Nominally forbidden transitions in the interband optical spectrum of quantum dots

We calculate the excitonic optical absorption spectra of $(\mathrm{In},\mathrm{Ga})\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ self-assembled quantum dots by adopting an atomistic pseudopotential approach to the single-particle problem followed by a configuration-interaction approach to the many-body problem. We find three types of allowed transitions that would be naively expected to be forbidden: (i) transitions that are parity forbidden in simple effective mass models with infinite confining wells (e.g., $1S\text{−}2S$, $1P\text{−}2P$) but are possible because of finite band offsets and orbital-mixing effects; (ii) light-hole-to-conduction-band transitions, enabled by the confinement of light-hole states; and (iii) transitions that show an enhanced intensity due to electron-hole configuration mixing with allowed transitions. We compare these predictions with results of eight-band $\mathbf{k}\bullet \mathbf{p}$ calculations as well as recent spectroscopic data. Transitions of types (i) and (ii) explain recently observed satellites of the allowed $P\text{−}P$ transitions.

Figure 1

Optical absorption spectrum of $X^0$ in a lens-shaped ${\mathrm{In}}_{0.6}{\mathrm{Ga}}_{0.4}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dot (base diameter $b=200\mathrm{\AA }$, height $h=20\mathrm{\AA }$) calculated (a) in the single-particle approximation [Eq. (2)] under in-plane ($\widehat{\mathbf{e}}\parallel \left[100\right]$; top) and out-of-plane ($\widehat{\mathbf{e}}\parallel \left[001\right]$; bottom) polarization; and (b) at the many-particle level [Eq. (4)] for unpolarized light. Energy is shown relative to (a) the single-particle gap ${\mathcal{E}}_0^{\left(e\right)}-{\mathcal{E}}_0^{\left(h\right)}=1333\mathrm{meV}$ and (b) the ground-state energy $E^{\left(0\right)}\left(X^0\right)=1309\mathrm{meV}$ of $X^0$.

Figure 2

Idem Fig. 1b for two ${\mathrm{In}}_{0.6}{\mathrm{Ga}}_{0.4}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dots that confine two (left panel) and three (right) shells of electron states, with heights $h=20$ and $30\mathrm{\AA }$, respectively, and base $b=252\mathrm{\AA }$.

Figure 3

(Color) (a) First (thick line) and second (thin line) strain-modified valence-band offsets along a line parallel to [001] that pierces a lens-shaped ${\mathrm{In}}_{0.6}{\mathrm{Ga}}_{0.4}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dot through its center. The dot size is $b=252\mathrm{\AA }$ and $h=20\mathrm{\AA }$. Position is measured in units of the lattice parameter of GaAs $\left(a_{\mathrm{Ga}\mathrm{As}}\right)$ and the energies are relative to the GaAs VBM $[E_v\left(\mathrm{Ga}\mathrm{As}\right)=-5.620\mathrm{eV}]$. (b) Wave functions of the first hole state with significant lh character for different ${\mathrm{In}}_{0.6}{\mathrm{Ga}}_{0.4}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ dots. Isosurfaces enclose 75% of the charge density, while contours are taken at $1\mathrm{nm}$ above the base. The energy ${\mathcal{E}}_j^{\left(h\right)}-E_v\left(\mathrm{Ga}\mathrm{As}\right)$ of the state appears in each panel.

Figure 4

Optical absorption spectrum of $X^0$ in two lens-shaped ${\mathrm{In}}_{0.6}{\mathrm{Ga}}_{0.4}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dots that confine two (left panel) and three (right) shells of electron states. In both cases, the spectra are calculated within a model single-particle (M-SP) [Eq. (2)] (a) and configuration-interaction (M-CI) approach [Eq. (4)] (b), assuming degenerate $1P$, $2D$, and $2S$ (in three-shell dot) electron and hole states but retaining the atomistic wave functions. Vertical scales are different for each dot.