DARK-BRIGHT EXCITONS MIXING IN ALLOYED INGAAS SELF-ASSEMBLED QUANTUM DOTS
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Dark-bright excitons mixing in alloyed InGaAs self-assembled quantum dots
Michał Zieliński1, ∗
1
Institute of Physics, Faculty of Physics, Astronomy and Informatics
Nicolaus Copernicus University, ul. Grudziądzka 5, 87-100 Toruń, Poland
(Dated: March 24, 2021)
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
arXiv:2103.12356v1 [cond-mat.mes-hall] 23 Mar 2021
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 We find that alloy randomness, by reducing the overall sym-
metry, leads to (i) non-vanishing matrix elements of the optical
The optically weakly active [1–5] exciton states confined in transition (optical/transition dipole moments), as well as (ii)
quantum dots [6, 7], also known as dark excitons (DEs), [8] non-negligible exchange integrals mixing bright and dark ex-
are promising candidates for applications in quantum infor- citonic configurations. There are thus two equally important
mation processing. [2, 9–13] Together with optically active sources of non-zero oscillator strength for the in-plane dark
bright excitons (BEs), electron-hole pairs with opposite spins excitonic emission. On the contrary, the out-of-plane (z) po-
far better studied for, e.g., entanglement generation [14–17], larized emission is dominated by the contribution from the
they may form important building blocks for future quantum matrix element of the optical transition only, which is in turn
devices. However, this applicability strongly depends on abil- governed by the valence-band mixing, in agreement with the
ity to understand and control the details of excitonic spectra, phenomenological understating of dark exciton states. There-
so-called “excitonic fine structure”. [8] On the one hand, the fore, contrary to the case of in-plane emission, the exchange
bright exciton spectrum is strongly determined by quantum- mixing does not play a role for z-polarized dark exciton emis-
dot symmetry properties [18–29] with extensive efforts [30– sion.
43] aimed at reduction of the bright exciton splitting (BES). While most of the quantum dots in ensembles reveal rather
On the other, the dark exciton fine structure has been studied weak dark exciton activity, we show that a mere alloy random-
to a far lesser degree. In particular, contrary to simplified ness triggers the dark exciton optical activity reaching up to
theoretical predictions [8], the dark exciton can gain a non- 1/6000 fraction of that for the bright exciton, without faceting
negligible optical activity, even without an external magnetic or shape elongation. [3, 10, 45] This conclusion is valid for
field, [19, 22] whereas origins of this luminescence remain both out-of-plane and in-plane polarizations. As optical ac-
largely unexplored. [44, 45] tivity of dark exciton is inversely proportional to its lifetime,
Accurate theoretical modeling of excitonic fine structure, a strong variation of the former translates to a pronounced
regarding both bright and dark excitons, still presents a se- dot-to-dot variations of the latter on a millisecond scale, while
rious challenge for approaches utilizing continuum-media the bright exciton lifetimes tend to be systematically close to
approximation, [19, 46, 47] and even for atomistic meth- 1 ns. Moreover, despite the overall cylindrical quantum-dot
ods. [10, 22, 48] In this work, we use the combination of shape, and despite strong alloying, the polarization properties
tight-binding (TB) and configuration-interaction (CI) meth- of both bright and dark excitons for in-plane emission reveal
ods, which proved its ability to find excitonic fine structure in hallmarks of a pronounced anisotropy due to the underlying
a good agreement with experiment. [22, 49, 50] To understand crystal lattice.
the role of disorder due to alloy randomness, we present an The paper is organized as follows: after a short theoreti-
extensive theoretical study of properties of bright and dark ex- cal introduction in Section II, we start with the discussion of
citons calculated for an ensemble of 300 alloyed In0.5 Ga0.5 As bright exciton spectra and polarization angle in Section III,
self-assembled quantum dots, each treated with the same atom- Throughout most of the paper we aim to bridge between atom-
istic resolution. istic 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 prop-
∗ mzielin@fizyka.umk.pl erties, and finally we study excitonic lifetimes in Section V.2
II. SYSTEMS AND METHODS
The calculation starts with finding atomic positions that min-
imize the total elastic energy, by using the valence force field
(VFF) method of Keating [51, 52]. Minimization of strain en-
ergy is performed with the conjugate gradient method. [53, 54]
Next, the piezoelectric potential [55–60] is calculated by ac-
counting 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
sp3 d5 s∗ tight-binding method accounting for d-orbitals and
spin-orbit interaction. [50, 61–63] The tight-binding calcula-
tion is performed on a smaller domain than the valence force
field calculation. [64, 65] The details regarding the sp3 d5 s? FIG. 1. Schematics of systems under consideration: (a) lens-shaped
tight-binding calculations for various nanostructures are dis- InAs/GaAs self-assembled quantum dot located on a wetting layer
cussed thoroughly in our earlier papers. [50, 53, 63, 66, 67] and (b) alloyed In0.5 Ga0.5 As quantum dots with the same dimen-
We note that eigenstates of the tight-binding Hamiltonian sions, however with 5 randomly generated samples (realizations) cor-
are doubly degenerate to due to time reversal-symmetry. [68] responding to the same average composition, yet different (random)
Therefore, for convenience, in several places we label tight atomic arrangement. See the text for details. Surrounding GaAs
binding electron ground states with up and down spin-indices, material is not shown.
i.e., e↑ and e↓ . However, we emphasize that due to spin-orbit
interaction, and low overall symmetry, due to underlying lattice ~
Eq. 1, Mc,v ≡ hΨc |~ · ~r |Ψv i is the optical dipole moment
with alloying, the spin in no longer a good quantum number,
matrix element calculated from single-particle tight-binding
and such labeling is only approximate. Similarly, we label
electron Ψc and hole Ψh wave functions respectively, for a
atomistically obtained hole ground states with double-arrows,
given polarization of light ~, e.g., Mez↑ ,h⇓ = he↑ |z|h⇓ i, for
i.e., h⇑ and h⇓ , even though due to valence-band mixing and
alloying they do not exactly correspond to heavy-hole ± 23 z-polarized (out-of-plane) light, and optical (transition) dipole
eigenstates. matrix element involving ground electron and hole states with
After the tight-binding stage of calculations, the excitonic opposite (quasi-)spins.
