Quantum confinement induced strain in quantum dots

We investigate quantum confinement induced strain in quantum dots. While the impact of mechanical strain on the electronic structure of quantum dots is well studied, the “reverse” effect remains relatively unexplored. Even in the complete absence of external stress, for very small sizes ($1–3\mathrm{nm}$ range), the electronic structure change due to quantum confinement may induce a strain in the quantum dot, which in turn will further alter the electronic structure. Despite the limitations of an envelope function approach for small sizes, a multiband analytical model is developed to make explicit the qualitative features of this phenomenon with physical interpretation in terms of acoustic polarons. We quantitatively predict the induced strain due to quantum confinement and the polaron binding energy for the example cases of Si and GaAs. The Si polaron binding energy calculated from the developed model compares favorably with both our density-functional and semiempirical atomistic calculations.

Figure 1

(Color online) Ground-state band energy shift for (a) the conduction band and (b) the valence band due to quantum confinement induced strain in a cuboidal QD.

Figure 2

(Color online) Distribution of strain dilation for cuboidal Si QD $(a=7\mathrm{\AA })$ on the $z=0$ plane (blue plane, gray in print) viewed from top. (a) Conduction band. (b) Valence band. No external strain is applied and the strain depicted is entirely due to quantum confinement.

Figure 3

Variation of the maximum strain with size $a$ for cuboidal QDs. (a) ${\epsilon }_{\mathit{max}}=\max \left(\mid {\epsilon }_{11}\mid \right)=\max \left(\mid {\epsilon }_{22}\mid \right)=\max \left(\mid {\epsilon }_{33}\mid \right)$. Results for both Si and GaAs are shown. (b) ${\epsilon }_{\mathit{max}}=\max \left(\mid {\epsilon }_{ij}\mid \right)$. Only GaAs results are plotted.

Figure 4

Comparison of the polaron binding energy calculated from $\mathbf{k}\bullet \mathbf{p}$ with quantum confinement induced strain (qs in the figure) and from atomistic simulations. (a) Cuboidal Si QD. (b) Spherical QD.

Figure 5

(Color online) Isosurface plot for the molecular orbitals (wave function) of $a=7\mathrm{\AA }$ cuboidal QD, lowest unoccupied molecular orbital (LUMO), and highest occupied molecular orbital (HOMO). Red (lighter gray in print) stands for positive value isosurface and blue (darker gray in print) stands for negative value isosurface. (a) The electron doped LUMO at the value of ∣0.015∣ not incorporating polaron formation. (b) The electron doped LUMO at the value of ∣0.015∣ accounting for polaron formation. (c) The hole doped HOMO at the value of ∣0.02∣ not incorporating polaron formation. (d) The hole doped HOMO at the value of ∣0.02∣ accounting for polaron formation.

Figure 6

Results from EFM for $a=7\mathrm{\AA }$ cubic QD showing the difference of the square of the magnitude of the wave function, ${\mid \psi \left(\mathbf{r}\right)\mid }_{qs}^2-{\mid \psi \left(\mathbf{r}\right)\mid }_{cl}^2$. $\psi {\left(\mathbf{r}\right)}_{qs}$ incorporates quantum confinement induced strain, while polaronic state is not accounted for in $\psi {\left(\mathbf{r}\right)}_{cl}$. The units are ${\mathrm{nm}}^{-3}$. Part (a) is the electron doped ground-state conduction band, while part (b) is the hole doped ground-state valence band.