Multi-excitons in GaAs/AlGaAs quantum dot arrays
Citation for published version (APA):de Groote, F. P. J. (2003). Multi-excitons in GaAs/AlGaAs quantum dot arrays. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR573935
DOI:
10.6100/IR573935
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CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN Groote, Frank Pieter Jannis de
Multi-Excitons in GaAs/AlGaAs Quantum Dot Arrays / by F.P.J. de Groote. – Eindhoven : Technische Universiteit Eindhoven, 2003. – Proefschrift.
ISBN 90-386-1715-1 NUR 926
Subject headings: semiconductors; excitons; multi-excitons; quantum dots; photoluminescence.
Trefwoorden: halfgeleiders; excitonen; multi-excitonen; fotoluminescentie. The work described in this dissertation was carried out at the COBRA Inter-University Research Institute on Communication Technology within the De-partment of Physics of the Eindhoven University of Technology. It was part of the research programme of the Dutch Foundation for Fundamental Research on Matter (FOM), which is financially supported by the Dutch Organization for the Advancement of Research (NWO).
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr. R.A. van Santen, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op
donderdag 4 september 2003 om 16.00 uur
door
Frank Pieter Jannis de Groote geboren te Nijmegen
prof.Dr. J.H. Wolter en
prof.Dr. C.M. Sotomayor Torres Copromotor:
1 Introduction 1
1.1 Short historic overview . . . 1
1.1.1 Device development . . . 1
1.1.2 Carrier confinement . . . 2
1.2 Quantum dots . . . 3
1.3 Quantum dot fabrication . . . 7
1.3.1 Stranski-Krastanov growth . . . 7
1.3.2 Hydrogen-assisted growth on a patterned substrate . . 8
1.4 Scope of this dissertation . . . 9
Bibliography . . . 11
2 Multi-exciton phenomena — Theory 17 2.1 Introduction . . . 17
2.2 Electronic structure . . . 19
2.3 Single-carrier confinement energies . . . 21
2.4 Coulomb and exchange interactions . . . 22
2.5 Filling of the quantum dot . . . 23
2.5.1 Complete carrier relaxation . . . 25
2.5.2 Incomplete carrier relaxation . . . 27
2.6 Conclusions . . . 30
Bibliography . . . 35
3 Multi-exciton phenomena — Experiments 39 3.1 Introduction . . . 39
3.2 Experimental set-up . . . 42 3.2.1 Excitation . . . 42 3.2.2 Sample mount . . . 45 3.2.3 Detection branch . . . 45 3.3 QD homogeneity . . . 47 3.4 Results . . . 51
3.4.1 Incomplete carrier relaxation . . . 53
3.5 Discussion . . . 55
3.5.1 Phonon bottleneck . . . 55
3.5.2 Fast hole relaxation . . . 56
3.6 Conclusions . . . 57
Bibliography . . . 59
Appendix A: Complete carrier relaxation . . . 63
4 Carrier diffusion in a corrugated quantum well 67 4.1 Introduction . . . 67
4.2 Experimental . . . 68
4.2.1 Confocal versus non-confocal arrangement . . . 71
4.2.2 Excitation beam alignment . . . 72
4.3 Diffusion and recombination — theory . . . 72
4.4 Model formulation and evaluation . . . 73
4.5 Determination of the diffusion constant . . . 78
4.6 Discussion . . . 84
4.6.1 Broad initial carrier distribution . . . 84
4.6.2 High ambipolar diffusion rate . . . 87
4.6.3 Effect of corrugation . . . 89
4.7 Conclusions . . . 89
Bibliography . . . 91
5 Time-correlated photoluminescence: a tool for carrier capture measurements 93 5.1 Introduction . . . 93
5.2 Theory . . . 95
5.2.1 Two-level rate equation . . . 95
5.2.2 Time-correlated photoluminescence . . . 97
5.2.3 Saturation effects . . . 99
5.3 Experimental . . . 102
5.3.1 Set-up . . . 102
5.3.2 Band structure . . . 102
5.3.3 Quantum well saturation . . . 103
5.4 Results . . . 105
5.4.1 Carrier capture and lifetime . . . 105
5.4.2 Relaxation . . . 107 5.5 Conclusions . . . 109 Bibliography . . . 111 Summary 113 Samenvatting 115 List of publications 117 Curriculum Vitae 119 Dankwoord 121
Introduction
1.1
Short historic overview
Over the past seven decades, semiconductor technology has evolved from an entirely new subject to an amazingly diverse and rich research field. Al-though semiconductor diodes exist since 1934, the technological breakthrough came with the invention of the point-contact transistor in 1947 (Bardeen and Brattain [1]), followed not long after by the junction transistor in 1948 (Shockley [2]). Today, the role that semiconductors play in modern life is exhaustive: there is virtually no electronic device operating without a semi-conductor component of some kind. With research for example dedicated to opto-electronics [3], ultrafast all-optical signal processing and quantum com-puting [4], the end of the technological advancements are by no means in sight.
1.1.1
Device development
Research is aimed at improving ‘traditional’ semiconductor components [5,6] on the one hand (higher speed, more powerful etc.), and developing new components [7] on the other. These fields partly overlap and share a common need for a thorough understanding of physical and material properties.
The first transistors were made of germanium (Ge) and later of silicon 1
(Si), the latter of which is the basis for most transistors and integrated cir-cuits. A shared disadvantage of these two materials is that they form in-direct semiconductors, which means that the bandgap minimum is not at k = 0. Consequently, the efficiency for light emission upon recombination of an electron-hole pair is very low. The use of direct semiconductors like GaAs has overcome this problem and has led to the development of impor-tant electro-optical devices like the light-emitting diode (LED) and the diode laser in 1962.
Novel production methods such as chemical beam epitaxy (CBE) and molecular beam epitaxy (MBE) made it possible to produce structures of unique purity and composition [8]. Using these techniques, the semiconductor composition can be changed within one atomic layer while maintaining near atomically flat interfaces. An important structure based on this production method is the quantum well (QW), consisting of one material sandwiched between two other materials.
