Excitons, biexcitons and trions in self-assembled (In,Ga)As/GaAs quantum dots: Recombination energies, polarization and radiative lifetimes versus dot height
Gustavo A. Narvaez
We calculate the height dependence of recombination energies, polarization and radiative lifetimes of the optical transitions of various excitonic complexes: neutral excitons ($X^0$), negatively- ($X^{-}$) and positively-charged ($X^{+}$) trions, and biexcitons ($XX^0$) in lens-shaped, self assembled In$_{0.6}$Ga$_{0.4}$As/GaAs quantum dots. By using an atomistic pseudopotential method combined with the configuration-interaction method, we predict the following. % (i) The recombination energy of the lowest transition of $X^-$ blue-shifts as height increases, whereas that of $X^+$ red-shifts. Remarkably, the recombination of $XX^0$ shows a red-shift at small heights, reaches a maximum shift, and then blue-shifts for taller dots. This feature results from the height dependence and relative magnitude of the inter-electronic direct Coulomb interaction. % (ii) Changes in dot height lead to a bound-to-unbound crossover for $X^-$, $X^+$ and $XX^0$. % (iii) When considering the $[110]$ and $[1\bar{1}0]$ directions, the lowest transitions of $X^0$ and $XX^0$ manifest $[110]$ vs $[1\bar{1}0]$ in-plane polarization anisotropy that switches sign as a function of height as well as alloy randomness. $X^-$ and $X^+$ show transitions with negligible polarization anisotropy regardless of height. % (iv) The ground state of $X^0$ is split in a low-energy pair that is forbidden (dark) and a high-energy pair that is allowed; thus, at $T=0\;{K}$ the radiative lifetime $\tau(X^0)$ is long ($\sim {ms}$) due to the dark exciton. On the other hand, at $T=10\;{K}$, $\tau(X^0)$ decreases moderately as height increases and its magnitude ranges from $2$-$3\,{ns}$. The ground state of $X^-$ and $X^+$, and that of $XX^0$ is allowed (bright); so, $\tau(X^-)$, $\tau(X^+)$ and $\tau(XX^0)$ are fast ($\sim {ns}$) even at $T=0\,{K}$. These radiative lifetimes depend weakly on height. In addition, $\tau(X^-)\sim\tau(X^+)\simeq 1.1\,{ns}$, while $\tau(XX^0)\simeq 0.5\,{ns}$. % We compare our predictions with available spectroscopic data.
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\documentclass[prb,twocolumn,aps,floatfix]{revtex4}          % Physical Review B

\usepackage{graphicx}% Include figure files
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\begin{document}

\preprint{}

\title{Excitons, biexcitons and trions in self-assembled (In,Ga)As/GaAs
  quantum dots: Recombination energies, polarization and radiative lifetimes
  versus dot height}

\author{Gustavo A. Narvaez}
\author{Gabriel Bester}
\author{Alex Zunger}

\affiliation{National Renewable Energy Laboratory, Golden, Colorado 80401, USA}

\date{\today}

\begin{abstract}
  We calculate the height dependence of recombination energies, polarization
  and radiative lifetimes of the optical transitions of various excitonic
  complexes: neutral excitons ($X^0$), negatively- ($X^{-}$) and
  positively-charged ($X^{+}$) trions, and biexcitons ($XX^0$) in lens-shaped,
  self assembled In$_{0.6}$Ga$_{0.4}$As/GaAs quantum dots. By using an
  atomistic pseudopotential method combined with the configuration-interaction
  method, we predict the following.
%
  (i) The recombination energy of the lowest transition of $X^-$ blue-shifts
  as height increases, whereas that of $X^+$ red-shifts. Remarkably, the
  recombination of $XX^0$ shows a red-shift at small heights, reaches a
  maximum shift, and then blue-shifts for taller dots. This feature results
  from the height dependence and relative magnitude of the inter-electronic
  direct Coulomb interaction.
%
  (ii) Changes in dot height lead to a bound-to-unbound crossover for $X^-$,
  $X^+$ and $XX^0$.
%
  (iii) When considering the $[110]$ and $[1\bar{1}0]$ directions, the lowest
  transitions of $X^0$ and $XX^0$ manifest $[110]$ vs $[1\bar{1}0]$ in-plane
  polarization anisotropy that switches sign as a function of height as well
  as alloy randomness. $X^-$ and $X^+$ show transitions with negligible
  polarization anisotropy regardless of height.
%
  (iv) The ground state of $X^0$ is split in a low-energy pair that is
  forbidden (dark) and a high-energy pair that is allowed; thus, at $T=0\;{\rm
    K}$ the radiative lifetime $\tau(X^0)$ is long ($\sim {\rm ms}$) due to
  the dark exciton. On the other hand, at $T=10\;{\rm K}$, $\tau(X^0)$
  decreases moderately as height increases and its magnitude ranges from
  $2$-$3\,{\rm ns}$. The ground state of $X^-$ and $X^+$, and that of $XX^0$
  is allowed (bright); so, $\tau(X^-)$, $\tau(X^+)$ and $\tau(XX^0)$ are fast
  ($\sim {\rm ns}$) even at $T=0\,{\rm K}$. These radiative lifetimes depend
  weakly on height. In addition, $\tau(X^-)\sim\tau(X^+)\simeq 1.1\,{\rm ns}$,
  while $\tau(XX^0)\simeq 0.5\,{\rm ns}$.
%
We compare our predictions with available spectroscopic data.  
\end{abstract}

\maketitle


%%%%%%%%%%%%%%%%
%              %
% Introduction %
%              %
%%%%%%%%%%%%%%%%
%
%
\section{Introduction}

Single-dot spectroscopy makes it possible to probe dot-to-dot changes in the
excitonic properties of self-assembled InGaAs/GaAs quantum dots.
\cite{smith_PRL_2005,ware_PhysicaE_2005,skolnick_ARMR_2004,guffarth_PhysicaE_2004,rodt_papers,urbaszek_PRL_2003,finley_PRB_2001,findeis_SSC_2000,landin_papers} 
Both single-particle and many-particle aspects of these properties depend
non-trivially on the quantum dots size and shape,\cite{books+review}
reflecting not only simple quantum-confinement physics, but also electronic
structure effects such as interband, intervalley, spin-orbit and
strain-induced state coupling. The description of these effects require an
atomistic multi-band approach.
\cite{wang_PRB_1999,wang_APL_2000,zunger_pss.b_2001,TB_method,sheng_PRB_2005}
%
Here we adopt a method that is based on screened pseudopotentials and the
configuration-interaction approach, and address the changes with height of
recombination (emission) energies, polarization and radiative lifetimes of
various neutral and charged excitons: the neutral exciton ($X^0$), negatively-
($X^{-}$) and positively-charged ($X^{+}$) excitons, and the biexciton
($XX^0$) in lens-shaped, self-assembled alloyed In$_{0.6}$Ga$_{0.4}$As/GaAs 
quantum dots. Our predictions compare reasonably well with available spectroscopic data.  We
also compare our findings in In$_{0.6}$Ga$_{0.4}$As/GaAs dots with those in
pure, non-alloyed InAs/GaAs dots.


%%%%%%%%%%%%%%%%%%%%
%                  
% Energetics of Xs 
%                  
%%%%%%%%%%%%%%%%%%%%
%
%
%
\section{Energetics of the monoexciton, charged trions and biexciton}

%%%%%%%%%%%
% Figure 1
%%%%%%%%%%%
%
\begin{figure*}[ht]
\includegraphics[width=10.0cm]{./Fig_1.eps}
\caption{{\label{Fig_1}} Quantitative many-particle, configuration-interaction
  low-lying energies and lowest optical transitions for (a) $X^0$, (b) $XX^0$,
  (c) $X^-$, and (d) $X^+$ in a lens-shaped In$_{\rm 0.6}$Ga$_{\rm
    0.4}$As/GaAs dot with base diameter $b=252\,${\AA} and height $h=75\,${\AA}. In
  each panel [(a)-(b)], the energy is shown relative to the ground-state
  energy of the corresponding exciton $\chi^q$ (=$X^0$, $X^-$, $X^+$, and
  $XX^0$). Two horizontally aligned dashes and dotted lines
  indicate, respectively, two-fold degenerate bright and dark levels. The
  dominant single-particle configuration of each excitonic level is indicated by
  $e^p_0h^{q}_jh^{q^{\prime}}_{j^{\prime}}$, where $p$, $q$ and $q^{\prime}$
  (=0, 1, 2) indicate, correspondingly, the occupation of the electron level
  $e_0$ and hole levels $h_j$ and $h_{j^{\prime}}$.  The energy of allowed
  transitions (vertical arrows) is indicated by $\omega_{if}$, where $i$ and
  $f$ indicate, respectively, the initial and final state upon recombination.
  Bright and dark excitonic levels are indicated, respectively, by solid and
  dotted lines. The fine-structure splittings are shown schematically.}
\end{figure*}

\subsection{Electronic structure of the excitonic manifolds}

We describe the basic electronic structure of the excitonic manifolds (Fig.
\ref{Fig_1}) before describing recombination processes. All excitonic states
are based on mixing and excitations of the single-particle states $\{h_0$,
$h_1$, $h_2$, $\cdots\}$ and $\{e_0$, $e_1$, $e_2$, $\cdots\}$, for holes and
electrons, respectively. These states are solutions to the single-particle
Schr\"odinger equation

\begin{equation}
\label{SP-effective_eq}
\left\{-\frac{1}{2}\nabla^2+V_{ext}({\bf R})+V_{scr}({\bf R})\right\}\psi_i={\cal E}_i\,\psi_i,
\end{equation}

\noindent where both the external (pseudo) potential $V_{ext}({\bf R})$ due to
the ion-ion and ion-electron interaction and the screening response to such
external potential $V_{scr}({\bf R})$ are expressed as a superposition of
screened atomic pseudopotentials

\begin{equation}
\label{pseudo}
V_{ext}({\bf R})+V_{scr}({\bf R})=V_{SO}+\sum_l\sum_{\alpha}v_{\alpha}[{\bf
  R}-{\bf R}^{(\alpha)}_l;{\rm Tr}(\widetilde\varepsilon)].
\end{equation}

