A temperature-compensated silicon micromechanical resonator with a quadratic temperature characteristic is realized by acoustic engineering. Energy-trapped resonance modes are synthesized by acoustic coupling of propagating and evanescent extensional waves in waveguides with rectangular cross section. Highly different temperature sensitivity of propagating and evanescent waves is used to engineer the linear temperature coefficient of frequency. The resulted quadratic temperature characteristic has a well-defined turn-over temperature that can be tailored by relative energy distribution between propagating and evanescent acoustic fields. A 76 MHz prototype is implemented in single crystal silicon. Two high quality factor and closely spaced resonance modes, created from efficient energy trapping of extensional waves, are excited through thin aluminum nitride film. Having different evanescent wave constituents and energy distribution across the device, these modes show different turn over points of 67 °C and 87 °C for their quadratic temperature characteristic.

After successful demonstration of oscillators with sub part per million instabilities,^{1,2} microelectromechanical systems (MEMS) are now aiming at the realization of integrated frequency references with sub part per billion instability. Such a significant improvement in stability would require continuous ovenization of the resonator at a constant temperature to effectively blind the device from ambient temperature variations that induce frequency fluctuations.^{3–5} For this purpose, it is desirable for the resonator to have a turn-over, i.e., a local extremum in the temperature characteristic of frequency, at a point slightly above the operating temperature range of the device (e.g., −55 °C to 85 °C for consumer applications). This paper presents a technique to create a turn-over in the temperature characteristic of silicon microresonators at a well-defined temperature point.

To compensate the temperature coefficient of frequency (*TCF)* of MEMS resonators, a large variety of device-level^{6–9} and system-level^{10} compensation techniques have been proposed. However, these techniques usually impose excessive power consumption or manufacturing complexities. $\u2009$ We demonstrate a single crystal silicon acoustic resonator implemented based on energy trapping of in-plane extensional wave in waveguides with rectangular cross section. In a similar technique to thickness-mode quartz and thin-film bulk acoustic wave resonators,^{11,12} acoustic energy trapping in this device is done through coupling propagating waves in the central section of the structure into evanescent waves at flanks. This is unlike conventional in-plane silicon bulk acoustic resonators (SiBAR) where the cavity is defined entirely through trenches that bound the geometry.^{13}

Dispersive characteristic of laterally propagating waves in waveguides with rectangular cross section is used for acoustic engineering of the device. Fig. 1 shows the dispersion behavior and mode shape of several eigen-waves in a rectangular waveguide aligned to ⟨100⟩ crystallographic direction of a (100) silicon substrate, for both evanescent and propagating waves.

The dispersion relation between the frequency (*f _{0}*) and wavenumber (

*K*) of laterally propagating waves is defined by Christoffel equation

_{X}^{14}considering the stress-free peripheral faces of the waveguide:

Here, *U* = (*Ux*, *Uy*, *Uz*) is the polarization vector. *K _{X}*,

*ω,*and

*ρ*are wave number, angular frequency, and waveguide mass density, respectively, and σ = [σ

_{mn}] is the stress tensor.

Such a relation results in both propagating and evanescent wave solutions with real and imaginary wavenumbers, respectively. While for the propagating solutions, the acoustic energy is uniform across the length of the waveguide, the energy of evanescent solutions decays exponentially along the waveguide axis (X-axis in Fig. 1).

The dispersion behavior of a rectangular waveguide can be tailored by changing it cross-sectional dimensions (i.e., W and H). This is shown in Fig. 1 for the first width-extensional branch (S_{1}) where slight variations of the width (W ± ΔW) results in frequency shift of the branch. Such a geometry dependence can be used to couple propagating and evanescent waves with different wavenumbers, at a specific frequency. This can be done by cascading waveguides with different dimensions, along their axis. In such a structure, a coupled solution may exist if displacement and strain continuity hold at transition interface between consecutive waveguides.^{15} The thermo-acoustic properties of such a coupled solution can then be defined as a superposition of the properties of constituent waveguides, weighted by their respective contribution in trapping acoustic energy.

Each (*f _{0}*,

*K*) solution on eigen-wave dispersion branches is a result of a unique interaction between transverse and longitudinal waves launched into the waveguide in a specific incident angle, while reflected and converted at stress-free boundaries. Therefore, considering anisotropic temperature characteristic of longitudinal and transverse waves in single crystal silicon, each solution has a specific temperature characteristic. Fig. 2 shows the simulated

_{X}*TCF*and turn-over temperature (

_{1}_{,2}*T*) of the S

_{Turn-Over}_{1}branch for N-doped silicon substrates with different resistivities.

When properly oriented with respect to crystallographic axes of doped single crystal silicon, evanescent waves can provide a highly different *TCF* compared to propagating waves. Furthermore, as opposed to propagating waves, evanescent waves show a large variation in temperature characteristics for different *K _{X}*. This is due to effective contribution of both shear and extensional wave components in the formation of evanescent solutions and highly different thermo-acoustic properties of these constituents.

^{17}

In this work, evanescent waves with highly positive *TCF _{1}* are used to design a temperature-compensated acoustic resonator with well-defined turn-over temperature. Several waveguides with different width, and hence different dispersion behavior, are cascaded in series. In such a structure, a solution—called a synthesized resonance mode hereafter—may exist arising from acoustic coupling of propagating and evanescent waves in constituent waveguides. When such a cascaded group of waveguides is flanked with sufficiently long evanescent-supporting sections, the energy will be confined mainly in the central waveguide leaving negligible concentration at flank terminations. Therefore, the anchoring of the cascaded group will not perturb vibration dynamics of the structure or diminish the generality of the waveguide assumption of the constituents—even though their lengths are finite.

