Abstract
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 x 1019 cm -3), arsenic (1.7 and 2.5 x 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- c>sub>12 is zero at n-type doping level of
n ~ 2 x 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 c 11-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
| Original language | English |
|---|---|
| Pages (from-to) | 1063-1074 |
| Journal | IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control |
| Volume | 61 |
| Issue number | 7 |
| DOIs | |
| Publication status | Published - 2014 |
| MoE publication type | A1 Journal article-refereed |