{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,15]],"date-time":"2025-11-15T16:57:53Z","timestamp":1763225873237,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2010,5,17]],"date-time":"2010-05-17T00:00:00Z","timestamp":1274054400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In this paper a methodology for the three dimensional (3D) modeling and simulation of the profile evolution during anisotropic wet etching of silicon based on the level set method is presented. Etching rate anisotropy in silicon is modeled taking into account full silicon symmetry properties, by means of the interpolation technique using experimentally obtained values for the etching rates along thirteen principal and high index directions in KOH solutions. The resulting level set equations are solved using an open source implementation of the sparse field method (ITK library, developed in medical image processing community), extended for the case of non-convex Hamiltonians. Simulation results for some interesting initial 3D shapes, as well as some more practical examples illustrating anisotropic etching simulation in the presence of masks (simple square aperture mask, convex corner undercutting and convex corner compensation, formation of suspended structures) are shown also. The obtained results show that level set method can be used as an effective tool for wet etching process modeling, and that is a viable alternative to the Cellular Automata method which now prevails in the simulations of the wet etching process.<\/jats:p>","DOI":"10.3390\/s100504950","type":"journal-article","created":{"date-parts":[[2010,5,17]],"date-time":"2010-05-17T11:13:22Z","timestamp":1274094802000},"page":"4950-4967","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":["Level Set Approach to Anisotropic Wet Etching of Silicon"],"prefix":"10.3390","volume":"10","author":[{"given":"Branislav","family":"Radjenovi\u0107","sequence":"first","affiliation":[{"name":"Institute of Physics, Pregrevica 118, 11080 Beograd, Serbia"}]},{"given":"Marija","family":"Radmilovi\u0107-Radjenovi\u0107","sequence":"additional","affiliation":[{"name":"Institute of Physics, Pregrevica 118, 11080 Beograd, Serbia"}]},{"given":"Miodrag","family":"Mitri\u0107","sequence":"additional","affiliation":[{"name":"Vin\u010da Institute of Nuclear Sciences, P.O. Box 522, 11001 Beograd, Serbia"}]}],"member":"1968","published-online":{"date-parts":[[2010,5,17]]},"reference":[{"key":"ref_1","unstructured":"Elwenspoek, M., and Jansen, H.V. (2004). Silicon Micromachining, Cambridge University Press."},{"key":"ref_2","unstructured":"Gosalvez, M.A. (Atomistic Modelling Of Anisotropic Etching of Crystalline Silicon, 2003). Atomistic Modelling Of Anisotropic Etching of Crystalline Silicon, PhD. Dissertation,."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1109\/84.846706","article-title":"Micromachining process simulation using a continuous cellular automata method","volume":"9","author":"Zhu","year":"2000","journal-title":"J. Microelectromech. Syst"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"S38","DOI":"10.1088\/0960-1317\/17\/4\/S03","article-title":"A cellular automaton-based simulator for silicon anisotropic etching processes considering high index planes","volume":"17","author":"Zhou","year":"2007","journal-title":"J. Micromech. Microeng"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"174101","DOI":"10.1063\/1.2800884","article-title":"An atomic level model for silicon anisotropic etching processes\u2014Cellular automaton simulation and experimental verification","volume":"91","author":"Zhou","year":"2007","journal-title":"Appl. Phys. Lett"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"S1","DOI":"10.1088\/0960-1317\/17\/4\/S01","article-title":"An atomistic introduction to anisotropic etching","volume":"17","author":"Gosalvez","year":"2007","journal-title":"J. Micromech. Microeng"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1088\/1367-2630\/9\/12\/436","article-title":"Step flow-based cellular automaton for the simulation of anisotropic etching of complex MEMS structures","volume":"9","author":"Xing","year":"2007","journal-title":"New J. Phys"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1088\/0960-1317\/10\/2\/311","article-title":"A new model for Si {100} convex corner undercutting in anisotropic KOH etching","volume":"10","author":"Schroder","year":"2000","journal-title":"J. Micromech. Microeng"},{"key":"ref_9","unstructured":"Hubbard, T.J. (1994). MEMS Design\u2013Geometry of silicon micromachining, PhD. Dissertation,."