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Semiconductor quantum dots
Quantum dots (QDs) are nanometer size structures in which the motion of carriers is restricted in all spatial directions. This 3D conflnement makes the energy spectrum of a single carrier quantized in shells of discrete levels. Due to this analogy with the energy spectrum of atoms, quantum dots are often called « artiflcial atoms ». In semiconductor QDs, the potential walls produced by a mate-rial barrier surrounding the QD plays the role of the nucleus Coulomb potential in atoms. Electrons occupy discrete energy levels and, analogically to atoms, excited and ionized states can be formed. Although there are many similarities between an isolated atom and a quantum dot, one should not forget about important dif-ferences, which come from the speciflc properties of solid state. Also, one has to be careful with « expanding » our intuitions from other low-dimensional structures. For example, when under excitation with light an electron is promoted from the valence band to the conduction band, the Coulomb interaction between this pro-moted electron and the hole left in the valence band plays a signiflcative correction to the single particle energy. Such a correlated electron-hole pair in a QD will be named an exciton in analogy to the case of bulk materials. The symbol X will be used to denote the exciton state. In the case of QDs, the name « exciton » has be-come quite conventional though not very meaningful, since in forming an excitonic state the electrostatic Coulomb interaction between carriers should be the leading term. This is deflnitely not the case in QDs where electron-hole correlation is still dominated by the barrier conflnement.
The most intensely studied QDs are made from III-V binary compounds: ar-senides (e.g. InAs/GaAs), nitrides (e.g. GaN/GaAlN), phosphides (e.g. InP/GaP). More recently, II-VI materials gained a large interest thanks to their light emission at higher temperature (compared to III-V materials). Telluride (e.g. CdTe/ZnTe) and selenide (e.g. CdSe/ZnSe) compounds have been among the most intensively investigated II-VI QDs.
Quantum dot structural properties
In this work we focus only on properties of self-assembled semiconductor quan-tum dots. A detailed description of the samples with QDs studied in our exper-iments will be given in Chapter 2. Here general properties of QD structure and growth are recalled. The growth method is based on self-induced rearrangement of the semiconducting epitaxial layers generally due to a strong lattice mismatch. For this reason, one cannot have full control over the QD ordering and geome-try. Even though growth conditions have obvious strong in°uence on QD sample properties, a post-growth treatment is often required to improve these proper-ties. Recently such techniques have been strongly developed by difierent research groups 1. One of the most important parameters of QD heterostructures is the QD density. Typically it varies from 107cm¡2 to 1012cm¡2. One can see in the Scanning Tunnelling Microscope (STM) view presented in Fig. 1.1 that dots can be spatially placed very close one to another, (at the 10 nm scale). Convenient optical investigation of an individual QD requires thus samples with a low QD density (below 109cm¡2). In Chapter 2 we will describe some experimental tools, which allow us to have an access to the micro-world and to perform measurements on individual zero-dimensional objects.
A self-assembled QD consists of » 103 ¡ 105 atoms of one material embedded in a host barrier material. The detailed information about QD atomistic structure like shape, size and composition can be determined from the scanning tunneling microscopy (STM) or Transmission Electron Microscopy (TEM) [28, 29]. STM analysis has been extensively carried out by P. M. Koenraad et al. [30] and re-vealed very interesting details of QD atomic structure. The (110) plane containing the growth direction [001] is a natural cleaving plane of the crystal ofiering access to the cross-section of QDs. Such images realized with atomic resolution clearly indicate the QD size of the order of a few nm (2 ¡5nm in height and 10 ¡30nm in diameter) and additionally evidence the lattice relaxation (see Fig. 1.2) around the dot. The latter can be directly related to the chemical composition of a dot. It turns out that during the growth process the barrier material interdifiuses into QDs. For InAs QDs grown on GaAs the Indium proflle is strongly asymmetric (the In content varies linearly from 60% at the bottom to 100% at the top [31]). The chemical com-position in real space corresponds to the conflnement potential proflle experienced by carriers. Strong intermixing of materials has thus very important consequences in electronic structure of the system and on spatial localization of electrons and holes [32]. Those efiects will be discussed later in Chapter 3. QDs made from dif-ferent compounds and grown at difierent conditions difier from each other, but there is a common feature: the lateral size is a few times larger than the vertical one. The second important characteristics coming out from morphology studies by STM is the presence of regular monoatomic layers below QDs (see Fig. 1.2). This builds up an ultra-thin quantum well which is called a wetting layer (WL), and whose two-dimensional density of states can strongly in°uence QD electronic and optical properties. First, QD states can couple to this 2D-continuum [33] leading to a dramatic change of the expected 0D absorption spectrum. Second, optical excitation above WL band edge produces excitons or uncorrelated electrons and holes not directly in QDs. This can be useful to investigate the saturation regime of QD population, but may appear as a major drawback in experiments on single QDs where generally the charge state or population has to be kept under precise control. For such studies, QDs with discrete excited levels well separated from the 2D continuum must be privileged.
