Review article: Geometric evolution laws for thin crystalline films: modeling and numerics

Communications in Computational Physics

Volumn 6, Issue 3, 2009, Pages 433-482

  Li, Bo ^a   Lowengrub, John ^b   Ratz, Andreas ^c   Voigt, Axel ^b,c,d  

^a UNIVERSITY OF CALIFORNIA AT SAN DIEGO   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c DRESDEN UNIVERSITY OF TECHNOLOGY   (Germany)

^d AALTO UNIVERSITY   (Finland)

Author keywords

Anisotropy; Chemical vapor deposition; Electrodeposition; Front tracking; Geometric evolution laws; Interface problems; Kinetics; Level set; Liquid phase epitaxy; Molecular beam epitaxy; Phase field

Indexed keywords

EID: 67650966931     PISSN: 18152406     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Review

References (140)

1. J. G. Amar and F. Family. Effects of crystalline microstructure on epitaxial growth. Phys. Rev. B, 54(20):14742-14753,1996.
2. R. Asaro and W.A. Tiller. Interface morphology development during stress corrosion cracking: Part I. via diffusion. Metall. Trans., 2:1789-1796, 1972.
3. M. Asta, J.J. Hoyt, and A. Karma. Method for computing the anisotropy of the solid-liquid interfacial free energy. Phys. Rev. Lett., 86:5530, 2001.
4. R. Backofen, A. Rätz, and A. Voigt. Equilibrium shapes of nanocrystals - computed using a phase field crystal model. J. Crystal growth, in review.
5. G.S. Bales and A. Zangwill. Morphological instability of a terrace edge during step flow growth. Phys. Rev. B, 41:5500, 1990.
6. E. Bänsch, P. Morin, and R.H. Nochetto. A finite element method for surface diffusion: the parametric case. J. Comput. Phys., 203:321-343,2005.
7. A.-L. Barabási and H. E. Stanley. Fractal Concepts in Surface Growth. Cambridge University Press, Cambridge, 1995.
8. 3. J. Comput. Phys., 227:4281-307,2008.
9. T. Biben, K. Kassner, and C. Misbah. Phase-field approach to three-dimensional vesicle dynamics. Phys. Rev. E, 72:049121, 2005.
10. T. Biben and C. Misbah. Tumbling of vesicles under shear flow with an advected-field approach. Phys. Rev. E, 67:031908, 2003.
11. F. Bornemann and C. Rasch. Finite-Element discretization of static Hamilton-Jacobi equations based on a local variational principle. Comput. Vis. Sci., 2006.
12. C. R. Brundle, C.A. Evans, Jr., and S. Wilson. Encyclopedia of Materials Characterization. Butterworth-Heinemann, 1992.
13. M. Burger, F. Haußer, C. Stöcker, and A. Voigt. A level set approach to anisotropic flow with curvature regularization. J. Comput. Phys., 225:183-205, 2007.
14. M. Burger, C. Stöcker, and A. Voigt. Finite element based level set method for higher order flows. J. Sci. Comput., in press.
15. W.K. Burton, N. Cabrera, and E.C. Frank. The growth of crystals and the equilibrium of their surfaces. Phil. Trans. Roy. Soc. London A, 243(866) :299-358, 1951.
16. R. E. Caflisch, W. E, M. F. Gyure, B. Merriman, and C. Ratsch. Kinetic model for a step edge in epitaxial growth. Phys. Rev. E, 59(6):6879-6887, 1999.
17. R. E. Caflisch and B. Li. Analysis of island dynamics for epitaxial growth of thin films. Multiscale Modeling and Simulation, 1(1):150-171, 2003.
18. R.E. Caflisch. Growth, structure and pattern formation for thin films. J. Sci. Comput., 37:3-17, 2008.
19. J.W. Cahn, CM. Elliott, and A. Novick-Cohen. The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature. Euro. J. Appl. Math., 7:287-301, 1996.
