Kinetic roughening and porosity scaling in film growth with subsurface lateral aggregation

We study surface and bulk properties of porous films produced by a model in which particles incide perpendicularly to a substrate, interact with deposited neighbors in its trajectory, and aggregate laterally with probability of order $a$ at each position. The model generalizes ballisticlike models by allowing attachment to particles below the outer surface. For small values of $a$, a crossover from uncorrelated deposition (UD) to correlated growth is observed. Simulations are performed in 1+1 and 2+1 dimensions. Extrapolation of effective exponents and comparison of roughness distributions confirm Kardar-Parisi-Zhang roughening of the outer surface for $a>0$. A scaling approach for small $a$ predicts crossover times as $a^{-2/3}$ and local height fluctuations as $a^{-1/3}$ at the crossover, independent of substrate dimension. These relations are different from all previously studied models with crossovers from UD to correlated growth due to subsurface aggregation, which reduces scaling exponents. The same approach predicts the porosity and average pore height scaling as $a^{1/3}$ and $a^{-1/3}$, respectively, in good agreement with simulation results in 1+1 and 2+1 dimensions. These results may be useful for modeling samples with desired porosity and long pores.

Figure 1

Illustration of the rules of particle aggregation in the SPD model. Deposited particles are gray and black squares, the latter indicating those particles in the outer surface. Incident particles are indicated as squares marked $A$ and $B$, and circles numbered from 1 to 7 indicate possible aggregation positions of those particles.

Figure 2

(a) Time evolution of the surface roughness in the SPD model with $a=0.5$ (red solid curve), $a=0.1$ (green dashed curve), and 0.025 (blue dotted curve). The dashed line has slope 1/3 of KPZ scaling. (b) Saturation roughness as a function of the lattice size for $a=0.5$ (red squares) and 0.1 (green triangles). The dashed line has slope 1/2 of KPZ scaling.

Figure 3

Effective roughness exponents as a function of $L^{-\mathrm{\Delta }}$ for $a=0.1$ (red triangles) with $\mathrm{\Delta }=0.72$ and $a=0.5$ (blue squares) with $\mathrm{\Delta }=0.52$.

Figure 4

Effective dynamical exponents as a function of $1/L$ for (a) $a=0.1$ and (b) $a=0.5$.

Figure 5

Two neighboring columns after some time of random growth. Deposited particles are gray and black squares, the latter indicating particles at the outer surface. The incident particle is the empty square, and circles indicate possible aggregation positions of this particle.

Figure 6

(a) Characteristic time $t_0$ and (b) saturation roughness $W_{\mathrm{sat}}$ in size $L=1024$ as a function of the stickness parameter. Solid lines are fits of the data for $a\le 0.05$.

Figure 7

Regions of size $48\times 48$ (in lattice units) of samples grown with stickness parameters (a) 0.1, (b) 0.01, (c) 0.001, and (d) 0.0001.

Figure 8

(a) Porosity as a function of the stickness parameter. The solid line is a linear fit of the data for $a\le 0.01$. (b) Average pore height as a function of the stickness parameter. Dashed lines have slopes $-0.15$ (right) and $-0.28$ (left).

Figure 9

Scaled roughness distributions in the steady states of the SPD model with $a=0.1$ (squares) and of the RSOS model (solid curve) in 2+1 dimensions with $L=256$.

Figure 10

Porosity as a function of the stickness parameter of the SPD model in 2+1 dimensions. The solid line is a linear fit of the data for $a\le 0.01$.