31 e15,11 e33,33 u3
(4)
(e15 e31 )u1,13 e15u3,11 e33u3,33 11,11 33,33 0
Similarly, for silicon which is a cubic crystal, from the equations of anisotropic elasticity,
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we have s s c11s u1,11 c44 u1,33 (c13s c44 )u3,13 s u1 s s c44 u3,11 c33s u3,33 (c44 c13s )u1,31 s u3
(5)
where the superscript ‘s’ is used to indicate the material constants of silicon. In the ZnO film, consider the possibility of the following waves: u1 A exp( x3 ) cos( x1 ) exp(it )
(6)
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u3 B exp( x3 ) sin( x1 ) exp(it )
C exp( x3 ) sin( x1 ) exp(it )
where A, B and C are undetermined constants. Substituting Eq. (6) into Eq. (4), we obtain three homogeneous linear equations for A, B and C. For the existence of nontrivial solutions, the determinant of the coefficient matrix of the equations has to vanish, which leads to a
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polynomial equation of degree six for . The six roots of the equation are denoted by ( m) (, ) , 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
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ratios, i.e., A:B:C= 1:α(m):β(m). Then the general solution to Eq. (4) in the form of Eq. (6) is 6
u1 A( m ) exp(
( m)
m 1
x3 ) cos( x1 ) exp(it )
6
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u3 A( m ) ( m ) exp(
( m)
m 1
x3 ) sin( x1 ) exp(it )
(7)
6
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A( m ) ( m ) exp( ( m ) x3 ) sin( x1 ) exp(it ) m 1
where A(m) represents six undetermined constants.
u1 F exp( x3 ) cos( x1 ) exp(it ) u3 G exp( x3 )sin( x1 ) exp(it )
(8)
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Similarly, for the silicon layer, we let
where F and G are undetermined constants. Substituting Eq. (8) into Eq. (5) gives a system of two homogeneous linear equations for F and G. For the existence of nontrivial solutions, we can obtain four roots of ( n ) with the corresponding amplitude ratios, i.e. F:G=1:γ(n). Then the general solution to Eq. (5) in the form of Eq. (8) can be written as 4
u1 F ( n ) exp( ( n ) x3 ) cos( x1 ) exp(it ) n 1
(9)
4
u3 F (n)
(n)
exp(
n 1
5
(n)
x3 ) sin( x1 ) exp(it )
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u1 H ( n ) exp( ' x3 ) cos( x1 ) exp(it ) (n)
n 1 4
(10)
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u3 H ( n ) ( n ) exp( '( n ) x3 ) sin( x1 ) exp(it ) n 1
where H(n) represents four undetermined constants, and ( n ) denotes amplitude ratios.
For the bottom electrode, the general solution to Eq. (5) in the form of Eq. (8) can be written as 4
n 1 4
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u1 J ( n ) exp( '' x3 ) cos( x1 ) exp(it ) (n)
u3 J ( n ) ( n ) exp( ''( n ) x3 ) sin( x1 ) exp(it ) n 1
(11)
where J(n) represents four undetermined constants, and ( n ) denotes amplitude ratios.
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For boundary and continuity conditions, since the electrodes are treated as elastic layers, the mechanical boundary conditions are free traction and the continuity conditions are
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continuous stresses and displacements. For different models, the electric boundary conditions are different. The model without top driving electrode represents open circuit; while the
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model with a top driving electrode represents short circuit. The electric boundary conditions correspond to different circuits. As a conclusion, boundary and continuity conditions are as
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follows.
