# Dark-bright excitons mixing in alloyed InGaAs self-assembled quantum dots

Michał Zieliński<EMAIL_ADDRESS>Institute of Physics, Faculty of
Physics, Astronomy and Informatics
Nicolaus Copernicus University, ul. Grudziądzka 5, 87-100 Toruń, Poland

###### Abstract

Quantum dots are arguably one of the best platforms for optically accessible
spin based qubits. The paramount demand of extended qubit storage time can be
met by using quantum-dot-confined dark exciton: a long-lived electron-hole
pair with parallel spins. Despite its name the dark exciton reveals weak
luminescence that can be directly measured. The origins of this optical
activity remain largely unexplored. In this work, using the atomistic tight-
binding method combined with configuration-interaction approach, we
demonstrate that atomic-scale randomness strongly affects oscillator strength
of dark excitons confined in self-assembled cylindrical InGaAs quantum dots
with no need for faceting or shape-elongation. We show that this process is
mediated by two mechanisms: mixing dark and bright configurations by exchange
interaction, and equally important appearance of non-vanishing optical
transition matrix elements that otherwise correspond to nominally forbidden
transitions in a non-alloyed case. The alloy randomness has essential impact
on both bright and dark exciton states, including their energy, emission
intensity, and polarization angle. We conclude that, due to the atomic-scale
alloy randomness, finding dots with desired dark exciton properties may
require exploration of a large ensemble, similarly to how dots with low bright
exciton splitting are selected for entanglement generation.

##  I Introduction

The optically weakly active [1, 2, 3, 4, 5] exciton states confined in quantum
dots [6, 7], also known as dark excitons (DEs), [8] are promising candidates
for applications in quantum information processing. [9, 2, 10, 11, 12, 13]
Together with optically active bright excitons (BEs), electron-hole pairs with
opposite spins far better studied for, e.g., entanglement generation [14, 15,
16, 17], they may form important building blocks for future quantum devices.
However, this applicability strongly depends on ability to understand and
control the details of excitonic spectra, so-called “excitonic fine
structure”. [8] On the one hand, the bright exciton spectrum is strongly
determined by quantum-dot symmetry properties [18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29] with extensive efforts [30, 31, 32, 33, 34, 35, 36, 37, 38,
39, 40, 41, 42, 43] aimed at reduction of the bright exciton splitting (BES).
On the other, the dark exciton fine structure has been studied to a far lesser
degree. In particular, contrary to simplified theoretical predictions [8], the
dark exciton can gain a non-negligible optical activity, even without an
external magnetic field, [19, 22] whereas origins of this luminescence remain
largely unexplored. [44, 45]

Accurate theoretical modeling of excitonic fine structure, regarding both
bright and dark excitons, still presents a serious challenge for approaches
utilizing continuum-media approximation, [46, 47, 19] and even for atomistic
methods. [48, 22, 10] In this work, we use the combination of tight-binding
(TB) and configuration-interaction (CI) methods, which proved its ability to
find excitonic fine structure in a good agreement with experiment. [49, 50,
22] To understand the role of disorder due to alloy randomness, we present an
extensive theoretical study of properties of bright and dark excitons
calculated for an ensemble of 300 alloyed In0.5Ga0.5As self-assembled quantum
dots, each treated with the same atomistic resolution.

We find that alloy randomness, by reducing the overall symmetry, leads to (i)
non-vanishing matrix elements of the optical transition (optical/transition
dipole moments), as well as (ii) non-negligible exchange integrals mixing
bright and dark excitonic configurations. There are thus two equally important
sources of non-zero oscillator strength for the in-plane dark excitonic
emission. On the contrary, the out-of-plane ($z$) polarized emission is
dominated by the contribution from the matrix element of the optical
transition only, which is in turn governed by the valence-band mixing, in
agreement with the phenomenological understating of dark exciton states.
Therefore, contrary to the case of in-plane emission, the exchange mixing does
not play a role for $z$-polarized dark exciton emission.

While most of the quantum dots in ensembles reveal rather weak dark exciton
activity, we show that a mere alloy randomness triggers the dark exciton
optical activity reaching up to 1/6000 fraction of that for the bright
exciton, without faceting or shape elongation. [3, 10, 45] This conclusion is
valid for both out-of-plane and in-plane polarizations. As optical activity of
dark exciton is inversely proportional to its lifetime, a strong variation of
the former translates to a pronounced dot-to-dot variations of the latter on a
millisecond scale, while the bright exciton lifetimes tend to be
systematically close to 1 ns. Moreover, despite the overall cylindrical
quantum-dot shape, and despite strong alloying, the polarization properties of
both bright and dark excitons for in-plane emission reveal hallmarks of a
pronounced anisotropy due to the underlying crystal lattice.

The paper is organized as follows: after a short theoretical introduction in
Section II, we start with the discussion of bright exciton spectra and
polarization angle in Section III, Throughout most of the paper we aim to
bridge between atomistic results, and the effective treatment based on
valence-band mixing. When feasible these discussions are augmented with an
abridged statistical analysis. In Section IV we study in-plane emission form
dark excitons, including their polarization properties, and finally we study
excitonic lifetimes in Section V.

##  II Systems and methods

The calculation starts with finding atomic positions that minimize the total
elastic energy, by using the valence force field (VFF) method of Keating [51,
52]. Minimization of strain energy is performed with the conjugate gradient
method. [53, 54] Next, the piezoelectric potential [55, 56, 57, 58, 59, 60] is
calculated by accounting for both linear and quadratic contributions, with
piezoelectric coefficients from Ref. [57]. The single-particle spectra of
electrons and holes are obtained with the empirical $sp^{3}d^{5}s^{*}$ tight-
binding method accounting for $d$-orbitals and spin-orbit interaction. [61,
62, 63, 50] The tight-binding calculation is performed on a smaller domain
than the valence force field calculation. [64, 65] The details regarding the
$sp^{3}d^{5}s^{\star}$ tight-binding calculations for various nanostructures
are discussed thoroughly in our earlier papers. [63, 50, 53, 66, 67]

We note that eigenstates of the tight-binding Hamiltonian are doubly
degenerate to due to time reversal-symmetry. [68] Therefore, for convenience,
in several places we label tight binding electron ground states with up and
down spin-indices, i.e., $e_{\uparrow}$ and $e_{\downarrow}$. However, we
emphasize that due to spin-orbit interaction, and low overall symmetry, due to
underlying lattice with alloying, the spin in no longer a good quantum number,
and such labeling is only approximate. Similarly, we label atomistically
obtained hole ground states with double-arrows, i.e., $h_{\Uparrow}$ and
$h_{\Downarrow}$, even though due to valence-band mixing and alloying they do
not exactly correspond to heavy-hole $\pm\frac{3}{2}$ eigenstates.

After the tight-binding stage of calculations, the excitonic spectra [69] are
calculated with the configuration-interaction method described in detail in
Ref. [66]. The Hamiltonian for the interacting electron-hole pair can be
written in second quantization as follows [69]

$\hat{H}_{ex}=\sum_{i}E_{i}^{e}c_{i}^{\dagger}c_{i}+\sum_{i}E_{i}^{h}h_{i}^{\dagger}h_{i}\\\
-\sum_{ijkl}V_{ijkl}^{eh,\text{dir}}c_{i}^{\dagger}h_{j}^{\dagger}h_{k}c_{l}+\sum_{ijkl}V_{ijkl}^{eh,\text{exch}}c_{i}^{\dagger}h_{j}^{\dagger}c_{k}h_{l}$
(1)

where $E_{i}^{e}$ and $E_{i}^{h}$ are the single-particle electron and hole
energies obtained at the single-particle stage of calculations, respectively,
and $V_{ijkl}$ are Coulomb matrix elements (Coulomb direct and exchange
integrals) calculated according to the procedure given in Ref. [66]. More
details regarding the Coulomb matrix element computation for tight-binding
wave-functions can also be found in Refs. [70, 71] as well as in our recent
papers. [72, 73]

Finally, the optical spectra are found by calculating the intensity of
photoluminescence from the recombination of electron-hole pair using Fermi’s
golden rule

$\begin{split}I\left(\omega\right)=\Big{|}\sum_{c,v}C_{c,v}^{i}\matrixelement{\Psi_{c}}{\vec{\bm{\epsilon}}\cdot\vec{\bm{r}}}{\Psi_{v}}\Big{|}^{2}\delta\left(E_{i}-$\hbar$\omega\right)\\\
=\Big{|}\sum_{c,v}C_{c,v}^{i}M_{c,v}^{\vec{\bm{\epsilon}}}\Big{|}^{2}\delta\left(E_{i}-$\hbar$\omega\right)\end{split}$
(2)

where $E_{i}$ is the energy of initial $i$th state of the exciton,
$C_{v,c}^{i}$ are CI expansion coefficients for the $i$th state obtained by
solving Eq. 1,
$M_{c,v}^{\vec{\bm{\epsilon}}}\equiv\matrixelement{\Psi_{c}}{\vec{\bm{\epsilon}}\cdot\vec{\bm{r}}}{\Psi_{v}}$
is the optical dipole moment matrix element calculated from single-particle
tight-binding electron $\Psi_{c}$ and hole $\Psi_{h}$ wave functions
respectively, for a given polarization of light $\vec{\bm{\epsilon}}$, e.g.,
$M_{e_{\uparrow},h_{\Downarrow}}^{z}=\matrixelement{e_{\uparrow}}{z}{h_{\Downarrow}}$,
for $z$-polarized (out-of-plane) light, and optical (transition) dipole matrix
element involving ground electron and hole states with opposite (quasi-)spins.

Fermi’s golden rule allows for calculation of first-order radiative
recombination lifetime in a similar manner [74, 75, 76]

$\frac{1}{\tau_{i}}=\frac{4n\alpha\omega_{i}^{3}}{3c^{2}}\sum_{\vec{\bm{\epsilon}}=\vec{\bm{x}},\vec{\bm{y}},\vec{\bm{z}}}\Big{|}\sum_{c,v}C_{c,v}^{i}M_{c,v}^{\vec{\bm{\epsilon}}}\Big{|}^{2},$
(3)

where $\omega_{i}=E_{i}/\hbar$, $n=3.6$ [74] is the refractive index, $\alpha$
is the fine-structure constant, $c$ is the speed of light in the vacuum, $x$,
$y$ are in-plane, and $z$ is the out-of-plane radiative decay channel.
Finally, the average bright exciton radiative lifetime, assuming identical
occupation of two bright exciton states BE1 and BE2 (discussed later in the
text), and neglecting thermal effects is calculated as [75, 76]

$\frac{1}{\tau_{BE}}\approx\frac{1}{2}\left(\frac{1}{\tau_{BE1}}+\frac{1}{\tau_{BE2}}\right)$
(4)

with an analogous formula for the dark exciton.

Figure 1: Schematics of systems under consideration: (a) lens-shaped InAs/GaAs
self-assembled quantum dot located on a wetting layer and (b) alloyed
In0.5Ga0.5As quantum dots with the same dimensions, however with 5 randomly
generated samples (realizations) corresponding to the same average
composition, yet different (random) atomic arrangement. See the text for
details. Surrounding GaAs material is not shown.

