Docteur

Alban Lafuente-Sampietro

Thèse Dirigée

L Besombes

S Kuroda

Professeur, INSP, Rapporteur Mme Maria Chamarro

M Aristide Lemaître

M Bernhard Urbaszek

M Etienne Gheeraert

M Yasuhiro

Tokura Professeur

M Takashi

Suemasu Professeur

M Kazuaki

Pr. Takashi Suemasu Yasuhiro Tokura

Kazuaki Sakoda

Aurélie Laurent

Adrien Daubois

Amaury Josse

Céline " Nobody

" Gallien

Avatar Z Brown

THÈSE Pour obtenir le grade de

Keywords: Conclusion

I could not have nished

But the supervisors and the jury are only a part of the people surrounding you during a thesis. Some many more persons make the working environment, and the thesis would be quite dierent without them. I would like to thank rst the people of the NPSC team. I think I was not the most well integrated PhD student, due in part to my long absences caused by my double degree with a Japanese university, but also due to my oce being a bit far from the usual oces of the team, and my usual diculties to socialize. Nonetheless, it was always interesting to meet people of the team and discuss with them, be it about physics or other subjects. Among them, I want to extend a special thank to some PhD students: Valentin Delmonte, Mathieu Jeannin, Alberto Artioli, Thibault CRémel and Kimon Moratis, with whom I had quite a lot of discussion on a lot of subjects, and who where always interesting. In the permanent members of the team, I especially want to thank Joel Cibert, always smiling and helpful, and Henri Mariette, really supportive and always willing to discuss around a nice table in Japan.

As I said, my oce was a bit far from the main oces of the team NPSC.

It made the contact with my team a bit more dicult, but it also gave me the opportunity to meet a lot of PhD student from other teams. Among them, I especially want to thank Yann Périn, André Dias and Dayane De-Chouza-Chaves, with whom I shared an oce for two or three years. It was a really lively, and always fun to be here. We had some great discussion, and often some great laugh.

I will with no doubt miss the atmosphere of this open space. The arriving of Hawa Abdul-Lati in the oce was quite a good surprise too, and I want to thank her particularly. We bonded quite quickly, in part thank to our shared experience being double degree students between the university of Grenoble and the university of Tsukuba. This year and a half we shared seems way to short and it would be pleasure to meet again to continue our discussion. Speaking of encounter started in Grenoble thanks to my Japanese experience, I do not want to forget Ryogen Fujiwara. It was really nice to be able to discuss our two countries and to exchange about our experiences.

The Tsukuba side is not to be neglected. I meet several great persons during my time there. First, I want to thank the whole Kuroda team. I had some really great moments with them, and everyone have always been helpful every time I had a problem. They also were really accepting about the small errors I did in Japanese etiquette, and I was able to learn a lot thank to them. I especially want to thank the whole dot team in the Kuroda lab: Hayato Utsumi, Masahiro Sunaga and Kenji Makita. It was a pleasure to work with them, and also to meet them outside of work. Even though the language was sometimes a problem, we found ways to have nice discussions. I also want to deeply thank Ryo Ishikawa and Takuma Nakamura. They did a lot to help me while I was there, and to go through the Japanese administrative side. I could not have done anything without Introduction Building a quantum computer is one of the challenges of this century. The core component of such a computer is the qubit, the quantum bit. Instead of regular bits, which can take the states (values) |0 and |1 , the qubits, being quantum devices, can also be in a superposition of states, α|0 + β|1 . An important step in the realization of the computer is to nd a system to store and control these quantum states, which do not exist yet. The two main criteria for this system are its characteristic time, that must be long enough to do the operation and store the result, and the speed of preparation in a given state, determining the speed of each operation. Moreover, it has to be possible to build gate of one or two qubits. The system has also to be scalable, in order to be able to build quantum component with a large number of qubits.

Several approaches exist for the fabrication of a qubit, such as cold atoms, superconductors... A promising system for their realization is a single quantum dot (QD), nanometer-sized objects designed to conne carriers in all three dimensions.

This connement leads to a quantization of the carriers energy, akin to the energy levels of the electrons in an isolated atom.

Multiple methods exist to form such devices: gate trapping of single electron between electrodes, nanometer-sized grains formed by the precipitation of semiconductors in a solution (colloidal dots), thickness variation of a quantum well, strain relaxation of a semiconductor layer... I will focus in this thesis on the latter type of QDs, usually grown by Molecular Beam Epitaxy (MBE). They are formed by small islands, with a characteristic size of a few nanometers, of a small gap semiconductor inserted in a wide gap semiconductor. Well-known examples are InAs/GaAs (for III-V semiconductors), CdSe/ZnSe or CdTe/ZnTe (for II-VI semiconductors). More specically, in this thesis, I studied optically active QDs:

carriers can be injected in the QD by a laser excitation, and their relaxation occurs with the emission of a photon.

The spin of the carriers injected in a QD is a good candidate for the realization of a two level quantum system. For a single gate qubit, coherence time of the electrons as long as 1 µs was found [1]. Moreover, it has been demonstrated that QDs can be used to control electrically (for the gate qubit) or optically (for the v optically active dots) the spin of the injected carriers [2,3]. Finally, the optical preparation of the carrier spin state takes only a few nanoseconds. All of this makes the spin of carriers trapped in a QD a promising system for the fabrication of qubits [START_REF] Basdevant | Mécanique quantique[END_REF]. However, the dephasing time of an ensemble of QDs is a lot shorter than the coherence time of single QD, falling to about 10 ns [810]. This is too short to do any signicant data storage or processing.

Exiting the world of QDs, several systems were proposed to get longer spin coherence time, such as Nitrogen-Vacancy (NV) centers in diamond [START_REF] Fuchs | A quantum memory intrinsic to single nitrogenvacancy centres in diamond[END_REF] or atomic spins directly inserted in the semiconductors [START_REF] Pierre | Singledonor ionization energies in a nanoscale CMOS channel[END_REF]. In NV centers, electronic spin coherence time in the milliseconds range was found in ultrapure isotopically puried diamonds [13]. However, the preparation of the electronic spin of the NV center takes hundreds of nanoseconds, which would slow the calculations down [START_REF] Fuchs | A quantum memory intrinsic to single nitrogenvacancy centres in diamond[END_REF].

The same kind of coherence and manipulation time can be expected for the atomic spins.

Another approach comes from the Diluted Magnetic Semiconductors (DMS).

In these materials, a low density of magnetic atoms is inserted in the semiconductor lattice. The semiconductor keeps its conventional optical and electrical properties and new ones arise from the presence of the magnetic atoms. It was shown that there is a strong exchange interaction between the carriers and the magnetic atom spins. When inserting magnetic atoms in a quantum dot, the carriers are conned with them. Their interaction is enhanced, enabling the control of the magnetic atoms spins via the injected carriers. In this thesis, this reasoning is pushed to its limit, inserting a single magnetic atom in a QD, and controlling its spin optically. Such individual spins are promising for the implementation of emerging quantum information technologies in the solid state [1416]. They are expected to present many desirable features for the realization of spin qubits, such as reproducible quantum properties, stability, and potential scalability by coupling dots [START_REF] Koenraad | Single dopants in semiconductors[END_REF]. Thanks to their point-like character, a longer spin coherence time (compared to carriers' spins) can also be expected at low temperature. All of this makes single magnetic dopants in QDs a good candidate to store quantum information.

The control of the spin state of individual [1823] or pairs [24,25] of magnetic atoms has been demonstrated. The spin of a magnetic atom in a QD can be prepared by the injection of spin polarized carriers and its state can be read through the energy and polarization of the photons emitted by the QD [2628].

The insertion of a magnetic atom in a QD where the strain or the charge states can be controlled also oers degrees of freedom to tune the properties of the localized spin such as its magnetic anisotropy [29].

Tab. 1 lists some magnetic atoms that can be inserted in a semiconductor lattice. Each of those atoms has a unique set of electronic spin, nuclear spin and orbital momentum. For a given semiconductor structure, those properties vi change the magnetic atom behaviour. Each can be used for dierent applications, such as the realization of a spin mechanical qubit for the elements with an orbital momentum. The rst atom to have been inserted in a quantum dot is the Mn, rst in II-VI ( 2004) [START_REF] Besombes | Probing the spin state of a single magnetic ion in an individual quantum dot[END_REF], and then in III-V (2007) [START_REF] Kudelski | Optically probing the ne structure of a single mn atom in an inas quantum dot[END_REF]. Since then, other magnetic atoms have been embedded in II-VI QDs and studied: Co (2014) [30] and Fe (2016) [31]. They have not been inserted successfully in III-V semiconductors yet.

Mn in II-VI semiconductors has been widely studied in the last decades. In bulk semiconductors, its relaxation time was found to reach the milliseconds range, for vanishing Mn concentration [32,33]. Inserted in II-VI QDs, it was demonstrated that a single Mn spin could be optically prepared in a few tens of nanoseconds, depending on the laser power [START_REF] Gall | Dynamics and Optical contrôl of a single spin in a Quantum Dot[END_REF]. At the same time, a relaxation time of the Mn spin of a few microseconds was found [26]. The dynamic of a Mn spin was also probed in a positively charged QD, forming a hybrid spin by coupling with the resident hole [START_REF] Varghese | Dynamics of a Mn spin coupled to a single hole conned in a quantum dot[END_REF], and in a strain-free environment [START_REF] Besombes | Resonant optical pumping of a Mn spin in a strain-free quantum dot[END_REF].

Single Cr atom in a QD is also of particular interest: thanks to its orbital momentum, it should be very sensitive to strains. This opens new ways to manipulate the spin state of this magnetic atom without having to use optical excitation. It also opens the possibility to realize spin mechanical systems where the Cr is used as a qubit coupled to an oscillator. Cr could be used to probe the movement of the oscillator, cool the oscillator down or create non-classical states of the oscillator. Moreover, the Cr atom in a II-VI matrix presents no nuclear spin. There is therefore no hyperne interaction for Cr atom in a CdTe/ZnTe QD in 90% of the cases. This simplies the Cr spin structure and leads to an expected longer coherence time.

In this thesis, I will present a detailed study of the hole-Mn hybrid spin, and start the study of a single Cr atom in a QD. Those two systems are promising for the realization of spin qubit coupled to strains. Growth of the Mn-doped QDs was done in Grenoble, in the INAC-CNRS joined team NPSC, by Hervé Boukari.

The Cr-doped QDs were grown in Tsukuba, in the team of Pr. Shinji Kuroda, by

Hayato Ustumi, Masahiro Sunaga and myself. The optical study of the magnetic

QDs was performed at the Néel Institut in Grenoble, where an optical setup for the study of single quantum dots doped with a single magnetic atom had been vii developed.

This thesis is organized as follows:

Chapter I This chapter focuses on the theoretical background of this thesis. I begin to discuss the properties of a semiconductor crystal. This discussion is then used as a basis to present the physics of the QDs and their properties. Then the interaction between carriers and magnetic atom in a diluted magnetic semiconductor is presented. Particular attention is given to the interaction between the carriers and the two atoms studied in this thesis: the Mn and the Cr. I also show how the inclusion of these magnetic atoms in a crystal aects their spin energy structure. Finally, I present a short example of application of these theories on CdTe/ZnTe QDs doped with single Mn.

Chapter II The growth of Cr doped QDs was an important part of this thesis.

I present here the techniques used to grow the samples studied optically. I begin with a general explanation of the MBE process. I then explain how the Cr-doped QDs were grown and how they are formed using Stranski-Krastanov (SK) relaxation. I also present some tests that were done for the growth of two other kinds of Cr-doped samples: charge tunable sample, and strain-free dots formed by the thickness variation of a quantum well. For each kind of sample, I present basic optical characterization.

Chapter III I discuss in this chapter the dynamics of the hole-Mn hybrid spin.

I show that spin states form optical Λ-level systems. Two hole-Mn ground states are connected to one X + -Mn excited state via two transitions of op- posite polarizations. They were used to study the dynamics of the hole-Mn hybrid spin. A fast hole-Mn spin relaxation, in the nanoseconds range, caused by the interplay between acoustic phonons and the hole-Mn exchange interaction is evidenced. I also show that two X + -Mn level can be coupled by the carriers in three dimension, creating what is called the quantum dots (QDs). We describe the eect of the connement on the carriers, before explaining how they interact with each other inside the QD. We nish the section with the eect of the shape and strains anisotropy on the carriers and, thus, the photo-luminescence (PL) of the QDs.

In the second section, we introduce magnetic atoms in the semiconductor matrix and see how they interact with the semiconductor carriers. We begin describing the interaction between localized electrons on the outer shell of a magnetic atom, and carriers of interest in the semiconductor. We then present the carriers interaction with the two magnetic atoms studied in this thesis: Manganese (Mn)

and Chromium (Cr).

In the third section, it is shown how the insertion of the magnetic atom in the semiconductor matrix aects its spin. We begin by describing the case of the Mn, weakly coupled to the crystal eld. We continue with the case of Cr, which is strongly aected by the crystal eld and its modication by the strains.

We nish this chapter by applying the dierent concepts we described so far to a simple system: an exciton coupled to a Mn atom in a neutral quantum dot.

I.1 II-VI semiconductor quantum dots The external orbitals for the cation are s (4d 10 5s 2 for Cd, 3d 10 4s 2 for Zn) and p for the anion (4d 10 5s 2 5p 4 for Te). In the crystal, their orbitals hybridize to form the conduction band and the valence band of the semiconductor. It is the wellknown sp-hybridization. A good and more simple picture of this construction can be given considering the hybridization of two elements of the column IV of the periodic table. A single unit of the crystal contains 8 valence electrons, coming from the s and p levels of the ions. The s and p orbitals of these atoms hybridize to form 8 levels, 4 bonding and 4 anti-bonding.

The lowest band of the bonding levels, coming from s orbitals, will be lled by two valence electrons. Six will be taken to ll the three bonding bands of higher Since all the available electrons are used to ll the valence band, the conduction band is empty in the ground state. The lower energy band of the conduction band is formed by the anti-symmetric combination of the s orbitals. At higher energy, the anti-symmetric hybridization of p orbitals form three other bands.

The energy needed to excite one electron from the higher energy state of the valence band to the lower energy state of the conduction band is called the gap.

The gap is really important for the opto-electronic properties of the semiconductor.

In the ZnTe and CdTe cases, they are equal to E g,ZnT e = 2.40 eV and E g,CdT e = 1.60 eV at 5K [START_REF] Furdyna | Diluted magnetic semiconductors[END_REF].

The promotion of an electron to the conduction band leaves in the valence band a quasi-particule called a hole. The hole represents the absence of an electron, and has thus the opposite characteristic (spin, charge, wave vector...) as the missing electron. Aside from the promotion of an electron to the conduction band, holes can also be injected in the semiconductor via the introduction of p-type defect.

When growing two semiconductors of dierent band gap on top of each other, a band oset appears at the interface. This oset is distributed between the valence band and the conduction band. It can be distributed in three dierent ways, heterostructure. In this case, both the electrons and the holes are conned in the semiconductor of smaller gap (since holes have the opposite energies as the missing electrons, they are conned in the higher energy part of the valence band). For CdTe on ZnTe, the valence band oset is of about 0.1 meV [START_REF] Continenza | Electronic properties and valence-band oset of strained ZnTe/CdTe (001) superlattices[END_REF]. The conduction band then takes most of the band gap oset. The connement of the electrons in the conduction band is thus far more ecient than connement of the hole.

Band structure near the center of the Brillouin zone (k = 0)

The optical properties of direct gap semiconductors such as CdTe and ZnTe are controlled by their symmetry at the center of the Brillouin zone, corresponding to an electron wave vector k = 0. At this point, the conduction band has the T d group Γ 6 symmetry. As discussed, this band comes from the overlap of atomic s orbitals, meaning the conduction electron will have no orbital momentum, and a total angular momentum J = 1 2

.

The valence band is formed from p orbitals, presenting an orbital momentum L = 1. It couples at k = 0 with the electron spin S = 3/2 through the spin-orbit coupling. L and S are then not good quantum number anymore. We can show that J = L + S, however, commute the hamiltonian presented in the following sections. This composition gives two sub-bands of total angular momentum J = 1/2 and J = 3/2. In the T d group, the quadruplet J = 3/2 is of Γ 8 symmetry, and the doublet J = 1/2 is of Γ 7 symmetry. Those bands are split by the spin-orbit interaction, with an energy ∆ SO . The eigenstates J 2 , J z are chosen as a basis for the system at k = 0.

The conduction band and valence band states can now be written from the composition of the two uncoupled basis:

Γ 6 : u Γ 6 , 1 2 = | 1 2 , 1 2 = |S | ↑ u Γ 6 ,-1 2 = | 1 2 , - 1 2 = |S | ↓ (I.1) Γ 8 : u Γ 8 , 3 2 = | 3 2 , 3 2 = |1 | ↑ u Γ 8 , 1 2 = | 3 2 , 1 2 = 2 3 |0 | ↑ + 1 3 |1 | ↓ u Γ 8 ,-1 2 = | 3 2 , - 1 2 = 2 3 |0 | ↓ + 1 3 |1 | ↑ u Γ 8 ,-3 2 = | 3 2 , - 3 2 = | -1 | ↓ (I.2) Γ 7 : | 1 2 , 1 2 = 2 3 |1 | ↓ - 1 3 |0 | ↑ | 1 2 , - 1 2 = - 2 3 | -1 | ↑ + 1 3 |0 | ↓ (I.3) with: |1 = - |X + i|Y √ 2 |0 = |Z | -1 = |X -i|Y √ 2
where |X , |Y and |X are the wave function of the valence band top at k = 0.

They are calculated from the hybridization of the atomic orbital p X , p Y and p Z . Since ∆ SO 0.9 eV for CdTe and ZnTe, the Γ 7 band is far enough in the valence band to be able to limit our study to the interaction between the Γ 6 and Γ 8 bands. The electrons in the conduction band have a spin σ = 1

2

. The spin operator can therefore be written using the Pauli matrices. For a quantization along the growth axis of the semiconductor, noted z:

σ x = 1 2 0 1 1 0 ; σ y = 1 2 0 -i i 0 ; σ z = 1 2 1 0 0 -1 (I.4)
In the same way, we can dene the operators at the top of the valence band. For an angular momentum J = 3 2 quantized along the z axis, we have in the basis (3/2, 1/2, -1/2, -3/2):

J x =      0 √ 3 2 0 0 √ 3 2 0 1 0 0 1 0 √ 3 2 0 0 √ 3 2 0      ; J y = i      0 - √ 3 2 0 0 √ 3 2 0 -1 0 0 1 0 - √ 3 2 0 0 √ 3 2 0      J z =     3 2 0 0 0 0 1 2 0 0 0 0 -1 2 0 0 0 0 -3 2     (I.
O + = O x + iO y (I.6) O -= O x -iO y (I.7)
We are mainly interested in two light matter interaction in the semiconductor: absorption of a photon of energy equal to or higher than the band gap by an electron of the valence band, to reach the conduction band; or emission of photon by an electron relaxing from the conduction band to the valence band. However, angular momentum conservation rules forbid some transitions. In order to nd them, we consider the coupling between a conduction band electron |Ψ c and a valence band hole |Ψ v through, in the dipolar approximation, the hamiltonian H dip = -d.E with d the dipole operator and E the electric eld, that can also be written H dip = -e m p.A with p the electron momentum and A the potential vector [START_REF] Rosencher | Optielectronics[END_REF]. In this section, we go with the later. We then get: Near this band edge, we can describe the curvature of the energy E(k) of the band using an eective mass for the carrier on it:

Ψ v |H AF |Ψ c ∝ u Γ 8 ,Jz |p|u Γ 6 ,σz (
E {c,v} (k) = - (hk) 2 2m {c,v} (k) (I.9)
As we move away from the Γ point, the valence band split into two branches: the one with small curvature, meaning a high eective mass for the carriers on it, is called the heavy-hole (hh) band; the one presenting the highest curvature and smallest eective mass is called the light-hole (lh) band. One way to understand this evolution is to apply the k.p approximation, as proposed by Kane in 1957 [40]. This model gives an estimation of the electronic band structure starting from the exact solution of the Schrödinger equation at the center of the Brillouin. The hamiltonian to resolve is then :

p 2 2m 0 + U (r) |ψ n,k = E n,k |ψ n,k (I.10)
with n the band index, p = -ih∇, U (r) the potential of the crystal, m 0 the mass of a free carrier and |ψ n,k the Bloch wave, separated between a periodic part u n,k (r) and plane-wave part exp(ik.r) as follow:

|ψ n,k = u n,k (r) exp(ik.r) (I.11)
After development of the gradient of ψ n,k , it is possible to rewrite Eq. I.10 as a function of the periodic part only:

p 2 2m 0 + U (r) + h2 k 2 2m 0 + h m 0 k.p u n,k = E n,k u n,k (I.12)
To nd the solutions around k = 0, we develop the u n,k on the basis of the {u n,0 } n as:

u n,k = n c n u n ,0
Assuming that the u n,0 are known, we can calculate the matrix elements of Eq. I.12.

The resolution of this hamiltonian is often done in the books [START_REF] Voon | The k.p method[END_REF], and gives, taking into account the Γ 6 and Γ 8 bands only: Taking into account the coupling with the higher energy band, hh are found to have indeed a negative curvature [START_REF] Voon | The k.p method[END_REF]. Considering only two bands is too rough to really model the system. k = 0: the Luttinger hamiltonian

E c (k z ) = E c + h2 k 2 z 2m c E v,± 1 2 (k z ) = E v - h2 k 2 z 2m lh E v,± 3 2 (k z ) = E v + h2 k 2 z 2m 0 (I.13) with E c (respectively, E v )
Another way to obtain the matrix describing the Γ 8 band is to use symmetry consideration. Luttinger showed in 1956 [42] that the only Hamiltonian fullling the cubic symmetry is:

H L = - h 2 2m 0 γ 1 k 2 I 4 -2γ 2 i=x,y,z k 2 i J 2 i - 1 3 J 2 -2γ 3 (k x k y (J x J y + J y J x ) + c.p.) (I.14)
with γ 1 , γ 2 and γ 3 the Luttinger parameters, I 4 the 4 × 4 identity matrix, k a vector of the Brillouin zone, J the angular momentum operator with J x , J y and J z being 4 × 4 matric satisfying [J x , J y ] = iJ z and circular permutation (c.p), such as the one dened on Eq. I.5. This hamiltonian can be simplied using the parameters:

A = γ 1 + 5 2 γ 2 B = 2γ 2 C = 2(γ 3 -γ 2 ) (I.15)
The Luttinger hamiltonian can then be rewritten:

H L = - h 2 2m 0 (Ak 2 I 4 -B(k.J) 2 + C(k x k y (J x J y + J y J x ) + c.p.)) (I.16)
The B-term lifts the degeneracy of the Γ 8 band into two lh and hh sub-bands, and is invariant under arbitrary rotations. The C-term describes the warping of the valence band.

In the spherical approximation, the Luttinger hamiltonian has two eigenvalues, giving us the value of the lh and hh eective mass:

E hh = - h2 k 2 2m 0 (A -2.25B) -1 = - h2 k 2 2m 0 (γ 1 -2γ 2 ) -1 = - h2 k 2 2m hh E lh = - h2 k 2 2m 0 (A -0.25B) -1 = - h2 k 2 2m 0 (γ 1 + 2γ 2 ) -1 = - h2 k 2 2m lh (I.17)
We nd that the hh presents a negative curvature, as was found in other works [START_REF] Voon | The k.p method[END_REF].

The parameters and carriers eective masses are given in the Tab. I.1.

