Da-Peng Hao

Dynamic scaling behaviors of the discrete growth models on fractal substrates

The dynamic scaling behaviors of the Family model and the Etching model on different fractal substrates are studied by means of Monte Carlo simulations, so as to discuss the microscopic mechanisms influencing the dynamic behavior of growth interfaces by changing the structure of the substrates. The Sierpinski arrowhead, crab lattice and dual Sierpinski gasket are employed as the substrates of the growth. These substrates have same fractal dimensions (df???1.585), but with different morphologies. It is shown that the structure of the substrates can affect the dynamic scaling properties of the surfaces and interfaces. Although the standard Family?Vicsek scaling is still satisfied in describing the scaling behavior of the growth on fractal substrates, the original continuum equations are invalid. The dynamic behavior of the Family model satisfies the fractional Edwards?Wilkinson equation introduced by Lee and Kim, and the dynamic behavior of the Etching model implies that ??+?z?>?2, which is different from the analytical results of the Kardar?Parisi?Zhang equation. The fractal character of the substrate also affects the lateral behavior of the Etching growth. Interestingly, the same fractal dimensions lead to different scaling exponents. The scaling exponents of the growth models on fractal substrates are determined by not only the fractal dimensions of the substrates, but also the spectral dimensions. Fortunately, it seems that the fractal dimension and the spectral dimension are sufficient to determine the scaling exponents of the growth model on fractal substrates.

Dynamics of surface roughening in the space-fractional Kardar–Parisi–Zhang growth: numerical results

We numerically study the (1+1)-dimensional space-fractional Kardar–Parisi–Zhang (SFKPZ) equation describing surface roughening in the presence of anomalous diffusion based on the Riesz-type fractional derivative. To suppress the instability in the SFKPZ growth, the nonlinear term is replaced by an exponentially decreasing function. The dynamic scaling exponents in the different growth regions are numerically obtained. The results are consistent with the analytical results using the self-consistent expansion approach. We find that the SFKPZ model exhibits scaling properties that have weak dependence on the fractional orders. Our results also show that both finite-time and finite-size effects in the SFKPZ system are very weak in comparison with the normal KPZ equation.

Dynamic scaling behaviors of the Etching model on fractal substrates

The dynamic scaling behaviors of the Etching model on fractal substrates are studied by means of Kinetic Monte Carlo simulations. Surface width and distributions of relative extremal height in the saturated surfaces are calculated respectively. Our results indicate that the structure of the substrates can affect the dynamic scaling properties of the surfaces and interfaces. Specially, the saturated properties of the surface, i.e. the roughness exponents α, are mainly determined by the dynamic exponents of random walk zRW on fractal lattices. In addition, it is shown that the relative extremal height of the saturated surfaces can be fitted by Asym2Sig (asymmetric double sigmoidal) distribution.

A fractal langevin equation describing the kinetic roughening growth on fractal lattices

A fractal Langevin equation ( is the random walk exponent on the lattice) is proposed to describe the kinetic roughening growth on fractal substrates. The scaling relation can be obtained. Kinetic Monte Carlo simulations are carried out for Restricted Solid-on-solid model and Etching model growing on various fractal substrates, and the results prove this scaling relation.

CONFORMAL INVARIANCE OF CONTOUR LINES ON THE (2 + 1)-DIMENSIONAL RESTRICTED SOLID-ON-SOLID SURFACE

The contour lines of the saturated surface of the (2 + 1)-dimensional restricted solid-on-solid (RSOS) growth model are investigated by numerical method. It is shown that the calculated contour lines are conformal invariant curves with fractal dimension df = 1.34, and they belong to the universality class at large-scale limit, called the Schramm–Loewner evolution with diffusivity κ = 4. This is identical to the value obtained from the inverse cascade of surface quasigeostrophic (SQG) turbulence [Phys. Rev. Lett.98 (2007) 024501]. We also found that the measured fractal dimensions of contours on the (2 + 1)-dimensional RSOS saturated surfaces do not coincide well with that of SLE4 df = 1 + κ/8.

SCALING APPROACH TO ANOMALOUS SURFACE ROUGHENING OF THE (d+1)-DIMENSIONAL MOLECULAR-BEAM EPITAXY GROWTH EQUATIONS

To determine anomalous dynamic scaling of continuum growth equations, Lopez12 proposed an analytical approach, which is based on the scaling analysis introduced by Hentschel and Family.15 In this work, we generalize this scaling analysis to the (d+1)-dimensional molecular-beam epitaxy equations to determine their anomalous dynamic scaling. The growth equations studied here include the linear molecular-beam epitaxy (LMBE) and Lai–Das Sarma–Villain (LDV). We find that both the LMBE and LDV equations, when the substrate dimension d>2, correspond to a standard Family–Vicsek scaling, however, when d<2, exhibit anomalous dynamic roughening of the local fluctuations of the growth height. When the growth equations exhibit anomalous dynamic scaling, we obtain the local roughness exponents by using scaling relation αloc=α-zκ, which are consistent with the corresponding numerical results.

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.

Scaling behaviour of the time-fractional Kardar–Parisi–Zhang equation

The scaling behaviour of the time-fractional Kardar–Parisi–Zhang (TFKPZ) equation in (1 + 1) dimensions is investigated by scaling analysis and numerical simulations. The surface morphology and critical exponents with different fractional orders are obtained. The analytical results are consistent with the corresponding numerical solutions based on a Caputo-type fractional derivative. We find that, similar to the normal Kardar–Parisi–Zhang equation, anomalous behaviour does not appear in the TFKPZ model according to the scaling idea of local slope and numerical evidence. However, there exists significant finite-time effect of local scaling exponents in the TFKPZ system. Our results also imply that memory effects can affect the scaling behaviour of evolving fractional surface growth.

Influences of Fractal Substrate Structures on Dynamic Scaling Behaviors of Etching Model

The Etching model on various fractal substrates embedded in two dimensions was investigated by means of kinetic Mento Carlo method in order to determine the relationship between dynamic scaling exponents and fractal parameters. The fractal dimensions are from 1.465 to 1.893, and the random walk exponents are from 2.101 to 2.578. It is found that the dynamic behaviors on fractal lattices are more complex than those on integer dimensions. The roughness exponent increases with the increasing of the random walk exponent on the fractal substrates but shows a non-monotonic relation with respect to the fractal dimension. No monotonic change is observed in the growth exponent.