Renormalization of stochastic lattice models: epitaxial surfaces

Modeling of nonequilibrium surface growth by a limited-mobility model with distributed diffusion length

Kinetic Monte Carlo (KMC) simulations are a well-established numerical tool to investigate the time-dependent surface morphology in molecular beam epitaxy experiments. In parallel, simplified approaches such as limited mobility (LM) models characterized by a fixed diffusion length have been studied. Here we investigate an extended LM model to gain deeper insight into the role of diffusional processes concerning the growth morphology. Our model is based on the stochastic transition rules of the Das Sarma-Tamborena model but differs from the latter via a variable diffusion length. A first guess for this length can be extracted from the saturation value of the mean-squared displacement calculated from short KMC simulations. Comparing the resulting surface morphologies in the sub- and multilayer growth regime to those obtained from KMC simulations, we find deviations which can be cured by adding fluctuations to the diffusion length. This mimics the stochastic nature of particle diffusion on a substrate, an aspect which is usually neglected in LM models. We propose to add fluctuations to the diffusion length by choosing this quantity for each adsorbed particle from a Gaussian distribution, where the variance of the distribution serves as a fitting parameter. We show that the diffusional fluctuations have a huge impact on cluster properties during submonolayer growth as well as on the surface profile in the high coverage regime. The analysis of the surface morphologies on one- and two-dimensional substrates during sub- and multilayer growth shows that the LM model can produce structures that are indistinguishable to the ones from KMC simulations at arbitrary growth conditions.

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 et al. [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.

Optimal detrended fluctuation analysis as a tool for the determination of the roughness exponent of the mounded surfaces

We present an optimal detrended fluctuation analysis (DFA) and apply it to evaluate the local roughness exponent in nonequilibrium surface growth models with mounded morphology. Our method consists in analyzing the height fluctuations computing the shortest distance of each point of the profile to a detrending curve that fits the surface within the investigated interval. We compare the optimal DFA (ODFA) with both the standard DFA and nondetrended analysis. We validate the ODFA method considering a one-dimensional model in the Kardar-Parisi-Zhang universality class starting from a mounded initial condition. We applied the methods to the Clarke-Vvedensky (CV) model in 2+1 dimensions with thermally activated surface diffusion and absence of step barriers. It is expected that this model belongs to the nonlinear molecular beam epitaxy (nMBE) universality class. However, an explicit observation of the roughness exponent in agreement with the nMBE class was still missing. The effective roughness exponent obtained with ODFA agrees with the value expected for the nMBE class, whereas using the other methods it does not agree. We also characterize the transient anomalous scaling of the CV model and obtained that the corresponding exponent is in agreement with the value reported for other nMBE models with weaker corrections to the scaling.

Transition from compact to porous films in deposition with temperature-activated diffusion

We study a thin-film growth model with temperature activated diffusion of adsorbed particles, allowing for the formation of overhangs and pores, but without detachment of adatoms or clusters from the deposit. Simulations in one-dimensional substrates are performed for several values of the diffusion-to-deposition ratio R of adatoms with a single bond and of the detachment probability ε per additional nearest neighbor, respectively, with activation energies are E(s) and E(b). If R and ε independently vary, regimes of low and high porosity are separated at 0.075≤ε(c)≤0.09, with vanishingly small porosity below that point and finite porosity for larger ε. Alternatively, for fixed values of E(s) and E(b) and varying temperature, the porosity has a minimum at T(c), and a nontrivial regime in which it increases with temperature is observed above that point. This is related to the large mobility of adatoms, resembling features of equilibrium surface roughening. In this high-temperature region, the deposit has the structure of a critical percolation cluster due to the nondesorption. The pores are regions enclosed by blobs of the corresponding percolating backbone, thus the distribution of pore size s is expected to scale as s(-τ̃) with τ̃≈1.45, in reasonable agreement with numerical estimates. Roughening of the outer interface of the deposits suggests Villain-Lai-Das Sarma scaling below the transition. Above the transition, the roughness exponent α≈0.35 is consistent with the percolation backbone structure via the relation α=2-d(B), where d(B) is the backbone fractal dimension.

Asymptotic dynamic scaling behavior of the (1+1)-dimensional Wolf-Villain model

Extensive kinetic Monte Carlo simulations are presented for the Wolf-Villain model in (1+1) dimensions. Asymptotic dynamic scaling is found for lattice sizes L≥2048. The exponents obtained from our simulations, α=0.50±0.02 and β=0.25±0.02, are in excellent agreement with the exact values α=1/2 and β=1/4 for the one-dimensional Edwards-Wilkinson equation. Our findings explain the widespread discrepancies of previous reports for exponents of the Wolf-Villain model in (1+1) dimensions, and the results are also consistent with the theoretical predictions of López et al. [J. M. López, M. Castro, and R. Gallego, Phys. Rev. Lett. 94, 166103 (2005)].

