Tuning The Frequency-Temperature Response of MEMS Resonators Using Composite Eigenmodes

A method for designing a MEMS resonator device, the method comprising:

• selecting a desired Nth order temperature coefficient of frequency (TCFNdesired) for a composite eigenmode resonator element, wherein the composite eigenmode resonator element comprises at least two sub-regions, wherein each of the at least two sub-regions is configured to resonate with a respective eigenmode from a set of at least two eigenmodes and is configured to have a respective Nth order temperature coefficient of frequency (TCFNk);
• providing a first sub-region of the at least two sub-regions, wherein the first sub-region in isolation is configured to resonate with a first eigenmode from the set of at least two eigenmodes and has a particular dopant type, a particular doping concentration, and a first orientation with respect to the crystal axis, and wherein the first sub-region has a first TCFNk;
• providing a second sub-region of the at least two sub-regions, wherein the second sub-region in isolation is configured to resonate with a second eigenmode from the set of at least two eigenmodes and has the particular dopant type, the particular doping concentration, and a second orientation with respect to the crystal axis, and wherein the second sub-region has a second TCFNk;
• determining, for the composite eigenmode resonator element comprising the at least two sub-regions, an Nth order temperature coefficient of frequency of (TCFNtotal);
• and
• tuning TCFNtotal such that it is approximately equal to TCFNdesired,
• wherein tuning TCFNtotal comprises altering at least one of the first or second TCFNk such that either (i) the first TCFNk is greater than TCFNdesired and the second TCFNk is less than TCFNdesired or (ii) the first TCFNk is less than TCFNdesired and the second TCFNk is greater than TCFNdesired.