1.1.2
Carrier confinement
A key feature in these new materials is carrier confinement (we speak of confinement when electrons or holes are trapped in a region with typical di-mensions ranging from a few nanometer to several hundred nanometer). In a QW, unlike bulk, electrons and holes are confined to a two-dimensional re-gion and are only free to move in-plane. Using special production techniques, the confinement can be extended to a one-dimensional region (quantum wire, QWR), or even a zero-dimensional region (quantum dot, QD). The capability to confine carriers is based on a difference in bandgap between two semicon-ductor materials that are brought in contact with each other. Free electrons and holes preferentially stay in the material with the lowest bandgap, where their potential energy is minimal.
The idea of reducing dimensionality—leading to increasing confinement— is visualised in the semiconductor structures shown in figure 1.1. This reduc-tion of dimensionality is directly reflected in the density of states, as shown in figure 1.2.
bulk quantum well quantum wire quantum dot
z y
x
Figure 1.1: Various semiconductor structures with increasing confinement: bulk, quantum well, quantum wire, quantum dot.
energy
density of states
bulk quantum well quantum wire quantum dot
Figure 1.2: Density of states for bulk material, a quantum well, a quantum wire and a quantum dot.
1.2
Quantum dots
Of all semiconductor structures, QDs comprise the ultimate in carrier con-finement, trapping electrons and holes in all three spatial dimensions. It is this fundamental property that gives rise to the discrete energy spectrum (see figure 1.2) which distinguishes QDs from other semiconductor structures.
The position of energy levels in a QD depends primarily on the QD size. This can be used to generate light at specific wavelengths using, for example, photo- or electroluminescence. Applications based on this effect can be found in QD lasers [9–11], biological agent indicators [12,13], etc. It is a direct con-sequence of the discrete energy spectrum that, based on the Pauli principle, each QD energy level can only store up to two carriers. When more carriers are added at a high enough rate to a QD, for example by photoexcitation, a process known as state filling takes place: first the ground state and then the higher excited state levels become occupied. When more than one
electron-hole pair (commonly called an ‘exciton’) occupies the QD, Coulomb and exchange interactions between the carriers will alter the QD energy levels. This accounts for the appearance of additional spectral lines when higher energy levels get occupied [14–16].
The appearance of a biexciton transition is reported in many studies. A ground state biexciton has a lower energy than two separate single excitons due to the additional Coulomb interactions. The biexciton binding energy is negative, usually a few meV [15,17,18], resulting in a spectral line appearing below the single exciton ground state transition. In some cases, however, a positive binding energy is reported, see for example [19] or [20]. When the excitation density is increased, and more excitons occupy the QD, additional lines appear in the spectrum due to the recombination of tri-exciton and four-exciton complexes. This is clearly shown in the work of Bayer et al. [21], see figure 1.3. A tri-exciton can recombine to either a ground state biexciton or an excited biexciton. This results in two additional spectral features, located on either side of the ground state single exciton transition. A similar situation appears with the four-exciton complex, which can recombine to the ground state tri-exciton or to the excited tri-exciton. For an increasing number of excitons that occupy the QD, more lines appear in the spectrum.
A single QD system with up to six excitons is reported by Bayer et al. in [15], see figure 1.4. The QDs are degenerate, which leads to so-called ‘hidden symmetries’ governing the filling of QD energy levels, analogue to Hund’s rules for atoms. Hawrylak [22] developed a multi-exciton model which systematically describes the different Coulomb and exchange interactions. His model is capable to assign the different multi-exciton spectra for a degenerate QD filled with up to six excitons. This spectrum shows only a small number of PL lines. On the other hand, the PL spectra of a non-degenerate QD are considerably more complicated [16], and also the description in terms of s- and p-shells is no longer applicable [19]. In this dissertation we use an extension of Hawrylak’s model in order to describe our non-degenerate QDs. It is one of the implication of a discrete energy spectrum in QDs that
Figure 1.3:Data by Bayer et al. [21] showing PL spectra for increasing excitation density. X → 0 denotes the ground state exciton transition. The ground state biexciton transition is indicated by X2 → X. X3 and X4 mark tri- and four-exciton complexes, while an asterisk indicates an excited state.
an electron that relaxes from a high to a low level should loose an amount of energy exactly equal to the energy difference between those two levels (usually several tens of meV). An obvious and fast way for an electron to loose energy is via optical phonon emission. However, electrons are predicted to have a very slow relaxation rate [23, 24] due to the fact that the energy of an optical phonon (approximately 36 meV in GaAs) in general does not match the inter-level energy difference. So far, however, this so-called ‘phonon bottleneck’ is claimed to be observed in only a few publications [25–29].
A detailed analysis of the QD energy levels is difficult due to the QD size distribution, present within any group of QDs in virtually every sample. This size distribution causes so-called inhomogeneous broadening of the spectral
Figure 1.4: Spectral emission from a QD filled with up to six excitons (Bayer et al. [15]). The investigated QD is degenerate, which leads to a description in terms of s- and p-shells.
lines that show up in photoluminescence experiments, and obscures the ef-fects of Coulomb and exchange interactions within the QDs. Inhomogeneous broadening can be avoided by single-dot spectroscopy, usually performed by masking the sample or by applying the relatively new near-field optical microscopy technique (NSOM) [30]. For the lowest energy levels, these ex-periments show the effect of state filling, and the influence of Coulomb and exchange interactions on the position of spectral lines [21,31–37]. A practical problem is that a long integration time (up to several hours) and/or a high excitation power is necessary to obtain enough signal.
1.3
Quantum dot fabrication
It is difficult to produce high-density homogeneous QD material, i.e. material in which all QDs have exactly the same size and composition and thus the same emission spectrum. In 1998 N¨otzel et al. for the first time succeeded in making such homogeneous QDs [38], arranged in 1-dimensional arrays and extending over a length of several micrometers. These length scales are large enough to allow processing of the areas that show size homogeneity, which is an important requirement for device applications. The fabrication of this sample differs in many ways from Stranski-Krastanov growth, a method often applied to produce high-density semiconductor QD samples. Both methods are described below.
1.3.1
Stranski-Krastanov growth
In Stranski-Krastanov (SK) growth, a thin semiconductor layer (the so-called ‘wetting layer’) is grown on top of a semiconductor with a larger bandgap and a different lattice constant (the barrier layer). This difference in lattice constant induces strain between the two materials. Spontaneous strain relax-ation lowers the potential energy of this system, and leads to the formrelax-ation of islands out of the wetting layer. When covered with more barrier mate-rial, these islands form the quantum dots. SK-growth is schematically shown in figure 1.5. QDs grown by this process are usually called ‘self-assembled’ quantum dots (SAQDs).
