# Optical properties and dynamics of direct and spatially and momentum
indirect excitons in AlGaAs/AlAs quantum wells

Dabrówka Biegańska  Department of Experimental Physics, Faculty of Fundamental
Problems of Technology, Wrocław University of Science and Technology, Wybrzeże
Wyspiańskiego 27, 50-370 Wrocław, Poland<EMAIL_ADDRESS>Maciej Pieczarka  Department of Experimental Physics, Faculty of Fundamental
Problems of Technology, Wrocław University of Science and Technology, Wybrzeże
Wyspiańskiego 27, 50-370 Wrocław, Poland  Krzysztof Ryczko  Department of
Experimental Physics, Faculty of Fundamental Problems of Technology, Wrocław
University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370
Wrocław, Poland  Maciej Kubisa  Department of Experimental Physics, Faculty of
Fundamental Problems of Technology, Wrocław University of Science and
Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland  Sebastian
Klembt  Julius-Maximilians-Universität Würzburg, Physikalisches Institut and
Würzburg-Dresden Cluster of Excellence ct.qmat, Lehrstuhl für Technische
Physik, Am Hubland, 97074 Würzburg, Germany  Sven Höfling  Julius-Maximilians-
Universität Würzburg, Physikalisches Institut and Würzburg-Dresden Cluster of
Excellence ct.qmat, Lehrstuhl für Technische Physik, Am Hubland, 97074
Würzburg, Germany  Christian Schneider  Institute of Physics, University of
Oldenburg, D-26129 Oldenburg, Germany  Marcin Syperek  Department of
Experimental Physics, Faculty of Fundamental Problems of Technology, Wrocław
University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370
Wrocław, Poland<EMAIL_ADDRESS>
###### Abstract

We present an experimental study on optical properties and dynamics of direct
and spatially and momentum indirect excitons in AlGaAs/AlAs quantum wells near
the crossover between $\varGamma-$ and $X$-valley confined electron states.
The time-integrated photoluminescence experiment at $T=$
$4.8\text{\,}\mathrm{K}$ revealed three simultaneously observed optical
transitions resulting from (a) a direct exciton recombination, involving an
electron and a hole states both located in the $\varGamma$-valley in the
quantum well layer, and (b) two spatially and momentum indirect excitons,
comprising of the confined electron states in the $X$-valley in the AlAs
barrier with different effective masses and quantum well holes in the
$\varGamma$-valley. The measured spatial extent, density dependence and
temperature dependence of the structure photoluminescence revealed
characteristics necessary to pinpoint the states’ nature and provided their
characterization crucial in future device design. Temporal dynamics observed
in the time-resolved photoluminescence measurement showed the complexity of
the capture and recombination dynamics, largely affected by nonradiative
processes, what is additionally critical in the system’s use. This solid state
platform hosting both direct and indirect excitons in a highly tunable
monolithic system can benefit and underline the operation principles of novel
electronic and photonic devices.

## Introduction

A discovery and recent interest in two-dimensional (2D) materials spark
remarkable progress in research on excitons and the use of excitonic
properties in optoelectronic devices. Particular attention has been paid to
spatially indirect excitons with Coulomb-bound electrons and holes
characterized by spatially separated wavefunctions. The strong dipolar nature
of such excitations allows for the observation of new exciting phenomena,
including a dipolar excitonic insulator [1], high-temperature superfluidity
[2], and high-temperature Bose-Einstein condensation [3, 4, 5], among others.
Moreover, due to their high dipole moment, prolonged lifetime, high diffusion
length and bosonic nature, as well as good control of the exciton properties
due to their sensitivity to an electric field and dielectric environment,
indirect excitons can benefit and underline the operation principles of novel
electronic and photonic devices [6, 7]. In this new wave of exciting
discoveries, the attention can be brought back to the relatively mature
AlGaAs/AlAs quantum well (QW) system in which similar indirect excitons can be
generated in a highly controllable environment. Surprisingly, the knowledge on
the AlGaAs/AlAs QW system and its excitonic properties is limited in the
literature [8, 9, 10, 11, 12], despite nearly 45 years of progress in
semiconductor technology.

What is important is that this system offers fabrication capabilities that are
still unreachable in 2D layered semiconductors. The use of III-V
semiconductors, in particular with AlGaAs alloy materials in monolithically
integrated structures is highly beneficial due to the high quality of
epitaxial growth, precise control over the structure parameters, and small
lattice mismatches necessary to achieve high quality factors of optical
microcavities. They can be easily integrated into multilayer monolithic
structures and are suitable for a wide range of well-developed microelectronic
processing techniques. On the other hand, the important drawback in developing
useful systems and studying new excitonic-based physical phenomena in these
materials is a low exciton binding energy, particularly when compared with 2D
materials such as transition-metal dichalcogenides or 2D perovskites. The
typical energies on the order of a few millielectronvolt are too small
compared to the thermal energy at increased temperatures, which results in the
exciton’s effective dissociation, preventing an elevated-temperature
operation.

