Quasi-3D Model for Lateral Resonances on Homogeneous BAW Resonators

Abstract

Lateral modes are responsible for the in-band spurious resonances that appear on BAW resonators, degrading the in-band filter response. In this work, a fast computational method based on the transmission line matrix (TLM) method is employed to model the lateral resonances of BAW resonators. Using the precomputed dispersion curves of Lamb waves and an equivalent characteristic impedance for the TE₁ mode, a network of transmission lines is used to calculate the magnitude of field distributions on the electrodes. These characteristics are specific to the stack layer configuration. The model's implementation is based on nodal Y matrices, from which particle displacement profiles are coupled to the electric domain via piezoelectric constitutive relations. Consequently, the input impedance of the resonator is obtained. The model exhibits strong agreement with FEM simulations of FBARs and SMRs, and with measurements of several SMRs. The proposed model can provide accurate predictions of resonator input impedance, which is around 200 times faster than conventional FEM.

Keywords: BAW resonator; Film Bulk Acoustic Resonator; Lamb wave; solidly mounted resonator; spurious modes.

Figure 1

Dispersion curves of a 1.74 μm thickness ZnO plate. The curves for two symmetric modes TE₁, TS₂, and the evanescent mode, between their cutoff frequencies, are labeled on the plot.

Figure 2

Two-port Π-network for a transmission line.

Figure 3

Dispersion curve for the TE₁ mode of a 1.74 μm thickness ZnO plate. FEM simulations were performed to obtain the dispersion curve and then expression (7) was adjusted.

Figure 4

(a) Two-dimensional resonator in the xz plane. The x dimension is discretized in N elements; (b) equivalent Π network of a dispersive transmission line in the x direction.

Figure 5

(a) Magnitude and phase of the impedance of the 2D FEM ZnO resonator (blue), and the quasi-2D model (red); (b) magnitude and phase of the impedance of the 2D FEM AlN SMR (blue), and the quasi-2D model (red).

Figure 6

(a) Schematic of a TLM mesh for a square resonator. The lateral dimensions a and b, and the number of discretizations in each direction (N_x and N_y) are indicated; (b) equivalent Π network of a dispersive transmission line in the x and y direction. Four nodes interconnected by different directions of transmission lines are shown.

Figure 7

(a) Magnitude and phase of the impedance of the 3D FEM ZnO square resonator (blue), and the quasi-3D model (red); (b) magnitude and phase of the impedance of the 3D FEM ZnO rectangular resonator (blue), and the quasi-3D model (red); (c) standing wave pattern of |v_z| at the frequency of mode 311 and 131 in the square resonator; (d) standing wave pattern of |v_z| at the frequency of mode 311 in the rectangular resonator.

Figure 7

(a) Magnitude and phase of the impedance of the 3D FEM ZnO square resonator (blue), and the quasi-3D model (red); (b) magnitude and phase of the impedance of the 3D FEM ZnO rectangular resonator (blue), and the quasi-3D model (red); (c) standing wave pattern of |v_z| at the frequency of mode 311 and 131 in the square resonator; (d) standing wave pattern of |v_z| at the frequency of mode 311 in the rectangular resonator.

Figure 8

(a) Magnitude and phase of the impedance of the 3D FEM ZnO trapezoidal resonator (blue), and the quasi-3D model (red); (b) standing wave pattern of normalized |v_z| for the first four resonant modes of the trapezoidal resonator.

Figure 9

(a) Magnitude and phase of the impedance of the measured square SMR (A = 6400 µm²) (blue), and the quasi-3D model (red); (b) magnitude and phase of the Impedance of the measured square SMR (A = 12,900 µm²) (blue), and the quasi-3D model (red); (c) magnitude and phase of the impedance of for the measured rectangular SMR (A = 6400 µm²) (blue), and the quasi-3D model (red); (d) magnitude and phase of the impedance of for the measured rectangular SMR (A = 12,900 µm²).