Forbidden Island Heights in Stress-Driven Coherent Stranski-Krastanov Growth

The observed height distribution of clusters obtained in strained epitaxy has been often interpreted in terms of electronic effects. We show that some aspects can be explained classically by the interplay of strain and edge energies. We find that soft materials can transform directly from a monolayer into thicker islands by two-dimensional (2D) multilayer nucleation and growth. There is a critical thickness decreasing with the force constant of the material’s bonds. Thinner islands are thermodynamically forbidden, due to the insufficient stress relaxation upon clustering, particularly under tensile stress. At sufficiently large misfits, the barrier for 2D multilayer nucleation is significantly smaller than the barrier for subsequent single-layer nucleation. The effects are found to be quantitatively reasonable and offer a plausible explanation for the absence of thin islands and 2D growth of flattop islands usually attributed to quantum size effects.

Figure 1

Energy per atom of mono- and multilayer islands as a function of the total number of atoms. The misfit $\epsilon$ amounts to $-7\%$ and $\mu =2\nu =26$. The inset shows the dependence of the critical size $N_{1-X}$ (beyond which 1-ML islands become unstable against $X$-ML islands) on the force constant $\gamma$. The threshold thickness $X$ is denoted by the numbers at each point.

Figure 2

Phase diagram showing the stability ranges of islands of different heights (given by the numbers in MLs) in coordinates of the total number of atoms and the bulk strain energy per bond $\mathcal{E}=0.5\gamma {\epsilon }^2{a}^2$, for positive and negative values of the misfit and for $\mu =2\nu$ potentials.

Figure 3

Transformation curves representing the energy change in units of the bond energy $V_0$ as a function of the number of atoms transferred to the upper level for (a) $\epsilon =-0.07$ and (b) $\epsilon =-0.12$. The number of atoms in the initial monolayer island ($365\approx 19\times 19$) gives a complete truncated trilayer pyramid ($12\times 12+11\times 11+10\times 10$) and $\mu =2\nu =26$.

Figure 4

Nucleation barriers in units of $V_0$ as a function of the absolute value of the negative misfit, for mono- to trilayer islands. The number of atoms in the critical nucleus is given at each point. Also shown is the barrier for monolayer nucleation on top of the trilayer island. The total number of atoms is 365, and $\mu =2\nu =26$.