Study of Electronic Polarons in Polarization

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Magneto-Photoluminescence: Interband Transitions

During a photoluminescence (PL) experiment, electron-hole pairs are created by shin- ing a laser on a sample. In the case of our QD samples, the pairs are trapped in the InAs islands and will relax to the ground state of the system (se − sh). Finally, the pair recombines by emitting a photon, which we are able to detect with either a photomultiplier or a CCD (charge coupled device) camera.
In this thesis, three different types of PL measurements are used: non-resonant photoluminescence (NRPL), resonant photoluminescence (RPL) and photolumines- cence excitation (PLE).
In both NRPL and RPL measurements, the excitation energy of the laser is fixed, while detection is possible in a certain energy range. For NRPL, the excitation energy is fixed to be superior or equal to the GaAs energy gap. Initially, electron hole pairs are created in the GaAs lattice which eventually relax down into the quantum dot states. In this way, we are assured the creation of an electron hole pair in all the dots.
The resulting NRPL spectrum is therefore the sum of the contribution of all the dots. In the case of RPL, the excitation energy is lower than the GaAs gap and the InAs WL (see Fig. 1.5). As a result, electron hole pairs will only be created in QDs with discrete excited energy transitions that correspond to the given excitation energy.
Finally, unlike the two methods described above, in a PLE measurement the detection energy is fixed while the excitation energy is varied. The detection energy is chosen to correspond to the luminescence of certain dots in the sample. The excitation energy is then varied through a certain energy range. When an excitation energy coincides with an excited state transition in a QD, an electron hole pair is created.
The pair relaxes to the ground state and finally the electron and hole recombine and emit a photon. A signal is detected, at the chosen fixed detection energy, each time the excitation energy corresponds to an excited state transition of a dot. PLE, therefore, measures a signal from a subensemble of dots in the sample whose ground state energy corresponds to the chosen detection energy. A summary of the three PL methods is presented in Table 1.2.
All the PL data presented in this thesis was collected at the High Magnetic Field Laboratory in Grenoble, in collaboration with Francis Teran and Marek Potemski. A schematic of the setup for a magneto-RPL or NRPL experiment is shown in Fig. 1.10. An Ar+ laser is used for non-resonant excitation and a Ti:sapphire laser for resonant excitation. A chopper coupled to a lock-in amplifier allows the elimination of any optical noise. A system of optical fibers is used for the sample excitation and the collection of the PL signal. The sample is immersed in a liquid helium bath which is pumped to 4 K. A resistive magnet surrounds the cryostat such that a magnetic field up to 28 T can be applied along the sample growth axis. The emitted light from the sample is dispersed through a Jobin Yvon spectrometer and detected by a photomultiplier.

Strong or Weak Coupling

In bulk, 2D and 1D systems, the Fr¨ohlich Hamiltonian introduces an interaction be- tween two continuums of states: the broad continuum of the electronic states (several eVs) and the narrow continuum of the phonon states (several meVs). In this case, Fermi’s Golden rule can be applied and the irreversible relaxation of a carrier in a particular state, to a state of lower energy accompanied by the emission of one or several LO phonons is expected. The probability of finding the carrier in its initial state decays exponentially over time. This is called a weak coupling [13]. Now let’s consider a system of two discrete levels, whose energies without coupling cross. In the presence of a coupling term, the two levels “repel each other” and an anti-crossing is observed. The probability of finding the system, which was initially in the first state, in the second state oscillates over time. This is called a strong coupling [14].
In QD systems, the Fr¨ohlich Hamiltonian provokes an interaction between a dis- crete state (the electronic state of the QD) and a narrow continuum of phonon states. It can be shown that this interaction can result in either a weak coupling (as in the case of bulk materials) or a strong coupling (as in the case of a two discrete level system) [14, 10]. The type of coupling depends on the relationship between the width of the continuum, δ, and the strength of the coupling term, Veff (to be defined). If δ ≪ Veff a strong coupling takes place and if, on the contrary, δ ≫ Veff , a weak coupling is expected.
In the next section it will be demonstrated that for our samples, Veff is on the order of 4 meV. The coupling regime of the QD system can therefore be determined by comparing this Veff with the width of the phonon continuum in interaction with the carrier. The width of the phonon continuum of GaAs (InAs) spanning the first Brillouin zone is approximatively 6 meV (3 meV), as seen in Fig. 2.1. The extension of the continuum in interaction with the electronic states can be approximated by calculating the matrix element of the Fr¨ohlich Hamiltonian between two discrete QD states, hυi, ni~q|VF |υf , nf ~qi. The Fr¨ohlich Hamiltonian only directly couples states.