Nevertheless, several attempts have been made to overcome this issue. One of
the approaches employs the $\varGamma-X$ band mixing [13, 14, 9]. As GaAs is a
direct bandgap semiconductor at the $\varGamma$ point, while AlAs is
characterised by an indirect bandgap with the global minimum of the conduction
band in the $X$ point of the Brillouin zone, the AlxGa1-xAs alloy is
characterized by a direct-to-indirect bandgap transition at a specific
proportion of the two binary semiconductors of around $x=0.24$ [15, 16]. Near
the $\varGamma-X$ crossover, the quantum mechanical mixing of the $\varGamma-$
and the $X-$valley electrons was reported to manifest in large enhancement of
the donor activation energies and the exciton binding energies [17]. At the
same time, the nearly resonant spatially separated levels open a possibility
for indirect excitons to emerge.

Near-resonant levels of $\varGamma$ QW electrons and $X$ AlGaAs states were
studied in single and multi-QW structures [18, 19, 20], as well as in
superlattices [13, 14, 21, 22, 23, 24, 25], showing the possibility of
tunnelling [19] and interactions [14] between the confined electron states.
Such interacting states have yet to be studied in more complex structures,
such as enclosed microcavities, where complex dynamics seems crucial in the
structure design and band structure engineering.

In this report, we bring the III-V-based system back to the spotlight,
uncovering the properties that can underline the operation of exciton-based
photonic devices. We studied the optical properties of AlGaAs/AlAs QWs
initially designed for room-temperature polaritonics when embedded in the
optical microcavity, operating close to the $\varGamma-X$ coupling regime. We
characterize the system prior to the coupling with photonic modes. Using
photoluminescence experiments supported by theoretical calculations within the
effective-mass framework, we unravel the nature of multiple resonances present
in the structure spectra and interpret two of the lowest energy states as
spatially- and momentum-indirect X-excitons. We characterize the density and
temperature evolution of all the energy states, as well as their temporal
decay and spatial diffusion. Our work points to the complex exciton and
carrier dynamics in this system, hindered by nonradiative processes and
further affected by localization effects. Such detailed investigation on the
interplay between the energy states is important to understand the resulting
complex dynamics in the full structure such as microcavity, under the
additional interaction with photonic modes, particularly of the dipolar
excitons. Investigating the system in a wide range of temperatures is
necessary to carefully design and understand future high-temperature devices.

## Results

### Steady-state photoluminescence spectroscopy

Figure 1:

a) Schematic illustration of the investigated quantum well (QW) structure. b)
Low temperature ($T=4.8\ K$) photoluminescence (PL) spectra of the
Al0.20Ga0.80As /AlAs QWs registered at quasi-resonant excitation, $E_{exc}=$
$2.00\text{\,}\mathrm{eV}$, and low excitation power
$P_{exc}\approx$$10\text{\,}\mathrm{\SIUnitSymbolMicro W}$. c) Scheme of the
band structure of one period of the structure with the single-particle levels
marked with dashed lines the and optical transitions indicated with arrows.
The solid black line represents the edge of the valence band (VB), while solid
grey and green lines show the edges of the conduction band (CB) in the
$\varGamma$ and $X$ valley, respectively.

The structure under study is schematically depicted in Fig. 1a). It consists
of twelve $9\text{\,}\mathrm{nm}$-wide Al0.20Ga0.80As QWs, separated by
$4\text{\,}\mathrm{nm}$ AlAs barriers, distributed in three stacks of four,
comprising the active part of the structure. Good isolation of each QW,
provided by the barriers, ensures negligible coupling of confined states
between neighbouring QWs. The active part was monolithically integrated on a
distributed Bragg reflector (DBR), enhancing emission involving confined QW
states. More details on the sample can be found in the Methods section.

To probe the fundamental excitations in this QW system, we employ the steady-
state photoluminescence (PL) experiment (see Methods). Figure 1b) displays the
measured low-temperature ($T=$ $4.8\text{\,}\mathrm{K}$) PL spectrum of the
AlGaAs/AlAs QW, registered under a quasi-resonant energy excitation at
$E_{exc}=$ $2.00\text{\,}\mathrm{eV}$ and an excitation power of $P_{exc}=$
$10\text{\,}\mathrm{\SIUnitSymbolMicro W}$. The PL spectrum consists of three
well-resolved features, with enhanced visibility due to the DBR. The features
originate from the carrier recombination between QW-confined states. The
highest-energy and highest-intensity PL band located at
$1.8472\text{\,}\mathrm{eV}$ corresponds well to the Coulomb-correlated
electron-hole (exciton) recombination in the vicinity of the $\varGamma$
valley in the Brillouin zone. The transition involves the lowest-lying
$\varGamma$-electron state and the topmost $\varGamma$-hole state confined in
the AlGaAs QW. The assignment has been suggested by the energy match with the
calculations performed during the initial design of the sample [26], as well
as the PL band intensity dominating the spectrum. Therefore, it should reflect
the recombination of momentum- and spatially direct $\varGamma$-exciton
(annotated simply as $\Gamma$), having a high optical transition probability.

