Controlling surface morphologies by time-delayed feedback

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 (1986)] 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, 0.33], for a significant length of time.

Figure 1

(Color online) Scaling plots of the roughness $w(L,t)$ and height-height correlation function $C\left(l\right)\equiv ⟨h(l,t)h(0,t)⟩\sim l^{\alpha }$ (insets) for system sizes $L=1024$ (red/light gray) and $L=4096$ (black) and three choices of $\lambda$: (i) $\lambda =0$ (shifted by a factor 100), (ii) $\lambda =0.1$ (shifted by a factor 10), and (iii) $\lambda =0.25$. $\nu =0.1$, $D=0.5$ for all data sets; space discretization $\Delta x=1$ and time increment $\Delta t={10}^{-3}$. The green broken lines provide guides for the eyes.

Figure 2

(Color online) Roughness and control function (inset) evolution for digital time-delayed feedback control for strong (blue/dark gray), weak (red/light gray), and zero (black) initial nonlinearity ${\lambda }_0$. The desired effective growth exponent is set at ${\beta }_0=0.29$. To provide a comparison, the straight (green) lines have slopes 0.33 (dotted), 0.29 (dashed), and 0.25 (dash-dotted). All data sets are obtained with $\nu =0.1$, $D=0.5$, and $a=0.01$.

Figure 3

(Color online) Roughness and control function (inset) evolution for differential time-delayed feedback control for strong (blue/dark gray), weak (red/light gray), and zero (black) initial nonlinearity ${\lambda }_0$. The desired effective growth exponent is set at ${\beta }_0=0.29$. To provide a comparison, the straight (green) lines have slopes 0.33 (dotted), 0.29 (dashed), and 0.25 (dash-dotted). All data sets are obtained with $\nu =0.1$, $D=0.5$, and $K=0.02$.