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Fluidity and morphological stability of an amorphous thin film with radiation-induced defect kinetics

Tyler P. Evans

Eden Heyen

Phys. Rev. E 113, 055502 – Published 8 May, 2026

DOI: https://doi.org/10.1103/tyh5-fjh7

Abstract

It is common to model ion-irradiated amorphous thin films as if they were highly viscous fluids. In such models, one is frequently concerned with the ion-enhanced fluidity, a measure of the ability of the free interface to relax surface energy. Motivated by usual fluid dynamics problems, the ion-enhanced fluidity is near universally treated as a constant throughout the amorphous layer. However, for an irradiated thin film, the fluidity is ultimately caused by radiation-induced defect kinetics within the film, leading to regions with greater or lesser fluidity, and sensitive dependence on ion energy, species, flux, irradiation angle, temperature, and other experimental parameters. Here we develop and analyze a model of radiation-induced defect kinetics coupled to the continuum equations of a viscous thin film where sputtering-induced surface destabilization is neglected. Using realistic parameter values, we show that defect kinetics can meaningfully alter theoretical predictions of surface relaxation and ion-enhanced fluidity. Implications for aligning theoretical and experimental work, especially for surface nanopatterning of silicon induced by low-energy argon irradiation, are discussed.

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References (80)

  1. C. C. Umbach, R. L. Headrick, and K.-C. Chang, Spontaneous nanoscale corrugation of ion-eroded SiO2: The role of ion-irradiation-enhanced viscous flow, Phys. Rev. Lett. 87, 246104 (2001).
  2. M. Navez, D. Chaperot, and C. Sella, Microscopie electronique - etude de lattaque du verre par bombardement ionique, C. R. Hebd. Acad. Sci. 254, 240 (1962).
  3. W. L. Chan and E. Chason, Making waves: Kinetic processes controlling surface evolution during low energy ion sputtering, J. Appl. Phys. 101, 121301 (2007).
  4. J. Muñoz-García, L. Vázquez, M. Castro, R. Gago, A. Redondo-Cubero, A. Moreno-Barrado, and R. Cuerno, Self-organized nanopatterning of silicon surfaces by ion beam sputtering, Mater. Sci. Eng. R 86, 1 (2014).
  5. S. A. Norris and M. J. Aziz, Ion-induced nanopatterning of silicon: Toward a predictive model, Appl. Phys. Rev. 6, 011311 (2019).
  6. R. Cuerno and J.-S. Kim, A perspective on nanoscale pattern formation at surfaces by ion-beam irradiation, J. Appl. Phys. 128, 180902 (2020).
  7. R. M. Bradley and J. M. Harper, Theory of ripple topography induced by ion bombardment, J. Vac. Sci. Technol. A 6, 2390 (1988).
  8. G. Carter and V. Vishnyakov, Roughening and ripple instabilities on ion-bombarded Si, Phys. Rev. B 54, 17647 (1996).
  9. S. A. Norris, M. P. Brenner, and M. J. Aziz, From crater functions to partial differential equations: A new approach to ion bombardment induced nonequilibrium pattern formation, J. Phys.: Condens. Matter 21, 224017 (2009).
  10. S. A. Norris, J. Samela, L. Bukonte, M. Backman, D. F. K. Nordlund, C. Madi, M. Brenner, and M. Aziz, Molecular dynamics of single-particle impacts predicts phase diagrams for large scale pattern formation, Nat. Commun. 2, 276 (2011).
  11. S. A. Norris, J. Samela, K. Nordlund, M. Vestberg, and M. Aziz, Crater functions for compound materials: A route to parameter estimation in coupled-PDE models of ion bombardment, Nucl. Instrum. Methods Phys. Res. Sect. B 318, 245 (2014).
  12. A. S. Rudy and V. K. Smirnov, Hydrodynamic model of wave-ordered structures formed by ion bombardment of solids, Nucl. Instrum. Methods Phys. Res. Sect. B 159, 52 (1999).
