RSS feeds
РусскийEnglish
Issue 3, 2025 Retrieve →
Issues Authors PACS Subscription For authors
Search →

Issues

 / 

2022

 / 

July

←  →

Instruments and methods of investigation


Approaches to determining curvature of wafers by their topography

†  a, ‡  b,  a
a National Research University of Electronic Technology (MIET), sq. Shokina 1, Zelenograd, Moscow, 124498, Russian Federation
b Institute of Mathematical Problems of Biology, Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Prof. Vitkevich str. 1, Pushchino, Moscow Region, 142290, Russian Federation

We discuss peculiarities of the curvature analysis of’wafers considering the heterogeneity of their topography for a quantitative estimation and localization of irregularity, or for subsequent calculations of mechanical stresses. We analyze three approaches to calculating surface curvatures from digital elevation models. The first one is based on the analysis of wafer surface profiles using polynomial approximation; the calculation is based on the curvature radius determination of a curved line; mechanical stresses are calculated using the Stoney method. The second approach uses the second partial derivatives of an elevation function in the Cartesian or cylindrical coordinate systems to analyze irregular topography and then to calculate mechanical stresses. The third considers the entire wafer topography as a two-dimensional elevation matrix and uses the mathematical apparatus of differential geometry and the experience of geomorphometry to determine convex and concave areas of the surface as well as to perform a complete analysis of the surface curvature system. We demonstrate the implementation of these approaches on a wafer like a segment of a sphere and on a complex-shaped wafer.

Fulltext pdf (978 KB)
To the readers pdf (115 KB)
Fulltext is also available at DOI: 10.3367/UFNe.2021.10.039076
Correspondence should be addressed to † dedkova@ckp-miet.ru and ‡ iflor@mail.ru
Keywords: surface, topography, curvature, radius of curvature, deflection, mechanical stresses, deformation, Stoney equation, silicon wafer, optical profilometry, defectiveness, geomorphometry, digital elevation model, DEM
PACS: 68.35.Gy
DOI: 10.3367/UFNe.2021.10.039076
URL: https://ufn.ru/en/articles/2022/7/d/
001100230300004
2-s2.0-85165580531
2022PhyU...65..706D
Citation: Dedkova A A, Florinsky I V, Djuzhev N A "Approaches to determining curvature of wafers by their topography" Phys. Usp. 65 706–722 (2022)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Received: 29th, March 2021, revised: 5th, August 2021, 6th, October 2021

Оригинал: Дедкова А А, Флоринский И В, Дюжев Н А «Подходы к определению кривизны пластин по рельефу их поверхности» УФН 192 754–771 (2022); DOI: 10.3367/UFNr.2021.10.039076

References (112) ↓ Cited by (6) Similar articles (4)

