Effects of initial height on the steady-state persistence probability of linear growth models

Abstract

The effects of the initial height on the temporal persistence probability of steady-state height fluctuations in up-down symmetric linear models of surface growth are investigated. We study the (1+1)-dimensional Family model and the (1+1)- and (2+1)-dimensional larger curvature (LC) model. Both the Family and LC models have up-down symmetry, so the positive and negative persistence probabilities in the steady state, averaged over all values of the initial height h(0), are equal to each other. However, these two probabilities are not equal if one considers a fixed nonzero value of h(0). Plots of the positive persistence probability for negative initial height versus time exhibit power-law behavior if the magnitude of the initial height is larger than the interface width at saturation. By symmetry, the negative persistence probability for positive initial height also exhibits the same behavior. The persistence exponent that describes this power-law decay decreases as the magnitude of the initial height is increased. The dependence of the persistence probability on the initial height, the system size, and the discrete sampling time is found to exhibit scaling behavior.

Analyzing asymmetry in the Wolf-Villain model via the study of the persistence of various initial height fluctuations

“…The persistence probabilities of a specific h 0 , P ± S (h 0 ,t), are investigated when the strength of fluctuations is of interest. Previous researches [10,12] have found that the function of P ± S (h 0 ,t) are not power-law decay when fluctuations are very weak compared to the saturation width i.e. |h 0 |≪W sat .…”

“…For all discrete growth models, P ± S (t)∝ t -θ ± S , where θ ± S is the persistence exponent which indicates how long the fluctuations last. In up-down symmetric models, θ + S = θ -S [9,[12][13]18]; while in asymmetric models, θ + S ≠ θ -S [9][10]14,18].…”

“…From the figures, we can see qualitatively that the two plots are slightly different statistically implying that the WV model may have weak up-down asymmetry. It is unlike other prominent toy models such as the Das Sarma-Tamborenea (DT) model [7][8] which shows clear strong up-down asymmetric properties [9][10] and the Family model [11] which demonstrates definite symmetry properties [9,12]. One common method to study symmetry quantitatively in film morphologies is by measuring the skewness.…”

“…Another method to analyse this asymmetry is the study of persistence probabilities, which is the topic of this work. Persistence probabilities of height fluctuations [9][10][12][13][14][15][16][17][18] are probabilities that indicate how well the height fluctuations can maintain their initial properties. There are two types of persistence probabilities, which are the positive and the negative one.…”

“…The positive (negative) persistence probability is the probability that the height fluctuations h remain larger (smaller) than the initial height fluctuations h 0 . It has been shown that the positive and negative persistent probabilities are the same in symmetric growth models [9,[12][13]18], but they differ in growth model without the up-down symmetry [9][10]14,18].…”