  13. R. Cuerno, M. Castro, J. Muñoz-García, R. Gago, and L. Vásquez, Nanoscale pattern formation at surfaces under ion-beam sputtering: A perspective from continuum models, Nucl. Instrum. Methods Phys. Res. Sect. B 269, 894 (2011).
  14. M. Castro and R. Cuerno, Hydrodynamic approach to surface pattern formation by ion beams, Appl. Surf. Sci. 258, 4171 (2012).
  15. M. Castro, R. Gago, L. Vázquez, J. Muñoz-García, and R. Cuerno, Stress-induced solid flow drives surface nanopatterning of silicon by ion-beam irradiation, Phys. Rev. B 86, 214107 (2012).
  16. S. A. Norris, Stress-induced patterns in ion-irradiated silicon: Model based on anisotropic plastic flow, Phys. Rev. B 86, 235405 (2012).
  17. A. Moreno-Barrado, M. Castro, R. Gago, L. Vázquez, J. Muñoz-García, R. Redondo-Cubero, B. Galiana, C. Ballesteros, and R. Cuerno, Nonuniversality due to inhomogeneous stress in semiconductor surface nanopatterning by low-energy ion-beam irradiation, Phys. Rev. B 91, 155303 (2015).
  18. S. A. Norris, Stability analysis of a viscoelastic model for ion-irradiated silicon, Phys. Rev. B 85, 155325 (2012).
  19. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Instability (Courier, Chelmsford, 2013).
  20. S. A. Norris, J. C. Perkinson, M. Mokhtarzadeh, E. Anzenberg, M. J. Aziz, and K. F. Ludwig Jr., Distinguishing physical mechanisms using GISAXS experiments and linear theory: The importance of high wavenumbers, Sci. Rep. 7, 2016 (2017).
  21. J. M. Swenson and S. A. Norris, Swelling as a stabilizing mechanism in irradiated thin films, J. Phys.: Condens. Matter 30, 304003 (2018).
  22. J. Muñoz-García, R. Cuerno, and M. Castro, Stress-driven nonlinear dynamics of ion-induced surface nanopatterns, Phys. Rev. B 100, 205421 (2019).
  23. T. P. Evans and S. A. Norris, Swelling as a stabilizing mechanism in irradiated thin films II: Effect of swelling rate, J. Phys.: Condens. Matter 34, 325302 (2022).
  24. T. P. Evans and S. A. Norris, Swelling as a stabilizing mechanism in irradiated thin films: III. Effect on critical angle in a composite model, J. Phys.: Condens. Matter 35, 325302 (2023).
  25. T. P. Evans and S. A. Norris, Interfacial phase-change and geometry modify nanoscale pattern formation in irradiated thin films, J. Eng. Math. 146, 1 (2024).
  26. M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys. 65, 851 (1993).
  27. M. Cross and H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems (Cambridge University Press, Cambridge, 2009).
  28. M. A. Makeev and A.-L. Barabási, Ion-induced effective surface diffusion in ion sputtering, Appl. Phys. Lett. 71, 2800 (1997).
  29. M. A. Makeev, R. Cuerno, and A.-L. Barabási, Morphology of ion-sputtered surfaces, Nucl. Instrum. Methods Phys. Res. Sect. B 197, 185 (2002).
  30. R. M. Bradley, Exact linear dispersion relation for the Sigmund model of ion sputtering, Phys. Rev. B 84, 075413 (2011).
  31. J. Muñoz-García, M. Castro, and R. Cuerno, Nonlinear ripple dynamics on amorphous surfaces patterned by ion beam sputtering, Phys. Rev. Lett. 96, 086101 (2006).
  32. J. Muñoz-García, R. Cuerno, and M. Castro, Coupling of morphology to surface transport in ion-beam irradiated surfaces: Oblique incidence, Phys. Rev. B 78, 205408 (2008).
  33. J. Seo, H. Jeong, W. Kim, J. Muñoz-García, M. Castro, R. Cuerno, and J.-S. Kim, Surface nanopatterning of Si by ion beam irradiation with sub-sputter-threshold energy, Phys. Rev. B 112, 165408 (2025).
  34. D. A. Pearson and R. M. Bradley, Theory of terraced topographies produced by oblique-incidence ion bombardment of solid surfaces, J. Phys.: Condens. Matter 27, 015010 (2015).
  35. M. P. Gelfand and R. M. Bradley, One dimensional conservative surface dynamics with broken parity: Arrested collapse versus coarsening, Phys. Lett. A 379, 199 (2015).
