High-Q energy trapping of temperature-stable shear waves with Lamé cross-sectional polarization in a single crystal silicon waveguide

Micromechanical resonators are suitable candidates for low power and low profile oven-controlled oscillators due to their small size and wafer level vacuum packaging. Oven-controlled oscillators require continuous and controlled ovenization of the resonator at a constant temperature above the operating range (e.g., −55 °C to 85 °C for consumer applications).¹ 

Among different low loss acoustic platforms, single crystal silicon can provide unique advantages for realization of integrated oven-controlled oscillators: Besides serving as the low loss frequency reference, a silicon microresonator can simultaneously serve for self-temperature sensing and self-ovenization. While the former is demonstrated by monitoring the temperature-induced variations in electrical resistance² or acoustic properties of the device,³ the latter can be realized by passing electrical current through resonator body and via Joule heating.⁴ The temperature stability of such a self-sufficient oven-controlled reference is then defined by the resolution of the sensing and ovenization mechanisms, weighted by temperature sensitivity of the device around operating temperature. Therefore, device-level temperature compensation techniques are focusing on realization of silicon resonators with temperature characteristics that have a turnover above 85 °C.^5–8 

For this purpose, purely shear Lamé modes excited in properly designed plates aligned to ⟨100⟩ crystallographic direction of low resistivity single crystal silicon are desirable, since they provide largest turnover temperature compared to their longitudinal or flexural counterparts.^9,10 Lamé resonators are usually anchored through narrow tethers at nodal points of the vibration mode. This can make the self-ovenization of the structure at high temperatures impractical since excessive current density at narrow supporting tethers can result in their irreversible deformation or even melting. Furthermore, uniform ovenization of the device body can become challenging if current is flowing into the device at discrete points.¹¹ This might induce undesired perturbation in the vibration mode of the ovenized device and degrade its performance.

This letter presents a Lamé mode acoustic cavity that does not need discrete narrow tethers for mode confinement and energy trapping. Dispersive behavior of propagating and evanescent waves with Lamé mode cross-sectional polarization (called Lamé-X hereafter) is used to engineer the acoustic energy distribution and realize acoustic energy trapping with high quality factor (Q). Such an energy trapping confines the energy in the central region of the acoustic cavity and far from substrate. Therefore, the resulting structure can be uniformly self-ovenized to maintain high local-temperatures without any concern about performance or structural/material rigidity degradation.

Silicon rectangular waveguides support different laterally propagating/evanescent eigen-modes with different cross-sectional polarizations and dispersion behaviors (Fig. 1). The dispersion relation between the frequency and wavenumber of the guided waves can be defined by Christoffel wave equation¹² considering the stress-free peripheral faces of the waveguide.⁸ 

Opting for similar width and height (i.e., W = H in Fig. 1) for the waveguide aligned to ⟨100⟩ crystallographic axes of single crystal silicon, an eigen-mode exists with a Lamé cross-sectional polarization (S₁ branch in Fig. 1). Such a mode is well known for lower temperature sensitivity of frequency compared to flexural and extensional modes.⁹ More specifically, a turnover temperature above 85 °C can be achieved for Lamé eigen-modes with doping concentrations much lower compared to what required for other eigen-modes—such as flexural or extensional—to provide similar temperature characteristic. However, unlike extensional bulk acoustic/Lamb mode resonators, efficient energy trapping of Lamé-X mode is not practically feasible in rectangular parallelepiped structures. This is due to the absence of a nodal line in the cross-sectional polarization of the Lamé-X mode that prevents from non-destructive suspension of the cavity by narrow tethers.

In this work, an efficient energy trapping of purely shear Lamé-X eigen-mode is realized by exploiting its unique dispersive characteristic (Fig. 2) and integration of in- and out-of-plane capacitive transducers. Unlike extensional waves with simple dispersion types I and II,¹² Lamé-X dispersion characteristic has two local extremum (f, K_x) points that are referred to as higher and lower cut-off frequencies of the dispersion curve (f_c,high and f_c,low, respectively) hereafter. An electromechanical excitation of the waveguide at frequency f₀ that lies between higher and lower cut-off frequency (i.e., f_c,low < f₀ < f_c,high) can simultaneously induce both propagating and evanescent waves with Lamé-X polarization. The possible solutions are highlighted with stars on the S₁ dispersion curve in Fig. 2.

Depending on the placement and extension of electromechanical transducers, a solution A(x,y,z,t)—defining position/time dependent vibration amplitude—can be excited in the waveguide as

Here, A_i,1 and A_r,1&2 are the vibration amplitudes of the excited evanescent and propagating Lamé-X solutions at the frequency f₀ and wavenumbers K_i,1 and K_r,1&2, respectively, and Γ(y, z) is the vibration polarization of the Lamé-X mode.

The dispersive behavior of S₁ can be tailored by changing the cross-sectional dimensions of the waveguide.⁸ This characteristic can be used to engineer the energy distribution in the waveguide by acoustic coupling of propagating to evanescent waves with exponentially decaying energy.^8,13 For this purpose, several waveguides with different dimensions can be cascaded in series to form an acoustically engineered structure that efficiently trap the energy of a desired eigen-mode. This is schematically shown in Fig. 3 for Lamé-X eigen-mode, where a central waveguide with a width of W₀ is flanked on two sides by a cascade of differential waveguides with different widths (W_1–5) but similar height (H₀), and hence different dispersion behaviors.

