Electronics states of interdiffused quantum dots

We have developed a three-dimensional model for electronic states calculation of interdiffused quantum dots (QDs) with arbitrary shape by solving the BenDaniel-Duke’s equation in momentum space domain. The proposed model features several advantages such as automatic solution to the Fick’s diffusion equation, a relatively compact and efficient Hamiltonian matrix, and natural representation of a large array of QDs. Without considering the interdiffusion effect, our model yields good agreement with our references of $\mathrm{InAs}∕\mathrm{GaAs}$ QDs ground state energy calculation. We analyze the interdiffusion effect in QDs with various shapes of theoretical and practical interest like spherical, cubical, pyramidal, and lens shaped. We study the effect of QD size and aspect ratio to the blueshift profile due to interdiffusion. We found a similar blueshift profile in these QDs at almost any size that can be well approximated by $\mathrm{sech}\left(x\right)$ function. This model will serve as a valuable tool for QD band gap engineering based on the interdiffusion technique.

Figure 1

(a) A superlattice of pyramidal QDs with tetragonal unit cell. (b) Interdiffusion smears the alloy distribution resulting in band gap disordering in all directions. Band gap profiles for the noninterdiffused (left) and interdiffused (right) QD along the $x$ direction are shown. (c) Conduction band edge potential profile at $z=0$ plane for: noninterdiffused (left) and interdiffused (right) QD.

Figure 2

Convergence of electron ground state energies at increasing $N_{FC}$ and as a function of $a∕b$ for: (a) cubical QD, (b) pyramidal QD. Both QDs have the same volume and unit cell. The dotted line indicates asymptotic curve for $N_{FC}=\infty$.

Figure 3

Validation of our model (solid line) for noninterdiffused $\mathrm{InAs}∕\mathrm{GaAs}$ pyramidal QDs with the theoretical models from Cusack et al. (hollow squares) and Grundmann et al. (filled circles). The plots display ground state energies with respect to the unstrained GaAs conduction and valence band, respectively, for (a) electron and (b) heavy holes.

Figure 4

The interdiffusion effect on the ground state transitional energies for the four types of isovolume QDs $(V_{QD}=167\times {10}^3{\mathrm{\AA }}^3)$ with the same unit cell. The lens-shaped and pyramidal QD have aspect ratio $h∕b=0.5$ (Inset:) Normalized blueshift that follows an empirical $\mathrm{sech}\left(x\right)$ curve.

Figure 5

Variation of $\beta$ with respect to the QD size $R$ for the four types of QDs. The lens-shaped and pyramidal QD have aspect ratio $h∕b=0.5$. The volume ratio of the QD to the unit cell are kept constant (constant $c_{{\mathbf{K}}_{0,0,0}}$).

Figure 6

(a) Blueshift profile of isovolume lens shaped QDs with varying aspect ratio. The profile for thin lens QDs with aspect ratio $h∕b⩽0.2$ do not fit to $\mathrm{sech}\left(x\right)$ curve anymore and exhibits a kink indicated by an arrow (see text). (b) Lens-shaped QDs $\beta$ parameter at various size and aspect ratio.

Figure 7

Comparison of experimental data (circles) of interdiffused $\mathrm{InAs}∕\mathrm{GaAs}$ lens-shaped QDs from Ref. 15 and our calculation (solid curve) and its uncertainties (gray band). The diffusion lengths on the horizontal axis are deduced from the blueshift of the WL.