Theoretical Study of Excitonic Complexes in GaAs/AlGaAs Quantum Dots Grown by Filling of Nanoholes

In this work, a theoretical study of the electronic and the optical properties of a new family of strain-free GaAs/AlGaAs quantum dots (QDs) obtained by AlGaAs nanohole filling is presented. The considered model consists of solving the three-dimensional effective-mass Schrödinger equation, thus providing a complete description of the neutral and charged complex excitons’ fine structure. The QD size effect on carrier confinement energies, wave functions, and s-p splitting is studied. The direct Coulomb interaction impact on the calculated s and p states’ transition energies is investigated. The behaviour of the binding energy of neutral and charged excitons (X− and X+) and biexciton XX versus QD height is studied. The addition of the correlation effect allows to explain the nature of biexcitons often observed experimentally.


Introduction
Optimizing the optical qualities of quantum dots (QDs) remains a challenge for researchers. For this purpose, new GaAs/AlGaAs semiconductor QDs have been grown using a new technique based on filling self-assembled nanoholes obtained by local etching of droplets [1,2]. is method provides QDs with particular structural properties such as unstrained, uniform, not very dense, with high symmetry, and of different shapes with reference to conventional strained QDs. Due to these characteristics, the optical properties of unstrained QDs are interesting for various increasing high-performance applications in optoelectronic devices, namely, lasers and solar cells [3,4], in addition to quantum cryptography [5].
In this context, we provide a modelling of the GaAs/ AlGaAs QDs' shape reported by Heyn et al. [2], compatible with available atomic force microscopy (AFM) images. e theoretical study in this paper will focus on the dependence of the confinement energy of the carriers, for the ground state s and state p, on the QD height (h QD ). e calculations are performed by adapting a simple configuration based on BenDaniel and Duke, Hamiltonian [6], which requires much computation time. e obtained results will be compared to the literature experimental results for validation. In addition, our theoretical approach will be compared with Graf et al.'s [1] model. ereby, we will then estimate the effect of the QDs' size on the binding energy of the excitonic complexes in terms of direct Coulomb interaction and correlation effects. erefore, the final alignment of excitonic states will be explained.

Theoretical Model
e electronic states of a QD strongly depend on the shape chosen for the dot and its symmetry. In the literature, unstrained self-assembled QDs have been modelled by different shapes such as cone [7,8], pyramid with a square base [9,10], and lens [11,12]. e electronic structure of QDs has been calculated by adopting various approaches such as the pseudo-potential model proposed by Williamson and Zunger [13,14], the strong bond model suggested by Lee et al. [15], and the formalism of the envelope function at several bands by Stier et al. [16].
In this paper, calculations are limited to the formalism of the envelope function with one band. is method consists in choosing a large quantization box, containing the studied physical system (QD), and having all the possible symmetries of the system ( Figure 1). is quantization box is a cylinder of radius R and height H. ese dimensions are chosen such as R � 4r nanohole and H � 4h nanohole to avoid side effects. e parameters r and h are, respectively, the radius and height of the QD (Figure 1). e Hamiltonian of each electron (e) or hole (h) is written in the form where m * e(h) and V e(h) are, respectively, the effective mass of the electron or hole and the potential for conduction and valence offset. r e and r h are the position vectors of electrons and holes inside the cylinder. e offset potential depends on the geometry of the QD; thus, we consider that the confinement potential has a Gaussian shape, and we describe it as follows: To access the energy levels of QDs, we used the matrix method initially proposed by Marzin and Bastard [8]. Due to the cylindrical symmetry of the QDs, the wave function is expressed as a Fourier-Bessel series written as where A nml are the coefficients to be determined and Φ nml are periodic and orthogonal functions which are given by where n, m, and l are integers and J m is the Bessel function of order m and k nm r is its n th root. e matrix element is defined as with Ω being the cylinder volume. e energy levels and wave functions are calculated using the parameters listed in Table 1. e AlGaAs nanoholes are modelled by holes of the same shape and dimensions as those of [1]. Nanoholes of radius r nanohole � 30 nm and depth h nanohole � 16 nm have been considered in our calculations. e excitonic energies of ground state s and excited state p are, respectively, denoted by E s X and E p X . eir expressions are given by where E g is the GaAs QD band gap energy. E i and HH i are the confinement energies of the electron and hole (i � 1 for ground state s and i � 2, 3, 4, ..., for exited states p, d, f, . . ., respectively). J s(p) eh is the electron-hole Coulomb energy, in the s or p state, treated in the perturbative approach, and it is written as with ε r being the relative dielectric constant of GaAs [17] and Ω e and Ω h representing the volume of the cylinder associated with electrons and holes, respectively. e recombination energies of the neutral exciton (E X ), charged excitons (E X + , E X − ), and biexciton (E XX ) are calculated as follows [18]: e binding energies of the trions Δ X + , Δ X − and the biexciton Δ XX , without correlation effects, are estimated within the framework of the Hartree-Fock approximation and defined by 2 Advances in Condensed Matter Physics e new binding energies of the trions Δ cX + , Δ cX − and the biexciton Δ cXX , with correlation effects, are defined as follows [18]: with δ cX , δ cX + , δ cX − , and δ cXX being the correlation energies which are written in terms of W ij in the following form: with where U n i m j ,00 ij  Figure 2, we have presented the first five calculated squared electron wave functions for a GaAs QD of 12.9 nm height. e plots show that the electron and hole wave functions are highly localized inside the GaAs QD. In Figure 3, we have represented the variation of electron (E n ) and heavy hole (HH n ) confinement energies for the states s (n � 1) and p (n � 2) as functions of the QD height h QD . From Figure 3, the confinement energies are very sensitive to the QD height. We observe a progressive decrease of these energies when h QD increases.