Evidence of Electronic Polarons

In this section, the results of FIR magnetotransmission experiments measured on sam- ple N1 and conducted in the conditions described in Section 1.5.1 will be presented. Sample N1 was doped in order to obtain, on average, one electron per dot. The single electron will initially be found in the ground state of the QD system, |s, 0i. The allowed intraband energy transitions are therefore to the |p+, 0i and |p−, 0i states, as demonstrated in Section 1.5.3.

High Magnetic Fields Experiments

The validity of the above polaron model has been demonstrated in the thesis of J.N. Isaia and S. Hameau. We will present here results reported in these thesis. Figure 2.3 displays the magnetic dispersion results measured for sample N1 up to B = 28 T in unpolarized light. Four branches are observed with three incidences of anti-crossings: at 70 meV for a a magnetic field of 16 T, at 80 meV for B = 25 T and the top branch of an anti-crossing at 40 meV. We therefore find the signature behavior of a strong coupling regime.
Let us now attempt to theoretically reproduce the experimentally observed anti- crossings. We start by finding all the uncoupled QD states in the energy region of the experimental results, i.e. 30 – 90 meV above the ground state energy Es. Taking a dot with R=106 °A and h=23 °A, we find the five states shown in Fig. 2.4, whose energies are plotted as a function of magnetic field. The other parameters used are listed in the figure caption. There are three occasions for a strong coupling interaction to occur in such a system:
• The crossing at 24 T between the discrete energy state |p−, 0i and the one phonon continuum state |s, 1{~q}i.
• The crossing at 21 T between the discrete energy state |p+, 0i and the one phonon continuum state |p−, 1{~q}i.
• The crossing at 24 T between the one phonon continuum state |p−, 1{~q}i and the two phonon continuum state |s, 2{~q},{~q′}i.

Table of contents :

1 Presentation of Quantum Dots 
1.1 History and Motivation
1.2 Fabrication
1.3 Characteristics of Samples
1.4 Electronic States
1.4.1 Electronic Structure of an InAs/GaAs System
1.4.2 Calculation of Energy Levels
1.5 Investigation of Energy Levels
1.5.1 FIR Magnetospectroscopy: Intraband Transitions
1.5.2 Magneto-Photoluminescence: Interband Transitions
1.5.3 Coupling to Light
1.5.4 Coupling to a Constant Magnetic Field
1.6 Conclusion
2 Electronic Polarons 
2.1 Calculation of Polaron States
2.1.1 The Fr¨ohlich Hamiltonian
2.1.2 Strong or Weak Coupling
2.1.3 The Effective Potential
2.1.4 A Simple Example: one discrete state coupled to one continuum state
2.2 Evidence of Electronic Polarons
2.2.1 High Magnetic Fields Experiments
2.2.2 Study of Electronic Polarons in Polarization
2.3 Conclusion
3 Hole-LO Phonon Interaction 
3.1 Experimental Results
3.1.1 Study at 0 T
3.1.2 Be Impurities
3.1.3 Magnetotransmission Spectra
3.2 Hole Polaron Calculation
3.3 Comparison Theory/Experiment
3.3.1 Comparison of Magnetic Dispersion Curves in Polarization .
3.3.2 Oscillator Strength
3.3.3 Comparison of Magnetic Dispersion Curves in High Fields
3.4 Conclusion
4 Exciton-LO Phonon Interaction: PLE Experiments 
4.1 Experimental Results
4.1.1 PLE Spectroscopy
4.1.2 Study as a Function of Detection Energy
4.1.3 Study as a Function of Magnetic Field
4.2 Calculation of Excitonic Polarons
4.2.1 Fr¨ohlich Interaction
4.2.2 Coulomb Interaction
4.2.3 Calculation of Polaron States
4.3 Comparison Theory/Experiment
4.3.1 Comparison of Magnetic Dispersion Curves
4.3.2 Oscillator Strength
4.4 Conclusion
5 Exciton-LO Phonon Interaction: RPL Experiments 
5.1 RPL Spectroscopy
5.2 Experimental Results
5.2.1 Magneto-RPL Measurements
5.2.2 Study as a Function of Excitation Energy
5.3 Comparison with Polaron Model
5.3.1 Magnetic Field Dispersion Comparison
5.3.2 Oscillator Strength
5.3.3 High Energy Peak
5.4 Conclusion
A Atomic Wavefunctions


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