In matrix form, using the basis (u

Γ 8 ,+ 3 2 , u Γ 8 ,+ 1 2 , u Γ 8 ,-1 2 , u Γ 8 ,-3 2
), the Luttinger hamiltonian can be written as: [START_REF] Besombes | Probing the spin state of a single magnetic ion in an individual quantum dot[END_REF] with: In order to represent these strains and see their eects on the bands, we need to dene a hamiltonian representing them. Strains deform the structure, so let's begin the representation with a volume V = xu x + yu y + zu z , with (u x , u y , u z ) an orthonormal basis. This volume will transform into another one V = xu x + yu y + zu z , where, for ε ij an expansion of the vector i in the direction j:

H L = - h2 2m 0     a hh b Lutt c Lutt 0 b * Lutt a lh 0 c Lutt c * Lutt 0 a lh -b Lutt 0 c * Lutt -b * Lutt a hh     (I.
a hh = (γ 1 -2γ 2 )k 2 z + (γ 1 + γ 2 )k 2 a lh = (γ 1 + 2γ 2 )k 2 z + (γ 1 -γ 2 )k 2 b Lutt = -2 √ 3γ 3 (k x -ik y )k z c Lutt = - √ 3(γ 2 (k 2 x -k 2 y ) -2iγ 3 k x k y )
u x = (1 + ε xx )u x + ε xy u y + ε xz u z u y = ε yx u x + (1 + ε yy )u y + ε yz u z u z = ε zx u x + ε zy u y + (1 + ε zz )u z (I.20)
They are small deformation of the lattice, so we choose |ε ij | 1. Such transformation can be decomposed in a symmetric part, the strain tensor, and an antisymetric one. We note the strain tensor ε, dened such as:

ε ii = ε ii (I.21) ε ij = 1 2 ε ij + ε ji (I.22)
In the linear regime, the strain tensor ε is proportional to the stress tensor σ, where σ ij describe a force parallel to i applied on a surface perpendicular to j. Therefore, σ ii will describe an elongation or compression stress, while σ ij (i = j) represents a shear stress. Since these tensor are symmetric, we can reduce the number of coecient from nine to six: σ xx , σ yy , σ zz , σ xy = σ yx , σ xz = σ zx and σ yz = σ zy . Since x, y and z are physically equivalent, as well as xy, xz and yz, only two diagonal coecient are needed: C 11 and C 44 . In the linear regime and for a cubic crystal, we can now write the Hooke's law:

        σ xx σ yy σ zz σ xy σ xz σ yz         =         C 11 C 12 C 12 0 0 0 C 12 C 11 C 12 0 0 0 C 12 C 12 C 11 0 0 0 0 0 0 2C 44 0 0 0 0 0 0 2C 44 0 0 0 0 0 0 2C 44                 ε xx ε yy ε zz ε xy ε xz ε yz         (I.23)
These coecient coupling strains in a direction to a force in the same direction are obviously positive.

When the aforementioned cube is compressed in one direction (e.g. ε zz < 0), it will expand in the other direction in order to minimize elastic energy (ε xx , ε yy > 0 in the example). If we do not allow strain in these other directions (ε xx = ε yy = 0), a stress in the x and y directions had to be applied to keep the cube from expanding in these directions (σ xx , σ yy < 0 in the example). We can therefore physically expect C 12 > 0.

The strain hamiltonian can be constructed noticing that the strain tensor ε induces a shift in the energy band, and that any ε ij has the same symmetry as k i k j .

The hamiltonian should then be formarly identical to the Luttinger hamiltonian.

In the Γ 8 subspace, we can then use the Luttinger Hamiltonian, written in Eq. I.14, replacing the k i k j by ε ij . We obtain the Bir-Pikus Hamiltonian by replacing the γ j parameters by the Bir-Pikus parameters a ν , b and d for the four levels at the top of the valence band [START_REF] Voon | The k.p method[END_REF]:

H BP = a ν ε + b i ε ii J 2 i - 1 3 J 2 + 2d √ 3 i>j ε ij {J i J j } (I.24) with ε = T r(ε) = ε xx + ε yy + ε zz and {J i J j } = 1 2 (J i J j + J j J i )
The a ν term, called the hydrostatic term, shifts the Γ 8 energy. In case of nonequal ε ii (shear strains), the b term will split the two Γ 8 sub-bands as did a k = 0 in the Luttinger hamiltonian. The d term, the pure shear strain (i.e ε ij with i = j), has the same eect on the Γ 8 band. For CdTe, Bir-Pikus parameter are a ν = 0.91 eV, b = -1.0 eV and d = -4.4 eV [START_REF] Adachi | Properties of group IV, II-V and II-VI semiconductors[END_REF].

One can notice that the Bir-Pikus hamiltonian is completely independant from k, meaning that the valence band hamiltonian of a strain semiconductor is simply the sum of the Luttinger hamiltonian H L (Eq. I.14) and the Bir-Pikus hamiltonian H BP (Eq. I.24).

Let see how this apply to a CdTe layer deposited on a ZnTe layer. As previously, we dene z as the growth direction. Since both semiconductors crystallize in a cubic lattice, the strains are the same in the x and y direction. We can then write the strains in the xy plane:

ε xx = ε yy = ε =
a ZnT e -a CdT e a CdT e (I.25)

In the z direction, however, no stress applies: the crystal is free to expand in this direction in order to reduce the elastic energy. Therefore, we can write σ zz = 0 and, according to Hooke's law in Eq. I.23:

σ zz = C 12 ε xx + C 12 ε yy + C 11 ε zz = 0 (I.26)
Using equality I.25, we can then deduce:

ε zz = - 2C 12 C 11 ε = - 2C 12 C 11
a ZnT e -a CdT e a CdT e (I.27)

Since we grow CdTe over a ZnTe substrate, the CdTe lattice is compressed in the plane, i.e. ε < 0. Since C 11 , C 12 > 0 and ε < 0 for CdTe over ZnTe (see Eq. I.25), one can easily deduce that ε zz > 0. In the hypothesis of no defect created by the lattice mismatch, all the other strain terms are equal to zero. We can then decompose this strain into two component: a hydrostatic part ε hyd describing the volume variation without breaking the cubic symmetry, and a shear part ε sh introducing an anisotropy, breaking this symmetry:

ε hyd = 1 3 (ε xx + ε yy + ε zz )I 3 (I.28) ε sh = ε -ε hyd (I.29)
One can notice that T r(ε hyd ) = T r(ε) = ε. Since in the case of a hydrostatic compression, such as for CdTe over ZnTe, ε hyd < 0, we then have ε < 0 and, according to the Bir-Pikus hamiltonian (Eq. I.24), the gap of CdTe increase.

Seeing that ε ij = 0 for i = j, we can rewrite the Bir-Pikus hamiltonian without the shear strain term. Moreover, since J 2 = J 2

x + J 2 y + J 2 z and ε xx = ε yy = ε , we can simplify this hamiltonian to:

H BP,biax = a ν εI 4 + b 3 (ε -ε zz )(J 2 x + J 2 y -2J 2 z ) (I.30)
And, since we are in the valence band with J = 3 2 and J 2 x + J 2 y + J 2 z = J(J + 1)I 4 , we can simplify the Bir-Pikus hamiltonian to its nal form in the case of biaxal strains:

H BP,biax = a ν ε + 5 4 b(ε -ε zz ) I 4 -b(ε -ε zz )J 2 z (I.31)
Using Eq. I.25 and I.27, we can easily calculate ε -ε zz . Since J z |n = n|n , we can nd the hh/lh splitting: We nd that, in a fully strained CdTe layer over a ZnTe substrate, the hh band is 300 meV above the lh one, and we could thus neglect the lh bad in rst approximation. However, to describe the valence band in QDs, one should take into account more complex eects, such as a coupling of the conned heavy hole with ground state light holes in the barriers [START_REF] Michler | Single quantum dots fundamentals, applications and new concepts[END_REF], or the the eective reduction of hh/lh splitting due to supercoupling via a dense manifold of hh like QD states lying between the conned heavy hole and light hole levels [START_REF] Luo | Supercoupling between heavy-hole and light-hole states in nanostructures[END_REF]. These eects can drastically change the hh/lh eective splitting, creating energy levels between lh and hh ones and will be needed to understand the dynamics of a hole coupled to an atomic atom presented in Chap. III and V.

∆ lh = E ± 3 2 -E ± 1 2 = -2b 1 + 2C 12 C 11
I.1.3 The quantum dot: conning the carrier in 3 dimensions Embedding a semiconductor in another one of larger band gap connes spatially the carriers in one or multiple directions. Using the procedure we will describe in Chap. II, we can create nanometre size islands of CdTe in a ZnTe lattice, eectively conning electrons in all three directions. This lead to a quantization of the carriers energy levels and a discretization of the optical properties. This connement being analogue to the Coulomb interaction of an isolated atom, such a structure is often dubbed "articial atom". The hole and the electron also interact through the Coulomb interaction, consisting of an attractive term, shifting energy levels, and an exchange interaction (discussed in Sec. I.1.4). The electron-hole system has a hydrogen-like behaviour and is called an exciton. The eects of connement on the carrier are generally described in the envelop function formalism. To dene these envelop functions, we develop the carriers wave-functions on all the Bloch states:

Ψ(r) = n,k c n,k ψ n,k = n,k c n,k u n,k (r) exp(ik.r) (I.33)
Since we are in a conned environment, we can consider only the states around k = 0. Since we consider the band extrema, we neglect for this part the inter-band wave function mixing and use the eective-mass approximation. We can then limit the expansion of Bloch state to an expansion on the u n,0 (r) exp(ik.r), with n = Γ 6 for the conduction band and n = Γ 8 for the valence band. The sum on Γ 8 is equivalent to a sum on the J z = {± 3 2 , ± 1 2 }. We can then write:

Ψ c (r) k u Γ 6 ,0 c Γ 6 ,k (r) exp(ik.r) = u Γ 6 ,0 F e (r) (I.34) Ψ v (r) Jz={± 3 2 ,± 1 2 },k u Γ 8 ,Jz c Jz,k (r) exp(ik.r) = Jz={± 3 2 ,± 1 2 } u Γ 8 ,Jz F Jz (r) (I.35)
with F e (r) = k c k exp(ik.r) the electron envelop function and

F Jz (r) = c Jz,k exp(ik.r), J z = {± 3 2 , ± 1 2 } the
( HL + HBP + V h (r))      F + 3 2 (r) F + 1 2 (r) F -1 2 (r) F -3 2 (r)      = E h      F + 3 2 (r) F + 1 2 (r) F -1 2 (r) F -3 2 (r)      (I.37)
with HL and HBP the hole hamiltonians. For those hamiltonians, we have to take the opposite of the electron hamiltonian dened in Eq. I.14. In HL , the k-terms transform into a gradient of the envelop function with the form i∇. For simplicity, the ˜will be dropped in the next equations. The derivation of the eective mass approximation can be found in reference [START_REF] Luttinger | Motion of electrons and holes in perturbed periodic elds[END_REF].

As pointed out in the end of Sec. I.1.2, the gap between lh and hh is of about 300 meV, wide enough to neglect the lh contribution in rst approximation. This is called the heavy hole approximation, decoupling the four dierential equations dened in Eq. I.37. Only the ground states | ± 3 2 are considered, with the eective mass given by the diagonal term of H L , noted m h, in the plane and m h,z along the growth axis. The spin operator J x , J y and J z can then be redened in the heavy-hole space as j x , j y , j z , written with the Pauli matrices like σ x , σ y and σ z .

Even with those two approximations, the problem is not solvable analytically.

However, it is possible for some chosen potentials. Let's consider a lens like quantum dot, with a radius in the xy plane, noted R, much larger than its height L z . We can therefore dene two dierent harmonic oscillators: a 2D oscillator V c,v (R) in the plane, and a 1D oscillator V c,v (z) along the growth axis:

V c,v (R) = 4∆E c,v R 2 L 2 z (I.38) V c,v (z) = 4∆E c,v z 2 L 2 z (I.39)
with ∆E c,v the band oset between the two conduction (resp. valence) bands.

The potential of the whole quantum dot will then be V

c,v (r) = V c,v (R) + V c,v (z).
Separating the potential in those two parts means we are searching for solution of the form F (z, R, θ) = χ(z)φ n,m (R, θ), with θ the angle between the position vector and the x axis. We write the characteristic spatial width σ and characteristic frequency ω of the 2D harmonic oscillator felt by the hole:

Σ h R = h m h, ω h R (I.40) ω h R = 8∆E v m h, L 2 R (I.41)
We can write the same equality along z replacing ρ by z and m h, by m h,z . The same can be done for electron, replacing the m h, or m h,z by m e and E v by E c .

We can nd in textbook such as ref. [START_REF] Basdevant | Mécanique quantique[END_REF] the eigenstates of a harmonic oscillator from which we can deduce the eigenstates for the ground state (GS) and the rst two degenerated excited states. The rst excited state is found to have an angular momentum l z = ±1, and is then noted Exc, ±1. The envelop functions and energy are then found to be:

F GS c,v (z, R, θ) = 1 ( √ πΣ z ) 1 2 exp - z 2 2Σ 2 z 1 ( √ πΣ R ) 1 2 exp - R 2 2Σ 2 R (I.42) E GS e,h = hω e,h z + ω e,h R 2 (I.43) F Exc,±1 c,v (z, R, θ) = 1 ( √ πΣ z ) 1 2 exp - z 2 2Σ 2 z 1 ( √ πΣ R ) 1 2 exp - R 2 2Σ 2 R R σ R exp(±iθ) (I.44) E Exc,±1 e,h = hω e,h z + 3ω e,h R 2 (I.45)
We see that these energy levels are quantized in a way looking like an isolated atom. In reference to the atomic notation, the ground state, lower energy level, is noted S and the two rst degenerated level are noted P , even though atomic p-states usually are 3 fold degenerated.

One remarkable feature of the envelop functions is that both GS and the two rst excited states present the same envelop along the z axis. The cause is directly the symmetry of the QD: since L z L R , ω e,h z ω e,h R , and since E osc. harmo. = (n + 1 2 )hω, the next possible envelop function along the z axis is at higher energy than the next one in the plane. This geometry is also responsible for the 2 fold degeneracy of the P -states.

Both the GS and the excited states are once again degenerated due to the spin of the electron and the hole. The electron is in the conduction band with the Γ 6 symmetry: its spin along the z axis is σ z = ± 1 2

(noted | ↑ for + 1 2 and | ↓ for -1
enough from the band edge to be negligible, the hole spin can only take the values

J z = ± 3 2 (noted | ⇑ for + 3 2 and | ⇓ for -3 2
). As pointed ahead, the hole is dened with the opposed characteristic of the missing electron. For instance, a hole | ⇓ corresponds to the absence of a valence electron

Ψ v (r) = u Γ 8 , 3 2 (r)F 3 2 (r).
From the selection rules found in Sec. I.1.1, we can see that the only excitons able to recombine optically in the hh approximation are either formed by

• an electron of spin σ z = -1 2 and a hole of angular momentum J z = + 3 2 (exciton of total angular momentum X z = +1, recombining in σ+ polarization) • an electron of spin σ z = + 1 2 and a hole of angular momentum J z = -3 2 (exciton of total angular momentum X z = -1, recombining in σpolarization).

Excitons of total angular moment X z = ±2 (electron spin σ z = ± 1 2 with hole spin J z = ± 3 2 exists, but cannot recombine optically.

The addition of envelop functions in the carriers wave function adds another selection term:

| Ψ v |p|Ψ c | 2 = | F v |F c | 2 | u Γ 8 ,Jz |p|u Γ 6 ,σz | 2 (I.46)
The rst term is the overlap of the envelop function, making sure the hole and the electron are of the right state. For instance, a transition between a S state of the valence band and a P state in the conduction band is then forbidden. The second term is the same as the one studied earlier, and from which we deduced the selection rules.

Approximating the QD potential as a harmonic potential usually overestimate the connement, and thus the single-particle energy. But the wave-functions found in this chapter can still be used as trial wave-functions for variational calculations in other potential, in order to calculate the correct energy level.

I.1.4 Electron-hole exchange in quantum dots

Electrons are fermions, and thus are subjected to the Pauli exclusion principle.

Being charged particles, they also interact with each other via the Coulomb interaction. Both of those have to be considered to write the interaction between the carriers in the semiconductor. It was shown by Wardzy«ski et al. [48] that the interaction between a conduction electron and all the electrons of the valence band can be written as an interaction between the considered electron and the corresponding hole. It can be separated into two terms: the direct Coulomb interaction, independent of the particles spins, and the exchange Coulomb interaction.

In excitons, the direct Coulomb term is attractive, as classically expected from an electric interaction between two opposite charges. However, more complex systems can exist in a semiconductor: charged excitons X + (hole-hole-electron com- plex) and X -(hole-electron-electron complex), or biexciton X 2 (two excitons of op- posite total angular momentum). Higher order multi-exciton or charged biexciton might also exists but they are not discussed in this thesis. In those multi-excitons, the total Coulomd interaction and the connement make the system stable.

Taking into account the symmetry of the crystal, Bir and Pikus demonstrated [START_REF] Pikus | Exchange interaction in excitons in semiconductors[END_REF] that the exchange hamiltonian between an electron of the conduction band and hole in the valence band can be decomposed in two dierent components: i) For an electron and a hole in the same Brillouin zone, the short-range exchange interaction has to be considered. It can be written:

H sr eh = I sr eh σ.j + i=x,y,z b exch i σ i j 3 i (I.47)
The rst term lift the degeneracy between exciton of total angular moment X = 2 and X = 1. The second one takes into account the reduction of symmetry in a cubic lattice and gives the dark states a ne structure. This splitting is expected to be much smaller than the lift induced by I sr eh , but has never been observed experimentally in bulk semiconductor.

ii) Carriers in dierent Brillouin-zone might be aected by the long-range exchange interaction. In bulk semiconductors, the long range term doesn't aect the bright exciton at k = 0. Since the radiative recombination we study occurs at k = 0 in the studied semiconductors, the eects of the long range interaction are not visible via this probe.

In a quantum dot, the connement of the carrier leads to a better overlap of the wave function and thus greater short range exchange energies. Moreover, in an anisotropic potential, such as the one of Stransky-Krastanov dots (see Sec. II.2), the long-range interaction mixes the bright excitons, splitting them in two levels. It is usually written as a δ 1 term in the exchange hamiltonian.

Taking into account all these eects, we can the write the total exchange hamiltonian in the heavy hole exciton subspace (

X z = | + 2 , | + 1 , | -1 , | -2 ): H eh = 1 2     δ 0 0 0 δ 2 0 -δ 0 δ 1 exp(-2iφ 1 ) 0 0 δ 1 exp(2iφ 1 ) -δ 0 0 δ 2 0 0 δ 0     (I.48)
with δ 0 = 3 2 I eh , representing the splitting between dark and bright exciton, I eh < 0 being the electron-hole exchange constant; δ 1 the splitting between the bright exciton states; and δ 2 the ne structure of the dark exciton states. The value of δ 0 is controlled both by long-range and short-range interaction, and is typically about 1 meV in CdTe/ZnTe. δ 1 only appears in anisotropic QDs and is induced by the long-range interaction, varying between a few tens and a few hundreds of µeV. Finally, δ 2 primarily arise from the short-range interaction.

Calculating the eigenstate of the hamiltonian I.48, we nd that the long range interaction induce a linear polarization of the PL of the optically active states along ϕ 1 and ϕ 1 + 90 • as follows:

|π ϕ 1 = 1 √ 2 (exp(-iϕ 1 )| + 1 + exp(iϕ 1 )| -1 ) (I.49) |π ϕ 1 +90 • = 1 √ 2 (exp(-iϕ 1 )| + 1 -exp(iϕ 1 )| -1 ) (I.50)
where ϕ 1 = π 4

corresponds to a polarization of the emission along the 110 axis.

This model works well for quantum dots with an elongated lens shape (C 2v symmetry), the shape taken for ideal QDs. However, more realistic self-assembled QDs can have symmetries which can deviate quite substantially from the idealized shapes of circular or ellipsoidal lenses. For a C s symmetry (truncated ellipsoidal lens), additional terms coupling the dark and the bright excitons have to be included in the electron-hole exchange Hamiltonian. Following ref.

[50], the general form of the electron-hole exchange hamiltonian in the heavy-hole exciton basis for a low symmetry quantum dot (C s ) and a polarization along 110 is: It is clear for the neutral species (X, X 2 ) that the angle between the polarization of the two lines is dierent from 90 • . Moreover, the charged species (X + , X -) present linear polarization dependencies. In those systems, the presence of two carriers (hole for X + , electron for X -) with opposite spin cancel out the electron- hole exchange interaction. Their linear polarization dependencies arise then from another phenomena.

H eh = 1 2     δ 0 e -iπ/
In order to understand the linear polarization dependency, we have include the light hole contribution. Looking at the general form of the Luttinger hamiltonian I.18, we see that it mixes heavy hole and light hole through its non-diagonal term b Lutt and c Lutt . The Bir-Pikus hamiltonian I.24 presents the same symmetry and the same form as the Luttinger hamiltonian and thus also induces a coupling between the lh states and the hh states. In general, we can write the hamiltonian describing the inuence of shape and strain on the valence structure in the (|

3 2 , + 3 2 , | 3 2 , + 1 2 , | 3 2 , - 1 2 , | 3 2 , - 3 2
) basis as:

H V BM =     p + q s r 0 s * p -q 0 r r * 0 p -q -s 0 r * -s * p + q     (I.52)
The induced hh/lh mixing is called Valence Band Mixing (VBM).

Supposing a VBM only caused by strain anisotropy, its parameters can be written as function of the Bir-Pikus parameters and the crystal strain ε ij (i, j = x, y, z). The VBM parameters then writes [START_REF] Voon | The k.p method[END_REF]:

p = a v T r(ε) (I.53) q = b ε zz - ε xx + ε yy 2 (I.54) r = b √ 3 2 (ε xx -ε yy ) -idε xy (I.55) s = d(ε xz -iε yz ) (I.56)
The splitting between the hh states and the lh states can now be calculated in function of the Bir-Pikus parameters:

∆ lh = E ± 3 2 -E ± 1 2 = (p + q) -(p -q) = 2b ε zz - ε xx + ε yy 2 (I.57)
If we now suppose a system with pure in-plane strain anisotropy (r = 0 and s = 0), for an origin of the energy at the top of the valence band, i.e. the hh band, we can rewrite the VBM hamiltonian in the same basis as above as:

H in plane V BM =     0 0 ρ s exp(-2iθ s ) 0 0 ∆ lh 0 ρ s exp(-2iθ s ) ρ s exp(2iθ s ) 0 ∆ lh 0 0 ρ s exp(2iθ s ) 0 0     (I.58)
with ρ s the strain coupling amplitude and θ s the angle between axis of the strain induced anisotropy in the QD plane and the x axis. . With this notation, in the limit of weak VBM, we can now rewrite the ground state of the holes as pseudo-spin in order to take the hh/lh mixing into account:

| ⇑ ∝ | + 3 2 - ρ s ∆ lh exp(2iθ s )| - 1 2 (I.59) | ⇓ ∝ | - 3 2 - ρ s ∆ lh exp(-2iθ s )| + 1 2 (I.60) J+ = ρ s ∆ lh 0 -2 √ 3 exp(-2iθ s ) 0 0 (I.61) J-= ρ s ∆ lh 0 0 -2 √ 3 exp(2iθ s ) 0 (I.62) Jz =    3 2 0 0 - 3 2    (I.63)
J± are the ladder operators, ipping the hole spin, whereas Jz return the spin value. This last operator shows these states are mainly hh. This pseudo spin description is usually enough to understand the eect of the VBM, and we will use it to study how it modies the emission of the quantum dot.

