Design rules for temperature compensated degerately n-type-doped silicon MEMS resonators

The first- and second-order temperature coefficients and the total temperature-induced frequency deviation of degenerately n-type-doped silicon resonators are modeled. Modeling is based on finite element modelling-based sensitivity analysis of various resonator geometries combined with the experimental results on doping-dependent elastic constants of n-type-doped silicon. The analysis covers a doping range from 2.4 × 10¹⁷ to 7.5 × 10¹⁹ cm^-3. Families of resonance modes that can be temperature compensated via n-type doping are identified. These include bulk modes, such as the width/length extensional modes of a beam, Lame /square extensional modes of a plate resonator, as well as flexural and torsional resonance modes. It is shown that virtually all resonance modes of practical importance can reach zero linear temperature coefficient of frequency when correctly designed. Optimal configurations are presented, where a total frequency deviation of ~150 ppm can be reached. The results suggest that full second-order temperature compensation familiar from AT cut quartz is not possible in silicon resonators with doping below 7.5 × 10¹⁹ cm^-3. However, an analysis relying on extrapolated elastic constant data suggests the possibility of full second-order temperature compensation for a wide range of resonance modes when doping is extended beyond 10²⁰ cm^-3.