spectra [69] are calculated with the configuration-interaction Fermi’s golden rule allows for calculation of first-order ra-
method described in detail in Ref. [66]. The Hamiltonian diative recombination lifetime in a similar manner [74–76]
for the interacting electron-hole pair can be written in second 1 4nαωi3 X X 2
i ~
quantization as follows [69] = Cc,v Mc,v , (3)
τi 3c2 c,v
~=~
x,~
y ,~
z
Eie c†i ci + Eih h†i hi
X X
Ĥex =
i i where ωi = Ei /~, n = 3.6 [74] is the refractive index, α is the
X eh,dir † †
X eh,exch † † fine-structure constant, c is the speed of light in the vacuum,
− Vijkl ci hj hk cl + Vijkl ci hj ck hl (1) x, y are in-plane, and z is the out-of-plane radiative decay
ijkl ijkl
channel. Finally, the average bright exciton radiative lifetime,
where Eie and Eih are the single-particle electron and hole en- assuming identical occupation of two bright exciton states BE1
ergies obtained at the single-particle stage of calculations, re- and BE2 (discussed later in the text), and neglecting thermal
spectively, and Vijkl are Coulomb matrix elements (Coulomb effects is calculated as [75, 76]
direct and exchange integrals) calculated according to the pro- 1 1
1 1
cedure given in Ref. [66]. More details regarding the Coulomb ≈ + (4)
matrix element computation for tight-binding wave-functions τBE 2 τBE1 τBE2
can also be found in Refs. [70, 71] as well as in our recent with an analogous formula for the dark exciton.
papers. [72, 73] Using the above formalism, we perform calculations for an
Finally, the optical spectra are found by calculating the inten- ensemble of alloyed In0.5 Ga0.5 As lens-shaped quantum dots
sity of photoluminescence from the recombination of electron- located on a wetting layer (Fig. 1), with low overall (shape
hole pair using Fermi’s golden rule + alloyed lattice) C1 symmetry. Each quantum-dot system is
X
i
2 treated separately by full VFF/TB/CI calculation. The height
I (ω) = Cc,v hΨc |~ · ~r |Ψv i δ (Ei − ~ω) of all quantum dots is equal to 3 nm, whereas the diameter
c,v is 25.4 nm. Each quantum dot is located on a 0.6 nm (1 lat-
2
(2)
=
X
i
Cc,v ~
Mc,v δ (Ei − ~ω) tice constant) thick wetting layer. A quantum-dot system with
c,v
such dimensions (in particular in a non-alloyed InAs/GaAs
variant), has been studied thoroughly in literature (including
i
where Ei is the energy of initial ith state of the exciton, Cv,c are our own work [50, 63]), as a model of a typical self-assembled
CI expansion coefficients for the ith state obtained by solving quantum dot. For comparison, in several places we discuss as3
well results obtained for C2v InAs quantum, with no gallium
addmixture (no alloying; upper part of Fig. 1). However, our
focus is on the alloyed system, by considering 50% addmix-
ture of barrier material in the dot region, i.e., an In0.5 Ga0.5 As
quantum dot in the GaAs surrounding, with uniform (on av-
erage) 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 FIG. 2. (a) Histogram illustrating distribution of average composition
variations (from ideal 50% gallium content) for a family of 300 alloyed
To study effects of alloy randomness, we focus on uniform quantum dots studied in the text; (b) exciton ground-state energy as
composition profile, rather than on effects of spatial changes function of composition difference and alloy randomness. See the
in the overall composition. [77–79] To model random alloy text for details.
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 ap- states in such approaches is thus constructed from the above-
proach will thus generate microscopic configurations that dif- mentioned, idealized single-particle states, and characterized
fer significantly in the atomic arrangement, while maintaining by their angular momentum projections
(on average) 50% gallium content. A more careful inspection
reveals however, that such procedure may in fact result in ran- 3 1 3 1
|±1i = ± , ∓ , |±2i = ± , ± , (5)
dom realizations that differ from average composition within 2 2 2 2
approximately ±1% range, as shown in Fig. 2 (a). Therefore,
a generated ensemble of quantum dots is a subject to (a) ran- where the first value in the ket corresponds to the hole, and
domness due to different atomic arrangement corresponding to the second one to the electron. Matrix elements of the
the same composition, plus (b) small (± ∼1%) fluctuations of optical transition are given as: [81] M± 32 ,∓ 21 ≡ M±1 =
average composition. As shown in Fig. 2 (b) the ground state M (ex ∓ iey ) , M±2 = 0, where ex , ey are in-plane compo-
excitonic energy of the considered ensemble of quantum dots nents of e polarization vector, and M is independent of the
is altered by both effects. Changes in average composition (x light polarization. Therefore |±1i are optically active, whereas
axis in Fig. 2) correspond to a change of excitonic energy by |±2i are optically inactive states.
approximately 8 meV per 1% of gallium content, consistent Electron-hole exchange Hamiltonian is expanded in basis
with 10 meV estimate that can be obtained from virtual crys- of these states, and in the order |1i , |−1i , |2i , |−2i can be
tal approximation (VCA) with bulk InGaAs parameters taken written as [8]
from Ref. [80] However, changes on the vertical axis, which
are also on a scale of several meV’s, are related purely due to δ0 δ1 0 0
alloy randomness, that is going beyond the VCA. Importantly, 1 δ1 δ0 0 0
Hexch = , (6)
whatever the cause, the exciton ground energy varies within 2 0 0 −δ0 δ2
relatively small ranges, namely less then 1.3% from the av- 0 0 δ2 −δ0
erage of approximately 1257.6 meV. Therefore, as a rule of
thumb, any spectral quantity that changes substantially more where δ0 describes the dark-bright exciton splitting (“isotropic
than 2% when going from a sample to sample (such as the BES electron-hole exchange interaction”), δ1 is responsible for
and the DES discussed in the following) cannot be attributed the bright doublet splitting, and since it is related to distor-
to a change in averaged composition, but to alloy randomness tions from idealized symmetry, it is often called “anisotropic
(or changes in the microscopic atomic arrangement). electron-hole exchange interaction”, and finally δ2 is respon-
In a standard treatment, the excitonic fine structure is typ- sible for the dark exciton splitting. For C3v , and D2d high-
ically [8, 81] addressed by starting from idealized single- symmetry quantum dots δ1 = 0, i.e., bright exciton splitting
particle states, with electron spin states |± 21 i, and the hole vanishes. [20] For C3v systems additionally δ2 = 0. Nonethe-
ground states of heavy-hole character, associated with the pro- less, for realistic alloyed C1 quantum dots both δ1 6= 0 and
jection of the total hole angular momentum: |± 32 i. It is δ2 6= 0, and δ1 δ2 . Similar properties are shared by C2v
important to reemphasize that |± 12 i are not exactly equal to quantum dots, hence C2v systems are often treated as good
e↑ /e↓ tight-binding eigenstates, which are not eigenfuctions approximations to real system, although such approximation
of spin, but are labeled with pseudo-spin index instead. The often fails, [83, 84] as also discussed in this work.
