Rev. Roum. Sci. Techn. – Électrotechn. et Énerg., 55, 1, p. 90–99, Bucarest, 2010 FIELD MODELS OF POWER BAW RESONATORS MIHAI MARICARU 1 , FLORIN CONSTANTINESCU 1 , ALEXANDRE REINHARDT 2 , MIRUNA NIŢESCU 1 , AURELIAN FLOREA 1 Key words: Power BAW resonators, 3D models. A simplified 3 D model, taking into account the longitudinal propagation only, has been used for the validation of material parameters values. The results, obtained building the stack layer by layer, agree very well with those computed using the analytical 1D multi- layer Mason model. A 3D model taking into account the lateral propagation too, considering various approximations for the mechanical boundary conditions has been used, showing that the simulated lateral modes are more accentuated than the measured ones. Some aspects of the future research are pointed out. 1. INTRODUCTION The piezoelectric effect occurs in materials for which an externally applied elastic strain causes a change in electric polarization which produces a charge and a voltage across the material. The converse piezoelectric effect is produced by an externally applied electric field, which changes the electric polarization, which in turn produces an elastic strain. The most known piezoelectric material is quartz crystal. Many other natural crystalline solids as Rochelle salt, ammonium dihydrogen phosphate, lithium sulphate, and tourmaline as well as some man-made crystal as gallium orthophosphate, aluminium nitride (AlN), and langasite exhibit piezoelectric properties. A lot of artificial ceramics as barium titanate, lead titanate, lead zirconate titanate (PZT), potassium niobate, lithium niobate, and lithium tantalate have similar properties. In the last years the bulk acoustic wave (BAW) AlN resonators emerged as a very efficient solution for mobile communications filters due to the possibility to be integrated at a relatively low cost together with CMOS circuits in systems on a chip and systems in a package. The electromechanical constitutive equations of a linear piezoelectric material are: 1 Dept. of Electrical Engineering, “Politehnica” University, Bucharest, Romania. 2 CEA, LETI, Minatec, Grenoble, France.
Transcript
Rev. Roum. Sci. Techn. – Électrotechn. et Énerg., 55, 1, p. 90–99, Bucarest, 2010
FIELD MODELS OF POWER BAW RESONATORS
MIHAI MARICARU1, FLORIN CONSTANTINESCU1, ALEXANDRE REINHARDT2, MIRUNA NIŢESCU1, AURELIAN FLOREA1
Key words: Power BAW resonators, 3D models.
A simplified 3 D model, taking into account the longitudinal propagation only, has been used for the validation of material parameters values. The results, obtained building the stack layer by layer, agree very well with those computed using the analytical 1D multi-layer Mason model. A 3D model taking into account the lateral propagation too, considering various approximations for the mechanical boundary conditions has been used, showing that the simulated lateral modes are more accentuated than the measured ones. Some aspects of the future research are pointed out.
1. INTRODUCTION
The piezoelectric effect occurs in materials for which an externally applied elastic strain causes a change in electric polarization which produces a charge and a voltage across the material. The converse piezoelectric effect is produced by an externally applied electric field, which changes the electric polarization, which in turn produces an elastic strain.
The most known piezoelectric material is quartz crystal. Many other natural crystalline solids as Rochelle salt, ammonium dihydrogen phosphate, lithium sulphate, and tourmaline as well as some man-made crystal as gallium orthophosphate, aluminium nitride (AlN), and langasite exhibit piezoelectric properties. A lot of artificial ceramics as barium titanate, lead titanate, lead zirconate titanate (PZT), potassium niobate, lithium niobate, and lithium tantalate have similar properties.
In the last years the bulk acoustic wave (BAW) AlN resonators emerged as a very efficient solution for mobile communications filters due to the possibility to be integrated at a relatively low cost together with CMOS circuits in systems on a chip and systems in a package.
The electromechanical constitutive equations of a linear piezoelectric material are: 1 Dept. of Electrical Engineering, “Politehnica” University, Bucharest, Romania. 2 CEA, LETI, Minatec, Grenoble, France.