\noindent Here, $V_{SO}$ is a non-local spin-orbit
interaction;\cite{williamson_PRB_2000} $v_{\alpha}$ is a screened
pseudopotential for atom of type $\alpha$ that depends on strain, and it has
been fitted to {\em bulk} properties of GaAs and InAs, including bulk band
structures, experimental deformation potentials and effective masses, as well
as local-density-approximation (LDA)-determined band
offsets.\cite{williamson_PRB_2000} The single-particle Shr\"odinger equation
[Eq. (\ref{SP-effective_eq})] includes not only quantum-confinement effects
(as in simple, one-band particle-in-a-box models), but also multi-band
coupling (light hole, heavy hole, conduction); inter-valley ($\Gamma$-$X$-$L$)
coupling; and spin-orbit coupling. Strain effects are present through the
relaxation, via a valence force field,\cite{williamson_PRB_2000} of the atomic
positions $\{{\bf R}^{(\alpha)}_l\}$ within the simulation supercell (quantum
dot+GaAs-matrix); and directly ``felt'' by the potential $v_{\alpha}[{\bf
  R}-{\bf R}^{(\alpha)}_l;{\rm Tr}(\widetilde\varepsilon)]$.  States $\{h_0$,
$h_1$, $h_2$,$\cdots\}$ and $\{e_0$, $e_1$, $e_2$,$\cdots\}$ form a basis for
the excitonic states.  We indicate the dominant single-particle configuration
of each excitonic level by
$e^p_ie^{p^{\prime}}_{i^{\prime}}h^{q}_jh^{q^{\prime}}_{j^{\prime}}$, where
$p$, $p^{\prime}$, $q$, and $q^{\prime}$ (=0, 1, 2) indicate, correspondingly,
the occupation of the electron levels $e_i$ and $e_{i^{\prime}}$, and hole
levels $h_j$ and $h_{j^{\prime}}$. Figure \ref{Fig_1} illustrates the
excitonic manifolds using a $75\,${\AA}-tall lens-shaped In$_{\rm
  0.6}$Ga$_{0.4}$As/GaAs quantum dot (base diameter $b=252\,${\AA}), as
obtained from configuration-interaction calculations based on the
pseudopotential single-particle description.


%
{\em Monoexciton.} The monoexciton $X^0$ has a ground-state
$|\Psi^{(0)}(X^0)\rangle$ created by occupying $h_0$ and $e_0$, denoted
$e^1_0h^1_0$ [Fig \ref{Fig_1}(a)].  However, configuration-interaction\cite{franceschetti_PRB_1999} (CI)
also mixes into $|\Psi^{(0)}(X^0)\rangle$ other states such as
$e^1_0h^1_1,e^1_0h^1_2,\cdots$. This is done by expanding the monoexciton
states $\{|\Psi^{(\nu)}(X^0)\rangle\}$ ($\nu=0,1,2,\cdots$) in a basis of
Slater determinants (configurations) $\{|\Phi(X^0)\rangle\}$ constructed in
the subspace of $N_e$ and $M_h$ electron and hole confined single-particle
levels, respectively:

\begin{equation}
\label{psi.nu.x0}
|\Psi^{(\nu)}(X^0)\rangle=\sum_{\kappa}C^{(\nu)}_{\kappa}(X^0)\,|\Phi_{\kappa}(X^0)\rangle, 
\end{equation}
  
\noindent where $C^{(\nu)}_{\kappa}(X^0)$ are the CI coefficients 
and $\kappa$ is a composite index that labels each Slater determinant.
%
\noindent Since $e_0$ and $h_0$ are each two-fold degenerate due to spin,
$e^1_0h^1_0$ has four-fold degeneracy at the single-particle level of Eqs.
(\ref{SP-effective_eq}) and (\ref{pseudo}). We next allow Coulomb 
electron-electron and hole-hole interactions. Direct Coulomb 
is given by $J^{\,(\mu\mu)}_{ij;ji}$ and exchange by $J^{\,(\mu\mu)}_{ij;ij}$, 
with the Coulomb scattering matrix elements given by 

\begin{widetext}
\begin{equation}
\label{J_eq}
J^{\,(\mu\mu^{\prime})}_{ij;kl}=\int\int {\rm d}{\bf R}{\rm d}{\bf
  R}^{\prime}\frac{\left[\psi^{(\mu)}_i({\bf
  R})\right]^{*}\left[\psi^{(\mu^{\prime})}_j({\bf
  R}^{\prime})\right]^{*}\left[\psi^{(\mu^{\prime})}_k({\bf
  R}^{\prime})\right]\left[\psi^{(\mu)}_l({\bf R})\right]}{\epsilon({\bf R},{\bf
  R}^{\prime})|{\bf R}-{\bf R}^{\prime}|}.   
\end{equation}
\end{widetext}

\noindent Here $\mu,\mu^{\prime}=e,h$; and $\epsilon({\bf R},{\bf R}^{\prime})$ 
is a microscopic, phenomenological dielectric constant.\cite{resta_PRB_1977} 
We also allow electron-hole direct Coulomb interaction $J^{\,(eh)}_{ij;ji}$ and
electron-hole exchange 

\begin{widetext}
\begin{equation}
\label{K_eq}
K^{\,(eh)}_{ij;kl}=\int\int {\rm d}{\bf R}{\rm d}{\bf
  R}^{\prime}\frac{\left[\psi^{(h)}_i({\bf
  R})\right]^{*}\left[\psi^{(e)}_j({\bf
  R}^{\prime})\right]^{*}\left[\psi^{(e)}_k({\bf R})\right]\left[\psi^{(h)}_l({\bf
  R}^{\prime})\right]}{\epsilon({\bf R},{\bf R}^{\prime})|{\bf R}-{\bf R}^{\prime}|}.
\end{equation}
\end{widetext}

%
\noindent Inclusion of these Coulomb interactions splits $e^1_0h^1_0$ into 
four distinct levels:\cite{bester_PRB_2003,bayer_PRB_2002} The lowest two are
spin-forbidden (``dark'') in the absence of spin-orbit coupling, and the
highest two are allowed (``bright''). The bright-dark splitting of the
ground-state levels shown in Fig. \ref{Fig_1}(a) is $84\,{\rm \mu eV}$. The
magnitude of this splitting increases up to $178\,\mu{\rm eV}$ for a
$20\,${\AA}-tall dot.  At $T\sim 0\,{\rm K}$ only the dark states are
populated, thus, the transition to the ground state $e^0_0h^0_0$ is
long-lived. Figure \ref{Fig_1}(a) (shaded area) shows transitions
$\omega_{20}$ and $\omega_{30}$ from the bright states to $e^0_0h^0_0$. The
low-lying excited states of the monoexciton correspond to excitations of the
hole, i.e. $e^1_0h^1_1$, $e^1_0h^1_2$ and $e^1_0h^1_3$, due to the much
smaller spacing of hole single-particle energy levels compared to that of the
electrons. Thus, the spacing of the excited states fingerprint the hole energy
level structure.  Figure \ref{Fig_1}(a) shows the first twelve excited states
that are derived from $e^1_0h^1_1$, $e^1_0h^1_2$ and $e^1_0h^1_3$. Each of
these states are four-fold degenerate and their fine structure is shown
schematically.  We find that three out of the four levels that arise from
$e^1_0h^1_2$ are optically allowed. In particular, one of the levels,
indicated as $\bot$, emits light that is polarized along $[001]$, while the
remaining two emit in-plane polarized light. These excited levels become
optically forbidden as the dot becomes flatter.


%% XX^0
%
{\em Biexciton.} In contrast to the four levels comprising the monoexciton,
the biexciton $XX^0$ has a singly-degenerate, {\em closed-shell} ground state
($e^2_0h^2_0$) that is bright. Thus, even at $T\sim 0\,{\rm K}$ both emissions
$\omega_{03}$ and $\omega_{02}$ [Fig.  \ref{Fig_1}(b); shaded area] of the
biexciton are fast ($\sim {\rm ns}$). The biexciton has a non-trivial ladder
of excited states that is determined primarily by the relative magnitude of
the direct Coulomb hole-hole interaction and the single-particle energy
splittings of the hole states. The ladder begins with states derived from
$e^2_0h^1_0h^1_1$ at about $6\,{\rm meV}$ above the ground state and follows
with $e^2_0h^1_0h^1_2$ and $e^2_0h^1_0h^1_3$ at about $11\,{\rm meV}$ and
$16\,{\rm meV}$, respectively.  Due to the two-fold Kramers degeneracy of the
hole levels, $e^2_0h^1_0h^1_1$ is four-fold degenerate at the single-particle
level, these states split into four distinct states due to hole-hole and
electron-hole exchange [Fig \ref{Fig_1}(b)]. Similarly, the four states in
$e^2_0h^1_0h^1_2$ split in two groups of two. In this case, remarkably, the
splitting is about twice as big as that in $e^2_0h^1_0h^1_1$ and nearly five
times bigger than in $e^2_0h^1_0h^1_3$ ($\sim 500\,\mu{\rm eV}$).  Similarly
to the monoexciton case, the lowest split-off pair in $e^2_0h^1_0h^1_2$ is
optically active and emits light polarized along $[001]$ [$\bot$, Fig.
\ref{Fig_1}(b)].  Further, these states become darker as the dot becomes
flatter, being forbidden at $h=20\,${\AA}.  The subsequent excited state in
the ladder derives from the singly-degenerate closed-shell state $e^2_0h^2_1$
and is closely-spaced with $e^2_0h^1_0h^1_3$.  Note that albeit the splitting
between these states is small, the state derived from the closed-shell
configuration $e^2_0h^2_1$ is relatively ``inert'' in that it mixes weakly
with other states. In particular the weight of this configuration in the CI
expansion is $\sim 83\%$.  Below, we discuss the energy, polarization and
lifetime of $\omega_{03}$ and $\omega_{02}$.