Confinement Energy and Wave Function. In
is behaviour is also observed for high excited states (n > 2) of the GaAs QD (Table 2). e electrons' confinement energies are higher than those of heavy holes due to their lower effective mass compared to that of holes.

Excitonic Energies.
It is known that exciton formation in a QD is different than in a bulk crystal. By comparing the QD size to the Bohr radius, it is possible to define three regimes: strong confinement regime, weak confinement regime, and intermediate confinement regime. In our calculations, we will employ the strong confinement approximation, where the electron-hole Coulomb interaction is considered as a small perturbation against the single-particle terms in the   Hamiltonian [9]. In our case, the nanoholes have a lateral shape with a Gaussian profile, their depth is typically 16 nm, and their radius is around 30 nm. e GaAs QDs have a lateral size similar to that of the nanoholes, but their height can vary between 3 and 12 nm. erefore, in view of the 15 nm exciton Bohr radius in GaAs, the confinement along the growth axis (z-axis) is stronger than the lateral confinement in the (xy) plane. Yet, the wave function spatial dependence in the growth direction and lateral direction is only slight, but it is significantly affected by the Coulomb interaction. us, J eh can be considered as a disturbance [9,19]. In Table 3 [20]. e impact of the direct Coulomb interaction is more pronounced for state s than for state p because of the small spatial extension of the wave function of state s compared to that of state p. From Table 3, we attribute the decrease of |J s(p) eh |, for h QD < 6.4 nm, to the overlapping integrals of the wave functions associated with the electrons and holes of states s and p. e minimum recovery is obtained for h QD � 6.4 nm. Similar behaviour has been observed in selfassembled InAs/GaAs QDs [20,21].
To investigate the accuracy of the numerical approach, the emission energies of states s and p are compared to those calculated by Graf et al. [1] and experimental data reported by Heyn et al. [2]. In Figure 4, we have represented the theoretical and experimental variations of the neutral exciton transition energies for the ground (E 1 X ) and first excited (E 2 X ) states as a function of h QD . In our approach, we neglected the N body effects. is choice is justified based on the results of Heyn et al. [2], who showed that photoluminescence (PL) peaks shift slightly towards red by about 2 meV when the excitation power was increased. is value remains very low compared to the confinement and Coulomb energies. Graf et al. [1] theoretically studied the optical properties of GaAs QDs. eir approach is based on the eight-band k.p model, considering N body effects. Although their model is sophisticated, they neglected the band curvature of the top surface of the QD and considered it flat.
is has a direct impact on the confinement energies of the carriers. However, an agreement is obtained between our theoretical results and experimental data reported by Heyn et al. as shown in Figure 4. is agreement demonstrates the suitability of our modelling approach.

Excitonic Complexes.
e binding energies of the excitonic complexes in this type of QDs have been calculated. ereby, more particular interest is given to the neutral exciton X, the biexciton XX, and the positively X + and negatively X − charged excitons. ese excitons have been the subject of several theoretical and experimental studies for different types of self-assembled QDs such as InGaAs/GaAs [21,22] and InAs/InP [23]. In these high-confinement systems, the bond energies can be determined via micro-PL spectroscopy on single dots to study the effect of size and shape on the bond energies. Unlike self-assembled InAs QDs, the addition of correlation effect, in strain-free GaAs QDs, helps to explain the binding nature of the biexciton, often observed. Figure 5 shows the variations of the correlation energies and the bond energies of the complex excitons, with and without correlation effect, as a function of QD height h QD . e degree of correlation is specific to excitonic complexes and is sensitive to the variation of h QD . e correlation effect for a biexciton is larger due to the higher number of charge carriers involved.
We also obtain |δ c (X)| < |δ c (X − )| < |δ c (X + )| < |δ c (XX)|, in agreement with the prediction established by Schliwa et al. [22]. By comparing the bond energies without and with correlation effect, we observe the formation of bound excitonic states. Indeed, we underline a transition from an unbound state to a bound state for the biexciton and for the negatively charged exciton when the correlation effect is introduced. e binding energies of the biexciton obtained via our model are between − 2.78 and − 5.01 meV. ese values are higher in magnitude than the experimental values (− 1.3 to − 2 meV) available in the literature [6,24] for GaAs QDs based on nanoholes' shape and size. However, our results agree with those of self-assembled III-V QDs, where it is well known that the binding energy of the biexciton varies between 1 and 6 meV [25,26]. e comparison between experimental and theoretical data is complex given the lack of atomic force microscopy data for the studied GaAs QDs. Our study provides a comprehensive understanding of the correlation and size effects on the final alignment of excitonic states in strain-free GaAs QDs.

Conclusion
We have theoretically studied GaAs/AlGaAs QDs obtained by filling AlGaAs nanoholes with shape and profile provided by AFM measurements. e effect of size on carrier confinement, wave functions, and s-p splitting has been studied. e confinement energies of electrons and holes are very sensitive to the quantum dot height h QD . e excitonic energies of the s and p states are calculated. e impact of the direct Coulomb interaction is more pronounced for state s than for p due to the small spatial extension of the wave function of state s compared to that of state p. An agreement is obtained between our theoretical results and experiment data of the literature, which indicates the suitability of our modelling approach. e behaviour of the binding energy of charged excitons (X − and X + ) and biexciton XX has been studied. e addition of the correlation effects allowed the formation of bound excitonic states as expected by experiments, and they are sensitive to the QD height. Our study provides a comprehensive understanding of the correlation and size effects on the final alignment of excitonic states in strainfree GaAs QDs.

Data Availability
e data used to support the findings of this study are included within the article.