The presence of two PL bands located energetically below the $\Gamma$ exciton
is initially surprising. They are centred at energies of
$1.8007\text{\,}\mathrm{eV}$ and $1.7843\text{\,}\mathrm{eV}$ respectively,
redshifted by nearly $47\text{\,}\mathrm{meV}$ and $63\text{\,}\mathrm{meV}$
from the direct $\Gamma$ exciton transition. Large energy separation with
respect to $\Gamma$ excludes their identification as charged-excitons or
multi-exciton recombination since these are typically characterized by
sub-$10\text{\,}\mathrm{meV}$ binding energies [27, 28]. The defect-assisted
recombination would likely appear in a spectrum as a single broad transition
instead of two clearly resolved bands, resembling the excitonic transitions in
the AlGaAs/AlAs QW [29]. Therefore, we attribute the observed spectral
features to the recombination of excitons involving the $X$-valley electrons
and $\varGamma$-valley holes, observed previously in other AlGaAs
nanostructures[11, 20, 10]. We further investigate the nature of these
transitions in the following sections.

### Calculations

calc To elucidate the nature of the two optical transitions energetically
below the $\Gamma$ exciton and to provide arguments for their identification,
the QW band structure has been calculated within the effective mass
approximation. Fig. 1c) shows the band alignment for Al0.20Ga0.80As QW
material and the AlAs barriers in the conduction (CB) and valence (VB) bands,
including the electrons from both the $\varGamma$ and $X$ valleys of the
Brillouin zone. It is important to note that while in the $\varGamma$ valley
the CB profile (Fig. 1c), solid grey line) creates a confinement potential for
electrons in the QW material, the confinement for $X$-valley electrons (Fig.
1c), solid green line) is created in the AlAs barrier [11, 30, 31]. Holes can
be confined only within the $\varGamma$-valley in the QW material [11]. For
clarity, only the topmost heavy hole ($hh$) band is presented, and unstrained
materials are considered, in good agreement with the experimental
implementation.

The band alignment is the basis for calculations of the single-particle
confined states for $\varGamma$ electrons ($e_{1}$) and holes ($hh_{1}$), and
$X$ electrons ($x_{x,y}$, $x_{z}$), as schematically depicted in Fig. 1c). The
presence of two $X$-electron confined states results from the $X$-electron
effective mass anisotropy, well known for the AlAs material [32, 33, 13]. As
the quantization occurs in the growth direction perpendicular to the QW plane
(the $z$-direction, [001] in the crystal structure), two $X$-electron masses
need to be taken into account in calculations, in contrast to the isotropic
$\varGamma$ conduction band [31, 13]. This results in a doublet of states for
transverse, $x_{x,y}$, and longitudinal, $x_{z}$, electrons in the $X$ valley.
Their masses differ by a factor of $0.2268$ in the AlAs layer (with a higher
value of the longitudinal mass) [34], yielding the energy separation of
approximately $28\text{\,}\mathrm{meV}$ between the calculated energy levels
in the doublet. Our calculations do not include strain effects that can lead
to further state separation [35].

$\varGamma$ and $X$ electrons in confined states can interact with $\varGamma$
holes confined in the QW layer via Coulomb interactions, forming spatially and
momentum direct and indirect excitons. In our calculations, we estimated the
Coulomb corrections to single-particle energies using the Rydberg formula (see
Methods). This simplified approach is likely to be an overestimation of the
actual electron-hole Coulomb coupling for the indirect excitons due to the
spatial separation between carriers [36, 34]. On the contrary, the binding
energy of the $\Gamma$ exciton is underestimated due to the two-dimensional
nature of the QW states [31]. Additionally, the $\varGamma-X$ coupling
(neglected in our calculations) may affect the exciton energies [17]. More
accurate evaluations of the exciton energies require complex self-consistent
calculations, which are beyond the scope of this work [37, 36, 38].
Nevertheless, our simplified approach results in good agreement with
experimental observations, as discussed below.

The calculated band structure and the energies of the optical transitions are
confronted with the results of the PL measurement in Table 1. The data shows
that the Coulomb-corrected energy difference between the $e_{1}$ and the
$hh_{1}$ states in the $\varGamma$-band corresponds very well to the spectral
position of the most intensive PL band identified as $\Gamma$. As expected,
the type-I confinement and the direct nature of the transition in the real and
in the momentum space are linked to its high optical intensity. More
importantly, the calculated energies of transitions involving the $x_{x,y}$
and $x_{z}$ states in the $X$-valley and the $\varGamma$-valley heavy hole
state (marked as $X_{X,Y}$, and $X_{Z}$ in Fig. 1c)) are in good agreement
with the PL band energies observed below the $\Gamma$. Both $X_{X,Y}$, and
$X_{Z}$ transitions are nominally indirect in the real and momentum space,
which should lead to a low transition probability. However, several effects
can increase this value. It is important to note that as the translational
symmetry is broken in the growth direction ($z$), the $X_{Z}$ recombination is
allowed without phonon assistance, regardless of its indirect character in the
momentum space, due to the weakening of the momentum selection rules [39, 35,
25, 31]. Translational invariance of the Hamiltonian (hence the momentum
selection rules) should not be broken in the plane of the structure, resulting
in the need for phonon presence for the $X_{X,Y}$ transition to occur [39, 35,
25, 31]. The excitonic states can therefore be described as pseudo-direct in
the case of $X_{Z}$ and indirect in the case of $X_{X,Y}$. Notably, both
remain indirect in the real space.