Using the above formalism, we perform calculations for an ensemble of alloyed
In0.5Ga0.5As lens-shaped quantum dots located on a wetting layer (Fig. 1),
with low overall (shape + alloyed lattice) $C_{1}$ symmetry. Each quantum-dot
system is treated separately by full VFF/TB/CI calculation. The height of all
quantum dots is equal to 3 nm, whereas the diameter is 25.4 nm. Each quantum
dot is located on a 0.6 nm (1 lattice constant) thick wetting layer. A
quantum-dot system with such dimensions (in particular in a non-alloyed
InAs/GaAs variant), has been studied thoroughly in literature (including our
own work [63, 50]), as a model of a typical self-assembled quantum dot. For
comparison, in several places we discuss as well results obtained for $C_{2v}$
InAs quantum, with no gallium addmixture (no alloying; upper part of Fig. 1).
However, our focus is on the alloyed system, by considering 50% addmixture of
barrier material in the dot region, i.e., an In0.5Ga0.5As quantum dot in the
GaAs surrounding, with uniform (on average) composition profile. We consider
300 random samples (lower part of Fig. 1), i.e., 300 different random
realizations of the same alloyed lens-shaped quantum dot, with nominally the
same average composition yet different atomic arrangement on a microscopic
scale.

##  III Results

To study effects of alloy randomness, we focus on uniform composition profile,
rather than on effects of spatial changes in the overall composition. [77, 78,
79] To model random alloy fluctuations, for each cation site a random number
(uniformly distributed within 0 to 1 range) is generated, and based on that Ga
atom is replaced with In with 0.5 probability. This approach will thus
generate microscopic configurations that differ significantly in the atomic
arrangement, while maintaining (on average) 50% gallium content. A more
careful inspection reveals however, that such procedure may in fact result in
random realizations that differ from average composition within approximately
$\pm 1$% range, as shown in Fig. 2 (a). Therefore, a generated ensemble of
quantum dots is a subject to (a) randomness due to different atomic
arrangement corresponding to the same composition, plus (b) small
($\pm\sim$1%) fluctuations of average composition. As shown in Fig. 2 (b) the
ground state excitonic energy of the considered ensemble of quantum dots is
altered by both effects. Changes in average composition ($x$ axis in Fig. 2)
correspond to a change of excitonic energy by approximately 8 meV per 1% of
gallium content, consistent with 10 meV estimate that can be obtained from
virtual crystal approximation (VCA) with bulk InGaAs parameters taken from
Ref. [80] However, changes on the vertical axis, which are also on a scale of
several meV’s, are related purely due to alloy randomness, that is going
beyond the VCA. Importantly, whatever the cause, the exciton ground energy
varies within relatively small ranges, namely less then 1.3% from the average
of approximately 1257.6 meV. Therefore, as a rule of thumb, any spectral
quantity that changes substantially more than 2% when going from a sample to
sample (such as the BES and the DES discussed in the following) cannot be
attributed to a change in averaged composition, but to alloy randomness (or
changes in the microscopic atomic arrangement).

Figure 2: (a) Histogram illustrating distribution of average composition
variations (from ideal 50% gallium content) for a family of 300 alloyed
quantum dots studied in the text; (b) exciton ground-state energy as function
of composition difference and alloy randomness. See the text for details.
Figure 3: Bright exciton splitting (a), dark-bright splitting (b), and dark
exciton splitting (c) as a function of difference from average composition
($x$ axis), and alloy randomness (distribution of values along $y$ axis). See
the text for more details.

In a standard treatment, the excitonic fine structure is typically [8, 81]
addressed by starting from idealized single-particle states, with electron
spin states $\ket*{\pm\frac{1}{2}}$, and the hole ground states of heavy-hole
character, associated with the projection of the total hole angular momentum:
$\ket*{\pm\frac{3}{2}}$. It is important to reemphasize that
$\ket*{\pm\frac{1}{2}}$ are not exactly equal to $e_{\uparrow}/e_{\downarrow}$
tight-binding eigenstates, which are not eigenfuctions of spin, but are
labeled with pseudo-spin index instead. The same applies the
$h_{\Uparrow}/h_{\Downarrow}$ hole states as well. Nevertheless,
$\ket*{\pm\frac{1}{2}}$, $\ket*{\pm\frac{3}{2}}$ states are of vital
importance for defining effective Hamiltonians. [8, 82] To this end the
excitonic basis of four states in such approaches is thus constructed from the
above-mentioned, idealized single-particle states, and characterized by their
angular momentum projections

$\left|\pm 1\right>=\left|\pm\frac{3}{2},\mp\frac{1}{2}\right>,\left|\pm
2\right>=\left|\pm\frac{3}{2},\pm\frac{1}{2}\right>,$ (5)

where the first value in the ket corresponds to the hole, and the second one
to the electron. Matrix elements of the optical transition are given as: [81]
$M_{\pm\frac{3}{2},\mp\frac{1}{2}}\equiv M_{\pm 1}=M\left(e_{x}\mp
ie_{y}\right),M_{\pm 2}=0$, where $e_{x},e_{y}$ are in-plane components of
$\mathbf{e}$ polarization vector, and $M$ is independent of the light
polarization. Therefore $\left|\pm 1\right>$ are optically active, whereas
$\left|\pm 2\right>$ are optically inactive states.

Electron-hole exchange Hamiltonian is expanded in basis of these states, and
in the order $\left|1\right>,\left|-1\right>,\left|2\right>,\left|-2\right>$
can be written as [8]

$\displaystyle\mathbf{H}_{\mathrm{exch}}$
$\displaystyle=\frac{1}{2}\begin{bmatrix}\delta_{\mathrm{0}}&\delta_{1}&0&0\\\
\delta_{1}&\delta_{\mathrm{0}}&0&0\\\ 0&0&-\delta_{\mathrm{0}}&\delta_{2}\\\
0&0&\delta_{2}&-\delta_{\mathrm{0}}\end{bmatrix},$ (6)

where $\delta_{0}$ describes the dark-bright exciton splitting (“isotropic
electron-hole exchange interaction”), $\delta_{1}$ is responsible for the
bright doublet splitting, and since it is related to distortions from
idealized symmetry, it is often called “anisotropic electron-hole exchange
interaction”, and finally $\delta_{2}$ is responsible for the dark exciton
splitting. For $C_{3v}$, and $D_{2d}$ high-symmetry quantum dots
$\delta_{1}=0$, i.e., bright exciton splitting vanishes. [20] For $C_{3v}$
systems additionally $\delta_{2}=0$. Nonetheless, for realistic alloyed
$C_{1}$ quantum dots both $\delta_{1}\neq 0$ and $\delta_{2}\neq 0$, and
$\delta_{1}\gg\delta_{2}$. Similar properties are shared by $C_{2v}$ quantum
dots, hence $C_{2v}$ systems are often treated as good approximations to real
system, although such approximation often fails, [83, 84] as also discussed in
this work.

Since the off-diagonal block consists of zero’s, there is no coupling between
bright and dark manifolds, and they can be treated separately, with bright
exciton states given as symmetric/anti-symmetric combinations of basis states

$\displaystyle\ket*{BE1}$
$\displaystyle=\left(\ket*{1}-\ket*{-1}\right)/\sqrt{2}$ (7)
$\displaystyle\ket*{BE2}$
$\displaystyle=\left(\ket*{1}+\ket*{-1}\right)/\sqrt{2},$ (8)

with analogous formulas for the dark exciton. Corresponding eigenenergies are
given as

$\displaystyle E_{BE1}$ $\displaystyle=\left(\delta_{0}-\delta_{1}\right)/2$
(9) $\displaystyle E_{BE2}$
$\displaystyle=\left(\delta_{0}+\delta_{1}\right)/2,$ (10)

For sake of comparison, atomistic results (calculated in a basis of 144 CI
configurations) can be “recast” to spectra of Eq.6. For non-alloyed, lens-
shapes system of $C_{2v}$ symmetry we found from atomistic simulations, that
$\delta_{1}>0$, thus in such a case anti-symmetric $\ket*{BE1}$ would be the
lower-energy bright excitonic state. Moreover, $\ket*{BE1}$ is linearly
polarized along $[1\overline{1}0$ direction, and $\ket*{BE2}$ is linearly
polarized along $[110]$. Notably, there is no $z$-polarized emission allowed.
For an alloyed system polarization properties are more complicated, and will
be discussed in the following part of the paper.

The bright and dark exciton splittings calculated atomistically correspond to
$\left|\delta_{1}\right|$, $\left|\delta_{2}\right|$ of the simple model
respectively, which are shown in Fig. 3. Excitonic fine structure does not
show any clear trend with respect to average composition fluctuations, yet, it
strongly varies due to alloy randomness. [85, 22] Bright exciton splitting
[Fig. 3 (a)] varies from zero to over 20 $\mu$eV, with average value of 7.6
$\mu$eV, and standard deviation of 3.6 $\mu$eV. The dark-bright splitting
[Fig. 3 (b)] has the average value of 184 $\mu$eV, whereas the dark exciton
splitting varies from 0.28 to 1.48 $\mu$eV, with mean value of 0.66 $\mu$eV
and standard deviation equal to 0.16 $\mu$ eV. Again, since we do considered
cylindrical quantum dot shape shape [84] both BES and DES are in good
agreement with experimental values, although DES results are generally smaller
than observed in the experiment. [9] Nevertheless, we note that this range of
DE energy level splittings corresponds to the precession periods [1] of a
coherent superposition of DE eigenstates varying from to 2.8 to 14.5 ns, (with
average of 6.3 ns), thus again overlapping with typical experimental values.
[9, 86]

To further analyze polarization properties of bright excitons, while still
assuming no dark-bright exciton coupling, one can use the Hamiltonian for the
bright manifold only, derived based on group-theoretical [42] arguments:

$\displaystyle\mathbf{H}$ $\displaystyle=\begin{bmatrix}E+\delta&\kappa\\\
\kappa&E-\delta\end{bmatrix},$ (11)

where $E$ is the reference excitonic energy, $\delta$ (with no subscript) is
responsible for the bright exciton splitting in the $C_{2v}$ case due to
lattice (and shape) anisotropy, whereas $\kappa$ determines the fraction of
the BES related to contribution due to lowering of symmetry caused by
alloying. Moreover, $\kappa$ is responsible for the coupling between the two
bright states, leading to the rotation [42] of the emission lines from the
perfect crystal directions. This polarization angle $\theta$ is given as:
$\mathrm{tan}\left(\theta\right)=\frac{\delta}{\kappa}\pm\sqrt{1+\left(\delta/\kappa\right)^{2}}$,
with the overall bright exciton splitting given as
$\delta_{\mathrm{1}}=2\sqrt{\delta^{2}+\kappa^{2}}$, and conversely

$\kappa=-\frac{\delta_{1}}{2}\mathrm{sin}\left(2\theta\right),\;\delta=\frac{\delta_{1}}{2}\mathrm{cos}\left(2\theta\right)$
(12)

with bright eigenstates given as

$\displaystyle\ket*{BE_{1}}$
$\displaystyle=-\ket*{\mathrm{sin}\left(\theta\right)}+\ket*{\mathrm{cos}\left(\theta\right)}$
(13) $\displaystyle\ket*{BE_{2}}$
$\displaystyle=\ket*{\mathrm{cos}\left(\theta\right)}+\ket*{\mathrm{sin}\left(\theta\right)}$
(14)

Figure 4: (a) $\delta$ and $\kappa$ contribution to the BES as function of
excitonic ground state energy. Histogram (b) of $\delta$ and $\kappa$ with
mean $\kappa$ (contribution due to alloyin) value close to zero, mean $\delta$
(contribution due to lattice anisotropy) different from zero. See the text for
details.

