Spontaneous Pattern Formation Induced by Ion Bombardment of Binary Compounds R. Mark Bradley Department of Physics, Colorado State University, Fort Collins, Colorado 80523, USA Patrick D. Shipman Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523, USA (Received 22 June 2010; revised manuscript received 12 August 2010; published 1 October 2010) A theory is developed that explains the genesis of the strikingly regular hexagonal arrays of nanoscale mounds that can form when a flat surface of a binary compound is subjected to normal-incidence ion bombardment. We find that the species with the higher sputter yield is concentrated at the peaks of the nanodots and that hysteretic switching between the flat and the hexagonally ordered state can occur as the sample temperature is varied. Surface ripples are predicted to emerge for a certain range of the parameters. DOI: 10.1103/PhysRevLett.105.145501 PACS numbers: 81.16.Rf, 05.45.a, 68.35.Ct, 79.20.Rf Bombarding a solid surface with a broad ion beam produces a remarkable variety of self-assembled nanoscale patterns [1]. The spontaneous emergence of these patterns is not just fascinating in its own right, since ion bombard- ment may prove to be an important tool in the fabrication of cutting-edge nanostructures. The first type of pattern formation to be discovered was the periodic height modulations or ‘‘ripples’’ that often develop when the nominally flat surface of a solid is subjected to oblique-incidence ion bombardment [2]. According to the widely accepted Bradley-Harper (BH) theory [3], ripples are a result of a surface instability caused by the curvature dependence of the sputter yield. In the BH theory, a solid surface subject to normal- incidence ion bombardment (NIIB) is also unstable. Since the growth rate of the unstable Fourier modes is independent of the direction of the wave vector, it was expected that NIIB would produce a rough, unstructured surface. It therefore came as a considerable surprise when experiments by Facsko et al. revealed that NIIB of the binary compound GaSb can result in the formation of nanoscale mounds or ‘‘nanodots’’ arranged in a hexagonal array of astonishing regularity [4]. Well-ordered hexagonal nanodot arrays can also be produced by oblique-incidence ion bombardment of InP if the sample is rotated while it is bombarded [5]. These observations are not just of aca- demic interest: Ion bombardment provides a fast and re- producible means of producing a nearly regular array of quantum dots on a semiconductor surface in a single process step. In the experiments of Facsko et al., the nanodot size distribution was sharply peaked and the dot arrays had short-range hexagonal order (SRHO) that extended over six or more lattice spacings. These observations strongly suggest that there is a narrow band of unstable wave- lengths, according to the modern theory of pattern forma- tion [6]. By contrast, all ripple wavelengths that exceed a critical value are unstable in the linear BH theory and in theories that add nonlinear terms to the BH equation of motion [7,8]. In this Letter, we advance a theory for the pattern for- mation that occurs when the initially flat surface of a binary compound is subjected to NIIB. We demonstrate analyti- cally that there is a narrow band of unstable wavelengths and that nanodot arrays with SRHO emerge spontaneously for a certain range of the parameters. Since the hexagonal ordering in the surface height is mirrored in the variations of composition at the surface, NIIB could be used as a tool to simultaneously achieve nanoscale patterning of the sur- face topography and composition. We also find that hyste- retic switching between the flat and hexagonally ordered states can occur as the sample temperature is varied. Finally, our theory makes the exciting prediction that normal-incidence bombardment will produce surface rip- ples for a certain range of the parameters. When an ion beam impinges on a binary compound, generally one of the components is preferentially sput- tered, yielding a surface layer of altered stoichiometry. During the formation of hexagonal arrays of nanodots by NIIB of GaSb, for example, a Ga excess of 30 at. % developed at the surface [9]. The coupling between this altered surface layer and the topography is crucial to the formation of hexagonal arrays of nanodots in our theory. The pioneering work on this coupling is due to Shenoy, Chan, and Chason [10]. We will extend the theory of Shenoy, Chan, and Chason to include the key physical effect that can lead to a narrow band of unstable wave- lengths—momentum transfer from the incident ions to atoms at the surface produces surface atomic currents [11]. In addition, nonlinear terms must be added to the linear theory of Shenoy, Chan, and Chason if hexagonal ordering is to occur. Two theories for the formation of hexagonal arrays of mounds on elemental materials have previously been intro- duced. In the theory of Facsko et al., a linear, highly non- local term is added to the usual Kuramoto-Sivashinsky PRL 105, 145501 (2010) PHYSICAL REVIEW LETTERS week ending 1 OCTOBER 2010 0031-9007= 10=105(14)=145501(4) 145501-1 Ó 2010 The American Physical Society