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1090\/conm\/238\/03552","article-title":"The geometry of Wulff crystal shapes and its relations with Riemann problems","volume":"238","author":"Peng","year":"1999","journal-title":"Contemp. Math"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/0021-9991(88)90002-2","article-title":"Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations","volume":"79","author":"Osher","year":"1988","journal-title":"J. Comp. Phys"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1006\/jcph.2000.6657","article-title":"Evolution, implementation, and application of level set and fast marching methods for advancing fronts","volume":"169","author":"Sethian","year":"2001","journal-title":"J. Comp. Phys"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1109\/66.554505","article-title":"An overview of level set method for etching, deposition and lithography development","volume":"10","author":"Sethian","year":"1996","journal-title":"IEEE Trans. Semiconductor Dev"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1023\/A:1008036829907","article-title":"A level-set approach to 3DReconstruction from range data","volume":"29","author":"Whitaker","year":"1998","journal-title":"Int. J. Comp. Vision"},{"key":"ref_15","unstructured":"Sethian, J. (1998). Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Fluid Mechanics, Computer Vision and Materials Sciences, Cambridge University Press."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Osher, S., and Fedkiw, R. (2002). Level Set Method and Dynamic Implicit Surfaces, Springer-Verlag.","DOI":"10.1007\/b98879"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0924-4247(97)01658-0","article-title":"Characterization of orientation-dependent etching properties of single-crystal silicon\u2014effects of KOH concentration","volume":"64","author":"Sato","year":"1998","journal-title":"Sensor. Actuator. A-Phys"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"S38","DOI":"10.1088\/0960-1317\/17\/4\/S03","article-title":"A cellular automaton-based simulator for silicon anisotropic etching processes considering high index planes","volume":"17","author":"Zhou","year":"2007","journal-title":"J. Micromech. Microeng"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"213102","DOI":"10.1063\/1.2388860","article-title":"Non-convex Hamiltonians in 3D level set simulations of the wet etching of silicon","volume":"89","year":"2006","journal-title":"Appl. Phys. Lett"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4233","DOI":"10.1016\/j.tsf.2009.02.007","article-title":"3D simulations of the profile evolution during anisotropic wet etching of silicon","volume":"517","year":"2009","journal-title":"Thin Solid Films"},{"key":"ref_21","first-page":"623","article-title":"Level set methods and their applications in image science","volume":"1","author":"Tsai","year":"2003","journal-title":"Comm. Math. Sci"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/S0021-9991(02)00016-5","article-title":"Shock capturing, level sets, and PDE based methods in computer vision and image processing: a review of Osher's contributions","volume":"185","author":"Fedkiw","year":"2003","journal-title":"J. Comput. Phys"},{"key":"ref_23","unstructured":"Evans, L. (1998). Partial Differential Equations, American Mathematical Society."},{"key":"ref_24","unstructured":"Available online: http:\/\/www.itk.org (Accessed date: 06. April 2010.)."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.cpc.2005.09.010","article-title":"Sparse field level set methodfor non-convex Hamiltonians in 3D plasma etching profile simulations","volume":"174","author":"Lee","year":"2006","journal-title":"Comput. Phys. Commun"},{"key":"ref_26","unstructured":"Buerger, M. (1956). Elementary Crystalography, John Wiley & Sons, Inc."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/S0924-4247(99)00264-2","article-title":"Differences in anisotropic etchingproperties of KOH and TMAH solutions","volume":"80","author":"Shikida","year":"2000","journal-title":"Sensor. Actuat. A-Phys"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"560","DOI":"10.4310\/AJM.1997.v1.n3.a6","article-title":"The Wulff shape as the asymptotic limit of a growing crystalline inteface","volume":"1","author":"Osher","year":"1997","journal-title":"Asian J. Math"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"R111","DOI":"10.1088\/0960-1317\/17\/10\/R01","article-title":"Fabrication techniques of convex corners in a (100)-silicon wafer using bulk micromachining: a review","volume":"17","author":"Pal","year":"2007","journal-title":"J. Micromech. Microeng"},{"key":"ref_30","unstructured":"Horn, A., Schroeder, H., Obermeier, E., and Wachutka, G. Simulation of orientation-dependent etching of Silicon using a new step flow model of 3D structuring. San Diego, CA, USA."},{"key":"ref_31","unstructured":"Available online: http:\/\/www.intellisensesoftware.com\/modules\/IntelliEtch.html (Access date: 06. April 2010.)."},{"key":"ref_32","unstructured":"Nguyen, Q.D. (2007). Electrochemistry in anisotropic etching of silicon in alkaline solutions\u2014A kinematic wave analysis, PhD. Dissertation,."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/10\/5\/4950\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:02:27Z","timestamp":1760220147000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/10\/5\/4950"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,5,17]]},"references-count":32,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2010,5]]}},"alternative-id":["s100504950"],"URL":"https:\/\/doi.org\/10.3390\/s100504950","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2010,5,17]]}}}