Exciton and negative trion dissociation by an external electric field in vertically coupled quantum dots

We study the Stark effect for an exciton confined in a pair of vertically coupled quantum dots. A single-band approximation for the hole and a parabolic lateral confinement potential are adopted which allows for the separation of the lateral center-of-mass motion and consequently for an exact numerical solution of the Schrödinger equation. We show that for intermediate tunnel coupling the external electric field leads to the dissociation of the exciton via an avoided crossing of bright and dark exciton energy levels which results in an atypical form of the Stark shift. The electric-field-induced dissociation of the negative trion is studied using the approximation of frozen lateral degrees of freedom. It is shown that in a symmetric system of coupled dots the trion is more stable against dissociation than the exciton. For an asymmetric system of coupled dots the trion dissociation is accompanied by a positive curvature of the recombination energy line as a function of the electric field.

Figure 1

Exciton energy spectrum as a function of external electric field $F$ for barrier thickness $b=2\mathrm{nm}$ (a), $b=4\mathrm{nm}$ (b), $b=7\mathrm{nm}$ (c). The area of the dots is proportional to the recombination probability. The insets in (a) and (b) show zooms of regions marked by rectangles.

Figure 2

(a), (c) contour plots of wave functions at the axis $\rho =0$ of the system and (b) probability density integrated over the lateral degrees of freedom ${\int }_0^{\infty }d\rho \rho {\mid \chi (\rho ,z_e,z_h)\mid }^2$ for different values of the electric field for barrier thickness $b=2\mathrm{nm}$ (a), (b) and $b=7\mathrm{nm}$ (c). Lower plots correspond to lower energies. Shaded area show the quantum wells for the electron and for the hole. Dashed line shows the nodal surface of the wave function.

Figure 3

Electron (dotted line) and hole (solid line) charge accumulated in the left quantum dot as function of the electric field for different barrier thicknesses. Inset shows the dipole moment as function of the field.

Figure 4

Stark effect for the asymmetric system of quantum dots at $b=6\mathrm{nm}$. In (a) the electron confinement is symmetric and the left dot for the hole is shallower by $3\mathrm{meV}$. In (b) the hole confinement is symmetric and the left dot for the electron is shallower by $3\mathrm{meV}$. The insets in (a) and (b) show a schematic drawing of the vertical confinement for $F=0$.

Figure 5

(a) Contour plots of the wave functions at the axis $\rho =0$ corresponding to the energy levels shown in Fig. 4a. (b) similar as for Fig. 4b. Higher plots correspond to higher energies.

Figure 6

Stark effect for the asymmetric hole confinement of Fig. 4a for $b=4.5\mathrm{nm}$ (a) and $b=3\mathrm{nm}$ (b). The area of the dots shows the recombination probability.

Figure 7

The exact results solid lines and the results with the frozen degree of freedom for equal hole and electron confinement energies (dashed lines) and with equal hole and electron confinement lengths (dotted line) for the parameters of Fig. 4b.

Figure 8

(a) Difference of the ground-state trion energy and the electron ground state (trion recombination energy with respect to GaAs energy gap—solid lines) and the exciton ground-state energy (exciton recombination energy—dashed lines). The curves are labeled by the barrier thickness $b$ in nanometers. Inset shows the difference of the exciton and trion energy lines. (b) Electron (dotted lines) and hole (solid lines) charge on the left side of the origin as function of the electric field. Inset shows the dipole moment.

Figure 9

Probability density integrated over the vertical coordinate of one of the electrons (right panel) and the vertical coordinate of the hole (left panel) for $b=6\mathrm{nm}$ and different values of the electric field $F$. The shaded areas show the positions of the quantum dots.

Figure 10

(a) The trion recombination energies as functions of the electric field (solid lines) for a pair of coupled quantum dots. The curves are labeled by the barrier thickness in nanometers. Right dot has a width of $6\mathrm{nm}$ and the width of the left dot is $4\mathrm{nm}$. Dotted line shows the exciton recombination energy for $b=10\mathrm{nm}$. (b) The electron (solid lines) and the hole (dashed lines) charge accumulated in the right dot for $b=4$ and $10\mathrm{nm}$