Phys. Rev. Lett. Phys. Rev. A Phys. Rev. B Phys. Rev. C Phys. Rev. D Phys. Rev. E Phys. Rev. ST AB Rev. Mod. Phys. Phys. Rev. (Series I) Phys. Rev. Volume: Page/Article:

Phys. Rev. A 40, 2064–2077 (1989)

[Issue 4 – 15 August 1989 ]

[ Previous article | Next article | Issue 4 contents ]

Microstructure and surface scaling in ballistic deposition at oblique incidence

Joachim Krug

Theoretische Physik, Universität München, Theresienstrasse 37, 8000 München 2, Federal Republic of Germany

Paul Meakin

Central Research and Development Department, E.I. du Pont de Nemours Company, Wilmington, Delaware 19880-0356

Scaling properties of two-dimensional ballistic deposits grown at near-grazing angles of incidence are investigated analytically and numerically. We map the problem onto a system of coalescing Brownian particles and derive exact values for the static and dynamic surface exponents, zeta =1 and z=2, and for the exponents characterizing the self-affine columnar microstructure, tau =(4/3, nu =(2/3, and nu _perp=(1/3. This implies that the average column width increases as the square root of the deposit thickness. The distribution of surface step heights and the angular variation of the deposit density are also obtained analytically. The predictions are confirmed by large-scale computer simulations. Qualitative arguments are given to explain the slow crossover behavior at intermediate angles of incidence, which leads to apparently continuously varying scaling exponents. The substructure exponents for deposits grown at normal incidence are derived from a general scaling relation. We find tau =(7/5, nu d||=(3/5, and nu perp=(2/5, in agreement with previous numerical work.

©1989 The American Physical Society

URL: http://link.aps.org/abstract/PRA/v40/p2064
DOI: 10.1103/PhysRevA.40.2064
PACS: 05.40.+j, 68.55.-a, 81.30.-t

[ Previous article | Next article | Issue 4 contents ]