2 Field models of power BAW resonators 91
T
,
,E
S
c e
e
= −
= + ε
T S E
D S E (1)
where: T is the stress vector; D is the electric flux density vector; S is the strain vector; E is the electric field vector; cE is the elasticity matrix (evaluated at constant electric field); e is piezoelectric stress matrix (eT denotes the transpose of e matrix), and εS is the dielectric matrix (evaluated at constant mechanical strain).
The elasticity matrix is a (6 × 6) anisotropic symmetric matrix cE given at a constant electric field. The cD elasticity matrix (considered at a constant electric flux density) can be given also:
S
EDeccε
+=2
. (2)
The piezoelectric stress matrix e is (6 × 3) and relates the electric field vector E in the order x, y, z to the stress vector T in the order x, y, z, xy, yz, xz. The piezoelectric strain matrix dE (at constant electric field) can be obtained from:
EE dce ⋅= . (3)
The dielectric matrix εS, uses the electrical permittivity at a constant strain and can be described in orthotropic or anisotropic form [5]. The dielectric matrix at constant stress εT can also be given:
deTST−ε=ε . (4)
The most convenient numerical procedure for writing the electromechanical field equations is the finite element method (FEM) which uses spatial discretization of the whole domain [1]. A typical structure of a BAW resonator is given in Fig. 1 together with the mechanical and electrical boundary conditions (BC). This implies that FEM encounters inherent difficulties in dealing with open boundary field problems, as the problem domain needs to be truncated to keep the size finite. Truncation inevitably introduces an artificial boundary and, consequently, a modelling error resulting from an approximation of the BC on this boundary. Considering acoustic waves, the truncation of the model causes reflections of the wave on the artificial boundaries. Placing infinite elements along the artificial boundary on the side of the continuum has been suggested as a solution to this problem. The infinite elements strive to implement an ideal absorbing boundary condition, such that a wave incident on the boundary would not reflect back. Instead of infinite elements, one may simply introduce regions at the boundaries of the model where the attenuation of the material increases from zero to a given finite value. Since the increase of the attenuation is gradual, there is no abrupt
change in the materials properties which would give rise to reflections of the wave. With a sufficiently high attenuation, the amplitude of the wave entering the region will decay rapidly such that there is no reflection. This solution has the benefit that it can be readily applied without any need for special FEM elements [2].
SiN Mo
AlN Mo
SiOC SiN
SiN
Si substrate
SiOC
SiOC
top electrode, V2=0V
bottom electrode, V1=1 V
symmetry BC: ux=0 (yOz), uy=0(xOz)
xyz
lateral BC: ux=0 , uy=0
dV/dn=0
Fig. 1 – BAW resonator with boundary conditions (electric potential V1 = 1V and V2 = 0V,
dV/dn = 0, displacements ux, uy, uz).
A possible solution is to combine FEM with another method which is used to model the semi-infinite region. An example of such a modelling technique is the FEM/BEM (boundary-element-method) formalism, which is used in the modelling of surface-acoustic wave devices [2, 7].
Starting from coupled electromechanical equations four types of solutions are possible [3]: Static Analysis, Mode-Frequency Analysis, Harmonic Analysis, and Transient Analysis.
A linear electromechanical field model of an AlN piezoelectric resonator is developed in this paper. The values of the material parameters on the longitudinal direction are validated in Section 2 by comparing the FEM results with the Mason multilayer model results and with small signal measurements. Various models for FEM analysis including transversal wave propagations are proposed and checked in Section 3.
As the longitudinal propagation (along the z axis) is playing a key role in the operation of a BAW resonator, we have chosen to validate the associated material parameters using a simplified 3D FEM model. A part of them haven’t been measured directly for the case in point, being taken from literature.
The well-known Mason multi-layer model provides an analytical solution for the mechanical waves on z axis only [5].
The material parameters which must be known for using this model are given in Table 1 [8].
Table 1
Material properties used for analytical model simulations
Mat. No.
