Composition Maps in Self-Assembled Alloy Quantum Dots

Nanoscale variations in composition arising from the competition between chemical mixing effects and elastic relaxation can substantially influence the electronic and optical properties of self-assembled alloy quantum dots. Using a combination of finite element and quadratic programming optimization methods, we have developed an efficient technique to compute the equilibrium composition profiles in strained quantum dots. We find that the composition profiles depend strongly on the morphological features such as the slopes and curvatures of their surfaces and the presence of corners and edges as well as the ratio of the strain and chemical mixing energy densities. More generally, our approach provides a means to quantitatively model the interplay among the composition variations, the temperature, the strain, and the shapes of small-scale lattice-mismatched structures.

Figure 1

(a) Composition profiles in axially symmetric quantum dots of identical size, but with shallow ($\theta =15°$) and steep sidewalls ($\theta =30°$). The composition profiles are obtained with $F_0=-0.2$ and $\overline{c}=0.5$. (b) The 3D rendering of the shapes of the quantum dots in (a) upon etching with a selective chemical agent that dissolves regions of dot whose composition $c$ exceeds 65%. The segregation indices [Eq. (3))] for the steep and shallow dots are 0.177 and 0.051, respectively.

Figure 2

Compositional phase diagram showing degree of segregation, $\Phi$ as a function of the parameter $F_0$, and the shape of the quantum dot represented by the angle of sidewalls, $\theta$. Even when the thermodynamic mixing energy favors mixing ($F_0<0$), the relaxation of strain for quantum dots with steep sidewalls results in the alloy segregation within the quantum dot. Similarly, when thermodynamics favors phase separation ($F_0>0$), complete decomposition is not observed. The average composition of the quantum dot $\overline{c}$ is 0.5.

Figure 3

Variation of the energy per unit volume for quantum dots with shallow and steep sidewalls ($\theta =15°$ and $\theta =45°$, respectively) as a function of their size (normalized by the characteristic volume, $V_0=[\gamma /(M{ϵ}_m^2){]}^3$). The total energy of the decomposed dots (dotted lines) is lower than the energy of dots with uniform composition, $\overline{c}=0.5$ (bold lines). However, the reduction in the energy is greater for the steeper dot, resulting in smaller critical size for transition in shape. The surface energies for shallow and steep sidewalls are assumed to be identical and the mixing parameter $F_0=-0.2$.

Figure 4

Equilibrium composition profiles in axisymmetric quantum dots with (a) “dome” shape, the angles of the sidewalls being 30° and 15°, and (b) a truncated-cone shape with a sidewall angle of 30°. While the composition profiles are similar near the base, larger strain relaxation in the regions near the corners results in a greater segregation in the apex of the dome-shaped quantum dot. The composition profiles are obtained for $F_0=-0.2$ and $\overline{c}=0.5$.