Faceting instability in the presence of wetting interactions: A mechanism for the formation of quantum dots

A mechanism for the formation of quantum dots on the surface of thin solid films is proposed, not associated with the Asaro-Tiller-Grinfeld instability caused by epitaxial stresses. This mechanism, free of stress, involves instability of the film surface due to strong anisotropy of the surface energy of the film, coupled to wetting interactions between the film and the substrate. According to the mechanism, the substrate induces the film growth in a certain crystallographic orientation. In the absence of wetting interactions with the substrate, due to a large surface-energy anisotropy, this orientation would be thermodynamically forbidden and the surface would undergo a long-wave faceting (spinodal decomposition) instability. We show that wetting interactions between the film and the substrate can suppress this instability and qualitatively change its spectrum, leading to the damping of long-wave perturbations and the selection of the preferred wavelength at the instability threshold. This creates a possibility for the formation of stable regular arrays of quantum dots even in the absence of epitaxial stresses. This possibility is investigated analytically and numerically, by solving the corresponding nonlinear evolution equation for the film surface profile, and analyzing the stability of patterns with different symmetries. It is shown that, near the instability threshold, the formation of stable hexagonal arrays of quantum dots is possible. With the increase of the supercriticality, a transition to a square array of dots or the formation of spatially localized dots can occur. Different models of wetting interactions between the film and the substrate are considered and the effects of the wetting potential anisotropy are discussed. It is argued that the mechanism can provide a new route for producing self-organized quantum dots.

Figure 1

Sketch of dispersion curves defined by (22) for (a) ${\sigma }^2∕\left(4\nu W_{01}\right)>1$, (b) ${\sigma }^2∕\left(4\nu W_{01}\right)=1$, and (c) ${\sigma }^2∕\left(4\nu W_{01}\right)<1$.

Figure 2

(a) Parameter regions where a planar film surface is unstable (above the corresponding curves) for ${\alpha }_w=6.0$ (dashed line), ${\alpha }_w=3.0$ (solid line) and ${\alpha }_w=1.0$ (dashed-dotted line). (b) Parameter regions where weakly nonlinear periodic surface structures are stable (above the corresponding curves) for different values of ${\alpha }_w$.

Figure 3

Parameter regions where a planar film surface is unstable (above the solid line) and where stable periodic structures can form near the instability threshold (near the solid line, above the dashed line): (a) $\epsilon =0.1$, ${\gamma }_s{\delta }^2∕\nu =0.5$; (b) $\epsilon =0.1$, ${\gamma }_s{\delta }^2∕\nu =2.0$; (c) $\epsilon =0.2$, ${\gamma }_s{\delta }^2∕\nu =0.5$; and (d) $\epsilon =0.4$, ${\gamma }_s{\delta }^2∕\nu =0.5$.

Figure 4

Stationary numerical solutions of Eq. (26) with the wetting potential (12) showing stationary surface structures for $h_0=3.0\mathrm{nm}$, $(\zeta =2.0)$, $a=0.17\mathrm{J}{\mathrm{m}}^{-2}$, and (a) $w=6.8\times {10}^8\mathrm{J}{\mathrm{m}}^{-3}(w_{01}=0.24)$; (b) $w=2.8\times {10}^8\mathrm{J}{\mathrm{m}}^{-3}(w_{01}=0.1)$; and (c) $w=2.8\times {10}^7\mathrm{J}{\mathrm{m}}^{-3}(w_{01}=0.01)$.

Figure 5

Different stages of coarsening of the initial periodic structure, yielding the formation of localized dots divided by a thin wetting layer—the numerical solution of Eq. (26) with the wetting potential (12); $h_0=1.5\mathrm{nm}$, $w=8.2\times {10}^6\mathrm{J}{\mathrm{m}}^{-3}$ $(\zeta =1.0,w_{01}=0.1)$, $a=0.22\mathrm{J}{\mathrm{m}}^{-2}$.

Figure 6

Spatial patterns described by (54) with different values of $\Theta ={\theta }_1+{\theta }_2+{\theta }_3$.

Figure 7

Formation of spatially regular arrays of dots: numerical solutions of Eq. (15) for [001] surface orientation. (a) Stationary hexagonal array of equal-size dots, $h_0=1.5\mathrm{nm}$, $w=1.89\times {10}^7\mathrm{J}{\mathrm{m}}^{-3}$, $a=6.6\mathrm{J}{\mathrm{m}}^{-2}$ $(w_{01}=0.23,\zeta =1.0)$. (b) Stationary square array of equal-size dots, $h_0=1.5\mathrm{nm}$, $w=8.2\times {10}^5\mathrm{J}{\mathrm{m}}^{-3}$, $a=11.1\mathrm{J}{\mathrm{m}}^{-2}$ $(w_{01}=0.01,\zeta =1.0)$; Other parameters are the same as in Figs. 4, 5 and $b=0$.

Figure 8

Formation of localized dots via coarsening: numerical solutions of Eq. (15) for [001] surface orientation. (a) Nearly hexagonal array of dots (initial stage); (b) spatially localized dots divided by a thin wetting layer (late stage); $h_0=1.5\mathrm{nm}$, $w=8.2\times {10}^6\mathrm{J}{\mathrm{m}}^{-3}$, $a=0.22\mathrm{J}{\mathrm{m}}^{-2}$ $(w_{01}=0.1,\zeta =1.0)$. Other parameters are the same as in Figs. 4, 5 and $b=0$.

Figure 9

Formation of localized dots: numerical solutions of Eq. (15) for [111] surface orientation showing different stages of coarsening of initial regular hexagonal array of triangular pyramids. The parameters are the same as in for the case shown in Fig. 8, except $b=0.44\mathrm{J}{\mathrm{m}}^{-2}$.

Figure 10

Evolution of dots accompanied by ballistic deposition: numerical solutions of Eq. (15) in 2D with the additional constant deposition term. (a) Hexagonal array of equal-size dots (initial stage); (b) larger dots with pyramidal shape (later stage); $h_0=1.5\mathrm{nm}$, $w=8.2\times {10}^6\mathrm{J}{\mathrm{m}}^{-3}$, $a=0.22\mathrm{J}{\mathrm{m}}^{-2}$ $(w_{01}=0.1,\zeta =1.0)$, deposition rate $V=0.2\mathrm{nm}∕\min$. Other parameters are the same as in Fig. 1 and $b=0$.