%% X^- and X^+
%
{\em Trions.} The negatively- ($X^-$) and positively-charged ($X^+$) trions
have ground states that are bright, two-fold degenerate and arise from
occupying $e^2_0h^1_0$ and $e^1_0h^2_0$, respectively. Due to
configuration-interaction mixing, the ground states of $X^-$ and $X^+$ mix
with $e^2_0h^1_1$ and $e^1_0h^1_0h^1_1$, respectively.  As in the case of
$XX^0$, even at $T\sim 0\,{\rm K}$ the recombination $\omega_{00}$ of $X^-$
[Fig.  \ref{Fig_1}(c)] into state $e^1_0h^0_0$ and that of $X^+$ [Fig.
\ref{Fig_1}(d)] into state $e^0_0h^1_0$ is fast. For both trions, while there
are only two dipole-allowed transitions in the absence of spin-orbit coupling,
we predict four allowed transitions.  The first few excited states of $X^-$
correspond to occupying states derived primarily from $e^2_0h^1_1$,
$e^2_0h^1_2$ and $e^2_0h^1_3$ [Fig.  \ref{Fig_1}(c)].  (Naturally, these
states are mixed in with other states of $X^-$.) These excited states are
two-fold degenerate and lie, correspondingly, about $10\,{\rm meV}$, $13\,{\rm
  meV}$ and $20\,{\rm meV}$ above the ground state, as shown in Fig.
\ref{Fig_1}(c). Note the similarity of the low-lying excited states energy
spacing with that in $X^0$. In addition, note that one of the two states
derived from $e^2_0h^1_2$ is optically active and emits light polarized along
$[001]$ [$\bot$ in Fig.  \ref{Fig_1}(c)]. Again, these states become optically
forbidden for a $20\,${\AA}-tall dot.  The excited states of $X^+$ are more
complex and present a high density of states [Fig. \ref{Fig_1}(d)]; a
consequence of the closely spaced ($\sim 5\,{\rm meV}$) single-particle hole
levels: In contrast to $X^-$, all low-lying excited states show fine-structure
splitting due to electron-hole exchange.  The first excited states at about
$5\,{\rm meV}$ corresponds to occupying $e^1_0h^1_0h^1_1$, due to the two-fold
degeneracy of each of these single-particle states, there are eight excited
states that split in two groups of four, due to hole-hole exchange.  Occupying
$e^1_0h^1_0h^1_2$ leads to the next eight excited states around $10\,{\rm
  meV}$, which are also split in two quartets.  In this case, there is an
optically active (polarization $\Vert\,[001]$) pair of excited states that
belong to the lower-energy quartet [$\bot$, Fig. \ref{Fig_1}(d)]. As in the
other excitons, these states become dark as the dot height decreases.  The
next ten excited states about $14\,{\rm meV}$ are a mixture of eight
configurations derived from $e^1_0h^1_0h^1_3$ and two from $e^1_0h^2_1$.
Because $e_0$ is {\em half-filled}, $e^1_0h^2_1$ is very ``reactive'' in the
sense that it heavily mixes via configuration interaction. Note that this is
significantly different from the case of configuration $e^2_0h^2_0$ in the
biexciton [Fig.  \ref{Fig_1}(b)].  Higher excited states correspond to
occupying $e^1_0h^1_1h^1_2$ and $e^1_0h^1_1h^1_3$. Below, we discuss the
energy, as well as polarization and lifetime of $\omega_{00}$ for $X^-$ and
$X^+$.



%%%%%%%%%%%%%%%%%%%%%%%%
%
% Recombination energies
%
%%%%%%%%%%%%%%%%%%%%%%%%
%
%
\subsection{Recombination energies}
\label{recomb.sec}

We define the recombination energy upon recombination of an electron-hole pair
in $\chi^q$ as the difference between the total energies of the initial state
$|\Psi^{(i)}(\chi^q)\rangle$ and the final state
$|\Psi^{(f)}(\chi^q-1)\rangle$. Namely,

\begin{equation}
\label{omega_if.eq}
{
\begin{array}{rcl}
\omega_{i0}(X^0)&=&E^{(i)}(X^0) \\
\omega_{if}(X^-)&=&E^{(i)}(X^-)-{\cal E}^{\,(e)}_f \\
\omega_{if}(X^+)&=&E^{(i)}(X^+)+{\cal E}^{\,(h)}_f \\
\omega_{if}(XX^0)&=&E^{(i)}(XX^0)-E^{(f)}(X^0). 
\end{array}
}
\end{equation}

\noindent where $i$ and $f$ label an initial and final state,
respectively; ${\cal E}^{\,(e)}_f$ and ${\cal E}^{\,(h)}_f$ are, respectively,
the single-particle energy of the electron and hole in the final state; and
$E^{(\nu)}(\chi^q)$ is the multi-particle, configuration-interaction energy of
state $|\Psi^{(\nu)}(\chi^q)\rangle$.  [In Eq. (\ref{omega_if.eq}), the energy
of the ground state of the system in the absence of excited electron-hole
pairs is taken to be zero.]  Optical experiments like photoluminescence probe
electron-hole recombination transitions that are allowed
(bright).\cite{cardona_book}
%
Figure \ref{Fig_1}(a)-(d) indicates with arrows the lowest bright
recombination transitions of exciton $\chi^q$, for a $75\,${\AA}-tall
lens-shaped In$_{\rm 0.6}$Ga$_{\rm 0.4}$As/GaAs dot with base diameter
$b=252\,${\AA}. The final states upon electron-hole recombination are
presented in the shaded area of Fig.  \ref{Fig_1}.  Figure \ref{Fig_2} shows
the recombination energies $\omega_{00}(X^-)$, $\omega_{00}(X^+)$ and
$\omega_{03}(XX^0)$ calculated at the many-particle, configuration-interaction
level as a function of dot height. These recombination energies are shown
relative to the lowest recombination energy $\omega_{20}(X^0)$ of the
monoexciton (the latter energy is shown in the top axis of Fig.  \ref{Fig_2}).
Thus, the results correspond to {\em spectroscopic shifts} that are currently
measured by several groups. Prominent features:


%%%%%%%%%%%%
% Figure 2
%%%%%%%%%%%%
%
\begin{figure}[htb]
\includegraphics[width=8.5cm]{./Fig_2.eps}
\caption{{\label{Fig_2}} (a) Recombination energies $\omega_{00}(X^-)$ and
  $\omega_{00}(X^+)$, and $\omega_{03}(XX^0)$ (see text and Fig. \ref{Fig_1})
  in In$_{\rm 0.6}$Ga$_{\rm 0.4}$As/GaAs quantum dots (base $b=252\,$\AA) with
  different heights. The energy is shown as a spectroscopic shift relative to
  $\omega_{20}(X^0)$ [see Fig. \ref{Fig_1}(a)]. (b) Screened direct Coulomb
  interaction $J^{\,(\mu\mu^{\prime})}_{00}$ versus dot height. At
  $h=50\,${\AA} a nearly ``symmetric'' regime [$J^{\,(eh)}_{00}\sim
  J^{\,(hh)}_{00}\sim J^{\,(ee)}_{00}$] is attained.}
\end{figure}



%
(i) The recombination energy $\omega_{00}(X^{-})$ blue-shifts as height
increases; in contrast, $\omega_{00}(X^{+})$ red-shifts. These trends have
been explained by Bester and Zunger in Ref. \onlinecite{bester_PRB_2003b} by
adopting the Hartree-Fock approximation.

%
(ii) For the flattest dot ($h=20\;${\AA}) the ordering
$\omega_{00}(X^+)>\omega_{20}(X^0)>\omega_{00}(X^-)$ of the emission energies
and the relative magnitude of the spectroscopic shifts
$\omega_{00}(X^+)-\omega_{20}(X^0)<\omega_{20}(X^0)-\omega_{00}(X^-)$ agree
with photoluminescence (PL) data.
\cite{ware_PhysicaE_2005,skolnick_ARMR_2004,guffarth_PhysicaE_2004,finley_PRB_2001}
%
These relationships are explained at the Hartree-Fock level of approximation,
as in Ref. \onlinecite{bester_PRB_2003b}, in which one derives

\begin{eqnarray}
\omega_{20}(X^0)&=&[{\cal E}^{\,(e)}_0-{\cal E}^{\,(h)}_0]-J^{\,(eh)}_{00},\\
\omega_{00}(X^+)&=&[{\cal E}^{\,(e)}_0-{\cal E}^{\,(h)}_0]+J^{\,(hh)}_{00}-2\,J^{\,(eh)}_{00},\\
\omega_{00}(X^-)&=&[{\cal E}^{\,(e)}_0-{\cal E}^{\,(h)}_0]+J^{\,(ee)}_{00}-2\,J^{\,(eh)}_{00}.
\end{eqnarray}

\noindent Then, the relationship $J^{\,(hh)}_{00}>J^{\,(eh)}_{00}>J^{\,(ee)}_{00}$ that
holds at $h=20\,${\AA} [Fig. \ref{Fig_2}(b)] reveals the ordering of the
emission energies and the magnitude of the spectroscopic shifts.
%
Regarding the magnitude of the spectroscopic shifts,
$\omega_{00}(X^+)-\omega_{20}(X^0)=2.5\;{\rm meV}$ and
$\omega_{00}(X^-)-\omega_{20}(X^0)=-3.7\;{\rm meV}$ agree reasonably well with
(a) $2\;{\rm meV}$ and $-6\;{\rm meV}$, respectively, that Ware and coworkers
have recently observed in an InAs/GaAs dot (size unspecified) with monoexciton
emission at $\sim 1.268\;{\rm eV}$ (Ref. \onlinecite{ware_PhysicaE_2005});
also with (b) $\omega_{00}(X^-)-\omega_{20}(X^0)=-5\;{\rm meV}$ measured by
Smith {\em et al.} in an (In,Ga)As/GaAs dots (size unspecified) with emission
energy at $\sim 1.319\;{\rm eV}$ (Ref. \onlinecite{smith_PRL_2005}); and with
(c) the value of $-5.8\;{\rm meV}$ for $\omega_{00}(X^-)-\omega_{20}(X^0)$
observed by Finley and coworkers in an (In,Ga)As/GaAs dots [$b=(230\pm
70)\;${\AA}, $h=(25\pm 10)\;${\AA}] with emission at $\sim 1.263\;{\rm eV}$
(Ref.  \onlinecite{finley_PRB_2001}).
 