Model Eq.11 allows to decompose the bright exciton splitting into two
contributions, due to anisotropy ($\delta$) and alloying ($\kappa$)
respectively, which are shown in Fig. 4. The $\kappa$ mean value,
$\left<\kappa\right>$, is close to zero indicating lack of any directional
character of alloying, whereas $\delta$ has mean value of -2.32 $\mu$eV, which
is due lattice anisotropy. This result is in vivid contrast to atomistic
pseudopotential calculations, [85] where $\left<\delta\right>$ is different
from zero only for elongated quantum dots. This result is however consistent
with systematic difference between empirical tight-binding and empirical
pseudopotential approaches for $C_{2v}$ systems, [50] as well BES reported
experimentally. [48]

$\kappa$ and $\delta$ have standard deviations $\sigma$ of $2.66~{}\mu$eV, and
2.30 $\mu$eV, respectively. Since both $\kappa$ and $\delta$ are apparently
independent, and have distribution that can be quite well described by the
normal distribution [Fig. 4 (a)], with
$\sigma=\sigma_{\delta}\approx\sigma_{\kappa}=2.6~{}\mu$eV, and
$\left<\delta\right>$ not exceeding $\sigma$, one can expect that
$\delta_{\mathrm{1}}=2\sqrt{\delta^{2}+\kappa^{2}}$, will follow Rayleigh
distribution, [87, 88] (i.e., essentially a $\chi$ distribution with two
degrees of freedom):

$P\left(\delta_{1}\right)=\frac{\delta_{1}}{4\sigma^{2}}\mathrm{exp}\left(-\frac{\delta_{1}^{2}}{8\sigma^{2}}\right),$
(15)

where the denominator is multiplied by a factor of 4 (with respect to the
original Rayleigh distribution), due to 2 occurring in the definition of
$\delta_{\mathrm{1}}$, i.e,
$\delta_{\mathrm{1}}=2\sqrt{\delta^{2}+\kappa^{2}}$. Fig. 5 (a) shows a
histogram of BES values with Eq. 15 plotted with dotted black line revealing a
rather good fit, despite not strictly fulfilling the assumption of
$\langle\delta\rangle=0$, and with $\sigma=2.66$, not taken as fitting
parameter. Thus, as a result of $\delta_{\mathrm{1}}$ having a two-component
character its distribution varies from the normal distribution, but resembles
$\chi$ with k=2. Rayleigh distribution is derived for the system of two-
dimensions, however the mathematical form of the Rayleigh distribution is
identical with Wigner surmise for the one-dimension energy levels problem [88,
85]

$P\left(s\right)=\frac{\pi}{2}s\,\mathrm{exp}\left(-\frac{\pi}{4}s^{2}\right),$
(16)

with $s=\delta_{1}/\expectationvalue{\delta_{1}}$, and where
$\expectationvalue{\delta_{1}}=$ 7.60 $\mu$eV was calculated as an average BES
of all 300 samples. The plot of Eq. 16 is also shown in Fig. 5 (a) with a
dashed line, compared with the histogram of BES values. We note that Eq.15 and
Eq. 16 have identical mathematical forms, whereas the difference between plots
on Fig. 5 (a) is related to difference in parameters describing “variances” of
the both distributions (i.e.,
$\langle\delta_{1}\rangle\neq\sqrt{2\pi}\sigma$).

Since Eq. 16 corresponds to the standard level-spacing distribution of the
Gaussian orthogonal ensemble in random matrix theory, Gong et al. [85] argue
this sort of statistical dependence reveals strong repulsion between excitonic
levels, and the presence of multiplicative $s$ in Eq. 16 reduces the
probability of finding quantum dots with small BES.

Figure 5: Histogram of (a) the bright exciton splitting (with dotted/dashed
line corresponding to Rayleigh/Wigner distributions respectively), and (b)
dark exciton splitting (the dashed line is a fit to a normal distribution).

As a result of this sort of behavior, we find that over 50% of samples has the
BES within 5–10 $\mu$eV, with only in 3.4% of cases with BES larger than 15
$\mu$eV, and importantly only 1.7% cases with the BES smaller than 1 $\mu$eV.
BES smaller than the line-width (of approximately 1 $\mu$eV) is important in
the context of entanglement generation. Results presented here may look
counter-intuitive, since cylindrical quantum dots seem to be good candidates
for small BES values, yet are consistent with earlier findings. [85]

For comparison Fig. 5 (b) shows a histogram of the DES values revealing normal
distribution with the mean of 0.657 and standard deviation of 0.16 $\mu$eV.
Thus, the DES is order of magnitude smaller than the BES, however, in systems
studied here it is virtually impossible to find a quantum dot with a vanishing
dark exciton splitting.

Finally, we note that mixing with higher shells increases both BES and DES.
Should only the s-shell be included in the atomistic calculation
(corresponding to $4\times 4$ CI Hamiltonian) the mean BES value would be 4
$\mu$eV, and DES mean value 0.53 $\mu$eV, with no substantial qualitative
difference in histograms. For comparison, atomistic results obtained for the
s-shell only are presented in the Appendix A.

Statistical analysis performed for $\kappa$ and $\delta$ can be performed for
the bright exciton polarization angle as well, with results shown in Fig. 6,
where the histogram obtained from the current atomistic calculation is
compared with a statistical model (dashed line) from Supplementary Information
of Ref. [89]. The lower energy bright exciton state (BE1) tends be polarized
along the [1$\overline{1}$0] direction (rotated by 90$\degree$ from [110]),
whereas the higher-energy one (BE2) prefers polarization along the [110]
direction. Therefore, despite substantial alloying (and overall cylindrical
shape) polarization angle dependence has well-defined maxima corresponding to
the vicinity of crystal axis. This is again in stark contrast to empirical
pseudopotential results [89] that predict pronounced polarization angle
distribution maxima only for elongated dots. This difference can be traced
back to $\delta$ distribution as shown earlier (Fig. 4): in atomistic tight-
binding calculation $\langle\delta\rangle\neq 0$, contrary to the empirical
pseudopotential method. This is consistent with the latter systematically
underestimating lattice BES contribution, origin of which remains to a large
degree unexplored. [48]

Figure 6: Histograms of polarization angle for lower (BE1) and higher (BE2)
energy bright exciton states with maxima corresponding to $[110]$ and
$[1\underline{1}0]$ crystal axes (0 and 90 degrees respectively). The dashed
lines are calculated using approach from Ref. [85] See the text for details.

Model of Eq. 11 assumes the angle between bright exciton to be exactly equal
to 90$\degree$. To go beyond that, and further study polarization properties,
the exchange Hamiltonian can explicitly account for phases of electron-hole
exchange interaction [81, 90]

$\displaystyle\mathbf{H}_{\mathrm{exch}}$
$\displaystyle=\frac{1}{2}\begin{bmatrix}\delta_{\mathrm{0}}&\delta_{1}e^{-2i\theta_{1}}&0&0\\\
\delta_{1}e^{2i\theta_{1}}&\delta_{\mathrm{0}}&0&0\\\
0&0&-\delta_{\mathrm{0}}&\delta_{2}e^{-2i\theta_{2}}\\\
0&0&\delta_{2}e^{2i\theta_{2}}&-\delta_{\mathrm{0}}.\end{bmatrix},$ (17)

where $\theta_{1}$ describes the rotation of polarization axis (around the
growth axis $z$) by the angle $\theta_{1}$ with respect to the fixed axes,
usually taken as crystal axis $[110]$. The eigenstates have the following form

$\displaystyle\ket*{BE_{1,2}}$ $\displaystyle=\left(\ket*{1}\pm
e^{-2i\theta_{1}}\ket*{-1}\right)/\sqrt{2}$ (18)
$\displaystyle\ket*{DE_{1,2}}$ $\displaystyle=\left(\ket*{1}\pm
e^{-2i\theta_{2}}\ket*{-1}\right)/\sqrt{2}$ (19)

By comparing with atomistic calculations, we find that non-alloyed $C_{2v}$
system $2\theta_{1}=\pi$, and $2\theta_{2}=\frac{3}{2}\pi$ indicating $\pi/2$
phase difference between dark and bright manifolds.

While Eq. 17 allows understanding rotation of quantum dot principal axes with
respected to crystal axes, it still by definition predicts exactly vanishing
polarization anisotropy, [90] i.e.,
$C=\left(I_{max}-I_{min}\right)/\left(I_{max}+I_{min}\right)=\left(I_{BE2}-I_{BE1}\right)/\left(I_{BE2}+I_{BE1}\right)=0$
since magnitudes of emission intensities of both bright excitons are exactly
equal.

To go beyond this picture, and allow for better agreement with the experiment,
one can account for valence-band mixing, which occurs in realistic quantum
dots, due to shape asymmetry and lattice anisotropy, leading to an addmixture
of light-hole component

$\left|\pm\widetilde{1}\right>=\sqrt{1-\beta^{2}}\left|\pm\frac{3}{2},\mp\frac{1}{2}\right>+\beta
e^{\pm 2i\psi}\left|\mp\frac{1}{2},\mp\frac{1}{2}\right>,$ (20)

where $\beta$ and $\psi$ are amplitude and phase of mixing with light-hole
exciton $\left|\mp\frac{1}{2},\mp\frac{1}{2}\right>$, and tilde symbol was
used to distinguish from the basis with no band-mixing included. A Similar
formula is used for dark states

$\left|\pm\widetilde{2}\right>=\sqrt{1-\beta^{2}}\left|\pm\frac{3}{2},\pm\frac{1}{2}\right>+\beta
e^{\pm 2i\psi}\left|\mp\frac{1}{2},\pm\frac{1}{2}\right>.$ (21)

Exchange Hamiltonian expressed in the basis
$\ket*{\pm\widetilde{1}},\ket*{\pm\widetilde{2}}$ maintains the block diagonal
structure of Eq. 6, and its eigenstates yet with tilded states replacing
$\ket*{\pm 1},\ket*{\pm 2}$. $\beta$ is responsible for polarization
anisotropy [90, 91]

$C\left(\beta\right)=\frac{2\beta\sqrt{3\left(1-\beta^{2}\right)}}{3-2\beta^{2}},$
(22)

Polarization anisotropy is naturally present in atomistic calculation, and for
a $C_{2v}$ system it reaches 2.9% with emission intensity for $\ket*{BE2}$
along $[110]$ larger then for $\ket*{BE1}$ along $[1\overline{1}0]$ direction.
Recasting atomistic $C_{2v}$ results on valence-band-mixed $4\times 4$
Hamiltonian gives $\beta$ value of 2.5%. Moreover, for $C_{2v}$ quantum dots
mixing phase $\psi=0$, and emission lines remain orthogonal. [90, 82]

For alloyed systems, both $\beta$ and $\psi$ determine an angle between
excitonic lines, which in principle can be different from 90$\degree$.

Figure 7: Histogram of (a) $\beta$ light-hole content for ensemble of alloyed
quantum dots, and (b) in-plane angle difference between lower (BE1) and higher
(BE2) energy bright exciton states. The dashed lines are plotted assuming
normal distribution. Despite alloying $\beta$ content remains rather small,
with the angle between bright excitons close to 90 degrees. See the text for
more details.

Fig. 7 (a) shows a histogram of $\beta$ values for alloyed systems studied in
this paper, whitch appears to be normally distributed around zero light-hole
content, with both positive and negative light-hole mixing amplitudes
possible, and most of the cases corresponding to $|\beta|<1\%$. Moreover, as
shown in Fig. 7 (b) valence-band mixing allows for angle between polarizations
of bright excitonic lines different from the exact 90$\degree$, however for
the vast majority of cases this difference does not exceed 1$\degree$,
rendering Eq. 11 a valid approximation for the considered ensemble of quantum
dots.

