Das Sarma Tamborenea Research Articles

Persistence probabilities of height fluctuation in thin film growth of the Das Sarma–Tamborenea model

The dependence of the steady-state persistence probability on initial height fluctuation $$\left( {h_{0} } \right)$$ in the (1 + 1)-dimensional Das Sarma–Tamborenea (DT) model, which is up–down asymmetric, is investigated. Our results show that the positive and negative persistence probabilities for fixed $$h_{0}$$ are not equal to each other. The positive persistence probability for negative initial height $$\left( {P_{ + } \left( { - \left| {h_{0} } \right|} \right)} \right)$$ as well as the negative persistence probability for positive initial height $$\left( {P_{ - } \left( { + \left| {h_{0} } \right|} \right)} \right)$$ shows power-law decay with time if $$\left| {h_{0} } \right|$$ is larger than the saturated interface width $$\left( {W_{\text{sat}} } \right)$$ of the model. The persistence exponent decreases when $$\left| {h_{0} } \right|$$ increases. The (2 + 1)-dimensional DT model is studied for comparison. We also find a relation that shows how the persistence probability scales with the initial height, the system size and the discrete sampling time. The relation works for models with and without up–down symmetry.

Effects of surface diffusion length on steady-state persistence probabilities

The dependence of the steady-state persistence probabilities (Ps) and exponents (θs) on surface diffusion length (l) for four discrete growth models is investigated. The persistence exponents which describe the decay of the persistence probabilities, the probabilities of the average of all initial height (h0), are increased as l is increased for all models. The results of one-dimensional Family ((1+1)-Family) and Das Sarma-Tamborenea ((1+1)-DT) models with kinetically rough film surface show the decrease of the growth exponent (β) with l. The l>1 results preserve the relation . In contrast, β is observed to increase with l for the two-dimensional larger curvature ((2+1)-LC) and Wolf-Villain ((2+1)-WV) models with mounded morphology. Our results show that the relation is not valid in l>1 cases in models with mounded surfaces. The persistence probabilities of a specific value of initial height (Ps(h0)) for l>1 are found to behave differently between mounded and kinetically rough models.

Schramm–Loewner evolution theory of the asymptotic behaviors of (2+1)-dimensional Das Sarma–Tamborenea model

A mass of theoretical analysis and numerical calculation on (2+1)-dimensional Das Sarma–Tamborenea model have been carried out over the years. A problem about asymptotical behaviors that this model belongs to which universality classes is still interesting and controversial. The Schramm–Loewner evolution theory is an innovative point of view to describe the conformal invariance of contour lines of saturated rough surfaces. In this paper, we used SLE theory to analyze the conformal invariance of the (2+1)-dimensional DT model. A noise reduction technique was also utilized to get better scaling behavior. The results show that the contour lines of (2+1)-dimensional DT model satisfy the property of conformal invariance and belongs to κ=4 universality class. The relation df=1+κ∕8 is also satisfied for the fractal dimension and diffusion coefficient of the contour lines.

Effects of a kinetic barrier on limited-mobility interface growth models.

The role played by a kinetic barrier originated by out-of-plane step edge diffusion, introduced by Leal etal. [J. Phys.: Condens. Matter 23, 292201 (2011)JCOMEL0953-898410.1088/0953-8984/23/29/292201], is investigated in the Wolf-Villain and Das Sarma-Tamborenea models with short-range diffusion. Using large-scale simulations, we observe that this barrier is sufficient to produce growth instability, forming quasiregular mounds in one and two dimensions. The characteristic surface length saturates quickly indicating a uncorrelated growth of the three-dimensional structures, which is also confirmed by a growth exponent β=1/2. The out-of-plane particle current shows a large reduction of the downward flux in the presence of the kinetic barrier enhancing, consequently, the net upward diffusion and the formation of three-dimensional self-assembled structures.

Local roughness exponent in the nonlinear molecular-beam-epitaxy universality class in one dimension.