%
(iii) As height increases, $\omega_{03}(XX^0)$ red-shifts at small heights,
reaches a maximum shift of nearly $-2\;{\rm meV}$ at $h\sim 50\;${\AA}, and
then it moderately blue-shifts for taller dots. In addition, at $h=50\;${\AA}
the emission energy of $XX^0$ coincides with that of $X^{+}$. As in (ii),
these results are explained at the Hartree-Fock level, which predicts

\begin{eqnarray}
\label{wxx0}
\omega_{03}(XX^0)-\omega_{20}(X^0)&=&J^{\,(ee)}_{00}+J^{\,(hh)}_{00}-2\,J^{\,(eh)}_{00},
\\
\label{wx+}
\omega_{03}(XX^0)-\omega_{00}(X^+)&=&J^{\,(ee)}_{00}-J^{\,(eh)}_{00}.
\end{eqnarray}   

\noindent Here, we have neglected the small ($\sim 1$-$6\;\mu{\rm eV}$) splitting of
the monoexciton bright states. Then, by analyzing the height dependence of the
direct Coulomb interactions $J^{\,(ee)}_{00}$, $J^{\,(hh)}_{00}$ and
$J^{\,(eh)}_{00}$ [Fig. \ref{Fig_2}(b)] we find that Eq. (\ref{wxx0}) predicts
the observed height dependence of $\omega_{03}(XX^0)$, although the actual
magnitude of the emission is not quantitatively predicted due to
correlations.\cite{shumway_PRB_2001,narvaez_PRB_tobepublished} In addition,
Eq. (\ref{wx+}) reveals the coincidence of the emission of $XX^0$ and that of
$X^+$ as a result of balancing the magnitudes of $J^{\,(ee)}_{00}$ and
$J^{\,(eh)}_{00}$ at $h=50\,${\AA}. The latter balance arises from the similar
degree of localization of $\psi^{(e)}_0$ and $\psi^{(h)}_0$ within the
dot.\cite{narvaez_condmat}

%
(iv) $\omega_{03}(XX^0)-\omega_{20}(X^0)=-2.0\;{\rm meV}$ at $h=50\;${\AA}
agrees with the value of $-2.0\pm 0.1\;{\rm meV}$ for $XX^0$ measured by
Finley and coworkers in PL experiments in an (In,Ga)As/GaAs dot [$b=(230\pm
70)\;${\AA} and $h=(25\pm 10)\;${\AA}] with exciton ground-state emission at
$\sim 1.345\;{\rm eV}$ (Ref.  \onlinecite{finley_PRB_2001}). In addition, this
result of $-2.0\;{\rm meV}$ for the biexciton shift agrees remarkably well
with the value of $-2\;{\rm meV}$ measured by Rodt {\em et al} in InAs/GaAs
dots ($b=100-200\;${\AA}, height unspecified) with monoexciton emission
energies ranging between $1.260\;{\rm eV}$ and $1.295\;{\rm eV}$ (Ref.
\onlinecite{rodt_papers}); and also with the measured shift of $-2.3\;{\rm
  meV}$ observed by Urbaszek and coworkers in (In,Ga)As/GaAs dots (size
unspecified) with monoexciton emission energy of $1.294\;{\rm eV}$ (Ref.
\onlinecite{urbaszek_PRL_2003}).
%
The value of about $-1.7\,{\rm meV}$ for $\omega_{03}(XX^0)-\omega_{20}(X^0)$
that we find at $h=35\,${\AA} and $65\,${\AA} agrees well with the value of
$-1.6\,{\rm meV}$ observed by Bayer and coworkers in (In,Ga)As/GaAs dots
[$b=(500\pm 30)\;${\AA}] with exciton emission at $1.428\,{\rm eV}$ (Ref.
\onlinecite{bayer_PRB_1998}). Our results also agree satisfactorily with the
value of $-2.7\;{\rm meV}$ measured by Findeis and coworkers in dots (size
unspecified) with monoexciton emission energy at $1.284\;{\rm eV}$ (Ref.
\onlinecite{findeis_SSC_2000}).


%%%%%%%%%%%%%%%%%%%
%
% Binding energies
%
%%%%%%%%%%%%%%%%%%%
%
%
%
\subsection{Binding energies}
\label{binding.mp}


The binding energies of excitons $\chi^q$ are defined as

\begin{equation}
\label{binding.eqs_CI}
{
\begin{array}{rcl}
\Delta(X^0)&=&\big[{\cal E}^{\,(e)}_0-{\cal E}^{\,(h)}_0\big]-E^{(0)}(X^0) \\
\Delta(X^-)&=&\big[{\cal E}^{\,(e)}_0+E^{(0)}(X^0)\big]-E^{(0)}(X^-) \\
\Delta(X^+)&=&\big[-{\cal E}^{\,(h)}_0+E^{(0)}(X^0)\big]-E^{(0)}(X^+) \\
\Delta(XX^0)&=&2E^{(0)}(X^0)-E^{(0)}(XX^0). 
\end{array}
}
\end{equation}

\noindent Exciton $\chi^q$ is said to be {\em bound} when the binding energy
$\Delta(\chi^q)$ is positive. Conversely, $\Delta(\chi^q)<0$ implies the
exciton is unbound. Note that the binding energy is defined with respect to
the ground-state energy of {\em dissociated} excitonic complexes.  For
instance, the binding energy of $X^0$ is defined with respect to the energy of
a non-interacting electron-hole pair. In turn, the binding energy of the
biexciton $XX^0$ is defined with respect to the total energy of two
non-interacting monoexcitons [Eq. (\ref{binding.eqs_CI})].



%%%%%%%%%%%
% Figure 3
%%%%%%%%%%%
%
\begin{figure}[htb]
\includegraphics[width=8.5cm]{./Fig_3.eps}
\caption{{\label{Fig_3.mp}} Binding energy $\Delta(\chi^q)$
  [Eq. (\ref{binding.eqs_CI})] of $X^0$, $X^-$ and $X^+$, and $XX^0$ as a
  function of dot height. The ground-state energy $E^{(0)}(X^0)$ of $X^0$ is
  also shown (top axis). For $X^-$, $X^+$, and $XX^0$, height drives a
  bound-to-unbound crossover.}
\end{figure} 


Figure \ref{Fig_3.mp} shows $\Delta(X^0)$, $\Delta(X^-)$, $\Delta(X^+)$ and
$\Delta(XX^0)$ as well as $E^{(0)}(X^0)$ as a function of dot height. 
%
$\Delta(X^0)$ decreases with increasing height, and it is well approximated by
$J^{eh}_{00}$ as correlation effects are relatively small ($\sim 2\;{\rm
  meV}$). The height dependence of the binding energy of $X^-$, $X^+$, and
$XX^0$ follows the height dependence of the spectroscopic shifts shown in Fig.
\ref{Fig_2}(a). This is so because the bright-dark splitting for the
monoexciton $X^0$ is small $\sim 80$-$180\;{\rm \mu eV}$. As expected from the
spectroscopic shifts results [Fig.  \ref{Fig_2}(a)], the height dependence of
the binding energy is qualitatively different for each exciton $\chi^q$.
%
(i) $\Delta(X^-)$ is bound ($\sim 4\;{\rm meV}$) for the
flattest dot ($h=20\;${\AA}) and it decreases almost linearly, becoming
unbound for dots taller than $h=65\;${\AA}.
%
(ii) In contrast, $\Delta({X^+})$ is unbound for the flattest dot and
increases up to $3\;{\rm meV}$, becoming bound slightly below $h=35\;${\AA}.
%
(iii) $\Delta(XX^0)$ does not depend monotonically on the
gap, reaching a maximum around $h=50\;${\AA}. 
%
In addition, $XX^0$ is bound [$\Delta(XX^0)>0$] for all heights above
$20\;${\AA} while unbound for the flattest dot ($h=20\;${\AA}). The latter is
due to an interplay between Hartree-Fock and correlation contributions to
binding, $\Delta_{\rm HF}(XX^0)$ and $\delta(XX^0)$, respectively, that
results in correlation being insufficient to bind $XX^0$. Namely,

\begin{eqnarray}
\label{Correlation}
\Delta(XX^0)&=&\Delta_{\rm HF}(XX^0)+\delta(XX^0) \\
& = & \{2J^{\,(eh)}_{00}-[J^{\,(ee)}_{00}+J^{\,(hh)}_{00}]\}+\delta(XX^0) \nonumber \\
 &=& -1.6\;{\rm meV}+1.4\;{\rm meV} \nonumber \\ 
&=& -0.2\;{\rm meV}, \nonumber
\end{eqnarray}

\noindent It should be noted that Rodt and coworkers\cite{rodt_papers}
observed (in photoluminescence) a bound-unbound crossover for $XX^0$ as the
monoexciton emission energy of InGaAs/GaAs quantum dots decreased. Those
authors also calculated the binding energy of $XX^0$, using an 8-band ${\bf
  k}\cdot{\bf p}$ model, and suggested that the reduction of correlation
effects was responsible for unbinding $XX^0$ as the gap
increased.\cite{rodt_papers}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Polarization and optical transitions
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
\section{Polarization anisotropy of optical transitions}


When an electron-hole pair in exciton $\chi^q$ recombines optically, the
transition is characterized by both the transition energy
$\omega_{if}(\chi^q)$ [Eq.  (\ref{omega_if.eq})] and the transition dipole
matrix element

\begin{equation}
\label{dipole.element}
{M}^{(\hat{\bf e})}_{if}(\chi^q)=\langle\Psi^{(f)}(\chi^q-1)|\hat{\bf e}\cdot{\bf
  p}|\Psi^{(i)}(\chi^q)\rangle . 
\end{equation}

\noindent Here, ${\bf p}$ is the electron momentum and $\hat{\bf e}$ is
the polarization vector of the electromagnetic field.\cite{cardona_book}


%%% polarization and anisotropies %%%
%
The dipole matrix elements ${\bf M}^{(\hat{\bf e})}_{if}(\chi^q)$ [Eq.
(\ref{dipole.element})] depend on the polarization vector $\hat{\bf e}$; so,
it is natural to quantify what is the degree of polarization anisotropy
between different polarizations $\hat{\bf e_{1}}$ and $\hat{\bf e_{2}}$.
Therefore, we introduce the recombination (emission or pholuminescence)
intensity spectrum of exciton $\chi^q$ for polarization $\hat{\bf e}$,

\begin{equation}
I^{(\hat{\bf e})}(\omega,T;\chi^q)=\sum_{i,f}\big|{
    M}^{(\hat{\bf
    e})}_{if}(\chi^q)\big|^2\,P_{i}(T;\chi^q)\,\delta[\omega-\omega_{if}(\chi^q)].
\end{equation} 

\noindent Here, 

\begin{equation}
\label{bolzmann.n}
P_i(T;\chi^q)={\cal N}\exp\{-[E^{(i)}(\chi^q)-E^{(0)}(\chi^q)]/k_B\,T\}
\end{equation}

\noindent is the occupation (Boltzmann) probability of the 
initial state $|\Psi^{(i)}(\chi^q)\rangle$ at temperature $T$; ${\cal N}$ is a
normalization constant such that $\sum_{i}P_i(T;\chi^q)=1$ and $k_B$ is the
Boltzmann constant. Then, as {\em in-plane} polarizations $\hat{\bf
  e}_1=[110]$ and $\hat{\bf e}_2=[1\bar 10]$ have been probed extensively, we
introduce the in-plane polarization anisotropy parameter $\lambda$,

\begin{equation}
\label{lambda_eq}
\lambda(\omega,T;\chi^q)=\frac{I^{([110])}(\omega,T;\chi^q)-I^{([1\bar 10])}(\omega,T;\chi^q)}{I^{([110])}(\omega,T;\chi^q)+I^{([1\bar 10])}(\omega,T;\chi^q)}.
\end{equation}

\noindent Thus, $\lambda=1$ indicates an optical
transition that is {\em fully} polarized along the $[110]$ direction, while 
$\lambda=-1$ indicates one {\em fully} polarized along $[1\bar 10]$.


