Second-layer nucleation in coherent Stranski-Krastanov growth of quantum dots

We have studied the monolayer-bilayer transformation in the case of the coherent Stranski-Krastanov growth. We have found that the energy of formation of a second-layer nucleus is largest at the center of the first-layer island and smallest on its corners. Thus nucleation is expected to take place at the corners (or the edges) rather than at the center of the islands as in the case of homoepitaxy. The critical nuclei have one atom in addition to a compact shape, which is either a square of $i\times i$ or a rectangle of $i\times (i-1)$ atoms, with $i>1$ an integer. When the edge of the initial monolayer island is much larger than the critical nucleus size, the latter is always a rectangle plus an additional atom, adsorbed at the longer edge, which gives rise to a new atomic row in order to transform the rectangle into the equilibrium square shape.

Figure 1

Locations of second-layer nuclei. From top to bottom and from left to right: initial $20\times 20$ monolayer island; 13-atoms second-layer cluster nucleated at the terrace center, at an island edge, and at an island corner of the initial monolayer island. The color scale denotes the height of the considered atom and has been represented using the atomeye software.[37] This height is measured above the level of the corresponding crystallographic plane, but a constant fraction of the interlayer distance has been added in order to better distinguish atoms from different levels. The height is biggest at edges and corners due to the atoms climbing up on their neighbours underneath due to strain relaxation. The lattice misfit is $-7\%$.

Figure 2

Transformation curves (total energy as a function of the number $N_2$ of atoms transferred to the second ML) for (a) positive ($+3.5\%$) and (b) and negative ($-12.0\%$) values of the misfit. A potential with $\mu =16$ and $\nu =14$ was used in the simulations and an initial monolayer island containing $20\times 20=400$ atoms was considered. The second-layer islands are formed at the center, at one of the edges, and at one corner of the initial, monolayer island.

Figure 3

Heights of the nucleation barriers in units of $V_0$ as a function of the value of the lattice misfit (main plot: positive misfits; insert: negative misfits). The figures at each point denote the number of atoms, $n^{*}$, in the critical nucleus. The values for compressed islands were calculated for $\mu =2\nu =12$, those for expanded islands for $\mu =16$ and $\nu =14$. A cluster size of 400 atoms was considered.

Figure 4

Closeup of the transformation curve for a lattice misfit of $+3.58\%$, $\mu =16$, $\nu =14,$ and a finite cluster of a size of 400 atoms (circles). Also shown (squares) is the transformation curve for a similar situation for which an infinitely large cluster was simulated by not removing atoms from the first level but by correcting instead for the binding energy at the half-crystal position at the center of an atomic row (see text for details). The thick arrow marks the absolute maximum of the transformation curve. The remaining arrows show that local maxima of the latter curve tend to be higher for $i\times (i-1)+1$ atoms, with $i$ an integer, than for $i\times i+1$ atoms. The insert shows a selected region of the plot of the barrier height vs lattice misfit for a potential with $\mu =2\nu =12$ containing also the critical nucleus sizes for both types of configurations. These data confirm the appearance of only (rectangle +1)-type islands for the simulated infinite islands.