The clear vertical asymmetry of QDs has been flrst considered theoretically by M. Grundmann et al. (Ref. [35]) for band structure calculations in the framework of k¢p formalism. More recently, an atomistic approach based on pseudo-potentials method has been also applied by G. Bester and A. Zunger [34] to take into account the exact symmetry (in particular at atomic scale) of QDs. Detailed calculations allowed them to compare contributions of difierent efiects into the level structure of electrons and holes, as they naturally take into account the shape of potential barriers as well as the role of atomistic interfaces (of lower symmetry) or of strain and piezo-induced efiects. It turns out that the piezoelectricity of the strained materials must be taken into account in order to explain some optical properties of self-assembled QDs. An example of calculations is shown in Fig. 1.3. The efiective potential due to the strain varies from the bottom to the top of a QD and a quadrupolar potential of piezo-charge clearly in°uences the preferential alignment of the carriers along crystallographic axes of the lattice. Note that this contributes also as a perturbation of C2v symmetry (meaning that directions [110] and [¡110] are not equivalent) even in the case of a square based pyramid-like QD, since the quadrupolar potential has difierent strength on top and bottom of the QD. Although in experiments all listed efiects contribute together and only a combined efiect is observed, theoretical predictions about the signs and the magnitude of the shape-, interface-, strain-, and piezo-electric -efiects are obviously useful to chose an external perturbation to modify the optical properties of QDs.
More detailed description of the properties of quantum dots requires the dis-cussion of their symmetry. We put emphasis on this aspect in the next Section, because it is essential for this work.
QD electronic states: a simplifled description
In order to understand how the actual symmetry of a QD impacts on its optical properties, we flrst need to describe in more detail the electronic states. The single particle Hamiltonian for an electron or a hole is composed of two terms: kinetic energy and the potential energy (including spin-orbit interaction). The flrst one is invariant under all unitary operations acting on spatial coordinates, while the sec-ond re°ects the actual symmetry of the QD system and thus determines the point group symmetry of the total Hamiltonian (see e.g. Ref. [36]). The investigated III-V and II-VI compounds crystallize in zinc-blend structure with symmetry Td, but as soon as a nanostructure is grown with an anisotropy between the growth direction (say parallel to [001] axis ) and the lateral directions, the symmetry point group reduces to D2d. This would be the case for a square based parallepipedic QD (neglecting the presence of the WL). In such a case the in-plane directions [110] and [-110] transform into each other by the fourfold roto-inversion 2. Actually, this is never the case, as in addition to the vertical/horizontal anisotropy, the obvious lack of symmetry along the growth axis leads to the lower symmetry point group C2v. Due to this reduced symmetry, the projection of the total angular momentum operator Jz along z is no longer a good quantum number for the QD eigenstates. As will be discussed further, the optical properties get thus anisotropic for a light propagating along z. Nevertheless, it remains very useful to use a basis of Jz eigen-states while treating as a perturbation the terms of the QD potential leading to C2v. This is the general framework which will be generally used in this thesis. Let us write more explicitly the electron (hole) wavefunctions “en;l(hz);jz forming this basis in the efiective mass approximation (EFA): “n;le(hz);jz = Fn;le(hz)(~r)un;je(hz)(~r) (1.1)
The indexes lz and jz refer respectively to the angular momentum projection on z of the envelope function Flez( h)(~r) and of the crystal Bloch function uejz(h)(~r), the latter assumed to be the same for barrier and QD materials, while n is a subband index required to describe the valence band components originating from the three degenerate X; Y; Z-like atomic orbitals. The envelope functions Fn;le(hz) are the solutions of the single band EFA Hamiltonian respectively for the electron (e) band and the three hole (h) bands indexed by n. In analogy to atoms they correspond to wave functions of symmetry S, P , D,… In this basis, the QD potential of D2d symmetry couples states with constant total angular momentum lz + jz. As in interband optical transitions the oscillator strength is weighted by the envelope function overlap j R Fle (~r)Flh0(~r) j2= –lz;lz0 this coupling turns out to be of rather weak importance. In contrast, even though the coupling of C2v symmetry has a smaller magnitude it will reveal more crucial because of a drastic change of optical selection rules. In most of the situations discussed further, it is thus enough to fo-cus on the Bloch part of the eigenstates associated with the envelope of S (lz = 0) character. Under this assumption, the flnal basis of convenient use may be reduced to the spin state basis for electrons (i.e. u+1e =2 • j+ 1 i = j « i, u¡e 1=2 • j¡ 1 i = j #i), of heavy-hole band (j = 3=2) and light-hole while for holes only the Bloch states 3 z § band (jz = §1=2) can be retained . The corresponding basis is recalled below, with a double arrow * to represent the hole spin degree of freedom.
Exciton flne structure of actual quantum dots
When an electron-hole pair is photocreated or captured from the wetting layer in the QD, it forms a strongly correlated two-particle state. This complex named in the following exciton is described by an Hamiltonian which is the sum of both single particle Hamiltonians plus the Coulomb interaction between the particles. The latter decomposes into a direct term which amounts to a few tens of meV , and an exchange term due to Pauli principle and indiscernibility between conduction and valence electron (although they belong to difierent bands). The exchange term is much smaller in magnitude, but as it acts like a spin-dependent interaction; it in°uences dramatically the structure of the exciton ground state.
