Determination of doping and temperature-dependent elastic constants of degenerately doped silicon from MEMS resonators

Elastic constants c₁₁, c₁₂, and c₄₄ of degenerately doped silicon are studied experimentally as a function of the doping level and temperature. First-and second-order temperature coefficients of the elastic constants are extracted from measured resonance frequencies of a set of MEMS resonators fabricated on seven different wafers doped with phosphorus (carrier concentrations 4.1, 4.7, and 7.5 × 10¹⁹ cm^-3), arsenic (1.7 and 2.5 × 10¹⁹ cm^-3), or boron (0.6 and 3 × 10¹⁹ cm^-3). Measurements cover a temperature range from -40°C to +85°C. It is found that the linear temperature coefficient of the shear elastic parameter c₁₁ - c₁₂ is zero at n-type doping level of n ~ 2 × 10¹⁹ cm^-3, and that it increases to more than 40 ppm/K with increasing doping. This observation implies that the frequency of many types of resonance modes, including extensional bulk modes and flexural modes, can be temperature compensated to first order. The second-order temperature coefficient of c₁₁ - c₁₂ is found to decrease by 40% in magnitude when n-type doping is increased from 4.1 to 7.5 × 10¹⁹ cm^-3. Results of this study enable calculation of the frequency drift of an arbitrary silicon resonator design with an accuracy of ±25 ppm between the calculated and real(ized) values over T = -40°C to +85°C at the doping levels covered in this work. Absolute frequency can be estimated with an accuracy of ±1000 ppm.