The advantage of SK-growth is that no ‘special’ preparation or patterning of the substrate is needed to produce quantum dots, and that the QD den-sity and size can be more or less manipulated by control of growth conditions such as initial wetting layer thickness, growth temperature and growth in-terruption. A major disadvantage is that the spontaneous formation of QDs leads to a size distribution of these QDs, which causes spectral features to be inhomogeneously broadened. This complicates a detailed analysis of the processes taking place inside the QDs, and limits the use in practical
appli-barrier layer
deposition of wetting layer
’island’ formation
barrier layer
Figure 1.5: Schematic representation of Stranski-Krastanov growth of quantum dots. The QDs (here shown as small pyramidal ‘islands’) form spontaneously out of the deposited wetting layer due to strain relaxation.
cations. Further, the QDs are randomly distributed over the wetting layer.
1.3.2
Hydrogen-assisted growth on a patterned
substrate
The sample grown by N¨otzel and coworkers is grown by MBE on a lithograph-ically patterned GaAs (311)A substrate with a 50 nm thick GaAs buffer layer. It consists of a 3 nm thick GaAs QW, sandwiched between two 50 nm thick Al0.7Ga0.3As barrier layers. The structure is overgrown with a 20 nm thick
GaAs layer.
During growth of the QW, quantum wires are formed along the mesa sidewalls in the [01¯1] direction, with a distance between the wires of approxi-mately 300 µm. When atomic hydrogen is injected in the MBE chamber, the corrugation of the (311) growth surface is enhanced. This highly directional corrugation forms along the [¯233] direction, perpendicular to the wire direc-tion, and effectively chops up the quantum wire into 1-dimensional arrays of quantum dots. The resulting sample structure is illustrated in figure 1.6. The quantum dot dimensions are approximately 40 × 40 nm squared, and 6 nm high, as determined by AFM. In an array, the QD density is 1.5 · 105 cm−1.
Background doping is negligible at a level of < 1013cm−3. A special property
as compared to SAQDs is that this sample is inherently strain-free, since the difference in lattice constance between GaAs and AlGaAs is < 0.15% (5.643 ˚A versus 5.651 ˚A at 5 K).
Figure 1.6:Impression of the sample structure, taken from [38], and investigated in this dissertation. The fast growing sidewall forms a quantum wire, which is then effectively cut into quantum dots by atomic hydrogen enhanced corrugation. The GaAs QW is sandwiched between two AlGaAs barriers. The sample is overgrown by a GaAs buffer layer. There is also a GaAs buffer layer between the substrate and the rest of the structure.
1.4
Scope of this dissertation
As discussed in paragraph 1.3, the QD sample grown by N¨otzel and coworkers is a new type of QD material that distinguishes itself in some important aspects from many other commonly used QD samples. Due to the absence of strain in the QDs, the arrangement of the QDs in 1-dimensional arrays, and the property of local homogeneity in QD size, this sample is an ideal candidate for optical research.
In this dissertation, for the first time, we have analysed micro-photolu-minescence (µ-PL) spectra taken from a group of homogeneously sized QDs. The lack of inhomogeneous broadening effects in these spectra, together with
the observation of excited states even at low excitation density, allows for a detailed analysis of the QD energy levels. Using a multi-excitonic model, which is discussed in chapter 2, the µ-PL spectrum presented in chapter 3 is explained in terms of Coulomb and exchange interactions between the carriers that occupy the QD. This is done not only for the ground state but also for the excited states. State filling with up to six electron-hole pairs is investigated. The observation of excited states at low excitation density could indicate the presence of a phonon bottleneck.
It is a standard procedure to determine the carrier transport properties of a semiconductor structure, either electrically (e.g. by a Hall measurement), or optically (e.g. with a time-of-flight technique). Only a few reports, however, discuss carrier transport in corrugated semiconductor structures, and these mainly focus on quantum wires [39]. In our sample the highly directional and pronounced corrugation is an intrinsic property of the QW. Since the QW is adjacent to the QDs, and carrier excitation usually takes place in the QW, it is important to know the diffusion behaviour of the carriers before they are captured by the QD arrays. In chapter 4, we present time-resolved µ-PL measurements which yield the ambipolar carrier diffusion constant as well as the carrier recombination time in the corrugated QW. We also investigate the influence of the corrugation anisotropy on the diffusion constant.
Time-correlated photoluminescence (TCPL) is a technique that has been successfully applied to measure temporally resolved luminescence decay in semiconductors, see for example [40]. In chapter 5 we investigate whether this technique can also be used to determine the carrier capture time from a QW into QDs, an application that has to our knowledge not been described before. An analysis of the relevant processes is given, leading to a two-level rate equation model which incorporates saturation of the QW. By analysing the experimental data, an upper limit for the QD carrier capture time is extracted.
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Multi-exciton phenomena —
Theory
2.1
Introduction
When a quantum dot is occupied with more than one electron-hole pair, the Coulomb and exchange interactions that arise between the carriers will in-fluence the excitonic energy levels [1–4]. For QD applications that depend on the discrete nature of the energy spectrum, the consequences of an excitonic energy level shift are significant, and a theoretical description of the processes that play a role in these energy level shifts is very desirable.
In this chapter we discuss multi-excitonic processes inside quantum dots. In our approach we do not assume that the carrier relaxation rate is much faster than the carrier recombination rate, an approach different from what is mostly reported in literature (e.g. [5, 6]). This means that we also take into account higher energy levels, even when only one or two electron-hole pairs occupy the QD. Our approach is general enough to be applied to non-degenerate QDs. We will use it in chapter 3 to analyse the µ-PL spectra obtained from our new type of QD material.
The starting point of many publications is the QD shape, where usu-ally a high degree of symmetry is assumed. Amongst others, spherical [7],
(truncated) pyramidal [8, 9], lens- and brick-shaped [2, 3] QDs are reported and used for calculations. Although theoretical models get increasingly ad-vanced [10], the problem remains that in reality the exact QD shape is very difficult to determine, because the dots are buried inside the sample. Destruc-tive methods such as cross-sectional STM are often used to investigate the QD shape [11]. These methods also show that upon investigation of strained SAQDs, the QD material bulges out of the cleavage plane and thus the QD shape is altered from the buried QD. It is therefore a challenge to directly apply theoretical calculations to experimentally obtained data.