same applies the h⇑ /h⇓ hole states as well. Nevertheless, Since the off-diagonal block consists of zero’s, there is
|± 12 i, |± 23 i states are of vital importance for defining effec- no coupling between bright and dark manifolds, and they
tive Hamiltonians. [8, 82] To this end the excitonic basis of four can be treated separately, with bright exciton states given as4
FIG. 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.
symmetric/anti-symmetric combinations of basis states To further analyze polarization properties of bright excitons,
√ while still assuming no dark-bright exciton coupling, one can
|BE1i = ( |1i − |−1i) / 2 (7) use the Hamiltonian for the bright manifold only, derived based
√ on group-theoretical [42] arguments:
|BE2i = ( |1i + |−1i) / 2, (8)
E+δ κ
with analogous formulas for the dark exciton. Corresponding H= , (11)
κ E−δ
eigenenergies are given as
where E is the reference excitonic energy, δ (with no subscript)
EBE1 = (δ0 − δ1 ) /2 (9) is responsible for the bright exciton splitting in the C2v case
EBE2 = (δ0 + δ1 ) /2, (10) due to lattice (and shape) anisotropy, whereas κ determines
the fraction of the BES related to contribution due to lowering
For sake of comparison, atomistic results (calculated in a of symmetry caused by alloying. Moreover, κ is responsible
basis of 144 CI configurations) can be “recast” to spectra of for the coupling between the two bright states, leading to the
Eq.6. For non-alloyed, lens-shapes system of C2v symmetry rotation [42] of the emission lines from the perfect crystal
we found from atomistic simulations, that δ1 > 0, thus in directions.
q This polarization angle θ is given as: tan (θ) =
such a case anti-symmetric |BE1i would be the lower-energy δ
±
2
1 + (δ/κ) , with the overall bright exciton splitting
κ √
bright excitonic state. Moreover, |BE1i is linearly polarized
along [110 direction, and |BE2i is linearly polarized along given as δ1 = 2 δ 2 + κ2 , and conversely
[110]. Notably, there is no z-polarized emission allowed. For δ1 δ1
an alloyed system polarization properties are more compli- κ=− sin (2θ) , δ = cos (2θ) (12)
cated, and will be discussed in the following part of the paper. 2 2
The bright and dark exciton splittings calculated atomisti- with bright eigenstates given as
cally correspond to |δ1 |, |δ2 | of the simple model respectively,
which are shown in Fig. 3. Excitonic fine structure does |BE1 i = − |sin (θ)i + |cos (θ)i (13)
not show any clear trend with respect to average composi- |BE2 i = |cos (θ)i + |sin (θ)i (14)
tion fluctuations, yet, it strongly varies due to alloy random-
ness. [22, 85] Bright exciton splitting [Fig. 3 (a)] varies from Model Eq.11 allows to decompose the bright exciton split-
zero to over 20 µeV, with average value of 7.6 µeV, and standard ting into two contributions, due to anisotropy (δ) and alloying
deviation of 3.6 µeV. The dark-bright splitting [Fig. 3 (b)] has (κ) respectively, which are shown in Fig. 4. The κ mean value,
the average value of 184 µeV, whereas the dark exciton split- hκi, is close to zero indicating lack of any directional character
ting varies from 0.28 to 1.48 µeV, with mean value of 0.66 µeV of alloying, whereas δ has mean value of -2.32 µeV, which is
and standard deviation equal to 0.16 µ eV. Again, since we do due lattice anisotropy. This result is in vivid contrast to atom-
considered cylindrical quantum dot shape shape [84] both BES istic pseudopotential calculations, [85] where hδi is different
and DES are in good agreement with experimental values, al- from zero only for elongated quantum dots. This result is how-
though DES results are generally smaller than observed in the ever consistent with systematic difference between empirical
experiment. [9] Nevertheless, we note that this range of DE en- tight-binding and empirical pseudopotential approaches for
ergy level splittings corresponds to the precession periods [1] C2v systems, [50] as well BES reported experimentally. [48]
of a coherent superposition of DE eigenstates varying from to κ and δ have standard deviations σ of 2.66 µeV, and
2.8 to 14.5 ns, (with average of 6.3 ns), thus again overlapping 2.30 µeV, respectively. Since both κ and δ are appar-
with typical experimental values. [9, 86] ently independent, and have distribution that can be quite5
FIG. 4. (a) δ and κ contribution to the BES as function of excitonic FIG. 5. Histogram of (a) the bright exciton splitting (with dot-
ground state energy. Histogram (b) of δ and κ with mean κ (contri- ted/dashed line corresponding to Rayleigh/Wigner distributions re-
bution due to alloyin) value close to zero, mean δ (contribution due spectively), and (b) dark exciton splitting (the dashed line is a fit to a
to lattice anisotropy) different from zero. See the text for details. normal distribution).