20. J.W. Cahn and D.W. Hoffman. A vector thermodynamics for anisotropic surfaces ii. curved and faceted surfaces. Acta Metall. Mater., 22:1205-1214, 1974.
21. J.W. Cahn and J.E. Taylor. Linking anisotropic sharp and diffuse surface motion laws via gradient flows. J. Stat. Phys., 77:404-423, 1994.
22. J.W. Cahn and J.E. Taylor. Surface motion by surface-diffusion. Acta Metall, 42:1045-1063, 1994.
23. P. Cermelli and M.E. Jabbour. Multispecies epitaxial growth on vicinal surfaces with chemical reactions and diffusion. Phil. Trans. Roy. Soc. London A, 461:3483-3504, 2005.
24. A. Chatterjee and D.G. Vlachos. An overview of spatial microscopic and accelerated kinetic monte carlo methods. J. Computer-Aided Mater. Des., 14:253-308, 2007.
25. U. Clarenz, E Haußer, M. Rumpf, A. Voigt, and U. Weikard. On level set formulations for anisotropic fourth order geometric evolution problems. In A. Voigt, editor, Multiscale modeling in epitaxial growth, pages 227-237. Birkhäuser, 2005.
26. J.-M. Debierre, A. Karma, F. Celestini, and R. Guerin. Phase-field approach for faceted solidification. Phys. Rev. E, 68:041604,2003.
27. K. Deckenick, G. Dziuk, and CM. Elliott. Computation of geometric partial differential equations. Acta Numer., pages 1-94, 2005.
28. K. Deckenick, G. Dziuk, and CM. Elliott. Fully discrete finite element approximation for anisotropic surface diffusion of graphs. SIAM J. Numer. Anal, 43:1112-1138, 2005.
29. A. DiCarlo, M.E. Gurtin, and P. Podio-Guidugli. A regularized equation for anisotropic motion-by-curvature. SIAM J. Appl. Math., 52(4):1111-1119, 1992.
30. M. Droske and M. Rumpf. A level set formulation for Willmore flow. Interf. Free Bound., 6:361-378,2004.
31. Q. Du, C Liu, and X. Wang. A phase field formulation of the Willmore problem. Nonlin., 18:1249-1267,2005.
32. G. Dziuk. An algorithm for evolving surfaces. Numer. Math., 58:603-611, 1991.
33. S. F. Edwards and D. R. Wilkinson. The surface statistics of a granular aggregate. Proc. Roy. Soc. A, 381:17-31, 1982.
34. G. Ehrlich and E.J. Hudda. Atomic view of surface diffusion: tungsten on tungsten. J. Chem. Phys., 44:1039-1099, 1966.
35. CM. Elliott and H. Garcke. Existence results for diffuse surface motion laws. Adv. Math. Sci. Appl., 7:465-488, 1997.
36. H. Emmerich. Advances of and by phase-field modeling in condensed-matter physics. Adv. Phys., 57:1-87, 2008.
37. J.W. Evans, PA. Thiel, and M.C Bartelt. Morphological eovlution during epitaxial thin film growth: Formation of 2d islands and 3d mounds. Surf. Sci. Reports, 61:1-129, 2006.
38. LC. Evans, H.M. Soner, and P.E. Souganidis. Phase-transitions and generalized motion by mean-curvature. Comm. Pure. Appl. Math., 45:1097-1123, 1992.
39. X. Feng and A. Prohl. Numerical analysis of the allen-cahn equation and approximation for mean curvature flow. Numer. Math., 94:33-65, 2003.
40. E. Fried and M. Gurtin. A unified treatment of evolving interfaces accounting for small deformations and atomic transport with emphasis on grain-boundary and epitaxy. Adv. Appl. Mech., 40:1-178, 2004.
41. R. Ghez and S. S. Iyer. The kinetics of fast steps on crystal surfaces and its application to the molecular beam epitaxy of silicon. IBM J. Res. Develop., 32:804-818, 1988.
42. E. De Giorgi. Some remarks on T-convergence and least square methods. In G. Del Masso and G.F. Dell'Antonio, editors, Progress in Nonlinear Differential Equations and their Applications, volume 5, pages 135-142. Birkhäuser, 1991.
