A tunable ferroelectric based unreleased RF resonator

This paper introduces the first tunable ferroelectric capacitor (FeCAP)-based unreleased RF MEMS resonator, integrated seamlessly in Texas Instruments’ 130 nm Ferroelectric RAM (FeRAM) technology. The designs presented here are monolithically integrated in solid-state CMOS technology, with no post-processing or release step typical of other MEMS devices. An array of FeCAPs in this complementary metal-oxide-semiconductor (CMOS) technology’s back-end-of-line (BEOL) process were used to define the acoustic resonance cavity as well as the electromechanical transducers. To achieve high quality factor (Q) of the resonator, acoustic waveguiding for vertical confinement within the CMOS stack is studied and optimized. Additional design considerations are discussed to obtain lateral confinement and suppression of spurious modes. An FeCAP resonator is demonstrated with fundamental resonance at 703 MHz and Q of 1012. This gives a frequency-quality factor product \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \cdot Q = 7.11 \times 10^{11}$$\end{document}f⋅Q=7.11×1011 which is 1.6× higher than the most state-of-the-art Pb(Zr,Ti)O3 (PZT) resonators. Due to the ferroelectric characteristics of the FeCAPs, transduction of the resonator can be switched on and off by adjusting the electric polarization. In this case, the resonance can be turned off completely at ±0.3 V corresponding to the coercive voltage of the constituent FeCAP transducers. These novel switchable resonators may have promising applications in on-chip timing, ad-hoc radio front ends, and chip-scale sensors.

searching the eigenmodes along the first Brillouin zone, we can obtain the dispersion relations of acoustic waves in the CMOS stack.
Limited by the technology, a maximum of 5 metal layers are used in the process. While lithographic dimensions can be changed within the constraints of the design rules, materials and thicknesses of the layers are predetermined by the technology. Practically speaking, the first metal layer must be reserved for electrical routing to the transducers, and must therefore be omitted from the BEOL reflector design. Based on these design constraints, we optimize the following parameters to maximize energy confinement in the resonance cavity:  Mode optimization begins with a range of devices using 5 metal layers at the BEOL. Apart from the first metal layer, all the other layers are continuous across the FeCAP area ( Figure S1(a)). It is assumed that the length of the trapizoidal FeCAP is 1.4 μm and the length of the lattice constant a is 5 μm. The dispersion relation, obtained through eigenmode analysis in COMSOL, is shown in Fig. S1(b). The blue and red lines present the longitudinal and shear sound lines, respectively. Several discrete modes exist below the shear sound line, indicating modes which are slow enough to be guided laterally in the CMOS stack. However, according to the stress distribution of these guided modes along = / , only the modes at 400.9 MHz and 479.5MHz are coupled efficiently to the FeCAP transducer. These mode shapes also indicate the energy confined in the FeCAP is coupled with the metal layers above it, corresponding to acoustic loss.
Additionally, the localized modes are very close to the shear sound line, indicating an opportunity for scattering into the bulk Si corresponding to lower quality factor.
To push the strain energy downwards closer to the FeCAP, the number of metal layers is reduced to two.
The schematic of the device is shown in Fig. S2(a), with corresponding dispersion relation in Fig. S2(b).
Five localized modes are visible along = / below shear sound line. Their displacement field is plotted in Fig. S2(c). In this case, only the mode at 433.4 MHz results in a large overlap between the strain energy and the transducer area. This is a necessary characteristic for high efficiency electromechanical coupling for drive and sense. However, the energy is still coupled with part of the SiO 2 layers and it is not fully confined within the FeCAP area. To simplify the optimization process taking into account technology limits, the length of the FeCAP is fixed at 1.4 μm. Sweeping the geometry of the metal layers, it is found that a metal length of 600 nm provides the best confinement. To further enhance the energy confinement, the authors investigated the effect of decreasing the unit cell length a. This is because, the sound line is expressed as = • , where is the acoustic wave velocity of a given material and is the wavenumber. Normalizing k with and we obtain = 2 = 2 .
Thus, the slope of the sound line is inversely proportinal with a. By decreasing a, the resonance mode frequency is increased and when a=2 um (lithographic limit of the techonology), and the distance between the confined mode and the sound line reaches maximum. This provides lower probability of mode scattering into the bulk Si and indicates highest energy confinement.

Fig. S3
Dependence of the acoustic mode frequency on the changing lattice constant a. As lattice constant reduces, the distance between the localized mode and the sound line increases.
To conclude, the metal width of 600 nm for metals 1 and 2, with lattice cosntant a = 2 um provides a combination of large acoustic confinment with maximum overlap of elastic energy in the FeCAP transducer, within the lithography constraints of the TI E035 technology. The corresponding dispersion relation is shown in Fig. 2 of the manuscript.