Valence band mixing, expressed via the non-zero $\beta$ content, has its
effect on the dark manifold as well, since

$\left|\pm\widetilde{2}\right>=\sqrt{1-\beta^{2}}\left|\pm\frac{3}{2},\pm\frac{1}{2}\right>+\beta
e^{\pm 2i\psi}\left|\mp\frac{1}{2},\pm\frac{1}{2}\right>,$ (23)

and since matrix elements of the optical transition [81, 82] for the light-
hole component are given as: $M_{\mp\frac{1}{2},\pm\frac{1}{2}}\propto e_{z}$,
dark excitons $\ket*{\pm\widetilde{2}}$ get optical activity due to hole band
mixing, which is proportional to $\beta$, i.e., $|M_{\widetilde{\pm
2}}|^{2}=\beta^{2}\left[1\mp\mathrm{cos}\;2(\theta_{2}-2\psi)\right]$. For
$C_{2v}$ system $\psi=0$, [82] and since from atomistic calculation:
$2\theta_{2}=\frac{3}{2}\pi$, and $\delta_{2}>0$, only one of dark exciton
states (higher energy, DE2) is optically active, which allows the DE emission
perpendicular to the growth direction to be detected, and the other (lower
energy, DE1) remains fully dark in agreement with experimental findings. [92]

Moreover, group-theoretical arguments indicate that the optical dipole moment
(matrix elements of the optical transition) for the dark exciton can be non-
zero even for pure heavy-hole exciton, i.e., $\left|M_{DE2}\right|^{2}\propto
e_{z}$ even if $\beta=0$, whereas DE1 remains optically in-active, i.e.,
$\left|M_{DE1}\right|^{2}=0$. Thus, the single-particle dipole matrix element,
for $z$-polarized light, calculated in the basis of tight-binding states:
$|M^{z}|=|M^{z}_{e\uparrow,h\Uparrow}|=|M^{z}_{e\downarrow,h\Downarrow}|\propto|\beta
M_{LH}+M_{HH}|$, and therefore could be interpreted as having light- and
heavy-hole components.

Figure 8: Histogram of (a) modules of out-of-plane single-particle optical
dipole moment
$|M^{z}|=|M^{z}_{e\uparrow,h\Uparrow}|=|M^{z}_{e\downarrow,h\Downarrow}|$, and
(b) its squared moduli. The dashed lines are calculated assuming probability
distributions discussed in the text, suggesting the two-component character of
the optical dipole moment.

It not straightforward to perform such decomposition for atomistically
obtained states. We have thus tried an approximate attempt by applying
vertical electric field, aiming to use the field to reduce polarization
anisotropy of bright exciton to zero, for which case, based on Eq. 22, one can
assume $\beta\sim 0$. In such a situation $|M^{z}|$ should only have the
heavy-hole component, which could be estimated in this way. By performing such
analysis for the $C_{2v}$ system we found
$M_{LH}=-0.042~{}$\mathrm{\SIUnitSymbolAngstrom}$$ and
$M_{HH}=0.051~{}$\mathrm{\SIUnitSymbolAngstrom}$$.

Figure 9: Histogram of $z$-polarized (out-of-plane) optical spectra for lower
(DE1) and higher (DE2) dark exciton states for cases with (a) optical dipole
moment artificially set to zero ($M_{e\uparrow,h\Uparrow}=0$ and
$M_{e\downarrow,h\Downarrow}=0$), and with (b) optical dipole moment included
in the calculation, and (c) calculation performed in a basis including $s$,
$p$, and $d$ shells of 144 excitonic configurations. The $z$-polarized dark
exciton activity thus stems from non-zero optical dipole moment (due to
valence-band-mixing), whereas the exchange mixing plays a negligible role, and
higher-shells play a rather small role as well. See the text for more details.

Therefore, both contributions can have comparable magnitudes, and can have
opposite signs, thus partially cancel each other. Despite its approximated
character, we have performed a similar study for 5 alloyed systems as well,
and came to similar conclusions. To further analyze $|M^{z}|$ distributions in
the considered ensemble of alloyed system, we show its histogram in Fig. 8
(a), and then $|M|^{2}$ histogram in Fig. 8 (b). $|M^{z}|$ distribution can be
well fit to $\chi$ distribution with two degrees (k=2) of freedom, which is
mathematically equivalent to Rayleigh distribution/Wigner surmise discussed
earlier (i.e., $\sim x\,\mathrm{exp}\left(-\frac{\pi}{4}x^{2}\right)$, with
$x=|M^{z}|/\langle|M^{z}|\rangle$ shown as dotted line on Fig. 8 (a).
Therefore, it is expected that, $|M^{z}|^{2}$ would be well described by
$\chi^{2}$ distribution, again with two degrees of freedom, corresponding to
the exponential distribution $\sim\mathrm{exp}\left(-x\right)$. In other
words, Fig. 8 does indirectly support two (light- and heavy-hole) component
character of single-particle dipole moment $|M^{z}|$. Aside from speculations
regarding heavy- and light- hole contributions, since $M^{z}$ is a complex
number, Rayleigh distribution is expected [88] when real and imaginary
components are independently and identically (Gaussian) distributed (with
equal variance and zero mean). In such case, the absolute value of the complex
number is Rayleigh-distributed, as apparently shown in Fig. 9, thus supporting
the two-component character of $M_{z}$ irrespective from its origin.

Finally, after an analysis of single-particle transition matrix elements, we
can study many-body dark exciton optical spectra, shown for $z$-polarization
in Fig. 9. Should the single-particle optical dipole moments be artificially
neglected, i.e., $|M_{e\uparrow,h\Uparrow}|=|M_{e\downarrow,h\Downarrow}|=0$,
both dark excitons would not reveal any significant optical activity. The
residual, weak activity [Fig. 9 (a)] may be attributed to the dark-bright
mixing term (due to exchange interaction) present in the atomistic calculation
that will be discussed in the following part of the paper. In the case of
$z$-polarized dark exciton emission exchange mixing apparently does not play
any significant role. However, the inclusion of non-zero optical matrix
element leads to substantial increase in $z$-polarized emission as seen in
Fig. 9 (b). This increase is particularly important for the higher-energy DE2
states, for which it can reach over 0.006
$$\mathrm{\SIUnitSymbolAngstrom}$^{2}$, i.e., approximately 1/6000 fraction of
bright exciton emission merely due to alloying. DE1 states gains some optical
activity as well, which is however much smaller, and what seem to be a
property “inherited” from $C_{2v}$ system since both amplitude and phase of
valence band mixing i.e., $\beta$ (and apparently $\psi$) remain rather small
in the alloyed system. Notably, further extension of the CI basis by inclusion
of higher shells [Fig. 9 (c)] does not seem to affect dark exciton optical
spectra, indicating the dominant contribution is from the optical dipole
moment, and not due to configuration mixing.

In a phenomenological treatment, several authors further extend valence band
mixing approach by accounting for a mixing term between heavy- and light-hole
states with parallel spins:

$\displaystyle\left|\pm\accentset{\approx}{1}\right>=$
$\displaystyle\sqrt{1-\beta^{2}-\gamma^{2}}\left|\pm\frac{3}{2},\mp\frac{1}{2}\right>$
$\displaystyle+\beta e^{\pm
2i\psi}\left|\mp\frac{1}{2},\mp\frac{1}{2}\right>+\gamma e^{\pm
2i\xi}\left|\pm\frac{1}{2},\mp\frac{1}{2}\right>$ (24)

with a similar formula for $\ket*{\pm\accentset{\approx}{2}}$ dark states, and
where $\gamma$ and $\xi$ are the amplitude and the phase of mixing, and the
double tilde symbol is used to distinguish from previously considered cases.
This approach has been used to study the $z$-polarized emission of bright
excitons, [90] with $\gamma$ parameter being far more difficult to extract
from the experiment (or atomistic calculations) than $\beta$, and thus we will
not aim for doing that. However, it was also shown recently [45] that mixing
$\gamma$ term induces off-diagonal Hamiltonian matrix elements (exchange
integrals) responsible for the dark-bright exciton mixing:

$\displaystyle\begin{split}\mathbf{H}_{\mathrm{exch}}&=\frac{1}{2}\begin{bmatrix}\delta_{\mathrm{0}}&\delta_{1}e^{-2i\theta_{1}}&\delta_{11}&\delta_{12}\\\
\delta_{1}e^{2i\theta_{1}}&\delta_{\mathrm{0}}&-\delta_{12}&-\delta_{11}\\\
\delta_{11}^{*}&-\delta_{12}^{*}&-\delta_{\mathrm{0}}&\delta_{2}e^{-2i\theta_{2}}\\\
\delta_{12}^{*}&-\delta_{11}^{*}&\delta_{2}e^{2i\theta_{2}}&-\delta_{\mathrm{0}}\end{bmatrix}\\\
&=\frac{1}{2}\begin{bmatrix}\mathbf{H}_{\mathrm{BB}}&\mathbf{H}_{\mathrm{BD}}\\\
\mathbf{H}_{\mathrm{DB}}&\mathbf{H}_{\mathrm{DD}}\end{bmatrix}\end{split}$
(25)

where $\mathbf{H}_{\mathrm{DB}}$ denotes dark-bright coupling for brevity,
whereas $\delta_{11}$, and $\delta_{12}$ are responsible for mixing of bright
and dark configuration states differing by the electron and the hole spin
state respectively, i.e., $\delta_{11}$ mixes $\ket*{\accentset{\approx}{1}}$
with $\ket*{\accentset{\approx}{2}}$, and $\delta_{12}$ mixes
$\ket*{-\accentset{\approx}{1}}=$ with $\ket*{-\accentset{\approx}{2}}$. With
exception to phase factors, an identical Hamiltonian can be also derived in a
phenomenological manner, by assuming the rotation of quantum-dot axes
(angular-momentum quantization axes) from the ideal crystal axes. [44]
Virtually identical Hamiltonian is also derived from atomistic calculations
for $C_{s}$ quantum dots, where low shape symmetry induces mixing between
bright and dark states. [10]

Figure 10: Histogram of (a) moduli (absolute values) of $\delta_{11}$, and
$\delta_{12}$ dark-bright mixing integrals, and (b) their arguments (phases).
See the text for details. Figure 11: Histograms of (a) moduli of in-plane
single-particle optical dipole moments for light polarized in [110] and
[1$\overline{1}$0] directions, i.e., $|M^{[110]}_{e\uparrow,h\Uparrow}|$ and
$|M^{[1\overline{1}0]}_{e\downarrow,h\Downarrow}|$, (b) their squared moduli.
The dashed lines are calculated assuming probability distributions discussed
in the text.

Interestingly, we also found identical structure of (lowest four states)
Hamiltonian from atomistic calculations for currently studied alloyed quantum
dot despite their cylindrical symmetry, and lack of faceting. Therefore, not
only shape deformation can induce bright-dark exciton exchange coupling, but
breaking the symmetry on the scale of local atomic arrangement can lead to the
same effect. Moreover, despite overall $C_{1}$ symmetry, the dark-bright
coupling block derived from atomistic calculation, in a basis of tight-binding
states
$\ket*{e_{\uparrow},h_{\Uparrow}},\ket*{e_{\downarrow},h_{\Downarrow}}|$ is
given as:

$\displaystyle\mathbf{H}_{\mathrm{DB}}$
$\displaystyle=\begin{bmatrix}\delta_{11}&\delta_{12}\\\
-\delta_{12}^{*}&-\delta_{11}^{*}\end{bmatrix},$ (26)

where the only difference from phenomenological Hamiltonian Eq. 26 (expressed
in double-tilde basis) is related to complex conjugations in the second row.