well described by the normal distribution [Fig. 4 (a)], with smaller than 1 µeV. BES smaller than the line-width (of ap-
σ = σδ ≈ σκ = 2.6 √ µeV, and hδi not exceeding σ, one proximately 1 µeV) is important in the context of entanglement
can expect that δ1 = 2 δ 2 + κ2 , will follow Rayleigh dis- generation. Results presented here may look counter-intuitive,
tribution, [87, 88] (i.e., essentially a χ distribution with two since cylindrical quantum dots seem to be good candidates for
degrees of freedom): small BES values, yet are consistent with earlier findings. [85]
For comparison Fig. 5 (b) shows a histogram of the DES
δ12
δ1 values revealing normal distribution with the mean of 0.657
P (δ1 ) = exp − 2 , (15)
4σ 2 8σ and standard deviation of 0.16 µeV. Thus, the DES is order of
magnitude smaller than the BES, however, in systems studied
where the denominator is multiplied by a factor of 4 (with re- here it is virtually impossible to find a quantum dot with a
spect to the original Rayleigh distribution),
√ due to 2 occurring vanishing dark exciton splitting.
in the definition of δ1 , i.e, δ1 = 2 δ 2 + κ2 . Fig. 5 (a) shows a Finally, we note that mixing with higher shells increases
histogram of BES values with Eq. 15 plotted with dotted black both BES and DES. Should only the s-shell be included in
line revealing a rather good fit, despite not strictly fulfilling the the atomistic calculation (corresponding to 4 × 4 CI Hamil-
assumption of hδi = 0, and with σ = 2.66, not taken as fitting tonian) the mean BES value would be 4 µeV, and DES mean
parameter. Thus, as a result of δ1 having a two-component value 0.53 µeV, with no substantial qualitative difference in
character its distribution varies from the normal distribution, histograms. For comparison, atomistic results obtained for the
but resembles χ with k=2. Rayleigh distribution is derived for s-shell only are presented in the Appendix A.
the system of two-dimensions, however the mathematical form Statistical analysis performed for κ and δ can be performed
of the Rayleigh distribution is identical with Wigner surmise for the bright exciton polarization angle as well, with results
for the one-dimension energy levels problem [85, 88] shown in Fig. 6, where the histogram obtained from the cur-
π π rent atomistic calculation is compared with a statistical model
P (s) = s exp − s2 , (16) (dashed line) from Supplementary Information of Ref. [89].
2 4
The lower energy bright exciton state (BE1) tends be polarized
with s = δ1 / hδ1 i, and where hδ1 i = 7.60 µeV was calculated along the [110] direction (rotated by 90° from [110]), whereas
as an average BES of all 300 samples. The plot of Eq. 16 is the higher-energy one (BE2) prefers polarization along the
also shown in Fig. 5 (a) with a dashed line, compared with the [110] direction. Therefore, despite substantial alloying (and
histogram of BES values. We note that Eq.15 and Eq. 16 have overall cylindrical shape) polarization angle dependence has
identical mathematical forms, whereas the difference between well-defined maxima corresponding to the vicinity of crystal
plots on Fig. 5 (a) is related to difference in parameters √
describ- axis. This is again in stark contrast to empirical pseudopo-
ing “variances” of the both distributions (i.e., hδ1 i =6 2πσ). tential results [89] that predict pronounced polarization angle
Since Eq. 16 corresponds to the standard level-spacing dis- distribution maxima only for elongated dots. This difference
tribution of the Gaussian orthogonal ensemble in random ma- can be traced back to δ distribution as shown earlier (Fig. 4):
trix theory, Gong et al. [85] argue this sort of statistical depen- in atomistic tight-binding calculation hδi =6 0, contrary to the
dence reveals strong repulsion between excitonic levels, and empirical pseudopotential method. This is consistent with the
the presence of multiplicative s in Eq. 16 reduces the probabil- latter systematically underestimating lattice BES contribution,
ity of finding quantum dots with small BES. As a result of this origin of which remains to a large degree unexplored. [48]
sort of behavior, we find that over 50% of samples has the BES Model of Eq. 11 assumes the angle between bright exciton
within 5–10 µeV, with only in 3.4% of cases with BES larger to be exactly equal to 90°. To go beyond that, and further
than 15 µeV, and importantly only 1.7% cases with the BES study polarization properties, the exchange Hamiltonian can6
FIG. 7. Histogram of (a) β 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 β content
remains rather small, with the angle between bright excitons close to
FIG. 6. Histograms of polarization angle for lower (BE1) and higher 90 degrees. See the text for more details.
(BE2) energy bright exciton states with maxima corresponding to
[110] and [110] crystal axes (0 and 90 degrees respectively). The
dashed lines are calculated using approach from Ref. [85] See the formula is used for dark states
text for details.
E p
2
3 1 ±2iψ 1 1
±2 = 1 − β ± , ±
e + βe ∓ ,± . (21)
2 2 2 2
explicitly account for phases of electron-hole exchange inter-
action [81, 90] Exchange Hamiltonian expressed in the basis |±e 1i , |±e2i
maintains the block diagonal structure of Eq. 6, and its eigen-
δ1 e−2iθ1
δ0 0 0
states yet with tilded states replacing |±1i , |±2i. β is respon-
1 δ1 e2iθ1 δ0 0 0
Hexch = , (17) sible for polarization anisotropy [90, 91]
2 0 0 −δ0 δ2 e−2iθ2
0 0 δ2 e2iθ2 −δ0 .