Additionally, it is known that in the energetic vicinity of the electron
states of the $\varGamma$ and $X$ valleys in the Al-rich AlGaAs semiconductor
compounds, the $x_{x,y}$ and $x_{z}$ states can have an admixture of the
$e_{1}$ states, which increases the transition probability (this effect is not
included in our calculations) [37, 9, 31, 30]. What is more, the $X$-valley
electrons can be shifted towards the AlGaAs/AlAs heterointerface due to the
Coulomb interaction with holes confined in the QW layer, the localization at
interface inhomogeneities (caused by chemical content fluctuations), and local
electric field or strain fluctuations, which increases the overlap integral
with the $\varGamma$ valley hole states [9, 19, 39]. Interface inhomogeneities
are expected to have the strongest influence on the $x_{z}$ electron states,
as their effective mass is the largest [13] (effect visualized in our spatial
diffusion measurements, described in the next section).

Exciton |  PL peak energy $(4.8\ K)$ |  Calculated single-level transition energy $E$ |  Calculated exciton binding energy $E_{Ry}$ |  Calculated exciton recombination energy $E-E_{Ry}$
---|---|---|---|---
$\Gamma(e_{1}-hh_{1})$ |  1.8472 eV |  1.8521 eV |  4.8 meV |  1.8474 eV
$X_{X,Y}(x_{x,y}-hh_{1})$ |  1.8007 eV |  1.8130 eV |  9.5 meV |  1.8062 eV
$X_{Z}(x_{z}-hh_{1})$ |  1.7843 eV |  1.7847 eV |  6.8 meV |  1.7752 eV
Table 1:  Measured and calculated energies of the three investigated optical
transitions, including the energies calculated from single-particle states
($E$), calculated excitonic correctons ($E_{Ry}$) and the transition energies
at $T=$ $4.8\text{\,}\mathrm{K}$.

### Excitation power-dependent photoluminescence

Figure 2:  a) Excitation power-dependent photoluminescence (PL) spectra at
$E_{exc}=$ $2.00\text{\,}\mathrm{eV}$ and $T=$ $4.8\text{\,}\mathrm{K}$. b)
Change of the PL peak position with excitation power. c) Evolution of the PL
intensity (represented as fitted peak area) of three respective bands as a
function of $P_{exc}$. Dashed lines show the power-law fits to the
experimental points at low excitation powers (in the linear growth regime),
with slopes of $\gamma=$ $1.009(5)$, $0.997(5)$ and $0.99(1)$ for $\Gamma$,
$X_{X,Y}$ and $X_{Z}$ respectively.

The PL emission bands are examined as a function of the excitation power,
$P_{exc}$, providing additional information on the nature of observed emission
and the underlying optical transitions. Figure 2a) presents the spectra
evolution with increasing $P_{exc}$. Three PL maxima, corresponding to the
$\Gamma$, $X_{X,Y}$, and $X_{Z}$ excitons, were fitted with Gaussian profiles
and the extracted energies are shown in Fig. 2b). One can observe that the
middle-energy peak (corresponding to the $X_{X,Y}$ exciton) becomes
unresolvable from the spectra under the strongest pumping. In contrast to the
other two states, its energy blueshifts as a function of excitation power,
what can be attributed to its dipolar nature and the repulsive interactions
between quasiparticles, as was observed before for spatially indirect
excitons[11, 40, 41]. Contrarily, both $\Gamma$ and $X_{Z}$ exciton energies
present a small redshift with increasing excitation power, a behaviour not
expected for indirect states. It is important to note that in the case of the
$X_{Z}$ excitons, their ground state nature makes them more prone to
inhomogeneities, which affect their lifetime and transport properties. Higher
probability of the transition due to the weakened momentum selection rules,
shorter decay and higher localization can hinder their build-up, necessary for
the repulsive interactions to be observed. Small redshift of the $\Gamma$ and
the $X_{Z}$ PL bands at high excitation powers indicate the sample heating
under the strongest pumping.

Figure 2c) shows the spectrally integrated emission intensity of the three
investigated bands as a function of $P_{exc}$, plotted in the log-log scale.
Experimental points within the low excitation power range were fitted with the
power law function $I\propto P_{exc}^{\gamma}$, where the $\gamma$ value
depends on the recombination mechanism [42, 43]. For the defect-assisted mono-
molecular recombination of photo-injected electrons or holes the coefficient
is expected to be lower than 1. However, when the recombination is governed by
the mixture of free and bound exciton annihilation, $1<\gamma<2$. Our fitting
procedure yielded almost the same $\gamma$ parameter for all three PL bands.
Extracted values $\gamma=$ $1.009(5)$, $0.997(5)$ and $0.99(1)$ for $\Gamma$,
$X_{X,Y}$ and $X_{Z}$ respectively, clearly suggest that all the observed
optical transitions have excitonic character in the considered low excitation
power range. The superlinear growth at increased $P_{exc}$ (see Fig. 2c)
suggests a dissociation of excitonic complexes due to the sample heating (as
indicated by a small energy redshift) and screening of Coulomb interactions
when subject to the strongest pumping (hence at the largest carrier densities)
[44].

