[Based JMEMS paper: "Temperature Dependence of the Elastic Constants of Doped Silicon"
The frequency-temperature (f-T) dependence of monocrystalline silicon resonators is strongly affected when the silicon lattice is heavily doped with either p- or n-type dopants. This dependence is also dependent on the resonant mode shape and orientation: some common resonators and their f-T dependence on the doping is shown in . The code available on this page allows for the prediction of the f-T characteristic of an arbitrary resonator in heavily doped silicon when operated in the linear regime.
Since the frequency of an acoustic silicon resonator is strongly dependent on the elastic constants, the f-T dependence of a resonator can be predicted using knowledge of the elastic constants and their temperature dependences. From work done in our group  and other groups (VTT , CSEM , and IBM ), the doping and temperature dependence of the elastic constants of silicon have been extracted: p-type up to 1.75e20cm-3
, and n-type up to 7.5e19cm-3
. These extracted values (up to 2nd order, as shown below in Fig. 1, with least-squares fits to 4th order polynomials) can be used in conjunction with finite element analysis to simulate frequency-temperature dependences of doped silicon resonators.
Figure 1. The doping dependence of the elastic constants of silicon (first row) and
their temperature coefficients (second row: first order, third row: second order).
With fits to the elastic constants and their temperature coefficients across the different dopings, the f-T dependence of any doping level within the given range can be predicted for arbitrary resonant mode shapes using finite element analysis. COMSOL with MATLAB LiveLink is used here and the code is provided in the download section below.
1] The GUI provided allows for several settings to be chosen. Pick the wafer crystal orientation, doping type and concentration, and input your COMSOL model of the geometry. Enter the expected resonant frequency and apply the boundary conditions on the surfaces.
 Eldwin J. Ng, Vu A. Hong, Yushi Yang, Chae Hyuck Ahn, Camille L. M. Everhart, and Thomas W. Kenny, "Temperature Dependence of the Elastic Constants of Doped Silicon," Journal of Microelectromechanical Systems, Vol. 24, No. 3, pp. 730-741, Jun 2015.
 A. Jaakkola, M. Prunnila, T. Pensala, J. Dekker, and P. Pekko, "Determination of Doping and Temperature Dependent Elastic Constants of Degenerately Doped Silicon from MEMS Resonators," arXiv: 1401.1363 [cond-mat.mtrl-sci], 2014.
 C. Bourgeois, E. Steinsland, N. Blanc, and N. F. deRooij, "Design of Resonators for the Determination of the Temperature Coefficients of Elastic Constants of Monocrystalline Silicon," IEEE International Frequency Control Symposium 1997, pp. 791-799, 1997.
 J. J. Hall, "Electronic Effects in Elastic Constants of N-Type Silicon," Physical Review, vol. 161, pp. 756-761, 1967.