Electronic structure of nanocrystal quantum-dot quantum wells

The electronic states of $\mathrm{Cd}\mathrm{S}∕\mathrm{Cd}\mathrm{Se}∕\mathrm{Cd}\mathrm{S}$ colloidal nanocrystal quantum-dot quantum wells are studied by large-scale pseudopotential local density approximation (LDA) calculations. Using this approach, we determine the effects of CdS core size, CdSe well thickness, and CdS shell thickness on the band-edge wave functions, band-gap, and electron-hole Coulomb interactions. We find the conduction-band wave function to be less confined to the CdSe well layer than predicted by $\mathbf{k}\bullet \mathbf{p}$ effective-mass theory, which accounts for the previous underestimation of the electron $g$ factor.

Figure 1

(Color online) Top (cross section) and side view of the valence-band maximum (VBM) wave-function isosurface for a $2.9\text{−}\mathrm{nm}$-diam CdS core, surrounded by a 2 ML CdSe well and a 2 ML CdS shell.

Figure 2

Average radial probability distribution of the conduction-band minimum and valence-band maximum, respectively, of (a,b) varying CdS shell thickness for a $2.2\text{−}\mathrm{nm}$-diam CdS with 1 ML CdSe well, and (c,d) varying CdS core diameter for a 1 ML CdSe well and 2 ML CdS shell. The boundaries between the different semiconductor materials are indicated by the vertical lines in the schematic between the CBM and VBM plots. Only a small change in the VBM is observed when increasing the number of shell monolayers from one to two; further increases do not alter the VBM distribution. In contrast, the CBM is not well localized to the quantum well, due to the small conduction-band effective mass. This behavior is qualitatively similar to that predicted by effective-mass theory (Refs. 16, 18). Increasing the core diameter moves the localized VBM state outward, and the delocalized CBM widens to fill the nanocrystal.

Figure 3

Average radial probability distribution of the conduction-band minimum (CBM) and valence-band maximum (VBM), respectively, of (a,b) 1 ML CdS shell, (c,d) 2 ML CdS shell, (e,f) 3 ML CdS shell QDQW nanocrystals, each with a $2.2\text{−}\mathrm{nm}$-diam CdS core. The CdSe well thicknesses are indicated as dotted, dash-dotted, dashed, and solid lines, for 1, 2, 3, and 4 ML thicknesses, respectively, with the boundaries indicated as in Fig. 2.

Figure 4

Comparison of the conduction-band minimum (CBM) wavefunction obtained using the $\mathbf{k}\bullet \mathbf{p}$ effective-mass approximation (Refs. 16, 18) (EMA) to the present local-density approximation (LDA) results. The QDQW consists of a $3.4\text{−}\mathrm{nm}$-diam CdS core, 2 ML CdSe well, and a $1.6\text{−}\mathrm{nm}$-thick CdS shell, shown schematically above the figure. The dotted line shows a pure CdS nanocrystal of the same total radius as the QDQW, indicating the delocalized nature of the LDA CBM, as compared to that of the EMA.

Figure 5

(Color online) Histogram of the bond lengths in the $3.4\text{−}\mathrm{nm}$-diam CdS core, $1.6\text{−}\mathrm{nm}$-thick CdS shell QDQW as a function of CdSe well thickness (in monolayers). The vertical dotted lines show the unstrained Cd-S $\left(2.519\mathrm{\AA }\right)$ and Cd-Se $\left(2.633\mathrm{\AA }\right)$ bond lengths for the pure semiconductor materials.

Figure 6

Band gap for the $\mathrm{Cd}\mathrm{S}∕\mathrm{Cd}\mathrm{Se}∕\mathrm{Cd}\mathrm{S}$ QDQWs, as a function of the total number of radial monolayers of the nanocrystal. The leftmost point indicates a $2.2\text{−}\mathrm{nm}$-diam CdS core, surrounded by a 1 ML CdSe well, and a 1 ML CdS shell. The notation $x∕y∕z$ indicates the number of core, well, and shell monolayers, respectively. One monolayer is approximately $0.4\mathrm{nm}$ for both materials. Moving to the right shows the addition of monolayers to these layers. Solid lines connect changes in the shell, dotted lines connect changes in the well, and dashed lines connect changes to the core.

Figure 7

(a) Electron-hole Coulomb interaction, and (b) approximated exciton energy for the $\mathrm{Cd}\mathrm{S}∕\mathrm{Cd}\mathrm{Se}∕\mathrm{Cd}\mathrm{S}$ QDQWs, as a function of the total number of radial monolayers of the nanocrystal. Changes in core, well, and shell thickness are indicated as described in Fig. 6.