  36. G. H. Kinchin and R. S. Pease, The displacement of atoms in solids by radiation, Rep. Prog. Phys. 18, 1 (1955).
  37. C. A. Davis, A simple model for the formation of compressive stress in thin films by ion bombardment, Thin Solid Films 226, 30 (1993).
  38. W. L. Chan and E. Chason, Stress evolution and defect diffusion in Cu during low energy ion irradiation: Experiments and modeling, J. Vac. Sci. Technol. A 26, 44 (2008).
  39. Y. Ishii, C. Madi, M. J. Aziz, and E. Chason, Stress evolution in Si during low-energy ion bombardment, J. Mater. Res. 29, 2942 (2014).
  40. S. E. Orchard, On surface levelling in viscous liquids and gels, Appl. Sci. Res. 11, 451 (1963).
  41. H. Hofsäss, O. Bobes, and K. Zhang, Argon ion beam induced surface pattern formation on Si, J. Appl. Phys. 119, 035302 (2016).
  42. H. Hofsäss, Model for roughening and ripple instability due to ion-induced mass redistribution [addendum to H. Hofsäss, Appl. Phys. A 114 (2014) 401, "Surface instability and pattern formation by ion-induced erosion and mass redistribution"], Appl. Phys. A 119, 687 (2015).
  43. N. Kalyanasundaram, M. C. Moore, J. B. Freund, and H. T. Johnson, Stress evolution due to medium-energy ion bombardment of silicon, Acta Mater. 54, 483 (2006).
  44. T. P. Evans and S. A. Norris, Coupling free-surface geometry and localized ion dose for continuum models of radiation-induced nanopatterning, arXiv:2510.01503.
  45. P. Myint, K. F. Ludwig, L. Wiegart, Y. Zhang, A. Fluerasu, X. Zhang, and R. L. Headrick, Nanoscale dynamics during self-organized ion beam patterning of Si. I. Ar+ bombardment, Phys. Rev. B 103, 195423 (2021).
  46. P. Myint, K. F. Ludwig, L. Wiegart, Y. Zhang, A. Fluerasu, X. Zhang, and R. L. Headrick, Nanoscale dynamics during self-organized ion beam patterning of Si. II. Kr+ bombardment, Phys. Rev. B 103, 195424 (2021).
  47. T. P. Evans, Matlab code and data for “Fluidity and morphological stability of an amorphous thin film with radiation-induced defect kinetics” Zenodo, Geneva (2026), https://doi.org/10.5281/zenodo.19475106.
  48. T. P. Evans and S. A. Norris, Viscous flow model of ion-induced pattern formation: Consistency between theory and experiment, arXiv:2505.06430.
  49. J. F. Ziegler, J. P. Biersack, and U. Littmark, The Stopping and Range of Ions in Matter (Pergamon, New York, 1985).
  50. J. F. Ziegler and J. P. Biersack, SRIM-2000.40 (IBM, Yorktown, 1984-2000).
  51. P. D. Parry, Target heating during ion implantation, J. Vac. Sci. Technol. 13, 622 (1976).
  52. K. B. Albaugh, A. F. Puttlitz, and P. E. Cade, Target heating in ion implantation, J. Vacuum Sci. Technol. B 8, 134 (1990).
  53. W. Wolfer and M. Ashkin, Stress-induced diffusion of point defects to voids, Scr. Metall. 7, 1175 (1973).
  54. W. G. Wolfer and M. Ashkin, Stress-induced diffusion of point defects to spherical sinks, J. Appl. Phys. 46, 547 (1975).
  55. A. L. Kozlovskiy, D. B. Borgekov, M. V. Zdorovets, I. E. Kenzhina, and D. I. Shlimas, Study of radiation damage kinetics in dispersed nuclear fuel on zirconium dioxide doped with cerium dioxide, J. Compos. Sci. 7, 277 (2023).
  56. G. Z. Moldabayeva, A. L. Kozlovskiy, E. I. Kuldeyev, A. K. Syzdykov, and N. S. Buktukov, Efficiency of using nitride and oxy-nitride coatings for protection against high-temperature oxidation and embrittlement of the surface layer of steel structures, ES Mater. Manuf. 24, 1129 (2024).
  57. M. Garcia, R. Martínez-Cervantes, J. Rickards, J. Cañetas-Ortega, J. Morales-Morales, L. de la Vega, and L. Rodríguez-Fernández, Ion-induced surface effects on soda-lime, amorphous silica (aSiO2) and silicon (100) substrates with 1.5 MeV Cu ion implantation, Nucl. Instrum. Methods Phys. Res. Sect. B 550, 165304 (2024).