In order to do so, we begin to consider the emission of the negatively charged state X -. Since, as explained earlier, the exchange interaction is zero in such systems, because of the opposite spin of the two electrons, it will allow us to focus on the eects of the VBM. We can ignore the envelop function, testing mainly the overlap of the carriers wave function and thus not aecting the polarization of the emission. We write the polarization of the detection e = cos(α)e x + sin(α)e y . We then can nd the oscillator strength of the transition: For the charged states X + and X -, only the VBM leads to this linear polarization. However, in X and X 2 , the VBM and the long range exchange interaction are in competition for the polarization of the emission. The strain tends to polarize linearly the PL along θ s and θ s + 90 • , whereas the long range exchange interaction favour linear emission along ϕ 1 and ϕ 1 + 90 • . This explains that the angle between the two linearly polarized exciton lines is not equal to 90 • . Moreover, the valence band mixing results in a ne structure splitting through the short range exchange interaction that can either enhance or decrease the ne structure splitting due to the long range exchange interaction. In order to illustrate our point, we consider only the isotropic part of the short range electron-hole exchange interaction described in Eq. I.47:

Ω(α) ∝ | ↑ | cos(α)p X + sin(α)p Y | ↑↓ ⇑ | 2 = 1 + 1 3 ρ s ∆ lh 2 + 2 √ 3 ρ s ∆ lh cos(2(θ s -α)) (I.
H sr,iso eh = I eh σ.J (I. [START_REF] Gaj | Relation of magneto-optical properties of free excitons to spin alignment of mn 2 + ions in cd 1-x mn x te[END_REF] where 3 2 I eh gives the energy splitting between bright and dark excitons due to the short range exchange interaction. The coupling between the bright states | ↓ ⇑ and | ↑ ⇓ through H sr,iso eh can be calculated using the electron spin ladder operator dened in I.1.1 and the pseudo-spin ladder operator dened in I.61 and I.62:

↓ ⇑|H sr,iso eh | ↑ ⇓ = 1 2 √ 3 I eh ρ s ∆ lh exp(-2iθ s ) (I.66)
Hence, the valence band mixing through the short range exchange interaction splits the bright states into two linearly polarized states along axis dened by the strain angle θ s . The competition between this eect and the long range exchange interaction results in an angle between the two linearly polarized states dierent from 90 I.2 Exchange interaction between carrier and a magnetic atom I.2.1 Exchange interaction in diluted magnetic semiconductors

We are interested in thesis to introduce a low density of either Manganese (Mn)

or Chromium (Cr) atoms in the crystal. A semiconductor doped this way is called Diluted Magnetic Semiconductor (DMS). The magnetic atoms interact with the semiconductor electrons via its localized electrons on its outside d shell, via the exchange interaction. Specic properties of the DMS arise from this interaction.

In this thesis, we are at the limit of what is considered a DMS: the density of magnetic atom (Mn and Cr) is about the same of the density of QDs, in order to statistically get a single magnetic atom in some of the QDs.

In this section, we will express the interactions between the dierent electrons of the semiconductor as "Heisenberg" interactions:

H Heisenberg = Iσ.S (I.67)

with I the interaction constant, σ the electron spin and S the spin of the magnetic atom. This interaction represents the Pauli exclusion principle through the interaction between two spins. Since the same form will be used for all the interactions in this section, the dierent physical processes will be represented by dierent interaction constants I.

Hamiltonian of a DMS

Let's rst dened the wave functions considered in a DMS.

H BO = i p 2 i 2m c + V c (r i ) + 1 2 i,j e 2 4πε 0 |r i -r j | (I.68)
The rst term is a single particle hamiltonian, taking into account the kinetic energy of the electron and the crystal potential V c (r i ) felt by the electron at the position r i . This potential includes the impurities' potential. The nal term represents the Coulomb interaction between the electrons.

We can rewrite this hamiltonian using second quantication. We dene the destruction (resp. creation) operator of a particle in the conduction band at the wave vector k and the spin σ as a k,σ (resp. a † k,σ ). In the same way, we dene the destruction (resp. creation) operator of the electronic level of an impurity as a d,S (resp. a † d,S ). Assuming the number of electrons on the d orbital of the considered magnetic atom does not change, the hamiltonian I.68 then becomes:

H SQ = k,σ E k a † k,σ a k,σ + S E d a † d,S a d,S + k,k U k,k a † k,σ a k ,σ + k,σ,S V kd (a † k,σ a d,S + a † d,S a k,σ ) + 1 2 i,j,m,n V i,j,m,n a † i a † j a n a m (I.69) = H 0 + H d + V d + H hyb + H Coulomb with E k , E d , U k,k , V kd and V i,j,m,n the interactions constant. The constant electron
number assumption is good enough for the picture we want to draw since most of the spin-driven interactions do not induce a change of this number.

H 0 represents the energy of the unperturbed wave function of the semiconductor, with E k the energy of an electron with the wave vector k.

H d is the same as H 0 but for an electron of the d orbital of the considered magnetic impurity, with E d the energy of an electron on this orbital. It is included in H 0 in the following.

V d represents the impurity potential, allowing the semiconductor electrons to scatter on it. In CdTe, both Mn and Cr are isoelectric impurities, replacing Cd atoms. Their modication to the host semiconductor potential is negligible [58].

Therefore, |ψ k remain a good description for the electrons of the semiconductor.

H hyb , also called the Anderson hamiltonian, mixes the semiconductor states with the states of the impurities. This mixing leads to an exchange interaction between an electron of the semiconductor and one of the d orbital of an impurity.

H Coulomb is a two particles hamiltonian which represents the direct and the exchange Coulomb interaction between the electrons of the semiconductor and of the impurity. i, j, m and n each represents a full wave function, both spatial and spin part, and can be either an electron of the semiconductor or one of the impurities.

The Coulomb interaction in DMS

We rst focus on the action of H Coulomb . It can be separated in three dierent terms depending on the value of i, j, m and n as illustrated in Fig. I.7. Let's begin with the interaction noted J 1 on the diagram, between two states belonging to the continuum of the semiconductors. This is the hamiltonian H eh introduced in Sec. I.1.4.

The next interaction we consider is the one of two electrons from a localized atom. It represents internal transitions of the atom, given by the Hund rule. It is written: Hund rule is veried, with electrons rst lling all orbital states with parallel spin before adding an electron to an orbital with second one, with opposed spin.

H U = d,S,S U a † d,S a † d,S a d,S
The third interaction is the one between an electron from the magnetic atom and an electron from the semiconductor. Similarly to the carriers of the bulk, it can be separated in two terms that will be developed in the next paragraphs: a direct Coulomb interaction between the two electrons, and an exchange interaction arising from the fermionic nature of electrons.

The direct Coulomb interaction can be written:

K = + k,σ,σ K k a † k,σ a † d,σ a d,σ a k,σ (I.71) with K k = drdr |ψ k (r)| 2 e 2 4πε 0 |r -r| |Φ d (r )| 2
It is clear that the spin σ (resp. σ ) of the k electron (resp. d electron) are not involved in this interaction: they are not changed by it. The wave vector k is also not aected by this interaction. The direct Coulomb interaction therefore only acts on the total energy of the system. The origin of the energy axis can be redene to ignore it.

The second term, the exchange interaction, written in second quantication, reads:

J = + k,k ,σ,σ I ex kk a † k ,σ a † d,σ a k,σ a d,σ = - k,k ,σ,σ I ex kk a † k ,σ a k,σ a † d,σ a d,σ (I.72)
with

I ex kk = drdr ψ * k (r)ψ * k (r ) e 2 4πε 0 |r -r| Φ * d (r)Φ * d (r ) (I.73)
As can be seen on Eq. I.72, this interaction exchange the spin σ and σ of both electrons, as suggested by its name. Eq. I.73 shows that the spin interaction comes from a Coulomb interaction between two fermions.

We dene:

σ z kk = a † k,σ a k ,σ -a † k,-σ a k ,-σ σ + kk = a † k,σ a k ,-σ σ - kk = a † k,-σ a k ,σ (I.74)
Considering now that this interaction does not change the number of electrons on the d orbital of the considered magnetic atom, we can write the interaction as a Kondo hamiltonian [60]:

H sd = - k,k I ex kk σ k,k .S (I.75)
Since I ex k,k is positive, the negative sign in front of the Kondo hamiltonian shows that the energy minimum is reached when the spins of both electrons are aligned, and is therefore ferromagnetic.

We can write the total hamiltonian I.69, breaking the hamiltonian H Coulomb into its dierent parts:

H SQ = H 0 + H d + H hyb + H eh + H U + H sd (I.76)

Orbital hybridization

We now have two hamiltonians to model the exchange interaction between the impurity and the semiconductor electrons: H sd and H hyb . The rst one was put in Heisenberg form in the previous section. This section will focus on the hybridization. Its constant V kd can be written as [61]:

V kd = 1 √ N drΦ * d (r -R d )H HF (r)ψ k (r) (I.77)
with N the number of primitive cells in the crystal and H HF (r) the Hartree-Fock hamiltonian for a single electron.

H hyb = k,k V kd V k d 1 E k -(E d + U ) + 1 E k -(E d + U ) - 1 E k -E d - 1 E k -E d a † k ,σ a k,-σ a † d,S-2σ a d,S = - k,k I hyb kk a † k ,σ a k,-σ a † d,S-2σ a d,S (I.78)
This form results from a perturbative approach and supposes that the magnetic atom d levels are far from the band edge. This assumption works well for Mn.

However, the Cr ground state is at the edge of the valence band. Therefore,

knowing the sign of E k -E d or E k -E d is more dicult [63].
In order to give an idea of the construction of the hybridization, we will focus on the Mn case. The case of the Cr will be discussed more in details in Sec. I.2.2. Supposing the coupling occurs between two electrons with a close k (k k ), we can rewrite I hyb kk as:

I hyb kk = 2|V kd | 2 U (E k -E d )(E k -(E d + U ) = -2|V kd | 2 U (E k -E d )(E d + U -E k ) (I.79)
For a magnetic atom with a ground state deep in the valence band, as it is the case for the Mn atom, U and E k -E d are both positive, while E k -(E d + U ) is negative (see Fig. I.8). Thus, I hyb kk is negative, and leads to an anti-ferromagnetic coupling for the Mn atom.

Exchange constants in DMS

With this last transformation, it is now possible to use a Heisenberg type spin hamiltonian for all the exchange interactions instead of a hamiltonian mixing wave functions. The only dierence between the interactions is in the exchange constant:

I ex kk for the exchange, and I hyb kk for the hybridization.

Using the same hypothesis done on Sec. I.4 of small k value, and the value of V kd presented in Eq. I.77, we can rewrite the exchange constant: It has been shown that these results can be used both in the conduction band and in the valence band of semiconductors [START_REF] Bhattacharjee | Virtual bound state model for the exchange interaction in semimagnetic semiconductors such as Cd 1-x Mn x te[END_REF].

I hyb 00,{c,v} = -2 U (E {c,v} (0) -E d )(E d + U -E {c,v} (0)) |V kd | 2 (I.
In the conduction band, the orbitals are s, and so we will write the interaction I sd . However, since s orbitals have a spherical symmetry, there is no hybridization contribution [START_REF] Bhattacharjee | Virtual bound state model for the exchange interaction in semimagnetic semiconductors such as Cd 1-x Mn x te[END_REF]. The expression is then pretty easy:

I sd = I ex 00,c (I.82)
As discussed earlier, this lead to a ferromagnetic coupling between the conduction band electrons and the magnetic atom.

The valence band is formed by the p orbital of the semiconductor matrix, as discussed in Sec. I.4. We then write I pd as the sum of the hybridization and the exchange contributions:

I pd = I hyb 00,v + I ex 00,v (I.83)
The two interactions are in competition in the valence band, with the exchange interaction inducing a ferromagnetic coupling, and the hybridization inducing an anti-ferromagnetic one. The nal sign of the interaction depends on the material, and is more discussed in Sec. I.2.2.

However, we saw that, for the carriers of the valence band (electrons or holes), the good quantum number is the total angular momentum J and not the pure spin σ anymore. We have then to use J instead of σ for carriers of the valence band. And since J = 3/2 and the exchange constant I pd has been dened for σ = 1/2, the hole exchange hamiltonian have to be written using I pd /3 [START_REF] Bhattacharjee | Virtual bound state model for the exchange interaction in semimagnetic semiconductors such as Cd 1-x Mn x te[END_REF].

The interaction between the semiconductors carriers and one magnetic atom in the Heisenberg notation therefore reads:

H SQ = H 0 + H eh -I sd σ.S - I pd 3 J.S = H 0 + H eh + H sd + H pd (I.84)
Since a DMS contain a small percentage of magnetic atoms, we can write the full hamiltonian by summing the exchange interaction between carriers and a magnetic atom on all the sites with a magnetic atom. We nally get:

H DM S = H 0 + H eh - i I sd (R i )σ.S i - i I pd (R i )J.S i (I.85)
This can be further simplied with two approximations. Once again, let's begin with the conduction band. Since a conduction electron sees a lot of dierent atomic sites, we can work with the mean value of the magnetic atoms spins, S , instead of their individual value S i . This is the mean eld approximation, the magnetic atoms being seen as an eective magnetic eld. And for the same reason, we can consider the electron interaction with each site of the crystal multiplied by the probability x of being occupied by a magnetic atom, instead of summing only on the magnetic atoms positions. This is the virtual crystal approximation. We can then rewrite:

i I sd (R i )σ.S i = x R I sd (R)σ. S (I.86)
Projecting along the quantization axis, we just replace σ. S by σ z S z . Since the atoms are seen as a magnetic eld, they induce a splitting ∆E c between the two spin values of conduction electron, |σ z = ±1/2 :

∆E c = -N 0 xασ z S z (I.87)
with α ∝ I 00 sd the interaction constant between the impurity's and the conduction band's Bloch function at k = 0, and N 0 the number of cell per volume.

The same considerations can be done for the holes in the valence band. The splitting found for the heavy holes reads:

∆E v = -N 0 x β 3 J z S z (I.88)
with β ∝ I 00 pd the interaction constant between impurity's and the valence band's Bloch function at k = 0.

Interactions for k = 0

To be complete with the analysis, we should also take into account the connement due to the quantum dot. This means the wave vector k of the carriers can be dierent from 0, leading to small perturbative eect on H sd and H pd . This was done in details by Laurent Maingault in the Chap. I.3 of his PhD thesis [59]. It is shown that the hamiltonian changed as follow:

H sd (R) = -ασ.S F c (R) -A 2 ∂ 2 F c ∂z 2 (R) + ∂ 2 F c ∂ρ 2 (R) 2 -βσ.S (C 2 -B 2 ) ∂ 2 F c ∂z 2 (R) 2 + C 2 ∂ 2 F c ∂ρ 2 (R) 2 (I.89) H pd (R) = -βJ.S|F v (R) -V kd F v (R)| 2 (I.90) with F c (R) (resp. F v (R

I.2.2 Insertion of Mn or Cr atom in a semiconductor lattice

We will consider in the section the interaction between single magnetic atoms and the carrier trapped in quantum dots. Specically, we will be interested in the interaction with a Manganese or a Chromium atom. This does not change signicantly the picture we drew in the previous section. However, for the interaction between the semiconductor electrons and the magnetic atom d electrons, we will have to also take into account the overlap between the magnetic atom and the carriers.

For a magnetic atom A at the position R, we dene the exchange constant:

I eA = -α|F c (R)| 2 (I.91) I hA = - β 3 |F v (R)| 2 (I.92)

Mn in CdTe

Mn has an electronic structure [Ar]3d 5 4s 2 . When inserted into CdTe, it substitutes a Cd atom ([Kr]4d 10 5s 2 ). Both atoms are isolectronics and thus, as Cd, Mn bonds with the neighbouring Te atoms with its two s electrons of its outer shell. In the semiconductor matrix, it is then Mn 2+ that has to be considered, with an electronic structure [Ar]3d

5 . Since the d orbital is ve time degenerated (doubled when taking spin into account), it is half-full in the case of the Mn. Following

Hund's rule, the spin of those electron are all aligned, leading to a total electronic spin S M n = 5/2 and a total orbital momentum L = 0.

Using the exchange constant dened in Eq. I.91 and I.92, we can rewrite the hamiltonian I.84:

H cM n = H eM n + H hM n = I eM n σ.S M n + I hM n J.S M n (I.93)
We ignore H 0 here since it only shift the energy of the system without aecting the spins exchange interactions. We can then write its spin operator as we did for the carriers in Sec. I.4:

S M n,x =          0 √ 5 2 0 0 0 0 √ 5 2 0 √ 2 0 0 0 0 √ 2 0 3 2 0 0 0 0 3 2 0 √ 2 0 0 0 0 √ 2 0 √ 5 2 0 0 0 0 √ 5 2 0          S M n,y =          0 -i √ 5 2 0 0 0 0 √ 5 2 0 -i √ 2 0 0 0 0 √ 2 0 -3i 2 0 0 0 0 3 2 0 -i √ 2 0 0 0 0 √ 2 0 -i √ 5 2 0 0 0 0 √ 5 2 0          S M n,z =         5 2 0 0 0 0 0 0 3 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 -1 2 0 0 0 0 0 0 -3 2 0 0 0 0 0 0 -5 2         (I.94)
In the conduction band, the electrons s orbitals are orthogonal to the d orbital of the Mn atom. No hybridization can then occur. Only the Coulomd interaction remains, leading to a standard ferromagnetic interaction between conduction band electrons and Mn electronic spin. Conrming this, N 0 α = 0.22 ± 0.01 eV was measured [START_REF] Gaj | Relation of magneto-optical properties of free excitons to spin alignment of mn 2 + ions in cd 1-x mn x te[END_REF].

The deduction is a bit harder to work out in the valence band. Valence electrons p orbitals are not orthogonal to Mn electrons d orbital, meaning there is a competition between the standard ferromagnetic exchange interaction and the pd hybridization. It turns out that the hybridization is stronger than Coulomb exchange interaction for Mn in II-VI semiconductor, leading to an anti-ferromagnetic interaction between holes and Mn electronic spin [60]. For Cd x Mn 1-x Te, N 0 β = -0.88 ± 0.01 eV was measured, conrming this anti-ferromagnetism of the interaction. [START_REF] Gaj | Relation of magneto-optical properties of free excitons to spin alignment of mn 2 + ions in cd 1-x mn x te[END_REF].

Cr in CdTe

Cr as an electronic structure [Ar]3d 4 4s 2 . Cr is also isolectronic with the Cd and thus is introduced in the lattice in substitution to this atom, bonding with its two s electrons and thus being inserted as [Ar]3d 4 . Following Hunds rule for the Cr d orbital, with nd that Cr in CdTe has a total electronic spin S Cr = 2 and a total orbital momentum L = 2.

As done with the Mn, we can rewrite the hamiltonian I.84 using the exchange interaction dened in the introduction of this section:

H cCr = H eCr + H hCr = I eCr σ.S Cr + I hCr J.S Cr (I.95)
Once again, we ignore H 0 since it did not aect the spins exchange interaction.

We can also write the Cr spin operators as follows:

S Cr,x =          0 1 0 0 0 1 0 3 2 0 0 0 3 2 0 3 2 0 0 0 3 2 0 1 0 0 0 1 0          S Cr,y = i          0 -1 0 0 0 1 0 -3 2 0 0 0 3 2 0 -3 2 0 0 0 3 2 0 -1 0 0 0 1 0          (I.96) S Cr,z =       2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -2      
In the conduction band, the situation is the same as the Mn: the conduction electrons s orbital are orthogonal to the Chromium d orbital, and the hybridization is then zero. Only the Coulomb interaction remains and induces a ferromagnetic coupling between the electrons of the band and the Cr electronic spin. N 0 α was never measured in Cd x Cr 1-x Te, but most magnetic atoms in II-VI semiconductor presents a value between 0.2 eV and 0.3 eV. It is then generally assumed that, in II-VI semiconductor, N 0 α ≈ 0.2 eV [START_REF] Mac | p-d exchange interaction in Crbased diluted magnetic semiconductors[END_REF]. We also chose to use this value. results were coherent with an interaction of opposite signs for lh and hh [START_REF] Wojtowicz | Excitons in novel diluted magnetic semiconductor quantum structures[END_REF].

The evolution suggest a ferromagnetic interaction for hh and antiferromagnetic for lh. N 0 β was found positive (ferromagnetic) in Cd x Cr 1-x S [START_REF] Herbich | Magnetization and exciton spectroscopy of the diluted magnetic semiconductor cd 1-x cr x s[END_REF] and in all the studied Zn based II-VI semiconductors, assuming N 0 α = 0.2 eV [60]. Especially, in ZnTe, for N 0 α = 0.2, a large ferromagnetic N 0 β = +3.6 ± 1.2 eV was found [START_REF] Mac | p-d exchange interaction in Crbased diluted magnetic semiconductors[END_REF].

Since there is almost no valence band oset between ZnTe and CdTe and that the transition metal impurity levels does not signicantly changes between compounds with common anions [START_REF] Kossut | Handbook of magnetic materials[END_REF], it could be reasonable to assume that N 0 β in bulk CdTe is positive, giving a ferromagnetic hole-Cr interaction. However, any parameter that can change the position of the relative energies of the top of the valence band and of the d orbitals, such as strain or the connement, can aect the sign of this interaction.

I.3 Fine and hyperne structure of a magnetic atom in II-VI semiconductors I.3.1 Mn atom in II-VI semiconductors

We saw that the Mn d orbital is the outside orbital for Mn in CdTe and is halflled, with 5 electrons on it. Following Pauli exclusion principle, this gives the Mn atom in a CdTe lattice an angular momentum L = 0.

We get from Hund's rule that adding an electron to an half-lled orbital has a high energy cost. The lowest energy excited states are therefore the ipping of an electron. For a free atom, it leads to a rst excited state with an angular moment L = 4. Following the spectroscopic notation of these states (2S+1) L, the ground state is noted

6 S (L = 0, S = 5
2 ) and the rst excited state is

4 G (L = 4, S = 3 2 ).
Higher energy excited states also exist, but have no inuence in our study. We can safely neglect them. When inserted in the semiconductor matrix of T d symmetry, the interaction with the crystal eld will lift the degeneracies of the free atom level. Those new levels are written with the appropriate group theoritical transformation properties.

The 6 S ground state is symmetrical and non-degenerated, except for spin. Thus, it is not aected by the crystal eld and is noted 6 A 1 . On the other hand, 4 G, the rst excited state, has an orbital degeneracy and therefore is aected by the crystal eld. It split into four dierent states: 4 T 1 , three-fold degenerated ; 4 T 2 , three-fold degenerated ; 4 E, two-fold degenerated ; and 4 A 1 , not degenerated. It has been shown by calculations [7072] that the eect of the crystal eld is to pull the two 4 T levels closer to the ground state, while 4 E and 4 A 1 are weakly aected by the crystal eld.

For a free atom, the transition between the ground state with S = 5 2 and any of the excited states, with S = 3 2 is forbidden by the ∆S = 0 rule and the parity selection ones. However, in a II-VI lattice, the spin-orbit interaction and the absence of inversion symmetry relax those rules. The transitions between 6 A 1 and the dierent states coming from the 4 G splitting are allowed.

6 A 1 → 4 T 1
is the lowest energy of those transitions and therefore the most signicant. It corresponds to approximately 2.2 eV. This transition can kill the luminescence of a semiconductor: the electron-hole recombination ips a d electron instead of emitting a photon. However, this transition is unlikely to happen. For high concentration of Mn, it will signicantly reduces the photoluminescence of the sample. However, we work at a really low concentration of Mn atoms in order to have QD with only one of them. This phenomena shouldn't then have a noticeable eect on our samples.

6 A state of the Mn 2+ has no orbital degeneracy and thus is not aected by the T d crystal eld nor by the reduction of its symmetry by the biaxial strain.

However, the spin degeneracy is lifted by a combination of spin-orbit interaction and reduced symmetry of the crystal eld. In the presence of biaxial strain, this will results in a splitting the spin level according to D 0 S 2 z [START_REF] Qazzaz | Electron paramagnetic resonance of Mn 2+ in strained-layer semiconductor superlattices[END_REF], with D 0 the magnetic anisotropy and S z the Mn spin quantized along the z axis. Due to the null Mn orbital momentum (L = 0), the magnetic anisotropy is however weak and the spin splitting is only of a few tens of µeV, depending on the strains.

To describe in detail the spin energy levels of a Mn atom, one has also to consider its nuclear spin is I = 5 2 for all its isotopes. This nuclear spin couples to the 3d electrons via the hyperne interaction. The whole hamiltonian of an electronic spin coupled to the nuclear spin of a Mn atom in a strained layer grown along [001] axis is known from electron paramagnetic resonance in CdTe/ZnTe superlattices [START_REF] Qazzaz | Electron paramagnetic resonance of mn2+ in strained-layer semiconductor superlattices[END_REF] and reads: Bottom: schema of the ne and hyperne structure of a Mn atom in a strained II-VI QD. The inuence of the magnetic anisotropy D 0 , the in-plane anisotropy E and the hyperne interaction A are represented.