References

1. H. König and G. Helwig, Optik 6, 111 (1950).
2. H. J. Leamy, G. H. Gilmer, and A. G. Dirks, in Current Topics in Materials Science, edited by E. Kaldis (North-Holland, New York, 1980), Vol. 6; H. J. Leamy and A. G. Dirks, J. Appl. Phys. 49, 3430 (1978) [ SPIN][ INSPEC].
3. M. J. Vold, J. Colloid Sci. 14, 168 (1959).
4. D. Henderson, M. H. Brodsky and P. Chaudhari, Appl. Phys. Lett. 25, 641 (1977) [ INSPEC].
5. A. G. Dirks and H. J. Leamy, Thin Solid Films 47, 219 (1977) [ INSPEC].
6. J. M. Nieuwenhuizen and H. B. Haanstra, Philips Tech. Rev. 27, 87 (1966).
7. N. G. Nakhodkin and A. I. Shaldervan, Thin Solid Films 10, 109 (1972) [ INSPEC].
8. T. Hashimoto, K. Okamoto, K. Hara, M. Kamiya and H. Fujiwara, Thin Solid Films 91, 145 (1982) [ INSPEC].
9. O. Geszti, L. Gosztola and E. Seyfried, Thin Solid Films 136, L35 (1986) [ INSPEC].
10. P. Meakin, P. Ramanlal, L. M. Sander and R. C. Ball, Phys. Rev. A 34, 5091 (1986).
11. P. Meakin, Phys. Rev. A 38, 994 (1988).
12. T. A. Witten, Jr., and L. M. Sander, Phys. Rev. Lett. 47, 1400 (1981).
13. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1982).
14. D. Bensimon, B. Shraiman and S. Liang, Phys. Lett. 102A, 238 (1984).
15. P. Meakin, J. Colloid Interface Sci. 105, 240 (1985).
16. R. C. Ball and T. A. Witten, J. Stat. Phys. 36, 873 (1984) [ INSPEC].
17. F. Family and T. Vicsek, J. Phys. A 18, L75 (1985).
18. M. Kardar, G. Parisi and Y. C. Zhang, Phys. Rev. Lett. 56, 889 (1986) [SPIRES].
19. P. Meakin, J. Phys. A 20, L1113 (1987) [ INSPEC].
20. J. Krug, Phys. Rev. A 36, 5465 (1987).
21. J. Krug and H. Spohn, Phys. Rev. A 38, 4271 (1988).
22. J. Krug and H. Spohn, Europhys. Lett. 8, 219 (1989) [ INSPEC].
23. P. Ramanlal and L. M. Sander, Phys. Rev. Lett. 54, 1828 (1985).
24. S. Lichter and J. Chen, Phys. Rev. Lett. 56, 1396 (1986).
25. A. Mazor, D. J. Srolovitz, P. S. Hagan and B. G. Bukiet, Phys. Rev. Lett. 60, 424 (1988); D. J. Srolovitz, A. Mazor and B. G. Bukiet, J. Vac. Sci. Technol. A 6, 2371 (1988) [ INSPEC].
26. M. Plischke and Z. Rácz, Phys. Rev. A 32, 3825 (1985).
27. R. Lipowsky and M. E. Fisher, Phys. Rev. Lett. 56, 472 (1986).
28. Z. Rácz and M. Plischke, Phys. Rev. A 31, 985 (1985).
29. R. Jullien and R. Botet, J. Phys. A 18, 2279 (1985) [ INSPEC].
30. H. van Beijeren, R. Kutner and H. Spohn, Phys. Rev. Lett. 55, 2026 (1985).
31. Z. Rácz and T. Vicsek, Phys. Rev. Lett. 51, 2382 (1983).
32. M. Matsushita and P. Meakin, Phys. Rev. A 37, 3645 (1988).
33. S. Redner and A. Coniglio, J. Phys. A 15, L273 (1982) [ INSPEC].
34. D. Dhar, J. Phys. A 21, L893 (1988) [ INSPEC].
35. D. Forster, D. R. Nelson and M. J. Stephen, Phys. Rev. A 16, 732 (1977).
36. D. E. Wolf and J. Kertész, Europhys. Lett. 4, 651 (1987) [ INSPEC].
37. D. Liu and M. Plischke, Phys. Rev. B 38, 4781 (1988).
38. R. Baiod, D. Kessler, P. Ramanlal, L. Sander and R. Savit, Phys. Rev. A 38, 3672 (1988).
39. T. Halpin-Healy, Phys. Rev. Lett. 62, 442 (1989); T. Nattermann (unpublished).
40. M. Bramson and D. Griffeath, Ann. Prob. 8, 183 (1980).
41. M. Bramson and D. Griffeath, Z. Wahrscheinlichkeitstheorie verw. Gebiete 53, 183 (1980).
42. C. R. Doering and D. ben-Avraham, Phys. Rev. A 38, 3035 (1988).
43. J. L. Spouge, Phys. Rev. Lett. 60, 871 (1988); J. Phys. A 21, 4183 (1988) [ INSPEC].
44. H. Kondoh, M. Matsushita and Y. Fukuda, J. Phys. Soc. Jpn. 56, 1913 (1987) [ INSPEC]; H. Takayasu, I. Nishikawa and H. Tasaki, Phys. Rev. A 37, 3110 (1988).
45. K. Kang and S. Redner, Phys. Rev. A 30, 2833 (1984).
46. P. Meakin and R. Jullien, J. Phys. (Paris) 48, 1651 (1987) [ INSPEC].
47. Table of Integrals, Series and Products, edited by I. S. Gradshteyn and I. M. Ryzhik (Adademic, New York, 1980).
48. J. Kertész and D. E. Wolf, J. Phys. A 21, 747 (1988) [ INSPEC].
49. M. Matsushita, Y. Hayakawa and Y. Sawada, Phys. Rev. A 32, 3814 (1985).
50. W. Kley (private communication).
51. A. A. Chernov, Usp. Fiz. Nauk 73, 277 (1961); Kristallografiya 7, 895 (1962),; Modern Crystallography III: Crystal Growth, Vol. 36 of Springer Series in Solid State Sciences (Springer-Verlag, Berlin, 1984).
52. D. E. Wolf, J. Phys. A 20, 1251 (1987) [ INSPEC].
53. S. Liang and L. P. Kadanoff, Phys. Rev. A 31, 2628 (1985).

[ Previous article | Next article | Issue 4 contents ]