Controlling surface morphologies by time-delayed feedback

Abstract

We propose a method to control the roughness of a growing surface via a time-delayed feedback scheme. The method is very general and can be applied to a wide range of nonequilibrium growth phenomena, from solid-state epitaxy to tumor growth. Possible experimental realizations are suggested. As an illustration, we consider the Kardar-Parisi-Zhang equation ͓Phys. Rev. Lett. 56, 889 ͑1986͔͒ in 1 + 1 dimensions and show that the effective growth exponent of the surface width can be stabilized at any desired value in the interval ͓0.25, 0.33͔, for a significant length of time. DOI: 10.1103/PhysRevB.75.233414 PACS number͑s͒: 68.35.Rh, 05.10.Gg, 02.30.Ks, 02.60.Cb The control of unstable states in chaotic or patternforming nonlinear dynamical systems has attracted much interest recently. 1,2 Time-delayed feedback control 3 has been especially successful in stabilizing a variety of dynamic and spatial structures, including noise-induced oscillations and patterns found, e.g., in semiconductor nanostructures. [4][5][6][7] Here, we propose to apply these control techniques to a completely new class of dynamical phenomena, namely, farfrom-equilibrium surface growth. [8][9][10] The goal is to stabilize desired surface characteristics, such as spatiotemporal height-height correlations or the surface roughness, during the growth process. Even if such control can only be sustained in a finite window of time, its experimental potential is undiminished since the growth process can simply be terminated when the desired characteristics have been achieved, thanks to today's precise in situ characterization capabilities. Moreover, paradigmatic growth models, such as the KardarParisi-Zhang ͑KPZ͒ equation, 11 find applications in many diverse areas of science, e.g., thin film growth, 12-15 fluctuating hydrodynamics, 16 driven diffusive systems, [17][18][19] tumor growth in biophysics, [20][21][22] propagating fire fronts, 23 and econophysics. 24 Therefore, broad implications can be expected if methods from control theory can be successfully implemented in this vast context.In this Brief Report, we provide an exploration of these ideas. We choose the most promising type of control, timedelayed feedback, and study its effects on the KPZ equation. 11 Specifically, we attempt to control the effective dynamic growth exponent ␤ associated with the roughness of the growing surface. By implementing two realizations of the control scheme, we will see below that we can, indeed, stabilize ␤ in a range of values between the two universal limits, 1 / 4 and 1 / 3, over at least one to two decades in time. In the following, we will use the language of surface growth, but our findings are just as relevant in the context of all other applications of the KPZ equation. For instance, in recent work on tumor growth, 20 it has been shown that an efficient method to influence the proliferation of tumor cells at the border is coupled to the growth exponents and the universality class of growth. The authors suggest and establish a therapy which rests on th...