##  IV Dark exciton in-plane optical activity

To further study the role of dark-bright exciton mixing, in Fig. 10 (a) we
show histograms of $\delta_{11}$ and $\delta_{12}$. The absolute value of
$\delta_{12}$ can be quite well fit to $x\,\mathrm{exp}\left(-x^{2}\right)$,
which is again equivalent to $\chi$ distribution with k=2, suggesting two-
component character of $\delta_{12}$ integral. This is consistent with Ref.
[44], where it was interpreted in terms of two effective tilt angles (one
related to the electron, and second to the hole) leading to the dark-bright
mixing. This is also coherent with approach of Ref. [45] where $\delta_{12}$
is induced by the presence of both $\beta$ and $\gamma$ mixing terms. In the
treatment employed in Ref. [45] $\delta_{11}$ depends on $\gamma$ only. One
can thus expect its statistical distribution to differ from that found for
$\delta_{12}$, and in fact this is visible in Fig. 10 (a), where one can fit
$\delta_{11}$ to $x\,\mathrm{exp}\left(-x\right)$, resembling Gamma
distribution with k=1. Whereas this sort of analysis is speculative,
$\delta_{11}$ undoubtedly varies from 0 to 2.7 $\mu$eV with mean (average) of
0.5 $\mu$eV and $\delta_{12}$ varyies from 0 to to 3.05 $\mu$eV with the mean
of 1 $\mu$eV. Therefore, $\delta_{11}$ and $\delta_{12}$ have magnitudes
comparable with $\delta_{1}$, the dark exciton splitting, and thus a priori
cannot be neglected when considering excitonic fine structure of an alloyed
system. Moreover, both BES and DES will be renormalized by the presence of
dark-bright coupling [45, 44]. However, since the correction is equal to
$\delta_{11}\delta_{12}/\delta_{0}$, the effect of dark-bright mixing on both
BES and DES is negligible for the system considered in this work, as we have
$\delta_{0}\gg\delta_{11,12}$.

For $C_{2v}$ quantum dots, by symmetry, both dark excitons are strictly
optically inactive, when considering the in-plane polarization. For $C_{1}$
quantum dot however, there are two channels, through which dark excitons can
gain in-plane optical activity (see the Appendix C for more details). One of
them is related to the dark-bright exchange interaction, which was already
mentioned above, and in which the dark exciton can gain optical activity from
the admixture of bright configurations. The second channel is related to a
non-vanishing dipole moment for in-plane ($x$,$y$) polarized light
$M^{x,y}_{e\uparrow,h\Uparrow}\neq 0$, and
$M^{x,y}_{e\downarrow,h\Downarrow}\neq 0$, similarly to the $M^{z}$ dipole
moment discussed earlier.

Fig. 11 shows histograms of in-plane dipole moment matrix elements along two
experimentally relevant crystal axes, [110] and [1$\overline{1}$0].
Importantly, $|M|^{[1\overline{1}0]}$ has substantially larger magnitude than
$|M|^{1\overline{1}0]}$, thus despite alloying and cylindrical shape of
quantum dots in the ensemble, the lattice anisotropy still plays an important
role with average
$\langle|M|^{[1\overline{1}0]}\rangle=0.06304~{}$\mathrm{\SIUnitSymbolAngstrom}$$
about 3 times larger than $\langle|M|^{[110]}\rangle=0.02061$.

We note as well that $|M|^{[110]}$ is larger than the one for the
$z$-polarized case, i.e., $\langle|M|^{001]}\equiv\langle|M|^{z}\rangle$
(shown earlier in Fig. 8) This may be understood since quantum-dot lateral
dimensions are substantially larger than the in growth ($z$) directions.
However, similarly to $|M^{z}|$, $|M|^{110}]$ can be very well approximated by
$\chi$ distribution with k=2, and with $|M|^{2}$ [Fig. 12 (b)] showing a
$\chi^{2}$ dependence, with k=2, i.e., the exponential distribution.

Figure 12: Histogram of in-plane optical spectra (emission intensity) for
lower (DE1) and higher (DE2) dark exciton states with (a) optical single-
particle dipole moment included in the calculation, but exchange dark-bright
mixing terms neglected ($\delta_{11}=\delta_{12}=0$) in the Hamiltonian, (b)
optical dipole moment artificially set to zero ($|M_{e\uparrow,h\Uparrow}=0$
and $M_{e\downarrow,h\Downarrow}=0$), but exchange mixing accounted for, (c)
both contributions included, and (d) as in (c) but with calculation performed
in a basis including $s$, $p$, and $d$ shells of 144 excitonic configurations.
Note different scales, as well as presence of breaks on vertical axes.
Apparently, both non-zero optical dipole moments, and exchange mixing play
equally important roles, and none of these contributions can be neglected, as
it would overestimate the optical activity. See the text for more details.

Since in-plane single-particle dipole moment matrix elements are non-zero, as
are the dark-bright exchange mixing terms in the Hamiltonian, it is
instructive to study how these terms individually affect many-body dark
exciton optical spectra. To this end, in Fig. 12 (a) we show the results with
dark-bright mixing artificially neglected, and with single-particle dipole
moment being the only term inducing dark exciton in-plane optical activity. In
this case DE2 has much larger optical activity than DE1, similarly to
previously considered out-of-plane emission. Alternatively, exchange mixing
can be artificially neglected and optical dipole moments accounted for, as in
Fig. 12 (b). Such a situation produces very similar optical spectra to the
first case, with DE2 having substantially larger optical activity then DE1,
despite a different mechanism of luminescence.

Figure 13: Histogram of in-plane polarization angle for lower (DE1) and higher
(DE2) dark exciton states with (a) optical single-particle dipole moment
included in the calculation, but exchange dark-bright mixing terms neglected
($\delta_{11}=\delta_{12}=0$) in the Hamiltonian, (b) optical dipole moment
artificially set to zero ($|M_{e\uparrow,h\Uparrow}=0$ and
$M_{e\downarrow,h\Downarrow}=0$), but exchange mixing accounted for, (c) both
contributions included, and (d) as in (c) but with calculation performed in a
basis including $s$, $p$, and $d$ shells of 144 excitonic configurations.
Again, both non-zero optical dipole moments, and exchange mixing play equally
important roles, and none of these contributions can be neglected. See the
text for more details.

Interestingly, when both terms are included [Fig. 12 (c)], they appear to
partially compensate each other, with the dark exciton optical activity on
average an order of magnitude smaller compared to (a) and (b) cases, thus
comparable with the out-of-plane emission. A notable difference to z-polarized
emission can however be observed, since both DE1 and DE2 have (on average)
similar optical activity.

Therefore, reduction of symmetry due to alloying must be accounted for in both
transition dipole moment and exchange matrix elements on equal footing.
Failing to do so leads to a serious overestimation of the dark exciton in-
plane optical activity. This could be understood in the spirit of Ref. [44]
where the reduction of symmetry is related to rotation of quantum dot axes.
This new quantization axis, or rotation of angular momenta, must be included
simultaneously in the calculation of single-particle dipole moment, and two-
body Coulomb and exchange integrals.

Moreover, we find that for $C_{1}$ alloyed quantum dots it is impossible to
fit atomistic spectra to the effective $4\times 4$ Hamiltonian (formally
resembling Eq. 25), yet with idealized matrix element of the optical
transition (i.e., vanishing in-plane optical dipole moments), and with
$\delta_{11}$, $\delta_{12}$ being the only factors responsible for the dark-
exciton optical activity. Such procedure was indeed possible for $C_{s}$
systems, [10] however for $C_{1}$, since both dipole moments and Coulomb
matrix are affected by allowing, one would need to include two additional
fitting parameters in the Hamiltonian ($\delta_{21}$ and $\delta_{22}$
integrals) to further differentiate coupling of bright and dark states, what
does not have a formal justification. Thus, we again conclude that calculation
of dipole moments and Coulomb matrix elements in alloyed system must be
performed on equal footing, unless some other factor (speculatively large
quantum dot shape deformation/anisotropy) dominates over alloying.

Further inclusion of higher shells beyond the $4\times 4$ CI treatment does
not change the overall picture [Fig. 12 (d)] considerably. In this case,
despite complicated origin of dark exciton activity (dipole moments, exchange
mixing, and CI) the frequency distribution of optical activity can be quite
well fit with a single exponent for DE2 states, and with two exponents for
DE1. Interestingly, optical activity larger than 0.002
$$\mathrm{\SIUnitSymbolAngstrom}$^{2}$ seems to be dominated by that coming
from DE1 states. So for several samples, the combined effect of non-zero
dipole moments, and dark-bright mixing leads noticeably to more pronounce
emission from the lower energy dark exciton state, although this is a rather
subtle effect. Such activity would be in agreement with experimental results,
[2], although here we do not aim at comparison with a particular experiment.
Due to alloy randomness, the dark exciton in-plane activity reaches 1/6000
fraction of the bright exciton activity, with the average of 1/40000, thus
considerably smaller that observed when a shape distortion [10] is the source
of dark excitonic optical activity. Nevertheless, we can conclude that a mere
reduction of overall symmetry due to alloy randomness is able to trigger non-
negligible in-plane polarized optical activity of dark excitons in self-
assembled InGaAs quantum dots.

Finally, it is worth to conduct a similar analysis for polarization angle of
the dark exciton in-plane emission, as shown in Fig. 13. With either dark-
bright exchange mixing artificially neglected [Fig. 13 (a)] or dipole moments
set to zero [Fig. 13 (b)], the angle of polarization of both dark excitons
tends to prefer 90$\degree$ rotation from [110], i.e., polarization along
[1$\overline{1}$0] direction. This is particularly pronounced for DE2 states.
Therefore, with either of contributions accounted for, whilst the other being
neglected, the in-plane spectra of dark excitons somewhat resemble the out-of-
plane ($z$) polarized spectra with DE2 having dominant optical activity.

However, the situation is apparently different with both contributions
simultaneously included, as shown in Fig. 13 (c), where DE1 (in particular)
and DE2 (to a lesser degree) preferable polarization is along [110]
(polarization angle 0$\degree$). This is somewhat opposite to experimental
observation, where the dominant emission is from the lower dark exciton state,
following polarization from the lower energy bright exciton state. However, a
caution must be exercised since experiments are unavoidably performed on non-
ideal, low shape symmetry QDs. Our atomistic results [10] for a quantum dot
with a facet, and thus broken shape symmetry, indeed indicated a very good
agreement with a particular experiment [2, 9]. Thus, a deviation from
cylindrical base very likely must be included when one aims for a comparison
with a particular experimentally grown system, and here we focus on alloy
randomness effect only.

Since both DE1 and DE2 appear to have their polarization preferably along the
[110] direction, it is instructive to show the histogram of angles between
polarizations of DE1 and DE2 states (for their in-plane emission) as shown in
Fig. 14. Contrary to bright excitons [shown earlier in Fig. 5 (b)], where the
polarization directions strongly tend to be orthogonal (i.e., the angle
between polarizations close to 90$\degree$), the dark exciton states tend to
be polarized along the same direction, however due to alloy randomness
virtually any angle between these two emission lines is possible. Again,
accounting for shape elongation or faceting in a particular experimental
realization, together with alloy randomness could very likely change these
statistics.