For the model with a top driving electrode, at the top of the composite plate, where
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x3 h f ht hb / 2 , the boundary conditions are
T31 (h f ht hb / 2) 0, T33 (h f ht hb / 2) 0
(12)
At the interface between the top electrode and the film layer, where x3 h f hb / 2 , the continuity conditions are u1[(hb / 2 h f ) ] u1[(hb / 2 h f ) ], u3 [(hb / 2 h f ) ] u3 [(hb / 2 h f ) ] T31[(hb / 2 h f ) ] T31[(hb / 2 h f ) ], T33 [(hb / 2 h f ) ] T33 [(hb / 2 h f ) ]
(h / 2 h ) 0 b
f
6
(13)
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At the interface between the bottom electrode and the film layer, where x3 hb / 2 , the continuity conditions are u1[(hb / 2) ] u1[(hb / 2) ], u3 [( hb / 2) ] u3 [( hb / 2) ] T31[(hb / 2) ] T31[(hb / 2) ], T33 [(hb / 2) ] T33 [(hb / 2) ]
(14)
(h / 2) 0 b
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At the interface between the bottom electrode and the silicon layer, where x3 hb / 2 , the continuity conditions are
u1[(hb / 2) ] u1[(hb / 2) ], u3 [(hb / 2) ] u3 [(hb / 2) ]
T31[(hb / 2) ] T31[(hb / 2) ], T33 [(hb / 2) ] T33 [(hb / 2) ]
(15)
At the bottom of the composite plate, where x3 hs hb / 2 , the traction-free boundary
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conditions are
T31 (hs hb / 2) 0, T33 (hs hb / 2) 0
(16)
For the model without top driving electrode, Eqs. (12) and (13) are replaced by T31 (h f hb / 2) 0, T33 (h f hb / 2) 0
(17)
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D3 (h f hb / 2) 0
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The other equations are the same as those for electroded plates. Finally, we substitute Eq. (7) and Eqs. (9)-(11) into Eqs. (12)-(16) for an electroded plate,
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or into Eqs. (14)-(17) for an unelectroded plate. This results in eighteen homogeneous linear equations for the undetermined constants of A( m) , F ( n ) , H(n) and J(n) in the electroded plates
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and fourteen homogeneous linear equations for the undetermined constants of A( m) , F ( n ) and J(n) in the unelectroded plates. For the existence of nontrivial solutions, the determinant of the
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coefficient matrix of these equations has to vanish, which determines the dispersion relations of the waves. The nontrivial solutions of A( m) , F ( n ) , H(n) and J(n) give the displacement and potential fields of the waves. In the previous work [19], since the electrodes were simplified as inertia layers, the number of homogeneous linear equations was ten for both electroded and unelectroded plates, which was much simpler than the problem under question here.
3. NUMERICAL RESULTS AND DISCUSSION
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In our calculation, the size parameters for the models are set as,
h f 15 106 m, hs 5 106 m, hb 2 107 m, ht 3.1614 107 m The thicknesses of the film and the silicon substrate layer are adopted to correspond with the real size of FBAR, which is in the range of decade microns. The material of the film is zinc oxide; the material of the substrate layer is silicon; the top electrode is crystal aluminum; and
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the bottom electrode is crystal gold. The thickness of the top electrode is adopted to ensure that the mass ratio between the top electrode and the ZnO film is 0.01. Detailed material parameters can be found in the book [20]; and the dimensionless quantities are unique in our calculation, which are all listed as follows, 1
c 'ij cij / c44 , e 'ij eij / c44 33 2 , 'ij ij / 33 1
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' (h f h s ), ' （ ZnO∕c44）2 (h f h s ) ' ∕ ZnO , F ' F , G ' G，J ' J
(18)
h '=h /(h h ), H ' H , I ' I , K ' K f
s
1
A ' A, B ' B, C ' C 33 / c44 2
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where the superscript ‘′’ is used to indicate the dimensionless parameters. 3.1. Dispersion curves
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In this part, we compare the calculated dispersion curves with the results obtained by the simplified inertial electrode model in [19]. The two sets of results are plotted in the same
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figure in order to observe the slight difference between each other.
Fig. 3. Dispersion curves of electroded plate in Fig. 2 (b)
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Fig. 4.
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Dispersion curves of unelectroded plate in Fig. 2 (a)
Figures 3 and 4 show the dispersion curves of electroded plates and unelectroded plates, respectively. Each of them has two kinds of deriving models: one is the accurate elastic
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electrode model which is proposed here; the other is the simplified inertial electrode model which has been analyzed in [19]. From Figs. 3 and 4, it can be seen that the dispersion curves of these two kinds of deriving models are consistent with each other in both electroded and unelectroded plates, with only a subtle difference.