We use a different approach, in which the QD PL spectrum is taken as the starting point. The single-exciton spectral lines are used as a basis for our analysis. Coulomb and exchange interactions are treated as adjustable parameters, the values of which can be determined from the position of addi-tional spectral lines. The dimensions of the dots in x-, y- and z-direction are roughly estimated from the position of the spectral lines and compared to ex-isting AFM data [12]. However, an important characteristic of our approach is that the exact QD shape is irrelevant.
The theory described in the next paragraphs treats the interaction be-tween carriers in a QD as perturbations to the single-carrier confinement energy levels. We restrict ourselves to uncharged QDs only: the number of electrons and holes in each QD are assumed to be equal. Every electron-hole pair added to the QD introduces additional interaction terms. We derive expressions for the total energy of the QD system, on which the spectral position of PL lines depends. In chapter 3, we determine the values for the multi-carrier interaction terms by comparison of the theoretical results with experimentally obtained data. We discuss two ways in which carriers occupy the QD energy levels: 1) complete carrier relaxation, where all carriers re-lax to the lowest available energy level, and 2) incomplete carrier rere-laxation, where excited states are allowed. The latter situation is expected to occur in the case of a phonon bottleneck.
The ‘exciton’ concept in quantum dots
In bulk semiconductor material, free electrons and holes can reduce their en-ergy with a few meV by forming bound pairs, called excitons. The reduction in energy is called the exciton binding energy. In a system with reduced di-mensionality, such as quantum wells or wires, excitons can still exist and will be kept together by the exciton binding energy. In quantum dots, however, the formation of an electron-hole pair is artificially imposed by an external barrier potential, confining both the electrons and holes to the same spatial region. The binding energy in this case is typically of the order of 10 meV to 20 meV. In literature, an electron-hole pair inside a QD is still commonly referred to as an exciton, and we use both the term ‘exciton’ as well as ‘electron-hole pair’ throughout this dissertation.
2.2
Electronic structure
The energy states of a quantum dot containing electrons and holes can be calculated by solving Schr¨odinger’s equation Hφ = Eφ, where φ is the total wavefunction that describes all electrons and holes within the QD system. The Hamiltonian for a multi-carrier system is given by:
H = −X i∈[e] ¯h2 2m∗ e∇ 2 i − X i∈[h] ¯h2 2m∗ h ∇2i + X i,j∈[e,h] qiqj 4πε0εr|~ri− ~rj| , (2.1)
where qi and qj are the carrier electrical charges. The electron and hole
ef-fective masses are given by m∗
e and m∗h, and ~ri and ~rj represent the carrier
positions. Since it is difficult to solve Schr¨odinger’s equation for this Hamil-tonian we use the Hartree-Fock approximation, where a total wavefunction is constructed by linear combination of the single-carrier wavefunctions. In or-der to maintain the property of antisymmetry upon exchange of two electrons
or two holes, the wavefunction is written as products of Slater determinants: φ = √1 Ne! ϕe1↑(~r1) . . . ϕeNe↑(~r1) ... . .. ... ϕe1↑(~rNe) . . . ϕeNe↑(~rNe) ·√1 Ne! ϕe1↓(~r1) . . . ϕeNe↓(~r1) ... . .. ... ϕe1↓(~rNe) . . . ϕeNe↓(~rNe) ·√1 Nh! ϕh1↑(~r1) . . . ϕhNh↑(~r1) ... . .. ... ϕh1↑(~rNh) . . . ϕhNh↑(~rNh) ·√1 Nh! ϕh1↓(~r1) . . . ϕhNh↓(~r1) ... . .. ... ϕh1↓(~rNh) . . . ϕhNh↓(~rNh) , (2.2) where the determinants are separated into spin-up (↑) and spin-down (↓) parts. Ne and Nh are the number of electrons and holes, respectively. The
single-carrier wavefunctions ϕi are chosen orthogonal, so that hϕi|ϕji = δij.
To calculate the energy expectation values E we evaluate the solution φ of Schr¨odinger’s equation: E = hφ|H|φi/hφ|φi. In the Hamiltonian defined in equation (2.1), the first two terms produce the single-carrier energy levels, uncorrected for multi-carrier interaction. The third term includes all the two-body e-e, h-h and e-h interactions. In our calculations, we treat this last term as a first-order perturbation to the single-carrier energy levels. The expectation value of the last part of equation (2.1), which is a two-body operator of the form 1/ |~ri− ~rj|, takes a simple form when evaluated with
the Hartree-Fock wavefunction defined in equation (2.2). The simplification is due to the orthogonality of the single-carrier wavefunctions. We get:
hφ| 1 |~ri− ~rj||φi = Z Z d~ri,σid~rj,σjϕ ∗ i,σi(~ri,σi)ϕ ∗ j,σj(~rj,σj) 1 ~ri,σi− ~rj,σj ϕi,σi(~ri,σi)ϕj,σj(~rj,σj) − Z Z d~ri,σid~rj,σjϕ ∗ i,σi(~ri,σi)ϕ ∗ j,σj(~rj,σj) 1 ~ri,σi− ~rj,σj ϕi,σi(~rj,σj)ϕj,σj(~ri,σi). (2.3) Complex conjugates are indicated with ‘*’. The first term in the right-hand side of this expression is called the Coulomb interaction, the last term is the
exchange interaction. Notice that in the last part of this last term the particle positions ~ri and ~rj are exchanged. This part of the expression is non-zero only
for identical particles (e-e or h-h) with identical spin (σi = σj).
The perturbation theory is accurate only when the sum of all Coulomb and exchange energies is much smaller than the sum of the electron and hole confinement energies. In our case, the confinement energy of the electron-hole pair turns out to be typically of the order of 200 meV, while the sum of Coulomb and exchange interactions is usually less than 20 meV. We therefore conclude that the perturbation approach is allowed.