Material Density ρ [kg/m3]
Longitudinal velocity [m/s]
c33 [GPa]
ε33 [pF/m]
e33 [C/m2]
1 Mo 10000 6600 435.6 3 SiOC 1500 2400 10.14 4 SiN 2700 9300 233.523 5 Si 2330 8400 164.4 6 AlN 3300 11000 399.3 82.6 1.5
For a 3D simulation some additional material parameters are used for numerical simulations. It is assumed that all layers, except AlN, are isotropic. The following parameter values have been used:
• Si Major Poisson's ratios: PRXY = PRYZ = PRXZ = 0.26, Young's Modulus Ex = Ey= Ez = 134.36 GPa;
• SiN: PRXY = PRYZ = PRXZ = 0.29, Ex = Ey = Ez = 178.20 GPa; • SiOC: PRXY = PRYZ =PRXZ = 0.22, Ex = Ey = Ez = 8.88 GPa; • Mo: PRXY = PRYZ = PRXZ = 0.31, E x = Ey = Ez = 314.26 GPa. The top-down order of resonator layers is: SiN, Mo, AlN, Mo, SiOC, SiN,
SiOC, SiN, SiOC, Si (substrate), where the layers between the bottom Mo electrode and the substrate form the Bragg mirror (Fig. 1).
The resonator transversal cross-section is a square with a 150 µm edge. Mechanical boundary conditions: zero displacement constraints for all directions imposed on the bottom surface of the last block (only for configuration 10).
Only one 3-D 20-node brick element is considered in all transversal sections (the transversal waves are practically neglected). Each layer is meshed longitudinally with a number of brick elements, that have the height less than 1/8
of the wavelength, and this number is increased until we reach the stability of the resonance and anti-resonance frequency values. This stable solution is reached in general for elements with heights about 1/(16÷25) of the wavelength. The solution is obtained with the so called “strong coupling analysis” of ANSYS [4], meaning that only one system including all electromechanical field equations is used.The harmonic analysis results have been computed adding layer by layer and are given in Table 2. The resonance frequencies (Fr – series resonance frequency, Fa –parallel resonance frequency) are compared to those obtained using the 1D Mason multilayer model [5].
Table 2
Resonance frequencies computed by harmonic analysis
Stack configuration Resonance frequencies
(FEM simulation) [GHz]
Resonance frequencies
(Mason model) [GHz]
Config. 1: only AlN
Fr = 4.568 Fa = 4.702
Fr = 4.567 Fa = 4.701
Config. 2: Config. 1 + Mo atop the AlN layer
Fr = 3.093 Fa = 3.180
Fr = 3.092 Fa = 3.180
Config. 3: Config. 2 + Mo under the AlN layer
Fr = 2.138 , Fa = 2.208
Fr = 2.136 Fa = 2.206
Config. 4: Config. 3 + SiN atop the stack
Fr = 2.037 , Fa = 2.103
Fr = 2.029 Fa = 2.095
Config. 5: Config. 4 + SiOC under the stack
Fr1 = 1.850 Fa1 = 1.895 Fr2 = 2.408 Fa2 = 2.440
Fr1= 1.809 Fa1= 1.840 Fr2 = 2.293 i = 2.326
Config. 6: Config. 5 + SiN under the stack
Fr = 2.043 , Fa = 2.107
Fr = 2.037 , Fa = 2.100
Config. 7: Config. 6 + SiOC under the stack
Fr1 = 2.033 Fa1 = 2.090 Fr2 = 2.185 Fa2 = 2.193
Fr1 = 1.995 Fa1 = 2.013 Fr2 = 2.071 Fa2 = 2.117
Config. 8: Config. 7 + SiN under the stack
Fr = 2.042 , Fa = 2.106
Fr = 2.037 Fa = 2.100
Config. 9: Config. 8 + SiOC under the stack
Fr = 2.042 , Fa = 2.106
Fr1 = 2.026 Fa1 = 2.030 Fr2 = 2.041
Fa2 = 2.101 Config. 10 ( real resonator): Config. 9 + Si under the stack
Fr = 2.042 , Fa = 2.106
Fr = 2.037 Fa = 2.100
A good agreement between ANSYS numerical results and the analytical
Mason multilayer model can be observed, except Configuration 9. The frequency characteristic computed with ANSYS agrees with the measured one (Figs. 2, 3).