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The vibration modes of the first six order curves are also shown in Figs. 3 and 4. From the first order to the sixth order, the vibration modes are respectively the flexural mode, the
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extensional mode, the first order thickness-shear mode, the second order thickness-shear mode, the thickness-extensional mode and the third order thickness-shear mode. The results
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are also consistent with the ones obtained in the simplified inertial electrode model [19]. Such
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a calculation comparison on general dispersion curves shows the validity of the current elastic electrode model to some extent. The calculation of these modes will be discussed later in
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detail. Here, it is noted that if the ZnO film is replaced by AlN, dispersion curves similar to Figs. 3 and 4 can be obtained, but the cut-off frequency of the second order thickness-shear mode is above that of the thickness-extensional mode. 3.2. Vibration modes For the research of vibration modes, the distributions of displacements u1 and u3 along the boundaries of the plates are both calculated by arbitrarily picking a point on the dispersion curve of each order. For example, the point of 0.1 on the first order to the sixth order is chosen to calculate the displacements u1 and u3. 9
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(b)
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(a)
(d)
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(c)
(e)
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(f) Fig. 5. The vibration distributions of the first six modes: (a) Flexural mode, (b) Extensional mode, (c) First order thickness-shear mode, (d) Second order thickness-shear mode, (e) Thickness-extensional mode, (f) Third order thickness-shear mode
Fig. 6. The distribution of u1 for the first order curve on the boundary
x1 10π
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Figure 5 shows the vibration distributions of the first six order modes along the
boundaries. As the dimensionless quantities in Eq. (18), h '=h / (h f hs ) , the thickness of the model, is in the range of -0.255~0.7708 after dimensionless process. The thickness ranging from -0.005 to 0.005 represents the bottom electrode; the thickness ranging from -0.005 to -0.255 represents the silicon substrate; the thickness ranging from 0.005 to 0.755 represents the piezoelectric film; and the thickness ranging from 0.755 to 0.7708 represents the top electrode. The length of x1 is a cycle of -10π~10π due to the selected wavenumber 0.1. 11
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In Fig. 5a, the distributions of displacements are bent on the top and bottom boundaries. On the left and right boundaries, the displacements seem to vanish. Actually, the thickness of the composite layer is much smaller than its size in the direction of x1, thus the ratio between coordinates x1 and x3 is very big in Fig. 5, which makes it difficult to observe the tiny displacements in the direction of x1 on the left and right boundaries. The normalized displacements on the right boundary x1 10π in Fig. 5a are individually shown in Fig. 6. It
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can be seen that the distributions of displacements are in an oblique straight line, which represents the slant of cross section due to the bend and satisfies the plane cross-section assumption. It is reasonable for the composite layer to bend as a Bernoulli-Euler beam due to the big slenderness ratio.
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In Fig. 5b, it is obvious that the composite layer is pulled along x1 within the range of 0~10π and compressed along x1 within the range of -10π~0 . Meanwhile, the composite layer
shrinks and extends along x3 in the corresponding ranges due to Poisson effect. It is obviously the extensional mode. While in Fig. 5e, the composite layer is pulled and compressed along the
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thickness direction. Thus, it is clearly the thickness-extensional mode.
In Fig. 5c, the top boundary shifts in the negative direction of x1, while the bottom
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boundary shifts in the positive direction of x1. They have opposite directions and the distributions along thickness are curves with one null point. It is the so-called first order
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thickness-shear mode. In Fig. 5d, the direction of the composite layer’s shifting changes twice from the top boundary to the bottom boundary, and the distribution curves along thickness
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have two null points. It is the so-called second order thickness-shear mode. Similarly, the one shown in Fig. 5f is the so-called third order thickness-shear mode.
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Compared with the results in the simplified inertial electrode model [19], it can be
concluded that the two kinds of deriving models lead to the same modes except for a subtle difference in the frequency. 3.3. Quantitative discussion on cut-off frequency As mentioned above, the simplified inertial electrode model is simple but inaccurate to deal with the layered structure of FBAR, while the elastic electrode model is accurate but complicated. If the difference between the frequencies calculated using the two models are sufficiently small, the simplified inertial electrode model may replace the complicated elastic 12
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electrode model completely. Therefore, it is necessary to calculate the cut-off frequencies using both models and make a detailed comparison. Table 1 and Table 2 show the dimensionless cut-off frequencies of some selected modes calculated using both the elastic electrode model and the inertial electrode model for electroded plate (Fig. 2(b)) and unelectroded plate (Fig. 2(a)), respectively. It can be seen from Table 1 and Table 2 that the cut-off frequency difference changes a lot for different
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modes. For the thickness-shear modes, the cut-off frequency difference increases quickly from the first order to the third order. Compared with the results of the corresponding modes between both electroded plate and unelectroded plate, the cut-off frequency differences in thickness-shear modes do not change much, while the cut-off frequency difference in the
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thickness-extensional (TE) mode changes greatly. Besides, FBARs normally operate in the TE mode. Thus, it is necessary to study if the cut-off frequency difference in the TE mode between the two models would change further when the mass ratio of the top electrode to the film changes.