  1. Dyuzhev N A et al Nanotechnol. Russia 12 426 (2017); Dyuzhev N A et al Ross. Nanotekhnol. 12 (7-8) 97 (2017)
  2. Gusev E E et al Nano- Mikrosist. Tekh. 19 331 (2017)
  3. Florinsky I V Digital Terrain Analysis In Soil Science And Geology 2nd ed. (Amsterdam: Elsevier. Academic Press, 2016)
  4. Kirillovskii V K, Tochilina T V Opticheskie Izmereniya (Optical Measurements) Pt. 1 Izmereniya Geometricheskikh Parametrov (Measurements Of Geometric Parameters) 2nd ed., ster. (St. Petersburg: ITMO, 2015)
  5. Kirillovskii V K Opticheskie Izmereniya (Optical Measurements) Pt. 3 Funktsional’naya Skhema Pribora Opticheskikh Izmerenii. Tipovye Uzly. Opticheskie Izmereniya Geometricheskikh Parametrov (Functional Diagram Of An Optical Measurement Device. Standard Units. Optical Measurements Of Geometric Parameters) (St. Petersburg: ITMO, 2005)
  6. Malacara D (Ed.) Optical Shop Testing (New York: Wiley, 1978); Translated into Russian, Malacara D (Ed.) Opticheskii Proizvodstvennyi Kontrol’ (Moscow: Mashinostroenie, 1985)
  7. Djuzhev N A et al IOP Conf. Ser. Mater. Sci. Eng. 189 012019 (2017)
  8. Yun H M, Chao L P, Hsu J S Appl. Mech. Mater. 121-126 4295 (2012)
  9. Poelma R H et al J. Micromech. Microeng. 21 065003 (2011)
  10. Dunn M L, Zhang Y, Bright V M J. Microelectromech. Syst. 11 372 (2002)
  11. Pilipenko V A et al Prib. Metody Izmer. (1) 71 (2011)
  12. Egorov G P "Mekhanicheskie napryazheniya v metallicheskikh plenkakh pri magnetronnom osazhdenii (Mechanical stresses in metallic films under magnetron deposition)" Ph.D Thesis (Phys.-Math. Sci.) (Moscow: MISIS, 2018)
  13. Sen’ko S F, Zelenin V A Prib. Metody Izmer. 9 74 (2018)
  14. Sen’ko S F, Sen’ko A S, Zelenin V A Dokl. Beloruss. Gos. Univ. Inform. Radioelektron. (5) 12 (2018)
  15. Freund L B, Suresh S Thin Film Materials. Stress, Defect Formation, And Surface Evolution (Cambridge: Cambridge Univ. Press, 2003)
  16. Vaudin M D, Kessler E G, Owen D M Metrologia 48 201 (2011)
  17. Rosakis A J et al Thin Solid Films 325 42 (1998)
  18. van Dijk L et al Proc. SPIE 10145 101452L (2017)
  19. Dong X et al Opt. Express 19 13201 (2011)
  20. Dong X et al Exp. Mech. 53 959 (2013)
  21. Zhang C et al Exp. Mech. 56 1123 (2016)
  22. Stoney G G Proc. R. Soc. Lond. A 82 172 (1909)
  23. Aivazyan G E Izv. Nats. Akad. Nauk Resp. Armeniya Gos. Inzh. Univ. Armenii Ser. Tekh. Nauk 53 63 (2000)
  24. Dedkova A A, Kireev V Yu, Makhiboroda M A Nanostrukt. Matem. Fiz. Model. 20 (2) 23 (2020)
  25. Tabenkin A N, Tarasov S B, Stepanov S N Sherokhovatost’, Volnistost’, Profil’. Mezhdunarodnyi Opyt (Roughness, Waviness, Profile. International Experience) (St. Petersburg: Izd. Politekhnicheskogo Univ., 2007)
  26. Glang R, Holmwood R A, Rosenfeld R L Rev. Sci. Instrum. 36 7 (1965)
  27. Novak A V, Novak V R, Dedkova A A, Gusev E E Semiconductors 52 1953 (2018); Novak A V, Novak V R, Dedkova A A, Gusev E E Izv. Vyssh. Uchebn. Zaved. Elektron. 22 138 (2017)
  28. Bigl S et al Materials 10 1287 (2017)
  29. Dobrynin A V Tech. Phys. Lett. 23 709 (1997); Dobrynin A V Pis’ma Zh. Tekh. Fiz. 23 (18) 32 (1997)
  30. Markochev V M, Egorov G P Zavod. Lab. Diagnost. Mater. 84 (3) 61 (2017)
  31. Ohlidal I et al Proc. SPIE 5527 139 (2004)
  32. Mallik A, Stout R, Ackaert J IEEE Trans. Components Packag. Manuf. Technol. 4 240 (2014)
  33. Strunin V I, Khudaibergenov G Zh Omskie Nauchnye Chteniya - 2018. Materialy Vtoroi Vserossiiskoi Nauchnoi Konf., Omsk, 10 - 15 Dekabrya 2018 G. (Omsk Scientific Readings - 2018. Proc. Of The Second All-Russian Scientific Conf., Omsk: December 10-15, 2018, Exec. Ed. T F Yashchuk) (Omsk: Omsk. Gos. Univ. im. F.M. Dostoevskogo, 2018) p. 644
  34. Picciotto A et al Appl. Surf. Sci. 256 251 (2009)