p
2β 3 (1 − β 2 )
C (β) = , (22)
where θ1 describes the rotation of polarization axis (around 3 − 2β 2
the growth axis z) by the angle θ1 with respect to the fixed Polarization anisotropy is naturally present in atomistic cal-
axes, usually taken as crystal axis [110]. The eigenstates have culation, and for a C2v system it reaches 2.9% with emission
the following form intensity for |BE2i along [110] larger then for |BE1i along
√ [110] direction. Recasting atomistic C2v results on valence-
|BE1,2 i = |1i ± e−2iθ1 |−1i / 2 (18)
−2iθ2
√ band-mixed 4 × 4 Hamiltonian gives β value of 2.5%. More-
|DE1,2 i = |1i ± e |−1i / 2 (19) over, for C2v quantum dots mixing phase ψ = 0, and emission
By comparing with atomistic calculations, we find that non- lines remain orthogonal. [82, 90]
alloyed C2v system 2θ1 = π, and 2θ2 = 32 π indicating π/2 For alloyed systems, both β and ψ determine an angle be-
phase difference between dark and bright manifolds. tween excitonic lines, which in principle can be different from
While Eq. 17 allows understanding rotation of quan- 90°. Fig. 7 (a) shows a histogram of β values for alloyed
tum dot principal axes with respected to crystal axes, it systems studied in this paper, whitch appears to be normally
still by definition predicts exactly vanishing polarization distributed around zero light-hole content, with both positive
anisotropy, [90] i.e., C = (Imax − Imin ) / (Imax + Imin ) = and negative light-hole mixing amplitudes possible, and most
(IBE2 − IBE1 ) / (IBE2 + IBE1 ) = 0 since magnitudes of of the cases corresponding to |β| < 1%. Moreover, as shown
emission intensities of both bright excitons are exactly equal. in Fig. 7 (b) valence-band mixing allows for angle between
To go beyond this picture, and allow for better agreement polarizations of bright excitonic lines different from the exact
with the experiment, one can account for valence-band mixing, 90°, however for the vast majority of cases this difference does
which occurs in realistic quantum dots, due to shape asymme- not exceed 1°, rendering Eq. 11 a valid approximation for the
try and lattice anisotropy, leading to an addmixture of light- considered ensemble of quantum dots.
hole component Valence band mixing, expressed via the non-zero β content,
has its effect on the dark manifold as well, since
E p 3 1 1 1
±e1 = 1 − β2 ± , ∓ + βe±2iψ ∓ , ∓ , (20)
E p 3 1 1 1
2 2 2 2 ±e 2 = 1 − β2 ± , ± + βe±2iψ ∓ , ± , (23)
2 2 2 2
where β and ψ are amplitude and phase of mixing with light-
hole exciton ∓ 12 , ∓ 12 , and tilde symbol was used to distin- and since matrix elements of the optical transition [81, 82]
guish from the basis with no band-mixing included. A Similar for the light-hole component are given as: M∓ 12 ,± 12 ∝7
two degrees of freedom, corresponding to the exponential dis-
tribution ∼ exp (−x). 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 iden-
tically (Gaussian) distributed (with equal variance and zero
mean). In such case, the absolute value of the complex num-
ber is Rayleigh-distributed, as apparently shown in Fig. 9, thus
supporting the two-component character of Mz irrespective
from its origin.
FIG. 8. Histogram of (a) modules of out-of-plane single-particle Finally, after an analysis of single-particle transition ma-
optical dipole moment |M z | = |Me↑,h⇑
z z
| = |Me↓,h⇓ |, and (b) its trix elements, we can study many-body dark exciton optical
squared moduli. The dashed lines are calculated assuming probability spectra, shown for z-polarization in Fig. 9. Should the single-
distributions discussed in the text, suggesting the two-component particle optical dipole moments be artificially neglected, i.e.,
character of the optical dipole moment. |Me↑,h⇑ | = |Me↓,h⇓ | = 0, both dark excitons would not re-
veal any significant optical activity. The residual, weak activity
[Fig. 9 (a)] may be attributed to the dark-bright mixing term
ez , dark excitons |±e 2i get optical activity due to hole (due to exchange interaction) present in the atomistic calcula-
2
band mixing, which is proportional to β, i.e., |M±2 f| = tion that will be discussed in the following part of the paper. In
2
β [1 ∓ cos 2(θ2 − 2ψ)]. For C2v system ψ = 0, [82] and the case of z-polarized dark exciton emission exchange mixing
since from atomistic calculation: 2θ2 = 32 π, and δ2 > 0, apparently does not play any significant role. However, the in-
only one of dark exciton states (higher energy, DE2) is opti- clusion of non-zero optical matrix element leads to substantial
cally active, which allows the DE emission perpendicular to increase in z-polarized emission as seen in Fig. 9 (b). This
the growth direction to be detected, and the other (lower en- increase is particularly important for the higher-energy DE2
ergy, DE1) remains fully dark in agreement with experimental 2
states, for which it can reach over 0.006 Å , i.e., approximately
findings. [92] 1/6000 fraction of bright exciton emission merely due to al-
Moreover, group-theoretical arguments indicate that the op- loying. DE1 states gains some optical activity as well, which
tical dipole moment (matrix elements of the optical transition) is however much smaller, and what seem to be a property “in-
for the dark exciton can be non-zero even for pure heavy-hole herited” from C2v system since both amplitude and phase of
2
exciton, i.e., |MDE2 | ∝ ez even if β = 0, whereas DE1 re- valence band mixing i.e., β (and apparently ψ) remain rather
2
mains optically in-active, i.e., |MDE1 | = 0. Thus, the single- small in the alloyed system. Notably, further extension of the
particle dipole matrix element, for z-polarized light, calcu- CI basis by inclusion of higher shells [Fig. 9 (c)] does not seem
lated in the basis of tight-binding states: |M z | = |Me↑,h⇑
z
|= to affect dark exciton optical spectra, indicating the dominant
z
|Me↓,h⇓ | ∝ |βMLH + MHH |, and therefore could be inter- contribution is from the optical dipole moment, and not due to
preted as having light- and heavy-hole components. It not configuration mixing.
straightforward to perform such decomposition for atomisti- In a phenomenological treatment, several authors further
cally obtained states. We have thus tried an approximate at- extend valence band mixing approach by accounting for a
tempt by applying vertical electric field, aiming to use the field mixing term between heavy- and light-hole states with parallel
to reduce polarization anisotropy of bright exciton to zero, for spins:
which case, based on Eq. 22, one can assume β ∼ 0. In such
a situation |M z | should only have the heavy-hole component,
which could be estimated in this way. By performing such ≈
E p
2 2
3 1
±1 = 1 − β − γ ± , ∓
analysis for the C2v system we found MLH = −0.042 Å and 2 2
MHH = 0.051 Å. Therefore, both contributions can have
±2iψ 1 1 ±2iξ 1 1
comparable magnitudes, and can have opposite signs, thus + βe ∓ ,∓ + γe ± ,∓ (24)
2 2 2 2
partially cancel each other. Despite its approximated charac-
ter, we have performed a similar study for 5 alloyed systems as ≈
well, and came to similar conclusions. To further analyze |M z | with a similar formula for |±2i dark states, and where γ and
distributions in the considered ensemble of alloyed system, we ξ are the amplitude and the phase of mixing, and the double
show its histogram in Fig. 8 (a), and then |M |2 histogram in tilde symbol is used to distinguish from previously considered
Fig. 8 (b). |M z | distribution can be well fit to χ distribution cases. This approach has been used to study the z-polarized
with two degrees (k=2) of freedom, which is mathematically emission of bright excitons, [90] with γ parameter being far
equivalent to Rayleigh distribution/Wigner surmise discussed more difficult to extract from the experiment (or atomistic
earlier (i.e., ∼ x exp − π4 x2 , with x = |M z |/h|M z |i shown
calculations) than β, and thus we will not aim for doing that.