### Temperature-dependent photoluminescence

temp

Figure 3: a) Temperature-dependent photoluminescence (PL) spectra at
$E_{exc}=$ $2.00\text{\,}\mathrm{eV}$ and $P_{exc}=$
$150\text{\,}\mathrm{\SIUnitSymbolMicro W}$. b) Temperature-induced energy
shift of the PL bands. Points show the band energies extracted from fitting
the spectra in a) with Gaussian profiles, open points mark the energies
extracted from the spectra maximal intensity values and stars indicate the
excitonic transitions calculated within the effective mass approximation.
Dashed curves are the Varshni fits to the temperature dependences above
$100\text{\,}\mathrm{K}$.

To further characterize the studied system’s excitonic features, we look at
the temperature evolution of the PL spectra in the wide temperature range,
with results presented in Fig. 3. As the temperature $T$ increases, the energy
of all resonances redshifts, as seen in Figs. 3a)-3b). At
$T\approx$$80\text{\,}\mathrm{K}$, the $X_{X,Y}$ transition becomes
unresolvable, while the neighbouring $X_{Z}$ starts to dominate the spectrum.
We highlight the fact that spectra presented in Fig. 3a) are not normalized,
and the intensity of $X_{Z}$ emission strongly increases at
$T>$$100\text{\,}\mathrm{K}$, reaching the maximum at $\sim
200\text{\,}\mathrm{K}$. We hypothesize that the intensity increase originates
from the rising phonon density, making both the radiative transition from the
$X_{Z}$ exciton and the nonradiative transitions between $e_{1}$, $x_{x,y}$,
and $x_{z}$ electron levels more probable. Relaxation of electrons to the
$x_{z}$ state can be additionally increased by the decrease in energy
separation between the $\Gamma$ and $X$ states with temperature (due to the
different temperature variation of the $\varGamma$ and $X$ energy gaps in
AlGaAs and AlAs layers respectively [45, 25]), as well as by the quench of the
$X_{X,Y}$ emission, making the inter-level transfer more effective. However,
we also note, that absorption and effectiveness of the excitation strongly
differ with temperature as the energy gaps shift, with $E_{exc}$ crossing the
band gap energy of the Al0.40Ga0.60As spacer in the structure at $T$$\sim
220\text{\,}\mathrm{K}$, while the pumping energy in our experiment was set
constant ($E_{exc}=$ $2.00\text{\,}\mathrm{eV}$). Hence, no definite
conclusions based on the intensity evolution can be made. We note that the
efficient high-temperature PL highlights the potential use of the structure in
photonic devices at elevated temperatures, possibly up to room temperature.

Additionally, temperature dependences of the transition energies were fitted
using the Varshni formula: [46]
$E_{\mathrm{t}}(T)=E_{\mathrm{t}}(0)-\alpha T^{2}/\left(T+\beta\right)$ where
$E_{t}$ is the transition energy, and $\alpha$ and $\beta$ are fitting
constants related to the used semiconductor materials. The fitting results are
presented in Fig. 3b) with dashed lines. Extracted $\alpha$ and $\beta$
constants ($\alpha=$
$7.97\text{\times}{10}^{-4}\text{\,}\mathrm{eV}\text{\,}{\mathrm{K}}^{-2}$ and
$\beta=$ $399\text{\,}\mathrm{K}$ for the $\Gamma$ state, and the $\alpha=$
$6.71\text{\times}{10}^{-4}\text{\,}\mathrm{eV}\text{\,}{\mathrm{K}}^{-2}$ and
$\beta=$ $425\text{\,}\mathrm{K}$ for the $X_{Z}$ state) match well the
reported material parameters of GaAs and AlAs [45, 47]. Curves were fitted to
the experimental points above $100\text{\,}\mathrm{K}$, as at low temperatures
the dependences deviate from the Varshni model. Transition energies lower than
the theoretical values point to the likely additional localization effect and
Coulomb correlations. At low temperatures, both $\Gamma$ and $X_{Z}$ excitons
can be trapped, e.g. by spatial inhomogeneities of the QW interfaces or the
chemical composition fluctuations, acting as localization centres, when the
thermal energy is insufficient to overcome the energy potential minima [20,
48]. Localisation effect is also affecting the diffusion of the excitonic
complexes, showing the limited spatial extent of the low-temperature PL
emission of both $\Gamma$ and $X_{Z}$ states, contrasted with the increase of
the $X_{Z}$’s diffusion at elevated temperatures, as presented and described
in section diffusion, below. Localization energies suggested by the difference
between the Varshni curves and the measured exciton emission at low
temperatures are around $6\text{\,}\mathrm{meV}8\text{\,}\mathrm{meV}$.

Finally, we estimate the expected exciton recombination energies at room
temperature, using the same methods as described in the calc section. In
contrast to previous estimations [26], we verify the system to be indirect
(with the type-II transition from the quantized X-valley state in the barrier
to the first heavy hole QW energy state as the lowest energy transition) in
the whole studied temperature range up to the room temperature. At
$300\text{\,}\mathrm{K}$, the exciton energies estimated from our calculations
are $1.751\text{\,}\mathrm{eV}$, $1.730\text{\,}\mathrm{eV}$ and
$1.6992\text{\,}\mathrm{eV}$ for the $\Gamma$, $X_{X,Y}$ and $X_{Z}$
transitions respectively, matching nearly perfectly the PL resonances observed
in the experiment (see Fig. 3b), open stars).

