Excitonic properties of strained triple quantum-ring molecules

The tunneling coupling in three vertically stacked (In,Ga)As/GaAs quantum rings is investigated. With increasing inter-ring separation $\left(d\right)$, we find that the nonuniform strain results into a crossing of the lowest-energy electron states. Strain is also responsible for an increase in the ground electron energy above the level in the single quantum ring. The ground hole energy level exhibits decrease when $d$ decreases, which is typical for antibonding states in an unstrained structure. These effects lead to a local maximum in the dependence of the ground-state exciton energy on $d$. Our theoretical results compare well with recent photoluminescence measurements but deviate considerably from the calculations for flat bands in quantum-ring molecules. We conclude that the nonuniform character of the strain distribution gives rise to a peculiar exciton hybridization in self-assembled quantum-ring molecules.

Figure 1

(Color online) The effective potentials for $d=5\text{nm}$ (solid lines) and $d=1.5\text{nm}$ (dashed lines) for: (a) the electrons in the conduction band, (b) the heavy holes, and (c) the light holes. (d) Variation of the hydrostatic strain in the center of the central (solid line) and satellite rings (dashed line). The cross section of the rings is shown in the inset.

Figure 2

(Color online) The variations in the electron even (solid lines) and odd (dashed line) energy levels with inter-ring distance. The upper inset shows a detail of this dependence on the range $d\ge 10\text{nm}$, while the lower inset shows the level splitting in the flat-band structure. The dotted line in the upper inset denotes the energy of the ground state in a single quantum ring.

Figure 3

(Color online) (a) The even (solid lines) and the odd (dashed line) hole levels as functions of the inter-ring separation. The eigenenergy in the single quantum ring is displayed as the horizontal dotted line. The inset shows the level splitting in the unstrained structure. (b) The wave functions of the $1s^{+}$, $1s^{-}$, and $2s^{+}$ states.

Figure 4

(Color online) The variation in the ground $1S^{+}$ exciton energy with inter-ring separation (solid line). The experimental energies are shown by symbols. The energy level in the single quantum ring (dotted line) is close to the experimental energy, and the tunneling-dominated and strain-dominated regimes are indicated. The inset displays the similar dependence for the flat-band structure.