{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T11:22:53Z","timestamp":1769080973132,"version":"3.49.0"},"reference-count":41,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2018,7,4]],"date-time":"2018-07-04T00:00:00Z","timestamp":1530662400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>MicroElectroMechanical Systems (MEMS) resonators are attracting increasing interest because of their smaller size and better integrability as opposed to their quartz counterparts. However, thermal drift of the natural frequency of silicon structures is one of the main issues that has hindered the development of MEMS resonators. Extensive investigations have addressed both the fabrication process (e.g., introducing heavy doping of the silicon) and the mechanical design (e.g., exploiting proper orientation of the device, slots, nonlinearities). In this work, starting from experimental data published in the literature, we show that a careful design can help reduce the thermal drift even when slots are inserted in the devices in order to decrease thermoelastic losses. A custom numerical code able to predict the dynamic behavior of MEMS resonators for different materials, orientations and doping levels is coupled with an evolutionary optimization algorithm and the possibility to find an optimal mechanical design is demonstrated on a tuning-fork resonator.<\/jats:p>","DOI":"10.3390\/s18072157","type":"journal-article","created":{"date-parts":[[2018,7,4]],"date-time":"2018-07-04T12:23:02Z","timestamp":1530706982000},"page":"2157","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["Analysis of Frequency Stability and Thermoelastic Effects for Slotted Tuning Fork MEMS Resonators"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1607-5827","authenticated-orcid":false,"given":"Valentina","family":"Zega","sequence":"first","affiliation":[{"name":"Department of Civil and Environmental Engineering, Politecnico di Milano, 20133 Milano, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7100-1321","authenticated-orcid":false,"given":"Attilio","family":"Frangi","sequence":"additional","affiliation":[{"name":"Department of Civil and Environmental Engineering, Politecnico di Milano, 20133 Milano, Italy"}]},{"given":"Andrea","family":"Guercilena","sequence":"additional","affiliation":[{"name":"Department of Civil and Environmental Engineering, Politecnico di Milano, 20133 Milano, Italy"}]},{"given":"Gabriele","family":"Gattere","sequence":"additional","affiliation":[{"name":"STMicroelectronics, AMG R&amp;D, 20010 Cornaredo, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2018,7,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Lam, C.S. (2008, January 2\u20135). A review of the recent development of MEMS and crystal oscillators and their impacts on the frequency control products industry. Proceedings of the IEEE Ultrasonics Symposium, Beijing, China.","DOI":"10.1109\/ULTSYM.2008.0167"},{"key":"ref_2","unstructured":"SiTime (2018, June 09). SiT1532 Ultra-Small 32 kHz Oscillator. Available online: www.sitime.com."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1832","DOI":"10.1109\/JMEMS.2015.2443379","article-title":"Design rules for temperature compensated degenerately n-type doped silicon MEMS resonators","volume":"24","author":"Jaakkola","year":"2015","journal-title":"J. Microelectromech. Syst."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Mussi, G., Bestetti, M., Zega, V., Frangi, A., Gattere, G., and Langfelder, G. (2018, January 21\u201325). Resonators for real-time clocks based on epitaxial polysilicon process: A feasibility study on system-level compensation of temperature drifts. Proceedings of the IEEE Micro Electro Mechanical Systems (MEMS), Belfast, UK.","DOI":"10.1109\/MEMSYS.2018.8346654"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Ng, E.J., Wang, S., Buchman, D., Chiang, C.-F., Kenny, T.W., Muenzel, H., Fuertsch, M., Marek, J., Gomez, U.M., and Yama, G. (2012, January 3\u20137). Ultra-stable epitaxial polysilicon resonators. Proceedings of the Solid-State Sensors, Actuators and Microsystems Workshop, Hilton Head Island, SC, USA.","DOI":"10.31438\/trf.hh2012.73"},{"key":"ref_6","unstructured":"Hsu, W. (2008, January 19\u201321). Resonator miniaturization for oscillators. Proceedings of the IEEE International Frequency Control Symposium, Honolulu, HI, USA."