Fig. 2 – Small signal frequency characteristic of a power BAW resonator computed
taking into account all layers.
Fig. 3 – Measured small signal frequency characteristic of a power BAW resonator.
The resonance peaks are more accentuated because damping has not been taken into account in the FEM model.
3. A MODEL FOR THE LONGITUDINAL AND TRANSVERSAL WAVE PROPAGATION
Only a quarter of the resonator structure (as is it seen in the transversal plane) has been taken into account in order to reduce the computational effort required by 3D FEM simulations of the whole stack (Figs. 4, 5). Mechanical boundary
96 Mihai Maricaru et al. 7
conditions (zero displacement constraints for transversal directions, ux = 0 or uy = 0) and dV/dn = 0 boundary condition (for AlN only) are imposed on the symmetry planes.
Fig. 5 – 3D simulations (ndiv = 8), lateral boundaries with ux = 0, uy = 0, uz = 0.
The mesh discretization for longitudinal direction is the same as in previous Section. For lateral directions we used a discretization of ndiv x ndiv brick elements. In order to establish a more realistic model for transversal wave propagation we extend all layers by ext in both lateral directions (because the abrupt end of the
8 Field models of power BAW resonators 97
mesh is far from the real structure). For this new volume the mesh is much less dense. Some results obtained using this approach are presented in Fig. 6, Fig. 7, and Fig. 8.
Fig. 6 – 3D simulations for lateral extended stack (ndiv = 8 and 8 divisions for lateral extended side with ext = 150 µm) with lateral boundaries free.
Fig. 7 – 3D simulations for lateral extended stack (ndiv = 8 and 8 divisions for lateral extended side with ext = 150 µm) when lateral boundaries have zero displacement constraints for all directions.
The real structure contains many resonators, and the lateral extension can be taken up to the middle of the distance between two neighbour resonators. In this case symmetry boundary conditions have been considered on lateral boundaries, the result being shown in Fig. 8.
98 Mihai Maricaru et al. 9
Fig. 8 – 3D simulations for lateral extended stack (ndiv = 8 and 8 divisions for lateral extended side
with ext = 150 µm, lateral boundaries are symmetry planes.)
4. CONCLUSIONS
In order to validate a part of material parameters values, a simplified 3D model, built with only one brick element in the xy plane, has been used. This model reproduces the propagation along the z axis and its results, obtained building the stack layer by layer, agree well with those computed using the analytical 1D multi-layer Mason model. The frequency characteristic computed with this model shows a good agreement with the measured small-signal characteristic, also.
The lateral propagation is taken into account using more elements in the xy plane. Simulation results show that the lateral modes are unstable and more accentuated than the measured results. This effect can be produced by ill-conditioned meshing (in order to save CPU time, a small number of lateral divisions has been considered) and approximate material parameters on transversal directions (these values were taken from literature). Another source of errors can be that all layers, except AlN, have been considered isotropic, the parameters on lateral directions being considered equal to those on the longitudinal direction. Taking into account lossless materials may lead to this kind of results, also. The identification of the coefficients describing mechanical losses for each layer starting from measurements performed on a stack is very difficult.
Various approximations for the mechanical boundary conditions have been considered, the closest to the real situation being symmetry axes both in the resonator structure and between adjacent resonators.
If we assume that the cause for material deterioration is the high level of the local stress in resonator layers, then the regions in which material deteriorates
10 Field models of power BAW resonators 99
correspond to the regions of maximum local stress. The FEM simulation using 3D models can be a basis for a new approach to the power durability problem, if it is associated with experimental identification of the material failure zones.
Future work will be devoted to the identification of mechanical loss coefficients and to the implementation of a nonlinear mechanical characteristic aimed to reproduce both the amplitude-frequency and the intermodulation effects.
Received on 9 August 2008
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