Moreover, there is still little experimental data regarding dark exciton
spectra, since measurements are usually performed on a single quantum-dot
sample only. Our calculation indicates, that for alloyed quantum dots one can
cherry-pick a sample with DE1 polarized along [110] or [1$\overline{1}$0],
with DE1 being the dominant line, or vice-versa. Therefore, more experimental
research on the subject is needed.

Figure 14: Histogram of in-plane angle difference between the lower (DE1) and
the higher (DE2) energy dark exciton states. The dashed line is plotted
assuming a normal distribution. Contrary to bright excitons the dark excitonic
lines tend to have the same polarization angle, yet with a very broad
distribution. See the text for more details.

##  V Radiative lifetimes

Figure 15: Histograms of bright exciton states BE1 and BE2 optical activities,
showing very similar and relatively narrow distributions. Dashed lines are
plotted assuming a normal distribution. Figure 16: Histogram of (a) bright
exciton radiative lifetime (b) dark exciton radiative lifetime for the
ensemble of alloyed quantum dots studied in this work. The inset shows
lifetime of DE1 and DE2 states separately. The dashed lines are plotted
assuming a normal distribution for the bright exciton, and inverse-gamma
distribution for the dark exciton. See the text for more details.

In Section IV we have studied the dark exciton optical activity, therefore for
comparison Fig. 15 shows histogram of both bright exciton states optical
activities which are apparently normally distributed around approximately 39.5
$$\mathrm{\SIUnitSymbolAngstrom}$^{2}$, corresponding to mean optical dipole
moment of 0.63 nm, with rather small variance corresponding to dipole moments
from 0.6 to 0.66 nm, thus very close to typical experimental results (e.g.,
0.59 nm determined in Ref. [74]).

Finally, Fig. 16 shows radiative lifetimes of bright and dark excitons
calculated with use of Eq. 3 and Eq. 4. Reported lifetimes are similar for
both bright excitons, hence $\tau_{BE}\approx\tau_{BE1}\approx\tau_{BE2}$ with
again apparently normal-like distribution and with the mean value of 0.93 nm,
thus comparable to experimentally and theoretically reported values of
approximately 1 ns, thus again close to typical quantum dot results. [74]
Despite its simplicity such result needs a comments, since lifetime is
inversely proportional to optical activity (Eq. 2 and Eq. 3), thus the inverse
of normal distribution on Fig. 15 will lead to (in principle quite
complicated) inverse Gaussian distribution [93]
$\propto\sqrt{\frac{\lambda}{x^{3}}}exp(-\frac{\lambda\left(x-\mu\right)^{2}}{2x\mu^{2}})$.
However, since the bright exciton optical activity has much larger mean value
(approximately 40 $$\mathrm{\SIUnitSymbolAngstrom}$^{2}$) than its standard
deviation (of approximately 1 $$\mathrm{\SIUnitSymbolAngstrom}$^{2}$), thus
the inverse Gaussian distribution resembles regular Gaussian distribution. Fit
to this dependence is shown on Fig. 16 (a) as black dashed line, with
$\lambda=1541$ and $\mu=0.9284$ ns, and such large $\lambda$ being the reason
[93] why the inverse Gaussian distribution appears very similar to the regular
normal distribution.

The distribution of dark exciton lifetimes is different, with average (mean)
$\tau_{DE}$ value equal to 39 ms, and different distribution. Since emission
intensity (Eq. 2) and lifetimes (Eq. 3) are inversely proportional, the weak
dark exciton optical intensity, as shown earlier in Eq. 9 and Eq. 13,
corresponds to long lifetimes. Moreover since dark exciton optical activity
resembles the exponential $\mathrm{exp}(-x/\expectationvalue{x})$
distribution, mathematically equivalent to Gamma distribution [94] with k=1,
then the inverse should have the distribution in the following form
$\frac{1}{x^{2}}\mathrm{exp}\left(-\expectationvalue{x}/x\right)$ shown as a
dashed line in Fig. 16 (b). Despite it over-simplicity, the inverse gamma
distribution qualitatively well describes the distribution of dark exciton
lifetime with no quantum dots with lifetime below 6 ms and with a long tail of
quantum dots with lifetimes reaching up to 300 ms. Such extremely long
lifetime should be taken with great care, since we consider only the radiative
contribution, obtained within tight-binding Hamiltonian and with other
approximations included. [66, 95] We should also note that contrary to bright
excitons (and as a direct result of differences in their optical spectra), the
dark exciton states DE1 and DE2 reveal some difference of radiative lifetimes
[inset of Fig. 16 (b)]. Namely, the mean lifetime of DE2 state is equal to 41
ms, whereas for DE1, including the lowest 95% cases, it is equal to 76 ms with
a long tail [not shown in Fig. 16 (b)] extending up to seconds (with average
of 390 ms of the 5% highest-value subset), thus corresponding to virtually
optically non-active state, where other processes would likely dominate over
radiative recombination. Since $\tau_{DE2}\gg\tau_{DE1}$, the overall DE
radiative lifetime is limited by recombination through emission from DE2.
Additionally, generally one observes that $\tau_{DE}\approx\tau_{DE2}$, which
is consistent with the above-mentioned finding that in an ensemble of quantum
dots considered in this work higher energy dark exciton state has (on average)
larger optical activity for both out-of-plane and in-plane polarizations,
although a sizable number of quantum dots has dominant emission through DE1
state, and short DE1 lifetime, with a strong variation within the ensemble.

##  VI Summary

To summarize, changes in the local atomic arrangement in an alloyed self-
assembled quantum dot can trigger not only substantial fluctuation is BES and
DES, but can also lead to a non-negligible in-plane optical activity of dark
excitons. Whereas the out-of-plane emission from dark exciton states is
possible for higher-symmetry $C_{2v}$ quantum dots, and it origins from the
valence-band mixing and lattice anisotropy, the in-plane emission in $C_{2v}$
systems is forbidden by symmetry. Contrarily, in an alloyed $C_{1}$ system
dark excitons can emit in-plane polarized light. There are two contributions
to this luminescence. One is related to non-vanishing matrix elements of the
optical transition dipole moment, and the second is related to exchange mixing
of bright and dark configurations in the configuration-interaction
Hamiltonian. Dark exciton optical spectra can significantly vary in the
ensemble, from nanostructures having virtually vanishing oscillator strength,
to quantum dots with a substantial 1/6000 fraction of the bright exciton
intensity. Our results indicate that, apart from shape-elongation and presence
of facets, alloying must be accounted for in accurate modeling of quantum dot
systems used a as building blocks of novel quantum devices.

###### Acknowledgements.

The author acknowledges the support from the Polish National Science Centre
based on Decision No. 2018/31/B/ST3/01415. The author would like to thank
Michał Gawłeczyk for reviewing the manuscript and valuable comments.

##  Appendix A BES and DES for s-shell only

Whereas Fig. 3 in the main text shows BES and DES histograms for CI treatment
including $s$, $p$, and $d$ shells, for comparison Fig. 17 shows the (a) BES
and (b) DES calculated when accounting for the s-shell only. In other words,
Fig. 17 (a) and (b) shows histograms of $\delta_{1}$ and $\delta_{2}$ exchange
integrals, using notation of Eq. 6.

Figure 17:  Histograms of (a) the bright exciton splitting, and (b) dark
exciton splitting calculated when accounting for the $s$ shell only in the
configuration-interaction procedure.

##  Appendix B Phases of dark and bright exchange integrals

Fig. 18 shows the histogram of exchange interaction phases $\theta_{1}$ and
$\theta_{2}$ angles using the notation of Eq. 17. Should alloying be neglected
(in the $C_{2v}$ case), $\phi_{1}=2\theta_{1}$ would be equal exactly
180$\degree$, and $\theta_{2}$ to 270$\degree$ revealing 90$\degree$ phase
difference between dark and bright exciton. As seen in Fig. 18 when alloying
is accounted for, this relation is held approximately, with strong bright
exciton phase randomization.

Figure 18: Histogram of (a) $2\theta_{1}$ and (b) $2\theta_{2}$ angles using
the notation of Eq. 17.

##  Appendix C Two channels of dark excitons optical activity

For DE states, Eq. 3 for the $s$-shell case only, and in-plane ($x$, $y$)
polarizations, (and skipping energy index for simplicity) takes the form

$\begin{split}I^{\mathrm{s-shell}}=\Big{|}&C_{e\downarrow,h\Uparrow}M^{x,y}_{e\downarrow,h\Uparrow}+C_{e\uparrow,h\Downarrow}M^{x,y}_{e\uparrow,h\Downarrow}+\\\
&C_{e\uparrow,h\Uparrow}M^{x,y}_{e\uparrow,h\Uparrow}+C_{e\downarrow,h\Downarrow}M^{x,y}_{e\downarrow,h\Downarrow}\Big{|}^{2},\end{split}$
(27)

where $C$’s are expansion coefficients from diagonalization of the $4\times 4$
CI Hamiltonian expressed in the $s$-shell basis.

When exchange mixing terms are neglected ($\mathbf{H}_{\mathrm{BD}}=0$) this
formula is simplified with only with dark exciton optical activity possible if
$M^{x,y}_{e\uparrow,h\Uparrow}\neq 0$, and
$M^{x,y}_{e\downarrow,h\Downarrow}\neq 0$, thus due to non-vanishing optical
dipole moment corresponding to nominally dark configurations:

$I^{\mathrm{s-shell}}_{\mathrm{dipole}}=\Big{|}C_{e\uparrow,h\Uparrow}M^{x,y}_{e\uparrow,h\Uparrow}+C_{e\downarrow,h\Downarrow}M^{x,y}_{e\downarrow,h\Downarrow}\Big{|}^{2}.$
(28)

On the other hand, when exchange mixing terms are accounted for, while in-
plane optical dipole moments are neglected, i.e.:
$M^{x,y}_{e\uparrow,h\Uparrow}=M^{x,y}_{e\downarrow,h\Downarrow}=0$, dark
exciton optical activity is possible via an admixture of oscillator strengths
from bright configurations ($C_{e\downarrow,h\Uparrow}\neq 0$ or
$C_{e\uparrow,h\Downarrow}\neq 0$)

$I^{\mathrm{s-shell}}_{\mathrm{exch}}=\Big{|}C_{e\downarrow,h\Uparrow}M^{x,y}_{e\downarrow,h\Uparrow}+C_{e\uparrow,h\Downarrow}M^{x,y}_{e\uparrow,h\Downarrow}\Big{|}^{2}.$
(29)

Both effects apparently play comparable roles, as discussed in the main text,
and must be accounted for on equal footing by using Eq. 27 for the $s$-shell
case only, or by using Eq. 2 when $s$, $p$, and $d$ shells are accounted for.

