Effects of film growth kinetics on grain coarsening and grain shape

We study models of grain nucleation and coarsening during the deposition of a thin film using numerical simulations and scaling approaches. The incorporation of new particles in the film is determined by lattice growth models in three different universality classes, with no effect of the grain structure. The first model of grain coarsening is similar to that proposed by Saito and Omura [Phys. Rev. E 84, 021601 (2011)], in which nucleation occurs only at the substrate, and the grain boundary evolution at the film surface is determined by a probabilistic competition of neighboring grains. The surface grain density has a power-law decay, with an exponent related to the dynamical exponent of the underlying growth kinetics, and the average radius of gyration scales with the film thickness with the same exponent. This model is extended by allowing nucleation of new grains during the deposition, with constant but small rates. The surface grain density crosses over from the initial power law decay to a saturation; at the crossover, the time, grain mass, and surface grain density are estimated as a function of the nucleation rate. The distributions of grain mass, height, and radius of gyration show remarkable power law decays, similar to other systems with coarsening and particle injection, with exponents also related to the dynamical exponent. The scaling of the radius of gyration with the height $h$ relative to the base of the grain show clearly different exponents in growth dominated by surface tension and growth dominated by surface diffusion; thus it may be interesting for investigating the effects of kinetic roughening on grain morphology. In growth dominated by surface diffusion, the increase of grain size with temperature is observed.

Figure 1

Illustration of the rules of the deposition models, with black vertical arrows indicating the incidence point: (a) Family: red arrows indicate possible steps to lower columns; (b) RSOS: accepted and rejected aggregation attempts are shown; (c) LADP: possible first steps of the incident particle are indicated; red arrows show steps that lead to permanent aggregation, and blue arrows show steps to points in which diffusion may continue (up to a maximum of $G$ steps).

Figure 2

Application of the rules for selecting the grain label at five aggregation positions. Colors indicate the labels of previously aggregated particles. The new particle label is (A) red or blue with equal probability; (B) new label; (C) brown; (D) green with probability $2/3$ and yellow with probability $1/3$; and (E) green.

Figure 3

(a)–(c) Cross sections of lateral size 32 of initial deposits; (d)–(f) cross sections of lateral size 64 of the upper layers of deposits with height $\approx 400$, for the indicated deposition models and with grain nucleation only at the substrate.

Figure 4

Top views with lateral size 64 of deposits with $\approx 400$ layers for the indicated models and with grain nucleation only at the substrate.

Figure 5

Density of surface grains as a function of time in the model of grain nucleation only at the substrate and deposition with Family (red squares), RSOS (green crosses), LADP $G=20$ (blue triangles), and LADP $G=50$ (magenta stars) models. The solid lines have the slopes predicted for each growth class.

Figure 6

Square radius of gyration as a function of the height $z$ in the model of grain nucleation only at the substrate: (a) Family (red squares) and RSOS (green crosses) deposition; dashed lines are linear fits, with slopes 1.03 and 1.10, respectively (expected values: 1 and 1.24); (b) LADP deposition with $G=20$ (blue triangles) and $G=50$ (magenta stars); dashed lines are linear fits for $z\ge {10}^3$, with slopes 0.68 and 0.73, respectively (expected value 0.60).

Figure 7

(a)–(c) Cross sections of lateral size 64 of the upper layers of deposits with height $\approx 400$; (d)–(f) show top views of the same size, for the indicated deposition models and grain nucleation at all heights with $p={10}^{-3}$.

Figure 8

Density of surface grains as a function of time for grain nucleation at all heights, with $p={10}^{-3}$ (squares), $p={10}^{-4}$ (triangles), and $p={10}^{-5}$ (crosses) and the deposition models: (a) Family; (b) RSOS; (c) LADP with $G=20$; and (d) LADP with $G=50$.

Figure 9

Scheme of a film section with two grains formed in a time difference $\delta t$.

Figure 10

Scaled distributions of grain mass for grain nucleation at all heights, with $p={10}^{-3}$ (red squares), $p={10}^{-4}$ (green triangles), and $p={10}^{-5}$ (blue crosses), and the deposition models indicated in the plots. The solid lines have the theoretically predicted slopes for the power law decays. For LADP models with $p={10}^{-5}$, the bumps of the plots at $pM\sim {10}^3$ are consequences of the limited simulation time, which interrupted the growth of the largest grains.

Figure 11

Square radius of gyration as a function of grain height for the model with nucleation at all heights, with the same symbols of Fig. 10. The solid lines have the theoretically predicted slopes.