2.3
Single-carrier confinement energies
Calculating energy levels of a particle-in-a-box system with impenetrable walls is a textbook example of quantum mechanics, see for example House [13]. This simple model is used to estimate the electron and hole confinement energies. If we take box dimensions (a, b, c), the single-carrier confinement energies Ee for electrons and Eh for holes are given by:
E(nxnynz) i = ¯h2π2 2m∗ i n2 x a2 + n2 y b2 + n2 z c2 . (2.4)
Here, i ∈ [e, h] and m∗
i is either the electron or hole effective mass. We have
used m∗
e = 0.067meand m∗h = 0.5me[14] throughout this and the next
chap-ter. The quantum numbers in x-, y- and z-direction are indicated with nx,
ny and nz (1, 2, 3, . . . ). The corresponding electron and hole wavefunctions
are sines: ϕ(nxnynz) i = r 2 a r 2 b r 2 csin nxπ a x sinnyπ b y sinnzπ c z , (2.5) with x ∈ [0 . . . a], y ∈ [0 . . . b] and z ∈ [0 . . . c]. Since in our dots c a, b we usually regard levels (nxnynz) with nz = 1, since any higher value for
nz would bring us out of range of the low-lying energy levels relevant in our
system. In this dissertation, when an energy level is denoted by two quantum numbers (nxny), we automatically take nz = 1. The indices p or q are used
(nxnynz) (111) (121) (211) (221) (112) (131)
a)
QD base surfaceb
)
(13) (22) (12) (11) (21)Figure 2.1: a) Electron energy levels in an idealised quantum dot, with quan-tum numbers (nxnynz) labelling the levels. The QD z-dimension is assumed much smaller than the x- and y-dimensions, lifting level (112) to higher energy.
b) A schematic representation of some of the wavefunction amplitudes, projected in the xy-plane. Extremes in amplitude are shown in white and black. Quantum numbers (nxny) are indicated.
2.4
Coulomb and exchange interactions
When more than one carrier occupies the QD, Coulomb and exchange in-teractions are taken into account as first-order perturbations to the single-carrier energy states, as described in paragraph 2.2. The magnitude of these interactions depend on the overlap of the carrier wavefunctions, and their spins.
We follow a model originally developed by Hawrylak [15]. Coulomb and exchange contributions Vc,ijp/q and Vx,ijp/q are calculated by integration over the appropriate carrier wavefunctions in states p and q, acted upon by a two-body operator Vij of form:
Vij =
qiqj
4πε0εr|~ri− ~rj|
. (2.6)
The carriers are either electrons or holes: i, j ∈ [e, h], and qi and qj are the
by: Vc,ijp/q = hϕpiϕqj|Vij|ϕpiϕ q ji notation: −−−−−→ hp, q|Vc|p, qi, Vx,ijp/q = hϕpiϕqj|Vij|ϕqiϕ p ji notation: −−−−−→ hp, q|Vx|q, pi. (2.7) Notice that Vc and Vx differ only in the exchange of carriers. Coulomb terms
apply to e-e, h-h and e-h interaction, while exchange terms apply to e-e and h-h interactions only, with the additional restriction that the spins of the carriers must be identical. In the abbreviated notation, wavefunctions are indicated only by the levels p or q that they represent. In an idealised QD with impenetrable walls, Vc and Vx are independent on the electron and hole
effective masses.
We now give an expression for the total energy of the QD system. Let’s first consider the case where we have one electron-hole pair in energy level (11), a situation known as the QD single-exciton (1X) ground state. The system energy is now equal to the sum of the QD semiconductor material bandgap Eg, the electron and hole confinement energies Ee11 and Eh11, and
the e-h Coulomb interaction Vc,eh11/11:
E1X= Eg+ Ee11+ Eh11+ V 11/11
c,eh . (2.8)
Figure 2.2a shows a diagram of the relevant 1X energy contributions. A biexciton (2X) ground state system is shown in figure 2.2b. The energy of a system in this state is twice that of a single exciton system, plus four additional Coulomb interactions: one e-e repulsion, one h-h repulsion and two e-h attractions,
E2X = 2E1X+ Vc,ee11/11+ V 11/11 c,hh + 2V
11/11
c,eh . (2.9)
No exchange interaction terms are included in these two cases, since there are no identical carriers with identical spin state present.
2.5
Filling of the quantum dot
We distinguish two different ways of filling a QD, depending on whether carriers relax to their lowest possible energy state before they recombine, or
Figure 2.2:a) A single-exciton and b) a biexciton system. All carriers are in the ground state (11). In figure a), Eg is the bandgap and Ee11 and Eh11 the electron and hole confinement energies. The Coulomb attraction Vc,ee11/11, indicated by an ellipse, lowers the total energy of the electron-hole pair. In figure b), four additional Coulomb terms appear, shown as thick black lines: two attractive (e-h) and two repulsive (e-e and h-h). The arrows indicate the carrier spins.
whether they recombine before relaxation takes place. The former situation greatly simplifies the observed PL spectrum, since in this case the number of possible initial states is reduced. The recombination energies that can be observed in a PL experiment are determined by subtracting the calculated final state energy from the calculated initial state energy.
We assume that recombination can only take place between an electron and a hole that occupy levels with the same quantum numbers. For these levels, the electron and hole wavefunctions overlap so that a transition can take place. If the electron is in level (nxny) = (11) while the hole is in
level (12), the overlap of wavefunctions is greatly reduced. This results in a long lifetime and a low transition rate. However, there is a significant overlap between, for example, an electron in state (13) and a hole in state (11). These transitions are not considered in this analysis, although they might appear in the PL spectrum. In order to evaluate the PL spectrum we need to know the various energy states of our QD system. First we consider the situation
where all carriers relax to their lowest allowed position. Subsequently, we investigate the situation where excited initial states are allowed.
2.5.1
Complete carrier relaxation
For practical reasons we limit ourselves to a non-degenerate QD with three confinement levels and an occupation with up to 6 excitons. There are six possible initial states (numbered 1X through 6X), depending on the number of excitons that actually occupy the QD. If the single exciton (1X) state re-combines, the final state is the vacuum state (v) with energy 0, where the QD is empty. A single line appears in the PL spectrum, at energy E1X− 0.
Upon recombination of the biexciton (2X) state, the 1X state is the only allowed final state. Here, also, a single line is contributed to the PL spec-trum, at energy E2X− E1X. The two processes are shown in the top part of
figure 2.5. The Coulomb term Vc11/11 is commonly called the biexciton
bind-ing energy and is frequently encountered in literature. Reported values are usually negative [16, 17], but positive values have also been reported [9, 18].
The situation becomes more complex for the tri-exciton recombination, since now there are two excited final states (labelled 2X∗t and 2X∗s), in
addition to the 2X ground state. Superscript ‘s’ (for ‘singlet’) indicates a state where the spins are antiparallel. Superscript ‘t’ (for ‘triplet’) indicates a state where the spins are parallel. In the latter case, an extra exchange energy contribution has to be taken into account. Three emission lines appear in the PL spectrum, with energies E3X− E2X, E3X− E2X∗t and E3X− E2X∗s.
