Binding of electrons, holes, and excitons in symmetric strained $\mathrm{In}\mathrm{P}∕{\mathrm{In}}_{0.49}{\mathrm{Ga}}_{0.51}\mathrm{P}$ triple quantum-dot molecules

The electron, hole, and exciton spectra in the strained quantum-dot molecule consisting of three vertically arranged type-II $\mathrm{In}\mathrm{P}∕{\mathrm{In}}_{0.49}{\mathrm{Ga}}_{0.51}\mathrm{P}$ self-assembled quantum dots are modeled by the $\mathbf{k}\bullet \mathbf{p}$ theory. For the sake of simplicity, we consider dots of cylindrical shape, but take into account the anisotropy of the strain through the continuum mechanical model. For thick spacers, the strain leads to an upward shift of the lowest energies in all explored electron shells, but for spacers thinner than, say, the coupling length, the quantum mechanical coupling prevails, and downward shifts are observed. The magnitudes of both the energy shift and the coupling length vary with the quantum-dot height. For the holes, the interplay of strain and mixing enables binding at larger distances than for the electrons. The overlap of the hole clouds is basically established by means of the light holes, which are confined by the strain in the spacer between the dots and may efficiently couple the heavy-hole states, which are localized inside the quantum dots. Similar to electrons, the exciton lowest-energy states of different angular momenta, as computed by an exact-diagonalization approach, exhibit overshoots on the single-quantum-dot levels. Good agreement is found with experiment on the spatial location of electrons and holes in the triple-quantum-dot molecules.

Figure 1

(a) Sketch of a triple-dot molecule (TDM) composed of stacked InP dots (indicated by the shaded areas) of radius $R$ and height $h$, symmetrically arranged with respect to the $z=0$ plane and separated by the InGaP spacer of thickness $d$. The simulation area is given by the large cylinder. The distribution of the grid points used in the strain calculation for $h=3\mathrm{nm}$, $R=8\mathrm{nm}$, and $d=6\mathrm{nm}$ along the $z$ direction (b) and along the $\rho$ direction (c).

Figure 2

The angular averaged effective potentials in the triple-dot molecule (TDM) composed of $3\text{−}\mathrm{nm}$-high quantum dots, along the $z$ and $\rho$ axes, are shown for $d=2\mathrm{nm}$ (dashed lines) and $d=10\mathrm{nm}$ (dotted lines). The SQD case (solid line) is added for comparison. The left panel shows variations of the effective potentials along the $z$ axis: (a) the CB, (b) the hh band, and (c) the lh band. The right panel shows the effective potentials along the $\rho$ axis: (d) the CB, (e) the hh band, and (f) the lh band.

Figure 3

Variations of the extrema of the effective potentials in the three bands along $(\rho =0,z)$ as function of the spacer thickness. The stack consists of $3\text{−}\mathrm{nm}$-high quantum dots. (a) The CB maximum is shown by the dashed line in the upper part, while the CB minimum is displayed by the solid line in the lower part. (b) The maxima of the hh (solid line) and lh (dashed line) effective potentials. (c) The minima of the hh (solid line) and lh (dashed line) effective potentials.

Figure 4

The splitting of the CB energy levels in the TDM, for (a) $h=2\mathrm{nm}$, (b) $h=3\mathrm{nm}$, and (c) $h=4\mathrm{nm}$. The even- and odd-electron states are indicated by the solid and dashed lines, respectively. The values of the coupling length in different shells are indicated by the arrows. The shaded areas represent the energy continuum.

Figure 5

The VB levels in TDM’s as a function of the spacer thickness for (a) $S_{3∕2}^{\pm }$ states, (b) $S_{1∕2}^{\pm }$ states, and (c) the highest-energy VB states of $S_{3∕2}^{\pm }$, $S_{1∕2}^{\pm }$, and $P_{5∕2}^{\pm }$ symmetries. (d) $f_h$ of the ground hole state as it varies with the thickness of the spacer. The dot height is $3\mathrm{nm}$, and the radius is $8\mathrm{nm}$. The highest-energy SQD states and the different regimes of hole localization for the $1S_{3∕2}^{+}$ and $1S_{1∕2}^{+}$ states are explicitly labeled. The shaded area on the right shows the position of the energy levels in a SQD. The ground state of holes is denoted by solid dots (●) if it is even and by open circles (엯) if it is odd.

Figure 6

The PD’s of the highest energy $S_{3∕2}$ VB triplet in the TDM for $h=3\mathrm{nm}$ and $d=1$, 3, 6, 10, and $20\mathrm{nm}$: (a) the $1S_{3∕2}^{+}$ state, (b) the $1S_{3∕2}^{-}$ state, and (c) the $2S_{3∕2}^{+}$ state. The PD distribution of holes in the $1S_{3∕2}^{+}$ SQD state is shown to the right of the three panels. GS denotes the ground states, and the dot boundaries are indicated by the dashed lines.

Figure 7

The PD distribution for the three highest-energy $f_h=1∕2$ VB states in the TDM, composed of $h=3\mathrm{nm}$ high quantum dots, as function of the spacer thickness, for the same five values of $d$ as in Fig. 6: (a) the $1S_{1∕2}^{+}$ state, (b) the $1S_{1∕2}^{-}$ state, and (c) the $2S_{1∕2}^{+}$ state. The ground state is indicated by GS and the dot boundaries by the dashed lines.

Figure 8

Variations of the energies of the even-exciton states in TDM’s with the spacer thickness. The lowest energy $Q_1^{\pm }$ and $Q_2^{\pm }$ exciton quartets are shown. The solid and thick dashed lines show $Q_1^{+}$ and $Q_2^{+}$ energy levels, respectively, while the dotted and thin dashed lines display the energies of the $Q_1^{-}$ and $Q_2^{-}$ quartets, respectively.

Figure 9

The spacer-thickness-induced variations in the exciton 2DPD, for (a) the $1Q_1^{+}$ quartet and (b) the $1Q_2^{-}$ quartet. Five values of the spacer thickness are selected to show the evolution of the probability density in the TDM composed of $3\text{−}\mathrm{nm}$-high quantum dots. For each state, the distribution of the PD in the SQD is displayed in a separate figure to the right of the main panels. The interfaces between the dots and the matrix are indicated by the dashed lines.

Figure 10

The dependence of the relative contributions of the different VB’s in the ground even-exciton states in the TDM: (a) $1Q_1^{+}$ and (b) $1Q_2^{-}$ states. The height (radius) of each quantum dot is $3\mathrm{nm}$ $\left(8\mathrm{nm}\right)$. The hh parts are indicated by the solid lines, and the dashed lines indicate the lh part, while the so parts are shown by the dotted lines. The horizontal lines in the gray areas denote the corresponding SQD values.

Figure 11

Variations of the ground exciton energies with $d$, computed for three values of the quantum-dot height, are compared with the experimental results shown by the boxes. The box at the right of the main figure is the transition energy in the SQD.