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3952","DOI":"10.1103\/PhysRevB.2.3952","article-title":"Temperature dependence of the elastic constants","volume":"2","author":"Varshni","year":"1970","journal-title":"Phys. Rev. B"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1063","DOI":"10.1109\/TUFFC.2014.3007","article-title":"Determination of doping and temperature dependent elastic constants of degenerately doped silicon from MEMS resonators","volume":"61","author":"Jaakkola","year":"2014","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"730","DOI":"10.1109\/JMEMS.2014.2347205","article-title":"Temperature Dependence of the Elastic Constants of Doped Silicon","volume":"24","author":"Ng","year":"2015","journal-title":"J. Microelectromech. Syst."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Mirzazadeh, R., Saeed Eftekhar, A., and Mariani, S. (2016). Micromechanical characterization of polysilicon films through on-chip tests. Sensors, 16.","DOI":"10.3390\/s16081191"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Shin, D.D., Heinz, D.B., Kwon, H.-K., Chen, Y., and Kenny, W. (2018, January 26\u201329). Lateral diffusion doping of silicon for temperature compensation of MEMS resonators. Proceedings of the IEEE International Symposium on Inertial Sensors and Systems (INERTIAL), Lake Como, Italy.","DOI":"10.1109\/ISISS.2018.8358144"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Chen, Y., Shin, D.D., Flader, I.B., and Kenny, T.W. (2017, January 22\u201326). Tri-mode operation of higly doped silicon resonators for temperature compensated timing references. Proceedings of the IEEE 30th International Conference on Micro Electro Mechanical Systems (MEMS), Las Vegas, NV, USA.","DOI":"10.1109\/MEMSYS.2017.7863620"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"153502","DOI":"10.1063\/1.4964832","article-title":"Exploiting nonlinear amplitude-frequency dependence for temperature compensation in silicon micromechanical resonators","volume":"109","author":"Defoort","year":"2016","journal-title":"Appl. Phys. Lett."},{"key":"ref_14","unstructured":"Hsu, W.-T., Clark, J.R., and Nguyen, T.-C. (2000, January 11\u201313). Mechanically temperature compensated flexural-mode micromechanical resonators. Proceedings of the International Electron Devices Meeting, San Francisco, CA, USA."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1109\/TUFFC.2017.2667501","article-title":"Temperature compensation of silicon Lam\u00e9 resonators using etch holes: theory and design methodology","volume":"64","author":"Luschi","year":"2017","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_16","unstructured":"Zadeh, S.A.G., Saha, T., Allidina, K., Nabki, K., and El-Gamal, M.N. (2010, January 1\u20134). Electronic temperature compensation of clamped-clamped beam MEMS resonators. Proceedings of the 53rd IEEE International Midwest Symposium on Circuits and Systems, Seattle, WA, USA."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/j.ijsolstr.2016.02.038","article-title":"Semi-analytical and numerical estimates of anchor losses in bistable MEMS","volume":"92\u201393","author":"Frangi","year":"2016","journal-title":"Int. J. Solids Struct."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"632","DOI":"10.1109\/JMEMS.2017.2686650","article-title":"Near vacuum gas damping in MEMS: Simplified modeling","volume":"26","author":"Fedeli","year":"2017","journal-title":"J. Microelectromech. Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"890","DOI":"10.1109\/JMEMS.2016.2584699","article-title":"Near vacuum gas damping in MEMS: numerical modeling and experimental validation","volume":"25","author":"Frangi","year":"2016","journal-title":"J. Microelectromech. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1109\/JMEMS.2009.2016287","article-title":"Thermoelastic Damping in Hollow and Slotted Microresonators","volume":"18","author":"Prabhakar","year":"2009","journal-title":"J. Microelectromech. Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2723","DOI":"10.1007\/s00542-015-2652-y","article-title":"Effects of slots on thermoelastic quality factor of a vertical beam MEMS resonator","volume":"22","author":"Asadi","year":"2016","journal-title":"J. Microsyst. Technol."},{"key":"ref_22","first-page":"927","article-title":"Impact of geometry on thermoelastic dissipation in micromechanical resonant beams","volume":"15","author":"Candler","year":"2006","journal-title":"J. Microsyst. Technol."},{"key":"ref_23","first-page":"471","article-title":"Quality factor in trench-refilled polysilicon beam resonators","volume":"15","author":"Abdolvand","year":"2006","journal-title":"J. Microsyst. Technol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1109\/JMEMS.2009.2039697","article-title":"What is the Young\u2019s modulus of silicon?","volume":"19","author":"Hopcroft","year":"2010","journal-title":"J. Microelectromech. Syst."