### Exciton diffusion

diffusion

Figure 4: Spatial extent of the photoluminescence intensity, showing diffusion
profiles of the three studied excitons, measured at low excitation power
($P_{exc}=$ $10\text{\,}\mathrm{\SIUnitSymbolMicro W}$) and the temperature of
$T=$ $4.8\text{\,}\mathrm{K}$ (solid lines) and $T=$ $140\text{\,}\mathrm{K}$
(dashed lines). The $X_{X,Y}$ exciton is not visible at elevated temperatures.

The assignment of the observed QW transitions is further supported by the
measured spatial profiles of the time-integrated PL emission at low
temperatures and low excitation power density. The spatial extent of the PL
signal of the three observed transitions at $T=$ $4.8\text{\,}\mathrm{K}$ is
displayed in Fig. 4 (solid lines). One can see a clear difference between the
emission profiles of the studied states. The $X_{X,Y}$ excitons (the middle-
energy-peak in Fig. 1b)) travel further away from the excitation spot than the
$\Gamma$ excitons before recombining, with their emission profile extending
for more than $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The $X_{Z}$
excitons are characterized by the smallest in-plane diffusion profile at low
temperatures, even slightly smaller than that of the $\Gamma$ excitons.
Extended diffusion profiles are commonly observed for indirect excitons,
having high momentum, and often being characterized by long radiative
lifetimes, thus propagating macroscopic distances before recombination [49, 4,
50]. Several factors can influence the diffusion, including the in-plane
effective mass, phonon density, exciton-phonon interaction and spatial
localization of the states, as well as the efficiency of a non-radiative
recombination [51, 52, 49]. The need for the phonon assistance in the
$X_{X,Y}$ transition affects its recombination probability, hence the
diffusion length, explaining its stark distinction from the other two
profiles. In the case of the $X_{Z}$ excitons, the aforementioned breaking of
the momentum-selection rules makes them more similar to the $\Gamma$ excitons,
what is visualized in the narrower diffusion profile. Moreover, having the
largest effective mass in the z-direction, their transport properties might be
influenced by the spatial inhomogeneities of the QW interfaces, which act as
localization centres [4, 53], further affecting the diffusion.

The localization effect is clearly visualized by studying the diffusion
profiles at elevated temperatures (experimental curves measured at $T=$
$140\text{\,}\mathrm{K}$ shown in Fig. 4 with dashed lines). The diffusion of
the $X_{Z}$ state largely increases at elevated temperature and becomes
significantly broader spatially than that of the $\Gamma$ state, which remains
narrow. The observation can be linked to the nature of an indirect state and
the localization. Once the thermal activation overcomes the localization
energy from local potential minima, the diffusion range of the $X_{Z}$ exciton
is expected to be longer than that of the direct excitons. On the other hand,
it remains narrower than the $X_{X,Y}$ transition at low temperatures due to
the difference in selection rules and in the effective mass. Full
characterization of the temperature dependence is discussed in the previous
section, where more localization effects become apparent. Overall, observed
differences in the diffusion profiles support our interpretation of the nature
of the direct and indirect excitons, as similar behaviour has been observed
before in this material system [35].

### Time-resolved photoluminescence

Figure 5: a) Temporal evolution of the photoluminescence (PL) intensity of the
three studied transitions recorded at low temperature $T=$
$4.8\text{\,}\mathrm{K}$ and low excitation power $P_{exc}=$
$100\text{\,}\mathrm{\SIUnitSymbolMicro W}$. Solid lines show monoexponential
decay curves, fitted to experimental data (points), with the time constants of
$11\text{\,}\mathrm{ps}$, $17\text{\,}\mathrm{ps}$ and
$19\text{\,}\mathrm{ps}$ for $X_{X,Y}$, $X_{Z}$ and $\Gamma$ exciton
respectively. b) Temporal PL decays of three investigated transitions with
increasing temperature. Excitation energy ($E_{exc}=$
$2.00\text{\,}\mathrm{eV}$) and power ($P_{exc}=$
$150\text{\,}\mathrm{\SIUnitSymbolMicro W}$) were set constant throughout the
series.

Finally, we explore the exciton recombination dynamics in the system. Figure
5a) shows the temporal evolution of the PL intensity for the $\Gamma$,
$X_{X,Y}$ and $X_{Z}$ transitions, following a $\sim 200\text{\,}\mathrm{fs}$
excitation pulse at low temperature ($T=$ $4.8\text{\,}\mathrm{K}$) and low
excitation power ($P_{exc}=$ $100\text{\,}\mathrm{\SIUnitSymbolMicro W}$).