^{18}

Fig. 3 compares the mode shape and simulated *T _{Turn-Over}* of an ideal (i.e., infinitely long) SiBAR with two different coupled waveguides, all aligned to ⟨100⟩ axes of 0.0011 Ω . cm single crystal silicon.

The displacement mode shape of the coupled waveguides can be defined in different sections as

In order to calculate the contribution of each eigen-wave in temperature characteristic of the vibration mode, acoustically coupled system can be simplified to mechanically coupled harmonic oscillators with lumped equivalent mass and spring defined at the interface (i.e., X = L_{1,2}).^{19} Therefore, *TCF _{i}* (

*i*= 1, 2) of the coupled vibration mode (

*TCF*) can be defined as

_{i,}_{C}where *TCF _{i,j}* and

*m*(

_{eq,j}*j*= 1, 2) are the

*TCF*and equivalent mass of the constituent eigen-waves at waveguide transitional face and can be calculated using the displacement mode shape.

_{i}^{20,21}Having

*TCF*in (3), the

_{i,C}*T*of the coupled waveguides 1 and 2 (i.e.,

_{Turn-Over}*T*,

_{Turn-Over,i}*i*= 1, 2) can be approximated from the superposition of that of the constituents, weighted by their effective contribution in vibration mode

Fig. 4 shows the SEM image of the silicon resonator designed to serve as a test vehicle to reproduce coupled waveguides coupled modes. The device is composed of a central section with a width of W_{0} = 50 *μ*m that is surrounded by flanks with gradually changing widths. Flanks are designed to facilitate excitation of two synthesized modes with closely similar mode shapes to the trapped modes in coupled waveguides 1 and 2 in Fig. 3. The structure is finally clamped to the substrate through a tether with a width chosen to support no solution, either propagating or evanescent, at resonance frequency. This is done to reduce extension of acoustic field into the substrate, therefore increasing the quality factor (*Q*). A thin aluminum nitride (AlN) film is used to provide electromechanical transduction to the silicon microstructure. Wide tethers facilitate integration of several electrically isolated electrodes on the top surface of AlN, which in turn facilitates independent excitation of different synthesized modes.^{18}

Fig. 5 shows the mode shape of the synthesized modes in designed resonator in comparison with a SiBAR of the same length, anchored through narrow tethers at nodal points of the mode shape.

The electrode configuration of AlN transducer facilitate efficient excitation of both synthesized modes. Fig. 6 shows the frequency response of the device providing two modes with high *Q*. The larger *Q* of the mode 2 is a result of its smaller energy concentration in the flanks, and hence lower energy leakage, which can be observed in the mode shapes of Fig. 5.

Fig. 7 shows the COMSOL finite element simulated as well as measured temperature characteristics of the two modes of the device, in comparison with a regular SiBAR implemented on the same substrate. The devices are all implemented on a silicon substrate with a resistivity of 0.001(±%10) Ω . cm. Temperature characterizations are done in an environmental chamber with a temperature accuracy of ±0.5 °C and under a 1 mTorr vacuum.

The slight discrepancy between simulation and experimental results is due to the difference between doping concentration of silicon substrate compared to what assumed for material properties in finite element simulations as well as the variations in substrate thickness (20 *μ*m ± 1 *μ*m, as indicated by the substrate-providing vendor). Highly different temperature characteristic of the synthesized modes is a result of their different constituent propagating and evanescent waves. Specifically, in the synthesized mode 2, effective contribution of the evanescent wave with large wavenumber—and hence highly positive *TCF _{1}*—has resulted in a

*T*exceeding 85 °C, which facilitates implementation of an oven-controlled stable frequency reference.

_{Turn-Over}^{22}The larger turn-over temperatures of the synthesized modes in implemented device compared to designed values is due to slightly lower resistivity of the actual silicon substrate compared to the simulation assumptions. Furthermore, the

*T*can be tailored to a well-defined value by proper acoustic engineering, which includes design for evanescent wavenumber and relative energy distribution between propagating and evanescent fields. Fig. 8 shows the temperature characteristic of the

_{Turn-Over}*Q*for the modes/devices of Fig. 7, over the temperature range of −40 °C to 80 °C.

The temperature induced degradation in the *Q* of the vibration modes in acoustically engineered cavity and SiBAR is a result of an increase in dissipative thermo-acoustic interactions at higher temperatures.^{23} Slight irregularities in the *Q* characteristic trend for the modes in acoustically engineered cavity can be attributed to the change of evanescent wavenumber in the coupled solutions with temperature variations. Such a change results in different energy distribution profile in the device flanks and induces unexpected variations in acoustic energy leakage into the substrate.

In conclusion a thermo-acoustic engineering technique, based on acoustic coupling of propagating and evanescent waves, is introduced. The temperature characteristics of propagating and evanescent waves in the width-extensional eigen-wave branch is extracted through numerical simulations. The application of such characteristics in the turn-over temperature of coupled waveguides is presented through analytical derivation and experimentally verified through device implementation and characterization. *T _{Turn-Over}*s of 67 °C and 87 °C are measured for two synthesized modes in a single device. Such a device can serve as the frequency reference for implementation of highly stable ovenized oscillators.

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