H M n = AI.S + 1 6 a(S 4 x + S 4 y + S 4 z - 1 5 S(S + 1)(3S 2 + 3S -1)) +D 0 (S 2 z - 1 3 S(S + 1)) + E(S 2 x -S 2
The rst term is the hyperne interaction between the Mn nuclear spin I and its 5d electrons spin S, coupling two consecutive Mn spin states through an electronnuclei ip op. The hyperne constant of the Mn was found to be A ≈ 0.71 µeV [START_REF] Causa | Spin-lattice coecients of mn2+ in iivi compounds[END_REF]. The second term results from the crystal cubic symmetry and mixes dierent S z of the Mn spin. According to ref. [START_REF] Causa | Spin-lattice coecients of mn2+ in iivi compounds[END_REF], a = 0.32 µeV.

The last line contains terms linked to the strain state at the Mn position: the magnetic anisotropy caused by bi-axial strain D 0 and the anisotropy of strain in the xy plane E. Because of the partial relaxation of strain in the self-assembled QD we used, the value of D 0 may vary between 0 µeV for a strain-free QD, to 12 µeV for a fully strain CdTe layer matched on ZnTe. Typical values around 7 µeV are usually observed in CdTe/ZnTe QDs [START_REF] Jamet | Spin dynamics of a mn atom in a semiconductor quantum dot under resonant optical excitation[END_REF][START_REF] Goryca | Coherent precession of an individual 5/2 spin[END_REF] The Jahn-Teller distortion further reduces the symmetry to D 2d and leads to a splitting of the 5 T 2 ground state into a 5-fold degenerate 5 B 2 orbital singlet and a 5 E orbital doublet .

The ground state orbital singlet 5 B 2 is further split by the spin-orbit interaction.

In a strain free crystal, it was found that the ground state splitting can be described by the spin eective Hamiltonian [78]:

H Cr,CF = D 0 S 2 z + 1 180 F [35S 2 z -30S(S + 1)S 2 z + 25S 2 z ] + 1 6 a[S 4 1 + S 4 2 + S 4 3 ] (I.98)
with the Cr spin S = 2 and |D 0 | |a|, |F |. In the following, we will use a = 0 and F = 0. The x, y, z principal axes were found to coincide with the cubic axes (1,2,3) giving rise to three identical sites, each given by I.98 but with the z axis of each along a dierent cubic axis (1,2,3 with S i dened as:

S θ = S 2 z - 1 2 [S 2 x + S 2 y ] S = 1 2 √ 3[S 2 x -S 2 y ] S ξ = S y S z + S z S y S η = S x S z + S z S x S ζ = S x S y + S y S x (I.100)
and e i dened similarly as:

e θ = ε zz - 1 2 [ε xx + ε yy ] e = 1 2 √ 3[ε xx -ε yy ] e ξ = ε yz + ε zy e η = ε xz + ε zx e ζ = ε xy + ε yx e A = ε xx + ε yy + ε zz (I.101)
As shown in Sec. I.1.2, we can write for a at self-assembled quantum dot with dominant large biaxial strain:

ε xx = ε yy = ε ε zz = -2 C 11 C 12
meV ) extracted from ref. [78]. c 1 c 2 c 3 c 4 c 5 -0.25±2 +4.9 ±2 -1.25±0.5 +4.9±2 +3.7±1.25
We can now simplify the hamiltonian I.99, rst reducing it to the active term in our case:

H Cr,ε = c 1 e A S θ + c 2 e θ S θ
Replacing now e A , e θ and S θ by their value given in I.100 and I.101, and using the equalities given in Sec. I.1.2, we can rewrite the strain controlled part of the spin

Hamiltonian as H Cr,ε , depending only on ε :

H Cr,ε = 3 2 ε [2c 1 (1 - C 12 C 11 ) -c 2 (1 + 2 C 12 C 11 )] S 2 z = D 0 S 2 z (I.102)
where we can estimate D 0 ≈ 1 ± 0.6 meV from the values of the spin-strain coupling coecients in CdTe (Tab. I.2). This is about 100 times stronger than the value found in Mn QDs, as stated on Sec. I.3.1. One should note the QDs could be partially relaxed and may contain a signicant amount of Zn. This could signicantly change the spin-strain coupling coecients of the Cr atom.

An anisotropy of the strain in the quantum dot plane (001) with principal axis along [010] or [100] axes (ε xx = ε yy and ε xy = ε yx = 0) would aect the Cr ne structure through the tetragonal coecient c 3 . This coupling can be described by an additional term in the spin-strain Hamiltonian

H Cr,ε ⊥ = 3 4 c 3 (ε xx -ε yy ) (S 2 x -S 2 y ) = E (S 2 x -S 2 y ) (I.103)
This anisotropy term E couples Cr spin states separated by two units and in particular S z =+1 to S z =-1 which are initially degenerated. It could be exploited to induce a large strain mediated coherent coupling between a mechanical oscillator and the Cr spin [81].

We can now group I.102 and I.103 in order to write the complete hamiltonian of an isolated Cr in a CdTe/ZnTe QD with anisotropic strains :

H Cr,ε = D 0 S 2 z + E(S 2 x -S 2 y ) (I.104)
This large sensitivity of the Cr spin to local strains makes the Cr a particularly interesting qubit for the development of hybrid spin-mechanical systems [8284].

I.4 The X-Mn system

In order to illustrate the concepts explained in this chapter, we will present a quick study of a well known system: the exciton coupled to a single Manganese atom.

In order to simplify the study, let's consider a QD containing a single Mn atom with no shape or strain anisotropy. When an exciton is injected in the QD, it will interact with the magnetic atom spin in addition to the hole-electron interaction already taking place for an exciton. The hamiltonian of the system reads:

H XM n = H eh + H eM n + H hM n (I.105)
We use in this part the heavy-hole approximation, ignoring the eect of the nondiagonal term of the band hamiltonian.

In the heavy hole subspace, in the σ z , j z , S z basis, all the hamiltonians describing an interaction with the hole are diagonal. Therefore, the only non-diagonal element in the hamiltonian I.105 describe electron-Mn spin-ips. The hole having no part in this interaction, those spin-ips couple a bright state to a dark state, separated by δ 0 , as dened in Sec. I.1.4. The strength of this interaction is of the same magnitude as I eM n , found to be in the 100 µeV range in our quantum dots [START_REF] Varghese | Dynamics of a Mn spin coupled to a single hole conned in a quantum dot[END_REF]. Since distance between the bright and dark states is about one order of magnitude higher than the strength of the interaction, we neglect it.

We can now consider the hamiltonian I.105 to be diagonal. The interaction of the Mn spin with electron is ferromagnetic with I eM n -100 µeV, while its interaction with the hole is anti-ferromagnetic with I hM n 250 µeV. The carriers act on the Mn spin as an eective magnetic eld oriented along the growth axis. This lift the degeneracy of the Mn spins in a sextuplet. For the bright exciton, with anti-parallel electron and hole spins, the splitting is the widest. Each Mn spin state is separated from the closest one by 1 2 (3I hM n -I eM n ), for a total energy span of 5 2 (3I hM n -I eM n ). For the dark exciton states, each of them are separated The bright and dark excitons are split by the electron-hole exchange interaction is

given by δ 0 = -3 2 I eh ≈ 1 meV, with I eh -750 µeV. All these splitting are a lot larger than the one induced by the crystal eld and the hyperne interaction.

One can note that the system is symmetric under inversion of the quantization axis z. Excitons with opposed spins coupled to opposed Mn spins will then be degenerated. For instance, |S z = +5/2, X z = -1 share the same energy level as |S z = -5/2, X z = +1 . The whole energy level structure is summarize in Fig.

I.13.
The fundamental state of the QD is an isolated Mn atom in the dot. In our QDs, we usually nd D 0 7 µeV [START_REF] Jamet | Spin dynamics of a mn atom in a semiconductor quantum dot under resonant optical excitation[END_REF][START_REF] Goryca | Coherent precession of an individual 5/2 spin[END_REF], negligible compared to the exciton-Mn interactions. Since the other ne structure parameters of the Mn are weaker than D 0 , we can neglect their contribution to the spectra. Therefore, the fundamental state of the QD is six time degenerated. The recombination does not aect the Mn spin. Each emission line is thus the superposition of two optical transitions with opposed circular polarization (recombination of an exciton X z = ±1). In a given polarization, each line corresponds then to a single Mn spin: the spectra is a picture of the statistic state of the Mn spin when the exciton recombines. This simple model is however incomplete. More thorough investigation of the system were done in [51]. The eects of the VBM, and the strain and shape anisotropy on the system was studied in [56]. A complete study of the spin dynamics of the system was also done by Claire Le Gall in her PhD thesis [START_REF] Gall | Dynamics and Optical contrôl of a single spin in a Quantum Dot[END_REF].

Chapter II We will focus in this chapter on the growth done at Tsukuba by Molecular Beam Epitaxy (MBE). We begin with some general explanation on the MBE process and the dierent tools that are used in it. We then go to the growth of the self-assembled Cr-doped QDs, explaining the preparation of the substrate, the growth, and discussing the rst characterization of the samples. In the last section, we present two other kinds of samples we grew: samples with the possibility of applying an electric eld on them, and the strain-free dots. For each of them, we explain the growth process and present basic characterization.

II.1 The Molecular Beam Expitaxy

The samples we used are grown using epitaxy. It consists on depositing atoms or molecules on the surface of the sample. Each elements are deposited on top of each other, and the lattice parameter of the substrate in the plane is kept as long as no defect appears in the material. The elements condensate on the surface, where they can diuse before getting adsorbed, or re-evaporate in the gas. The substrate temperature is tuned in order to maximise the mobility of the elements, to get a at surface, while minimizing the re-evaporation. To achieve such a growth, we used a process called Molecular Beam Epitaxy (MBE), in which ux of elements are evaporated toward the sample, where each atom or molecule is deposited and crystallizes. This is done using cells of elements heated to control their evaporation or until they reach sublimation.

The MBE process was rst devised at the end of the 1960s [85]. This method oers a good control on the growth, which makes it useful for the development of nanostructures. Depositing the materials layer by layer gives the possibility to grow really thin structure, and the transition between two materials can be made really abrupt, with the possibility to go from one material to another over a single monolayer (ML).

The elements are kept in Knudsen cells, which consist of a crucible of highmelting-point material wrapped in Tungsten lament acting as heater. A shutter is put in front of them to stop the element ux. During the growth, the shutter is opened to let the element travel to the substrate. The growth by MBE is a slow process: it can take more than a hour to get a layer 1 µm thick. In order to avoid contamination, Ultra High Vacuum, in the order of 10 -8 Pa, is needed. Under this condition, the mean free path of the gas is long compared to the distance to the sample (km for the mean free path of gas, when the distance between the substrate and the cell is of the order of 50 cm). This process is illustrated in Incident electrons have a wave vector k i = 2π/λ e , with λ e the electron wavelength, typically 30 or 40 pm for an electron at 30-40 keV. Since only scattered diraction is considered, the diracted wave vector k f as the same module as the incident one k i . The Ewald's sphere has then a radius equal to ||k i ||. If we had a perfect 2D crystal surface, the reciprocal lattice seen by the incident electrons would be innite lines. Their intersection with the Ewald sphere would give points. However, the crystal may present some defects and neither the gun nor the detector are perfect, leading to a broadening of the Ewald's sphere surface.

The intersection between the line and the sphere occurs on a given length, and therefore the diraction pattern of the electrons presents lines instead of points.

The growth of QDs makes the surface rough at the scale of the length of coherence of the beam. The electrons can interact with more layers while passing through the dots. This can be seen on the diraction pattern, where lines become points. II.2 Self-assembled CdTe/ZnTe quantum dots doped with single Cr atoms

II.2.1 Substrate preparation

The self-assembled Cr-doped QDs were grown on ZnTe(100) substrates. Two cleaning methods were tested for the substrate: i) etching of the substrate in a Bromide solution (Br

2 -C 2 H 5 OH)
ii) exposition of the substrate to a hydrogen radical plasma.

The etching process was done in four steps. All of them, except the etching in Bromide-ethanol, occured in an ultrasonic cleaning device vibrating the sample at 43 kHz for 3 minutes. We began with a cleaning in acetone, followed by one in ethanol. The third step was etching in Bromide-ethanol, with 3% of Bromide, for 1 minute. We rinsed the sample in methanol. Once rinsed, we keep the sample in ethanol until mounting it on the sample holder. The transfer from the ethanol to the MBE chamber takes a few minutes, and some contamination might occur during this time.

In order to avoid contamination during the transfer, another type of cleaning of the surface was tried: using hydrogen radical (H * ) to remove the impurity at the surface. The substrate was rinsed four times, rst in acetone, then ethanol and then water, for a duration of 5 min with sonication at 43 kHz, and nally 5 more minutes in water with sonication at 23 kHz. Once the sample was clean, it was mounted on the sample holder and loaded in the MBE system. The hydrogen radicals were formed by a RF power source operating at 13.6 MHz and at a power of 300 W. The substrate is exposed to the plasma for 15 min at 400

• C. The evolution of the surface was monitored with the RHEED: its pattern presented diraction lines after the exposition to H * plasma.

Most of the samples were cleaned using the Br-ethanol etching. Cleaning by H * radicals was tested later during my PhD, and only a few containing Cr-doped dots. Among the samples presented in this thesis, the only sample presented cleaned with the later technique is the charge control sample, dot390.

II.2.2 Stranski-Krastanov quantum dots growth

The self-assembled dots are formed using Stranski-Krastanov growth. A material with a dierent lattice parameter than the substrate is deposited. This lattice parameter dierence creates strains in the grown layer. Growing over the critical thickness, the layer may relax in dierent fashion, depending on the material. In Stranski-Krastanov growth, the relaxation occurs via the formation of island, the QDs, on the layer. QDs formed by this method are called Stranski-Krastanov dots (SK dots). For CdTe/ZnTe, the critical thickness is h c = 6.5 MLs [86].

However, the dots do not form naturally in CdTe/ZnTe: if left as is, dislocation will form in the layer to relax the strains. In order to form the dots, a layer of amorphous Tellurium has to be deposited on the surface of the sample, and then evaporated [55].

The CdTe QDs layer is grown by Atome Layer Epitaxy (ALE), or Migration-Enhanced Epitaxy (MEE). While in classical use of MBE, the elements of the material are co-deposited on the substrate, in ALE, each cell is open one after the other, in sequence. Between each opening, the sample is left under vacuum in order to relax the surface. Such a growth is auto-regulated: when only one element is open, only a given quantity of material will be deposited and then the growth will stop until the other element is also deposited. The deposited quantity of material before the growth stop depends only on the substrate temperature. This allows for a really ne control of the growth and of the thickness of the grown layer. A full cycle corresponds to opening each cell once.

The targeted uxes chosen for the growth of the CdTe/ZnTe QDs are presented The relative ux of the elements has no inuence on the ALE process. We only have to be careful to deposit enough of each element to saturate the surface at each opening. We therefore chose to use the same ux for both Cd and Te [88]. Taking a temperature between those two limits allows for a small uncertainty on the substrate temperature while keeping a really good control on the growth of the sample.

The growth during ALE was monitored via the intensity of the specular spot.

Focusing on the lowest angle reected spot, called the specular spot, one can see small variations in the reected intensity during the growth. This is the same mechanism as the one presented in Therefore a really small ux has to be chosen, with a BEP of the magnitude of 10 -9 Pa, which is of the same order of magnitude as the main chamber pressure.

It cannot therefore be reliably measured using the gauge pressure. In order to

II.2.3 Optical characterization

The samples were studied in Grenoble, at the Néel Institute. The characterization of the samples was done in two steps. First, we took macro-photoluminescence spectra, on a large energy range, typically between 1.8 and 2.3 eV, with a laser set at 2.9 eV. This allows us to test the PL of the sample. As described in Sec. I.3.2, if the Cr concentration is too high, it may kill the PL of the dot layer, and thus it will not be seen in the macro-PL.

If luminescence from the sample is seen in macro-PL, the sample is studied The samples where QDs doped with a single Cr atom were found are listed in Tab. II.2. The Cr concentration in the sample was estimated using the Cr ux. The Cr composition is given as percentage of the cation site occupancy (Cd 1-x Cr x Te).

We see that optimal samples were found for a large range of concentration. The probability to nd good Cr-doped dots in dot338 and dot363 is about the same.

However, dot363, with the higher Cr targeted concentration, presents a lot of broad peaks. All samples grown with a Cr concentration higher than 0.20% had a luminescence too weak to be studied properly. We can estimate that the good range of concentration is between 0.05% and 0.20%. Some more test have to be done in order to ne tune the Cr composition.

II.3 Charge tunable samples and strain-free samples

II.3.1 Charge tunable samples

Samples with contact on the back and on the surface were grown in order to control the charge of the studied QD via the application of an electric eld. The growth was done on a p-doped ZnTe substrate to have a conductive back contact on the sample. The growth of the SK dots was done with the procedure described in the previous section. The substrates of the charge tunable samples were cleaned using H * plasma. Thinner buer and cap layers were necessary in order to be able to apply a stronger electric eld on the dot layer. We chose thicknesses between 150 nm and 200 nm for the buer, and of 110 nm for the cap.

Optical characterization

In order to test the application of bias voltage in the sample, we glued two electrodes with silver lacquer. A SIL was mounted on top of the sample. We looked

The main result of the experiment is the appearance of dierent excitonic species depending on the applied electric eld. The spectrum is dominated by the neutral state of the quantum dot for an applied electric eld of V = -1.5 V (Fig. II.8 (c)). At this voltage, the neutral excitonic species X and X 2 are the strongest. Charged excitons still appear because of charge variations occuring close to the QD, injecting electrons or holes inside it. Further lowering the bias voltage, the probability of the dot to be negatively charged increases. Neutral excitonic species disappear and the peaks of the negatively charged species (X -, X -and X -2 ) become dominant. A spectrum taken at V = -8 V is presented on Fig. II.8 (d). One can notice that, X -is split into two group of two peaks, No Cr-doped quantum dots were found in dot390. Some dots looking like dots doped with a single Cr atom were observed, but detailed investigation shows no sign of magnetic atom in them. The properties of such QDs are discussed in more details in Sec. IV.3.

II.3.2 Strain-free quantum dots

In SK dots, large strains remain in the QD layer, and we have no control on them.

As will be discussed later, remaining strains may prevent the observation of all the spin states of the Cr atom and limit the life-time of the Cr spin. We decided therefore to grow strain-free dot to get rid of those limits. They are formed by the thickness uctuations of a CdTe quantum well in Cd 0.7 Mg 0.3 Te barriers, grown on a CdTe substrate. These uctuations form steps localizing the carriers, acting as For the growth of strain-free samples, we began to raise the substrate temperature to 300

• C and waited for a few seconds to remove all the deposited Te. We then heated the sample to 360 As discussed in Sec. I.4, the presence of a magnetic atom splits the emission of the exciton into several peaks, the number depending on the spin of the magnetic atom. Such complex was not found in the strain-free samples. This may be caused by a problem in the Cr incorporation. The Cr is possibly not well embedded in the QW layer. More tests have to be done, depositing the Cr at dierent moment of the growth.

Another possible cause of this absence could lie in the absence of strain. In SK dots, the presence of in-plane biaxial strains increases the probability of the Jahn-Teller deformation to occur along the z axis. In strain-free dots, all directions are equivalent, and we expect two thirds of the Cr-doped dots to be undetectable because they are quantized in the plane, creating no splitting visible in our setup.

In order to increase the probability for the Cr to be quantized along the z axis, it is proposed to slightly strain the quantum well. This might be done by growing the sample on Cd 0.96 Zn 0.04 Te instead of CdTe. CdTe and Cd 0.96 Zn 0.04 Te have a lattice mismatch of only 0.3%, so the strain created in the lattice should remain weak. They should however be enough to increase the probability of a deformation along the z axis, making more dots visible in our experiments.

Conclusion

We saw in this chapter how we grew the samples studied in the next chapters. The samples were characterized in Grenoble. SK dots with single Cr atom embedded were found and studied. New Cr concentrations were tested in Tsukuba after the feedbacks from Grenoble. More tests have to be done to increase the probability of nding dots doped with a single Cr atom.

First steps toward new kind of samples were also done: strain-free QD with single Cr, and SK dots with single Cr embedded in a eld eect structure. Localized carrier emission was detected in the former, and we successfully applied bias voltage on the later. However, for both, no Cr-doped dots were found. More tests have to be done to optimize their growth.

Chapter III Spin dynamics in Mn-doped positively charged quantum dots In this chapter, we look at the spin dynamics of a Mn spin coupled to a single hole in a CdTe/ZnTe QD. We begin presenting the energy structure of this system.

We show that Λ-level systems are formed between the hybrid spin states formed by the hole-Mn system in the ground state, and the electron-Mn states in the excited state. Those systems can be addressed independently with a resonant laser and are used to study the dynamics of the hole-Mn hybrid spin.

In the second section, we study the dynamics of the h-Mn hybrid spin through two experiments: auto-correlation of the resonant PL and resonant optical pumping experiments. We identied an ecient relaxation channel for this hybrid spin arising from the interplay of the exchange interaction between the hole and the magnetic atom, and the coupling to accoustic phonons. We also show that the optical Λ-systems are connected through inecient spin-ips that can be enhanced under weak transverse magnetic eld. The dynamics of the resonant PL of a p-doped magnetic QD is well described by a complete rate equation model.

In the last section, we look at the electron-Mn coherent dynamics, directly observed in the time resonant PL. Quantum beats reecting the coherent transfer of population between e-Mn spin states, mixed by an anisotropic strain in the plane of the quantum dot, are clearly observed. We highlight that this strain induced coherent coupling is tunable with an external magnetic eld. When a hole is trapped in a QD containing a single Mn, the spin structure is controlled by the hole-Mn exchange interaction that reads:

H ex hM n = I hM n S • J (III.2)
with I hM n the exchange energy between the hole and the Mn and J the hole spin operator. In the presence of heavy-hole/light-hole mixing, J, represented in the basis of the two low energy heavy-hole states, is related to the Pauli matrices by

J z = 3 2 τ z and J ± = ξτ ± with ξ = -2 √ 3e -2iθ ρ c /∆ lh , τ + = τ
x +iτ y and τ -= τ x -iτ y , where τ x , τ y , τ z are the Pauli matrices. ρ c is the coupling energy between heavy holes and light holes separated by an eective energy splitting ∆ lh . θ is the angle relative to the [110] axis of the principal axis of the anisotropy (shape and/or strain) responsible for the heavy-hole/light-hole mixing [52,97]. The matrices J z and J ± are described in more details in Eq. I.61 to I.63. For a weak valence band mixing, the hole-Mn energy levels are mainly controlled by I hM n S z J z and form 12 eigenstates organized in 6 doublets with well-dened S z and J z . These states are labelled |S z , J z . |M, M z , we highlighted the weight of each recombination path in a given polarization (σ-(red) and σ+ (blue)). This weight is directly linked to the intensity of each peak. In the center, the dierent possible recombination paths for M = 3 and M = 2 are illustrated, with a schema of the resulting spectra below.

When an e-h pair is optically created, a positively charged exciton X + forms with the hole trapped in the QD via the bias voltage, coupled with the Mn atom.

The two holes associate with anti-parallel spins. Thus, in the excited state, the system is dominated by the electron-Mn exchange interaction:

H ex eM n = I eM n S.σ

(III.3)
with, as usual, σ the electron spin and I eM n the e-Mn exchange energy. This interaction is isotrope, and results in a ground state septuplet of total spin M = 3 and a vefold degenerated manifold of total spin M = 2, for a total of twelve electron-Mn states. In the absence of perturbations, the energy levels of each of They will therefore be detected in only one polarization. This leads to six dierent PL lines in each polarization, each of them being linked to a given hole-Mn ground state.

To illustrate the association between the e-Mn and the h-Mn states, let's consider the recombinations in σ+ polarization, meaning we consider the annihilation of the e-h pair | ↓ e , ⇑ h . We have an excited state of the form |M, M z = α|S z , ↓ e + β|S z -1/2, ↑ e and a nal state after the e-h annihilation |S z , ⇓ h , which is an eigenstate of the hamiltonian H h-M n . The intensity of the optical transition is given by the overlap M, M z |S z , ↓ e , which is the Clebsh-Gordan coecient appearing in the composition of a spin 1/2 with a spin 5/2, here noted α for the initial state associated with the σ+ transition. Let's focus on the transition linked with the high energy nal state | -5/2, ⇓ h . In order to have this state as a nal state, the initial state has to be | ⇑ h , ⇓ h × | -5/2, ↓ e . Only the excited state |3, -3 have this e-Mn state, this transition being forbidden from the M = 2 quintuplet. Since |3, -3 is a pure e-Mn state, this transition has the highest optical transition weight.

