Effect of the lattice misfit on the equilibrium shape of strained islands in Volmer-Weber growth

We have studied the effect of the misfit on the equilibrium shape of three-dimensional pyramidal islands grown on a foreign substrate in the case of incomplete wetting (Volmer-Weber mode of growth). By means of atomistic simulations using anharmonic interaction potentials, we find that tensile islands have smaller aspect ratios compared with compressed islands owing to their better adhesion to the substrate. The average strains of consecutive layers decrease faster with thickness in compressed than in tensile islands. The strains decrease rapidly with thickness, with the consequence that above a certain height, the upper layers of the pyramid become practically unstrained and do not contribute to a further reduction in the top base. As a result, the truncated pyramids are not expected to transform into full pyramids. Our results are in good agreement with experimental observations in different systems.

Figure 1

(Color online) Epitaxial fcc(100) island of $-7\mathrm{\%}$ misfit on a rigid substrate. It has the shape of a truncated square pyramid with $30\times 30$ atoms in the base plane and a height of 4 MLs. The color scale denotes the level of hydrostatic strain as given by the atomic local strain tensor (Ref. 19) and has been represented using the ATOMEYE software (Ref. 20). Strain is highest at the center of the island and at the bottom layer and it is relaxed at island edges and at the topmost layer.

Figure 2

(Color online) Dependence of the total energy per atom as a function of the number of atoms $N$ in the pyramid for islands with different heights varying through 1 ML. As seen islands with $N<300$ have lowest energy when they are 4 MLs high whereas the lowest energy height is 5 MLs at $N>300$. The wetting parameter is $\varphi =0.2$, the lattice misfit $f$ is zero, and a rigid substrate was assumed in the simulations.

Figure 3

(Color online) Misfit dependence of the aspect ratio $h/n$ for different values of the wetting function $\varphi =0.0$, 0.03, and 0.10 and a total number of atoms in the pyramid of (a) $N=250$ and (b) $N=1000$. A rigid substrate was assumed in the calculations. The arrows mark the appearance of misfit dislocations in the growing islands.

Figure 4

Misfit dependence of the adhesion parameter, ${\varphi }^{\prime }\left(f\right)$, for compressed and tensile islands for the case of $\varphi =0.0$ (SK mode of growth). Results are shown for different numbers $S$ of substrate layers allowed to relax: circles for $S=0$ (i.e., rigid substrate) and squares for $S=3\text{ML}$. The islands are 5 ML high and contain a total amount of $N=990$ atoms.

Figure 5

(Color online) Distribution of the lateral bond strains in the center of the crystal and in the layers of the substrate underneath for misfits of (a) $f=4\mathrm{\%}$ and (b) $f=-4\mathrm{\%}$. Strains are referred to the lattice parameters of a crystal and b substrate, respectively. The wetting parameter due to the different chemical nature of substrate and adsorbate is $\varphi =0.03$, islands of 5 ML height and 990 atoms were considered and 5 ML of the substrate were allowed to relax.

Figure 6

Average strains of consecutive layers in compressed and tensile pyramids with $+4.0\mathrm{\%}$ and $-4.0\mathrm{\%}$ lattice misfit, respectively. Islands are 5 ML high and contain 990 atoms, the wetting parameter is $\varphi =0.03$ and 5 ML of the substrate were allowed to relax.

Figure 7

Strain energy of consecutive layers in compressed and tensile pyramids with $+4.0\mathrm{\%}$ and $-4.0\mathrm{\%}$ lattice misfit, respectively. Islands are 5 ML high and contain 990 atoms, the wetting parameter is $\varphi =0.03$ and 3 ML of the substrate were allowed to relax.