{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:01:39Z","timestamp":1760234499435,"version":"build-2065373602"},"reference-count":57,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,5,26]],"date-time":"2021-05-26T00:00:00Z","timestamp":1621987200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["19-72-10076"],"award-info":[{"award-number":["19-72-10076"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The analytical solution for the displacements of an anisotropic piezoelectric material in the uniform electric field is presented for practical use in the \u201cglobal excitation mode\u201d of piezoresponse force microscopy. The solution is given in the Wolfram Mathematica interactive program code, allowing the derivation of the expression of the piezoresponse both in cases of the anisotropic and isotropic elastic properties. The piezoresponse\u2019s angular dependencies are analyzed using model lithium niobate and barium titanate single crystals as examples. The validity of the isotropic approximation is verified in comparison to the fully anisotropic solution. The approach developed in the paper is important for the quantitative measurements of the piezoelectric response in nanomaterials as well as for the development of novel piezoelectric materials for the sensors\/actuators applications.<\/jats:p>","DOI":"10.3390\/s21113707","type":"journal-article","created":{"date-parts":[[2021,5,26]],"date-time":"2021-05-26T21:56:44Z","timestamp":1622066204000},"page":"3707","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Piezoresponse in Ferroelectric Materials under Uniform Electric Field of Electrodes"],"prefix":"10.3390","volume":"21","author":[{"given":"Artur","family":"Udalov","sequence":"first","affiliation":[{"name":"School of Natural Sciences and Mathematics, Ural Federal University, 620000 Ekaterinburg, Russia"}]},{"given":"Denis","family":"Alikin","sequence":"additional","affiliation":[{"name":"School of Natural Sciences and Mathematics, Ural Federal University, 620000 Ekaterinburg, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3432-7610","authenticated-orcid":false,"given":"Andrei","family":"Kholkin","sequence":"additional","affiliation":[{"name":"School of Natural Sciences and Mathematics, Ural Federal University, 620000 Ekaterinburg, Russia"},{"name":"Department of Physics & CICECO\u2014Aveiro Institute of Materials, University of Aveiro, 3810-193 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"728","DOI":"10.1134\/S0020441216050043","article-title":"An Electromechanical X-ray Optical Element Based on a Hysteresis-Free Monolithic Bimorph Crystal","volume":"59","author":"Blagov","year":"2016","journal-title":"Instrum. 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