%%%%%%%%%%%
% Figure 4
%%%%%%%%%%%
%
\begin{figure}[bt]
\includegraphics[width=8.5cm]{./Fig_4.eps}
\caption{{\label{Fig_4}}In-plane polarization anistropy $\lambda$ 
  [Eq. (\ref{lambda_eq})] for the lowest optical transitions of (a) $X^0$, (b)
  $XX^0$, (c) $X^-$, and (d) $X^+$. (See Fig. \ref{Fig_1}.) The transitions of
  $X^0$ and $XX^0$ are polarized along $[110]$ and $[1\bar 10]$, and the
  degree of this polarization strongly depends on height. For $X^-$ and $X^+$,
  the transitions are not polarized neither along $[110]$ nor $[1\bar 10]$.}
\end{figure}

\subsection{In-plane polarization anisotropy of the lowest optical transitions 
of $X^0$, $X^-$, $X^+$, and $XX^0$}

% results
%
Figure \ref{Fig_4} shows the in-plane polarization anisotropy $\lambda$ for
the lowest optical transitions of (a) $X^0$, (b) $XX^0$, (c) $X^-$, and (d)
$X^+$ as a function of dot height. Two features are prominent:


(i) The bright transitions $\omega_{20}(X^0)$ and $\omega_{30}(X^0)$ are
polarized and the polarization anisotropy depends on height; see Fig.
\ref{Fig_4}(a).  Similarly, the lowest transitions of the biexciton
$\omega_{03}(XX^0)$ and $\omega_{02}(XX^0)$ are polarized and the degree of
polarization also depends on height. These transitions correspond to the {\em
  decay} into the two bright states of the monoexciton [Fig.  \ref{Fig_1}(d)].
Clearly, the biexciton transitions inherit the polarization of the monoexciton
transitions. For both $X^0$ and $XX^0$, we see that $\lambda$ switches sign as
a function of height.  In particular, at $h=50\,${\AA} the transitions of
$X^0$ present no in-plane polarization anisotropy [see arrow in Figs.
\ref{Fig_4}(a)] because the bright doublet formed by $|\Psi^{(2)}(X^0)\rangle$
and $|\Psi^{(3)}(X^0)\rangle$ is degenerate, which according to our
calculation results in $I^{([110])}(\omega_{20})\simeq I^{([1\bar
  10])}(\omega_{20})$.

%
(ii) The lowest optical transitions of $X^-$ and $X^+$ have degenerate
transition (recombination) energies [Fig. \ref{Fig_1}(b) and (c)] and,
according to our calculations, this results in $I^{([110])}(\omega_{00})\simeq
I^{([1\bar 10])}(\omega_{00})$ and, thus, negligible in-plane polarization
anisotropy regardless of height.




%
In addition to (i) and (ii), we find that (iii) the in-plane polarization
anisotropy of the lowest transitions of $X^0$ and $XX^0$ depends dramatically
on the dot's alloy randomness (disorder realization); as shown in Figure
\ref{Fig_5.new} for seven alloy realizations in a $35\,${\AA}-tall dot. For a
given transition in both $X^0$ and $XX^0$, $\lambda$ changes sign depending on
alloy randomness. Further, while some alloy realizations like $4$ and $5$
result in transitions nearly fully polarized ($\lambda\sim 99\%$), others such
as $2$, $6$ and $7$ present small anisotropy ($\lambda\sim 20\%$).


%%%%%%%%%%%%%%%%
% Figure 5 
%%%%%%%%%%%%%%%%
%
\begin{figure}[htb]
\includegraphics[width=8.5cm]{./Fig_5.eps}
\caption{{\label{Fig_5.new}}In-plane polarization $\lambda$ for the lowest
  optical transitions of (a) $X^0$ and (b) $XX^0$ for different realizations
  of alloy randomness in a $35\,${\AA}-tall In$_{\rm 0.6}$Ga$_{\rm
    0.4}$As/GaAs dot (base $b=252\,${\AA}). For both $X^0$ and $XX^0$,
  $\lambda$ depends dramatically on alloy realization, changing not only its
  sign but its magnitude too.}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%
%
% Radiative lifetimes
%
%%%%%%%%%%%%%%%%%%%%%%
%
%
%
\section{Radiative recombination lifetimes}

The {\em characteristic} radiative lifetime $\tau_{if}(\chi^q)$ of a transition
$|\Psi^{(i)}(\chi^q)\rangle\rightarrow|\Psi^{(f)}(\chi^q-1)\rangle$ follows
from both the magnitude of the dipole matrix element of the transition
$\big|{\bf M}^{(\hat{\bf e})}_{if}(\chi^q)\big|^2$ and the recombination
energy $\omega_{if}$.\cite{dexter_book} Namely,

%
\begin{equation}
\label{intrinsic.lifetime}
\frac{1}{\tau_{if}(\chi^q)}=\frac{4}{3}\,\left(\frac{e^2}{m_0^2\,c^3\,\hbar^2}\right)n\,\omega_{if}(\chi^q)\sum_{\hat{{\bf
      e}}=\hat{x},\hat{y},\hat{z}}\big|{\bf M}^{(\hat{\bf e})}_{if}(\chi^q)\big|^2.
\end{equation}


\noindent Here, $e$ and $m_0$ are the charge and mass of the electron, 
respectively, and $c$ is the velocity of light in vacuum. 
%
In addition, the refractive index $n$ of the dot material accounts
for the material's effects on the photon emission. The linear dependence of
$1/\tau_{if}(\chi^q)$ on refractive index is applicable only when considering
dot and matrix materials with similar dielectric constants, as it is the case
in InGaAs/GaAs dots. In a more general case, more complicate dependences have
been proposed.\cite{thranhardt_PRB_2002} Note that the characteristic
radiative lifetime [Eq. (\ref{intrinsic.lifetime})] does not depend on
temperature nor on the occupation probability of the initial state
$|\Psi^{(i)}(\chi^q)\rangle$, as it is a characteristic property of the
transition $|\Psi^{(i)}(\chi^q)\rangle\rightarrow|\Psi^{(f)}(\chi^q-1)\rangle$. On the
other hand, the actual radiative lifetime $\tau(\chi^q)$ of exciton $\chi^q$
depends both on the probability $n_i$ of having the initial states
$|\Psi^{(i)}(\chi^q)\rangle$ of exciton $\chi^q$ occupied and the number of
final states $|\Psi^{(f)}(\chi^q-1)\rangle$ of exciton $\chi^q-1$ available
for recombination, as well as on the characteristic radiative lifetimes
$\tau_{if}(\chi^q)$. We calculate $\tau(\chi^q)$ from

\begin{equation}
\label{lifetime}
\frac{1}{\tau(\chi^q)}=\sum_f\sum_i\,n_i\frac{1}{\tau_{if}(\chi^q)}.
\end{equation} 

\noindent Here, $\sum_i\,n_i=1$ and $n_i=n_i(T;\chi^q)$ where $T$ is the 
temperature of the system. In general, the calculation of $n_i$ involves
solving a system of rate equations.\cite{dekel_PRB_2000} However, in the case
that the intra-level relaxation of exciton $\chi^q$ is much faster than the
radiative lifetimes, $n_i$ is given by the Boltzmann weight $P_i(T;\chi^q)$
[Eq.  (\ref{bolzmann.n})].


\subsection{Radiative lifetime $\tau(\chi^q)$ of the lowest optical
  transitions of $X^0$, $X^-$, $X^+$, and $XX^0$}

{\em Monoexciton.} At $T=0\,{\rm K}$, and assuming fast non-radiative
relaxation to the dark ground-state of $X^0$, the radiative lifetime of the
monoexciton $\tau(X^0)$ equals the characteristic radiative lifetime
$\tau_{00}(X^0)$ and it is long ($\sim {\rm ms}$). At finite temperatures, all
four levels [Fig. \ref{Fig_1}(a)] of the $X^0$ ground state are thermally
populated and emit light with their own characteristic lifetime [Eq.
(\ref{intrinsic.lifetime})].  For temperatures such that the lowest four
monoexciton states are occupied, while the occupation of excited states is
negligible, we calculate the radiative lifetime from Eq. (\ref{lifetime}) and
find

\begin{equation}
\label{tau.HT}
\tau(X^0)=4\,\left[\frac{\tau_{20}(X^0)\,\tau_{30}(X^0)}{\tau_{20}(X^0)+\tau_{30}(X^0)}\right]\simeq 2\,\tau_{20}(X^0).
\end{equation}

\noindent Here, we have used a result of our calculations that predict
$\tau_{20}(X^0)\simeq\tau_{30}(X^0)$ regardless of dot height. In addition, we
have neglected the long-lived recombinations $\omega_{00}$ and $\omega_{10}$.
Figure \ref{Fig_5}(a) shows the characteristic $\tau_{20}(X^0)$ versus dot
height.  We find that this lifetime depends weakly on height. A calculation of
$\tau(X^0)$ at $T=10\,{\rm K}$ predicts a moderate decrease in the monoexciton
radiative lifetime as height increases [Fig.  \ref{Fig_5}(b)]. It should be
noted that $\tau(X^0)$ is actually bigger than the approximate value of
$2\tau_{20}(X^0)$ [Eq. (\ref{tau.HT})] due to the actual occupation
probability $n_i(10\,{\rm K},X^0)$ of the initial states. By measuring
time-resolved photoluminescence in InAs/GaAs dots ($b=120$-$220\;${\AA},
height unspecified) with monoexciton emission ranging from $1.2$-$1.3\;{\rm
  eV}$, Karachinsky and coworkers have recently observed the opposite
trend,\cite{karachinsky_APL_2004} i. e. $\tau(X^0)$ increases with dot size
from $1\;{\rm ns}$ in a dot with emission energy of $1.31\;{\rm eV}$ to $\sim
3\;{\rm ns}$ in one with emission energy of $1.25\;{\rm eV}$. The values of
$\tau(X^0)$ we predict for tall dots ($h=65\;${\AA} and $75\;${\AA}) agree
satisfactorily with the value of $1\;{\rm ns}$ measured by Buckle {\em et al}
for an InAs/GaAs dot ($b\sim 120\;${\AA}, $h\sim\;30${\AA}) with gap
$1.131\;{\rm eV}$ (Ref.  \onlinecite{buckle_JAP_1999}) at a temperature of
$6\;{\rm K}$. Further, our predictions also agree well with the value of
$1.55\;{\rm ns}$ extracted from time-resolved photoluminescence experiments at
$10\,{\rm K}$ performed in InAs/GaAs dots ($b=200\;{\AA}$, $h=20\;${\AA}) by
Bardot and coworkers.\cite{bardot_condmat_2005}


