[ Previous / Next Abstract | Issue Table of Contents | Bottom of Page ]

Phys. Rev. B 64, 205404 (2001) (13 pages)

Full Text:  [  PDF (219 kB)   GZipped PS    Order Document  ]

Rate-equation approach to island size distributions and capture numbers in submonolayer irreversible growth

Mihail N. Popescu,¹ Jacques G. Amar,² and Fereydoon Family¹

¹Department of Physics, Emory University, Atlanta, Georgia 30322
²Department of Physics & Astronomy, University of Toledo, Toledo, Ohio 43606

(Received 14 December 2000; revised 25 April 2001; published 19 October 2001)

We present a quantitative rate-equation approach to irreversible submonolayer growth on a two-dimensional substrate. Our method explicitly takes into account the existence of a denuded ("capture") zone around every island and the correlations between the size of an island and the corresponding average capture zone. The evolution of the capture-zone distributions is described by a set of Voronoi-area evolution equations, which are coupled to the usual rate equations for the island densities through local rates of monomer capture. The combined set of equations leads to a fully self-consistent calculation of the size- and coverage-dependent capture numbers. The resulting predictions for the average capture-zone and capture-number distributions are in excellent agreement with experimental results and Monte Carlo simulations. As a result, the corresponding scaled island size distributions and their dependence on coverage and deposition rate are also accurately predicted in the precoalescence regime. ©2001 The American Physical Society

URL: http://link.aps.org/abstract/PRB/v64/e205404
DOI: 10.1103/PhysRevB.64.205404
PACS: 81.15.Aa, 68.55.-a, 68.35.Bs, 82.40.Bj        Additional Information

References

1. J. Y. Tsao, Material Fundamentals of Molecular Beam Epitaxy (World Scientific, Singapore, 1993).
   
2. Y. W. Mo, J. Kleiner, M. B. Webb, M. G. Lagally, Phys. Rev. Lett. 66, 1998 (1991).
   
3. J. A. Stroscio, D. T. Pierce, and R. A. Dragoset, Phys. Rev. Lett. 70, 3615 (1993);
   J. A. Stroscio and D. T. Pierce, Phys. Rev. B 49, 8522 (1994).
   
4. H. J. Ernst, F. Fabre, and J. Lapujoulade, Phys. Rev. B 46, 1929 (1992).
   
5. R. Q. Hwang, J. Schroder, C. Gunther, and R. J. Behm, Phys. Rev. Lett. 67, 3279 (1991);
   R. Q. Hwang and R. J. Behm, J. Vac. Sci. Technol. B 10, 256 (1992).
   
6. W. Li, G. Vidali, and O. Biham, Phys. Rev. B 48, 8336 (1993).
   
7. E. Kopatzki, S. Gunther, W. Nichtl-Pecher, and R. J. Behm, Surf. Sci. 284, 154 (1993). [INSPEC]
   
8. G. Rosenfeld, R. Servaty, C. Teichert, B. Poelsema, and G. Comsa, Phys. Rev. Lett. 71, 895 (1993).
   
9. J.-K. Zuo and J. F. Wendelken, Phys. Rev. Lett. 66, 2227 (1991);
   J.-K. Zuo, J. F. Wendelken, H. Durr, and C.-L. Liu, ibid. 72, 3064 (1994).
   
10. D. D. Chambliss and R. J. Wilson, J. Vac. Sci. Technol. B 9, 928 (1991);
    D. D. Chambliss and K. E. Johnson, Phys. Rev. B 50, 5012 (1994).
    
11. Q. Jiang and G. C. Wang, Surf. Sci. 324, 357 (1995) (ScienceDirect). [INSPEC]
    
12. F. Tsui, J. Wellman, C. Uher, and R. Clarke, Phys. Rev. Lett. 76, 3164 (1996).
    
13. T. R. Linderoth, S. Horch, E. Laegsgaard, I. Stensgaard, and F. Besenbacher, Phys. Rev. Lett. 78, 4978 (1997).
    
14. J. A. Venables, G. D. Spiller, and M. Hanbucken, Rep. Prog. Phys. 47, 399 (1984); [INSPEC]
    J. A. Venables, Philos. Mag. 27, 697 (1973); [INSPEC]
    Phys. Rev. B 36, 4153 (1987).
    