While the thickness of all the waveguides is fixed by the substrate thickness, changing the width of the waveguides not only shifts the frequency of the Lamé-X dispersion curve but also slightly changes the trend/shape of the curve. This is a result of slight deviation from square cross-sectional shape, which induces distortion in the vibration polarization from purely shear Lamé and contaminates it with extensional stress field. Although the polarization distortion and consequent stress field contamination influence the temperature characteristic, the effect is negligible since the energy distribution is engineered to be maximum in the central waveguide that has a square cross section. Such an energy distribution engineering has been realized by increasing the relative length of the central waveguide versus flanks.

When properly designed—through proper length of the waveguides—to hold for the continuity of boundary conditions at transitional faces,¹² a synthesized vibration mode with a frequency f_res may exist created from the acoustic coupling of propagating and evanescent waves in constituent waveguides (shown by stars in Fig. 3). With an increase in the width, the local solutions in differential waveguides gradually converge to a purely evanescent solution. Finally, at a specific width of W₄, where f₀ = f_c,high, acoustic energy distribution decays exponentially since only evanescent solution can be excited. Further increase in the width of the waveguide reflects the energy back in to the cavity since no guided wave solution, neither propagating nor evanescent, exists in wider waveguides. This is shown in Fig. 3 through Δf_Gap, where f₀ is higher than f_c,high for W₅ branch. The design procedure includes optimization of the slope of width variation to guarantee proper acoustic coupling as well as minimum acoustic energy distribution in flanks where the waveguide is anchored to the substrate.

Fig. 4 (left) schematically shows the acoustically engineered structure and its COMSOL-simulated Lamé-X vibration mode. Release holes, required for practical implementation of the device through wet-etch release, are considered in design optimization procedure. Fig. 4 (right) shows the simulated normalized energy distribution of the vibration mode in a cut-line along its length, showing maximum and minimum energy distribution in central section and anchors, respectively.

The Lamé-X resonators are fabricated on a 40 ± 1 μm-thick arsenic-doped silicon substrates with resistivity of 0.02 Ω cm and 0.005 Ω cm using the HARPSS™ process.¹⁴ This process integrates 270 nm lateral and 300 nm out-of-plane capacitive gaps around the device to provide multiple transduction ports. Fig. 5 shows a SEM image of the device as well as a simple planar Lamé mode square resonator.

The in- and out-of-plane capacitive ports can be used for differential sensing of the Lamé-X mode to reduce feed-through signal of narrow gaps,¹⁵ or simultaneous and independent excitation of two different modes for self-temperature monitoring purpose.³ The transmission frequency response of the device is measured in different two-port configurations, as summarized in Fig. 6. A high Q of ∼43 k is measured for the device operating in pressure level of ∼1 Torr, with a polarization voltage of 35 V, resulting in an f × Q of ∼3.3 × 10¹², which is an order of magnitude higher compared to piezoelectrically transduced extensional acoustically engineered waveguides,⁸ and close to the intrinsic f × Q upper limit in ⟨100⟩ silicon resonators.¹⁶ Also, slightly different resonance frequency in two transduction schemes is a result of different electrical spring softening effects^17,18 for top and side capacitive transducers that have different gap sizes and transduction areas.

Fig. 7 compares the temperature characteristic of the devices implemented in low and high resistivity substrates. While a linear TCF of ∼27.3 ppm/ °C is measured for the high resistivity device, a parabolic temperature characteristic is measured for low resistivity device with a turnover temperature at 41 °C and 15× improvement in overall frequency drift in the temperature range of −20 °C to 80 °C. The planar Lamé resonator implemented on the same low resistivity substrate shows a parabolic temperature characteristic as well, with a turnover at 45 °C. While having similar polarization with respect to silicon crystallographic axes the temperature characteristic of the two Lamé devices should be identical, coupling the propagating shear waves to evanescent waves in flanks of the X-Lamé device is responsible for a slight difference in temperature characteristics. This is due to highly different temperature characteristic of evanescent and propagating waves.⁸ 

Having wide anchoring flanks connecting the acoustic cavity to the substrate, a Lamé-X resonator can be uniformly ovenized by passing large DC currents through its body. Fig. 8 compares the frequency of the high resistivity device as a function of self-ovenization current amplitude (for a constant ambient temperature of 40 °C) with frequency drift induced by ambient temperature variation (for zero self-ovenization current). ∼2100 ppm of frequency tuning is achieved for ovenization current of 45 mA, when the device operated in an environmental chamber at 40 °C. This corresponds to the device local temperature of ∼120 °C. The device Q reduced from ∼43k at 40 °C to ∼39k by temperature or current elevation to 130 °C or 45 mA, as a result of increased interaction between acoustic waves and thermal phonons at higher temperatures.^16,19

In order to evaluate the effect of self-ovenization on long-term frequency stability, the device is operated continuously over 10 days in an environmental chamber kept at 40 °C, while continuously passing a DC current of 45 mA through its body.

Fig. 9 shows the frequency variations over this time period. A burn-in frequency drop of 3 ppm is followed by a stable performance (±0.5 ppm), which is limited by the accuracy of the environmental chamber temperature control unit.

This work was supported by DARPA TIMU program through SSC pacific Contract No. # N66001-11-C-4176.