},{"key":"ref_25","unstructured":"Jaakkola, A. (2016). Piezoelectrically Transduced Temperature Compensated Silicon Resonators for Timing and Frequency Reference Applications. [Ph.D. Thesis, Aalto University]."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"756","DOI":"10.1103\/PhysRev.161.756","article-title":"Electronic effects in the elastic constants of n-type silicon","volume":"161","author":"Hall","year":"1967","journal-title":"Phys. Rev."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1063\/1.333965","article-title":"Precise determination of lattice parameter and thermal expansion coefficient of silicon between 300 and 1500 K","volume":"56","author":"Okada","year":"1984","journal-title":"J. Appl. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/0038-1101(67)90069-X","article-title":"Thermal conductivity of silicon, germanium, III-V compounds and III-V alloys","volume":"10","author":"Maycock","year":"1967","journal-title":"Solid-State Electron."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1109\/JMEMS.2008.924253","article-title":"Temperature dependence of quality factor in MEMS resonators","volume":"17","author":"Kim","year":"2008","journal-title":"J. Microelectromech. Syst."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Corigliano, A., Ardito, R., Comi, C., Frangi, A., Ghisi, A., and Mariani, S. (2018). Mechanics of Microsystems, Wiley.","DOI":"10.1002\/9781119053828"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1162\/106365603321828970","article-title":"Reducing the Time Complexity of the Derandomized Evolution Strategy with Covariance Matrix Adaptation (CMA-ES)","volume":"11","author":"Hansen","year":"2003","journal-title":"Evol. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1162\/106365601750190398","article-title":"Completely Derandomized Self-Adaptation in Evolution Strategies","volume":"9","author":"Hansen","year":"2001","journal-title":"Evol. Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/j.chemolab.2015.08.020","article-title":"Particle swarm optimization (PSO). A tutorial","volume":"149","author":"Marini","year":"2015","journal-title":"Chemom. Intell. Lab. Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/j.cam.2004.07.034","article-title":"Genetic algorithms for modelling and optimisation","volume":"184","author":"McCall","year":"2005","journal-title":"J. Comput. Appl. Math."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Auger, A., and Hansen, N. (2005, January 2\u20135). Performance Evaluation of an Advanced Local Search Evolutionary Algorithm. Proceedings of the IEEE Congress on Evolutionary Computation, Scotland, UK.","DOI":"10.1109\/CEC.2005.1554903"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1109\/TEVC.2008.924423","article-title":"A method for handling uncertainty in evolutionary optimization with an application to feedback control of combustion","volume":"13","author":"Hansen","year":"2009","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/14685240802441126","article-title":"Evolutionary optimization of an anisotropic compliant surface for turbulent friction drag reduction","volume":"9","author":"Fukagata","year":"2008","journal-title":"J. Turbul."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1439","DOI":"10.1016\/j.asoc.2007.10.018","article-title":"Human-competitive lens system design with evolution strategies","volume":"8","author":"Gagne","year":"2008","journal-title":"Appl. Soft Comput."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"04018016","DOI":"10.1061\/AJRUA6.0000966","article-title":"Structural health monitoring sensor network optimization through Bayesian experimental design","volume":"4","author":"Capellari","year":"2018","journal-title":"ASCE-ASME J. Risk Uncertain. Eng. Syst. Part A Civ. Eng."},{"key":"ref_40","unstructured":"(2018, June 09). UQLAB. Available online: www.uqlab.com."},{"key":"ref_41","unstructured":"(2018, June 26). CMA-ES. Available online: http:\/\/cma.gforge.inria.fr\/cmaes_sourcecode_page.html."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/18\/7\/2157\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:11:19Z","timestamp":1760195479000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/18\/7\/2157"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,7,4]]},"references-count":41,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2018,7]]}},"alternative-id":["s18072157"],"URL":"https:\/\/doi.org\/10.3390\/s18072157","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,7,4]]}}}