We can repeat this operation with each h-Mn state, in both polarizations.

They can all be associated with both |2, M z and |3, M z initial states, with optical weights lying between 1/6 and 5/6. The only exception is the other high energy nal state, | + 5/2, ⇑ h , only associated with the initial state |3, +3 in σ- There are two things we neglected in the previous paragraphs: the Valence Band Mixing and the perturbative eect of the hole-Mn exchange interaction.

The VBM mixes the h-Mn states two by two, and induces population transfer between the dierent levels [START_REF] Varghese | Dynamics of a Mn spin coupled to a single hole conned in a quantum dot[END_REF]. For ρ s /∆ lh 1, the eect of this interaction is small both on the wave function and on the degeneracy of all the h-Mn doublets except the third, which is split. This is illustrated in The wave function of the hole in the ground state and of the charged exciton in the excited state are also perturbed by the h-Mn exchange interaction. This perturbation depends on the value of the exchange energy between the Mn spin S z and the hole spin J z . It can be represented using second-order perturbation theory by an eective spin hamiltonian [START_REF] Besombes | Resonant optical pumping of a Mn spin in a strain-free quantum dot[END_REF]98,99]:

H scat = -ηS 2 z (III.4)
with η > 0. This perturbation has to be taken into account twice in the excited state where there are two holes interacting with the Mn. It is responsible for the irregular energy spacing of the X + -Mn spectra, that can be seen on Including the eect of a magnetic eld B, the total hamiltonian of the ground state (h-Mn) can be written:

H hM n = H ex hM n + H scat + H CF,M n + H mag,GS = I hM n S.J -ηS 2 z + D 0 S 2 z + E(S 2 x -S 2 y ) + g M n µ B S.B + g h µ B J.B (III.5)
and, for the excited state (X + -Mn):

H X + -M n = H ex eM n + 2H scat + H CF,M n + H mag,exc = I eM n S.σ -2ηS 2 z + D 0 S 2 z + E(S 2 x -S 2 y ) + g M n µ B S.B + g e µ B σ.B (III.6)
The full energy structure given by those hamiltonian is presented in (a). We see three sharp absorption lines, labelled (1), ( 2) and ( 3). These lines are strongly σpolarized, cross-polarized with the excitation, except (1) which is unpolarized. Those peaks appear when the laser is in resonance with specic QD transitions. We can associate a given high energy transition to each of the low energy peaks.

These hole-Mn transitions are characteristic Λ-level systems linking one excited state (e-Mn) to two ground states (h-Mn). All the e-Mn state can be written as

|M, M z = α|S z , ↓ e + β|S z -1/2, ↑ e .
The rst e-Mn state recombines towards the hole-Mn state |S z , ⇓ h , emitting a photons in σ+ polarization, while |S z -1/2, ↓ e recombines towards |S z -1/2, ⇑ h , emitting a photon in σpolarization. When an exciton is injected in the QD, the electron and Mn spins will be able to ip op between those two states.

Using

|3, +1 = 1 √ 6 ( √ 4| + 1/2, ↑ e + √ 2| + 3/2, ↓ e ) (III.7) |3, +2 = 1 √ 6 ( √ 5| + 3/2, ↑ e + √ 1| + 5/2, ↓ e ) (III.8) |2, +2 = 1 √ 6 ( √ 1| + 3/2, ↑ e - √ 5| + 5/2, ↓ e ) (III.9)
The energy splitting between the resonant absorption and the emission corresponds to the splitting between the two hole-Mn ground states of the Λ system. It is given by 4 × (3/2)I hM n (≈2.1 meV for the QD in III.2 Hole-Mn spin dynamics under resonant excitation III.2.1 Cycling in and escaping from the Λ-level system

These Λ systems can be used to probe the dynamics of the hybrid h-Mn spin.

Under resonant excitation of one of the branch, a fast optical pumping controlled by the generation rate and the radiative lifetime of the excited state is expected for an isolated Λ system: the population should be stored in the level which is not excited and the resonant PL should vanish. In the case of X + -Mn, the PL intensity observed under resonant excitation of the high energy branch of the Λ systems is similar to the PL intensity obtained under non-resonant excitation. This suggests a very inecient optical pumping of the hole-Mn spin and an ecient spin-ip mechanism which links the two ground states of the Λ systems. given by the second order correlation function of the resonant PL intensity. To access it, a Handury Brown and Twiss (HBT) setup was exploited with a resolution of about 0.8 ns to perform photon-correlation measurements. Under our experimental conditions, with photon count rates of a few kHz, the measured photon pair time distribution yields, after normalization, to the second order correlation function g (2) (τ ) of the PL intensity. Permanent magnet mounted on a translation stage were used to apply a weak magnetic eld in Faraday (magnetic eld parallel to the laser direction) or Voight (magnetic eld perpendicular to the laser direction) congurations.

For the three resonant excitation conditions reported in Fig. III.5, g (2) (τ ) is mainly characterized by a large photon bunching with a full width at half maximum (FWHM) in the 20 ns range. The amplitude of the bunching reaches 9 for line (2)

and is slightly weaker for the two other lines. This large bunching, reecting an intermittency in the emission of the QD, is not sensitive to a longitudinal magnetic eld B z except for an excitation on (1).

The presence of a photon bunching is surprising at rst: under resonant excitation of an isolated Λ system, an anti-bunching of the resonant PL controlled by the transfer time between the two ground states is indeed expected for a single emitter. For X + -Mn, the observed short anti-bunching (dip near zero delay, better evidenced in In the presence of a transfer process connecting the two hole-Mn ground states in a nanosecond time-scale (τ f f ), the photon bunching can be explained by leaks outside the resonantly excited Λ system. Under CW excitation, the population cycles inside the Λ system until a spin ip occurs and drives the carrier-Mn spin out of the Λ levels under investigation. The resonant PL is then switched o until multiple spin-ips drives back the carriers and Mn spin inside the Λ system under excitation. The selected QD line can be either in a ON (populated) or OFF (empty) state depending on the uctuations of the carrier and Mn spins. The amplitude of the bunching is then given by Γ Out /Γ In , the ratio of the transition rates from OFF to ON (Γ In ) and from ON to OFF (Γ Out ). An amplitude of g (2) (τ )

higher than 1 is expected for the multilevel system considered here where, after a spin relaxation, multiple spin ips are in average required to come back to the initial state (Γ In < Γ Out ). Thus the width of the bunching is a measurement of the escape time out of the Λ level system. We present these transitions in As the spin of the hole-Mn complex is highly anisotropic, with a large energy splitting induced by the exchange interaction I hM n S z J z , the weak transverse magnetic eld mainly aects the electron-Mn dynamics in the excited state of the charged QD. Indeed, the transverse magnetic eld couples the dierent electron-Mn states and induces a leak outside the resonantly excited Λ system. Both spin-ips within the hole-Mn (ground state) and the electron-Mn (excited state) systems can contribute to the "bunching signal".

The signicant eect of the weak transverse eld shows that the probability of presence in the excited state of the Λ system is large. This is consistent with the large excitation intensity used for these auto-correlation measurements which requires a high photon count rate.

A slight reduction of the width of the bunching signal is also observed with the increase of the excitation power (Fig. III.7 (a)). This shows that the leaks outside a given Λ system slightly increases with the probability of presence of the positively charged exciton in the QD.

III.2.2 Resonant optical pumping of the hole-Mn spin

Resonant optical pumping experiments were done to estimate how long it takes, after a spin-ip, for the hybrid hole-Mn spin to come back inside the resonantly excited Λ system. The experiments were performed using trains of circularly polarized light, prepared with an electro-optic or an acousto-optic modulator with a switching time of about 10 ns. The resonant PL was detected with a fast avalanche photodiode. As with the autocorrelations, a permanent magnet mounted on a translation stage were used to apply a weak magnetic eld in Faraday or Voight congurations.

A demonstration of resonant optical pumping of the hole-Mn system was rst As observed for the auto-correlation, the resonant pumping signal is also strongly within a few nanoseconds.

Another source of spin pumping can be the leak outside the resonantly excited Λ system. In this case, the speed of the pumping is controlled by the leakage time and, as observed experimentally, the pumping time is similar to the width of the photon bunching signal. This mechanism of pumping for the hole-Mn spin is conrmed by the transverse magnetic eld dependence. The acceleration of the optical pumping in transverse magnetic eld (Fig. III.9 (b)) has the same origin as the decrease of the width of the bunching signal. By mixing the dierent electron-Mn states, the transverse eld enhances the leakage probability out of the resonantly driven Λ system and decreases the corresponding optical pumping time.

III.2.3 Modelling of spin relaxation mechanism: hole-Mn ip-ops mediated by a lattice deformation

The observed large resonant PL amplitude of X + -Mn and its dynamics can be qualitatively explained if a fast (nanosecond) and ecient spin transfer mechanism connects the two hole-Mn ground states of each Λ system. This transfer time is noted τ f f . We propose a mechanism for the hole-Mn ip-op at low temperature resulting from a deformation induced exchange interaction [101,102]. We show here that hole-Mn states are eciently coupled via the interplay of their exchange interaction and the lattice deformation induced by acoustic phonons. We will focus in the following on the two hole-Mn states |+3/2; ⇑ h and |+5/2; ⇓ h in the ground states of the Λ system associated with the electron-Mn levels |3, +2 and |2, +2 . Similar results could be obtained with the hole-Mn ground states of the other Λ systems.

First, let us notice that the non diagonal term of the hole-Mn exchange interaction I hM n 2 (S + J -+ S -J + ) couples the heavy-holes (⇑ h , ⇓ h ) and light-holes (↑ h , ↓ h ) levels split by ∆ lh through a hole-Mn ip-op. We consider this interaction as a perturbation on the Mn heavy-hole level structure given by I hM n S z J z . As presented in Sec. I.1.5, we can rewrite the two perturbed ground states | + 3 2 ; ⇑ h and | + 5 2 ; ⇓ h of the Λ system as:

| + 5 2 ; ⇓ h = | + 5 2 ; ⇓ h - √ 15 2 I hM n ∆ lh | + 3 2 ; ↓ h | + 3 2 ; ⇑ h = | + 3 2 ; ⇑ h - √ 15 2 I hM n ∆ lh | + 5 2 ; ↑ h (III.10)
where we neglect the exchange energy shifts of the hole-Mn levels much smaller than ∆ lh .

Phonon-induced deformations comes into play via the o-diagonal terms of the Bir-Pikus Hamiltonian H BP , describing the inuence of strain on the valence band, as written in Eq. I.24. The strain produced by phonon vibrations couples the perturbed hole-Mn states | + 5/2 ⇓ h and | + 3/2 ⇑ h through the Hamiltonian term Table III.2: Material (CdTe or ZnTe) [START_REF] Adachi | Properties of group IV, II-V and II-VI semiconductors[END_REF] and QD parameters used in the calculation of the coupled hole and Mn spin relaxation time.

+ 5 2 ; ⇓ h |H BP | + 3 2 ; ⇑ h = 2 × - √ 15

CdTe

; ⇑ h , | + 5 2 ; ⇓ h , | + 3 2 ; ⇑ h , | + 3 2 ; ⇓ h } is H int = - √ 15 I hM n ∆ lh R * | + 5 2 ; ⇓ h + 3 2 ; ⇑ h | + H.c
(III.12)

The spin decay rates from | + 3/2; ⇑ h to | + 5/2; ⇓ h accompanied by the emission of an acoustic phonon is given by Fermi's golden rule

τ -1 f f = 2π h k + 5 2 ; ⇓ h ; ψ; n k + 1|H int | + 3 2 ; ⇑ h ; ψ; n k 2 × δ(hω 0 -hω k ) (III.13)
where hω 0 is the energy splitting between | + 5/2; ⇓ h and | + 3/2; ⇑ h , n k the number of phonons in mode k and ψ the orbital part of the hole wave function.

To evaluate the matrix element in Eq. III.13 we use the strain tensor components ij given by ij = 1 2 ∂u i ∂r j + ∂u j ∂r i

(III. [START_REF] Veldhorst | A two-qubit logic gate in silicon[END_REF] where u(r) is the local displacement eld. For an acoustic phonon, the quantized displacement eld can be written in the real space [102,103]:

u(r) = i k,λ h 2ρω k,λ N ν 0 e k,λ (b k,λ + b † -k,λ ) exp(ikr) (III.15)
where N is the number of unit cells in the crystal, ν 0 is the volume of a cell and ρ the mass density. b † k,λ (b k,λ ) is the creation (annihilation) operator of phonon in the mode (k, λ) of energy hω k,λ and e k,λ is the unit polarization vector. In zinc-blend crystals there are two transverse acoustic phonon branches λ = t 1 , t 2 and one longitudinal acoustic phonon branch λ = l. The polarization vectors of these phonons branches are given by [104]:

e k,l = k k = 1 k (k x , k y , k z ) e k,t 1 = 1 kk ⊥ (k x k z , k y k z , -k 2 ⊥ ) e k,t 2 = 1 k ⊥ (k y , -k x , 0) (III.16) with k ⊥ = k 2 x + k 2
y . Using Eq. III.14, III.15 and III.16, we calculate the matrix element in Eq. III.13:

|M k,λ | 2 = 15 I hM n ∆ lh 2 h 2ρω k,λ N ν 0 (n B (ω k,λ ) + 1) × 3b 2 4 (k x e x,λ -k y e y,λ ) 2 + d 2 4 (k x e y,λ + k y e x,λ ) 2 × |F λ (k)| 2 (III.17) with F λ (k) = ∞ -∞ d 3 rψ * (r) exp(ikrψ(r)) (III.18)
and n B (ω k,λ ) = 1/(e hω k,λ /K B T -1), the thermal phonon distribution function.

For a Gaussian hole wave function with in-plane and z-direction parameters l ⊥ and l z respectively (full width at half maximum 2 √ 2 ln 2l i ) we have

ψ(r) = 1 π 3/4 l ⊥ √ l z exp - 1 2 r ⊥ l ⊥ 2 + z l z 2 (III.19)
Then, the form factor F λ (k), which is the Fourier transform of |ψ(r)| 2 , becomes

F λ (k) = exp - 1 4 (l ⊥ k ⊥ ) 2 + (l z k z ) 2 (III.20)
Considering a linear dispersion of acoustic phonons ω k,λ = c λ k and in spherical coordinates with k = k(sin θ cos ϕ, sin θ sin ϕ, cos θ) Eq. III.13 reads:

τ -1 f f = λ 15 (2π) 2 I hM n ∆ lh 2 ω 0 c λ 3 1 2hρc 2 λ π 4 (3b 2 + d 2 ) × (n B (ω 0 ) + 1)) π 0 dθ sin θ|F λ (ω 0 , θ)| 2 G λ (θ) (III.21)
where we used the continuum limit ( k → V /(2π) 3 d 3 k with V = N ν 0 the crystal volume) and integrated over k and ϕ. The summation is taken over the acoustic phonon branches λ of corresponding sound velocity c λ . The geometrical form factors of each phonon branch G λ (θ) is given by QDs listed in Tab. III.2. The calculated relaxation time strongly depends on the energy separation between the hole-Mn levels hω 0 . This energy dependence is controlled by the size of the hole wave-function given by l ⊥ and l z . The estimated ip-op time is also strongly sensitive to the exchange induced mixing of the ground heavy-hole states with the higher energy light-hole levels.

G l (θ) = sin 4 θ G t 1 (θ) = sin 2 θ cos 2 θ G t 2 (θ) = sin 2 θ (III.
In the model presented here, this mixing is controlled by ∆ lh , the eective energy splitting between heavy-holes and light-holes. This parameter can describe more complex eects such as a coupling of the conned heavy-hole with ground state light-holes in the barriers [START_REF] Michler | Single quantum dots fundamentals, applications and new concepts[END_REF] or an eective reduction of heavy-hole/lighthole splitting due to a presence of a dense manifold of heavy-hole like QD states lying between the conned heavy-hole and light-hole levels [START_REF] Luo | Supercoupling between heavy-hole and light-hole states in nanostructures[END_REF]. From this modelling it is deduced that, for a hole conned in small Cd x Zn 1-x Te alloy QDs, the hole-Mn ip-op time τ f f can easily be bellow 2 ns for an eective heavy-hole/lighthole splitting ∆ lh = 15 meV and an energy splitting hω 0 in the meV range. The value of τ f f = 1.5 ns will be used in the following for the ground states of each Λ system.

III.2.4 Model of the carrier-Mn spin dynamics under resonant excitation

Using the level scheme presented in Fig. III.3 (a) for a positively charged Mn-doped QD and the estimated hole-Mn ip-op rates, we calculated the time evolution of the 24 × 24 density matrix describing the population and the coherence of the 12 electron-Mn states in the excited state and the 12 hole-Mn states in the ground state of a positively charged QD. In the Markovian approximation, the master equation which governs the evolution of can be written in a general form (Lindblad form) as:

∂ ∂t = -i h [H, ] + L (III.23)
where H is the Hamiltonian of the complete system hole-Mn, H hM n , and X + -Mn, H X + M n , as described in Eq. III.5 and III.6. In Eq. III.23, L describes the coupling or decay channels resulting from an interaction with the environment [START_REF] Jamet | Spin dynamics of a mn atom in a semiconductor quantum dot under resonant optical excitation[END_REF]105,106]. The population transfers from level j to level i in an irreversible process associated with a coupling to a reservoir is described by a Lindblad term of the form

L inc,j→i = Γ j→i 2 (2|i j| |j i| -|j j| -|j j| ) (III.24)
where Γ j→i is the incoherent relaxation rate from level j to level i. Such term can describe the radiative decay of the exciton (irreversible coupling to the photon modes) or the relaxation of the carriers or Mn spin (irreversible coupling to the phonon modes). It can also be used to describe the optical generation of an exciton in the low excitation regime where the energy shift induced by the coupling with the laser eld is neglected.

A pure dephasing (i.e. without exchange of energy with a reservoir) can also be introduced for the dierent spins and described by L deph,jj :

L deph,jj = γ jj 2 (2|j j| |j j| -|j j| -|j j| ) (III.25)
where γ jj is the pure dephasing rate of level j.

To identify the main spin relaxation channels responsible for the observed spin uctuations, we rst modelled the auto-correlation of the resonant PL using the full spin level structure of a p-doped magnetic QD. For a qualitative description of the observed spin dynamics, we used as an example the Mn-doped QD parameters listed in Tab. III.1 and reasonable order of magnitude for the spin relaxation times, listed in Fig. III.12.

We considered that the spin dynamics in the excited state is controlled by the time evolution of H X + M n , the generation rate of excitons γ g = 1/τ g and their radiative lifetime τ r = 0.3 ns. The coherence of the coupled electron-Mn spins is limited by a pure dephasing term T eM n 2 = 0.5 ns, extracted from the time resolved PL (see Sec. III.3).

For the hole-Mn system in the ground state, we take into account a spin relaxation time of the Mn in the exchange eld of the hole, τ M n , describing the relaxation channels involving a change of the Mn spin by one unit. This spin relaxation channel was introduced for a general description, however its characteristic time (in the µs range) is long compared to the time-scale of the dynamics considered in the resonant PL experiments and does not qualitatively aect the calculated time evolution.

Because of the presence of VBM, the spin ip of the hole independently of the Mn is expected to be more ecient. A spin ip time in the 10 ns range has been calculated for a hole in the exchange eld of a Mn [107,108]. A relaxation time of the hole spin around 5 ns has also been measured in negatively charged CdTe/ZnTe QDs in the absence of magnetic eld [109]. Possible spin ips of the hole by one unit with a characteristic time τ h = 10 ns were also included in the model. The phonon induced hole-Mn ip-ops, occurring at τ f f , are also introduced between the two hole-Mn ground states of each Λ system. For a general qualitative description, an additional pure dephasing time T hM n 2 was also included in the dynamics of the hole-Mn system with a Lindblad term in the form of Eq. III.25. We cannot extract this parameter from the experiments.

We chose T hM n 2 = 5ns, slightly longer than what was measured for electron-Mn, as the hole-Mn system is highly split and less sensitive to eective uctuating magnetic eld such as the one produced by nuclear spins [109,110].

The transition rates Γ γ→γ between the dierent hole-Mn states depend on their energy separation E γγ = E γ -E γ . Here we used Γ γ→γ = 1/τ i if E γγ < 0 and Γ γ→γ = 1/τ i e -E γγ /k B T if E γγ > 0 [107,111]. This accounts for a thermalization among the 12 hole-Mn levels with an eective spin temperature T . The optical excitation (τ g ), the exciton recombination (τ r ), the Mn spin relaxation (τ M n ), the hole spin relaxation (τ h ) and the phonon induced transfer time (τ f f ) produce an irreversible population transfer between the γ and γ levels and are described by Lindblad terms (Eq. III.24). To model the auto-correlation of the σ-PL intensity of the electron-Mn state |3; +2 under CW σ+ resonant excitation, we used the density matrix formal-ism and calculated the time evolution of ρ |+ 3 2 ;↑e (t) with the initial condition ρ |+ 3 2 ;⇑ h (0) = 1. The initial condition corresponds to the hole-Mn spin being in the state | + 3/2; ⇑ h just after the emission of a σphoton on the low energy branch of the Λ system. This is a slight approximation: in the presence of VBM, the two ground states of a given Λ system are not completely pure hole-Mn spin states but are slightly coupled by a hole-Mn ip-op induced by the exchange interaction H ex hM n . However, since the splitting between the | + 3/2; ⇑ h and | + 5/2; ⇓ h states

(∆ = 4 × (3/2)I hM n ) is large compared to the coupling term (W = √ 15 ρc ∆ lh I hM n ),
their coherent coupling is weak. With a large valence band mixing ρc ∆ lh = 0.1, as observed in the dot discussed in this paper, the population of the hole-Mn system initialized in the state | + 3/2; ⇑ h oscillates rapidly between the two corresponding hole-Mn ground states of the Λ system with a maximum amplitude of about 1.6% and an average population transfer eciency of 0.8% [112]. Under resonant excitation on the high energy branch of the the Λ system, the QD remains OFF more than 99% of the time. As we will see in the following, the contribution of this weak coherent population transfer to the auto-correlation signal is not signicant.

ρ |+ 3 2 ;↑e (t) obtained with the QD parameters listed in Tab. III.1 is presented in Fig. III.12 (a). This quantity has to be normalized by ρ |+ 3 2 ;↑e (∞)) to show the calculated autocorrelation signal. After a fast increase, the calculated population presents a maximum at short delay (t < 5 ns). This model is based on a large number of parameters, whose values cannot all be extracted precisely from the measurements. However, with the spin relaxation parameters given in the caption The width of the calculated bunching is controlled by all the spin-ip terms that can induce an escape out of the resonantly excited Λ system. At zero transverse magnetic eld, it is dominated by spin ips in the hole-Mn system, with a characteristic given by τ h . If we put the hole spin relaxation time as innite (τ h → ∞), ρ |+ 3 2 ;↑e (t) is only controlled by the evolution of the hamiltonian H X + -M n and the decoherence. The time evolution of | + 3/2; ↑ e , illustrated in The inuence of a transverse magnetic eld, B ⊥ , on the optical pumping tran- sient is also well described by this model. First, a signicant reduction of the pumping time is observed for a weak magnetic eld (B ⊥ = 0.2 T in Fig. III.13(b)).

As for the autocorrelation, this acceleration comes from the increase of the leakage out of the Λ system induced by the mixing of the electron-Mn states. Secondly, the transients obtained when switching the polarization from σ co to σ cross and from σ cross to σ co become identical for B ⊥ ≈ 0.4T, as observed in the experiments (Fig. III.9 (b)).