%%%%%%%%%%%
% Figure 6
%%%%%%%%%%%
%
\begin{figure}[htb]
\includegraphics[width=8.5cm]{./Fig_6.eps}
\caption{{\label{Fig_5}}(Color online) (a) Characteristic radiative lifetime $\tau_{if}(\chi^q)$ 
  [Eq. (\ref{intrinsic.lifetime})] of the lowest optical transition of $X^0$,
  $X^-$, $X^+$, and $XX^0$ as a function of dot height. For $X^0$ and $XX^0$,
  the characteristic lifetimes are nearly the same and depend weakly on
  height. A stronger dependence on height is found for $X^-$ and $X^+$. (b)
  $T=10\,{\rm K}$ radiative lifetime $\tau(\chi^q)$ [Eq. (\ref{lifetime})] of
  the lowest optical transition of $X^0$, $X^-$, $X^+$, and $XX^0$ as a
  function of dot height.  The lifetimes depend weakly on height for all these
  excitons. For $X^0$ and $XX^0$, the lifetimes are nearly the same. A similar
  trend is found for $X^-$ and $X^+$.  At all heights, the radiative lifetimes
  of $X^-$ and $X^+$ are about twice as big as that of $XX^0$.}  
\end{figure}


%%
%
% Biexciton
%
{\em Biexciton.} In contrast to the monoexciton, the biexciton ground state
is singly-degenerate and bright [Fig. \ref{Fig_1}(b)]. Thus, at $T=0\,{\rm
  K}$, the radiative lifetime is given from Eq. (\ref{lifetime}) as

\begin{equation}
\label{tau.XX0}
\tau(XX^0)=\left[\frac{\tau_{02}(XX^0)\,\tau_{03}(XX^0)}{\tau_{02}(XX^0)+\tau_{03}(XX^0)}\right]\simeq \frac{1}{2}\,\tau_{03}(XX^0).
\end{equation}

\noindent Similarly to the $X^0$ case, in Eq. (\ref{tau.XX0}) we have used the
relationship $\tau_{03}(XX^0)\simeq\tau_{02}(XX^0)$ that our calculations
predict, and we have neglected the long-lived dark recombination channels.
Figure \ref{Fig_5}(a) shows $\tau_{03}(XX^0)$ as a function of height.  We
find a weak dependence with height, as in the monoexciton case.  Remarkably,
we find that $\tau_{20}(X^0)\simeq\tau_{03}(XX^0)$ regardless of height. The
latter leads to the following relationship between the radiative lifetime of the
monoexciton and biexciton,

\begin{equation}
\tau(XX^0)\simeq\frac{1}{4}\,\tau(X^0).
\end{equation}

Figure \ref{Fig_5}(b) shows that at $T=10\,{\rm K}$ the calculated
$\tau(XX^0)$ depends weakly on height, changing by about $0.1\,{\rm ns}$ in
the entire range of heights studied. $\tau(XX^0)$ is in excellent agreement
with the value of $0.5\;{\rm ns}$ measured by Ulrich {\em et al} in an
(In,Ga)As/GaAs quantum dot ($b=150$-$200\;${\AA}, $h=10$-$20\;${\AA}) with an
exciton gap of $1.337\,{\rm eV}$ (Ref.  \onlinecite{ulrich_condmat_2004}).


%%%%%%%%%%%%%
%
% X^- and X^+
%
{\em Trions.} Both $X^-$ and $X^+$ have a two-fold degenerate ground state
that is bright [Figs. \ref{Fig_1}(c) and (d)]. In each of these excitons,
there are four lowest transitions; namely, $\omega_{00}(X^-)$ [Fig.
\ref{Fig_1}(c)], $\omega_{10}(X^-)$, $\omega_{01}(X^-)$, and
$\omega_{11}(X^-)$; and $\omega_{00}(X^+)$ [Fig. \ref{Fig_1}(d)],
$\omega_{10}(X^+)$, $\omega_{01}(X^+)$, and $\omega_{11}(X^+)$, for $X^-$ and
$X^+$, respectively. In turn, each of these transitions have a corresponding
characteristic radiative lifetime. For the latter, our calculations predict
$\tau_{00}(X^-)=\tau_{11}(X^-)$ and $\tau_{01}(X^-)=\tau_{10}(X^-)$ as well as
$\tau_{00}(X^+)=\tau_{11}(X^+)$ and $\tau_{01}(X^+)=\tau_{10}(X^+)$.
%
Thus, at $T=0\,{\rm K}$, the radiative lifetimes of $X^-$ and $X^+$ are given by

\begin{widetext}
\begin{eqnarray}
\tau(X^-)=\left\{\frac{1}{2}\left[\frac{1}{\tau_{00}(X^-)}+\frac{1}{\tau_{01}(X^-)}\right]+\frac{1}{2}\left[\frac{1}{\tau_{10}(X^-)}+\frac{1}{\tau_{11}(X^-)}\right]\right\}^{-1}=\frac{\tau_{00}(X^-)\tau_{01}(X^-)}{\tau_{00}(X^-)+\tau_{01}(X^-)},\nonumber\\
\\
\tau(X^+)=\left\{\frac{1}{2}\left[\frac{1}{\tau_{00}(X^+)}+\frac{1}{\tau_{01}(X^+)}\right]+\frac{1}{2}\left[\frac{1}{\tau_{10}(X^+)}+\frac{1}{\tau_{11}(X^+)}\right]\right\}^{-1}=\frac{\tau_{00}(X^+)\tau_{01}(X^+)}{\tau_{00}(X^+)+\tau_{01}(X^+)}.\nonumber\\
\end{eqnarray}
\end{widetext}

\noindent Figure \ref{Fig_5}(a) shows the height dependence of $\tau_{00}(X^-)$
and $\tau_{01}(X^-)$, and $\tau_{00}(X^+)$ and $\tau_{01}(X^+)$. These {\em
  characteristic} radiative lifetimes depend strongly and non-monotonically on
height. This non-monotonic dependence translates into a rather simple and
monotonic $\tau(X^-)$ and $\tau(X^+)$, as shown in Fig. \ref{Fig_5}(b). For
flat dots $\tau(X^-)\simeq\tau(X^+)$, whereas for taller dots these lifetimes
become slightly different. Our predicted $\tau(X^-)$ are in satisfactory
agreement with the value of $0.6\;{\rm ns}$ recently observed by Smith and
co-workers in (In,Ga)As/GaAs dots (size unspecified) with exciton ground-state
emission at $\sim 1.318\,{\rm eV}$ (Ref. \onlinecite{smith_PRL_2005}). We find
that the radiative lifetimes of the charged trions satisfy the relationship
$\tau(X^0)>\tau(X^-)\sim\tau(X^+)>\tau(XX^0)$.


%
\section{Comparison of $X^0$, $X^-$, $X^+$, and $XX^0$ in 
lens-shaped pure InAs/GaAs with alloyed (In,Ga)As/GaAs dots} 

For completeness, we briefly compare the binding and recombination energies,
polarization anisotropy, and radiative lifetimes in lens-shaped pure,
non-alloyed InAs/GaAs dots with In$_{0.6}$Ga$_{0.4}$As/GaAs dots.  Note that
Williamson, Wang, and Zunger have already compared results for $X^0$ for
several alloy profiles.\cite{williamson_PRB_2000} In Ref.
\onlinecite{narvaez_condmat} we predicted that hole localization takes place
at the dot-GaAs matrix interface as the height of these dots increases above
$35\;${\AA}.  Thus, we discuss here two flat dots ($h=20\;${\AA} and
$35\;${\AA}) with base $b=252\;${\AA}.

%
(i) Recombination energies are smaller in InAs/GaAs dots than in In$_{\rm
  0.6}$Ga$_{\rm 0.4}$As/GaAs dots with the same geometry. For instance,
$\omega_{20}(X^0)=1.078\;{\rm eV}$ and $0.987\;{\rm eV}$ for $h=20\;${\AA} and
$35\;${\AA}, respectively.

%
(ii) The spectroscopic shifts show the same trends with height in flat
InAs/GaAs as those in flat In$_{\rm 0.6}$Ga$_{\rm 0.4}$As/GaAs dots. So do the
binding energies $\Delta(X^-)$ and $\Delta(X^+)$, and $\Delta(XX^0)$. However,
there are two important differences bewteen the pure, non-alloyed InAs/GaAs
dots and their In$_{\rm 0.6}$Ga$_{\rm 0.4}$As/GaAs counterparts: (a) For the
$20\;$\AA-tall InAs/GaAs dot, while the binding energies still satisfy
$\Delta(X^-) > \Delta(XX^0) > \Delta(X^+)$, we find that $X^+$ and $XX^0$ are
{\em bound}, with $\Delta(X^+)=1.6\;{\rm meV}$ and $\Delta(XX^0)=1.5\;{\rm
  meV}$, respectively. This is so because at this height the InAs/GaAs dot is
in the nearly ``symmetric'' regime: $J^{(hh)}_{00}=25.6\;{\rm meV}\sim
J^{(ee)}_{00}=25.1\;{\rm meV} \sim J^{(eh)}_{00}=25.3\;{\rm meV}$, so the
Hartree-Fock component of the binding energy [see Eq. (\ref{Correlation}) for
$XX^0$ case] is much smaller than in the In$_{\rm 0.6}$Ga$_{0.4}$As/GaAs dot
with same height.  Thus, correlation becomes capable of binding $X^+$ and
$XX^0$ in the pure, non-alloyed dot.  (b) For the $35\;$\AA-tall InAs/GaAs
dot, we find {\em ordering reversal}, i.e.  $\Delta(X^+) > \Delta(XX^0) >
\Delta(X^-)$. In In$_{0.6}$Ga$_{0.4}$As/GaAs dots, this ordering is attained
at $h=50\;${\AA} (Fig. \ref{Fig_3.mp}).

