{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,18]],"date-time":"2026-09-18T16:32:21Z","timestamp":1789749141498,"version":"4.0.1"},"reference-count":37,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2026,8,28]],"date-time":"2026-08-28T00:00:00Z","timestamp":1787875200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["11872186"],"award-info":[{"award-number":["11872186"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12472148"],"award-info":[{"award-number":["12472148"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"award":["11872186"],"award-info":[{"award-number":["11872186"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]},{"award":["12472148"],"award-info":[{"award-number":["12472148"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Micromachines"],"abstract":"<jats:p>In this study, we compare four electrode models, i.e., solid model, added mass model, thin-film model, and a combining model of thin-film with added mass, for a 76.8 MHz AT-cut quartz resonator. Relative to the high-fidelity solid electrode benchmark, the combining model accurately captures the thickness-shear mode but overestimates displacement confinement due to mass loading and boundary effects. The thin-film model fails to replicate practical device states by ignoring mechanical mass and stiffness. In contrast, the added mass model optimally balances fidelity and efficiency, matching the benchmark\u2019s frequency and resistance predictions with a 43% reduction in computation time. Parametric studies further indicate that cut angle deviations (33\u00b0~37\u00b0) alter equivalent shear stiffness, while electrode mass variations (mass coefficient 0.99~1.02) modify vibrational inertia. Both of them are critical for frequency tuning. Our findings clarify the scope and limitations of these strategies, guiding high-precision simulations and designs for high-frequency resonators.<\/jats:p>","DOI":"10.3390\/mi17091023","type":"journal-article","created":{"date-parts":[[2026,8,28]],"date-time":"2026-08-28T12:27:46Z","timestamp":1787920066000},"page":"1023","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Comparative Performance of Electrode Models for Thickness-Shear AT-Cut Quartz Resonators"],"prefix":"10.3390","volume":"17","author":[{"given":"Hang","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Engineering Mechanics, School of Aerospace Engineering, Huazhong University of Science and Technology, Wuhan 430074, China"},{"name":"Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, Huazhong University of Science and Technology, Wuhan 430074, 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Sci. Semicond. Process."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"117542","DOI":"10.1016\/j.sna.2026.117542","article-title":"Advances in the preparation of quartz film resonators for high-frequency applications: A review","volume":"401","author":"Shi","year":"2026","journal-title":"Sens. Actuators A Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1109\/PROC.1966.4625","article-title":"State of the art\u2014Quartz crystal units and oscillators","volume":"54","author":"Gerber","year":"1966","journal-title":"Proc. IEEE"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1109\/OJCAS.2021.3093373","article-title":"Review of Resonator\u2019s Q-Factor Measurement with Focus on Design of Analog and Mixed Circuits for In-Situ Measurement","volume":"2","author":"Ming","year":"2021","journal-title":"IEEE Open J. Circuits Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1109\/58.20450","article-title":"The acceleration sensitivity of quartz crystal oscillators: A review","volume":"35","author":"Filler","year":"1988","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"113506","DOI":"10.1016\/j.ymssp.2025.113506","article-title":"Temperature drift Identification, modeling and combined structural self- and VCR\/LSTM algorithm compensation in quartz resonant accelerometer","volume":"241","author":"Xue","year":"2025","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"124014","DOI":"10.1088\/0957-0233\/20\/12\/124014","article-title":"Phononic crystals for liquid sensor applications","volume":"20","author":"Lucklum","year":"2009","journal-title":"Meas. Sci. Technol."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Huang, Z.G., and Lee, W.H. (2026). Frequency and Quality Factor Analysis of Loss Factor Addition to High-Frequency AT-Cut Quartz Resonators with Femtosecond Laser Drilling Electrode and Inverted Etching. Eng. Proc., 134.","DOI":"10.3390\/engproc2026134091"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3765","DOI":"10.1063\/1.337589","article-title":"Studies of stress compensated quartz resonators with ultralinear frequency-temperature responses","volume":"60","author":"Nakazawa","year":"1986","journal-title":"J. Appl. Phys."},{"key":"ref_10","unstructured":"Jin, J., Yang, F., and Hu, H.P. (2021). Research on Inverted-Mesa-Type Quartz Resonator. 2020 15th Symposium on Piezoelectrcity, Acoustic Waves and Device Applications (SPAWDA), IEEE."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"109465","DOI":"10.1016\/j.mssp.2025.109465","article-title":"Design strategy and micromachining technology of AT-cut high-frequency quartz resonators: A review","volume":"193","author":"Xu","year":"2025","journal-title":"Mater. Sci. Semicond. Process."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Zhang, L.M., Wang, S.Y., Xie, L.T., Ma, T.F., Du, J.K., Wang, J., and Yong, Y.K. (2018). Frequency-Temperature Relations of Novel Cuts of Quartz Crystals for Thickness-Shear Resonators. 2018 IEEE International Frequency Control Symposium (IFCS), IEEE.","DOI":"10.1109\/FCS.2018.8597466"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1007\/s10483-026-3360-9","article-title":"Temperature-induced frequency activity dips in AT-cut quartz crystal resonators","volume":"47","author":"Li","year":"2026","journal-title":"Appl. Math. Mech."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/0020-7683(66)90010-2","article-title":"Thickness-shear and flexural vibrations of partially plated, crystal plates","volume":"2","author":"Mindlin","year":"1966","journal-title":"Int. J. Solids Struct."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"895","DOI":"10.1016\/0020-7683(72)90004-2","article-title":"High frequency vibrations of piezoelectric crystal plates","volume":"8","author":"Mindlin","year":"1972","journal-title":"Int. J. Solids Struct."