Off-centering of hydrogenic impurities in quantum dots

We report exact numerically calculated ground state and binding energies of a hydrogenic donor impurity confined everywhere inside a spherical quantum dot (QD) surrounded by air or a vacuum. Finite spatial steplike potentials allowing the electronic density to partially leak outside the QD are considered. This model faces a divergence produced by the self-polarization potential at the position of the dielectric mismatch. We bypass it by replacing the edge steplike dielectric mismatch by a continuous variation within an extremely thin layer at this edge. A comprehensive study of several confining factors influencing electronic and binding energies is carried out and a highly nonadditive interplay is found. Our calculations show that within both the strong and weak confinement regimes we may be faced with three different behavior regimes. We call them low, intermediate, and high. In the low and intermediate behaviors, the mass, polarization, and self-polarization effects exert a very strong influence on the electron density distribution, so that perturbational estimations of the binding energy may not be appropriate even in the strong confinement regime. These low and intermediate behavior regimes are responsible for binding energy profiles not being monotonously decreasing vs off-centering. It is even theoretically possible to design systems with off-centering independent binding energies.

Figure 1

(a) Ground state binding energy vs impurity off-centering $z_I$ corresponding to a $R=3\mathrm{nm}$ spherical $\mathrm{Si}{\mathrm{O}}_2$ nanocrystal in air or vacuum $(m_o^{*}={ϵ}_o=1)$. Calculations have been performed without (S0, dotted line) and with the inclusion of the self-polarization contributions (S1, full line). $\mathrm{Si}{\mathrm{O}}_2$ parameters are specified in the text. (b) Same as (a) but with $V_0=10\mathrm{eV}$ and the self-polarization potential that is now calculated with ${ϵ}_i=4$ and ${ϵ}_0=1$.

Figure 2

(a) Exact and first-order perturbational estimations of binding energy $E_b$ vs off-centering $z_I$ for a $R=3\mathrm{nm}$ spherical QD in air or vacuum ($m_i^{*}=0.05$, $m_0^{*}=1$, ${ϵ}_i=4$, ${ϵ}_o=1$ and $V_0=0.9\mathrm{eV}$). S1 (full lines) include while S0 (dotted lines) exclude self-polarization contributions. The zeroth-order wave functions employed in the perturbational calculations are those of the impurity-free QD in the presence and absence of the self-polarization potential, respectively. (b) Same as (a) but with $V_0=10\mathrm{eV}$ and ${ϵ}_i=4$. (c) Same as (a) but with $V_0=10\mathrm{eV}$ and ${ϵ}_i=8$.

Figure 3

(a) Ground state binding energy vs spatial confinement $V_0$ corresponding to a $R=3\mathrm{nm}$, ${ϵ}_i=4$, $m_i^{*}=0.2$ spherical QD in air or vacuum $(m_o^{*}={ϵ}_o=1)$. Results for $z_I=0.0$, 0.4, 0.6, 0.8, and 0.9 are displayed following the order indicated by the auxiliary arrow. Full (dotted) lines correspond to calculations including (excluding) self-polarization effects. (b) Same as (a) but with ${ϵ}_i=8$. (c) Same as (a) but with ${ϵ}_i=16$.