Modes
elastic electrode model (R=0.01)
inertial electrode model (R=0.01)
Difference (ppm)
3.5240487 7.0911815 7.8170531 10.6752217
3.5305297 7.1325251 7.8348066 10.7790716
1839.08 5830.28 2271.12 9728.13
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Table 1 Dimensionless cut-off frequencies by different models for electroded plate
TS2 TE
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TS3
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Table 2 Dimensionless cut-off frequencies by different models for unelectroded plate Modes
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TS1 TS2 TE
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elastic electrode model (R=0)
inertial electrode model (R=0)
Difference (ppm)
3.5541280 7.1525560 8.1798774 10.7672421
3.5608000 7.1949549 8.2005978 10.8731797
1877.25 5927.80 2533.09 9838.88
Table 3 shows the dimensionless cut-off frequencies of the TE mode calculated using both the elastic electrode model and the inertial electrode model for different mass ratios of the top electrode to the film. It can be seen from Table 3 that the cut-off frequency difference (ppm) between the two models decreases more and more slowly when the mass ratio changes
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from 0.001 to 0.015, while the difference starts to increase more and more quickly when the mass ratio is over 0.015. In fact, the difference of the TE mode does not decrease significantly when the mass ratio changes from 0.001 to 0.015, and the minimum value of the difference is about 2261. In general, the single-digit ppm is acceptable for the design of resonators. Therefore, it is obvious that the simplified inertial electrode model leads to an unacceptable error, and is not suitable for calculating the working frequencies of FBARs. The simplified
modes.
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inertial electrode model can only be used for qualitative analysis, such as the determination of
Table 3 Dimensionless cut-off frequencies of TE mode by different models elastic electrode model
inertial electrode model
Difference (ppm)
0.001 0.002 0.005 0.010 0.012 0.015 0.016 0.018 0.020 0.025 0.030 0.050 0.080 0.100
7.87769437 7.87090978 7.85062925 7.81705309 7.80369399 7.78372433 7.77708501 7.76383052 7.75060639 7.71766480 7.68486701 7.55452106 7.35873647 7.22607038
7.89593449 7.88908811 7.86863010 7.83480656 7.82137376 7.80132903 7.79467546 7.78141053 7.76820210 7.73542972 7.70301518 7.57698067 7.39887978 7.28736896
2315.41 2309.56 2292.92 2271.12 2265.56 2261.73 2261.83 2264.35 2270.24 2301.85 2361.55 2973.00 5455.19 8482.98
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Mass ratios R
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4. CONCLUSION
In this paper, the accurate dispersion curves in a piezoelectric layered structure of FBARs are
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obtained using the elastic electrode model for both electroded and unelectroded layered plates. The vibration modes are calculated and verified as the flexural mode, the extensional mode, the first order thickness-shear mode, the second order thickness-shear mode, the thickness-extensional mode and the third order thickness-shear mode for dispersion curves of the first six orders. The differences of calculated cut-off frequencies between the current elastic electrode model and the simplified inertial electrode model are analyzed in detail. As a result, the vibration modes obtained by the two models correspond with each other, but the simplified inertial electrode model leads to an unacceptable error in the cut-off frequencies, 14
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and cannot be used for the accurate analysis of FBARs. ACKNOWLEDGEMENTS This work was supported by the National Natural Science Foundation of China (Nos. 11502108, 11232007, 51405225), the Natural Science Foundation for Distinguished Young Scholars of Jiangsu Province (Nos. BK20140037, BK20140808), the Fundamental Research
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Funds for Central Universities (No. NE2013101), the Program for New Century Excellent Talents in Universities (No. NCET-12-0625), and a project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). REFERENCES
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[1] R. Ruby, P. Bradley, et al., PCS 1900 MHz duplexer using thin film bulk acoustic resonators (FBARs), Electronics Letters 35(10) (1999) 794-795.
[2] K.M. Lakin, A review of the thin-film resonator technology, IEEE Microwave Magazine 4(4) (2003) 61-67.
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[3] S. Lucyszyn, Review of radio frequency microelectromechanical systems technology, IEE Proceedings Science Measurement and Technology 151(2) (2004) 93-103.
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