To understand this behaviour under B ⊥ , let us remember that we resonantly excite |3, +2 from | + 5/2, ⇓ h with σ+ light and excite |3, -2 from | -5/2, ⇑ h with σphotons. In both cases we detect the population of | + 3/2, ↑ e in σpolarization (see the excitation/detection conguration illustrated in the inset of The splitting between the two new eigenstates, formed by the mixing of |3, +1 and |3, -1 , in the µeV range, is much weaker than the width of the resonant laser used in our experiments (around 10 µeV) and the width of the PL lines (around 50 µeV). Under circularly polarized resonant excitation we either excite |3, +1 with σ+ photons or |3, -1 with σphotons. Under circularly polarized resonant excitation, the two Λ systems associated with |3, ±1 are simultaneously We saw in those last paragraphs that the strain anisotropy is important to understand the dynamic of the e-Mn states |3, ±1 . Its eect on them is illustrated in Fig. III.16. It prevents the system to be pumped on those states in the absence of longitudinal magnetic eld. We will see in the last section that this beating between the states |3, +1 and |3, -1 can be used to extract the value of the QD in-plane strain anisotropy.

III.3 Strain induced coherent dynamics of coupled carriers and Mn spins in a quantum dot

We exploited the Λ-level structure evidenced in Sec. III.1.2 to investigate the coherent dynamics of the e-Mn spin through the time evolution of the circular polarization rate, κ = (σ Cross -σ Co )/(σ Cross + σ Co ), of the resonant PL.

The conguration of the time resolved resonant experiment is summarized in To understand the details of this dynamics, we calculated the time evolution of the populations and coherence of the twelve X + -Mn states and the twelve hole- Mn states. We neglected here the hyperne coupling between the electronic and nuclear spins of the Mn and solved the master equation for the 24 × 24 density matrix numerically, including relaxation and pure dephasing processes in the Lindblad form, as presented in Sec. III.2.3.

For the initial condition in the calculation of the time evolution, we considered that a σ+ pulse on resonance with the high energy side of the Λ-level system (1) (see Fig. III.4) projects the system on the M = 3 electron-Mn subspace on all the levels that contain a | + 3/2, ↓ e component. In the absence of transverse magnetic eld and strain anisotropy term E, this excitation simply corresponds to an optical transition from the hole-Mn state | + 3/2, ⇓ h towards the electron-Mn state |3, +1 . With a weak transverse magnetic eld (typically lower than 0.5 T), a linear combination of the M = 3 states is created. At large transverse magnetic eld, one should consider possible mixing with the M = 2 states. Similarly, a σ+ pulse on (2) projects the system on the M = 3 electron-Mn subspace on the levels that contain a | + 5/2, ↓ e component, and a σ+ pulse resonant on (3) projects the system on the levels that contain a | + 5/2, ↓ e component in the M = 2 e-Mn subspace.

After this excitation, the circular polarization of the resonant photoluminescence is governed by the evolution of the spin of the electron. For instance, to compute the circular polarization rate of the emission after a resonant σ+ excitation on (1) (optical excitation from the hole-Mn state | + 3/2, ⇓ h to |3, +1 : the high energy branch of the Λ-system) we calculate the dierence between the density matrix elements ρ |+1/2,↑e (σ-recombination towards the hole-Mn state | + 1/2, ⇑ h : the low energy branch of the Λ-system) and ρ |-1/2,↓e (σ+ recombination towards the hole-Mn state | -1/2, ⇓ h : low energy branch of the Λ-system associated with |3, -1 ).

In the absence of magnetic eld, the period of the quantum beats observed for an excitation of |3, +1 depends only on the anisotropy term E. The experimental data can be well reproduced by the model with E = 1.8 µeV (Fig. III.17 (d)). A coherence time, T eM n 2 ≈ 0.6 ns, of the spin of e-Mn is extracted from the damping of the oscillations. For an excitation of |3, +2 and |2, +2 one can observe a slow decrease of the polarization rate which is also qualitatively reproduced by the model.

The coherent transfer of population depends both on the initial splitting of the e-Mn spin states and on the strength of the coupling E. The splitting between the e-Mn states can be tuned by a magnetic eld, B z , applied along the growth axis. In addition, a coupling between the e-Mn spin states M z can be induced by a magnetic eld, B x , applied in the QD plane. The experimental and calculated evolution of the polarization rate of the e-Mn states, |3, +1 , |3, +2 and |2, +2 , versus magnetic eld are presented in Fig. III.18. Under a longitudinal magnetic eld B z , the e-Mn states M z = ±1 are split and the inuence of E is progressively reduced. For an excitation on the |3, +1 state, the amplitude and period of the oscillations in the polarization rate reduce as B z increases: the resonant PL becomes cross-circularly polarized with a polarization rate constant during the lifetime of X + . A weak longitudinal magnetic eld stabilizes the spin of the e-Mn states |3, +2 and |2, +2 and their polarization rate remains constant during the lifetime of X + -Mn.

In a transverse magnetic eld B x , the quantum beats observed for an excitation of |3, +1 are accelerated and the measured circular polarization rate drops to zero as the period of the oscillations becomes smaller than the time resolution of the experimental setup. A higher transverse magnetic eld induces a slower oscillation of the polarization rate for the states |3, +2 and |2, +2 .

The observed magnetic eld dependence of the coherent dynamics of |3, +1 , |3, +2 and |2, +2 can be reproduced by the model with the exchange parameters the levels with an angular momentum M z = ±1 is lifted and the small E value is not enough to couple them. Therefore, the oscillations observed in the polarization disappear for a high enough magnetic eld.

In a transverse magnetic eld, the observed splitting of the e-Mn doublets strongly depend on the considered levels. The eld induced splitting results in this case from a mixing of the electron-Mn states with a dierence in total angular momentum ∆M z = ±1. These states are initially irregularly separated at zero eld, resulting in dierent coupling under a transverse magnetic eld.

Conclusion

This chapter focused on the details of the energy structure and the dynamics of the coupled carriers and Mn spins in a positively charged QD. Its ground state consists of hole-Mn hybrid spin states. The electronics properties of its excited state are dominated by the e-Mn interaction. This leads to an ensemble of optical Λ-level systems, opening the possibility to perform a coherent manipulation of the magnet formed by the coupled h-Mn spins with two resonant optical elds [110].

We used these Λ-level systems to study the spin dynamics of the hole-Mn and the electron-Mn hybrid spin in a positively charged Mn-doped CdTe/ZnTe QD.

We identied and ecient spin relaxation channel for the hybrid hole-Mn spin in a QD, via hole-Mn ip-ops. We propose that these ips-ops are caused by the interplay of the h-Mn exchange interaction and the h-Mn states coupling induced by acoustic phonons. The characteristic time of these ip-ops depends on the splitting between the two considered hole-Mn states, and are found in our Λ-system to be of the nanosecond range. This characteristic time permits to reproduce well the experiments and shows that these ip-ops are responsible for the large PL intensity observed under resonant excitation of the Λ-systems. Leaks out of these systems are possible and induce a large bunching in the resonant PL. They also are at the origin of the optical hole-Mn spin pumping observed under circularly polarized resonant excitation. The escape out of the excited Λ system can be enhanced by applying a transverse magnetic eld, mixing the e-Mn states in the excited state of the charged QD. All of this shows a fast hole-Mn dynamics that limits the hybrid spin uses for quantum information devices. However, the use of a dierent QD systems with a better hole connement and a larger hh-lh splitting could signicantly reduce the inuence of acoustic phonons on the hole-Mn spin and enhance their spin relaxation time.

Another interesting point was found for Mn in positively charged strained QDs probing the time dependence of the resonant PL polarization for each Λ-level system. Oscillations the Λ-level associated with |3, +1 were observed.

Oscillations were observed in the time resolved resonant PL of the peak associated with the states |3, +1 and |3, -1 . They evidence a change of polarization of the emitted photons. These oscillations are explained by the inuence of the strain symmetry on the electron-Mn dynamics. These results demonstrate the potential of magnetic QDs where one could exploit the intrinsic spin to strain interaction to coherently couple the spin of a magnetic atom to the motion of a nano-mechanical oscillator [82,113] and suggest some possible coherent mechanical spin-driving of a magnetic atom.

Coupling spins to a mechanical oscillator opens new ways to manipulate spin.

In the next chapter, we will study the case of a single Cr atom in a CdTe/ZnTe QD where strong spin to strain coupling is expected and evidenced by experiments.

Chapter IV Magneto-optical study of Cr-doped CdTe quantum dots The main goal of this thesis was to optically probe the spin of a single Cr atom in a semiconductor matrix. We saw in Sec. I.3.2 how the incorporation of a Cr atom in a semiconductor lattice aects its spin structure. This chapter is dedicated to demonstrate that a single Cr embedded in a CdTe/ZnTe QD can be probed optically, and to highlight the study of the carrier-Cr interactions.

We begin presenting the PL and the energy structure of the X-Cr complex.

We show that the exchange interaction between the carrier and the Cr is strong enough to see the eect of a single Cr spin in the QD. We discuss the evolution of the emission in temperature and present dierent excited states of the system.

Magneto-optical experiments conrm the energy structure extracted from the PL at zero magnetic eld, and suggests an anti-ferromagnetic coupling between hole and Cr spins. In the next section, we use the evolution of the QDs PL under magnetic eld in order to deduce the QD parameters, using a spin hamiltonian model including the strain induced ne structure of the magnetic atom, the exchange coupling with the carriers and the inuence of the reduced symmetry of the QDs on the electron-hole exchange interaction and on the VBM. In the last section, we present another kind of QDs found in Cr-doped samples that are characterized by three emission lines. We propose that this last feature is related to the presence of Cr atoms with uctuating charge (Cr + , Cr 2+ and Cr 3+ ) in the vicinity of the QD.

IV self-assembled QDs, the heavy-holes and light-holes are separated in energy by the biaxial strain and the connement. In a rst approximation, the ground state in such QD is a pure heavy-hole (J z = ±3/2) exciton and the exchange interaction with the Cr spin S is described by the spin Hamiltonian

H c-Cr = I eCr S • σ + I hCr S z J z (IV.1)
with σ the electron spin and J z the hole spin operator. I eCr and I hCr are, respectively, the exchange integrals of the electron and the hole spins with the Cr spin.

These exchange energies depend on the exchange constant of the Cr 3d electrons with the CdTe carriers and on the overlap of the Cr atom with the conned carriers. Even though the exchange interaction of the Cr spin with both electron and hole is ferromagnetic in bulk II-VI semiconductor studied until now [60,[START_REF] Mac | p-d exchange interaction in Crbased diluted magnetic semiconductors[END_REF][START_REF] Herbich | Magnetization and exciton spectroscopy of the diluted magnetic semiconductor cd 1-x cr x s[END_REF],

the hole-Cr interaction is supposed to be anti-ferromagnetic here. This does not change the structure of the PL at B = 0T. The only visible eect of the sign of the exchange interaction will be on the PL intensity distribution in the magneto-optics experiments. This choice of sign of the hole-Cr exchange interaction will be further discussed in Sec. IV.1.3.

For highly strained CdTe/ZnTe QDs with a weak hole connement, the strain induced energy splitting of the Cr spin D 0 S 2 z is much larger than the exchange energy with the conned carriers (D 0

|I hCr | > |I eCr |).
The exchange interaction with the exciton acts as an eective magnetic eld which further splits the Cr spins states S z = ±1 and S z = ±2. The resulting X-Cr energy levels are presented in Fig. IV.2. The exciton recombination does not aect the Cr atom and its spin is conserved during the optical transitions. Consequently, the large strain induced splitting of the Cr spin is not directly observed in the optical spectra. However, at low temperature, the Cr spin thermalizes on the low energy states S z =0 and S z =±1. This leads to a PL dominated by three contributions: a central line corresponding to S z = 0 and the two outer lines associated with S z = ±1 split by the exchange interaction with the carriers.

Most of the Cr-doped quantum dots exhibit a linear polarization dependence, as presented in Fig. IV.3. The central line (S z =0) is split and linearly polarized along two orthogonal directions. As in non-magnetic QDs, this results from a coupling of the two bright excitons | ± 1 by (i) the long-range e-h exchange interaction in a QD with an in-plane shape anisotropy [114] and/or (ii) the short range e-h exchange interaction in the presence of valence band mixing. This anisotropic e-h exchange energy mixes the bright exciton associated with the same Cr spin state, inducing an extra splitting between them. The mixing is maximum for the central pair of bright excitons (S z =0) which are initially degenerated. The outer lines are also slightly linearly polarized but the inuence of the e-h exchange interaction is attenuated by the initial splitting of the | ± 1 excitons induced by the exchange interaction with the Cr spin S z = ±1.

In order to identify the nature of the lower energy peak ((4) in Fig. IV.3(a)), we performed time resolved PL experiments on the emission peaks (1), ( 3) and (4).

The results, presented in Fig. IV.4, show that the line (4) presents a decay time about twice longer than the other lines. Such a long decay time is consistent with the radiative recombination of a dark exciton. Under normal circumstances, the recombination of such a state is forbidden. However, in QDs with a symmetry lower than C 2v (truncated ellipsoidal lens for example), the bright and dark exciton states can get mixed. It is then possible to observe a dark exciton emitting a photon [56,115]. Since it is initially a forbidden transition, the recombination will be less ecient and will thus take more time [116]. This hypothesis will be conrmed by the magneto-optical study of the dot presented in Since the Cr spin states S z = ±2 do not appear on the PL because they cannot be thermally populated at T = 5K, one could expect to see their emission at higher temperature. Fig. IV.5 presents the emission of two dots as a function of the temperature. With the increase of the temperature, we observe a signicant line broadening induced by the interaction with acoustic phonons [100]. In order to keep a signicant PL intensity and resolved PL lines, we limited our investigation to temperature below 40K. Even at this temperature, the PL peaks corresponding to S z = ±2 do not appear. However, the structure of the emission changes slightly with the temperature. The intensity of the outside peaks, associated with the states S z = ±1, decreases slightly faster than the intensity of the PL peak associated with S z = 0 when the temperature increases. This is an unexpected picture, since the higher temperature should allow the higher energy states S z = ±1 to be more populated by emptying the ground state. We will see in Sec. V.3.1 that there is a relaxation path from S z = ±1 to S z = 0 in the X-Cr via the interplay of the h-Cr exchange interaction and the coupling to acoustic phonons. This mechanism could be enhanced by the temperature, leading to this evolution in the PL peaks intensity. The rst feature of this peak is that, each of the peak here presents a slightly dierent resonant energy. Moreover, one can note that the order of appearance during the energy scan of the two central peaks is reversed compared to the external ones. For the outside peaks, the low energy one appears before the high energy one. This order is inverted for the central peaks. Such splitting inversion, was rst observed on QDs in GaAs quantum well [117]. It has been discussed by Takagahara [114] and is due to the electron-hole exchange interaction in anisotropic potential. When applying such a magnetic eld, the bright exciton X z = ±1 splits, leading to a σbranch shifting at low energy and a σ+ one shifting at high energy. This splitting can compensate the one induced by the exchange interaction with the Cr [119]. For QD1, this results in an anti-crossing of | + 1 and | -1 excitons due to the e-h exchange interaction around B z =6 T observed both in σ+ and σ- ). This anti-crossing arises from a mixing of the bright and dark excitons interacting with the same Cr spin state. When detecting in σpolarization, it corresponds to the mixing of the exciton states | -1 and | + 2 coupled to the Cr spin S z = +1. This dark/bright excitons coupling δ 12 , dened in Eq. I.51, is induced by the e-h exchange interaction in a conning potential with a symmetry lower than C 2v [115]. In such symmetry, the dark exciton acquires an in-plane dipole moment which leads to possible optical recombination at zero magnetic eld [56]. The oscillator strength of this "dark exciton" increases as the initial splitting between | -1 and | + 2 excitons is reduced by the magnetic eld.

To illustrate the inuence of the QD symmetry on the magneto-optical properties of X-Cr, we show in Fig. IV.9(b) the emission of a QD with a dierent strain or shape anisotropy (QD3). For QD1, the splitting of the central peak is not clear in the PL at 0T ( The PLs under magnetic eld of X-Cr and X 2 -Cr present a mirror symmetry. In particular, the dark/bright exciton mixing observed around B z = 2.5 T on the low energy side of the PL in σpolarization for X-Cr is observed on the high energy If one considers the ground state of X 2 as a spin-singlet (total spin 0), it cannot be split by the magnetic eld or the spin interaction part of the carriers-Cr hamiltonian. The creation of two excitons in the QD cancels the exchange interaction with the Cr atom. Thus, the PL of X 2 -Cr is controlled by the nal state of the optical transitions, i.e. the eigenstates of X-Cr, resulting in the observed mirror symmetry in the PL spectra.

The sign of the interaction between the Cr and the hole spin can be found by studying the evolution under magnetic eld of the relative intensity of each of the QD peaks. As shown in Fig. IV.2, for a given polarization, each peak can be linked to a Cr spin state. As discussed earlier, applying a magnetic eld lifts the degeneracy between the exciton states and allows to eciently select the polarization of the emission. In the σbranch, the high energy peak get brighter while the low energy one disappears for B z ≥ 8T. The situation is opposite in the σ+ branch, where the intensity concentrates on the lower energy peak associated with the bright exciton state.

We will focus in this analysis on the Cr spin states S z = ±1, associated with the two outside peaks. Under magnetic eld, S z = -1 shifts at lower energy, while S z = +1 shifts at higher energy, due to the Zeeman eect with g Cr ≈ 2. Therefore, at high enough magnetic eld, the recombination occurs mainly toward S z = -1. In σpolarization, the spin state |S z = -1, X z = -1 is detected. The PL intensity concentrates on the high energy peak, meaning that this state is at high energy. The same analysis can be done for an excitation in σ+ polarization, detecting the spin state |S z = -1, X z = +1 : the lower energy line is the most intense, meaning that this state is at low energy. This shows that, in Cr-doped QDs, the evolution of the peaks relative intensities under magnetic eld is consistent with an anti-ferromagnetic h-Cr exchange interaction. This situation is similar to the one observed in II-VI QDs doped by a single Mn atom, presented in Fig. IV.11. Mn is known to have an anti-ferromagnetic interaction with the hole in CdTe [START_REF] Gaj | Relation of magneto-optical properties of free excitons to spin alignment of mn 2 + ions in cd 1-x mn x te[END_REF]. It was shown in Ref.

[51] that, under magnetic eld, the intensity was mainly on the high energy side when detecting in σpolarization, and on the low energy side in σ+ polarization.

The same evolution was found in our other QDs, even if the evolution of the intensity distribution is less clear. All of this shows that hole-Cr exchange interaction is anti-ferromagnetic in self-assembled CdTe QDs. As was seen in Sec. I.2.2, the sign of this interaction strongly depends on the position of the Cr 3d level relative to the valence band top. This position can be aected by the strains and the connement. A more detailed modelling is therefore required to explain the hole-Cr anti-ferromagnetic coupling.

IV.2 Modelization of a Cr-doped QD

The system can be described by a spin eective hamiltonian, that we separate as follows:

H X-Cr = H Cr,ε + H cCr + H mag + H eh + H band + H scat (IV.2)
where:

H Cr,ε describes the ne structure of the Cr atom and its dependency on local strain, as presented in Eq. I.104. It is mainly drived by D 0 , the magnetic anisotropy. The in-plane strain anisotropy E has to be small (in the µeV range) in order for model to reproduce well the data (see Fig. IV.13 for the emission of a dot with a higher E).

H cCr describes the coupling of the electron and hole with the Cr spin, depending on I eCr , the ferromagnetic exchange integral of the electron-Cr spins, and I hCr , the anti-ferromagnetic exchange integral of the hole-Cr spins, as described in Eq. I.95.

H mag describes the eect of a magnetic eld, coupled to both the Cr and carrier spins by the Zeeman terms, depending on the g-factor of each of them and the Bohr magneton µ B , and including the diamagnetic shift of the electron-hole via the term γ:

H mag = g Cr µ B - → B . - → S + g e µ B - → B . - → σ + g h µ B - → B . - → J + γB 2 (IV.3)
H eh describes the short range and long range electron-hole exchange interaction, through the bright and dark exciton splitting δ 0 , the bright exciton coupling δ 1 , the dark exciton coupling δ 2 and the bright and dark exciton coupling δ 11 and δ 12 . All of these terms are described in Eq. I.51.

H band is the band Hamiltonian. It is written H band = E g + H V BM , with E g the band gap of CdTe and H V BM the valence band mixing, described in Eq. I.52.

H with η > 0. It is introduced here for a general description.

We considered the general case of QDs with a symmetry lower than C 2v (trun- cated ellipsoidal lens for example [115]), and took into account the inuence of this reduced symmetry on the valence band and on the e-h exchange interaction.

The population of the X-Cr spin states split by the large magnetic anisotropy and the carriers-Cr exchange interaction is described by a spin eective temperature This model also predicts an anti-crossing around B z = 5 T, noted (5), caused by an electron-Cr ip op, which is not seen in the experiments. Its position is controlled by D 0 and its intensity by I eCr . If D 0 > 3 meV, the anti-crossing (5) appears for B z > 11 T, outside of our magnetic eld range (-11 T < B z < +11 T). For lower value of D 0 , a low value of I eCr is needed to keep the anti-crossing smaller than the peak width, and thus not detectable experimentally. We chose this last solution in order to keep reasonable intensity ratio between the PL peaks (see below).

T ef f ,
Finally, the remaining tail of an anti-crossing, labelled (6), also appears at high magnetic eld in the σpolarization due to the coupling a bright and a dark exciton coupled to the Cr state S z = 0. Such anti-crossing is observed in some of the Cr-doped QDs like QD4 in Fig. IV.10 The magnetic anisotropy D 0 cannot be precisely extracted from the PL spectra. However, for D 0 < 2 meV, a wide anti-crossing due to a VBM induced hole-Cr ip-op between the | -1, +2 and the |0, -1 would appear for B z < 11 T on the central line in σpolarization. D 0 > 4 meV would put the Cr spin state S z = ±1 at high energy, increasing the population of S z = 0 and leading to a higher PL intensity of the central peak, associated with S z = 0, than observed experimentally. For these reasons, a value of D 0 in the range of 2 to 3 meV was chosen. However, even in this range, the intensity distribution of the PL cannot be perfectly reproduced: while the evolution under magnetic eld of the intensity ratio of the peaks is quite well predicted for high value of the magnetic eld, the S z = 0 state still presents a stronger emission at B z = 0 T than the one observed in the experiments. This dierence may be due to out of equilibrium phonons, generated during the optical excitation, that can heat the Cr spin, increasing the population of the Cr spin states S z = ±1.

Our model reproduces the data found experimentally with enough satisfaction.

It can now be used to understand the impact on the PL of the dierent parameters.

Especially, an interesting point is the inuence of the in-plane strain anisotropy E.

The results of the calculations are presented on Fig. IV.13. The QD emission at 0 T splits into six lines with some linear polarization dependence. This dependence is more marked when the value of E gets higher.

As discussed previously, the in-plane strain anisotropy term E couples two states close in energy and separated by two units of spin, such as the Cr spin states S z = +1 and S z = -1 in the QD ground state. This coupling splits the levels in two linearly polarized states. The observed peak splitting is caused by a splitting of the nal state of the recombination. On the magnetic eld map of the PL, this splitting appears as anti-crossings at B = 0 T, noted ( 9) and [START_REF] Braun | Direct observation of the electron spin relaxation induced by nuclei in quantum dots[END_REF] in polarized magnetic eld dependency of the PL intensity of a Cr-doped QD with a higher in-plane strain anisotropy (E = 100 µeV). The intensity variations of the PL in linear polarization are more marked, and the anti-crossings are wider and occur on a larger range of energy. A large E value is able to couple the states on a wider range of energy before and after they are actually brought in degeneracy.

The anti-crossings ( 7), ( 8), ( 9) and ( 10) merge, giving the complex structure shown in the magnetic eld PL map in Fig. IV.13 (b). This complex structure leads to an apparent diminution of the peak splitting at zero magnetic eld.

Most of the dot we found presented a vanishing anisotropy term E. The application of an electric eld through the Schottky gate attracts the hole toward the surface or the back of the sample, depending on the direction of the applied eld. This changes the hole interaction with the Cr ion. For a high bias voltage, the h-Cr Coulomb interaction becomes negligible, and the splitting induced by the variation of the charge of the ion disappears.

These variations of a charge close to the dot would also explain the dierence of splitting between the dierent excitonic complex. The application of a magnetic eld changes the excitonic species binding energy, changing their emission energy [122]. However, each species is aected dierently: the shift energy can be smaller, or even, for X -, the shift for a given electric eld is opposed to the one of other species. These dierent shifts of the dierent excitonic complexes lead to dierent apparent splittings.