We report local roughness exponents, α_{loc}, for three interface growth models in one dimension which are believed to belong to the nonlinear molecular-beam-epitaxy (nMBE) universality class represented by the Villain-Lais-Das Sarma (VLDS) stochastic equation. We applied an optimum detrended fluctuation analysis (ODFA) [Luis et al., Phys. Rev. E 95, 042801 (2017)2470-004510.1103/PhysRevE.95.042801] and compared the outcomes with standard detrending methods. We observe in all investigated models that ODFA outperforms the standard methods providing exponents in the narrow interval α_{loc}^{}∈[0.96,0.98] quantitatively consistent with two-loop renormalization group predictions for the VLDS equation. In particular, these exponent values are calculated for the Clarke-Vvdensky and Das Sarma-Tamborenea models characterized by very strong corrections to the scaling, for which large deviations of these values had been reported. Our results strongly support the absence of anomalous scaling in the nMBE universality class and the existence of corrections in the form α_{loc}^{}=1-ε of the one-loop renormalization group analysis of the VLDS equation.

Skewness and kurtosis of height distribution of thin films simulated by larger curvature model with noise reduction techniques

Time varying skewness (S) and kurtosis (Q) of height distribution of (2+1)-dimensional larger curvature (LC) model with and without noise reduction techniques (NRTs) are investigated in both transient and steady state regimes. In this work, effects of the multiple hit NRT (m>1 NRT) and the long surface diffusion length NRT (ℓ>1 NRT) on the surface morphologies and characteristics of S and Q are studied. In the early growth time, plots of S and Q versus time of the m>1 morphologies show pronounced oscillation indicating the layer by layer growth. Our results show that S=0 and Q<0 at every half layer while S=0 and Q>0 at every complete layer. The results are confirmed by the same plots of the results from the Das Sarma–Tamborenea (DT) model. The ℓ>1 LC model, on the other hand, has no evidence of the layer by layer growth mode due to the rapidly damped oscillation of S and Q. In the steady state, the m>1 and ℓ>1 NRTs affect weakly on the values of S and Q and the mounded morphologies of the film. This lead to the evidence of universality of S and Q in the steady state of the LC models with various m and ℓ. The finite size effect on the values of S and Q is found to be very weak in the LC model. By extrapolating to L→∞, we obtain SL→∞≈0.05 and QL→∞≈−0.62 which are in agreement with the NRTs results.

Roughness distribution of multiple hit and long surface diffusion length noise reduced discrete growth models

Abstract Conventionally, the universality class of a discrete growth model is identified via the scaling of interface width. This method requires large-scale simulations to minimize finite-size effects on the results. The multiple hit noise reduction techniques ( m > 1 NRT) and the long surface diffusion length noise reduction techniques ( l > 1 NRT) have been used to promote the asymptotic behaviors of the growth models. Lately, an alternative method involving comparison of roughness distribution in the steady state has been proposed. In this work, the roughness distribution of the (2 + 1)-dimensional Das Sarma–Tamborenea (DT), Wolf–Villain (WV), and Larger Curvature (LC) models, with and without NRTs, are calculated in order to investigate effects of the NRTs on the roughness distribution. Additionally, effective growth exponents of the noise reduced (2 + 1)-dimensional DT, WV and LC models are also calculated. Our results indicate that the NRTs affect the interface width both in the growth and the saturation regimes. In the steady state, the NRTs do not seem to have any impact on the roughness distribution of the DT model, but it significantly changes the roughness distribution of the WV and LC models to the normal distribution curves.

Large-scale numerical study on the dynamic scaling behavior of Das Sarma-Tamborenea model by employing noise reduction technique

By employing the noise reduction technique, extensive kinetic Monte Carlo simulations are presented for the Das Sarma-Tamborenea model in (1 + 1) and (2 + 1) dimensions on large length and long time scale. Surface width and height-height correlation function are calculated to estimate the global and local dynamic scaling behaviors in this model. Asymptotic dynamic scaling behaviors have been found. Normal scaling has been shown in (1 + 1) dimensions, and the values of global scaling exponents are below the one-loop renormalization calculation for the Lai-Das Sarma-Villain theory, which confirms the existence of higher-order corrections proposed by Janssen. Further, we have found that the DT model in (2 + 1) dimensions belongs to the Edwards-Wilkinson dynamic universality, in contrast to the (1 + 1)-dimensional universality class of this model. Our findings can explain the widespread discrepancies of previous reports for exponents of the Das Sarma-Tamborenea model both in (1 + 1) and (2 + 1) dimensions.