  58. P. Sigmund, Theory of sputtering. I. Sputtering yield of amorphous and polycrystalline targets, Phys. Rev. 184, 383 (1969).
  59. P. Sigmund, A mechanism of surface micro-roughening by ion bombardment, J. Mater. Sci. 8, 1545 (1973).
  60. M. Z. Hossain, J. B. Freund, and H. T. Johnson, Ion impact energy distribution and sputtering of Si and Ge, J. Appl. Phys. 111, 103513 (2012).
  61. G. Hobler, R. M. Bradley, and H. M. Urbassek, Probing the limitations of Sigmund's model of spatially resolved sputtering using Monte Carlo simulations, Phys. Rev. B 93, 205443 (2016).
  62. T. K. Chini, F. Okuyama, M. Tanemura, and K. Nordlund, Structural investigation of keV Ar-ion-induced surface ripples in Si by cross-sectional transmission electron microscopy, Phys. Rev. B 67, 205403 (2003).
  63. P. G. Drazin and W. H. Reid, Introduction to Hydrodynamic Stability (Cambridge University Press, Cambridge, 1985).
  64. S. A. Norris, Ion-assisted phase separation in compound films: An alternate route to ordered nanostructures, J. Appl. Phys. 114, 204303 (2013).
  65. P. M. Fahey, P. B. Griffin, and J. D. Plummer, Point defects and dopant diffusion in silicon, Rev. Mod. Phys. 61, 289 (1989).
  66. S. Coffa and S. Libertino, Room-temperature diffusivity of self-interstitials and vacancies in ion-implanted Si probed by in situ measurements, Appl. Phys. Lett. 73, 3369 (1998).
  67. C. S. Madi, Linear stability and instability patterns in ion bombarded silicon surfaces, Ph.D. thesis, Harvard University, 2011.
  68. J. C. Perkinson, C. S. Madi, and M. J. Aziz, Nanoscale topographic pattern formation on Kr+-bombarded germanium surfaces, J. Vac. Sci. Technol. A 31, 021405 (2013).
  69. J. C. Perkinson, Nanoscale pattern self- organization under ion bombardment, Ph.D. thesis, Harvard University, 2017.
  70. Y. Yamamura, Y. Itikawa, and N. Itoh, Angular Dependence of Sputtering Yields of Monatomic Solids (Institute of Plasma Physics, Nagoya University, Nagoya, 1983).
  71. Y. Yamamura, C. Mössner, and H. Oechsner, The bombarding-angle dependence of sputtering yields under various surface conditions, Radiat. Eff. Defects Solids 103, 25 (1987).
  72. M. Holmes-Cerfon, W. Zhou, A. L. Bertozzi, M. P. Brenner, and M. J. Aziz, Development of knife-edge ridges on ion bombarded surfaces, Appl. Phys. Lett. 101, 143109 (2012).
  73. M. Holmes-Cerfon, M. J. Aziz, and M. P. Brenner, Creating sharp features by colliding shocks on uniformly irradiated surfaces, Phys. Rev. B 85, 165441 (2012).
  74. B. Ziberi, F. Frost, T. Höche, and B. Rauschenbach, Ripple pattern formation on silicon surfaces by low-energy ion-beam erosion: Experiment and theory, Phys. Rev. B 72, 235310 (2005).
  75. B. Liedke, Ion beam processing of surfaces and interfaces: Modeling and atomistic simulations, Ph.D. thesis, Helmholtz Zentrum Dresden Rossendorf, 2011.
  76. G. B. Bronner and J. D. Plummer, Gettering of gold in silicon: A tool for understanding the properties of silicon interstitials, J. Appl. Phys. 61, 5286 (1987).
  77. A. Witvrouw and F. Spaepen, Viscosity and elastic constants of amorphous Si and Ge, J. Appl. Phys. 74, 7154 (1993).
  78. S. Vauth and S. G. Mayr, Relevance of surface viscous flow, surface diffusion, and ballistic effects in keV ion smoothing of amorphous surfaces, Phys. Rev. B 75, 224107 (2007).
  79. S. Vauth and S. G. Mayr, Ion bombardment induced smoothing of amorphous metallic surfaces: Experiments versus computer simulations, Phys. Rev. B 77, 155406 (2008).
  80. H. Hofsäss, Surface instability and pattern formation by ion-induced erosion and mass redistribution, Appl. Phys. A 114, 401 (2014).

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