This hypothesis of a Cr in the barrier causing these three peaks emission is currently studied more in details. A similar auto-correlation on a X-Cr PL line still presents a reduced coincidence rate near zero delay, but it is mainly characterized by a large photon bunching with a FWHM of about 20 ns. This large bunching reects an intermittency in the emission of a given line of the QD coming from uctuations of the Cr spin in a 10 ns timescale as it will be conrmed by cross-correlation measurements.

The amplitude of the bunching reaches 5 for line (1) and is slightly weaker for the lower energy lines (4) and ( 5). In a simple picture of blinking where the selected QD line can be either in a state ON (populated) or OFF (empty), the amplitude of the bunching is given by Γ OF F /Γ ON , with Γ ON the transition rates from OFF to ON, and Γ OF F the one from ON to OFF [125]. An amplitude of x -S 2 y ) in the Cr ne structure Hamiltonian [START_REF] Lafuente-Sampietro | Individual Cr atom in a semiconductor quantum dot: optical addressability and spin-strain coupling[END_REF].

However, even at low excitation power, the measured transfer time is not aected by a longitudinal magnetic eld (Fig. V.3(b)). This shows that for such QD the strain anisotropy term E is weak and is not the main parameter controlling the transfer time between the states S z = ±1. The spin transfer time is dominated by spin-ips induced by the exciton/Cr interaction.

This fast transfer time is an indication of ecient carrier-Cr spin ip-ops, an interaction with optically injected acoustic phonons, or both. However, it is hard to extract precise informations from the auto-correlation and cross-correlation experiments only. In order to delve into more details, we have to use more precise tools, such as the time evolution of a prepared state.

V To perform resonant optical pumping of the Cr spin, we developed a two wavelengths pump-probe experiment. A circularly polarized single mode laser (resonant pump) tuned on a X-Cr level is used to pump the Cr spin (i.e., empty the spin state under resonant excitation). Then, a second laser, tuned on an excited state of the QD (quasi-resonant probe), injects excitons independently of the Cr spin S z and drives the Cr to an eective spin temperature where the three ground states S z = 0, ±1 are populated [START_REF] Lafuente-Sampietro | Individual Cr atom in a semiconductor quantum dot: optical addressability and spin-strain coupling[END_REF]. By recording the PL of a X-Cr line in circular polarization under this periodic sequence of excitation, we can monitor the time evolution of the population of a given Cr spin state. probe pulse shows that the population of S z = -1 has been partially transferred to S z = 0 during the resonant pumping sequence. This will be described in more details in Sec. V.3.2.

A more direct way to probe the optical pumping speed and eciency is to The complementary evolution happens when the pump is on the low energy line (4) and detect on the high energy line (1). In this case, when the pump

V.3.1 Optical pumping induced by h-Cr ip-op

We saw in Sec. V.2 an evidence of a possible transfer between the Cr spin states S z = -1 and S z = 0, marked by the appearance of a transient at the beginning of the PL of the probe pulse taken on line (3) when the resonant pump was turned on. In order to study this transfer in more details, we performed resonant optical excitation of the state S z = -1, while detecting on an emission line associated with the state S z = 0. As noticed previously, during the probe pulse, the central peak presents a slow decrease, lasting for about 250 ns. This intensity decrease contrasts with the increase observed when detecting on the line (4): we see the repopulation of the state S z = -1 when detecting on line (4), and the population diminution of the state S z = 0 when detecting on line (3). It suggests that, when pumping the state S z = -1, some of the population has transferred to S z = 0. When heating with the quasi-resonant pulse, this state is partially emptied and a thermal equilibrium This transfer can be explained by a process similar as the one proposed for h-Mn, presented in Sec. III.2.3. When we excite resonantly the QD on the high energy state |S z = +1, X z = +1 , the system will usually cycle between the excited state and the ground state S z = +1, through photon absorption and emission. However, in the excited state, the hole spin can ip with a characteristic time τ h of a few ns, getting the system to the state |S z = +1, X z = -2 . Once there, a phonon-mediated hole-Cr spin ip-op can occur to reach the state |S z = 0, X z = +1 , with a characteristic time τ hCr . A similar process can occur for an excitation on |S z = -1, X z = -1 . The hole spin ip will then bring the system to |S z = -1, X z = +2 , from which a phonon-mediated hole-Cr spin ip-op can occur towards |S z = 0, X z = -1 . Those two processes are illustrated in Fig. V.11.

In order to estimate the relaxation time τ hCr between |S z = +1, X z = -2 and |S z = 0, X z = +1 , we used the same formalism as in Sec. III.2.3. The nondiagonal term of the hole-Cr exchange interaction mixes the heavy holes and the light hole excitons. The two perturbed exciton ground states can be written:

|0 | ⇑ h , ↓ e = |0 | ⇑ h , ↓ e - √ 18 2
I hCr ∆ lh | + 1 | ↑ h , ↓ e (V.1) | + 1 | ⇓ h , ↓ e = | + 1 | ⇓ h , ↓ e - √ 18 2
I hCr ∆ lh |0 | ↓ h , ↓ e (V.
2)

The eects of phonons are seen by coupling those two states through the Bir- Using the same steps done in Sec. III.2.3, we can calculate τ hCr using Fermi's Golden rule: A more intense laser injects a greater amount of phonons in the sample, heating the Cr spin more eciently. The pumping transient amplitude is thus higher for a more intense probe pulse.

τ -1 hCr = λ 18 (2π) 2 I hCr ∆ lh 2 ω 0 c λ 3 1 2hρc 2 λ π 4 (3b 2 + d 2 ) × (n B (ω 0 ) + 1)) π 0 dθ sin θ|F λ (ω 0 , θ)| 2 G λ (θ) (V.
We also studied the evolution of the pumping normalized transient intensity as a function of the distance between the the pump laser and the probe laser. In a simple picture, phonons are created in the sample in a sphere centred at the position of the probe laser, and we suppose an isotropic diusion of the phonons.

When the probed laser is moved away, the QD occupies a smaller surface on the phonons sphere. Less phonons will reach it, making the heating process less This normalized transient intensity at high laser power higher than for a heating pulse far from the dot may come from the higher density of phonons injected by a laser at the dot position.

These measurements show that the created non-equilibrium acoustic phonons lead to an increase of the Cr eective spin temperature without injecting carriers in the QD. This eective temperature is much higher than the lattice temperature and changes the spin dynamics of the magnetic atom. Another conrmation of this eect could be obtained by depositing a thin metal layer close to the QD. A laser From the time delay dependence of the amplitude of the transient, we deduce a Cr relaxation time τ Cr ≈ 1.7 µs at B= 0 T and T= 5 K. For such neutral QD meV, but not for the transfer from S z = +1, because ∆ +1,-1 ≈ 0 meV. However, the states S z = +1 and S z = -1 can be coupled by a small in-plane anisotropy term E [START_REF] Lafuente-Sampietro | Individual Cr atom in a semiconductor quantum dot: optical addressability and spin-strain coupling[END_REF]. A transfer between the two states is therefore also allowed. Another possible source of transfer comes from two phonons mechanism: the system rst transfers from S z = +1 to an excited state by the absorption of a phonon, and then relaxes toward S z = -1 by the emission of another phonon. Such two phonons mechanism can either be the Raman mechanism, using a virtual state as the excited state, or the Orbach mechanism, using an actual excited state of the system [132].

From this picture, we could expect two transient times: one between S z = 0 and S z = ±1, driven by the one phonon mechanism, and one between S z = +1 and S z = -1, driven by the in-plane strain anisotropy and the two phonons With the quasi-resonant probe, the system is brought to a non-equilibrium state distribution, at higher eective temperature than the lattice. This eective temperature increases the population of the states S z = ±1. Therefore, during the dark time, the eective temperature of the Cr spin decreases via transfers of population from S z = ±1 to S z = 0. This relaxation occurs via the one phonon process presented above. The reduction of the amplitude of the transient with the dark time would then solely probe this transfer time.

Two relaxation times are evidenced by those experiments and can be linked to the two mechanisms proposed above. The time in the 10 µs range τ |±1 ↔|0 could be associated to the single phonon process, and the one in the µs range τ |±1 ↔|∓1 could be associated to the two phonons processes. The one phonon process, between |S z = ±1 and |S z = 0 , is directly probed by the relaxation after a heating probe pulse (Fig. V.15 (a)). The relaxation after a pump pulse (Fg. V.15 (c)) shows at short dark time (τ dark < 10 µs) the relaxation due to the two phonons processes. A second, longer relaxation time should appear for longer dark time (tens of µs), signature of the one phonon process. However, we were not able to do the experiment at such a long time scale, due to low count rate obtained for a pump pulse alone.

More experiments are needed to conclude on those two times. The direct mechanism could be probed using the relaxation in the dark of the state S z = 0. A linearly polarized laser can be used to excite both the states S z = +1 and S z = -1, and transfer their populations toward S z = 0. The state of S z = 0 would be probed in cross-linear conguration during the resonant excitation. The evolution of the pumping transient with the time between two pulses τ dark would be a measure of the transfer between S z = 0 and S z = ±1 via the direct mechanism.

The transfer mechanism between S z = +1 and S z = -1 could be probed using other experiments. First, probing the Cr spin relaxation in the dark under magnetic eld would show whether this relaxation is driven by E or by the two phonon mechanisms: two levels need to be almost degenerate to be eciently coupled by E. Therefore its eect should disappear quickly in magnetic eld, whereas the eect of the two phonon mechanism would need a higher magnetic eld to disappear. Probing the evolution of τ Cr in temperature would conrm the role of phonons in the transfer mechanism. The two phonons mechanism should be inecient at low temperature, and become quickly important as the temperature rises.

V.5 Optical Stark eect on an individual Cr spin

Optical Stark eect can be used to control the energy of a given state of a QD, using a high intensity optical eld in resonance with a QD transition [133,134]. It has been demonstrated that this control can also been achieved in QD doped with a single Mn atom [START_REF] Gall | Optical stark eect and dressed exciton states in a Mn-doped CdTe quantum dot[END_REF]. In this case, each exciton-Mn level can be addressed and tuned individually. We propose in this section to demonstrate the same control for a Cr embedded in a QD.

When a laser is put on resonance with a QD transition, a coupling is created between the laser photons and the resonantly excited QD levels. The photons of the laser eld are coherently absorbed by the ground state and emitted by the excited state at a Rabi frequency Ω r , depending on the strength of the coupling between the laser eld and the transition dipole. For a n photon state of the control laser, the unperturbed states |Cr ×|n and |X -Cr ×|n-1 , degenerated at resonance, are no longer stationary solutions of the hamiltonian. Instead, stationary solutions In the dressed-atom formalism, the dressed states and the energy splitting are found using a quantum description of the QD levels and the laser eld. We make the assumption that the Rabi splitting Ω r and the laser detuning δ are small compared to the splitting between two PL lines of the quantum dot. We can then calculate the dressed states resulting from the interaction between the two levels of the QD addressed by the laser, noted |G for the ground state and |E for the excited state, and the resonant laser photons, neglecting the light-matter interaction of all the other levels. An illustration of such a system is given in In the absence of coupling between the two considered states, the Hamiltonian of this two level system is written: The light-matter interaction originates from the dipole interaction. In second quantication, in the rotating wave approximation, it is written:

V ≈ hg(p|E G| + p † |G E|) (V.6)
with g the electric dipole matrix element describing the coupling between the dipole of a QD transition and the mode ω L of the electromagnetic eld. The levels |G, n and |E, n -1 of a multiplet are coupled through stimulated emission (p † |G E|) and absorption (p|E G|) of a photon. The total hamiltonian for a given n photon state can be written:

H n =    hnω L hΩ r,n 2 hΩ r,n 2 h(ω EG + (n -1)ω L )    (V.7)
with Ω r,n = g √ n the Rabi frequency. We write the mean energy of the n photon state multiplet E 0 n = h 2 (ω EG + (2n -1)ω L ). The hamiltonian H n becomes:

H n = E 0 n I 2 + h 2 δ Ω r,n Ω r,n δ (V.8)
with δ = ω L -ω EG the detuning of the laser. Diagonalizing the hamiltonian, we nd the eigenstates |I, n and |II, n described at the beginning of this section:

|I, n = c|G, n -s|E, n -1 (V.9)

|II, n = c|G, n + s|E, n -1

(V.10) with c = 1 2 1 - δ Ω r,n , s = 1 2 1 + δ Ω r,n
, and Ω r,n = Ω 2 r,n + δ 2 the generalized Rabi opping frequency. We also nd the corresponding energies and the splitting of the states:

E I = E 0 n - h 2 Ω r,n (

Conclusion

The Cr spin dynamics was rst observed using autocorrelation and cross-correlation experiments, giving a relaxation time of the Cr spin under optical excitation in Using resonant optical pumping, we also showed that the Cr spin dynamics is strongly aected by optically generated phonons. The possibility to heat the Cr spin through non-equilibrium phonons was demonstrated.

The relaxation of the Cr spin in the dark was also probed using resonant optical pumping. Two times were found: one in the µs range, and another in the tens of µs range. Those two times could arise from two transfer mechanism between the Cr spin states: a transfer between S z = 0 and S z = ±1 via a single phonon, and one between S z = +1 and S z = -1 (also mixed by E) via two phonon mechanisms.

Several experiments were proposed to test this hypothesis and isolate those two transfer times.

We nally demonstrated the possibility to control the Cr spin state using optical Stark shift. It is possible to independently tune the energy of each Cr spin state. This energy tuning is particularly interesting for the control of the Cr spin states S z = ±1. These states could be eciently mixed by applying weak anisotropic in-plane strain through a ne structure term of the form E(S 2

x -S 2 y ) [START_REF] Lafuente-Sampietro | Individual Cr atom in a semiconductor quantum dot: optical addressability and spin-strain coupling[END_REF]. A relative shift of the energy of S z = +1 or S z = -1 by a resonant optical excitation would aect their coupling and consequently the Cr spin coherent dynamics.

Conclusion

Two systems were studied in this thesis: single Mn spin in positively charged QD, and single Cr atoms in neutral QDs. We have shown that both are strongly coupled to phonons and strains. This coupling opens new ways to probe and control the spin of the magnetic atoms, not only optically, but by changing the strain state at their position.

We showed that the Mn spin structure in a positively charged QD forms optical Λ-level systems between two hole-Mn ground states and one electron-Mn level. Those Λ systems were used to study the dynamic of the hybrid h-Mn spin. A spin relaxation time in the tens of nanoseconds range was found. A fast relaxation channel between the two hole-Mn ground states of the Λ systems was evidenced.

We proposed that this relaxation is caused by the interplay of the coupling to acoustic phonons and the hole-Mn exchange interaction. This results in a hole-Mn spin ip-op time of a few nanoseconds, reproducing well the experiments.

We also demonstrated that degenerated electron-Mn spin states separated by two units of spin are coupled through the in-plane strain anisotropy, and studied this strain induced coherent dynamics. These results demonstrate the potential of magnetic QDs where one could exploit the intrinsic spin to strain interaction to coherently couple the spin of a magnetic atom to the motion of a nano-mechanical oscillator [82,113] and suggest some possible coherent mechanical spin-driving of a magnetic atom.

For the rst time, the physics of a single Cr atom embedded in a II-VI QD was studied. Its spin was probed optically under magnetic eld. The energy structure of the Cr spin in a self-assembled QD was deduced from this experiment. The splitting caused by the strain induced magnetic anisotropy is strong enough to keep the states S z = ±2 from being thermally populated. Several anti-crossing characteristic from a Cr-doped QDs appears under magnetic eld. Several parameters of the dot can be extracted from these anti-crossings. The evolution of the PL under magnetic eld also evidences that the h-Cr coupling is anti-ferromagnetic, in contrary to what was expected from the study of Cr in other II-VI semiconductors.

We nally probed the dynamics of the Cr spin in a QD. Photon correlation experiments gave a spin relaxation time under optical excitation in the 10 nanosec-onds range. This number was then conrmed by resonant optical pumping experiments. We demonstrated that it was possible to prepare the Cr spin using this last technique. These experiments were also used to probe the relaxation of the Cr spin in the dark. A relaxation time in the microsecond range was found in this latter case. A strong inuence of the phonons on the Cr spin dynamics was also evidenced. Hole-Cr spin ip-ops mediated by acoustic phonons, similar to the hole-Mn ones, were evidenced in the excited state, ipping the Cr spin from S z = ±1 to S z = 0. The possibility of heating the Cr spin through phonon alone, without injecting any carrier in the dot, was also demonstrated. Finally, we demonstrated that it was possible to control the energy of a given Cr spin state using the optical Stark eect.

Even though we have a good picture of the Cr spin behaviour in a CdTe/ZnTe QD, many doors remain open for the study. As presented in Chapter V, the presence of a heating pulse before a dark time aects the Cr spin relaxation in the dark. It was supposed that it was caused by two dierent relaxation processes, mediated either by one or two phonons. Several experiments were proposed to test this hypothesis and probe those two mechanisms in Sec. V.4 and have yet to be realized.

The next important step is to prove the possibility to control the Cr spin with phonons. We propose to develop single spin Surface Acoustic Wave (SAW) devices to demonstrate a complete coherent mechanical driving of the spin of an individual Cr atom inserted in a II-VI semiconductor QD. SAW generated close to the surface of a sample can be used to make time variation of the in-plane strain at a controlled and precise frequency. To generate the SAW, Interdigital Transducer (IDT) on a piezo-electric layer deposited on samples with Cr-doped QDs will be used. SAW will be used to generate phonons tuned on resonance with the energy splitting of the Cr spin states S z = ±1, tuning the splitting between S z = +1 and S z = -1 with a magnetic eld. Introducing these SAW during the dark time of a pumping experiment such as the one presented in Sec. V.4, the resonance in the mechanical excitation of the S z = +1 and S z = -1 transition should be detected in the amplitude of the resonant optical pumping signal. This will permit to optically detect the mechanical coherent control of the Cr spin and mechanically determine its coherence time. This system forms a promising plateform to study the interaction of a spin qubit with SAW, which is proposed as an ecient quantum bus to couple dierent trails of qubits [136].

I. 1 . 1

 11 Band structure of CdTe and ZnTe CdTe and ZnTe are II-VI semiconductors, meaning they are composed of an anion from the column VI of periodic table (Te), and a cation from the column II (Cd or Zn). They both naturally crystallize in a zinc-blende structure when grown in Molecular Beam Epitaxy (MBE, see Chap. II for more informations on this technique). As shown in Fig. I.1 (a), in this structure, each species are organized in a face-centered lattice, one of them being shifted from the other by a quarter of the [111] diagonal. Each ion is then in a tetragonal environment. In other words, the zinc-blende structure is of the T d space-group.

Figure

  Figure I.1: (a) Zinc-blende crystal elementary cell. Both CdTe and ZnTe crystallize in this structure. (b) Bonding and anti-bonding state arising from the hybridization of s and p orbitals.

Figure

  Figure I.2: Dierent type of heterojunctions between two semiconductors.

  5) Finally, for any spin operator O (O = σ, J or any other angular momentum operator of this document), we can dene the ladder operator, ipping the considered spin by one unit, such as O + |O ∝ |O + 1 and O -|O ∝ |O -1 . They read in the general case as:

  I.8) Considering that | ↑ and | ↓ are orthogonal states, we can easily deduce the authorized transitions: • Between |u Γ 8 by p -= p x -ip y , corresponding to a σphoton absorption or emission. • Between |u Γ 8 ,-by p + = p x + ip y , corresponding to a σ+ photon absorption or emission. • Between |u Γ 8 ,-photon absorption or emission. • Between |u Γ 8 ,+ 1 2 and |u Γ 6 ,-1 2 (corresponding to a hole J z = -1 2 ) coupled via a σphoton absorption or emission. • Between |u Γ 8 ,∓ 1 2 and |u Γ 6 ,± 1 2 coupled via a π z photon absorption or emission. Those transitions are summarize on Fig. I.3, with their respective relative probability calculated from the oscillator strength of each of these transitions.

Figure I. 3 :

 3 Figure I.3: Selection rules for optical transitions between valence band and conduction band in a quantum well. They are presented in this conned environment in order to have a clear separation between the heavy holes (J z = ± 3 2) and the

Figure

  Figure I.4: CdTe band structure

  the energy of the conduction band (respectively, the valence band), m c the eective mass of the carrier in the conduction band and m lh the eective mass of the light hole. One can see that the splitting of the valence band separate the carrier with a spin J z = ± 3 2 (hh) from the one with a spin J z = ± 1 2 (lh). However, the hh presents a positive curvature in this results, meaning a positive eective mass. This contradict the schema given on Fig. I.4.

Figure I. 5 :

 5 Figure I.5: AFM image (AFM image (250 nm × 250 nm) of CdTe/ZnTe quantum dots before caping. The dot density is estimated to be in the 10 10 dots.cm -2 range.

  hole envelop functions. The eective mass approximation allows us to replace the periodic crystal potential and the free-electron kinetic energy by the eective hamiltonians representing the band extrema. Considering the eective mass is the same in CdTe and ZnTe, we can now work with the simple picture of an eective mass carrier with the envelop function dened in Eqs. I.34 and I.35, trapped in a potential V e (r) for the conduction band or V h (r) for the valence band, created by the band oset between the two semiconductors. We write the Schrödinger equations for these particles: h2 2m e ∆ F e (r) + V e (r)F e (r) = E e F e (r) (I.36)

Figure I. 6 :

 6 Figure I.6: PL intensities of the bi-exciton (X 2 ), the charged excitons (X + , X -) and the neutral exciton (X) of a CdTe/ZnTe QD as the function of the angle of the linearly polarized detection. To simplify the reading, the intensities were also plotted on polar graph (bottom). Picture taken from Yoan Léger PhD thesis [51].

  Figure I.7: Carrier interactions with no change of the number of electrons on the impurity, derived from the hamiltonian H Coulomb . Picture from Laurent Maingault PhD thesis [59].

Figure I. 8 :

 8 Figure I.8: Schema of the band structure and virtual transitions between valence band and conduction band. Picture taken from Laurent Maingault's PhD thesis [59]. E d is the ground energy of the d electrons of the magnetic atom; U is the energy needed for the magnetic atom to reach its rst excited state; E v (k) is the valence band energy; E g (k) is the conduction band energy

  )) the electron (resp. hole) envelop function, F v (R) the second derivative of the hole envelop function, and A 2 , B 2 , C 2 constant depending on the semiconductor lattice. For CdTe, A 2 = 10.3 Å -2 , B 2 = 0.781 Å -2 and C 2 = 19.8 Å -2 .

  In the valence band, the picture for Cr in CdTe is a lot more complicated and still open to discussions. The position of the d ground state energy relative to the semiconductor valence band is critical and can make the kinetic exchange interaction change sign (see Fig. I.8 and Eq. I.78 for a d level energy E d in the gap). Cr 3d level is almost at the same energy as the top of CdTe valence band, making it hard to decide whether it is in the gap or not. N 0 β was never directly measured in Cd x Cr 1-x Te and is challenging to nd theoritically for the reason stated in the previous paragraph [63]. Wojtowicz et al. measured the Zeeman spin splitting of 1S excitons in a quantum well. The

Figure I. 9 :

 9 Figure I.9: Schematic diagram of the eect of the crystal eld, biaxial strain and spin-orbit coupling on the Mn 2+ 3d 5 ground state ( 6 S). Eect of the crystal eld on the rst excited state ( 4 G) are also presented.

  Figure I.10: Top: schematic view of a strained QD containing a magnetic atom.

  , leading to the 40 µeV total splitting presented on Fig. I.10. An anisotropy of the strain in the QD plane can induce a mixing between dierent S z through the anisotropic crystal eld parameter E. It coupled two values of S z separated by two units of spin. In the absence of magnetic anisotropy (D 0 0), both the anisotropy of strain and the hyperne structure prevent the optical pumping of the Mn spin at zero magnetic eld. I.3.2 Cr atom in II-VI semiconductor Cr atoms are incorporated into II-VI semiconductors as Cr 2+ ions on cation sites forming a deep impurity level. 90% of Cr isotopes presents no nuclear spin, meaning we do not have to consider their hyperne structure. The ground state of a free Cr 2+ is 5 D with the orbital quantum number L=2 and a spin S=2 yielding a 25-fold degeneracy. In the crystal eld of T d symmetry of the tetrahedral cation site in zinc-blende crystal, the degeneracy is partially lifted (see Fig. I.3.2): the 5 D term splits into 15-fold degenerate orbital triplet 5 T 2 and 10-fold degenerate orbital doublet 5 E, separated by a gap ∆ ≈ 0.5 eV. As in the case of Mn, this transition introduce a path for non-radiative recombination and can kill the luminescence of the host semiconductor. We have then to keep the Cr concentration low in order for this transition to not become the main recombination path.