as dotted line on Fig. 8 (a). Therefore, it is expected that, However, it was also shown recently [45] that mixing γ term
|M z |2 would be well described by χ2 distribution, again with induces off-diagonal Hamiltonian matrix elements (exchange8
FIG. 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 (Me↑,h⇑ = 0 and Me↓,h⇓ = 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.
integrals) responsible for the dark-bright exciton mixing:
δ1 e−2iθ1 δ11
δ0 δ12
1 δ1 e2iθ1 δ0 −δ12 −δ11
Hexch = ∗ ∗
2 δ11 −δ12 −δ0 δ2 e−2iθ2
∗
δ12 ∗
−δ11 δ2 e2iθ2 −δ0 (25)
1 HBB HBD
=
2 HDB HDD
where HDB denotes dark-bright coupling for brevity, whereas
δ11 , and δ12 are responsible for mixing of bright and dark
configuration states differing by the electron and the hole spin
≈ ≈
state respectively, i.e., δ11 mixes |1i with |2i, and δ12 mixes FIG. 10. Histogram of (a) moduli (absolute values) of δ11 , and δ12
≈ ≈
|−1i = with |−2i. With exception to phase factors, an iden- dark-bright mixing integrals, and (b) their arguments (phases). See
tical Hamiltonian can be also derived in a phenomenologi- the text for details.
cal 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 Cs quantum dots, where low shape
symmetry induces mixing between bright and dark states. [10]
Interestingly, we also found identical structure of (lowest
four states) Hamiltonian from atomistic calculations for cur-
rently studied alloyed quantum dot despite their cylindrical
symmetry, and lack of faceting. Therefore, not only shape de-
formation can induce bright-dark exciton exchange coupling,
but breaking the symmetry on the scale of local atomic ar-
rangement can lead to the same effect. Moreover, despite
overall C1 symmetry, the dark-bright coupling block derived
from atomistic calculation, in a basis of tight-binding states
|e↑ , h⇑ i , |e↓ , h⇓ i | is given as:
FIG. 11. Histograms of (a) moduli of in-plane single-particle optical
δ11 δ12
HDB = ∗ ∗ , (26) dipole moments for light polarized in [110] and [110] directions,
−δ12 −δ11 [110] [110]
i.e., |Me↑,h⇑ | and |Me↓,h⇓ |, (b) their squared moduli. The dashed
where the only difference from phenomenological Hamilto- lines are calculated assuming probability distributions discussed in
nian Eq. 26 (expressed in double-tilde basis) is related to com- the text.
plex conjugations in the second row.9
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 δ11 and δ12 . The absolute
value of δ12 can be quite well fit to x exp −x2 , which is
again equivalent to χ distribution with k=2, suggesting two-
component character of δ12 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 δ12 is induced by the presence
of both β and γ mixing terms. In the treatment employed
in Ref. [45] δ11 depends on γ only. One can thus expect
its statistical distribution to differ from that found for δ12 ,
and in fact this is visible in Fig. 10 (a), where one can fit
δ11 to x exp (−x), resembling Gamma distribution with k=1.
Whereas this sort of analysis is speculative, δ11 undoubtedly
varies from 0 to 2.7 µeV with mean (average) of 0.5 µeV and
δ12 varyies from 0 to to 3.05 µeV with the mean of 1 µeV.
Therefore, δ11 and δ12 have magnitudes comparable with δ1 ,
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 [44, 45]. However, since FIG. 12. Histogram of in-plane optical spectra (emission intensity)
for lower (DE1) and higher (DE2) dark exciton states with (a) op-
the correction is equal to δ11 δ12 /δ0 , the effect of dark-bright
tical single-particle dipole moment included in the calculation, but
mixing on both BES and DES is negligible for the system exchange dark-bright mixing terms neglected (δ11 = δ12 = 0) in
considered in this work, as we have δ0 δ11,12 . the Hamiltonian, (b) optical dipole moment artificially set to zero
For C2v quantum dots, by symmetry, both dark excitons are (|Me↑,h⇑ = 0 and Me↓,h⇓ = 0), but exchange mixing accounted for,
strictly optically inactive, when considering the in-plane polar- (c) both contributions included, and (d) as in (c) but with calculation
ization. For C1 quantum dot however, there are two channels, performed in a basis including s, p, and d shells of 144 excitonic
through which dark excitons can gain in-plane optical activity configurations. Note different scales, as well as presence of breaks
(see the Appendix C for more details). One of them is related to on vertical axes. Apparently, both non-zero optical dipole moments,
and exchange mixing play equally important roles, and none of these
the dark-bright exchange interaction, which was already men-
contributions can be neglected, as it would overestimate the optical
tioned above, and in which the dark exciton can gain optical activity. See the text for more details.
activity from the admixture of bright configurations. The sec-
ond channel is related to a non-vanishing dipole moment for
x,y x,y
in-plane (x,y) polarized light Me↑,h⇑ 6= 0, and Me↓,h⇓ 6= 0,
z
similarly to the M dipole moment discussed earlier. tra. To this end, in Fig. 12 (a) we show the results with dark-
bright mixing artificially neglected, and with single-particle
Fig. 11 shows histograms of in-plane dipole moment matrix
dipole moment being the only term inducing dark exciton
elements along two experimentally relevant crystal axes, [110]
in-plane optical activity. In this case DE2 has much larger
and [110]. Importantly, |M |[110] has substantially larger mag- optical activity than DE1, similarly to previously considered
nitude than |M |110] , thus despite alloying and cylindrical shape out-of-plane emission. Alternatively, exchange mixing can be
of quantum dots in the ensemble, the lattice anisotropy still artificially neglected and optical dipole moments accounted
plays an important role with average h|M |[110] i = 0.06304 Å for, as in Fig. 12 (b). Such a situation produces very similar
about 3 times larger than h|M |[110] i = 0.02061. optical spectra to the first case, with DE2 having substantially
We note as well that |M |[110] is larger than the one for larger optical activity then DE1, despite a different mechanism
the z-polarized case, i.e., h|M |001] ≡ h|M |z i (shown earlier of luminescence.