  35. Stenzel O et al Thin Solid Films 517 6058 (2009)
  36. Szekeres A Fundamental Aspects Of Ultrathin Dielectrics On Si-based Devices (NATO Science Series, Vol. 47, Eds E Garfunkel, E Gusev, A Vul’) (Dordrecht: Springer, 1998) p. 65
  37. Bouaouina B et al Surf. Coat. Technol. 333 32 (2018)
  38. Klose Ph et al Int. J. Hydrogen Energy 42 22583 (2017)
  39. Seidl W M et al Surf. Coat. Technol. 347 92 (2018)
  40. Krawiec H et al Appl. Surf. Sci. 475 162 (2019)
  41. Sen’ko S F, Zelenin V A Prib. Metody Izmer. 9 254 (2018)
  42. Shinohara A et al Mater. Sci. Forum 941 2069 (2018)
  43. Liu D, Chen W J. Eng. Mater. Technol. 134 031002 (2012)
  44. Liu D Y, Chen W Q Mech. Res. Commun. 37 520 (2010)
  45. Wang T-G et al Mater. Sci. Eng. A 527 454 (2010)
  46. Zhang N-H, Chen J-Z Eur. J. Mech. A 28 284 (2009)
  47. Zhang X C et al Surf. Coat. Technol. 201 6715 (2007)
  48. Klein C A, Miller R P J. Appl. Phys. 87 2265 (2000)
  49. Benabdi M, Roche A A J. Adhesion Sci. Technol. 11 281 (1997)
  50. Kovalenko D A, Petrov V V Zh. Nano- Elektron. Fiz. 7 03036 (2015)
  51. Chou T-L, Yang S-Y, Chiang K-N Thin Solid Films 519 7883 (2011)
  52. Joseph S et al Proc. SPIE 9453 94530R (2015)
  53. But D K, Bychkov P S, Lychev S A Vestn. Perm. Nats. Issled. Politekh. Univ. Mekh. (1) 17 (2020)
  54. Bychkov P S, Lychev S A, But D K Vestn. Samar. Univ. Estestvennonauchn. Ser. 25 (4) 48 (2019)
  55. Dedkova A A Russ. J. Nondestruct. Testing 56 452 (2020); Dedkova A A Defektoskopiya (5) 52 (2020)
  56. Djuzhev N A et al IOP Conf. Ser. Mater. Sci. Eng. 289 012007 (2018)
  57. Jiang C et al Proc. SPIE 10932 109320K (2019)
  58. Mao C-L et al Acta Metrolog. 39 5628 (2018)
  59. Ardigo M R, Ahmed M, Besnard A Adv. Mater. Res. 996 361 (2014)
  60. Borgardt N I, Alekseev N V, Volkov R L Izv. Vyssh. Uchebn. Zaved. Elektron. (5) 91 (2011)
  61. Guzhov V et al Avtomatika Programmnaya Inzh. (2) 71 (2016)
  62. Loparev A V et al J. Opt. Technol. 79 371 (2012); Loparev A V et al Opt. Zh. 79 (6) 79 (2012)
  63. Dedkova A A, Makhiboroda M A Nanostrukt. Matem. Fiz. Model. 20 (2) 41 (2020)
  64. Zentsova E A Innovatsii, Kachestvo I Servis V Tekhnike I Tekhnologiyakh. V Mezhdunarodnaya Nauchno-prakticheskaya Konf., 04-05 Iyunya 2015 G. Sbornik Nauchnykh Trudov (Innovation, Quality And Service In Engineering And Technology. V International Scientific And Practical Conference, June 04±05, 2015. Collection Of Scientific Papers) (Kursk: Yugo-Zapadnyi Gos. Univ.) p. 150
  65. Kireev V Yu, Stolyarov A A Tekhnologii Mikroelektroniki. Khimicheskoe Osazhdenie Iz Gazovoi Fazy (Technologies Of Microelectronics. Chemical Vapor Deposition) (Moscow: Tekhnosfera, 2006)
  66. Ivanov A O, Tuzhilin A A "Lektsii po klassicheskoi differentsial’noi geometrii" ("Lectures on classical differential geometry"), http://dfgm.math.msu.su/files/IvaTuzTerm1-2017.pdf
  67. Skopenkov A B Osnovy Differentsial’noi Geometrii V Interesnykh Zadachakh (Fundamentals Of Differential Geometry In Interesting Problems) (Moscow: MTsNMO, 2009)
  68. Huang Y, Ngo D, Rosakis A J Acta Mech. Sinica 21 362 (2005)
  69. Kiyamov Kh G et al Izv. Kazan. Gos. Arkhitekturno-stroitel’nogo Univ. (1) 35 (2007)
  70. Seffen K A, Guest S D J. Appl. Mech. 78 011002 (2011)
  71. Seffen K A, McMahon R A Int. J. Mech. Sci. 49 230 (2007)
  72. Guest S D, Kebadze E, Pellegrino S J. Mech. Mater. Struct. 6 203 (2011)
  73. Eckstein E, Pirrera A, Weaver P M AIAA J. 54 1778 (2016)
  74. Seffen K A J. Appl. Mech. 83 021005 (2016)
  75. Sobota P M, Seffen K A R. Soc. Open Sci. 6 190888 (2019)
  76. Sobota P M, Seffen K A Proc. R. Soc. Lond. A 473 20170230 (2017)
  77. Wang M Q et al Int. J. Solids Struct. 49 1701 (2012)