%
(iii) In contrast to the findings in In$_{\rm 0.6}$Ga$_{\rm 0.4}$As/GaAs dots,
transitions $\omega_{20}(X^0)$ and $\omega_{30}(X^0)$ are fully polarized
along $[110]$ ($\lambda=1$) and $[1\bar 10]$ ($\lambda=-1$), respectively,
regardless of height. Consequently, $\omega_{03}(XX^0)$ and
$\omega_{02}(XX^0)$ are fully polarized along $[1\bar 10]$ and $[110]$,
respectively. These polarizations are expected from a dot with $C_{2v}$
symmetry, like a lens-shaped pure InAs/GaAs dot.

%
(iv) Radiative lifetimes $\tau(X^0)$, $\tau(X^+)$ and $\tau(X^-)$, and
$\tau(XX^0)$ are similar to those in In$_{\rm 0.6}$Ga$_{\rm
  0.4}$As/GaAs dots with the same geometry. For instance, $\tau(X^0)=2.8\;{\rm
  ns}$ and $\tau(XX^0)=0.7\;{\rm ns}$ for $h=35\;${\AA}; and
$\tau(X^0)=2.9\;{\rm ns}$ and $\tau(XX^0)=0.6\;{\rm ns}$ for $h=20\;${\AA}.



%%%%%%%%%%%
%         
% Summary 
%         
%%%%%%%%%%%
%
%
%
\section{Summary}

We have addressed the height dependence of recombination energies,
polarization and radiative lifetimes of the lowest optical transitions of the
neutral exciton ($X^0$), negatively- ($X^{-}$) and positively-charged
($X^{+}$) trions, and the biexciton ($XX^0$) in lens-shaped, self assembled
In$_{0.6}$Ga$_{0.4}$As/GaAs quantum dots. We have predicted the following.

%
(i) The recombination energy of the lowest transition of $X^-$, $X^+$ and
$XX^0$, correspondingly, $\omega_{00}(X^-)$, $\omega_{00}(X^+)$ and
$\omega_{03}(XX^0)$ shows qualitatively different behavior for each excitonic
complex.  Namely, $\omega_{00}(X^-)$ blue-shifts as height increases, whereas
that of $\omega_{00}(X^+)$ red-shifts. On the other hand, as height increases,
$\omega_{03}(XX^0)$ shows a red-shift at small heights, reaches a maximum
shift, and then blue-shifts for taller dots. This behavior is explained by the
height dependence and relative magnitude of $J^{\,(ee)}_{00}$,
$J^{\,(hh)}_{00}$ and $J^{\,(eh)}_{00}$.

%
(ii) The binding energies $\Delta(X^-)$, $\Delta(X^+)$ and $\Delta(XX^0)$
follow the height dependence of the emission spectroscopic shifts. Changes in
the dot height drives a bound-to-unbound crossover for each of these
complexes.

%
(iii) The in-plane polarization anisotropy $\lambda$ of the lowest transitions
of $X^0$ ($\omega_{20}$) and $XX^0$ ($\omega_{03}$) strongly depends on dot
height as well as on alloy randomness (disorder realization).  In contrast,
the lowest transitions of $X^-$ and $X^+$ present negligible $\lambda$
regardless of height.

%
(iv) The ground state of $X^0$ encompasses four states that split off in a
low-energy pair that is dark and a high-energy pair that is bright, with a
bright-dark splitting that increases as height decreases.  Thus, at $T=0\;{\rm
  K}$ the radiative lifetime $\tau(X^0)$ of $X^0$ is long.  On the other hand,
at $T=10\;{\rm K}$ both dark and bright states are populated; so, $\tau(X^0)$
becomes fast, moderately decreases as height increases, and its magnitude
ranges from $2$-$3\;{\rm ns}$.  In contrast, $\tau(X^-)$, $\tau(X^+)$ and
$\tau(XX^0)$ are fast even at $T=0\;{\rm K}$, as a consequence of these
excitons having ground states that are bright. These radiative lifetimes
depend weakly on height. Further, $\tau(X^-)\sim\tau(X^+)\simeq 1.1\;{\rm
  ns}$, while $\tau(XX^0)\simeq 0.5\;{\rm ns}$.

%
We have compared these predictions with available data and have found them in
satisfactory agreement. In addition, we compared with results in pure,
non-alloyed InGa/GaAs quantum dots.

%%%%%%%%%
%
% Thanks
%
%%%%%%%%%
%
\begin{acknowledgments}
  We thank Alberto Franceschetti (NREL) for valuable discussions. This work
  has been supported by U.S. DOE-SC-BES-DMS under contract No.
  DE-AC36-99GO10337.
\end{acknowledgments}

%%%%%%%%%%%%%%%%%%%%%%
% Bibliography begins 
%%%%%%%%%%%%%%%%%%%%%%
% 
\begin{thebibliography}{100}

\bibitem{smith_PRL_2005} J. M. Smith, P. A. Dalgarno, R. J. Warburton,
  A. O. Govorov, K. Karrai, B. D. Gerardot, and P. M. Petroff,
  Phys. Rev. Lett. {\bf 94}, 197402 (2005).

\bibitem{ware_PhysicaE_2005} M. E. Ware, A. S. Bracker, E. Stinaff, D. Gammon, D. Gershoni,
  and V. L. Korenev, Physica E {\bf 26}, 55 (2005).

\bibitem{skolnick_ARMR_2004} M. S. Skolnick and D. J. Mowbray,
  Annu. Rev. Mater. Res. {\bf 34}, 181 (2004).

\bibitem{guffarth_PhysicaE_2004} F. Guffarth, S. Rodt, A. Schliwa, K. P\"otschke, and D. Bimberg,
    Physica E {\bf 25}, 261 (2004).

\bibitem{finley_PRB_2001} J. J. Finley, A. D. Ashmore, A. Lema\^{i}tre,
  D. J. Mowbray, M. S. Skolnick, I. E. Itskevich, P. A. Maksym, M. Hopkinson,
  and T. F. Krauss, Phys. Rev. B {\bf 63}, 073307 (2001).

\bibitem{rodt_papers} S. Rodt, R. Heitz, R. L. Sellin, A. Schliwa, K. P\"otschke, and D. Bimberg, Physica E {\bf 21}, 1065 (2004); S. Rodt, R. Heitz, A. Schliwa, R. L. Sellin, F. Guffarth, and D. Bimberg, Phys. Rev. B {\bf 68}, 035331 (2003).

\bibitem{urbaszek_PRL_2003} B. Urbaszek, R. J. Warburton, K. Karrai,
  B. D. Gerardot, P. M. Petroff, and J. M. Garcia, Phys. Rev. Lett. {\bf 90},
  247403 (2003).

\bibitem{findeis_SSC_2000} F. Findeis, A. Zrenner, G. B\"ohm, and G. Abstreiter, Solid State 
Comm. {\bf 114}, 227 (2000).

\bibitem{landin_papers} L. Landin, M. S. Miller, M.-E. Pistol, C. E. Pryor, and L. Samuelson, Science
{\bf 280}, 262 (1998); L. Landin, M.-E. Pistol, C. Pryor, M. Persson,
L. Samuelson, and M. Miller, Phys. Rev. B {\bf 60}, 16640 (1999).

\bibitem{books+review} {\em Quantum Dot Heterostructures}, D. Bimberg,
  M. Grundmann and N. N. Ledenstov (Wiley, Chichester, 1999); 
{\em Nanostructures---Theory and Modeling}, C. Delerue and M. Lannoo
  (Springer-Verlag, Berlin, 2004); T. Takagahara, Phase Transitions {\bf 68},
  281 (1999);  {\em Optical Properties of Semiconductor Quantum Dots},
  U. Woggon (Springer-Verlag, Berlin, 1997)
   
\bibitem{wang_PRB_1999} L.-W. Wang and A. Zunger, Phys. Rev. B {\bf 59},
    15806 (1999).
  
\bibitem{wang_APL_2000} L. W. Wang, A. J. Williamson, A. Zunger, H. Jiang, and
  J. Singh, Appl. Phys. Lett. {\bf 76}, 339 (2000).
  
\bibitem{zunger_pss.b_2001}A. Zunger, phys. stat. sol. (b) {\bf 224}, 727
  (2001).

\bibitem{TB_method} S. Lee, O. L. Lazarenkova, P. von Allmen, F. Oyafuso, and
  G. Klimeck, Phys. Rev. B {\bf 70}, 125307 (2004); S. Lee, F. Oyafuso, P. von
  Allmen, and G. Klimeck, Phys. Rev. B {\bf 69}, 045316 (2004); R. Santoprete,
  B. Koiller, R. B. Capaz, P. Kratzer, Q. K. K. Liu, and M. Scheffler, Phys.
  Rev. B {\bf 68}, 235311 (2003). 

\bibitem{sheng_PRB_2005} W. Sheng, S.-J. Cheng, and P. Hawrylak, Phys. Rev. B {\bf 71}, 035316 (2005).
  
\bibitem{williamson_PRB_2000} A. J. Williamson, L.-W. Wang, and A. Zunger,
  Phys. Rev. B {\bf 62}, 12963 (2000).

\bibitem{franceschetti_PRB_1999} A. Franceschetti, H. Fu, L. W. Wang, and
  A. Zunger, Phys. Rev. B {\bf 60}, 1819 (1999).

\bibitem{resta_PRB_1977} R. Resta, Phys. Rev. B {\bf 16}, 2717 (1977).

\bibitem{bester_PRB_2003}G. Bester, S. Nair, and A. Zunger, Phys. Rev. B {\bf
    67}, 161306(R) (2003).
  
\bibitem{bayer_PRB_2002} M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A.
  Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. L. Reinecke, S.
  N. Walck, J. P. Reithmaier, F. Klopf, and F. Sch\"afer, 
Phys. Rev. B {\bf 65}, 195315 (2002).

\bibitem{cardona_book} {\em Fundamentals of Semiconductors---Physics and
  Materials Properties}, P. Y. Yu and M. Cardona (Springer, Berlin, 2001).
  
\bibitem{bester_PRB_2003b} G. Bester and A. Zunger, Phys. Rev. B {\bf 68},
  073309 (2003).

\bibitem{shumway_PRB_2001} J. Shumway, A. Franceschetti, and A. Zunger,
  Phys. Rev. B {\bf 63}, 155316 (2001).