The process is shown in figure 2.4.
There are two possible final states for recombination of a four-exciton state, five possible final states for recombination of a five-exciton state, and three possible final states for recombination of a six-exciton state. In figure 2.5, all possible recombinations for a multi-excitonic system, with fully relaxed initial states, are listed. We assume that after a recombination which leaves the system in an excited state, full relaxation takes place before the next recombination occurs. We can then derive expressions for the energies
Vc 2 X 1 X 1 X v energy [eV] 11/11
Figure 2.3:The biexciton binding energy Vc11/11causes a shift between the 1X → v and 2X → 1X spectral recombination lines. Notice that the graph only shows relative spectral positions. The ‘amplitudes’ shown in the graph do not relate to experimental intensities.
belonging to the various initial and final states. They are listed in table 2.1. In these equations, Vcp/q = Vc,eep/q + Vc,hhp/q + 2Vc,ehp/q accounts for the Coulomb
contribution between an electron-hole pair in an arbitrary state p and an electron-hole pair in an arbitrary state q. Vxp/q = Vx,eep/q + Vx,hhp/q is the
contri-bution of the exchange interaction between two electrons with same spin in state p and q, plus two holes with same spin in state p and q.
It is obvious that more lines appear in the spectrum when more excitons occupy the QD. As an example, the spectral position for PL lines of an imaginary QD are illustrated in figure 2.6, based on Coulomb and exchange contributions given in table 2.2. These values are chosen from experimental data presented in chapter 3. Six excitons are taken into account. Notice that 1) several doublets appear with nearly equal line spacing, and 2) several lines appear so close to each other that they could give rise to broadened structures in the spectrum.
t s 3 X 2 X * 3 X 2 X * 3 X 2 X 1 X * v Vx Vx 2 x Vc Vc Vc 2 X 1 X 1 X v energy [eV] 11/12 11/11 11/12 11/12 11/12
Figure 2.4: The addition of an extra electron-hole pair to the QD causes three additional PL lines to appear in the spectrum. The position of these three new lines is given with respect to the lines that are already present. The vertical line at the right indicates the transition 1X∗ → v, not visible in a system where all carriers are initially relaxed to the lowest possible energy state. Vc11/12and Vx11/12 are Coulomb and exchange energy contributions between level (11) and (12).
2.5.2
Incomplete carrier relaxation
In a similar manner we construct the possible recombination transitions for a system where we allow excited initial states. In such a system, the carrier relaxation time is longer than the exciton recombination time, for example because of the presence of a phonon bottleneck. A situation of excited states is presumably not present when the QD system is occupied by more than a few carriers, since inter-carrier scattering will reduce the carrier relaxation time. We therefore expect to see excited state recombination only for a QD occupied by one or two excitons. The resulting additional recombination possibilities are displayed in figure 2.7. The energies belonging to the initial states are already listed in table 2.1.
3X 2X 3X 2X* 4X 3X 3X 2X* 4X 3X* 5X 4X 5X 4X* 5X 4X* 5X 4X*’ 5X 4X*’ 6X 5X 6X 5X* 6X 5X*’ 1X v 2X 1X 21 12 11 11 12 21
electron energy levels
hole energy levels
t s
t s t s
Figure 2.5: Possible transitions of 3X, 4X, 5X and 6X states. These are all the possible recombinations for a system with three energy levels, where up to six excitons are relaxed to the system ground state. Notice that only 7 transitions remain when no more than four excitons occupy the QD.
Table 2.1: Energies belonging to all possible initial and final states encountered in a system which is initially fully relaxed.
fully re-laxed states state energy v 0 1X E1X 2X E2X= 2E1X+ Vc11/11 3X E3X= E2X+ E1X∗+ 2Vc11/12+ Vx11/12 4X E4X= E3X+ E1X∗+ 2Vc11/12+ Vc12/12+ Vx11/12 5X E5X= E4X+ E1X∗∗ + 2Vc11/21+ 2Vc12/21+ Vx11/21+ Vx12/21 6X E6X= E5X+ E1X∗∗ + 2Vc11/21+ 2Vc12/21+ Vc21/21 + Vx11/21+ Vx12/21 ex-cited states 1X∗ E1X∗ 1X∗∗ E1X∗∗ 2X∗t E2X∗t = E1X+ E1X∗+ Vc11/12+ Vx11/12 2X∗s E2X∗s = E1X+ E1X∗+ Vc11/12 3X∗ E 3X∗ = E2X∗t + E1X∗+ Vc11/12+ Vc12/12 4X∗t E4X∗t = E2X+ E1X∗+ E1X∗∗ + 2Vc11/12+ 2Vc11/21+ Vc12/21 + Vx11/12+ Vx11/21+ Vx12/21 4X∗s E4X∗s = E2X+ E1X∗+ E1X∗∗ + 2Vc11/12+ 2Vc11/21+ Vc12/21 + Vx11/12+ Vx11/21 4X∗0t E4X∗0t = E2X∗t+ E1X∗+ E1X∗∗ + Vc11/12+ Vc11/21+ Vc12/12 + 2Vc12/21+ Vx11/21+ Vx12/21 4X∗0s E4X∗0s = E2X∗t + E1X∗ + E1X∗∗ + Vc11/12+ Vc11/21+ Vc12/12 + 2Vc12/21+ Vx12/21 5X∗ E5X∗ = E2X+ E1X∗ + 2E1X∗∗ + 2Vc11/12+ 4Vc11/21+ 2Vc12/21 + Vc21/21+ Vx11/12+ 2Vx11/21+ Vx12/21 5X∗0 E5X∗0 = E2X∗t + E1X∗+ 2E1X∗∗ + Vc11/12+ 2Vc11/21+ Vc12/12 + 4Vc12/21+ Vc21/21+ Vx11/21+ 2Vx12/21
1.61 1.62 1.63 1.64 1.65 1.66 t s t s t s Vx Vx Vx Vc Vc Vc 6 X 5 X *' 6 X 5 X * 6 X 5 X 5 X 4 X 5 X 4 X * 5 X 4 X * 5 X 4 X *' 5 X 4 X *' 4 X 3 X 4 X 3 X * 3 X 2 X 3 X 2 X * 3 X 2 X * 2 X 1 X 1 X v energy [eV] 11/21 11/12 11/11 12/21 12/12 21/21
Figure 2.6:The position of spectral lines has been calculated based on the energy values given in table 2.2. Doublets (←→) with (nearly) equal line spacing appear, for example (6X → 5X, 5X → 4X) and (4X → 3X, 3X → 2X). The distance between doublets is often equal to a Coulomb or exchange energy. Other lines are so closely spaced (⊂⊃) that it is expected that experimentally they can probably not be resolved individually.
is allowed to be in an excited state, has three recombination possibilities. These lines are shown in figure 2.8, together with the ground state 2X → 1X transition. When the excited 2X-transitions are added to the spectrum, we obtain the situation shown in figure 2.9. Upon addition of more electron-hole pairs, we move back to the regime of fully relaxed initial states and obtain the situation already described in the previous paragraph.
