Hui Xia

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.

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.

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.