SURFACE DIFFUSION INCLUDING ADATOMS*

Communications in Mathematical Sciences

Volumn 4, Issue 1, 2006, Pages 1-51

  Burger, Martin ^a  

^a JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

Author keywords

Nanostructure Growth; Numerical Simulation; Surface Diffusion

Indexed keywords

EID: 33947668047     PISSN: 15396746     EISSN: 19450796     Source Type: Journal    
DOI: 10.4310/CMS.2006.v4.n1.a1     Document Type: Article

References (52)

1. [1] H. Amann, Linear and Quasilinear Parabolic Problems, Birkhäuser, Basel, 1995.
2. [2] A. Averbuch, M. Israeli and I. Ravve, Electromigration of intergranular voids in metal films for microelectronic interconnects, J. Comp. Phys., 186, 481-502, 2003.
3. [3] A. Averbuch, M. Israeli, N. Menachem and I. Ravve, Surface evolution in bare bamboo-type metal lines under diffusion and electric field effects, J. Comp. Phys., 188, 640-677, 2003.
4. [4] S. A. Alvarez and C. Liu, Motion of a closed curve by minus the surface Laplacian of curvature, Differ. Integral Equ., 13, 1583-1594, 2000.
5. [5] R. J. Asaro and W. A. Tiller, Surface morphology development during stress corrosion cracking: part I: via surface diffusion, Metall. Trans., 3, 1789-1796, 1972.
6. [6] E. Bansch, P. Morin and R. Nochetto, Surface diffusion of graphs: variational formulation, error analysis, and simulation, SIAM J. Numer. Anal., 42, 773-799, 2004.
7. [7] E. Baansch, P. Morin and R. Nochetto, Finite element methods for surface diffusion, Int. Ser. Num. Math., Birkhäuser, Basel, 147, 53—63, 2003.
8. [8] P. Baras, J. Duchon and R. Robert, Evolution d'une interface par diffusion de surface, Comm. Partial Diff. Eq., 9, 313-335, 1984.
9. [9] G. Bellettini, C. Mantegazza and M. Novaga, Singular perturbations of mean curvature flow, SNS Pisa, preprint, 2004.
10. [10] A. J. Bernoff, A. L. Bertozzi and T. P. Witelski, Axisymmetric surface diffusion: dynamics and stability of self-similar pinchoff, J. Stat. Phys., 93, 725-776, 1998.
11. [11] M. Burger, Numerical simulation of anisotropic surface diffusion with curvature-dependent energy, J. Comp. Phys., 203, 602-625, 2005.
12. [12] R. E. Caflisch and B. Li, Analysis ofisland dynamics in epitaxial growth ofthin films, Multiscale Model. Simul., 1, 150-171, 2003.
13. [13] J. W. Cahn and J. E. Taylor, Surface motion by surface diffusion, Acta Metall. Mater., 42, 1045-1063, 1994.
14. [14] D. Chopp and J. A. Sethian, Motion by intrinsic Laplacian of curvature, Interfaces and Free Boundaries, 1, 107-123, 1999.
15. [15] B. D. Coleman, R. S. Falk and M. Moakher, Space-time finite element methods for surface diffusion with applications to the theory of the stability of cylinders, SIAM J. Sci. Comp., 17, 1434-1448, 1996.
16. [16] J. Colin, J. Grilhé and N. Junqua, Transformation of a cubic precipitate to a butterfly shape due to localized instabilities, Acta Mater., 46, 1249-1255, 1998.
17. [17] F. Davi and M. E. Gurtin, On the motion of a phase interface by surface diffusion, J. Appl. Math. Phys., 41, 782-811, 1990.
18. [18] K. Deckelnick, G. Dziuk and C. M. Elliott, Fully discrete semi-implicit second order splitting for anisotropic surface diffusion of graphs, Isaac Newton Institute, Cambridge, preprint, 2003.
19. [19] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
20. [20] M. C. Delfour and J. P. Zoléesio, Shapes and Geometries. Analysis, Differential Calculus, and Optimization, SIAM, Philadelphia, 2001.
21. [21] E. DeGiorgi, Congetture riguardanti alcuni problemi di evoluzione. A celebration of J. F. Nash Jr., Duke Math. J., 81, 255-268, 1996.
22. [22] A. DiCarlo, M. Gurtin and P. Podio-Guidugli, A regularized equation for anisotropic motion by curvature, SIAM J. Appl. Math., 52, 1111-1119, 1992.
23. n: existence and com putation, SIAM J. Math. Anal., 33, 1228-1245, 2002.
24. [24] C. M. Elliott and H. Garcke, Existence results for diffusive surface motion laws, Adv. Math. Sci. Appl., 7, 465-488, 1997.
25. [25] J. Escher, U. F. Mayer and G. Simonett, The surface diffusion flow for immersed hypersurfaces, SIAM J. Math. Anal., 29, 1419-1433, 1998.
