Tuning of direct current bias-induced resonances in micromachined Ba0.3Sr0.7TiO3 thin-film capacitors

Abstract

Direct current ͑dc͒ bias-induced acoustic resonance phenomena have been observed in micromachined tunable thin-film capacitors based on paraelectric Ba 0.3 Sr 0.7 TiO 3 thin films. Both resonance and antiresonance frequencies are dc bias dependent. The antiresonance frequency is only weakly dc bias dependent and shifts slightly down with increasing dc bias. The resonance frequency shows a much stronger dependence on the applied dc bias than the antiresonance frequency, and also shifts down to lower frequencies with increasing bias. The resonance frequency shifted by 2.0% for a frequency of about 3 GHz and an applied field of 730 KV/cm. At the same time the effective electromechanical coupling constant k t,eff 2 increased up to 3.6%. The tuning of the resonance frequency depends on the tunability of the film permittivity and on the mechanical load on the piezoactive layer. The constitutive equations controlling the tuning of the resonance and antiresonance frequency were derived from the Landau theory using the P-expansion of its free energies. The theoretical predictions show that, in the considered system, the frequency of the first-order resonance ͑ / 2 resonance͒ will always shift down with increasing bias, whereas the antiresonance frequency might shift up or down with increasing bias. The sense of the shift of the antiresonance frequency depends on the sign and the magnitude of the corresponding coefficients of the tensors of linear and nonlinear electrostriction. The theoretical predictions corroborate the experimental observations very well.

Growth and high frequency characterization of Mn doped sol-gel PbxSr1−xTiO3 for frequency agile applications

“…8͑c͒ and is roughly linear as would be expected for electrostrictive behavior. An interesting feature is the observed maximum value of k 33 2 ϳ 0.04 which is significantly higher than the value for the same field, ϳ0.013, recently reported by Noeth et al 33 for Ba 0.3 Sr 0.7 TiO 3 . Although k 33 2 may be lower due to the slightly composite nature of the resonant structure used by Noeth et al, 33 it does show that the effect in PST would be potentially useful for tunable thin film bulk acoustic resonator ͑FBAR͒ applications.…”

“…However, to get a better fit, especially to the magnitude of the resonances, more accurate material properties and frequency dependent acoustic quality factors must be used. The resulting k 33 2 variation with bias field is shown in Fig. 8͑c͒ and is roughly linear as would be expected for electrostrictive behavior.…”

“…14,33 In this model, the well known formalism used for describing piezoelectric transducer operation has been modified to allow for field dependence of the effective piezoelectric and elastic constants and hence also of the electromechanical coupling coefficient ͑k 33 2 ͒ and acoustic velocity ͑V ac ͒-the parameters that appear directly in the equations.…”

“…Using solutions of the one dimensional plane wave boundary value problem as given by Rosenbaum 34 and Lakin, 35 a piezoelectric layer-in this case PST-with mechanically loaded end faces has the following electrical impedance: 33 2 / ͑1+k 33 2 ͒, = ␤d / 2, and ␤ = / V ac , where is the angular frequency, V ac is the PST acoustic velocity, d is the thickness of the PST film, ͑z t ͒, ͑z b ͒ z PST are the acoustic impedances at the top and bottom end faces and of the PST film, respectively, and k 33 2 is the longitudinal electromechanical coupling coefficient.…”

“…8͑a͒was obtained by making fine adjustments to the parameters, principally the acoustic velocities in Pt and Ni, until with these fixed it was only necessary to vary k 33 2 to obtain best fit for the different biases. In Fig.…”