Electrical tuning of dc bias induced acoustic resonances in paraelectric thin films

Abstract

A theoretical model for the dc bias dependence of induced acoustic resonances in paraelectric materials is presented. The field dependent piezoelectric constitutive equations were derived from the Landau free energy expansion with respect to the ferroelectric part of the polarization P. To derive the field dependent coefficients correctly, we demonstrate that it is important to take both linear and nonlinear electrostriction as well as the background permittivity into account. Two different resonator geometries, corresponding to the thickness excitation (TE) and the lateral field excitation (LFE) modes, are discussed and compared. In the TE mode the resonance frequency is expected to be much stronger dc bias dependent than the antiresonance frequency. In the LFE mode, both resonance and antiresonance frequencies may exhibit comparable dc bias dependences. In this case the antiresonance frequency shows a stronger tuning with increasing dc bias than the resonance frequency. We model the behavior of the field dependent acoustic resonances in BaxSr1−xTiO3 thin films addressing different compositions and orientations of the films. Our theoretical model corroborates the experimental results available in the literature.

Effect of mechanical loading on the tuning of acoustic resonances in Ba x Sr1−x TiO3 thin films

“…In contrast to the relative tunability, the tuning of the resonance frequency is strongly reduced for the device with a 700 nm thick SiO 2 layer. From our theoretical analysis we expect that the magnitude of the intrinsic electromechanical coupling factor k 2 BST in the BST layer should be the same for both devices since the layer shows the same relative tunability n r [11]. Thus, we conclude that the mechanical load leads to a strong reduction of k 2 ef f (exp) and also of the tuning of the acoustic resonances in the device.…”

“…1(a). The resonance frequency at E dc extrapolated to 0 kV/cm is 2.850 GHz at constant electric displacement D, which leads to a change of the sound velocity of the BST layer with dc bias [11]. The resonance frequency shows a larger shift of 66 MHz at the maximum dc electric field.…”

“…Like mentioned above the shift of the antiresonance frequency is due to the dc bias dependence of the elastic constant c D of the BST layer at constant D. The sound velocity of the BST layer at constant D is given by the relation [11]:…”

“…Thus, the dependence of k 2 BST on E dc has been determined. At this point we were able to check a prediction of the theory of non-loaded tunable TFBARs, specifically the relation [11]:…”

“…Both frequencies shift to lower frequencies with increasing dc bias. In a theoretical treatment of the electrical tuning of dc bias induced acoustic resonances in BST thin films, it was found that the tuning of resonance and antiresonance frequencies are a function of the dc bias dependent intrinsic electromechanical coupling factor k 2 BST of the BST layer [11]. Moreover, it was shown that the intrinsic electromechanical coupling factor k 2 BST and the relative tunability n r of the material permittivity are identical functions of the dc electric field.…”