Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-21T07:44:34.584Z Has data issue: false hasContentIssue false

Roughening by Ion Bombardment: A Stochastic Continuum Equation

Published online by Cambridge University Press:  03 September 2012

R. Cuerno
Affiliation:
Center for Polymer Studies and Dept. of Physics, Boston University, Boston, MA 02215
A.-L. Barabasi
Affiliation:
IBM T. J. Watson Research Center, P.O. Box 218, Yorktown Heigths, NY 10598
Get access

Abstract

In the context of linear cascade theory, we derive a stochastic nonlinear equation to describe the evolution and scaling properties of surfaces eroded by ion bombardment. The coefficients appearing in the equation are calculated explicitly in terms of the physical parameters characterizing the sputtering process. We find that transitions may take place between various scaling behaviors when experimental parameters such as the angle of incidence of the incoming ions or their average penetration depth, are varied.

Type
Research Article
Copyright
Copyright © Materials Research Society 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Dynamics of Fractal Surfaces, edited by Family, F. and Vicsek, T. (World Scientific, Singapore, 1991); P. Meakin, Phys. Rep. 235, 189 (1993); T. Halpin-Healey and Y.-C. Zhang, Phys. Rep. (in press); A.-L. Barabisi and H. E. Stanley, Fractal Concepts in Surface Growth, (Cambridge University Press, Cambridge, 1995).Google Scholar
[2] Eklund, E. A., Bruinsma, R., Rudnick, J. and Williams, R. S., Phys. Rev. Lett. 67, 1759 (1991); E. A. Eklund, E. J. Snyder and R. S. Williams, Surf. Sci. 285, 157 (1993).Google Scholar
[3] Krim, J., Heyvaert, I., Haesendonck, C. Van and Bruynseraede, Y., Phys. Rev. Lett. 70, 57 (1993).Google Scholar
[4] Chason, E., Mayer, T. M., Kellerman, B. K., McIlroy, D. T. and Howard, A. J., Phys. Rev. Lett. 72, 3040 (1994); T. M. Mayer, E. Chason and A. J. Howard, J. Appl. Phys. 76, 1633 (1994).Google Scholar
[5] Yang, H.-N., Wang, G.-C., and Lu, T.-M., Phys. Rev. B 50, 7635 (1994).Google Scholar
[6] Sputtering by Particle Bombardment, edited by Behrisch, R. (Springer-Verlag, Heidelberg 1981, 1983), Vols. 1, II.Google Scholar
[7] Kardar, M., Parisi, G. and Zhang, Y.-C., Phys. Rev. Lett. 56, 889 (1986).Google Scholar
[8] Moser, K., Wolf, D. E. and Kertész, J., Physica A 178, 215 (1991); T. Ala Nissila, T. Hjelt, J. M. Kosterlitz and O. Vemölöinen, J. Stat. Phys. 72, 207 (1993).Google Scholar
[9] Carter, G., Navingek, B. and Whitton, J. L. in Vol. II of Ref. [6], pp. 231269.Google Scholar
[10] Cuerno, R. and Barabási, A.-L., preprint; Barabási, A.-L. and Cuerno, R., unpublished.Google Scholar
[11] Sigmund, P., Phys. Rev. 184, 383 (1969); J. Mat. Sci. 8, 1545 (1973).Google Scholar
[12] Bradley, R. M. and Harper, J. M. E., J. Vac. Sci. Technol. A6, 2390 (1988).Google Scholar
[13] Herring, C., J. Appl. Phys. 21, 301 (1950); W. W. Mullins, J. Appl. Phys. 28, 333 (1957); in the context of stochastic models see D. E. Wolf and J. Villain, Europhys. Lett. 13, 389 (1990); S. Das Sarma and P. I. Tamborenea, Phys. Rev. Lett. 66, 325 (1991).Google Scholar
[14] Carter, G., in Erosion and Growth of Solids Stimulated by Atom and Ion Beams, edited by Kiriakidis, G., Carter, G. and Whitton, J. L., (Nijhoff, Hingham, 1986) pp. 7097; M. J. Nobes, pp.103-120.Google Scholar
[15] The θ = 0 case has to be treated separately and is discussed in Ref. [10].Google Scholar
[16] The terms -ν0 and ∂xh can be reabsorbed by a change of variables to a comoving frame, and do not affect the scaling properties.Google Scholar
[17] Kuramoto, Y. and Tsuzuki, T., Prog. Theor. Phys. 55, 356 (1977); G. I. Sivashinskxy, Acta Astronaut. 6, 569 (1979).Google Scholar
[18] Cuerno, R., Lauritsen, K. B. and Barabási, A.-L. (unpublished).Google Scholar
[19] Golubović, L. and Bruinsma, R., Phys. Rev. Lett. 66, 321 (1991); 67, 2747 (E) (1991).Google Scholar
[20] Wolf, D. E., Phys. Rev. Lett. 67, 1783 (1991).Google Scholar
[21] The relevance of the AKPZ equation to sputter erosion has been suggested by Bruinsma, R. in Surface Disordering: Growth, Roughening and Phase Transitions, edited by Jullien, R., Kertész, J., Meakin, P. and Wolf, D. E. (Nova Science, New York, 1992), pp. 9194.Google Scholar
[22] Bales, G. S., Bruinsma, R., Eklund, E. A., Karunasiri, R. P. U., Rudnick, J. and Zangwill, A., Science 249, 264 (1990).Google Scholar