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Yang, J.S. (2006). Analysis of Piezoelectric Devices, World Scientific.","DOI":"10.1142\/6156"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1016\/S0894-9166(12)60053-6","article-title":"Energy trapping of thickness-shear and thickness-twist modes in a partially electroded AT-cut quartz resonator","volume":"25","author":"He","year":"2012","journal-title":"Acta Mech. Solida Sin."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.ultras.2015.01.004","article-title":"Resonances and energy trapping in AT-cut quartz resonators operating with fast shear modes driven by lateral electric fields produced by surface electrodes","volume":"59","author":"Ma","year":"2015","journal-title":"Ultrasonics"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3409","DOI":"10.1143\/JJAP.41.3409","article-title":"Two-dimensional analysis of trapped-energy vibrations of AT-cut quartz resonators with large mass-loading","volume":"41","author":"Gong","year":"2002","journal-title":"Jpn. J. Appl. Phys."},{"key":"ref_20","first-page":"157","article-title":"Energy trapping of thickness-shear modes in inverted-mesa AT-cut quartz piezoelectric resonators","volume":"494","author":"Shi","year":"2016","journal-title":"Ferroelectrics"},{"key":"ref_21","unstructured":"Beerwinkle, A.D. (2011). Nonlinear Finite Element Modeling of Quartz Crystal Resonators. [Master\u2019s Thesis, Oklahoma State University]."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1088\/0964-1726\/14\/1\/028","article-title":"Energy-trapping analysis for the bi-stepped mesa quartz crystal microbalance using the finite element method","volume":"14","author":"Lu","year":"2005","journal-title":"Smart Mater. Struct."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Huang, X.H., Chen, Q., Pan, W., and Yao, Y. (2022). Advances in the mass sensitivity distribution of quartz crystal microbalances: A review. Sensors, 22.","DOI":"10.3390\/s22145112"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/j.apm.2019.03.014","article-title":"Vibration optimization of an infinite circular AT-cut quartz resonator with ring electrodes","volume":"72","author":"Zhu","year":"2019","journal-title":"Appl. Math. Model."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1109\/TUFFC.2012.2188","article-title":"Effects of mismatched electrodes on an ATCut quartz resonator","volume":"59","author":"He","year":"2012","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"9525809","DOI":"10.1109\/TIM.2025.3585205","article-title":"Metal-Film-Free Electrostatic Excitation and Detection Scheme for Maximizing the Q Factor of Quartz Resonators","volume":"74","author":"Feng","year":"2025","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1109\/TUFFC.2010.1409","article-title":"Modal analysis of practical quartz resonators using finite element method","volume":"57","author":"Yang","year":"2010","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"012017","DOI":"10.1088\/1742-6596\/2825\/1\/012017","article-title":"Analysis of amplitude and dissipation factor of AT-cut quartz crystal resonators over a wide temperature range","volume":"2825","author":"Zhang","year":"2024","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"890","DOI":"10.1109\/TUFFC.2020.3014887","article-title":"Modified modeling method of quartz crystal resonator frequency\u2013temperature characteristic with considering thermal hysteresis","volume":"68","author":"Deng","year":"2020","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"9278","DOI":"10.1109\/JSEN.2024.3361081","article-title":"A masterpiece of superior crystals: Quartz resonant pressure sensor\u2014A review","volume":"24","author":"Zhang","year":"2024","journal-title":"IEEE Sens. J."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Zheng, D.Z., Li, Z.L., Hu, C., Wang, Y.L., and Liu, B. (2025). A Frequency Calculation and Temperature Compensation System for Quartz Resonant Accelerometer Sensors. Fourth International Computational Imaging Conference (CITA 2024), SPIE.","DOI":"10.1117\/12.3057308"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Chen, J.F., Yong, Y.K., Kubena, R., Kirby, D., and Chang, D. (2015). Application of Electrode Stress for Improving Frequency-Temperature Behavior of UHF Quartz Resonators. 2015 IEEE International Ultrasonics Symposium (IUS), IEEE.","DOI":"10.1109\/ULTSYM.2015.0084"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/s10338-023-00384-1","article-title":"The multi-field coupled vibration analysis of AT-cut quartz crystal resonators with parallelism error","volume":"36","author":"Li","year":"2023","journal-title":"Acta Mech. Solida Sin."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1351","DOI":"10.1109\/58.949744","article-title":"Analysis of thickness modes of contoured doubly rotated, quartz resonators","volume":"48","author":"EerNisse","year":"2001","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1109\/TUFFC.2014.006473","article-title":"Experimental measurement of the frequency shifts of degenerate thickness-shear modes in a rotated Y-cut quartz resonator subject to diametrical forces","volume":"62","author":"Bao","year":"2015","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_36","first-page":"479","article-title":"Anisotropic acoustic attenuation with new measurements for quartz at room temperatures","volume":"293","author":"Lamb","year":"1966","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Sci."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Yang, J.S. (2005). An Introduction to the Theory of Piezoelectricity, Springer.","DOI":"10.1007\/b101799"}],"container-title":["Micromachines"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-666X\/17\/9\/1023\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,30]],"date-time":"2026-08-30T04:23:56Z","timestamp":1788063836000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-666X\/17\/9\/1023"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,28]]},"references-count":37,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2026,9]]}},"alternative-id":["mi17091023"],"URL":"https:\/\/doi.org\/10.3390\/mi17091023","relation":{},"ISSN":["2072-666X"],"issn-type":[{"value":"2072-666X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,28]]}}}