Patrick Shipman

Department of Mathematics
Colorado State University


When an energetic ion strikes a solid surface, one or more atoms can be ejected from the solid. Bombarding a flat solid surface with a broad ion beam therefore results in erosion of the solid. Naively, one would expect that the solid surface would simply remain flat as it is eroded. Instead, a remarkable variety of self-assembled nanoscale patterns can form, including periodic height modulations or "ripples," mounds arranged in hexagonal arrays of astonishing regularity, and arrays of sharply pointed conical protrusions. The spontaneous emergence of these patterns is not just fascinating in its own right, since in the future ion bombardment may prove to be an important tool in the fabrication of nanostructures.

At the present time, many aspects of the self-organization induced by ion bombardment are not fully understood. Our group is working closely with experimentalists to provide answers to many of the open questions in the field. We are using continuum modeling, the mathematical theory of pattern formation, and computer simulations in our research.

The first type of pattern formation to be discovered was the ripples that often develop when the nominally flat surface of a solid is eroded by oblique-incidence ion bombardment. According to the widely accepted Bradley-Harper theory [1], this is a result of a surface instability caused by the curvature dependence of the erosion velocity: The erosion velocity is greater in a trough than at a crest.

In 1999, experiments by Facsko et al. revealed that normal-incidence ion bombardment of the binary compound gallium antimonide can lead to the formation of nanoscale mounds or "nanodots" arranged in a beautifully ordered hexagonal array [2]. Recently, group leaders Mark Bradley and Patrick Shipman developed a theory that explains the genesis of these arrays [3,4,5]. In our theory, the coupling between the topography of the surface and a surface layer of altered composition is the key to the pattern formation. The figure below shows the results of a simulation of our model.

Patrick Shipman

[1] R. M. Bradley and J. M. E. Harper, J. Vac. Sci. Technol. A 6, 2390 (1988).

[2] S. Facsko, T. Dekorsy, C. Koerdt, C. Trappe, H. Kurz, A. Vogt, and H. L. Hartnagel, Science 285, 1551 (1999).

[3] R. M. Bradley and P. D. Shipman, Phys. Rev. Lett. 105, 145501 (2010).

[4] P. D. Shipman and R. M. Bradley, Phys. Rev. B 84, 085420 (2011).

[5] R. M. Bradley and P. D. Shipman, Appl. Surf. Sci. (to appear).


Numerical simulation of our model
of normal-incidence ion bombardment of a binary solid,
producing a hexagonal array of nanodots
starting from a white-noise initial condition

The Team

Mark Bradley, Professor of Physics

Victoria Rojo, Graduate Student, Physics

Kelly Mauser, Undergraduate Student, Physics

Patrick Shipman, Assistant Professor of Mathematics  

Francis Motta, Graduate Student, Mathematics 

Jaime Shinn, Graduate Student, Mathematics 

Selected Group Publications on Ion Bombardment of Solids

[12] R. M. Bradley and P. D. Shipman, "A Surface Layer of Altered Composition Can Play a Key Role in Nanoscale Pattern Formation Induced By Ion Bombardment," to appear in Appl. Surf. Sci.

[11] P. D. Shipman and R. M. Bradley, "Theory of Nanoscale Pattern Formation Induced by Normal-Incidence Ion Bombardment of Binary Compounds," Phys. Rev. B 84, 085420 (2011).

[10] R. M. Bradley, "Exact Linear Dispersion Relation for the Sigmund Model of Ion Sputtering," Phys. Rev. B 84, 075413 (2011).

[9] R. M. Bradley, "Theory of Nanodot and Sputter Cone Arrays Produced by Ion Sputtering with Concurrent Deposition of Impurities," Phys. Rev. B 83, 195410 (2011).

[8] R. M. Bradley, "Redeposition of Sputtered Material is a Nonlinear Effect," Phys. Rev. B 83, 075404 (2011).

[7] R. M. Bradley and P. D. Shipman, "Spontaneous Pattern Formation Induced by Ion Bombardment of Binary Compounds," Phys. Rev. Lett. 105, 145501 (2010).

[6] R. M. Bradley, "Dynamic Scaling of Ion-Sputtered Rotating Surfaces," Phys. Rev. E 54, 6149 (1996).

[5] R. M. Bradley and E.-H. Cirlin, "Theory of Improved Resolution in Depth Profiling with Sample Rotation," Appl. Phys. Lett. 68, 3722 (1996).

[4] J. M. E. Harper, S. E. Hornstrom, P. J. Rudeck, and R. M. Bradley, "Angle of Incidence Effects in Ion Beam Processing'', Proc. Mat. Res. Soc. 128, 269 (1989).

[3] R. M. Bradley, "Thin Film Modification by Off-Normal Incidence Ion Bombardment'', in Handbook of Ion Beam Processing Technology'', edited by J. J. Cuomo, S. M. Rossnagel and H. R. Kaufman (Noyes, Park Ridge, 1989), Chap. 15.

[2] R. M. Bradley and J. M. E. Harper, "Theory of Ripple Topography Induced by Ion Bombardment", J. Vac. Sci. Technol. A 6, 2390 (1988).

[1] R. M. Bradley, J. M. E. Harper and D. A. Smith, "Theory of Thin Film Orientation by Ion Bombardment during Deposition", J. Appl. Phys. 60, 4160 (1986).