Healing time for the growth of thin films on patterned substrates

The healing times for the growth of thin films on patterned substrates are studied using simulations of two discrete models of surface growth: the Family model and the Das Sarma–Tamborenea (DT) model. The healing time, defined as the time at which the characteristics of the growing interface are “healed” to those obtained in growth on a flat substrate, is determined via the study of the nearest-neighbor height difference correlation function. Two different initial patterns are considered in this work: a relatively smooth tent-shaped triangular substrate and an atomically rough substrate with single-site pillars or grooves. We find that the healing time of the Family and DT models on a L×L triangular substrate is proportional to Lz, where z is the dynamical exponent of the models. For the Family model, we also analyze theoretically, using a continuum description based on the linear Edwards–Wilkinson equation, the time evolution of the nearest-neighbor height difference correlation function in this system. The correlation functions obtained from continuum theory and simulation are found to be consistent with each other for the relatively smooth triangular substrate. For substrates with periodic and random distributions of pillars or grooves of varying size, the healing time is found to increase linearly with the height (depth) of pillars (grooves). We show explicitly that the simulation data for the Family model grown on a substrate with pillars or grooves do not agree with results of a calculation based on the continuum Edwards–Wilkinson equation. This result implies that a continuum description does not work when the initial pattern is atomically rough. The observed dependence of the healing time on the substrate size and the initial height (depth) of pillars (grooves) can be understood from the details of the diffusion rule of the atomistic model. The healing time of both models for pillars is larger than that for grooves with depth equal to the height of the pillars. The calculated healing time for both Family and DT models is found to depend on how the pillars and grooves are distributed over the substrate.

MODELING OF THIN FILM GROWTH ON A TILTED MISCUT SUBSTRATE: STATISTICAL PROPERTIES AND THE OPTIMUM GROWTH CONDITIONS

Most studies of thin film growth simulations are performed on flat substrates. However, in reality, a substrate is usually miscut leading to a vicinal surface with a small tilt. The goal of this work is to study effects of an initial configuration of a miscut substrate on the grown film. The Das Sarma–Tamborenea model with modified diffusion rules is used for the simulations. The modification is done to allow variation in the surface diffusion length and mobility of adatoms. The results show that the optimum conditions that lead to step-flow growth are long diffusion length and small step height.

Growth instability due to lattice-induced topological currents in limited-mobility epitaxial growth models

The energetically driven Ehrlich-Schwoebel barrier had been generally accepted as the primary cause of the growth instability in the form of quasiregular moundlike structures observed on the surface of thin film grown via molecular-beam epitaxy (MBE) technique. Recently the second mechanism of mound formation was proposed in terms of a topologically induced flux of particles originating from the line tension of the step edges which form the contour lines around a mound. Through large-scale simulations of MBE growth on a variety of crystalline lattice planes using limited-mobility, solid-on-solid models introduced by Wolf-Villain and Das Sarma-Tamborenea in 2+1 dimensions, we show that there exists a topological uphill particle current with strong dependence on specific lattice crystalline structure. Without any energetically induced barriers, our simulations produce spectacular mounds very similar, in some cases, to what have been observed in many recent MBE experiments. On a lattice where these currents cease to exist, the surface appears to be scale invariant, statistically rough as predicted by the conventional continuum growth equation.

Numerical study of anomalous dynamic scaling behaviour of (1+1)-dimensional Das Sarma–Tamborenea model

In order to discuss the finite-size effect and the anomalous dynamic scaling behaviour of Das Sarma–Tamborenea growth model, the (1+1)-dimensional Das Sarma–Tamborenea model is simulated on a large length scale by using the kinetic Monte–Carlo method. In the simulation, noise reduction technique is used in order to eliminate the crossover effect. Our results show that due to the existence of the finite-size effect, the effective global roughness exponent of the (1+1)-dimensional Das Sarma–Tamborenea model systematically decreases with system size L increasing when L > 256. This finding proves the conjecture by Aarao Reis[Aarao Reis F D A 2004 Phys. Rev. E 70 031607]. In addition, our simulation results also show that the Das Sarma–Tamborenea model in 1+1 dimensions indeed exhibits intrinsic anomalous scaling behaviour.