Langevin equations for competitive growth models

Langevin equations for several competitive growth models in one dimension are derived. For models with crossover from random deposition (RD) to some correlated deposition (CD) dynamics, with small probability p of CD, the surface tension ν and the nonlinear coefficient λ of the associated equations have linear dependence on p due solely to this random choice. However, they also depend on the regularized step functions present in the analytical representations of the CD, whose expansion coefficients scale with p according to the divergence of local height differences when p→0. The superposition of those scaling factors gives ν~p(2) for random deposition with surface relaxation (RDSR) as the CD, and ν~p, λ~p(3/2) for ballistic deposition (BD) as the CD, in agreement with simulation and other scaling approaches. For bidisperse ballistic deposition (BBD), the same scaling of RD-BD model is found. The Langevin equation for the model with competing RDSR and BD, with probability p for the latter, is also constructed. It shows linear p dependence of λ, while the quadratic dependence observed in previous simulations is explained by an additional crossover before the asymptotic regime. The results highlight the relevance of scaling of the coefficients of step function expansions in systems with steep surfaces, which is responsible for noninteger exponents in some p-dependent stochastic equations, and the importance of the physical correspondence of aggregation rules and equation coefficients.

Large-scale simulations of ballistic deposition: the approach to asymptotic scaling

Extensive kinetic Monte Carlo simulations are presented for ballistic deposition (BD) in (1+1) dimensions. Asymptotic scaling is found only for lattice sizes L≳2¹². Such a large system size for the onset of scaling explains the widespread discrepancies of previous reports for exponents of BD in one and likely higher dimensions. The exponents obtained from our simulations, α=0.499±0.004 and β=0.336±0.004, are in excellent agreement with the exact values α=½ and β=1/3 for the one-dimensional Kardar-Parisi-Zhang equation. Our findings make possible a more informed exploration of exponents for BD in higher dimensions, accurate estimates of which have proven to be elusive.

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.

Dynamic scaling in thin-film growth with irreversible step-edge attachment

We study dynamic scaling in a model with collective diffusion (CD) of isolated atoms in terraces and irreversible aggregation at step edges. Simulations are performed in two-dimensional substrates with several diffusion to deposition ratios R identical with D/F. Data collapse of scaled roughness distributions confirms that this model is in the class of the fourth-order nonlinear growth equation by Villain, Lai, and Das Sarma (VLDS) with negligible finite-size effects, while estimates of scaling exponents show some discrepancies. This result is consistent with the prediction of a recent renormalization group approach and improves previous numerical works on related models. The roughness follows dynamic scaling as W=Lalpha/R1/2f(xi/L), with correlation length xi=(Rt)1/z, where z is the dynamic exponent. We also propose a limited mobility (LM) model where the incident atom executes up to G steps before a new atom is adsorbed, and irreversibly aggregates at step edges. This model is also shown to belong to the VLDS class. The size of the plateaus in the film surface increases as G1/2 and the lateral correlation scales as G1/2t1/z. The time evolution of the roughness reproduces that of the CD model if an equivalent parameter G approximately R2/z is chosen. This suggests the possibility of using LM models with tunable diffusion length to simulate processes with simultaneous diffusion of many atoms. A scaling approach is used to justify exponent values and dynamic relations for both models, including the significant decrease of surface roughness as R or G increases.

Transient regimes and crossover for epitaxial surfaces

We apply a formalism for deriving stochastic continuum equations associated with lattice models to obtain equations governing the transient regimes of epitaxial growth for various experimental scenarios and growth conditions. The first step of our methodology is the systematic transformation of the lattice model into a regularized stochastic equation of motion that provides initial conditions for differential renormalization-group (RG) equations for the coefficients in the regularized equation. The solutions of the RG equations then yield trajectories that describe the original model from the transient regimes, which are of primary experimental interest, to the eventual crossover to the asymptotically stable fixed point. We first consider regimes defined by the relative magnitude of deposition noise and diffusion noise. If the diffusion noise dominates, then the early stages of growth are described by the Mullins-Herring (MH) equation with conservative noise. This is the classic regime of molecular-beam epitaxy. If the diffusion and deposition noise are of comparable magnitude, the transient equation is the MH equation with nonconservative noise. This behavior has been observed in a recent report on the growth of aluminum on silicone oil surfaces [Z.-N. Fang, Thin Solid Films 517, 3408 (2009)]. Finally, the regime where deposition noise dominates over diffusion noise has been observed in computer simulations, but does not appear to have any direct experimental relevance. For initial conditions that consist of a flat surface, the Villain-Lai-Das Sarma (VLDS) equation with nonconservative noise is not appropriate for any transient regime. If, however, the initial surface is corrugated, the relative magnitudes of terms can be altered to the point where the VLDS equation with conservative noise does indeed describe transient growth. This is consistent with the experimental analysis of growth on patterned surfaces [H.-C. Kan, Phys. Rev. Lett. 92, 146101 (2004); T. Tadayyon-Eslami, Phys. Rev. Lett. 97, 126101 (2006)].

Investigation of film surface roughness and porosity dependence on lattice size in a porous thin film deposition process

The dependence of film surface roughness and porosity on lattice size in a porous thin film deposition process is studied via kinetic Monte Carlo simulations on a triangular lattice. For sufficiently large lattice size the steady-state value of the expected film porosity has a weak dependence on the lattice size and the steady-state value of the expected surface roughness square varies linearly with lattice size. An analysis of the film morphology based on a stochastic partial differential equation description of the film surface morphology supports and explains the findings of the numerical simulations.

Geometric principles of surface growth

I introduce a new equation describing epitaxial growth processes. This equation is derived from a simple variational geometric principle and it has a straightforward interpretation in terms of continuum and microscopic physics. It is also able to reproduce the critical behavior already observed, mound formation and mass conservation, but however does not fit a divergence form as the most commonly spread models of conserved surface growth. This formulation allows me to connect the results of the dynamic renormalization group analysis with intuitive geometric principles, whose generic character may well allow them to describe surface growth and other phenomena in different areas of physics.