in Fig. 8) This may be understood since quantum-dot lateral Interestingly, when both terms are included [Fig. 12 (c)],
dimensions are substantially larger than the in growth (z) di- they appear to partially compensate each other, with the dark
rections. However, similarly to |M z |, |M |110 ] can be very exciton optical activity on average an order of magnitude
well approximated by χ distribution with k=2, and with |M |2 smaller compared to (a) and (b) cases, thus comparable with
[Fig. 12 (b)] showing a χ2 dependence, with k=2, i.e., the the out-of-plane emission. A notable difference to z-polarized
exponential distribution. emission can however be observed, since both DE1 and DE2
Since in-plane single-particle dipole moment matrix ele- have (on average) similar optical activity.
ments are non-zero, as are the dark-bright exchange mixing Therefore, reduction of symmetry due to alloying must be
terms in the Hamiltonian, it is instructive to study how these accounted for in both transition dipole moment and exchange
terms individually affect many-body dark exciton optical spec- matrix elements on equal footing. Failing to do so leads to a10
Further inclusion of higher shells beyond the 4 × 4 CI treat-
ment does not change the overall picture [Fig. 12 (d)] consid-
erably. In this case, despite complicated origin of dark exciton
activity (dipole moments, exchange mixing, and CI) the fre-
quency distribution of optical activity can be quite well fit
with a single exponent for DE2 states, and with two exponents
2
for DE1. Interestingly, optical activity larger than 0.002 Å
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 trig-
ger 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 polariza-
FIG. 13. Histogram of in-plane polarization angle for lower (DE1) tion angle of the dark exciton in-plane emission, as shown in
and higher (DE2) dark exciton states with (a) optical single-particle Fig. 13. With either dark-bright exchange mixing artificially
dipole moment included in the calculation, but exchange dark-bright
neglected [Fig. 13 (a)] or dipole moments set to zero [Fig. 13
mixing terms neglected (δ11 = δ12 = 0) in the Hamiltonian, (b)
optical dipole moment artificially set to zero (|Me↑,h⇑ = 0 and (b)], the angle of polarization of both dark excitons tends to
Me↓,h⇓ = 0), but exchange mixing accounted for, (c) both contri- prefer 90° rotation from [110], i.e., polarization along [110]
butions included, and (d) as in (c) but with calculation performed in direction. This is particularly pronounced for DE2 states.
a basis including s, p, and d shells of 144 excitonic configurations. Therefore, with either of contributions accounted for, whilst
Again, both non-zero optical dipole moments, and exchange mixing the other being neglected, the in-plane spectra of dark excitons
play equally important roles, and none of these contributions can be somewhat resemble the out-of-plane (z) polarized spectra with
neglected. See the text for more details. DE2 having dominant optical activity.
However, the situation is apparently different with both con-
tributions simultaneously included, as shown in Fig. 13 (c),
serious overestimation of the dark exciton in-plane optical ac- where DE1 (in particular) and DE2 (to a lesser degree) prefer-
tivity. This could be understood in the spirit of Ref. [44] where able polarization is along [110] (polarization angle 0°). This
the reduction of symmetry is related to rotation of quantum is somewhat opposite to experimental observation, where the
dot axes. This new quantization axis, or rotation of angular dominant emission is from the lower dark exciton state, fol-
momenta, must be included simultaneously in the calculation lowing polarization from the lower energy bright exciton state.
of single-particle dipole moment, and two-body Coulomb and However, a caution must be exercised since experiments are un-
exchange integrals. avoidably performed on non-ideal, low shape symmetry QDs.
Moreover, we find that for C1 alloyed quantum dots it is Our atomistic results [10] for a quantum dot with a facet, and
impossible to fit atomistic spectra to the effective 4 × 4 Hamil- thus broken shape symmetry, indeed indicated a very good
tonian (formally resembling Eq. 25), yet with idealized matrix agreement with a particular experiment [2, 9]. Thus, a devia-
element of the optical transition (i.e., vanishing in-plane opti- tion from cylindrical base very likely must be included when
cal dipole moments), and with δ11 , δ12 being the only factors one aims for a comparison with a particular experimentally
responsible for the dark-exciton optical activity. Such proce- grown system, and here we focus on alloy randomness effect
dure was indeed possible for Cs systems, [10] however for C1 , only.
since both dipole moments and Coulomb matrix are affected Since both DE1 and DE2 appear to have their polarization
by allowing, one would need to include two additional fitting preferably along the [110] direction, it is instructive to show
parameters in the Hamiltonian (δ21 and δ22 integrals) to fur- the histogram of angles between polarizations of DE1 and
ther differentiate coupling of bright and dark states, what does DE2 states (for their in-plane emission) as shown in Fig. 14.
not have a formal justification. Thus, we again conclude that Contrary to bright excitons [shown earlier in Fig. 5 (b)], where
calculation of dipole moments and Coulomb matrix elements the polarization directions strongly tend to be orthogonal (i.e.,
in alloyed system must be performed on equal footing, un- the angle between polarizations close to 90°), the dark exciton
less some other factor (speculatively large quantum dot shape states tend to be polarized along the same direction, however
deformation/anisotropy) dominates over alloying. due to alloy randomness virtually any angle between these11
FIG. 16. Histogram of (a) bright exciton radiative lifetime (b) dark
FIG. 14. Histogram of in-plane angle difference between the lower
exciton radiative lifetime for the ensemble of alloyed quantum dots
(DE1) and the higher (DE2) energy dark exciton states. The dashed
studied in this work. The inset shows lifetime of DE1 and DE2
line is plotted assuming a normal distribution. Contrary to bright
states separately. The dashed lines are plotted assuming a normal
excitons the dark excitonic lines tend to have the same polarization
distribution for the bright exciton, and inverse-gamma distribution
angle, yet with a very broad distribution. See the text for more details.
for the dark exciton. See the text for more details.
