Stark effect on the exciton spectra of vertically coupled quantum dots: Horizontal field orientation and nonaligned dots

We study the effect of an electric field on an electron-hole pair in an asymmetric system of vertically coupled self-assembled quantum dots taking into account their nonperfect alignment. We show that the nonperfect alignment does not qualitatively influence the exciton Stark effect for the electric field applied in the growth direction, but can be detected by application of a perpendicular electric field. We demonstrate that the direction of the shift between the axes of nonaligned dots can be detected by rotation of a weak electric field within the plane of confinement. Already for a nearly perfect alignment the two-lowest energy bright exciton states possess antilocked extrema as a function of the orientation angle of the horizontal field which appear when the field is parallel to the direction of the shift between the dot centers.

Figure 1

(Color online) The electron confinement potential [Eq. (2)] in meV, in the $y=0$ plane, for the offset between the axes of the dots $d=10\mathrm{nm}$ (equal to the dot radius $R$) and the vertical distance between the dot centers of $h=10\mathrm{nm}$. The height of both the dots is $2Z=4\mathrm{nm}$. The lower (negative $z$) dot has depth $V_l^e=518\mathrm{meV}$, the dot for positive $z$ has depth of $V_u^e=508\mathrm{meV}$.

Figure 2

The exciton spectra for the electric field oriented along the $z$ direction for a vertical distance between the dots centers $h=10\mathrm{nm}$ for (a) perfectly aligned dots $d=0$, (b) horizontal distance between the vertical symmetry axis of the dots $d=5\mathrm{nm}$, and (c) $d=10\mathrm{nm}$.

Figure 3

(Color online) The electron and hole probability density distributions for the ground state and the first excited bright state in the absence of the electric field for (a) $d=5\mathrm{nm}$ and $h=10\mathrm{nm}$, and (b) $h=8\mathrm{nm}$.

Figure 4

The exciton spectra for the electric field oriented along the $x$ direction for the vertical distance between the dots centers $h=10\mathrm{nm}$ for (a) perfectly aligned dots $d=0$, (b) horizontal distance between the vertical symmetry axis of the dots $d=5\mathrm{nm}$, and (c) $d=10\mathrm{nm}$. Single-particle densities for the bright energy levels labeled by numbers 1–4 in (b) are presented in Fig. 6.

Figure 5

(Color online) Electron and hole densities for three lowest-energy states for $d=0\mathrm{nm}$ and $h=10\mathrm{nm}$ taken within the plane of confinement of the lower, deeper dot $(z=-5\mathrm{nm})$. For the field of $F=20\mathrm{kV}∕\mathrm{cm}$ parallel to the $x$ axis (arrows at the electron and hole panels at the top-left side of the figure show the direction of the electric force acting on the electron and the hole). The energy levels of these states are displayed in Fig. 4a.

Figure 6

(Color online) The $y=0$ cross section of the electron and hole probability density distributions for the ground state and the first excited bright state for different values of the electric field in the $x$ direction (given at the top of the figure for each set of plots) for $d=5$ and $h=10\mathrm{nm}$. The selected bright states denoted by numbers 1–4, as in Fig. 4b. The contour plots for states 1 and 2 at $F=0$ were shown in Fig. 3.

Figure 7

The exciton spectra for the electric field oriented along the $x$ direction for a vertical distance between the dots centers $h=8\mathrm{nm}$ for (a) horizontal distance between the vertical symmetry axis of the dots $d=5\mathrm{nm}$ and (b) $d=10\mathrm{nm}$.

Figure 8

(Color online) The ground-state mean electron and hole position dependence on the horizontal electric field oriented along the $x$ direction for $h=8\mathrm{nm}$ (a) perfectly aligned dots and (b) offset of the dots axes of $d=5\mathrm{nm}$. The energy levels for parameters used in (b) are displayed in Fig. 7a.

Figure 9

The exciton spectra for the electric field $F=20\mathrm{kV}∕\mathrm{cm}$ oriented in the $x\text{−}y$ plane as function of the angle between the field and the $x$ axis $[\mathbf{F}=F\mathbf{(}\cos \left(\varphi \right),\sin \left(\varphi \right),0\mathbf{)}]$. The plots in the upper (lower) row correspond to $h=10\mathrm{nm}$ $(h=8\mathrm{nm})$. Figures (a), (b), (c), (d), (e), (f), and (g), (h) correspond to $d=2,5,10$ and $15\mathrm{nm}$, respectively. Electron and hole densities for energy levels of plot (d) are displayed in Fig. 10.

Figure 10

(Color online) The $y=0$ cross section of the electron (the panels marked by “e”) and hole (the panels marked by “h”) probability density distributions for the ground state (denoted by 1 in a square frame) and the first excited bright state (denoted by 2) for $d=5$ and $h=8\mathrm{nm}$. The corresponding energy levels are displayed and marked by the same numbers in Fig. 9d. The horizontal axis corresponds to $x$ variable, the vertical to $z$ variable. The electric field is kept at $\mid \mathbf{F}\mid =20\mathrm{kV}∕\mathrm{cm}$, each panel corresponds to a different orientation of the electric field within the $x\text{−}y$ plane given by $\varphi$ which is the angle between the $\mathbf{F}$ and the $x$ axis. The left plots in each column show the electron distribution and the right ones show the hole distribution The plots for states 1 and 2 at $F=0$ were shown in Fig. 3b (ground and excited states, respectively).