Smoothening in thin-film deposition on rough substrates

The evolution of the surface roughness $W$ of a thin film deposited on a rough substrate is studied with a model of temperature-activated adatom diffusion, irreversible lateral aggregation, and no step energy barrier, in which the main parameter is the ratio $R$ of diffusion and deposition rates. At sufficiently low temperatures $\left(R\lesssim 10\right)$, the average number of adatom steps after adsorption is very small, thus $W$ monotonically increases with time $t$ due to an approximately uncorrelated deposition at short times. If the temperature is not very low ($R\sim {10}^3$ or larger), smoothening occurs at short times and the Villain-Lai-Das Sarma (VLDS) growth equation governs the long time roughening, which is attained after a crossover time $t_c$ that increases with the correlation length ${\xi }_i$ of the substrate. Scaling arguments predict the dependence of $t_c$ on temperature and on the substrate production time and the scaling relation for the difference between the roughness of films deposited on rough and flat substrates, in good agreement with numerical results. The effect of temperature is not a direct extension of previous results on flat substrates because the short wavelength fluctuations delay the formation of terraces. For this reason, the effective energy obtained from the dependence of $t_c$ on $R$ is $40\%$ of the energy of activated adatom diffusion. A scaling law for the initial smoothening is proposed as $W/W_i=\mathrm{\Psi }\left(t/t_{c1}\right)$, with a crossover time $t_{c1}\equiv R^{-\theta }{{\xi }_i}^z$, where $W_i$ is the substrate roughness, $\theta \approx 0.4$, and $z$ is the VLDS dynamical exponent. It provides good data collapse if $W$ is not very small and is suggested to be tested experimentally.

Figure 1

(a) Growth rules of the E model: the attack of the solution at three columns is indicated by arrows in the upper panel, and the resulting configuration of the solid is shown in the lower panel, with removed particles indicated by crosses. (b) Growth rules of the WV model, in which the columns of particle incidence are indicated by thick arrows and aggregation positions, are indicated by thin arrows. The double arrow of the left case indicates that one of the NN columns will be randomly chosen.

Figure 2

Portion of cross section of lateral size 1024 of a deposit with 10 layers of atoms ($\tau =10$) deposited at $T=310\mathrm{K}$ over a rough substrate produced by the E model with $s=20$.

Figure 3

Roughness as a function of the film deposition time at temperatures 233 K ($R\approx 10$; blue squares), 290 K ($R\approx 4\times {10}^3$; black triangles), and 330 K ($R\approx 7\times {10}^4$; red circles). For each temperature, the lower curve is for deposition on flat substrates and the upper curve is for deposition on rough substrates produced by the E model with $s=100$.

Figure 4

Roughness as a function of the film deposition time at temperatures 233 K ($R\approx 10$; blue squares), 290 K ($R\approx 4\times {10}^3$; black triangles), and 330 K ($R\approx 7\times {10}^4$; red circles). For each temperature, the lower curve is for deposition on flat substrates and the upper curve is for deposition on rough substrates produced by the WV model with $s=3000$.

Figure 5

Correlation length ${\xi }_1$ as a function of the film deposition time at $310\text{K}\left(R\approx 1.7\times {10}^4\right)$ in three different substrates: the flat one (blue asterisks) and those produced by WV ($s=3000$) (red squares) and E ($s=100$) (black triangles) models. Error bars are also shown. The dashed lines have slope $0.3$.

Figure 6

Autocorrelation function as a function of the distance $r$ at several scaled times, for deposition at $T=233\text{K}\left(R\approx 10\right)$ on substrates produced by (a) the E model ($s=100$) and (b) the WV model ($s=3000$). The horizontal dashed lines indicate $\mathrm{\Gamma }\left(r,\tau \right)=0$.

Figure 7

Scaled square roughness change as a function of scaled time for deposition in various temperatures and substrates produced by the E model in different times $s$. Scaling exponents are $c=0.4$, $\kappa =2.04$, and $\theta =0.4$. The dashed line has slope $-0.6$.

Figure 8

Scaled square roughness change as a function of scaled time for deposition in various temperatures and substrates produced in different times $s$ by the WV model with (a) $m=1$ and (b) $m=10$. Scaling exponents are $c=0.4$, $\kappa =0.83$, and $\theta =0.4$. The dashed line has slope $-0.6$

Figure 9

(a) Terrace formation at a high temperature; (b) profile of a locally rough substrate, such as those produced by E or WV model showing a small lateral size of terraces; (c) increase of the lateral size of terraces on the rough substrates in a time interval $\mathrm{\Delta }t$ in which the average size increases from $l$ to $2l$; (d) adatom diffusion on terraces before reaching the step edges.

Figure 10

Reduced roughness as a function of (a) $\tau /t_{c1}$ and (b) $\tau /t_{c2}$ for deposition on substrates produced by the E model. The scaling exponents are $\theta =0.4$, $a=8.3$, and $z=3.33$.

Figure 11

Reduced roughness as a function of (a) $\tau /t_{c1}$ and (b) $\tau /t_{c2}$ for deposition on substrates produced by the WV model with $m=1$. The scaling exponents are $\theta =0.4$, $a=1.0$, and $z=3.33$.