2.6
Conclusions
We have given a theoretical description of the processes that influence the position of PL lines in the spectrum. A non-degenerate QD occupied with up to six excitons is considered. Detailed information about QD shape and
Table 2.2:Set of parameters chosen to produce the spectrum of figure 2.6. Values represent an imaginary QD, but are based on experimental data presented in chapter 3.
contribution energy [eV] contribution energy [eV] 1X → v 1.632 3X → 2X 1.646 Vc11/11 -0.003 Vc12/12 -0.003 Vc11/12 -0.002 Vc12/21 -0.001 Vc11/21 -0.002 Vx12/21 -0.004 Vx11/12 -0.004 5X → 4X 1.655 Vx11/21 -0.004 Vc21/21 -0.003 1X* v 1X** v 2X*t 1X 2X*s 1X 2X*t 1X* 2X*s 1X*
Figure 2.7: Diagram showing the possible excited state recombinations. Only ini-tial states with one or two electron-hole pairs are considered, occupying the two lowest energy levels.
size is not required. The theory can be applied equally well to single QDs, or to a group of homogeneously sized QDs.
The approach that we have developed is based on the spectral position of the ground state and excited single exciton recombination lines. Coulomb and exchange interactions, which are treated as adjustable parameters, then
1 X ** v 1 X * v Vc 2 X 1 X 1 X v energy [eV] 11/11
Figure 2.8: Lines appearing in a spectrum where the 1X-state is allowed to be excited. The ground state 2X → 1X transition is shifted over energy distance Vc11/11 with respect to the 1X → v transition.
s t s t 2 X * 1 X 2 X * 1 X Vx Vc Vx 2 X * 1 X * 2 X * 1 X * Vc 1 X ** v 1 X * v Vc 2 X 1 X 1 X v energy [eV] 11/12 11/11 11/12 11/12 11/12
Figure 2.9: Four additional spectral lines appear when excited 2X-transitions are allowed. A diagram of these transitions is shown in figure 2.7.
1.61 1.62 1.63 1.64 1.65 1.66 s Vx Vx Vc Vc 2 X * 1 X * 3 X 2 X * 3 X 2 X * 2 X * 1 X * 1 X ** v 1 X * v 4 X 3 X 3 X 2 X 2 X * 1 X 2 X * 1 X 2 X 1 X 1 X v energy [eV] 11/12 11/12 11/11 12/12 t t s t s
Figure 2.10: Using the same energy values shown in table 2.2, the position of PL lines in the spectrum has been calculated for the situation when up to four excitons occupy the dot, and incomplete carrier relaxation occurs. Not all excited initial states are considered. Only transitions with initially excited states as shown in figure 2.7 are added to the spectrum. Like in figure 2.6 doublets (←→) and possibly overlapping lines (⊂⊃) are indicated.
determine the position of the additional lines visible in the spectrum. By comparison to experimentally obtained data, the values for the Coulomb and exchange interactions are obtained, which is the subject of chapter 3.
Depending on whether we have the situation of complete or incomplete carrier relaxation, the spectrum will appear differently. When only a limited number of excitons occupy the QD, the major difference is the number of spectral lines. For a maximum of four excitons, only 7 lines will appear in the case of complete carrier relaxation, against 13 in the case of incomplete carrier relaxation, as can be seen by comparing figures 2.6 and 2.10. The difference is due to the number of allowed initial states, which is larger for the case of incomplete carrier relaxation. Incomplete carrier relaxation is only considered for an occupancy of the QD with one or two excitons.
The way in which the energy levels in a QD are occupied with electrons and holes has its direct consequence on the spectral position of lines in the PL spectrum. In particular, some spectral features repeat each time when a new energy level is occupied. We can observe this in the example given in figure 2.6, where similar pairs of lines (1X → v, 2X → 1X), (3X → 2X, 4X → 3X), and (5X → 4X, 6X → 5X) appear in the spectrum, with nearly equal line spacing. This behaviour can also be observed in the experimental data presented in chapter 3.
When higher energy levels get occupied, we observe that multiple lines appear at the low-energy side of the 1X → v transition. Some of these lines are so closely spaced that they can overlap and appear as single broad features in the spectrum.
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Multi-exciton phenomena —
Experiments
3.1
Introduction
Photoluminescence (PL) measurements are a standard tool for determination of energy levels, as well as the overall sample quality. Experiments performed on large ensembles of self-assembled quantum dots (SAQDs) by e.g. Hawrylak et al. [1] or Adler et al. [2] show two or three peaks in the PL spectrum, here named s-, p- and d -shell in analogy with atomic spectroscopy (see figure 3.1). These spectra clearly show state filling: for higher excitation power density the s-shell becomes completely occupied. Carriers then populate higher shells, first p, then d, which become visible in the spectra. However, multi-excitonic features like Coulomb and exchange interactions are not resolved in this type of spectra due to inhomogeneous broadening. The data give insight in the QD quality: the broad peaks in the PL spectra indicate that the QDs have a certain size distribution. This inhomogeneity in size is also clearly visible from AFM images of SAQDs [3], see figure 3.2.
Inhomogeneous broadening effects due to size fluctuations poses practical problems in determining the precise electronic properties of QDs. To over-come these problems, single-dot spectroscopy is often applied, which can be
Figure 3.1: State filling of quantum dot energy levels, measured by Hawrylak et al. [1]. In analogy with atomic spectroscopy, the energy levels are labelled s-, p-and d-shell. The broad peaks arise from inhomogeneous broadening, due to size fluctuations of the QDs.