26. [26] E. Fried and M. E. Gurtin, A unified treatment of evolving interfaces accounting for deforma tion and atomic transport with an emphasis on grain-boundaries and epitaxy, Advances in Applied Mechanics, 40, 1-177, 2004.
27. [27] H. Gao and W. D. Nix, Surface roughening of heteroepitaxial thin films, Annual Review of Materials Science 29, 173-209, 1999.
28. [28] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhaauser, Boston, 1984.
29. [29] A. A. Golovin and S. H. Davis, Effect of anisotropy on morphological instability in the freezing of a hypercooled melt, Physica D, 116, 363-391, 1998.
30. [30] A. A. Golovin, S. H. Davis and A. A. Nepomnyashchy, A model for facetting in a kinetically controlled crystal growth, Phys. Rev. E, 59, 803-825, 1999.
31. [31] M. E. Gurtin, M. E. Jabbour, Interface evolution in three dimensions with curvature-dependent energy and surface diffusion: interface-controlled evolution, phase transitions, epitaxial growth of elastic films, Arch. Rat. Mech. Anal., 163, 171-208, 2002.
32. [32] F. Hauler and A. Voigt, Weakly anisotropic surface diffusion, a numerical approach by para metric finite elements, CAESAR Research Center, Bonn, preprint, 2004.
33. [33] H. Heuser, Funktionalanalysis, Teubner, Stuttgart, 1992.
34. [34] F. Jonsdottir, L. B. Freund, Equilibrium surface roughness of a strained epitaxial film due to surface diffusion induced by interface misfit dislocations, Mech. Materials, 20, 337-349, 1995.
35. [35] Z. Li, H. K. Zhao and H. Gao, A numerical study of electro-migration voiding by evolving level set functions on a fixed Cartesian grid, J. Comp. Phys., 152, 281-304, 1999.
36. [36] J. L. Lions and E. Magenes, Non-Homogenous Boundary Value Problems and Applications, Vol. I, Springer, Berlin, Heidelberg, New York, 1972.
37. [37] J. L. Lions and E. Magenes, Non-Homogenous Boundary Value Problems and Applications, Vol. II, Springer, Berlin, Heidelberg, New York, 1972.
38. [38] V. Lods, A. Pietrus and J. M. Rakotoson, Mathematical study of the equation describing the evolution of the surface of a film, Asymptotic Analysis, 33, 67-91, 2003.
39. [39] U. F. Mayer, Numerical solutions for the surface diffusion flow in three space dimensions, Comput. Appl. Math., 20, 361-379, 2001.
40. [40] W. W. Mullins, Theory of thermal grooving, J. Appl. Phys., 28, 333-339, 1957.
41. [41] A. N. Norris, Equilibrium, stability and evolution of elastic plates under the combined effects of stress and surface diffusion, Q. J. Mech. Appl. Math., 52, 283-309, 1999.
42. [42] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd Edition, SIAM, Philadelphia, 2003.
43. [43] T. V. Savina, A. A. Golovin, S. H. Davis, A. A. Nepomnyashchy and P. W. Voorhees, On faceting of a growing crystal surface by surface diffusion, Phys. Rev. E, 67, 1-16, 2003.
44. [44] V. B. Shenoy and L. B. Freund, A continuum description of the energetics and evolution of stepped surfaces in strained nanostructures, J. Mech. Phys. Solids, 50, 1817-1841, 2002.
45. [45] V. B. Shenoy, A. Ramasubramaniam and L. B. Freund, A variational approach to nonlinear dynamics of nanoscale surface modulations, Surf. Sci.,529, 365-383, 2003.
46. [46] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, AMS, Providence, 1997.
47. [47] B. J. Spencer, P. W. Voorhees and J. Tersoff, Morphological instability theory for strained alloy film growth: the effect of compositional stresses and species-dependent surface mobilities on ripple formation during epitaxial film deposition, Phys. Rev. B, 64, 235-318, 2001.
48. [48] J. Stewart and N. D. Goldenfeld, Spinodal decomposition of a crystal surface, Phys. Rev. A, 46, 6505-6512, 1992.
49. [49] P. Smereka, Semi-implicit level set methods for curvature and surface diffusion motion, J. Sci. Comp., 19, 439-456, 2003.
50. [50] J. Stangl, V. Holy and G. Bauer, Structural properties of self-organized semiconductor nanos tructures, Rev. Mod. Phys., 76, to appear, 2004.
51. [51] J. E. Taylor, Some mathematical challenges in materials science, Bull. Amer. Math. Soc., 40, 69-87, 2002.
52. [52] C. Truesdell, The influence of elasticity on analysis: the classical heritage, Bull. Amer. Math. Soc., 9, 293-310, 1983.