Size-Dependent Fine-Structure Splitting in Self-Organized $\mathrm{InAs}/\mathrm{GaAs}$ Quantum Dots

A systematic variation of the exciton fine-structure splitting with quantum dot size in single $\mathrm{InAs}/\mathrm{GaAs}$ quantum dots grown by metal-organic chemical vapor deposition is observed. The splitting increases from $-80$ to as much as $520\mu \mathrm{eV}$ with quantum dot size. A change of sign is reported for small quantum dots. Model calculations within the framework of eight-band $\mathbf{k}·\mathbf{p}$ theory and the configuration interaction method were performed. Different sources for the fine-structure splitting are discussed, and piezoelectricity is pinpointed as the only effect reproducing the observed trend.

Figure 1

Energy scheme illustrating the effect of the exchange interaction on the exciton ($X$) bright states. $XX$ denotes the biexciton ground state. $|M⟩$ is the exciton total angular momentum, $\sigma \pm$ denotes circularly polarized light, and ${\pi }_{x/y}$ linearly polarized light with $x=[110]$ and $y=[1\overline{1}0]$. $D_{2d}$, $C_{4v}$, and $C_{2v}$ indicate the confinement potential symmetry.

Figure 2

Polarized spectra for two different QDs emitting at high and low energies are shown. The FSS is $-80$ and $420\mu \mathrm{eV}$, respectively. Gray lines in the lower panel originate from charged excitonic complexes not considered in this Letter.

Figure 3

Measured exciton fine-structure splitting (FSS) as a function of exciton recombination energy for 49 QDs. The gray curve depicts the QD ensemble luminescence with numbers denoting QD height in units of InAs monolayers. A clear trend of the FSS is observed. The data points extracted from Fig. 2 are indicated by red (gray) circles.

Figure 4

The piezoelectric potential calculated for (a) the InAs bulk and (b) the extrapolated $e_{14}$ value. The potential is shown for a lateral cut near the base of the QD (white square), which is modeled as a truncated pyramid having a base length of 12 nm.

Figure 5

Calculated exciton fine-structure splitting (FSS) values for a series of QDs with a square base including piezoelectricity (b1) and a series of QDs elongated in the $[1\overline{1}0]$ crystal direction in absence of piezoelectricity (b2). $C(n,n)$ denotes the number of electron and hole states included in the CI calculation. The structural assumptions of the modeled series are shown in (a1) and (a2).