Microelectromechanical devices for higher order passive temperature compensation and methods of designing thereof

A method for designing a MEMS resonator device with passive temperature-induced frequency drift compensation, the method comprising:

• determining, for a resonator element of the MEMS resonator device, an initial geometry and an initial orientation with respect to a crystal axis of silicon that cause the resonator element to have at least one associated eigenmode of vibration;
• selecting, from among the at least one associated eigenmode of vibration, a particular eigenmode;
• determining, for the resonator element with the particular eigenmode, a particular set of parameters including at least a combination of (i) a type of dopant and (ii) a dopant concentration that causes a closely temperature-compensated mode in which (i) an absolute value of a first order temperature coefficient of frequency of the resonator element is reduced to a first value that is below a threshold value and (ii) an absolute value of a second order temperature coefficient of frequency of the resonator element is reduced to about zero;
• modifying, for the resonator element with the particular eigenmode and the particular set of parameters, one or both of the initial geometry or the initial orientation of the resonator element such that the absolute value of the first order temperature coefficient of frequency of the resonator element is further reduced to a second value smaller than the first value;
• and
• positioning an anchor at a center of the resonator element such that the anchor extends in an out-of-plane direction relative to the resonator element to attach the resonator element to a support structure.