Modeling of Nonlinear Dynamics and Temperature Stability of Doped Silicon Microresonators
Bibliographic record
Abstract
Silicon is widely used as the device material for many micro resonators applied in timing and frequency referencing. One key disadvantage of silicon resonators compared to quartz resonators is their high thermal sensitivities. Doping silicon is a promising approach for temperature stability. Doped resonators operating at large deformation and finite strain amplitude often go to nonlinear regimes; therefore, nonlinear dynamics must be considered for adequately predicting the system behavior. In this article, the nonlinear vibration analysis is given for rectangular resonators operating in the Lamé mode, incorporating both the second- and third-order elastic constant (SOEC and TOEC) components. This article presents an analytic demonstration for the linear and nonlinear lumped mass system equivalent spring constants explicitly in terms of SOEC and TOEC. We show that, for a rectangular resonator in the Lamé mode, the first-order nonlinear spring constant would be an explicit expression in terms of TOEC components, which, for a square resonator, will be nullified. We show that there exist optimal doping levels where the anharmonic stiffness coefficient is minimized, implying the most dynamic stable vibrations. Furthermore, this article shows that there exists a tradeoff between dynamical and temperature-frequency stability in terms of the doping level.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".