Young's Modulus Resonator, with dimensions in mm. Hashed area shows the area where metal electrodes are deposited. Dashed line shows the upper extent of clamped region. These dimensions are those on the photolithographic mask and those of the top surface of the YMR; dimensions of the lower surface differ because of an etching method (see the text).
Photograph of Young's Modulus Resonator.
FEM rendering of the high-Q (AS2) mode of double-paddle oscillator. Color scale represents (exaggerated) displacement above (red) or below (blue) plane; black outline indicates equilibrium position. DPO is clamped to measuring apparatus in foot area below the horizontal line. Thickness and overall width are the same as for YMR.
FEM rendering of the high-Q mode of the YMR, termed the “Anti-symmetric Cantilever” or ASC mode. Color scale represents out-of-plane displacement.
Inverse quality factor (Q −1) versus temperature in K. Open diamonds: bare YMR; dashed line: typical DPO; dotted line: thermoelastic loss, from Eq. (3) .
Gradient scale and solid contour lines show the component of relative error δE f /E f related to film density and thickness as a function of film Young's modulus and density relative to silicon: E f /E s and ρ f /ρ s , respectively. Values are calculated from Eq. (17) , which presumes δ(t f /t f ) = δ(ρ f /ρ f ) = 0.01. Solid contour lines are spaced logarithmically with 1% and 10% errors emphasized and labeled with square boxes. The minimum such error, ΔE f /E f ≈ 0.0071, is shown by the dashed line. Error contours from thin film approximation are shown as dash-dotted lines for 100 nm films, and labeled with rounded boxes, for contour lines of δE f /E f = 0.0001, 0.001, and 0.01. Locations of (E f /E s , ρ f /ρ s ) values for several common film materials are also shown.
Measurements as a function of temperature of YMR with (solid squares) and without (open diamonds) 69.5 nm LPCVD SiN film: (a) Q −1; (b) frequency.
Frequency as a function of temperature of a YMR without (open diamonds) and with (solid squares) a 338.7 nm e-beam amorphous silicon (a-Si) film. The density of the film is 2100 kg/m3. Note the break in the frequency scale; scaling of each portion is identical.
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