Scaling of local interface width of statistical growth models

We discuss the methods to calculate the roughness exponent α and the dynamic exponent z from the scaling properties of the local roughness, which is frequently used in the analysis of experimental data. Through numerical simulations, we studied the Family, the restricted solid-on-solid, the Das Sarma–Tamborenea (DT) and the Wolf–Villain (WV) models in one- and two-dimensional substrates, in order to compare different methods to obtain those exponents. The scaling at small length scales do not give reliable estimates of α, suggesting that the usual methods to estimate that exponent from experimental data may provide misleading conclusions concerning the universality classes of the growth processes. On the other hand, we propose a more efficient method to calculate the dynamic exponent z, based on the scaling of characteristic correlation lengths, which gives estimates in good agreement with the expected universality classes and indicates expected crossover behavior. Our results also provide evidence of Edwards–Wilkinson asymptotic behavior for the DT and the WV models in two-dimensional substrates.

Rough growth fronts of evaporated gold films, compared with self-affine and mound growth models

The power spectra, S( q, t), of evaporated gold films at room temperature are quantitatively compared with the Das Sarma Tamborenea model. The experimental curves fit the theory. However, the noise parameter, deduced from the fitting, disagrees with the predicted value. The two-dimensional (2D) experimental height–height correlation functions of scanning tunnelling microscope (STM) images present a hexagonal anisotropy. This fact agrees with the existence of a mound structure in the three-dimensional STM images. In addition the 2D Fourier transform module shows the predicted maxima for a mound model.

Elementary local configurations, microscopic rules and Langevin equations for two models of molecular-beam epitaxy

Stochastic Langevin equations for the Wolf-Villain (WV) and Das Sarma-Tamborenea (DT) models are derived using the alternative method recently developed by Costanza (1997 Phys. Rev. E 55 6501) which avoids the complications arising in the calculation of the first moment of the transition rate required in the master equation approach. The calculations are compared with those recently published by Zhi-Feng Huang et al (1996 Phys. Rev. E 54 5935) obtaining the same results, as is expected. The microscopic rules are derived from the set of 243 elementary local configurations needed for the description of these two models and after using simple summation rules these were reduced to 16 for the DT model and to 25 for the WV model. The number of microscopic rules needed for the description of the present models are considerably larger than the previously solved models.

Growth equations for the Wolf-Villain and Das Sarma-Tamborenea models of molecular-beam epitaxy.

We derive the growth equations for the Wolf-Villain and Das Sarma- Tamborenea models based on the master-equation method. The Wolf-Villain model is shown to obey the conserved growth equation $\frac{\ensuremath{\partial}h}{\ensuremath{\partial}t}=\ensuremath{-}{\ensuremath{\nu}}_{4}{\ensuremath{\nabla}}^{4}h+{\ensuremath{\lambda}}_{22}{\ensuremath{\nabla}}^{2}{(\ensuremath{\nabla}h)}^{2}+\ensuremath{\Sigma}{n=1}^{\ensuremath{\infty}}{\ensuremath{\lambda}}_{12n+1}\ensuremath{\nabla}\ifmmode\cdot\else\textperiodcentered\fi{}{(\ensuremath{\nabla}h)}^{2n+1}+F+\ensuremath{\eta}$, which is expected to exhibit the scaling behavior of the Edwards-Wilkinson universality class. We find that the Das Sarma- Tamborenea model is governed by the Villain-Lai-Das Sarma equation $\frac{\ensuremath{\partial}h}{\ensuremath{\partial}t}=\ensuremath{-}{\ensuremath{\nu}}_{4}{\ensuremath{\nabla}}^{4}h+{\ensuremath{\lambda}}_{22}{\ensuremath{\nabla}}^{2}{(\ensuremath{\nabla}h)}^{2}+F+\ensuremath{\eta}$ in contrast to former results. The physical origin of the difference between these two similar models is also discussed.