Microscopic Dielectric Response Functions in Semiconductor Quantum Dots

We calculate and model the microscopic dielectric response function for quantum dots using first principle methods. We find that the response is bulklike inside the quantum dots, and the reduction of the macroscopic dielectric constants is a surface effect. We present a model for the microscopic dielectric function which reproduces well the directly calculated results and can be used to solve the Poisson equation in a nanosystem.

Figure 1

(a) Cross section of the change in the charge density of a 933-atom GaAs quantum dot due to a $\delta$-like perturbing potential. The full grid is only used in the small central region with significant $\Delta {\rho }_d(\mathbf{r})$. The contour plot at the base is for the total ${\rho }_d(\mathbf{r})$ of the QD. (b) Difference with respect to the bulk response. $a_0$ is the bulk lattice constant, $a_B=0.529\AA$ is the Bohr radius, and the $\Delta \rho$ shown in all the figures contains only one spin, thus is half of the total response.

Figure 2

Spherically averaged $\Delta \rho (\mathbf{r})$ for a 933-atom GaAs dot and bulk under different perturbations. (a) $\delta$ perturbation. (b) Coulomb perturbation truncated at $a_0$. (c) Coulomb perturbation truncated at $2a_0$. The vertical arrows indicate the dot radius $R_d$.

Figure 3

As in Fig. 2, but with a smaller 465-atom GaAs dot and the Coulomb perturbation truncated at $2a_0$. The Bulk and Dot are direct LDA-calculated results, and the Models are their corresponding model results according to Eq. (9). Note, in this case, the bulk and dot results are significantly different and our model predicts both of them accurately.

Figure 4

As in Fig. 2, but with a 465-atom GaAs dot under a (a) Coulomb perturbation truncated at $a_0$; (b) $\delta (r_2-a_0)$ perturbation; and (c) $\delta (r_2-1.3a_0)$ perturbation. The dotted line is the application of the model in Eq. (5). The insets show a close-up of the region close to the dot surface. The dashed-dotted line in (a) is the mask function $w(r)$ in Eq. (6).

Figure 5

Off center $\delta$-like perturbation (same amplitude as in Fig. 1) for the 465-atom GaAs quantum dot. Note the scale, the $\Delta {\rho }_{\mathrm{dot}}-\Delta {\rho }_{\mathrm{bulk}}$ is bigger than that in Fig. 1b. The thick dashed line in the lower panel contour plot indicates the boundary of the QD as the total QD charge density drops to 4% of its bulk average. The $R_d$ of this dot is $2.1a_0$. Only a small portion of the (110) cross section is shown here.