## References

  * Poem  _et al._ [2010] E. Poem, Y. Kodriano, C. Tradonsky, N. H. Lindner, B. D. Gerardot, P. M. Petroff, and D. Gershoni, Accessing the dark exciton with light, Nat. Phys. 6, 993 (2010).
  * Schwartz  _et al._ [2015a] I. Schwartz, E. R. Schmidgall, L. Gantz, D. Cogan, E. Bordo, Y. Don, M. Zielinski, and D. Gershoni, Deterministic writing and control of the dark exciton spin using single short optical pulses, Phys. Rev. X 5, 011009 (2015a).
  * Huo  _et al._ [2017] Y. H. Huo, V. Křápek, O. G. Schmidt, and A. Rastelli, Spontaneous brightening of dark excitons in gaas/algaas quantum dots near a cleaved facet, Phys. Rev. B 95, 165304 (2017).
  * Ohta  _et al._ [2018a] R. Ohta, H. Okamoto, T. Tawara, H. Gotoh, and H. Yamaguchi, Dynamic control of the coupling between dark and bright excitons with vibrational strain, Phys. Rev. Lett. 120, 267401 (2018a).
  * Sallen  _et al._ [2011] G. Sallen, B. Urbaszek, M. M. Glazov, E. L. Ivchenko, T. Kuroda, T. Mano, S. Kunz, M. Abbarchi, K. Sakoda, D. Lagarde, A. Balocchi, X. Marie, and T. Amand, Dark-bright mixing of interband transitions in symmetric semiconductor quantum dots, Phys. Rev. Lett. 107, 166604 (2011).
  * Jacak  _et al._ [1998] L. Jacak, P. Hawrylak, and A. Wojs,  _Quantum Dots_ (Springer, Berlin, 1998).
  * Bimberg  _et al._ [1999] D. Bimberg, M. Grundmann, and N. Ledentsov,  _Quantum Dot Heterostructures_ (Wiley, Chichester, 1999).
  * Bayer  _et al._ [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. Schäfer, Fine structure of neutral and charged excitons in self-assembled in(ga)as/(al)gaas quantum dots, Phys. Rev. B 65, 195315 (2002).
  * Schwartz  _et al._ [2015b] I. Schwartz, D. Cogan, E. R. Schmidgall, L. Gantz, Y. Don, M. Zieliński, and D. Gershoni, Deterministic coherent writing of a long-lived semiconductor spin qubit using one ultrafast optical pulse, Phys. Rev. B 92, 201201 (2015b).
  * Zieliński  _et al._ [2015] M. Zieliński, Y. Don, and D. Gershoni, Atomistic theory of dark excitons in self-assembled quantum dots of reduced symmetry, Phys. Rev. B 91, 085403 (2015).
  * McFarlane  _et al._ [2009] J. McFarlane, P. Dalgarno, B. Gerardot, R. Hadfield, R. Warburton, K. Karrai, A. Badolato, and P. Petroff, Gigahertz bandwidth electrical control over a dark exciton-based memory bit in a single quantum dot, Appl. Phys. Lett. 94, 093113 (2009).
  * Xu  _et al._ [2008] X. Xu, B. Sun, P. R. Berman, D. G. Steel, A. S. Bracker, D. Gammon, and L. Sham, Coherent population trapping of an electron spin in a single negatively charged quantum dot, Nat. Phys. 4, 692 (2008).
  * Ohta  _et al._ [2018b] R. Ohta, H. Okamoto, T. Tawara, H. Gotoh, and H. Yamaguchi, Dynamic control of the coupling between dark and bright excitons with vibrational strain, Phys. Rev. Lett. 120, 267401 (2018b).
  * Michler [2017] P. Michler,  _Quantum Dots for Quantum Information Technologies_ , Vol. 237 (Springer, 2017).
  * Benson  _et al._ [2000] O. Benson, C. Santori, M. Pelton, and Y. Yamamoto, Regulated and entangled photons from a single quantum dot, Phys. Rev. Lett. 84, 2513 (2000).
  * Stevenson  _et al._ [2006] R. M. Stevenson, R. J. Young, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, A semiconductor source of triggered entangled photon pairs, Nature 439, 179 EP (2006).
  * Prilmüller  _et al._ [2018] M. Prilmüller, T. Huber, M. Müller, P. Michler, G. Weihs, and A. Predojević, Hyperentanglement of photons emitted by a quantum dot, Phys. Rev. Lett. 121, 110503 (2018).
  * Bester  _et al._ [2003] G. Bester, S. Nair, and A. Zunger, Pseudopotential calculation of the excitonic fine structure of million-atom self-assembled $\mathrm{In}_{1-x}\mathrm{Ga}_{x}\mathrm{As}/\mathrm{GaAs}$ quantum dots, Phys. Rev. B 67, 161306 (2003).
  * Karlsson  _et al._ [2010] K. F. Karlsson, M. A. Dupertuis, D. Y. Oberli, E. Pelucchi, A. Rudra, P. O. Holtz, and E. Kapon, Fine structure of exciton complexes in high-symmetry quantum dots: Effects of symmetry breaking and symmetry elevation, Phys. Rev. B 81, 161307 (2010).
  * Singh and Bester [2009] R. Singh and G. Bester, Nanowire quantum dots as an ideal source of entangled photon pairs, Phys. Rev. Lett. 103, 063601 (2009).
  * Dupertuis  _et al._ [2011] M. A. Dupertuis, K. F. Karlsson, D. Y. Oberli, E. Pelucchi, A. Rudra, P. O. Holtz, and E. Kapon, Symmetries and the polarized optical spectra of exciton complexes in quantum dots, Phys. Rev. Lett. 107, 127403 (2011).
  * Zieliński [2019] M. Zieliński, Spectra of dark and bright excitons in alloyed nanowire quantum dots, Phys. Rev. B 100, 045309 (2019).
  * Zieliński [2013a] M. Zieliński, Excitonic fine structure of elongated inas/inp quantum dots, Phys. Rev. B 88, 155319 (2013a).
  * Zieliński [2019] M. Zieliński, From quantum dots to quantum dashes: Excitonic spectra of highly elongated inas/inp nanostructures, Phys. Rev. B 99, 205402 (2019).
  * Świderski and Zieliński [2017] M. Świderski and M. Zieliński, Atomistic theory of excitonic fine structure in inas/inp nanowire quantum dot molecules, Phys. Rev. B 95, 125407 (2017).
  * Zieliński [2012a] M. Zieliński, Influence of substrate orientation on exciton fine structure splitting of inas/inp nanowire quantum dots, Nanoscale Res. Lett. 7, 265 (2012a).
  * Gawełczyk  _et al._ [2017] M. Gawełczyk, M. Syperek, A. Maryński, P. Mrowiński, Ł. Dusanowski, K. Gawarecki, J. Misiewicz, A. Somers, J. P. Reithmaier, S. Höfling, and G. Sęk, Exciton lifetime and emission polarization dispersion in strongly in-plane asymmetric nanostructures, Phys. Rev. B 96, 245425 (2017).
  * Gawełczyk [2018] M. Gawełczyk, Excitons in Asymmetric Nanostructures: Confinement Regime, Acta Phys. Pol. A 134, 930 (2018).
  * Mrowiński  _et al._ [2016] P. Mrowiński, M. Zieliński, M. Świderski, J. Misiewicz, A. Somers, J. P. Reithmaier, S. Höfling, and G. Sęk, Excitonic fine structure and binding energies of excitonic complexes in single inas quantum dashes, Phys. Rev. B 94, 115434 (2016).
  * Abbarchi  _et al._ [2010] M. Abbarchi, T. Kuroda, T. Mano, and K. Sakoda, Fine structure splitting reduction in droplet epitaxy gaas quantum dots grown on (111) a surface, in  _J. Phys. Conf. Ser._, Vol. 245 (IOP Publishing, 2010) p. 012049.
  * Kors  _et al._ [2018] A. Kors, J. P. Reithmaier, and M. Benyoucef, Telecom wavelength single quantum dots with very small excitonic fine-structure splitting, Appl. Phys. Lett. 112, 172102 (2018).
  * Young  _et al._ [2005] R. Young, R. Stevenson, A. Shields, P. Atkinson, K. Cooper, D. Ritchie, K. Groom, A. Tartakovskii, and M. Skolnick, Inversion of exciton level splitting in quantum dots, Phys. Rev. B 72, 113305 (2005).
  * Langbein  _et al._ [2004] W. Langbein, P. Borri, U. Woggon, V. Stavarache, D. Reuter, and A. Wieck, Control of fine-structure splitting and biexciton binding in in x ga 1- x as quantum dots by annealing, Phys. Rev. B 69, 161301 (2004).
  * Bjork  _et al._ [2004] M. T. Bjork, C. Thelander, A. Hansen, L. Jensen, M. Larsson, L. R. Wallenberg, and L. Samuelson, Few-electron quantum dots in nanowires, Nano Lett. 4, 1621 (2004).
  * Borgström  _et al._ [2005] M. T. Borgström, V. Zwiller, E. Müller, and A. Imamoglu, Optically bright quantum dots in single nanowires, Nano Lett. 5, 1439 (2005).
  * Dalacu  _et al._ [2009] D. Dalacu, A. Kam, D. G. Austing, X. Wu, J. Lapointe, G. C. Aers, and P. J. Poole, Selective-area vapour–liquid–solid growth of inp nanowires, Nanotechnology 20, 395602 (2009).
  * Dalacu  _et al._ [2012] D. Dalacu, K. Mnaymneh, J. Lapointe, X. Wu, P. J. Poole, G. Bulgarini, V. Zwiller, and M. E. Reimer, Ultraclean emission from inasp quantum dots in defect-free wurtzite inp nanowires, Nano Lett. 12, 5919 (2012).
  * Bouwes Bavinck  _et al._ [2012] M. Bouwes Bavinck, M. Zieliński, B. J. Witek, T. Zehender, E. P. A. M. Bakkers, and V. Zwiller, Controlling a nanowire quantum dot band gap using a straining dielectric envelope, Nano Lett. 12, 6206 (2012).
  * Yanase  _et al._ [2017] S. Yanase, H. Sasakura, S. Hara, and J. Motohisa, Single-photon emission from inasp quantum dots embedded in density-controlled inp nanowires, Jpn. J. Appl. Phys. 56, 04CP04 (2017).
  * Bryant  _et al._ [2011] G. W. Bryant, M. Zieliński, N. Malkova, J. Sims, W. Jaskólski, and J. Aizpurua, Controlling the optics of quantum dots with nanomechanical strain, Phys. Rev. B 84, 235412 (2011).
  * Świderski and Zieliński [2019] M. Świderski and M. Zieliński, Electric field tuning of excitonic fine-structure splitting in asymmetric inas/inp nanowire quantum dot molecules, Phys. Rev. B 100, 235417 (2019).
  * Gong  _et al._ [2011] M. Gong, W. Zhang, G.-C. Guo, and L. He, Exciton polarization, fine-structure splitting, and the asymmetry of quantum dots under uniaxial stress, Phys. Rev. Lett. 106, 227401 (2011).
  * Trotta  _et al._ [2014] R. Trotta, J. S. Wildmann, E. Zallo, O. G. Schmidt, and A. Rastelli, Highly entangled photons from hybrid piezoelectric-semiconductor quantum dot devices, Nano lett. 14, 3439 (2014).
  * Don  _et al._ [2016] Y. Don, M. Zielinski, and D. Gershoni, The optical activity of the dark exciton, arXiv preprint arXiv:1601.05530 (2016).
  * Germanis  _et al._ [2018] S. Germanis, P. Atkinson, R. Hostein, C. Gourdon, V. Voliotis, A. Lemaître, M. Bernard, F. Margaillan, S. Majrab, and B. Eble, Dark-bright exciton coupling in asymmetric quantum dots, Phys. Rev. B 98, 155303 (2018).
  * Tomić and Vukmirović [2011] S. Tomić and N. Vukmirović, Symmetry reduction in multiband hamiltonians for semiconductor quantum dots: The role of interfaces and higher energy bands, J. Appl. Phys. 110, 053710 (2011).
