Kinetic Roughening of Ion-Sputtered Pd(001) Surface: Beyond the Kuramoto-Sivashinsky Model

Abstract

We investigate the kinetic roughening of Ar + ion-sputtered Pd(001) surface both experimentally and theoretically. In situ real-time x-ray reflectivity and in situ scanning tunneling microscopy show that nanoscale adatom islands form and grow with increasing sputter time t. Surface roughness, W (t), and lateral correlation length, ξ(t), follows the scaling laws, W (t) ∼ t β and ξ(t) ∼ t 1/z with the exponents β ≃ 0.20 and 1/z ≃ 0.20, for ion beam energy ε = 0.5 keV, which is inconsistent with the prediction of the Kuramoto-Sivashinsky (KS) model. We thereby extend the KS model by applying the Sigmund theory of sputter erosion to the higher order, O(∇ 4 , h 2 ), where h is surface height, and derive a new term of the form ∇ 2 (∇h) 2 which plays an indispensable role in describing the observed morphological evolution of the sputtered surface. 64.60.Cn, 79.20.Rf Recently, the observation of ordered nanostructures such as ripples and two-dimensional patterns on ion-sputtered surfaces has attracted much attention due to the demonstration of the possibility of fabrication of ordered nanoscale structures in a relatively easy and affordable way [1,2,3,4,5,6, 7]. Such experimental results have motivated extensive theoretical investigations aiming to understand the mechanism of the morphological evolution of ion-sputtered surfaces. A linear model, proposed by Bradley and Harper (BH) [8], has been successful in predicting the formation of the ripple structure. The wavelength, orientation and amplitude of the ripples can be predicted in terms of experimental parameters such as the incident angle of the ion beam and substrate temperature [9]. The BH theory, however, fails to explain a number of experimental observations such as the saturation of the ripple amplitude [10,11,12], or the appearance of kinetic roughening [13,14]. To remedy such shortcomings, the noisy Kuramoto-Sivashinsky (KS) equation [15,16] was introduced based on the Sigmund theory of sputter erosion [17]. In addition to the linear terms of the BH model, it contains a nonlinear term proportional to (∇h) 2 , known as the Kardar-ParisiZhang (KPZ) term [18], where h is surface height. Due to the nonlinear term, the surface roughness (or the ripple amplitude), which was growing exponentially with increasing sputter time in the linear model, changes to the type following a power law and eventually saturates to a constant value [19]. Although the KS model seems to be successful in offering insights for understanding the nonlinear behavior of sputtereroded surfaces, it has not been convincing yet because the detailed properties of the kinetic roughening predicted by the KS equation have not been fully tested experimentally. Kinetic roughening behavior is described by the scaling theory [20].The surface roughness of the sam-The roughness and growth exponents α and β are related via scaling relations to give the dynamic exponent, z = α/β, which determines the scaling of the saturation time with the system size L. Below, we will deal with the inverse dynamic expon...

Self-Organized Surface Nanopatterning by Ion Beam Sputtering

“…Indeed, such a higher order generalization has been performed [94]. However, the expressions of λ 1 and λ 2 in terms of Sigmund's parameters are such that the coefficients of the two nonlinearities in Eq.…”

Self-organized nanopatterning of silicon surfaces by ion beam sputtering

“…(9) [224,225]. This means that the h kcanc ðtÞ Fourier mode of the height which corresponds to wave-vector value k ¼ k canc ¼ ðz ð1Þ =z ð2Þ Þ 1=2 is linearly unstable; moreover, nonlinear terms cancel each other exactly for its equation of motion, so that its amplitude blows up exponentially, as a result of which the continuum description breaks down.…”

“…Actually, as has been recently discussed, [212,211], formula (4) relating the local surface velocity with collisional energy deposition is morphologically unstable at all length scales. Thus, mathematical features associated with its various linear and non-linear truncations, such as negative values of D eff [25,26], or cancellation modes [223], are probably nothing else than manifestations of this physical fact. All this indicates the need for a description that improves the present models, both from the physical and from the mathematical points of view.…”

“…Another possibility to go beyond the KS equation is to carry out the BH-type expansion to still a higher non-linear order, as indeed was done in [223], obtaining an interface equation that for normal incidence reads…”

Self-organized chains of nanodots induced by an off-normal incident beam

“…This term leads to a finite saturated surface ripple amplitude after a long time of evolution [16]. To account for kinetic roughening, the model was further improved by adding a conserved KPZ term, ∇ 2 (∇ h ) 2 , a higher-order term in Sigmund's theory [17]. These models necessarily generate ripples under off-normal bombardment because of anisotropic sputtering.…”