15. F. Family and P. Meakin, Phys. Rev. Lett. 61, 428 (1988).
    
16. J. G. Amar and F. Family, Phys. Rev. Lett. 74, 2066 (1995).
    
17. J. G. Amar, F. Family, and P. M. Lam, Phys. Rev. B 50, 8781 (1994).
    
18. J. G. Amar and F. Family, Thin Solid Films 272, 208 (1996) (ScienceDirect). [INSPEC]
    
19. J. G. Amar and F. Family, Surf. Sci. 382, 170 (1997) (ScienceDirect). [INSPEC]
    
20. G. S. Bales and D. C. Chrzan, Phys. Rev. B 50, 6057 (1994).
    
21. G. S. Bales and A. Zangwill, Phys. Rev. B 55, 1973 (1997).
    
22. J. A. Blackman and A. Wilding, Europhys. Lett. 16, 115 (1991). [INSPEC]
    
23. P. A. Mulheran and J. A. Blackman, Philos. Mag. Lett. 72, 55 (1995). [INSPEC]
    
24. J. A. Blackman and P. A. Mulheran, Phys. Rev. B 54, 11 681 (1996);
    P. A. Mulheran and J. A. Blackman, Surf. Sci. 376, 403 (1997) (ScienceDirect). [INSPEC]
    
25. P. A. Mulheran and D. A. Robbie, Europhys. Lett. 49, 617 (2000). [INSPEC]
    
26. C. Ratsch, A. Zangwill, P. Smilauer, and D. D. Vvedensky, Phys. Rev. Lett. 72, 3194 (1994).
    
27. M. C. Bartelt and J. W. Evans, Phys. Rev. B 46, 12 675 (1992).
    
28. M. C. Bartelt and J. W. Evans, J. Vac. Sci. Technol. A 12, 1800 (1994).
    
29. M. C. Bartelt and J. W. Evans, Surf. Sci. 344, 1193 (1995).
    
30. M. C. Bartelt and J. W. Evans, Phys. Rev. B 54, R17 359 (1996).
    
31. M. C. Bartelt, A. K. Schmid, J. W. Evans, and R. Q. Hwang, Phys. Rev. Lett. 81, 1901 (1998).
    
32. J. G. Amar, M. N. Popescu, and F. Family, Surf. Sci. 491, 239 (2001).
    
33. M. N. Popescu, J. G. Amar, and F. Family, Phys. Rev. B 58, 1613 (1998).
    
34. J. G. Amar, M. N. Popescu, and F. Family, Phys. Rev. Lett. 86, 3092 (2001).
    
35. M. von Smoluchowski, Z. Phys. Chem., Stoechiom. Verwandtschaftsl. 17, 557 (1916);
    ibid. 92, 129 (1917).
    
36. G. Zinsmeister, Thin Solid Films 2, 497 (1968);
    ibid. 4, 363 (1969); [INSPEC]
    ibid. 7, 51 (1971). [INSPEC]
    
37. V. Halpern, J. Appl. Phys. 40, 4627 (1969). [INSPEC]
    
38. E. M. Hendricks and M. H. Ernst, J. Colloid Interface Sci. 97, 176 (1984).
    
39. N. V. Brilliantov and P. L. Krapivsky, J. Phys. A 24, 4787 (1991). [INSPEC]
    
40. The exact term to account for the monomer-monomer collision rate in Eqs. (678) is (Rn/N₁)/ that leads to a nonlinear diffusion equation which may be solved numerically (see Ref. 41). In order to obtain a linear equation the mean-field approximation Rn₁/ is used (see also Ref. 20).
    
41. A. K. Myers-Beaghton and D. D. Vvedensky, Phys. Rev. B 42, 5544 (1990).
    
42. M. N. Popescu, J. G. Amar, and F. Family (unpublished).
    
43. The A dependence of B_A can be neglected because it is much smaller than that due to the terms depending on x_A.
    
44. P. A. Mulheran and J. A. Blackman, Phys. Rev. B 53, 10 261 (1996);
    M. C. Bartelt, C. R. Stoldt, C. J. Jenks, P. A. Thiel, and J. W. Evans, ibid. 59, 3125 (1999).
    
45. Coalescence effects (see Fig. 2) are also important starting at coverages around 0.18. While the KMC simulations naturally include coalescence, the rate equations do not include terms to account for such effects.
    
46. The condition a₀ + a₁ = 1 is required by the definition of the average capture number along with the normalization of the island-size distribution function.
    
47. A decreasing value of f(0) with increasing coverage for irreversible growth of extended islands has been seen previously experimentally (Ref. 3) as well as in KMC simulations. (Ref. 16).
    
48. Summing Eq. (36) over all s2, without any correction to ensure that the normalization condition dA G_s(;A) = N_s is satisfied, one obtains G(A) = (1/A²)H(A–1/N), where H(z) is the Heaviside step function. While this distribution satisfies the normalization condition dA G(A) = N, it has very little resemblance to the correct Voronoi-area distribution and leads to a divergent average Voronoi area.
    
49. D. D. Vvedensky, Phys. Rev. B 62, 15 435 (2000).
    

  The American Physical Society is a member of CrossRef.