43. A.A. Golovin, A.A. Nepomnyashchy, S.H. Davis, and M.A. Zaks. Convective Cahn-Hilliard models: From coarsening to roughening. Phys. Rev. Lett., 86:1550-1553, 2001.
44. L. Golubović. Interfacial coarsening in epitaxial growth models without slope selection. Phys. Rev. Lett., 78(l):90-93, 1997.
45. M.A. Grinfeld. Instability of the separation boundary between a non-hhydrostatically stressed elastic body and a melt. Sov. Phys. Dokl., 31:831-834, 1986.
46. M.E. Gurtin and M.E. Jabbour. Interface evolution in three dimensions with curvaturedependent energy and surface diffusion: Interface-controlled evolution, phase transitions, epitaxial growth of elastic films. Arch. Rat. Mech. Anal , 163:171-208, 2002.
47. F. Haufier and A. Voigt. A discrete scheme for regularized anisotropic surface diffusion, a 6th order geometric evolution equation. Interf. & Free Bound., 7:353-370, 2005.
48. F. Haußer and A. Voigt. Facet formation and coarsening modeled by a geometric evolution law for epitaxial growth. J. Crystal Growth, 275:e47-e51, 2005.
49. F. Haußer and A. Voigt. A numerical scheme for regularized anisotropic curve shortening flow. Appl. Math. Lett., 19:691-698, 2006.
50. F. Haußer and A. Voigt. A discrete scheme for parametric anisotropic surface diffusion. J. Sci. Comput., 30:223-235, 2007.
51. F. Haußer and A. Voigt. A geometeric ginzburg-landau theory for faceted crystals in 1d: from coarsening to chaos through a driving force. Phys. Rev. E, in review.
52. T. Haxhimali, A. Karma, F. Gonzales, and M. Rappaz. Orientation selection in dendritic evolution, nature materials, 5:660-604, 2006.
53. C. Herring. Some theorems on the free energies of crystal surfaces. Phys. Rev., 82:87-93, 1951.
54. C. Herring. Surface tension as a motivation for sintering. In W. E. Kingston, editor, The Physics of Powder Metallurgy, pages 143-179. McGraw-Hill, 1951.
55. C. Herring. The use of classical macroscopic concepts in surface energy problems. In R. Gomer and C. S. Smith, editors, Structure and Properties of Solid Surfaces, pages 2-72. Univ. of Chicago Press, 1953.
56. D. W. Hoffman and J. W. Cahn. A vector thermodynamics for anisotropic surfaces. I. Surf. Sci., 31:368-388, 1972.
57. J.M. Howe. Interfaces in Materials. Wiley-Interscience, 1997.
58. J.J. Hoyt, M. Asta, and A. Karma. Atomistic simulation methos for computing the kinetic coefficient in solid-liquid systems. Intreface Sci., 10:181-189, 2002.
59. D. Jamet and C. Misbah. Toward a thermodynamically consistent picture of the phase-field model of vesicles: Curvature energy. Phys. Rev. E, 78:031902, 2008.
60. H.-C. Jeong and E. D. Williams. Steps on surfaces: experiment and theory. Surface Sci. Reports, 34:171-294, 1999.
61. M.D. Johnson, C. Orme, A.W. Hunt, D. Graff, J. Sudijono, L.M. Sander, and B.G. Orr. Stable and unstable growth in molecular beam epitaxy. Phys. Rev. Lett., 64, 1994.
62. W.C. Johnson and W.I.D Alexander. Interfacial conditions for thermomechanical equilibrium in two-phase crystal. J. Appl. Phys., 59:2735,1986.
63. M. Kardar, G. Parisi, and Y. C. Zhang. Dynamic scaling of growing interfaces. Phys. Rev. Lett., 56:889-892, 1986.
64. D. Kessler, R.H. Nochetto, and A. Schmidt. Numerical analysis of the allen-cahn equation and approximation for mean curvature flow. Math. Model. Numer. Anal., 38:129-142, 2004.