Figure 3.2: AFM image and PL spectrum of an InAs/GaAs SAQD sample [3]. Due to fluctuation in QD size, the PL peak shows inhomogeneous broadening.
done in various ways. Masking of the sample, or the use of samples with an ex-tremely low QD density allow for single dot spectroscopy, but high excitation power or long integration times (up to several hours) are required to collect enough data [4, 5]. Other set-ups for performing single-dot spectroscopy in-clude etched and gated devices [6] and near-field scanning optical microscopy (NSOM) [7]. The general disadvantage is that these techniques require either extensive sample handling or the use of complicated instrumentation with extreme spatial resolution. An example of a single-dot PL spectrum, mea-sured by Hodeck et al. [8], is shown in figure 3.3. In this spectrum, three recombination lines are identified: the single exciton, the biexciton and the trion (two electrons and one hole).
Figure 3.3: Single QD PL spectrum measured by Hodeck et al. [8] using NSOM, at various excitation power densities and a temperature of 80 K. Three lines are identified: the ground state single exciton (X), the ground state biexciton (XX) and the trion (X*), which consists of two electrons and one hole.
In our experiments we circumvent the problems related to inhomogeneous broadening by making use of a new type of GaAs/AlGaAs QD material [9]. The QDs are arranged in 1-dimensional arrays, with a QD density of approx-imately 1.5 · 105 cm−1. Due to the 70% Al content in the barrier layers, the
barrier potential in growth direction is high. It turns out that the QDs that form these arrays exhibit an excellent homogeneity in size on a length scale of a few µm. This unique property allows for a precise determination of the energy levels by using an excitation spot size of a few µm in diameter, which greatly facilitates experimental procedures.
An important property of our QD sample is that it is inherently strain free. This means that it is free of induced band bending and strain-induced electric fields, which are present in many other types of QDs. This makes our QD sample an ideal candidate for investigation of confinement lev-els, and Coulomb and exchange interactions between the carriers occupying the QDs. Using the multi-exciton theory described in chapter 2, we describe in this chapter the PL experiments performed on this sample and explain the observed spectra.
3.2
Experimental set-up
The set-up for µ-PL measurements consists of three parts: the laser excita-tion source, the cryostat containing the sample, and the detecexcita-tion branch (monochromator and CCD array detector), see figure 3.4. A spatial resolu-tion of approximately 2 µm has been obtained by mounting the focussing lens inside the cryostat, directly in front of the sample. This lens also collects the PL light. External controls allow the lens to be scanned in all three direc-tions via a special spring system, which will be described later. The position of the excitation spot on the sample is monitored by a small CCD camera with an optical magnification of approximately 100×. This camera is also used to monitor the quality of focus while moving the excitation spot over the sample.
3.2.1
Excitation
The excitation source is a mode-locked dye laser, pumped by a mode-locked and frequency doubled Nd:YAG laser to which it is synchronised. The pulsed
laser sample x y z triple monochromator CCD neutral density filter diaphragm L CCD monitor beam expander
Figure 3.4:Set-up used for µ-PL measurements. The focussing and collecting lens L can be manually adjusted in x-, y- and z-direction by external controls. A triple monochromator followed by a 2-dimensional CCD array collects the spectra.
light beam has a photon energy of 2.119 eV (585 nm) and a repetition rate of 78 MHz. The excitation power is varied between 0.5 and 50 µW.
The use of a pulsed instead of a CW laser has consequences for the PL spectra. This has to do with the fact that the exciton lifetime is shorter than the pulse repetition rate. The result is that a QD, filled with several electron-hole pairs, goes through multiple radiative transitions, where the number of electron-hole pairs is reduced one by one. This means that all transitions, up to the final 1X → v transition, are visible in the spectrum even when the excitation takes place with high power density [10]. In contrast, a CW laser continuously refills the QD with new excitons through carrier capture. The situation where a single electron-hole pair occupies the QD therefore occurs only when the excitation power density is low.
The excitation density is an important parameter, since it controls the number of photoexcited carriers, and thus the average number of excitons within a QD. In our experiments we excite the GaAs quantum well. The number of photons incident on the sample surface by each pulse, Nphoton, is
given by: Nphoton= Eexc ¯hωphoton 1 frep , (3.1)
with Eexc the time-averaged laser beam power, ¯hωphoton the photon energy
quantum well: a fraction fR is reflected by the sample surface, a fraction fS
is reflected by each window interface, a fraction fBis lost through absorption
in the GaAs buffer layer, and only a fraction fW is absorbed by the quantum
well itself. The number of photons NQW,abs absorbed by the quantum well
is thus given by NQW,abs = Nphoton(1 − fR)(1 − fS)s(1 − fB)fW, with s the
number of window interfaces.
For normal incidence, the surface reflection factor fR is given by fR =
(ns− nv)/(ns+ nv), which is equal to 0.32 for a refractive index ns = 3.6 for
GaAs and nv = 1 for vacuum. Each window interface (four in total) reflects
a fraction fS = 0.04. The absorption in the GaAs buffer layer depends on
the thickness of this layer. In our sample, the buffer layer is only 50 nm thick, and absorption in this layer will be neglected (fB = 0). Blood [11]
has given a comprehensive theoretical discussion on the absorption in GaAs quantum wells, backed with experiments. He finds an absorption of the order of fW ≈ 10−2, independent of QW well thickness.
For the experiments described in this chapter we apply an excitation density between 16 and 1600 Wcm−2. With an excitation photon energy of
2.119 eV and a repetition rate frep equal to 78 MHz, this results in NQW,abs
ranging between approximately 1.1 · 102 and 1.1 · 104 absorbed photons per
pulse, corresponding to an exciton density between 4 · 109 and 4 · 1011 cm−2.
With a QD density of 1.5 · 105 cm−1 and a spot diameter of 2 µm,
approxi-mately 30 QDs are sampled simultaneously. We estimate that the average number of excitons per QD is as low as 0.25 for an exciton density of 4 · 109 cm−2.
The excitation density is said to be ‘low’ when the average exciton oc-cupancy per QD is less than 1, meaning that in our experiment it has to be kept below 64 Wcm−2. A ‘medium’ excitation density (with an average
occupancy between 1 to 5 excitons per QD) is between 64 and 320 Wcm−2,
while the ‘high’ excitation density regime (more than 5 excitons per QD) starts at 320 Wcm−2.
