Signatures of antibonding hole ground states in exciton spectra of vertically coupled quantum dots in an electric field

We study exciton energy spectra in a pair of vertically coupled self-assembled quantum dots in external electric field. We perform a systematic comparison of the four-band Luttinger Kohn modeling producing an antibonding hole ground state with the single valence-band approximation in which the hole ground state has the bonding character. We find that the single-band approximation remains relevant for description of the electric field dependence of the photoluminescence spectrum for interdot barrier thickness of 7 nm or larger. We explain that for thinner barriers the antibonding character of the hole orbital can be deduced from the ground state recombination probability as a function of the electric field.

Figure 1

(Color online) Vertical confinement potential for electron and hole in a pair of identical dots in presence of an electric field of $F=50\text{kV}/\text{cm}$ applied parallel to the symmetry axis of the system $\left(z\right)$.

Figure 2

(Color online) (a) Two lowest-energy levels of the hole for identical pair of quantum dots as calculated in the heavy hole single-band approximation [light gray (red) curves] as function of the interdot barrier thickness $b$. Black curves show the results obtained in the four-band KL Hamiltonian [Eq. (2)] and dark gray (blue) ones the results of the six-band Hamiltonian given by Eq. (A1). (b) Comparison of the results of the four-band modeling with the assumption of infinite potential well in the radial direction (red curve) and the results obtained in a model of a finite potential disklike quantum well of depth $V^h=530\text{meV}$. (c) Two upper curves show the contribution of the light hole to the ground state wave function as obtained in the four-band modeling (black curve). The contribution of the split-off (SO) valence band in the six-band modeling is also shown.

Figure 3

(Color online) Energy spectrum for a pair of identical dots in function of the electric field. The interdot barrier thickness $b=1.2$, 2.1, 4.1, and 7 nm is assumed in panels (a) and (e), (b) and (f), (c) and (g), and (d) and (h), respectively. Radius of the circles is set proportional to the recombination probability. The upper row of plots was obtained with the four-band description of the hole, and the lower row of plots by the heavy hole approximation. The insets in (a), (b), and (d) show the charge distribution of the carriers in the two dots. The areas of the circles is proportional to the electron (the right pair of circles) and hole (the left pair of circles) charge accumulated in the lower dot (the lower pair of circles) and in the upper dot (the upper pair of circles).

Figure 4

(Color online) Ground state recombination probability for a pair of identical dots separated by the barrier of (a) $b=2.1\text{nm}$ and (b) $b=4.1\text{nm}$. Black curves show the results obtained with the kp method and the red lines by the heavy hole model.

Figure 5

(Color online) Ground state and first-excited state wave functions plotted on the axis of the stack for ${\rho }_e={\rho }_h=0$ for a pair of identical dots separated by the barrier of (a)–(d) $b=2.1\text{nm}$ and (e)–(h) $b=4.1\text{nm}$ for $F=0$ and $F=50\text{kV}/\text{cm}$. The horizontal (vertical) axes correspond to the electron (heavy hole) $z$ coordinate. Red and blue colors correspond to opposite signs of the wave functions.

Figure 6

(Color online) Same as Fig. 3 but for a slightly asymmetric pair of quantum dots, the lower one has height $h_l=2\text{nm}$ and the upper $h_u=2.1\text{nm}$.

Figure 7

(Color online) Same as Fig. 4 but for the pair of nonidentical dots $h_l=2\text{nm}$, $h_u=2.1\text{nm}$. Barrier thickness $b=2.1\text{nm}$ is applied in (a) and $b=4.1\text{nm}$ in (b).

Figure 8

(Color online) Same as Fig. 3 but for strongly asymmetric pair of quantum dots, the lower one has height $h_l=2\text{nm}$ and the upper $h_u=2.5\text{nm}$.

Figure 9

(Color online) Same as Fig. 4 but for the pair of nonidentical dots $h_l=2\text{nm}$, $h_u=2.5\text{nm}$. Barrier thickness $b=2.1\text{nm}$ is applied in (a) and $b=4.1\text{nm}$ in (b).