  78. Ngo D et al Thin Solid Films 515 2220 (2006)
  79. Brown M A "Measuring stress in thin film - substrate systems featuring spatial nonuniformities of film thickness and/or misfit strain" Thesis in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Pasadena, CA: California Inst. of Technology, 2007)
  80. Leksin A Yu, Kutrovskaya S V Fizicheskie I Matematicheskie Printsipy Adaptivnoi Optiki (Physical And Mathematical Principles Of Adaptive Optics) (Vladimir: VlGU, 2015), Guidelines for laboratory training of VlSU students studying in the fields 12.02.05 ’Laser engineering and laser technologies’, 12.04.02 ’Optotechnics’, 12.04.05 ’Laser engineering and laser technologies’
  81. Tvorogov S "Sage: stroim polinomy Zernike (Sage: constructing Zernike polynomials)" https://sergeitvorogov.ru/archives/2376
  82. Zernike F Physica 1 689 (1934)
  83. Poznyak E G, Shikin E V Differentsial’naya Geometriya: Pervoe Znakomstvo (Differential Geometry: First Acquaintance) (Moscow: Izd. MGU, 1990)
  84. Kuznetsova E V Stroitel’naya Mekhanika. Izgib Plastin (Building Mechanics. Bending Of Plates) (Perm: Permskii Gos. Tekh. Univ., 2006), A tutorial
  85. Pogorelov A V Differentsial’naya Geometriya (Differential Geometry) (Moscow: Nauka, 1974)
  86. Machado G, Favier D, Chagnon G Exp. Mech. 52 865 (2012)
  87. Shary P A Math. Geol. 27 373 (1995)
  88. Shary P A, Sharaya L S, Mitusov A V Geoderma 107 1 (2002)
  89. Florinsky I V Prog. Phys. Geogr. Earth Environment 41 723 (2017)
  90. Mynatt I, Bergbauer S, Pollard D D J. Struct. Geol. 29 1256 (2007)
  91. Radzevich S P Formoobrazovanie Poverkhnostei Detalei (Osnovy Teorii, Formation Of Surfaces Of Parts (Fundamentals Of Theory)) (Kiev: Rastan, 2001)
  92. Ponomarev B B, Nguyen Shi Hien Vestn. Irkutsk. Gos. Tekh. Univ. 22 (4) 62 (2018)
  93. Nguyen Shi Hien Novaya Nauka Sovrem. Sost. Puti Razvitiya (12-4) 99 (2016)
  94. Pellis D, Pottmann H Advances In Architectural Geometry 2018 (Eds L Hesselgren) (Vienna: Klein Publ. GmbH, 2018) p. 34
  95. R-Optics, http://r-optics.ru/product/analizator-perednego-segmenta-glaza-pentacam-5002
  96. Grachev A V, Mukhamediev Sh A, Nikolaev V A Russ. J. Earth Sci. 2 (1) RJE00034 (2000)
  97. Pezzulla M et al Soft Matter 12 4435 (2016)
  98. Wang H, Nilsen E T, Upmanyu M J. R. Soc. Interface 17 20190751 (2020)
  99. Seffen K A, Maurini C J. Mech. Phys. Solids 61 190 (2013)
  100. Fernandes A, Maurini C, Vidoli S Int. J. Solids Struct. 47 1449 (2010)
  101. Hamouche W et al Meccanica 51 2305 (2016)
  102. Hamouche W et al Proc. R. Soc. Lond. A 473 20170364 (2017)
  103. Pike R J Int. J. Machine Tools Manuf. 41 1881 (2001)
  104. Pike R J Prof. Geogr. 53 263 (2001)
  105. Stepanenko V A, Okhotkin K G, Zakharov Yu V Differentsial’naya Geometriya (Differential Geometry) Pt. 1 Metodicheskie Ukazaniya Po Kursu ’Matematicheskii Analiz’ Dlya Studentov Fizicheskogo Fakul’teta I Mezhvuzovskogo Inzhenerno-fizicheskogo Otdeleniya (Guidelines For The Course ’Mathematical Analysis’ For Students Of The Faculty Of Physics And Interuniversity Engineering Physics Department) (Krasnoyarsk: Krasnoyarsk. Gos. Univ., 2000); Stepanenko V A, OkhotkK G, Zakharov Yu V http://sibsauktf.ru/courses/DiffGeom/
  106. Florinsky I V Int. J. Geogr. Inf. Sci. 16 475 (2002)
  107. Mitášová H, Mitáš L Math. Geol. 25 641 (1993)
  108. Favache A et al Rev. Sci. Instrum. 87 015002 (2016)
  109. Djuzhev N A et al Proc. SPIE 10224 1022428 (2016)
  110. Dedkova A A et al Nanostrukt. Matem. Fiz. Model. 17 (1) 51 (2017)
  111. Poelma R H et al J. Micromech. Microeng. 21 065003 (2011)
  112. Laconte J et al Microelectron. Eng. 76 219 (2004)
© 1918–2025 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions