Surface diffusion dewetting of thin solid films: Numerical method and application to $\mathrm{Si}∕{\mathrm{SiO}}_2$

A method has been developed to calculate and use a surface chemical potential which is valid in the large curvature regime for any surface energy function. It is applied to the solid-phase dewetting of a finite film with an initial rectangular profile and considers the surface diffusion mechanism. For an isotropic surface energy, the film aspect ratio and the adhesion energy between the film and the substrate are shown to be the main parameters that quantify the retraction, the breaking time, and the number of agglomerates. Moreover, it is found that mild surface energy anisotropy with an energy minimum in the horizontal plane postpones the mass detachment. Simple models of the $\gamma$-plots for the surface energy illustrate the influence of cusp points on the retraction profiles. Finally, the smooth and facetted experimental surfaces, that are observed in the $\mathrm{Si}∕{\mathrm{SiO}}_2$ system after $900°\mathrm{C}$ annealing under ${\mathrm{H}}_2$, are explained by a quite small anisotropy of the $\gamma$-plot.

Figure 1

Schematics of the dewetting of a thin film of thickness $H$ and length $2L$. Only half of the film is drawn due to symmetry consideration. The contact angle between the film and the substrate is described by $\alpha$. The initial edge (a) retracts (see $x_0$) and the profile forms a peak followed by a valley (b). The film evolves until peeling off in two (c) or three (d) agglomerates, or it rearranges in single agglomerate (e).

Figure 2

Sketch for the calculation of the surface chemical potential according to Eq. (5) (see text) at the point $M_i$ between $M_{i-1}$ and $M_{i+1}$, which is not positioned at the boundary with the substrate. $l_{i-1}$ $\left({\gamma }_{i-1}\right)$ and $l_i$ $\left({\gamma }_i\right)$ are the length (surface energies) of the segment $\left[M_{i-1}{M}_i\right]$ and $\left[M_i{M}_{i+1}\right]$. $h_{0,i}$ is the distance between $M_i$ and $\left(M_{i-1}{M}_{i+1}\right)$. $b_{1,i}$ and $b_{2,i}$ are respectively the distance between $M_{i-1}$ and $M_{i+1}$ to $P_i$, the orthogonal projection of $M_i$ on $\left(M_{i-1}{M}_{i+1}\right)$. $\delta V_i$ and $\delta G_i$ are respectively the increment of volume and energy of the surface after a displacement $\delta {\eta }_i$ of $M_i$ along $\left[P_i{M}_i\right]$. The inset shows the sketch of the calculation of the surface chemical potential at the point $M_0$ on the contact line.

Figure 3

Relative difference $(\alpha -\alpha y)∕\alpha y$ between the contact angle $\alpha$ and the equilibrium contact angle ${\alpha }_y$ as a function of time normalized by the breaking up time $t_c$. Simulations are carried out with $n$, $2n$, and $4n$ points $(n=75)$, an aspect ratio of $F=150$, and dimensionless adhesion energies corresponding to Young contact angle of $45°$ and $135°$. $\alpha$ converges to ${\alpha }_y$ at least four decades before $t_c$.

Figure 4

Surface profile simulated for an isotropic surface energy (see Sec. 4 for the numerical procedure). The initial state is shown in Fig. 1a with $F=L∕H=500$, the equilibrium contact angle is $160°$, and the time is $t=13000$. The insets zoom in on the second and third peaks. A fit of the profile with the function $z(x,t)=1+A\left(t\right)\exp [-kx∕x_p(t\left)\right]\cos [2\pi x∕x_p(t)+\phi ]$ is given with $A\left(13000\right)=6700$, $k=3.87$ (close to the value given by Srolovitz (Ref. 24): $2\pi ∕\sqrt{3}\approx 3.63$), $\phi =1.1$ rd and $x_p=48.33x_p\left(t\right)$ is found to vary as: $x_p\left(t\right)\approx 5.4t^{0.23}$.

Figure 5

Evolution of the minimum dimensionless film thickness $z_{min}$ (see Fig. 1) vs dimensionless time $t$ for a contact angle of $45°$ and for different aspect ratio $F$. The surface energy is considered to be isotropic. $F=500$ is close to the semi-infinite case. When $z_{min}$ reaches 0, the film breaks up, and when $z_{min}$ goes higher than 1, the dip is filled up and the matter rearranges in single agglomerate. $z_{min}\left(t\right)$ oscillates around the semi-infinite case with amplitude increasing with time, before the breakup or the coalescence. b) Minimum dimensionless film thickness $z_{min}$ vs dimensionless time $t$ for an aspect ratio $F=150$ and for different contact angle $\alpha$. Edge-coupling effect and breakup time appear earlier with an increase of the contact angle.

Figure 6

(Color online) Logarithm of the critical breakup time $t_c:\ln \left(tc\right)$ as the function of $\ln [1∕\sin (\alpha ∕2\left)\right]$ ($\alpha$: contact angle) for different aspect ratio $F$. The points correspond to numerical simulations for an isotropic surface energy, and the lines to empirical adjustments. Two regimes can be identified. The first one is the region where the edge-coupling effect is negligible. $t_c(F,\alpha )$ depends only on $F$, and can be fitted by: $\ln \left(t_c\right)=4.092\ln [1∕\sin (\alpha ∕2\left)\right]+9.4225$. In the second one, the edge coupling cannot be neglected and ${\mathrm{t}}_{\mathrm{c}}$ is depending both on $\alpha$ and $F$. $t_c$ is relatively well fitted by the general law: $\ln \left(t_c\right)=C_1∕\sin (\alpha ∕2)+C_2$, where $C_1=-0.328\ln \left(F\right)+2.23$ and $C_2=3.42\ln \left(F\right)-7.39$.

Figure 7

(Color online) Number of agglomerates calculated with an isotropic surface energy vs $1∕\sin (\alpha ∕2)$ ($\alpha$ is the contact angle) and aspect ratio $F$. The points correspond to the numerical simulations. The dashed lines show the boundaries for the formation of 1, 2, and 3 or more agglomerates. They can be fitted by $F=48.3∕\sin (\alpha ∕2)-4.33$ for 1–2 agglomerates and $F=123.4∕\sin (\alpha ∕2)-24.6$ for 2–3 and more agglomerates.

Figure 8

Ratio of anisotropic to isotropic breaking time $(t_{c,aniso}∕t_{c,iso})$ vs the corresponding isotropic Young angle $\alpha$ and the aspect ratio $F$. The anisotropic surface energy follows Eq. (7) with $\xi =0.01332$ and $\theta =\pi ∕8$. For large $F$ and $\alpha$, $t_c$ can be several times higher in the anisotropic case than in the isotropic one.

Figure 9

(Color online) Number of agglomerates calculated with an anisotropic surface energy (defined by Eq. (7) with $\xi =0.01332$ and ${\theta }_0=\pi ∕8$) as a function of $1∕\sin (\alpha ∕2)$ ($\alpha$ is the corresponding isotropic Young angle) and aspect ratio $F$. The points correspond to the numerical simulations. The boundaries between the numbers of agglomerates (solid lines) are shifted compared to the isotropic surface energy calculations (dashed lines taken from Fig. 7).

Figure 10

Surface profiles corresponding to a film retraction with an anisotropic surface energy without a cusp point. The eightfold $\gamma$-plots and equation are shown in the inset. Four orientations ${\theta }_0$ of the minimum in surface energy are drawn and discussed in the text. The corresponding isotropic Young angle $\alpha$ is $135°$ and dimensionless time is 3000. The film aspect ratio is $F=250$, i.e., long enough to avoid the edge-coupling effect. The initial sharp corner [see Fig. 1a] was at the abscissa $x=0$.

Figure 11

(Color online) $1.5\mathrm{\mu }\mathrm{m}\times 1.5\mathrm{\mu }\mathrm{m}$ AFM image (tapping mode) of the edge retraction of a film of Si on ${\mathrm{SiO}}_2$ after a $30\mathrm{mn}$ annealing at $950°\mathrm{C}$ (initial thickness $13.8\mathrm{nm}$). The first peak is $96\mathrm{nm}$ high and is followed by a dip. This profile is similar to the retraction simulations with an isotropic surface energy model.

Figure 12

(Color online) Image: $1\mathrm{\mu }\mathrm{m}\times 1\mathrm{\mu }\mathrm{m}$ AFM image (tapping mode) of the Si on ${\mathrm{SiO}}_2$ retraction after a $2\min$ annealing at $900°\mathrm{C}$ (the initial film thickness is $13.0\mathrm{nm}$). The peak is facetted and reaches $52\mathrm{nm}$ high. No dip appears behind the $\left(113\right)$ plane. Curve: Profile along $(0,x)$. It is similar to the retraction simulations with an anisotropic surface energy model.

Figure 13

Schematics of 2D network of atoms placed at the intersection of two dense planes $P_1$ and $P_2$. The disorientation angle between $P_1$ and $P_2$ is ${\theta }_{12}$. The distance between the $P_1$ planes is $a$ and between the $P_2$ planes is $b$. The disorientation angle of the vicinal surface is $\theta$ from the reference planes $P_1$. The terrace and step length are respectively $p$ and $q$. The distance between 2 consecutive steps is $l$.

Figure 14

Surface profiles corresponding to a film retraction with an anisotropic surface energy with cusp point. The inset shows the $\gamma$-plot with a 12-fold symmetry corresponding to the $\lfloor 1\overline{1}0\rfloor$ zone and to the $\left\{001\right\}$, $\left\{113\right\}$, $\left\{111\right\}$, and $\left\{110\right\}$ dense planes. $\gamma \left(\theta \right)$ is given by Eq. (10) between these dense plane orientations. ${\gamma }_{113}$ is varied from 0.94 to 1.03 and the other dense plane energies are fixed to 1.00. Note that the dense plane extension on this profile directly depends on the surface energy ratio.