  * Marquardt  _et al._ [2011] O. Marquardt, S. Schulz, C. Freysoldt, S. Boeck, T. Hickel, E. P. O’Reilly, and J. Neugebauer, A flexible, plane-wave based multiband k·p model, Opt. Quant. Electron. 44, 183 (2011).
  * Singh and Bester [2011] R. Singh and G. Bester, Effects of atomic ordering on the electronic and optical properties of self-assembled inxga1-xas/gaas semiconductor quantum dots, Phys. Rev. B 84, 241402 (2011).
  * Bryant  _et al._ [2013] G. W. Bryant, N. Malkova, and J. Sims, Mechanism for controlling the exciton fine structure in quantum dots using electric fields: Manipulation of exciton orientation and exchange splitting at the atomic scale, Phys. Rev. B 88, 161301 (2013).
  * Zieliński [2013b] M. Zieliński, Valence band offset, strain and shape effects on confined states in self-assembled inas/inp and inas/gaas quantum dots, J. Phys.: Condens. Matter 25, 465301 (2013b).
  * Keating [1966] P. N. Keating, Effect of invariance requirements on the elastic strain energy of crystals with application to the diamond structure, Phys. Rev. 145, 637 (1966).
  * Martin [1970] R. M. Martin, Elastic properties of zns structure semiconductors, Phys. Rev. B 1, 4005 (1970).
  * Jaskólski  _et al._ [2006] W. Jaskólski, M. Zieliński, G. W. Bryant, and J. Aizpurua, Strain effects on the electronic structure of strongly coupled self-assembled inas/gaas quantum dots: Tight-binding approach, Phys. Rev. B 74, 195339 (2006).
  * Saito and Arakawa [2002] T. Saito and Y. Arakawa, Electronic structure of piezoelectric in${}_{0}.2$ga${}_{0}.8$n quantum dots in gan calculated using a tight-binding method, Physica E 15, 169 (2002).
  * Bester  _et al._ [2006a] G. Bester, X. Wu, D. Vanderbilt, and A. Zunger, Importance of second-order piezoelectric effects in zinc-blende semiconductors, Phys. Rev. Lett. 96, 187602 (2006a).
  * Bester  _et al._ [2006b] G. Bester, A. Zunger, X. Wu, and D. Vanderbilt, Effects of linear and nonlinear piezoelectricity on the electronic properties of inas/gaas quantum dots, Phys. Rev. B 74, 081305 (2006b).
  * Beya-Wakata  _et al._ [2011] A. Beya-Wakata, P.-Y. Prodhomme, and G. Bester, First- and second-order piezoelectricity in iii-v semiconductors, Phys. Rev. B 84, 195207 (2011).
  * Tse  _et al._ [2013] G. Tse, J. Pal, U. Monteverde, R. Garg, V. Haxha, M. Migliorato, and S. Tomić, Non-linear piezoelectricity in zinc blende gaas and inas semiconductors, J. Appl. Phys. 114, 073515 (2013).
  * Caro  _et al._ [2015] M. A. Caro, S. Schulz, and E. P. O’Reilly, Origin of nonlinear piezoelectricity in iii-v semiconductors: Internal strain and bond ionicity from hybrid-functional density functional theory, Phys. Rev. B 91, 075203 (2015).
  * Świderski and Zieliński [2016] M. Świderski and M. Zieliński, Exact diagonalization approach for atomistic calculation of piezoelectric effects in semiconductor quantum dots, Acta Physica Polonica A 129, https://doi.org/10.12693/aphyspola.129.a-79 (2016).
  * Jancu  _et al._ [1998] J.-M. Jancu, R. Scholz, F. Beltram, and F. Bassani, Empirical spds* tight-binding calculation for cubic semiconductors: General method and material parameters, Phys. Rev. B 57, 6493 (1998).
  * Chadi [1977] D. J. Chadi, Spin-orbit splitting in crystalline and compositionally disordered semiconductors, Phys. Rev. B 16, 790 (1977).
  * Zieliński [2012b] M. Zieliński, Including strain in atomistic tight-binding hamiltonians: An application to self-assembled inas/gaas and inas/inp quantum dots, Phys. Rev. B 86, 115424 (2012b).
  * Lee  _et al._ [2004] S. Lee, F. Oyafuso, P. von Allmen, and G. Klimeck, Boundary conditions for the electronic structure of finite-extent embedded semiconductor nanostructures, Phys. Rev. B 69, 045316 (2004).
  * Zieliński [2012c] M. Zieliński, Multi-scale simulations of semiconductor nanostructures, Acta Phys. Pol. A 122, 312 (2012c).
  * Zieliński  _et al._ [2010] M. Zieliński, M. Korkusinski, and P. Hawrylak, Atomistic tight-binding theory of multiexciton complexes in a self-assembled inas quantum dot, Phys. Rev. B 81, 085301 (2010).
  * Díaz  _et al._ [2007] J. G. Díaz, G. W. Bryant, W. Jaskólski, and M. Zieliński, Theory of inp nanocrystals under pressure, Phys. Rev. B 75, 245433 (2007).
  * Dresselhaus  _et al._ [2007] M. S. Dresselhaus, G. Dresselhaus, and A. Jorio,  _Group theory: application to the physics of condensed matter_ (Springer Science & Business Media, 2007).
  * Michler [2003] P. Michler, ed.,  _Topics in Applied Physics_, Vol. 90 (Springer, New York, 2003).
  * Lee  _et al._ [2001] S. Lee, L. Jönsson, J. W. Wilkins, G. W. Bryant, and G. Klimeck, Electron-hole correlations in semiconductor quantum dots with tight-binding wave functions, Phys. Rev. B 63, 195318 (2001).
  * Schulz  _et al._ [2006] S. Schulz, S. Schumacher, and G. Czycholl, Tight-binding model for semiconductor quantum dots with a wurtzite crystal structure: From one-particle properties to coulomb correlations and optical spectra, Phys. Rev. B 73, 245327 (2006).
  * Różański and Zieliński [2016] P. T. Różański and M. Zieliński, Linear scaling approach for atomistic calculation of excitonic properties of 10-million-atom nanostructures, Phys. Rev. B 94, 045440 (2016).
  * Różański and Zieliński [2019] P. T. Różański and M. Zieliński, Efficient computation of coulomb and exchange integrals for multi-million atom nanostructures, Comput. Phys. Commun. 238, 254 (2019).
  * Karrai and Warburton [2003] K. Karrai and R. J. Warburton, Optical transmission and reflection spectroscopy of single quantum dots, Superlattices and Microstructures 33, 311 (2003).
  * Narvaez  _et al._ [2005] G. A. Narvaez, G. Bester, and A. Zunger, Excitons, biexcitons, and trions in self-assembled $(\mathrm{In}\text{,}\mathrm{Ga})\mathrm{As}/\mathrm{Ga}\mathrm{As}$ quantum dots: Recombination energies, polarization, and radiative lifetimes versus dot height, Phys. Rev. B 72, 245318 (2005).
  * de Oliveira  _et al._ [2012] E. L. de Oliveira, E. L. Albuquerque, J. S. de Sousa, G. A. Farias, and F. M. Peeters, Configuration-interaction excitonic absorption in small si/ge and ge/si core/shell nanocrystals, J. Phys. Chem. C 116, 4399 (2012).
  * Warming  _et al._ [2009] T. Warming, E. Siebert, A. Schliwa, E. Stock, R. Zimmermann, and D. Bimberg, Hole-hole and electron-hole exchange interactions in single inas/gaas quantum dots, Phys. Rev. B 79, 125316 (2009).
  * Liu  _et al._ [2000] N. Liu, J. Tersoff, O. Baklenov, A. L. Holmes, and C. K. Shih, Nonuniform composition profile in ${\mathrm{in}}_{0.5}{\mathrm{ga}}_{0.5}\mathrm{As}$ alloy quantum dots, Phys. Rev. Lett. 84, 334 (2000).
  * Williamson  _et al._ [2000] A. J. Williamson, L. W. Wang, and A. Zunger, Theoretical interpretation of the experimental electronic structure of lens-shaped self-assembled inas/gaas quantum dots, Phys. Rev. B 62, 12963 (2000).
  * Vurgaftman  _et al._ [2001] I. Vurgaftman, J. á. Meyer, and L. á. Ram-Mohan, Band parameters for iii–v compound semiconductors and their alloys, J. Appl. Phys. 89, 5815 (2001).
  * Ivchenko [2005] E. Ivchenko,  _Optical Spectroscopy of Semiconductor Nanostructures_, 2nd ed. (Alpha Science, 2005).
  * Tsitsishvili [2017] E. Tsitsishvili, Impact of heavy hole–light hole coupling on the exciton fine structure in quantum dots, Physica E 87, 161 (2017).
  * Singh and Bester [2010] R. Singh and G. Bester, Lower bound for the excitonic fine structure splitting in self-assembled quantum dots, Phys. Rev. Lett. 104, 196803 (2010).
  * Zieliński [2020] M. Zieliński, Vanishing fine structure splitting in highly asymmetric inas/inp quantum dots without wetting layer, Sci Rep 10, 10.1038/s41598-020-70156-1 (2020).
  * Gong  _et al._ [2014] M. Gong, B. Hofer, E. Zallo, R. Trotta, J.-W. Luo, O. G. Schmidt, and C. Zhang, Statistical properties of exciton fine structure splitting and polarization angles in quantum dot ensembles, Phys. Rev. B 89, 205312 (2014).
  * Heindel  _et al._ [2017] T. Heindel, A. Thoma, I. Schwartz, E. R. Schmidgall, L. Gantz, D. Cogan, M. Strauß, P. Schnauber, M. Gschrey, J.-H. Schulze, _et al._ , Accessing the dark exciton spin in deterministic quantum-dot microlenses, Apl Photonics 2, 121303 (2017).
  * Wojnar [2013] R. Wojnar, Rayleigh’s distribution, wigner’s surmise and equation of the diffusion, Acta Physica Polonica A 123, 624 (2013).
  * Mehta [2004] M. L. Mehta,  _Random matrices_ (Elsevier, 2004).
  * Gong  _et al._ [2008] M. Gong, K. Duan, C.-F. Li, R. Magri, G. A. Narvaez, and L. He, Electronic structure of self-assembled inas/inp quantum dots: Comparison with self-assembled inas/gaas quantum dots, Phys. Rev. B 77, 045326 (2008).
  * Tonin  _et al._ [2012] C. Tonin, R. Hostein, V. Voliotis, R. Grousson, A. Lemaitre, and A. Martinez, Polarization properties of excitonic qubits in single self-assembled quantum dots, Phys. Rev. B 85, 155303 (2012).
  * Léger  _et al._ [2007] Y. Léger, L. Besombes, L. Maingault, and H. Mariette, Valence-band mixing in neutral, charged, and mn-doped self-assembled quantum dots, Phys. Rev. B 76, 045331 (2007).
  * Smoleński  _et al._ [2012] T. Smoleński, T. Kazimierczuk, M. Goryca, T. Jakubczyk, Ł. Kłopotowski, Ł. Cywński, P. Wojnar, A. Golnik, and P. Kossacki, In-plane radiative recombination channel of a dark exciton in self-assembled quantum dots, Phys. Rev. B 86, 241305 (2012).
  * Chhikara [1988] R. Chhikara,  _The inverse Gaussian distribution: theory: methodology, and applications_ , Vol. 95 (CRC Press, 1988).
  * Papoulis [1965] A. Papoulis,  _Random Variables and Stochastic Processes_ (1965).
  * Sheng  _et al._ [2012] W.-d. Sheng, M. Korkusinski, A. D. Güçlü, M. Zielinski, P. Potasz, E. S. Kadantsev, O. Voznyy, and P. Hawrylak, Electronic and optical properties of semiconductor and graphene quantum dots, Front. Phys. 7, 328 (2012).