65. R. Kobayashi. Modeling and numerical simulations of dendritic crystal growth. Physica D, 63:410-23, 1993.
66. R. V. Kohn and X. Yan. Upper bounds on the coarsening rate for an epitaxial growth model. Commun. Pure Appl. Math., 56(11):1549-1564, 2003.
67. J. Krug. Four lectures on the physics of crystal growth. Physica A, 313:47-82, 2002.
68. J. Krug, M. Plischke, and M. Siegert. Surface diffusion currents and the universality classes of growth. Phys. Rev. Lett., 70(21):3271-3274, 1993.
69. J. Krug, P. Politi, and T. Michely. Island nucleation in the presence of step-edge barriers: Theory and applications. Phys. Rev. B, 61(20):14037-14046, 2000.
70. P. Kuhn, J. Krug, F. Hausßer, and A. Voigt. Complex shape evolution in electromigration driven single layer islands. Phys. Rev. Lett., 2005.
71. F.C. Larche and J.W. Cahn. Thermodynamic equilibrium of multiphase solids under stress. Acta Metall., 26:1579,1978.
72. PH. Leo and R.F. Sekerka. The effect of surface stress on crystal melt and crystal-crystal equilibrium. Acta Metall., 37:3139, 1989.
73. B. Li. High-order surface relaxation vs. the Ehrlich-Schwoebel effect. Nonlinearity, 19:2581-2603, 2006.
74. B. Li and J.-G. Liu. Thin film epitaxy with or without slope selection. European J. Appl. Math., 14(6):713-743, 2003.
75. B. Li and J.G. Liu. Epitaxial growth without slope selection: energetics, coarsening, and dynamic scaling. J. Nonlin. Sci., 14:429-151, 2004.
76. B. Li, A. Rätz, and A. Voigt. Stability of a circular epitaxial island. Physica D, 198:231-247, 2004.
77. F. Liu and H. Metiu. Stability and kinetics of step motion on crystal surfaces. Phys. Rev. E, 49(4):2601-2616, 1997.
78. P. Loreti and R. March. Propagation of fronts in a nonlinear fourth order equation. Euro. J. Appl. Math., 11:203-213, 2000.
79. S. Majaniemi and N. Provatas. personal communication. 2007.
80. M.A. Makeev. Self-organized quantum dot superstructures for nanoelectronic and optoelectronic applications. J. Nanoelectronics and Optoelectronics, 1:176-193, 2006.
81. D. Margetis and R. Kohn. Continuum relaxation of interacting steps on crystal surfaces in 2+1 dimensions. Multiscale Model. Sim., 5:729-758, 2006.
82. G. Marshall, P. Mocskos, H.L. Swinney, and J.M. Huth. Buoyancy and electrically driven convection models in thin-layer electrodeposition. Phys. Rev. E, 59:2157-2167, 1999.
83. G.B. Mcfadden. Phase-field models for solidification. Contem. Math., 306:107-145,2002.
84. G.B. McFadden, A.A. Wheeler, R.J. Braun, S.R. Coriell, and R.F. Sekerka. Phase-field models for anisotropic interfaces. Phys. Rev. E, 48:2016-2024, 1993.
85. T. Michely and J. Krug. Islands, mounds, and atoms: patterns and processes in crystal growth far from equilibrium. Springer, 2004.
86. D. Moldovan and L. Golubović. Interfacial coarsening dynamics in epitaxial growth with slope selection. Phys. Rev. E, 61(6):6190-6214, 2000.
87. C. Müller-Gugenberger, R. Spatschek, and K. Kassner. Comparison of phase-field models for surface diffusion. Phys. Rev. E, in press, 2008.
88. W.W. Mullins. 2-dimensional motion of idealized grain boundaries. J. Appl. Phys., 27:900-904, 1956.
89. W.W. Mullins. Theory of thermal grooving. J. Appl. Phys., 28:333-339, 1957.
90. S. Osher and J. Sethian. Front propagation with curvature-dependent speed - algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys., 79:12-49, 1988.