\bibitem{narvaez_PRB_tobepublished} G. A. Narvaez, G. Bester, and A. Zunger,
 Phys. Rev. B {\bf 72}, 041307(R) (2005). 

\bibitem{narvaez_condmat} G. A. Narvaez, G. Bester, and A. Zunger, 
J. Appl. Phys. {\bf 98}, 043708 (2005).

%
%
\bibitem{bayer_PRB_1998} M. Bayer, T. Gutbrod, A. Forchel, V. D. Kulakovskii,
  A. Gorbunov, M. Michel, R. Steffen, and K. H. Wang, Phys. Rev. B {\bf 58},
  4740 (1998).

\bibitem{dexter_book} D. L. Dexter in {\em Solid State Physics} (Academic
  Press, New York, 1958), vol. 6.

\bibitem{thranhardt_PRB_2002} A. Thr\"anhardt, C. Ell, G. Khitrova, and
  H. M. Gibbs, Phys. Rev. B {\bf 65}, 035327 (2002).

\bibitem{dekel_PRB_2000} E. Dekel, D. V. Regelman, D. Gershoni,
  E. Ehrenfreund, W. V. Schoenfeld, and P. M. Petroff, Phys. Rev. B {\bf 62},
  11038 (2000).

\bibitem{karachinsky_APL_2004} L. Ya. Karachinsky, S. Pellegrini, G. A. Buller,
  A. S. Sholnik, N. Yu. Gordeev, V. P. Evtikhiev, and V. B. Novikov,
  Appl. Phys. Lett. {\bf 84}, 7 (2004).

\bibitem{buckle_JAP_1999} P. D. Buckle, P. Dawson, S. A. Hall, X. Chen,
  M. J. Steer, D. J. Mowbray, M. S. Skolnick, and M. Hopkinson,
  J. Appl. Phys. {\bf 86}, 2555 (1999).


\bibitem{bardot_condmat_2005} C. Bardot, M. Schwab, M. Bayer, S. Fafard,
  Z. Wasilewski, and P. Hawrylak, Phys. Rev. B {\bf 72}, 035314 (2005).


\bibitem{ulrich_condmat_2004} S. M. Ulrich, M. Beyouncef, P. Michler, N. Baer,
  P. Gartner, F. Jahnke, M. Schwab, H. Kurtze, M. Bayer, S. Fafard, and
  Z. Wasilewski, and A. Forchel, Phys. Rev. B {\bf 71}, 235328 (2005).

%
\end{thebibliography}
%
%%% Bibliography ends %%%


\end{document} 


cond-mat0506714.extracted_bib
\begin{thebibliography}{100}

\bibitem{smith_PRL_2005} J. M. Smith, P. A. Dalgarno, R. J. Warburton,
  A. O. Govorov, K. Karrai, B. D. Gerardot, and P. M. Petroff,
  Phys. Rev. Lett. {\bf 94}, 197402 (2005).

\bibitem{ware_PhysicaE_2005} M. E. Ware, A. S. Bracker, E. Stinaff, D. Gammon, D. Gershoni,
  and V. L. Korenev, Physica E {\bf 26}, 55 (2005).

\bibitem{skolnick_ARMR_2004} M. S. Skolnick and D. J. Mowbray,
  Annu. Rev. Mater. Res. {\bf 34}, 181 (2004).

\bibitem{guffarth_PhysicaE_2004} F. Guffarth, S. Rodt, A. Schliwa, K. P\"otschke, and D. Bimberg,
    Physica E {\bf 25}, 261 (2004).

\bibitem{finley_PRB_2001} J. J. Finley, A. D. Ashmore, A. Lema\^{i}tre,
  D. J. Mowbray, M. S. Skolnick, I. E. Itskevich, P. A. Maksym, M. Hopkinson,
  and T. F. Krauss, Phys. Rev. B {\bf 63}, 073307 (2001).

\bibitem{rodt_papers} S. Rodt, R. Heitz, R. L. Sellin, A. Schliwa, K. P\"otschke, and D. Bimberg, Physica E {\bf 21}, 1065 (2004); S. Rodt, R. Heitz, A. Schliwa, R. L. Sellin, F. Guffarth, and D. Bimberg, Phys. Rev. B {\bf 68}, 035331 (2003).

\bibitem{urbaszek_PRL_2003} B. Urbaszek, R. J. Warburton, K. Karrai,
  B. D. Gerardot, P. M. Petroff, and J. M. Garcia, Phys. Rev. Lett. {\bf 90},
  247403 (2003).

\bibitem{findeis_SSC_2000} F. Findeis, A. Zrenner, G. B\"ohm, and G. Abstreiter, Solid State 
Comm. {\bf 114}, 227 (2000).

\bibitem{landin_papers} L. Landin, M. S. Miller, M.-E. Pistol, C. E. Pryor, and L. Samuelson, Science
{\bf 280}, 262 (1998); L. Landin, M.-E. Pistol, C. Pryor, M. Persson,
L. Samuelson, and M. Miller, Phys. Rev. B {\bf 60}, 16640 (1999).

\bibitem{books+review} {\em Quantum Dot Heterostructures}, D. Bimberg,
  M. Grundmann and N. N. Ledenstov (Wiley, Chichester, 1999); 
{\em Nanostructures---Theory and Modeling}, C. Delerue and M. Lannoo
  (Springer-Verlag, Berlin, 2004); T. Takagahara, Phase Transitions {\bf 68},
  281 (1999);  {\em Optical Properties of Semiconductor Quantum Dots},
  U. Woggon (Springer-Verlag, Berlin, 1997)
   
\bibitem{wang_PRB_1999} L.-W. Wang and A. Zunger, Phys. Rev. B {\bf 59},
    15806 (1999).
  
\bibitem{wang_APL_2000} L. W. Wang, A. J. Williamson, A. Zunger, H. Jiang, and
  J. Singh, Appl. Phys. Lett. {\bf 76}, 339 (2000).
  
\bibitem{zunger_pss.b_2001}A. Zunger, phys. stat. sol. (b) {\bf 224}, 727
  (2001).

\bibitem{TB_method} S. Lee, O. L. Lazarenkova, P. von Allmen, F. Oyafuso, and
  G. Klimeck, Phys. Rev. B {\bf 70}, 125307 (2004); S. Lee, F. Oyafuso, P. von
  Allmen, and G. Klimeck, Phys. Rev. B {\bf 69}, 045316 (2004); R. Santoprete,
  B. Koiller, R. B. Capaz, P. Kratzer, Q. K. K. Liu, and M. Scheffler, Phys.
  Rev. B {\bf 68}, 235311 (2003). 

\bibitem{sheng_PRB_2005} W. Sheng, S.-J. Cheng, and P. Hawrylak, Phys. Rev. B {\bf 71}, 035316 (2005).
  
\bibitem{williamson_PRB_2000} A. J. Williamson, L.-W. Wang, and A. Zunger,
  Phys. Rev. B {\bf 62}, 12963 (2000).

\bibitem{franceschetti_PRB_1999} A. Franceschetti, H. Fu, L. W. Wang, and
  A. Zunger, Phys. Rev. B {\bf 60}, 1819 (1999).

\bibitem{resta_PRB_1977} R. Resta, Phys. Rev. B {\bf 16}, 2717 (1977).

\bibitem{bester_PRB_2003}G. Bester, S. Nair, and A. Zunger, Phys. Rev. B {\bf
    67}, 161306(R) (2003).
  
\bibitem{bayer_PRB_2002} M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A.
  Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. L. Reinecke, S.
  N. Walck, J. P. Reithmaier, F. Klopf, and F. Sch\"afer, 
Phys. Rev. B {\bf 65}, 195315 (2002).

\bibitem{cardona_book} {\em Fundamentals of Semiconductors---Physics and
  Materials Properties}, P. Y. Yu and M. Cardona (Springer, Berlin, 2001).
  
\bibitem{bester_PRB_2003b} G. Bester and A. Zunger, Phys. Rev. B {\bf 68},
  073309 (2003).

\bibitem{shumway_PRB_2001} J. Shumway, A. Franceschetti, and A. Zunger,
  Phys. Rev. B {\bf 63}, 155316 (2001).


\bibitem{narvaez_PRB_tobepublished} G. A. Narvaez, G. Bester, and A. Zunger,
 Phys. Rev. B {\bf 72}, 041307(R) (2005). 

\bibitem{narvaez_condmat} G. A. Narvaez, G. Bester, and A. Zunger, 
J. Appl. Phys. {\bf 98}, 043708 (2005).

%
%
\bibitem{bayer_PRB_1998} M. Bayer, T. Gutbrod, A. Forchel, V. D. Kulakovskii,
  A. Gorbunov, M. Michel, R. Steffen, and K. H. Wang, Phys. Rev. B {\bf 58},
  4740 (1998).

\bibitem{dexter_book} D. L. Dexter in {\em Solid State Physics} (Academic
  Press, New York, 1958), vol. 6.

\bibitem{thranhardt_PRB_2002} A. Thr\"anhardt, C. Ell, G. Khitrova, and
  H. M. Gibbs, Phys. Rev. B {\bf 65}, 035327 (2002).

\bibitem{dekel_PRB_2000} E. Dekel, D. V. Regelman, D. Gershoni,
  E. Ehrenfreund, W. V. Schoenfeld, and P. M. Petroff, Phys. Rev. B {\bf 62},
  11038 (2000).

\bibitem{karachinsky_APL_2004} L. Ya. Karachinsky, S. Pellegrini, G. A. Buller,
  A. S. Sholnik, N. Yu. Gordeev, V. P. Evtikhiev, and V. B. Novikov,
  Appl. Phys. Lett. {\bf 84}, 7 (2004).

\bibitem{buckle_JAP_1999} P. D. Buckle, P. Dawson, S. A. Hall, X. Chen,
  M. J. Steer, D. J. Mowbray, M. S. Skolnick, and M. Hopkinson,
  J. Appl. Phys. {\bf 86}, 2555 (1999).


\bibitem{bardot_condmat_2005} C. Bardot, M. Schwab, M. Bayer, S. Fafard,
  Z. Wasilewski, and P. Hawrylak, Phys. Rev. B {\bf 72}, 035314 (2005).


\bibitem{ulrich_condmat_2004} S. M. Ulrich, M. Beyouncef, P. Michler, N. Baer,
  P. Gartner, F. Jahnke, M. Schwab, H. Kurtze, M. Bayer, S. Fafard, and
  Z. Wasilewski, and A. Forchel, Phys. Rev. B {\bf 71}, 235328 (2005).

%
\end{thebibliography}