All three transitions can be approximated by a mono-exponential decay with the
characteristic decay time constant of about $11\text{\,}\mathrm{ps}$ for the
$X_{X,Y}$ transition and $17\text{\,}\mathrm{ps}$ for the $X_{Z}$. We note
that such short decays are close to the temporal resolution of our detection
system (of $\sim 7\text{\,}\mathrm{ps}$), and are much smaller than the decay
times expected for indirect transitions [54, 9, 55, 39, 19, 56]. This
observation reveals the likely presence of nonradiative channels in our
system, dominating the measured dynamics and manifested by a shortening of the
PL decay. The $\Gamma$ exciton is characterized by a slightly larger $\tau$ of
around $19\text{\,}\mathrm{ps}$. However, its decay deviates from purely
monoexponential at early times, with noticeably longer rise time (see the blue
curve in Fig. 5a)). The elongated rise time for the $\Gamma$ exciton is
related to the inefficient capture of photo-generated carriers/excitons from
the barrier, which may also stem from non-radiative processes happening in
this layer, decreasing the overall capture probability. On the contrary, a
relatively short capture time for the $X$ valley can be driven by carrier
relaxation in the barrier, additionally enhanced by the carrier transfer from
the $\varGamma$ to the $X$ valley.

Furthermore, we studied the temporal evolution of the PL resonance intensity
when the temperature is increased and the results can be seen in Fig. 5b),
with corresponding time-integrated spectra presented in Fig. 3a). At elevated
temperatures, the decays of all resonances become multiexponential, with the
shorter component being on the same order of magnitude as the low temperature
and the longer component strongly increasing up to several nanoseconds. This
makes the total decay much longer, extending above the available
$2\text{\,}\mathrm{ns}$ temporal window. Importantly, the $X_{X,Y}$ is
characterised by the near resolution-limited decay for all temperatures at
which it is observed, which can be explained by the ultrafast transfer of
electrons to the true ground state, acting as an efficient nonradiative
channel for this state. As the transition probability of the $X_{X,Y}$ state
is lower due to the momentum-forbidden nature, the intra-level transitions
become more effective. It further visualises the complex dynamics within the
system, with possible transitions between closely neighbouring levels,
especially when phonons help to conquer the energy barriers and carry particle
momentum. Additionally, the temperatures corresponding to the stark elongation
of the $X_{Z}$ and $\Gamma$ emission ($\sim 80\div 100\ K$) match the
localization energies estimated from the transition energy evolution as
described in the section temp, pointing to the likelihood of localization
effects playing an additional role in the change of dynamics with temperature,
as it has been observed before [54].

## Summary and Discussion

In this work, we investigated fundamental optical excitations in the
AlGaAs/AlAs QWs near the regime of $\varGamma$-$X$ valley coupling for
confined electrons, using the time-integrated and time-resolved
photoluminescence. Our experimental characterization is supported by
theoretical calculations within the effective-mass framework.

The low-temperature PL experiments reveal three clearly resolved optical
transitions, which we interpreted as direct and momentum- and spatially-
indirect excitons. The strongest PL feature is attributed to the $\Gamma$
exciton recombination. It has been previously observed in the microcavity-
embbedded AlGaAs/AlAs QW [26] and its transition energy matches very well the
calculated one. It dominates the spectrum at low temperature and low photo-
excitation, as it is expected form a direct exciton transition of a high
transition probability.

The other two transitions below the $\Gamma$ exciton have not be thoroughly
investigated in similar QW structures. Our study suggests their indirect
nature and the $X$-valley-electrons origin. The middle energy transition,
pinpointed as $X_{X,Y}$, have features corresponding well to the momentum and
spatially indirect exciton, with the $x_{x,y}$-electron constituent. This
state is expected to be purely indirect in space and in momentum, hence
initially forbidden, yet allowed by a random potential which includes all
wavevectors [39]. It does not mix with the $\Gamma$ state by the structure
potential [39] (in contrast to $X_{Z}$). In our experiments this state is
characterized by the largely extended diffusion, and its energy blueshifts
with density, as it is expected from dipolar species, affected by repulsive
interactions. In both of those observations $X_{X,Y}$’s characteristics
clearly differ from the other two states, as one can expect from their
momentum-direct (or pseudo-direct) nature. At the highest excitation powers
and increased temperatures it becomes unresolvable from the spectra,
overwhelmed by the other two transitions. This can also be pinpointed to its
indirect nature, as the momentum-selection rules make its recombination the
least probable. With increased phonon influence, the $x_{x,y}$ electrons can
clearly relax or scatter into the other two states, preventing the $X_{X,Y}$
exciton radiative recombination (with $\Gamma-X_{X,Y}$ mixing allowed by the
potential fluctuations due to interface roughness [19]).

Finally, we have identified the ground state of this system as the $X_{Z}$
exciton, effectively recognizing the whole structure to be of the type-II
(spatially indirect) and pseudo-direct in momentum. Symmetry (the structure
potential) allows $X_{Z}$’s recombination without the assistance of phonons
and its mixing with the $\Gamma$ state [39, 19, 20]. Accordingly, in all of
our measurements its characteristics are similar to the highest $\Gamma$
state, e.g. it presents similar diffusion profile at low temperatures and
redshifts in energy at increased densities, due to the unintentional sample
heating. The largest effective mass makes it the lowest energy state in the
system [39]. It also makes it prone to the interface roughness and
fluctuations and becoming localized. One can clearly see this effect in its
observed narrow diffusion profile, the profile’s expansion at increased
temperatures, in the clear deviation from the Varshni curve at low
temperatures, but also in a significant increase of its luminescence and decay
time at elevated temperatures, as the excitons that are localized at low
temperature become mobile at elevated temperatures [20]. The temperature of
this increase in the decay constant corresponds well to the estimated
localization energy of around $7\text{\,}\mathrm{meV}$.