91. D. Peng, B. Merriman, S. Osher, H. Zhao, and M. Kang. A PDE based fast local level set method. J. Comput. Phys., 155:410-438,1999.
92. A. Pimpinelli and J. Villain. Physics of Crystal Growth. Cambridge University Press, 1998.
93. A. Pimpinelli, J. Villain, D. E. Wolf, J. J. Métois, J. C. Heyraud, I. Elkinani, and G. Uimin. Equilibrium step dynamics on vicinal surfaces. Surface Sci., 295:143, 1993.
94. P. Politi, G. Grenet, A. Marty, A. Ponchet, and J. Villain. Instabilities in crystal growth by atomic or molecular beams. Phys. Reports, 324:271-404, 2000.
95. P. Politi and A. Torcini. Coarsening in surface growth models without slope selection. J. Phys. A: Math. Gen., 33:L77-L82, 2000.
96. A. Rätz. PhD thesis, Universität Bonn, 2007.
97. A. Rätz, A. Ribalta, and A. Voigt. Surface evolution of elastically stressed film under deposition by a diffuse interface model. J. Comput. Phys., 214:187-208, 2006.
98. A. Rätz and A. Voigt. Higher order regularization of anisotropic geometric evolution equations in three dimensions. J. Theor. Comput. Nanosci., 3:560-564, 2006.
99. N. Reinecke and E. Taglauer. The kinetics of oxygen-induced faceting of Cu(115) and Cu(119) surfaces. Surf. Sci., 454:94-100, 2000.
100. M. Röger, personal communication, 2006.
101. M. Röger and R. Schätzle. On a modified conjecture of De Giorgi. Mathematische Zeitschrift, 254:675-714, 2006.
102. M. Rost and J. Krug. Coasening of surface structures in unstable epitaxial growth. Phys. Rev. E, 55(4):3952-3957, 1997.
103. M. Rost, P. Šmilauer, and J. Krug. Unstable epitaxy on vicinal surfaces. Surface Sci., 369:393-402, 1996.
104. J. Rubinstein and P. Sternberg. Nonlocal reaction diffusion-equations and nucleation. IMA J. Appl. Math., 48:249-254, 1992.
105. R. Rusu. An algorithm for elastic flows of surfaces. Interf. Free Bound., 7:229-239, 2005.
106. TV. Savina, A.A. Golovin, S.H. Davis, A.A. Nepomnyashchy, and P.W. Voorhees. Faceting of growing crystal surfaces by surface diffusion. Phys. Rev. E, 67:021606, 2003.
107. R. L. Schwoebel. Step motion on crystal surfaces II. J. Appl. Phys., 40:614-618, 1969.
108. R.L. Schwoebel and E.J. Shipsey. Step motion on crystal surfaces. J. Appl. Phys., 37:3682-3686, 1966.
109. R. Sekerka. Analytical criteria fo rmissing orientations on tree-dimensional equilibrium shapes. J. Cryst. Growth, pages 77-82, 2005.
110. M. Siegel, M.J. Miksis, and P.W. Voorhees. Evolution of material voids for highly anisotropic surface energy. J. Mech. Phys. Sol., 52:1319-1353, 2004.
111. M. Siegert. Ordering dynamics of surfaces in molecular beam epitaxy. Physica A, 239:420-427, 1997.
112. M. Siegert. Coarsening dynamics in crystalline thin films. Phys. Rev. Lett., 81(25):5481-5484, 1998.
113. M. Siegert and M. Plischke. Slope selection and coarsening in molecular beam epitaxy. Phys. Rev. Lett., 73:1517, 1994.
114. P. Smereka. Semi-implicit level set methods for curvature and surface diffusion motion. J. Sci. Comput., 19:439-456, 2003.
115. B. J. Spencer, P. W. Voorhees, and S. H. Davis. Morphological instability in epitaxiallystrained dislocation-free solid films: linear stability theory. J. Appl. Phys., 73:4955-4970, 1993.
116. B. J. Spencer, P. W. Voorhees, and S. H. Davis. Morphological instability in epitaxiallystrained dislocation-free solid films: nonlinear evolution. Phys. Rev. B, 47:9760-9777, 1993.