Finally, Fig. 16 shows radiative lifetimes of bright and
dark excitons calculated with use of Eq. 3 and Eq. 4. Re-
ported lifetimes are similar for both bright excitons, hence
τBE ≈ τBE1 ≈ τBE2 with again apparently normal-like dis-
tribution and with the mean value of 0.93 nm, thus compa-
rable 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 com-
ments, since lifetime is inversely proportional to optical activ-
ity (Eq. 2 and Eq. 3), thus the inverse of normal distribution
on Fig. 15 will lead to (in principle
q quite complicated) inverse
2
FIG. 15. Histograms of bright exciton states BE1 and BE2 optical Gaussian distribution [93] ∝ xλ3 exp(− λ(x−µ) 2xµ2 ). However,
activities, showing very similar and relatively narrow distributions. since the bright exciton optical activity has much larger mean
Dashed lines are plotted assuming a normal distribution. 2
value (approximately 40 Å ) than its standard deviation (of
2
approximately 1 Å ), thus the inverse Gaussian distribution re-
sembles regular Gaussian distribution. Fit to this dependence
two emission lines is possible. Again, accounting for shape
is shown on Fig. 16 (a) as black dashed line, with λ = 1541
elongation or faceting in a particular experimental realization,
and µ = 0.9284 ns, and such large λ being the reason [93]
together with alloy randomness could very likely change these
why the inverse Gaussian distribution appears very similar to
statistics.
the regular normal distribution.
Moreover, there is still little experimental data regarding
The distribution of dark exciton lifetimes is different, with
dark exciton spectra, since measurements are usually per-
average (mean) τDE value equal to 39 ms, and different distri-
formed on a single quantum-dot sample only. Our calculation
bution. Since emission intensity (Eq. 2) and lifetimes (Eq. 3)
indicates, that for alloyed quantum dots one can cherry-pick
are inversely proportional, the weak dark exciton optical in-
a sample with DE1 polarized along [110] or [110], with DE1
tensity, as shown earlier in Eq. 9 and Eq. 13, corresponds to
being the dominant line, or vice-versa. Therefore, more ex-
long lifetimes. Moreover since dark exciton optical activity
perimental research on the subject is needed.
resembles the exponential exp(−x/ hxi) distribution, math-
ematically equivalent to Gamma distribution [94] with k=1,
then the inverse should have the distribution in the following
V. RADIATIVE LIFETIMES
form x12 exp (− hxi /x) shown as a dashed line in Fig. 16 (b).
Despite it over-simplicity, the inverse gamma distribution qual-
In Section IV we have studied the dark exciton optical activ- itatively well describes the distribution of dark exciton lifetime
ity, therefore for comparison Fig. 15 shows histogram of both with no quantum dots with lifetime below 6 ms and with a long
bright exciton states optical activities which are apparently tail of quantum dots with lifetimes reaching up to 300 ms. Such
2
normally distributed around approximately 39.5 Å , corre- extremely long lifetime should be taken with great care, since
sponding to mean optical dipole moment of 0.63 nm, with we consider only the radiative contribution, obtained within
rather small variance corresponding to dipole moments from tight-binding Hamiltonian and with other approximations in-
0.6 to 0.66 nm, thus very close to typical experimental results cluded. [66, 95] We should also note that contrary to bright
(e.g., 0.59 nm determined in Ref. [74]). excitons (and as a direct result of differences in their optical12
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 recombi-
nation. Since τDE2 τDE1 , the overall DE radiative lifetime
is limited by recombination through emission from DE2. Ad-
ditionally, generally one observes that τDE ≈ τDE2 , which
is consistent with the above-mentioned finding that in an en-
semble of quantum dots considered in this work higher energy FIG. 17. Histograms of (a) the bright exciton splitting, and (b) dark
dark exciton state has (on average) larger optical activity for exciton splitting calculated when accounting for the s shell only in
both out-of-plane and in-plane polarizations, although a siz- the configuration-interaction procedure.
able 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 C2v quantum dots, and it ori-
gins from the valence-band mixing and lattice anisotropy, the
in-plane emission in C2v systems is forbidden by symmetry. FIG. 18. Histogram of (a) 2θ1 and (b) 2θ2 angles using the notation
Contrarily, in an alloyed C1 system dark excitons can emit of Eq. 17.
in-plane polarized light. There are two contributions to this
luminescence. One is related to non-vanishing matrix ele-
ments of the optical transition dipole moment, and the second
is related to exchange mixing of bright and dark configurations comparison Fig. 17 shows the (a) BES and (b) DES calculated
in the configuration-interaction Hamiltonian. Dark exciton when accounting for the s-shell only. In other words, Fig. 17
optical spectra can significantly vary in the ensemble, from (a) and (b) shows histograms of δ1 and δ2 exchange integrals,
nanostructures having virtually vanishing oscillator strength, using notation of Eq. 6.
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 Appendix B: Phases of dark and bright exchange integrals
building blocks of novel quantum devices.
Fig. 18 shows the histogram of exchange interaction phases
ACKNOWLEDGMENTS
θ1 and θ2 angles using the notation of Eq. 17. Should alloying
be neglected (in the C2v case), φ1 = 2θ1 would be equal
exactly 180°, and θ2 to 270° revealing 90° phase difference
The author acknowledges the support from the Pol- between dark and bright exciton. As seen in Fig. 18 when
ish National Science Centre based on Decision No. alloying is accounted for, this relation is held approximately,
2018/31/B/ST3/01415. The author would like to thank Michał with strong bright exciton phase randomization.
Gawłeczyk for reviewing the manuscript and valuable com-
ments.
Appendix A: BES and DES for s-shell only Appendix C: Two channels of dark excitons optical activity
Whereas Fig. 3 in the main text shows BES and DES his- For DE states, Eq. 3 for the s-shell case only, and in-plane
tograms for CI treatment including s, p, and d shells, for (x, y) polarizations, (and skipping energy index for simplicity)You can also read