Our clear identification and characterization of these states are crucial in a
careful design of future devices, where specific excitonic states are at the
core of interest. Our system offers a superb monolithic platform in which
three species of excitons, both direct and indirect, can be exploited. As a
clear example, such a QW system can be integrated inside a microcavity, where
the dipolar excitons coupled to the cavity field can provide new
functionalities and device applications.

## Methods

Methods

### Sample growth

The investigated structure was grown by molecular beam epitaxy on a nominally
undoped (001) GaAs substrate. The structure consists of twelve
$9\text{\,}\mathrm{nm}$-wide Al0.20Ga0.80As QWs, separated by
$4\text{\,}\mathrm{nm}$ AlAs barriers, distributed in three stacks of four.
The stacks were initially placed in a $\lambda/2$-AlAs cavity surrounded by
AlAs/Al0.40Ga0.60As distributed Bragg reflectors (DBRs), consisting of 28/24
mirror pairs in the bottom/top reflector, including $3\text{\,}\mathrm{nm}$
GaAs smoothing layers after each mirror pair. However, to characterize the QW
system, the top DBR has been etched away, so the QW part on a bottom DBR-
structure is studied in this work. The resulting structure is characterised by
an enhanced luminescence extraction efficiency. Additionally, it was verified
that no measurable signatures of remaining photonic modes in the spectral
vicinity of the QWs were observed. Thus, one can assume no photonic mode
coupling and focus on the QW emission. The structure with a top DBR mirror,
providing a full cavity system, has been presented elsewhere [26].

### Optical experiments

For the time-integrated (PL) and time-resolved photoluminescence (TRPL)
experiments, the structure was held in a helium-flow optical cryostat,
allowing for the sample temperature control in the range of
$4.8\text{\,}\mathrm{K}300\text{\,}\mathrm{K}$. The structure was excited by
$\sim 200\text{\,}\mathrm{fs}$-long laser pulses with the pulse central
wavelength of $620\text{\,}\mathrm{nm}$ ($E_{exc}=$
$2.00\text{\,}\mathrm{eV}$). The pulses were delivered by a synchronously-
pumped optical parametric oscillator with a $76\text{\,}\mathrm{MHz}$
repetition frequency, pumped with a Ti:Sapphire laser. The optical excitation
was focused on the sample surface via an infinity-corrected high-numerical-
aperture (NA) objective (NA=0.65). The PL/TRPL signal was then collected by
the same objective and directed to a $0.3\text{\,}\mathrm{m}$-focal length
monochromator for spectral resolution. The PL was registered by a thermo-
electrically-cooled Si-based electron-multiplied-CCD camera or by a Si-
cathode-based streak camera. The TRPL measurement system provided a time
resolution of $\sim 7\text{\,}\mathrm{ps}$. Spatial resolution in the time-
integrated spatial diffusion measurements was achieved using the two-
dimensional CCD camera [57, 58].

### Band structure calculations

The band structure calculations were performed within the effective mass
approximation. Due to the wide band gaps in the investigated materials the
conduction band was assumed parabolic near the $\Gamma$ and $X$ points and the
spin–orbit split-off (SO) energy is large enough to ignore the SO band. The
remaining valence bands (heavy and light hole) were described in the Luttinger
model [59]. The energies and wave functions of confined carriers were obtained
numerically by solving the Schrödinger equation using the method reported in
Ref. [60]. Standard material parameters [34] for GaAs and AlAs were used in
the calculation and linear interpolation was applied for Al0.20Ga0.80As. The
valence-band offset was calculated as $37\%$ of the difference in band gaps in
the well and barrier layers. The exciton binding energy was estimated from the
formula $E_{Ry}=\mu e^{4}/2\epsilon^{2}\hbar^{2}$, where $\mu$ is the in-plane
reduced effective mass and $\epsilon$ is the dielectric constant.

## Data availability

The datasets generated during and/or analysed during the current study are
available from the corresponding author on reasonable request.

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## Acknowledgements

D. B. and M. S. acknowledge financial support from the National Science Centre
Poland within the (Grant No. 2018/30/E/ST7/00648). The Würzburg group
acknowledges financial support by the German Research Foundation (DFG) under
Germany’s Excellence Strategy–EXC2147 “ct.qmat” (project id 390858490).

## Author contributions statement

D. B. conducted all the spectroscopic experiments and analysed the
experimental data, K.R. and M. K. performed the theoretical calculations, S.
K., S. H and C. S provided the QW structure, D. B., M. P., S.K., C. S, S. H.,
and M.S. analysed and discussed the results. D.B. wrote the first version of
the manuscript and prepared all figures. All authors reviewed the manuscript
to its final form.

## Additional information

The authors declare no competing interests.