117. B.J. Spencer. Asymptotic solutions for the equilibrium crystal shape with small corner energy regularization. Phys. Rev. E, 69:011603, 2004.
118. D. J. Srolovitz. On the stability of surfaces of stressed solids. Acta metall., 37(2):621-625, 1989.
119. C. Stöcker, S. Vey, and A. Voigt. AMDiS- adaptive multidimensional simulations: composite finite elements and signed distance functions. WSEAC Trans. Circ. Syst., 4:111-116, 2005.
120. C. Stöcker and A. Voigt. The effect of kinetics in the surface evolution of thin crystalline films. J. Cryst. Growth, 303:90-94, 2007.
121. J. A. Stroscio, D.T. Pierce, M.D. Stiles, A. Zangwill, and L.M. Sander. Coarsening of unstable surface features during Fe(001) homoepitaxy. Phys. Rev. Lett., 75:4246, 1995.
122. M. Sussman, E. Fatemi, P. Smereka, and S. Osher. An improved level-set method for incompressible two-phase flows. Comput. Fluids, 277:663-680, 1998.
123. A.P. Sutton and R.W. Balluffi. Interfaces in Crystalline Materials. Oxford University Press, 1997.
124. A. Szczepkowicz, A. Ciszewski, R. Bryl, C. Oleksy, C.-H. Men, Q. Wu, and T.E. Madey. A comparison of adsorbate-induced faceting on flat and curved crystal surfaces. Surf. Sci., 599:55-68, 2005.
125. J.E. Taylor. Mean curvature and weighted mean curvature. Acta Metall. Mater., 40:1475-1485, 1992.
126. T. Tiedje and A. Ballestad. Atomistic basis for continuum growth equation: Descrption of morphological evolution of gaas during molecular beam epitaxy. Thin Solid Films, 516:3705-3728, 2008.
127. S. Torabi, S. Li, A. Rätz, S. Wise, A. Voigt, and J. Lowengrub. Diffuse interface modeling of strongly anisotropic systems. Proc. Roy. Soc. A, in press.
128. Z.T. Trautt, M. Upmanyu, and A. Karma. Interface mobility from interface random walk. Science, 314:632-637, 2006.
129. J. Y. Tsao. Materials Fundamentals of Molecular Beam Epitaxy. Academic Press, 1993.
130. T. Uehara and R.F. Sekerka. Phase fieled simulations of faceted growth for strong anisotropy of kinetic coefficient. J. Cryst. Growth, 254:252-261, 2003.
131. S. Vey and A. Voigt. AMDiS: adaptive multidimensional simulations. Comput. Vis. Sci., 10:57-67, 2007.
132. J. Villain. Continuum models of crystal growthfrom atomic beams with and without desorption. J. de Phys., 1:19-12, 1991.
133. D. D. Vvedensky, A. Zangwill, C. N. Luse, and M. R. Wilby. Stochastic equations of motion for epitaxial growth. Phys. Rev. E, 48:852-862, 1993.
134. A.A. Wheeler. Phase-field theory of edges in an anisotropic crystal. Proc. Royal Soc. A, 462:3363-3384, 2006.
135. R. S. Williams and W. M. Tong. Kinetics of surface growth: Phenomenology, scaling, and mechanisms of smoothing and roughening. Ann. Rev. Phys. Chem., 45:401-438, 1994.
136. T.J. Willmore. Riemannian geoemerty. Oxford University Press, 1993.
137. S. Wise, J.-S. Kim, and J.S. Lowengrub. Solving the regularized, strongly anisotropic CahnHilliard equation by an adaptive nonlinear multigrid method. J. Comp. Phys., 226:414-446, 2007.
138. S. Wise, J. Lowengrub, J.S. Kim, K. Thornton, P.W Voorhees, and W.C. Johnson. Quantum dot formation on a strain-patterned epitaxial thin film. Appl. Phys. Lett., 87:133102, 2005.
139. D. E. Wolf and J. Villain. Growth with surface diffusion. Europhys. Lett., 13:389-394, 1990.
140. J.-J. Xu, Z. Li, J.S. Lowengrub, and H. Zhao. A level-set method for interfacial flows with surfactant. J. Comput. Phys., 212:590-616, 2006.