In lens- or pyramid-shaped self assembled quantum dots, the hole ground state has a pronounced heavy-hole character (above 90% according to Ref. [37]). As a result, to the flrst order the four possible spin conflgurations of a ground state exciton read:
The two states j § 1i are dipole-active (they can couple to light), and for this reason are called « bright » states. In experiments these states recombine radiatively giving rise to photoluminescence. The second pair j § 2i is optically non-active, therefore these states are called { « dark » states. However, these electron-hole states are not in general eigenstates of the total Hamiltonian because of the electron-hole exchange.
This gives rise to the exciton flne structure splitting (FSS) which is of central importance for this work. Formally, the corresponding Hamiltonian can be represented by the following expression [38]:
^ –0 e h –1 e h e h –2 e h e h
where the Pauli matrices ¾ie;h act on the spin components of the electron (e) or hole (h) respectively. Here, we used a §1=2 pseudo-spin to describe the QD hole ground states with angular momentum jz = ¤3=2. The quantities –0, –1, and –2 describe the exciton quartet flne structure as follows: –0 – between states of angular momentum jM j = 1 and jM j = 2 (or ¾ze + ¾zh=0), j–1j (i.e. FSS) – between the components of the optically active doublet (M = §1), and j–2j – between the dark states (M = §2). These parameters are determined by the quantum dot properties (size, shape, composition, strain fleld, etc). The arbitrary directions x, y of the Pauli matrices correspond here to the eigenaxes of the QD. We use the following deflnitions of the matrix representations of the angular momentum operators:
Figure 1.4. Scheme of the level splitting of the exciton state. From left to the right: evolution of level splitting after including difierent terms: flrst { splitting between light-and heavy-hole excitons, next (yellow shaded area) { splitting induced by lowering the symmetry to D2d symmetry (case of an isotropic QD), and last (red shaded area) { lowering to the C2v symmetry (case of an anisotropic QD) excitonic doublet in PL measured in two linear orthogonal polarization (from Ref. [39]).
of excitonic transitions. In the system of D2d symmetry bright states should not be split, whereas for dark states the degeneracy is already lifted. Note the char-acteristic relative energy scale: splitting between heavy- and light-holes ¢l¡h is around few tens of meV , between dark and bright states –0 » 1 meV , between two bright states –1 { from zero up to hundreds of „eV , and between two dark states –2 » 1„eV . We will concentrate mostly on the heavy-hole bright states. The other states will be considered only if they may mix with those states and in this way in°uence optical properties of the system.
Table of contents :
1 Introduction
1.1 Historical background
1.2 Semiconductor quantum dots
1.3 Quantum dot structural properties
1.4 QD electronic states: a simpli¯ed description
1.5 Exciton ¯ne structure of actual quantum dots
1.6 Polarization correlation in the \biexciton cascade »
1.7 QD-based source of entangled photons
1.8 Expression of the anisotropic exchange term
1.9 Aim of this work
2 Samples and Experimental setups
2.1 Samples
2.1.1 II-VI samples
2.1.2 III-V samples
2.2 Technology
2.2.1 Devices for in-plane electric ¯eld
2.2.2 Devices for vertical electric ¯eld
2.3 Experimental setups
2.3.1 Setup for measurements of II-VI materials
2.3.2 Setup for measurements of III-V materials
3 In°uence of electric ¯eld on quantum dots
3.1 In-plane electric ¯eld
3.1.1 PL intensity and linewidth
3.1.2 Fine structure splitting vs. in-plane ¯eld
3.2 Vertical electric ¯eld
3.2.1 Permanent vertical dipole
3.2.2 Direct Coulomb integrals
3.2.3 Fine structure splitting vs. vertical ¯eld
3.3 Local electric ¯eld
3.3.1 Transition identi¯cation
3.3.2 Fine structure vs. local ¯eld
3.4 Conclusion { FSS tuning
4 In°uence of magnetic ¯eld on quantum dots
4.1 Exciton Zeeman Hamiltonian
4.2 Faraday con¯guration
4.3 Voigt con¯guration
4.3.1 Theoretical model
4.3.2 Isotropic hole g factor (¯ = 0)
4.3.3 Anisotropic g factor (¯ 6= 0)
4.4 Experimental results
4.5 Orbital e®ects
4.6 Conclusions – FSS manipulation
4.7 Nuclear ¯eld
5 Toward Entanglement
5.1 Exciton spin dynamics in density matrix formalism
5.1.1 Time evolution of the exciton density matrix
5.1.2 Geometrical representation of spin precession
5.1.3 Polarization relaxation and conversion in cw experiments
5.2 Optical orientation/alignment of exciton
5.2.1 Optical orientation in QDs with tunable FSS
5.2.2 Optical orientation for di®erent excitations
5.3 Measurements of polarization conversion
5.3.1 Circular-to-linear conversion
5.3.2 Linear-to-circular conversion
5.3.3 Conversion of polarization for trions
5.4 Conclusion
6 Conclusions
