Process of irreversible nucleation in multilayer growth. I. Failure of the mean-field approach

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 66, Issue 3, 2002, Pages

  Politi, Paolo ^a   Castellano, Claudio ^b  

^a INFM   (Italy)

^b UNIVERSITÀ DI ROMA LA SAPIENZA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; DIFFUSION; EPITAXIAL GROWTH; FUNCTIONS; INTEGRATION; NUCLEATION; PROBLEM SOLVING; RANDOM PROCESSES;

DIFFUSING PARTICLES; MEAN-FIELD THEORY; MULTILAYER GROWTH; SPATIAL DISTRIBUTION;

DIMERS;

ARTICLE;

EID: 37649031484     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.66.031605     Document Type: Article

References (36)

1. M.A. Herman and H. Sitter, Molecular Beam Epitaxy (Springer, Heidelberg, 1996).
2. J.A. Venables, G.D.T. Spiller, and M. Hanbücken, Rep. Prog. Phys. 47, 399 (1984).
3. J.A. Venables, Introduction to Surface and Thin Film Processes (Cambridge University Press, Cambridge, 2000).
4. It is easy to realize that (Formula presented) should read (Formula presented) where (Formula presented) is the time an atom spends on each lattice site (Formula presented) is the substrate dimension).
5. J.G. Amar, F. Family, and P.-M. Lam, Phys. Rev. B 50, 8781 (1994).
6. J. Villain, A. Pimpinelli and D.E. Wolf, Comments Condens. Matter Phys. 16, 1 (1992).
7. A. Pimpinelli, J. Villain and D.E. Wolf, Phys. Rev. Lett. 69, 985 (1992).
8. P.J. Feibelman and T. Michely, Surf. Sci. 492, L723 (2001);
9. Surf. Sci.M. Kalff, P. Smilauer, G. Comsa, and T. Michely, 426, L447 (1999);Thomas Michely (private communication).
10. C.-M. Zhang, M.C. Bartelt, J.-M. Wen, C.J. Jenks, J.W. Evans, and P.A. Thiel, Surf. Sci. 406, 178 (1998);
11. L. Bardotti, C.R. Stoldt, C.J. Jenks, M.C. Bartelt, J.W. Evans, and P.A. Thiel, Phys. Rev. B 57, 12 544 (1998).
12. J.A. Stroscio, D.T. Pierce, and R.A. Dragoset, Phys. Rev. Lett. 70, 3615 (1993);
13. J.A. Stroscio and D.T. Pierce, Phys. Rev. B 49, 8522 (1994).
14. J. Tersoff, A.W. Denier van der Gon, and R.M. Tromp, Phys. Rev. Lett. 72, 266 (1994).
15. I. Elkinani and J. Villain, J. Phys. I 4, 949 (1994).
16. G. Ehrlich and F.G. Hudda, J. Chem. Phys. 44, 1039 (1966);
17. R.L. Schwoebel and E.J. Shipsey, J. Appl. Phys. 37, 3682 (1966);
18. J. Appl. Phys.R.L. Schwoebel, 40, 614 (1969);
19. K. Kyuno and G. Ehrlich, Surf. Sci. 383, L766 (1997).
20. See also Appendix B in: P. Politi, G. Grenet, A. Marty, A. Ponchet, and J. Villain, Phys. Rep. 324, 271 (2000).
21. P. Politi and J. Villain, Phys. Rev. B 54, 5114 (1996).
22. This has also been tested directly by changing the Zeno Model 1215 in such a way as to switch on/off nucleation on vicinal terraces [P. Politi and J. Villain (unpublished)].
23. C. Ratsch, M.F. Gyure, S. Chen, M. Kang and D.D. Vvedensky, Phys. Rev. B 61, R10598 (2000).
24. J. Rottler and P. Maass, Phys. Rev. Lett. 83, 3490 (1999);
25. S. Heinrichs, J. Rottler, and P. Maass, Phys. Rev. B 62, 8338 (2000).
26. J. Krug, P. Politi, and T. Michely, Phys. Rev. B 61, 14037 (2000).
27. J. Krug, Eur. Phys. J. B 18, 713 (2000).
28. P. Politi and C. Castellano, following paper, Phys. Rev. E 66, 031606 (2002).
29. C. Castellano and P. Politi, Phys. Rev. Lett. 87, 056102 (2001).
30. A. Pimpinelli and J. Villain, Physics of Crystal Growth (Cambridge University Press, Cambridge, 1998).
31. This is a very common assumption, but it may not be always appropriate. In fact the interaction between the landing atom and surface steps can steer the atom trajectory and create spatial inhomogeneities in the incoming flux of adatoms. See, S. van Dijken, L.C. Jorritsma, and B. Poelsema, Phys. Rev. Lett. 82, 4038 (1999).
32. In the discrete representation, if (Formula presented) is the deposition probability per time step we have (Formula presented) The definition of (Formula presented) as the average time of deposition implies the relation (Formula presented) so that (Formula presented) If (Formula presented) [i.e., (Formula presented) in the continuum picture] the prefactor is (Formula presented) and the quantity in square brackets is just e. Hence (Formula presented) coincides with Eq. (9).
33. K. Bromann, H. Brune, H. Röder, and K. Kern, Phys. Rev. Lett. 75, 677 (1995).
34. We take the single walker to be distributed uniformly at (Formula presented) (Formula presented) but a different choice changes (Formula presented) and (Formula presented) only slightly. For this reason the relations between W are (Formula presented) and (Formula presented) and (Formula presented) are given by the symbol (Formula presented)
35. B.D. Hughes, Random Walks and Random Environments (Clarendon Press, Oxford, 1995).
36. The function (Formula presented) has the maximum and the minimum on the border of the terrace [see V.I. Smirnov, A Course of Higher Mathematics (Oxford, Pergamon Press, 1964), Vol. 2]. If (Formula presented) is not identically equal to zero, let us suppose the maximum is positive (otherwise we can suppose the minimum negative) and apply the boundary condition (Formula presented) at the maximum. If (Formula presented) it implies that (Formula presented) increases in the inward direction and therefore it cannot be the maximum. If (Formula presented) and we apply the boundary condition at the maximum and the minimum, we find that they both vanish.