  Figure I.11: Schema of the energy level splitting of Cr 2+ at a tetragonal site (T d symmetry) with a crystal eld parameter ∆, a Jahn-Teller energy E J-T and a spin-orbit level spacing D 0 .

  ε where C 11 ≈ 5.4 10 10 Pa and C 12 ≈ 3.7 10 10 Pa are the elastic constants of CdTe [80]. For this strain conguration, the Cr ne structure is controlled by the spin-lattice coupling coecients c 1 (symmetric coecient) and c 2 (tetragonal coecients). Strain-coupling coecients estimated from EPR measurements in bulk Cr doped CdTe are listed in Tab. I.2.

Figure I. 12 :

 12 Figure I.12: Photoluminescence of the X, X and X 2 complex for (a) a QD with no magnetic atom and (b) a QD containing a single Mn atom.

Figure I. 13 :

 13 Figure I.13: Energy level of the ground state and the exciton in a Mn-doped QD. The levels are sketched as a function of the Mn spin S z . Dark states are represented in grey. Optical transitions from the bright states of the Mn complex results in six-line PL spectra, such as in Fig. I.12.

  Figure II.1: Schema of MBE cells and substrate during a growth. An electron gun is attached to the chamber in order to probe the surface of the sample.

Figure

  Figure II.2: (a) Construction of the Ewald sphere in the reciprocal space for an elastic scattering. For diraction to occur, the reciprocal lattice must lie on the Ewald sphere. (b) Change of the RHEED pattern from a at surface (left) to a rough one due to the formation of QDs (right). (c) Surface state during the growth of 2 MLs (left) and RHEED intensity for each step (right).

  We begin the growth by opening the Zn cell shutter at a substrate temperature of 355 • C. The substrate temperature was then raised to 400 • C in a Zn atmosphere in order to atten the surface for the growth. While it took several minutes to raise the substrate temperature, the growth was stopped by the auto-regulation of the ZnTe growth. When the substrate temperature reaches 400 • C, the Te shutter was also opened, in order to grow the ZnTe buer layer. This thick ZnTe layer keeps the dots far from the substrate surface, that can have some defects, and gives us a at surface [89]. The surface quality is checked by RHEED, presenting clear diraction lines (Fig. II.3 (b)). The substrate temperature was then lowered to 295 • C, the Zn cell being open until the temperature reach 355 • C. CdTe is grown by ALE, in which the choice of substrate temperature xes the quantity of material deposited at each cycle. It was found that 0.5 ML of CdTe is deposited at each cycle for a substrate temperature between 260 • C and 290 • C [88].

  Figure II.3: Left: Layer structure of the strain Cr-doped CdTe QDs samples. Right: RHEED pattern taken along the Te reconstruction line at dierent key moment of the growth: (a) before the growth of the ZnTe buer, (b) after the growth of the ZnTe buer, (c) end of the (Cd,Cr)Te layer deposition, (d) after the Te deposition, (e) after the Te evaporation (T substrate = 163 • C) and (f ) after the growth of the ZnTe cap.

Figure

  Figure II.4: RHEED oscillation for the ALE of the strained dots.

Figure

  Figure II.6: Opening and closing cycles of each cells used to grow either 6.5 MLs or 5.5 MLs of (Cd,Cr)Te for the formation of Cr-doped SK dots.

  photoluminescence (PL) in a low temperature optical microscope. The sample is scanned randomly. We judge the quality of the sample by the number of actual dots we found with single Cr embedded inside, and by the proportion of sharp emission peaks (a few tens of µeV) versus broad ones (in the meV range): if we saw mainly broad peaks, it suggests that the Cr concentration is too high.

Figure

  Figure II.7: Schema of the charge control structure, on a p-doped ZnTe substrate and with a thin, semi-transparent gold layer deposited on the surface.

  around 1917.5 meV and 1933 meV on Fig.II.8 (d). This structure is dominated by the electron-electron exchange interaction and has already been observed in InAs QDs[92]. On the contrary, applying a positive bias voltage increases the probability to detect positively charged exciton complexes, maximizing X + and X ++ intensity. This is shown on Fig.II.8 (b) for a bias voltage of V = +3 V. A structure of two doublet is also observed for X ++ . This shows that we can select eciently the charge of a quantum dot applying a bias voltage across the sample.

13

 13 QDs. We chose to use Cd 0.7 Mg 0.3 Te to keep a lattice parameter close enough to CdTe to grow thick enough barriers, and keeping enough gap dierence to localize the carriers. The needed ux for this growth are shown in Tab. II.3.The critical thickness of Cd 0.7 Mg 0.3 Te on CdTe is not exactly known. System- atic studies were done on CdTe/CdZnTe, and give an empirical law to calculate the critical thickness[93]. This law was used for dierent II-VI material and gives a good approximation of the dierent critical thicknesses, overestimating slightly the eects of strain. For CdTe/Cd 0.7 Mg 0.3 Te, we found a critical thickness h c = 130 nm. We chose to grow 40 nm below the quantum well (QW), and 90 nm above it, to keep the surface far from the quantum well.The strain-free samples were grown on a hybrid substrate formed by a thick TableII.3: Aimed ux for each cell during the growth of the strain-free samples. CdTe layer grown on GaAs. It was done on a fourth of a 2 inches GaAs substrate and protected by Te. The GaAs substrate was cleaned using H * plasma. A thin layer (about 7 nm) of ZnTe was rst grown at 415 • C in order to get the right crystal orientation when growing the CdTe layer. We then grew a CdTe layer of about 3 µm at 360 • C, to recover a good crystallinity of the CdTe layer [94]. The RHEED taken after the growth of the CdTe layer (Fig. II.9 (d)) shows sharp straight line, showing the recovery of a at surface.

Figure

  Figure II.9: Left: Layer structure of the hybrid substrate with its protective Te cap. Right: RHEED pattern taken at dierent key moments of the growth: (a) before H * cleaning of GaAs, (b) after H * cleaning of GaAs, (c) after the growth of the ZnTe layer, (d) after the growth of the CdTe layer.

Figure II. 11 :

 11 Figure II.11: (a) Macro-PL of a SFD5. (b) Examples of spectra taken under µ-PL in SFD5.

  Contents III.1 Mn in a positively charged CdTe quantum dot . . . . . 64 III.1.1 Spin structure of a positively charged Mn doped quantum dot . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 III.1.2 Resonant PL of X + -Mn . . . . . . . . . . . . . . . . . . 70 III.2 Hole-Mn spin dynamics under resonant excitation . . . 72 III.2.1 Cycling in and escaping from the Λ-level system . . . . 72 III.2.2 Resonant optical pumping of the hole-Mn spin . . . . . 75 III.2.3 Modelling of spin relaxation mechanism: hole-Mn ipops mediated by a lattice deformation . . . . . . . . . . 79 III.2.4 Model of the carrier-Mn spin dynamics under resonant excitation . . . . . . . . . . . . . . . . . . . . . . . . . . 83 III.3 Strain induced coherent dynamics of coupled carriers and Mn spins in a quantum dot . . . . . . . . . . . . . . 91Magnetic anisotropy depends on the local environment of magnetic atoms, and are a crucial properties for modern storage devices. We saw in Sec. I.3.1 that the Mn atom in neutral, self-assembled QD presents a small anisotropy of a few tens of µeV. It is expected that a Mn atom could develop a large anisotropy energy in the meV range when it is exchange coupled with a single conned heavy-hole spin[95]. The two low energy hole-Mn state should behave like an Ising spin system, forming an atomic ferromagnet. In addition the exchange induced Mn spin magnetic anisotropy could be electrically controlled by changing the Mn/hh overlap with a bias voltage applied across the QD.

III. 1

 1 Figure III.1: (a) Color scale plot of the PL intensity of one QD of sample M3085, embedding a single Mn inserted in Schottky structure, showing the emission of the neutral (X-Mn) and positively charged (X + -Mn) exciton as a function of energy and bias voltage. (b) Cut in the map shown in (a) fo V = 5.5 V

Figure

  Figure III.2: Electron-Mn spin and hole-Mn energy levels. For each e-Mn state

  those manifold are degenerated. Each of those states are labelled |M, M z . They are presented in Fig. III.2, as well as the resulting recombination from each of the manifold.There are ve M = 2 states, each being the anti-linear superposition of two e-Mn spin states with opposite electron spin. This lead to ve PL peaks in both circular polarization, that can each be associated with a hole-Mn ground state. One can notice that none of the |2, +2 states contains the states | + 5/2 | ↑ or | -5/2 | ↓ : they would have, respectively, a total spin M z = +3 or M z = -3, forbidden in M = 2 states. The transition towards the ground states | + 5/2 | ⇑ and | -5/2 | ⇓ are therefore impossible. Five of the seven M = 3 states are linear superposition of two e-Mn spin states with opposite electron spin. The states |3, +3 and |3, -3 , however, are pure states containing only, respectively, the e-Mn states | + 5/2 | ↑ or | -5/2 | ↓ .

  polarization. For a given polarization, we then have eleven resolved lines. The resulting PL for the X + -Mn is illustrated at the bottom of Fig. III.2 and can be seen as a superposition of two structures: six lines which intensities decrease with increasing their energy position (transitions from M = 3 states) and ves lines which intensities increase with increasing their energy position (transitions from M = 2 states).

Figure

  Figure III.3: (a) Energy levels of the ground (h-Mn) and excited (X + -Mn) states in a positively charged Mn-doped quantum dot as a function of their angular momentum (M z ). The levels in dotted lines corresponds to the h-Mn states | -1/2 | ⇑ h and | + 1/2 | ⇓ h coupled by the valence band mixing. Optical recombination towards these levels leads to linearly polarized lines. (b) Experimental (left) and calculated (right) color-scale plot of the linear polarization dependence of the PL of X + -Mn of the QD presented in Fig. III.1 at B = 0 T (top) and B ⊥ = 0.42 T (bottom). The parameters used in the calculation are listed in Tab. III.1.

  Fig. III.3 (a) by the two split doted levels. These states are the bonding and antibonding combinations of |S z = -1/2, ⇑ h and |S z = +1/2, ⇓ h . They are coupled via linearly polarized photons to the |2, 0 and |3, 0 e-Mn states, giving the four linearly polarized lines observed on Fig. III.3 (b).

  Fig. III.1 (b) and III.3 (b). Table III.1: Values of the parameters used in the model of the positively charged Mn-doped QD presented in Fig. III.1. I eM n , I hM n , ρs ∆ lh , θ, η and T ef f are used to model the linear polarization intensity map of Fig. III.3. The other parameters cannot be extracted from the PL measurements and values for typical Mn-doped QDs are chosen for the calculation of the spin dynamics presented in Sec. III.

  Fig. III.3, along with the linear polarization PL intensity map of X + -Mn. Values of I hM n , I eM n , ρ c /∆ lh and η for a given QD can be obtained by comparing the linear polarization dependence of the experimental PL data to the optical transition probabilities calculated with the discussed eective spin model (Fig. III.3 (b)) [35]. A Boltzmann distribution function P i eM n = e -E i eM n /k B T ef f / i e -E i eM n /k B T ef f with an eective spin temperature T ef f is used to describe the population of the emitting states (electron-Mn energy levels E i eM n ). The extracted parameters are listed in Tab. III.1 for the QD presented in Fig. III.1 and III.3. III.1.2 Resonant PL of X + -Mn Using a tunable continuous wave dye laser, a cross-polarized PLE map was acquired, detecting at low energy half on the X + -Mn spectra and scanning the higher energy lines in σ+ polarization. The results are presented in the inset of Fig. III.4

  Fig. III.2, we can assign transitions observed in Fig. III.4 (a) to an electron-Mn state. They correspond, for a σ+ laser, to the successive resonant excitation of the electron-Mn levels |3, +1 , |3, +2 and |2, +2 , as shown in Fig. III.4 (b). As presented in Fig. III.2, these states can be expressed as linear combinations of the Mn and electron spins |S z , σ z coupled by a ip-op:

  Figure III.4: (a) Non resonant (Non Res.) and resonant (Res.) PL of X + -Mn. Co and cross circularly polarized PL spectra are collected for three dierent energies of the CW resonant laser (green). Inset: intensity map of the cross-circularly polarized PL detected on the low energy side of X + -Mn as the CW laser is scanned through the high energy side. (b) Energy levels of X + -Mn and identication of the three resonances observed in (a) corresponding to the optical Λ systems associated with the e-Mn states |3, +1 , |3, +2 and |2, +2 .

Figure III. 5 :

 5 Figure III.5: Low temperature (T = 5 K) auto-correlation of the resonant PL for cross-circularly polarized excitation and detection of the electron-Mn states (a) |3, +1 , (b) |3, +2 and (c) |2, +2 .

  Fig. III.5 (b)) suggests a fast transfer time in the nanosecond range between the two ground states of the Λ systems.

Figure III. 6 :

 6 Figure III.6: Schema of the energy levels of the optical Λ system associated with the electron-Mn state |3, +2 extracted from the full level structure of a positively charged Mn-doped QD (Fig. III.4). The dierent processes discussed in this section are presented.

  Fig. III.6, on the Λ system associated with |3, +2 state.

Figure III. 7 :

 7 Figure III.7: Excitation power dependence (a) and transverse magnetic eld dependence (b) at T = 5 K of the auto-correlation of the resonant PL obtained for an excitation on the high energy branch of the Λ level system associated to the e-Mn state |2, +2 .

  done by exciting the high energy branch of the Λ systems with trains of resonant light, alternating the circular polarization and recording the circularly polarized PL of the low energy branch. As observed in Fig. III.8, for an excitation on resonance with the electron-Mn states |3, +2 or |2, +2 , switching the polarization of the excitation from co to cross circular produces a change of the PL intensity with two transients: rst, an abrupt PL increase, reecting the population change of the observed spin-polarized charged excitons; then a slower transient with a characteristic time of a few tens of nanoseconds, depending on the laser excitation power.The exponential decrease of the resonant PL intensity is the signature of an optical pumping of the hole-Mn spin: the hole-Mn state which is optically addressed is partially emptied when the population is transferred out of the excited Λ system. As presented in Fig.III.8, this pumping signal is not sensitive to a longitudinal magnetic eld B z except for an excitation of |3, ±1 where B z enhances the intensity dierence between co and cross circular polarization.The speed of the optical pumping increases with the excitation intensity. This is presented in Fig.III.9 (a) in the case of a resonant excitation of |3, ±2 with alternate circular polarizations. At high excitation intensity, the pumping time saturates to a value similar to the width of the bunching signal observed in the auto-correlation measurements.

Figure III. 8 :

 8 Figure III.8: Resonant optical pumping transients obtained at T = 5 K under circular polarization switching of the resonant excitation for the Λ systems associated with (a) |3, +1 , (b) |3, +2 and (c) |2, +2 at zero eld and under a weak longitudinal magnetic eld B z =0.23T. The insets present the corresponding states which are resonantly excited and detected in σpolarization.

  deformation dependent non-diagonal term of H BP [101, 102] as dened in Eq. I.53. b and d are given in Tab. III.2 for CdTe and ZnTe. The coupling of the hole-Mn states results from an interplay between the hole-Mn exchange interaction and the deformation: neither the exchange interaction nor the deformation perturbation alone can couple these states.

  . III.11, the eective Hamiltonian describing the interaction mechanism with phonons in the subspace {| + 5 2

  Figure III.11: Relaxation time τ f f , between the two hole-Mn ground states of the Λ system calculated with the parameters listed in Tab. III.2 at a temperature T = 7 K. The doted line shows the energy splitting of the hole-Mn states involved in the Λ systems considered here (Resonances (2) and (3) identied in Fig. III.4).

Figure III. 12 :

 12 Figure III.12: Calculated time evolution of ρ |+ 3 2 ,↑e (t) with the QD parameters listed in Tab. III.1 and, unless specied on the gure, τ r =0.3ns, τ M n =5 µs, τ h =10ns, τ g =0.25 ns, τ f f =1.5 ns, T hM n 2

of

  Fig. III.12, the width and the amplitude of the maximum are in good agreement with the photon bunching signals observed experimentally. The dependence of ρ |+ 3 2 ;↑e (t) on the excitation intensity, τ g , and transverse magnetic eld, B ⊥ , are also well reproduced by the model (Fig. III.12 (c) and (d) respectively).

  Fig. III.12 (a), becomes then much slower than observed experimentally (Fig. III.7). Suppressing the fast ip-op process connecting the two hole-Mn ground states (τ f f = ∞ in Fig. III.12 (b)) still produces a bunching as with the initial condition used in the calculation (ρ |+ 3 mediated relaxation to be the main transfer process in the Λ system.

Figure III. 13 :

 13 Figure III.13: Calculated resonant optical pumping transients for a σdetection and an excitation of |3, +2 and |3, -2 with modulated circular polarization. The QD parameters for the calculations are those listed in Tab. III.1 and τ r =0.3 ns, τ M n =5 µs, τ h =10 ns, T hM n 2

Fig

  Fig.III.8 (b)). If the states |3, +2 and |3, -2 are uncoupled, as it is the case at zero eld, we do not detect any light during the σexcitation. With a suciently large mixing of |3, +2 and |3, -2 induced by the transverse magnetic eld, for a σexcitation of |3, -2 , the population can be coherently transferred to |3, +2 during the charged exciton lifetime and σlight is detected after a recombination towards | + 3/2, ⇑ h (see Sec. III.3). In the optical pumping sequence, we can then observe, in σpolarization, a transient when the σ+ excitation empties the state | + 5/2, ⇓ h but also a similar transient when the σexcitation empties the state | -5/2, ⇑ h . The transverse magnetic eld dependence of the dierence of steady state intensity observed in σ co and σ cross polarization (inset of Fig. III.9 (b)) is also well reproduced by the model (inset of Fig. III.13 (b)). This depolarization curve is controlled by the anisotropy of the electron-Mn spin that is induced by η and D 0 [35]. Finally, as expected, suppressing τ f f from the model, a very weak average resonant PL and a fast optical pumping are obtained (Fig. III.13 (a), top curve).Including a dark time in the pumping sequence, we can numerically evaluate the time required for the hole-Mn spin to return to the ground state of the excited Λ system. The time evolution of the population of the hole-Mn state | + 5/2, ⇓ h initially prepared by a sequence of σ-/σ+ excitation resonant with |3; +2 (and |3; -2 ) is presented in Fig.III.14. When the optical excitation is switched o, after an abrupt jump due to the optical recombination of the charge exciton, the ground hole-Mn state | + 5/2, ⇓ h is repopulated in a timescale of about 100 ns, much shorter than the Mn spin relaxation time used in the model (τ M n = 5 µs). This relaxation in the dark is induced by the presence of VBM: in this case, H ex hM n couples two by two the dierent hole-Mn levels. This coupling induces a transfer

Figure

  Figure III.15: (a) Calculated time evolution of ρ |+ 1 2 ,↑e with ρ |+ 1 2 ,⇑ h =1 (hole-Mn spin in the state |+ 1 2 , ⇑ h after a σrecombination) for a resonant σ+ excitation of the coupled electron-Mn states |3, +1 and |3, -1 without and with a longitudinal magnetic eld. (b) Zoom on the oscillation appearing in the signal from | + 1 2 , ↑ e at the beginning of a resonant σ+ excitation. (c) Time evolution of ρ |+ 1 2 ,↑e under excitation with modulated circular polarization. The parameters used in the calculations are those of Fig. III.13. Zoom on the oscillation appearing at the beginning and the end of the σ+ pusle are presented in, respectively, (d) and (e).

Figure III. 16 :

 16 Figure III.16: Copy of Fig. III.3 with a highlight on the coupling induced by the strain anisotropy E(S 2x -S 2 y ). Optical Λ systems associated with |3, +1 and |3, -1 are presented.

Fig. III. 17

 17 Fig. III.17 (a). Circularly polarized and spectrally ltered 10 ps laser pulses are

Figure III. 17 :

 17 Figure III.17: (a) Conguration of the time resolved PL experiment for an excitation of the e-Mn state |3, +1 (pulsed laser in green). (b) Top panel: Time resolved resonant PL at T = 5K of |3, +1 with a σ+/σsequence of laser pulses and a detection in σ+ and σpolarization. Bottom panel: corresponding time dependence of the circular polarization rate κ = (σ --σ + )/(σ -+ σ + ). (c) Time dependence of the circular polarization rate of the resonant PL of the states |3, +1 (red), |3, +2 (black) and |2, +2 (blue). (d) Polarisation rates calculated with D 0 = 7µeV [35], T eM n 2 = 0.6ns, E = 1.8µeV , a radiative lifetime T r = 0.3ns, I eM n = -175 µeV, I hM n = 345 µeV., ρs ∆ lh = 0.09, η = 30 µeV, T ef f = 20 K.

Figure III. 18 :

 18 Figure III.18: (a) Inuence of a longitudinal (B z , red) and a transverse (B x , blue) magnetic eld on the time dependence of the circular polarization rate κ = (σ -σ + )/(σ -+ σ + ) of the resonant PL of |3, +1 , |3, +2 and |2, +2 . On the top left panel, curves are shifted by 0.5 for clarity. (b) Corresponding time dependence of the circular polarization rate calculated with g M n = 2, g e = -0.4, g h = 0.6 [35], and the parameters of Fig. III.17. The curves are shifted by 1 for clarity.

Figure III. 19 :

 19 Figure III.19: (Color line) Calculated energy of the electron-Mn states in a longitudinal magnetic eld, B z , and in a transverse magnetic eld, B x . The parameters used in the calculations are listed in Fig. III.18.

  Contents IV.1 Strained QDs containing an individual Cr atom . . . . 100 IV.1.1 Energy structure of X-Cr in a quantum dot . . . . . . . 100 IV.1.2 Excited states of a Cr-doped QD . . . . . . . . . . . . . 104 IV.1.3 Magneto-optics of QDs doped with a single Cr atom . . 108 IV.2 Modelization of a Cr-doped QD . . . . . . . . . . . . . . 113 IV.3 Charge uctuation of a Cr ion in the vicinity of the QDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

  .1 Strained QDs containing an individual Cr atom IV.1.1 Energy structure of X-Cr in a quantum dot Using the procedure described in Sec. II.2.2, we randomly incorporated Cr atoms in CdTe/ZnTe self-assembled (SK) QDs. The PL of individual QDs, produced by optical excitation with a dye laser tuned on resonance with an excited state of the dots, is studied by optical micro-spectroscopy.

Figure

  Figure IV.1: PL of X-Cr complex at low temperature (T=5K) for (a) QD1 (found in dot334), (b) QD2 (dot338) and (c) QD3 (dot334). Inset in (c) presents the PLE map of this QD, showing a sharp quasi-resonant state for an excitation at 2100 meV.

Figure IV. 2 :

 2 Figure IV.2: Illustration of the energy levels of the ground state (Cr), the bright exciton states (| ± 1 ) coupled to the spin of a Cr (X-Cr) and dominant PL transitions (σ+, σ-). The states S z = ±2 cannot be populated through thermalization, and thus their recombination channel are not shown on this schema.

  Fig. IV.8 and IV.12.

Figure

  Figure IV.4: Time-resolved PL of QD2 taken on two outside peaks at T = 5 K, attributed to S z = ±1 (noted (1) and (3) in Fig. IV.3(a)), and the lower energy one (noted (4)).

  Figure IV.5: Temperature evolution from T=5K to T=40K of (a) the PL of QD2 and (b) the PL of a QD with a good thermalisation on the low energy states (QD4).

Figure

  Figure IV.6: (a) QD2 X-Cr PLE map at T = 5 K in π cross polarization. Several excited states are highlighted. (b) Photoluminescence of QD2 X-Cr complex for an excitation at 2120 meV. (c) PLE scan detected on the lower energy peak, taken close to the QD emission energy, showing the phonon replica taken in π cross excitation/detection. The emission integrated intensity as a function of the laser energy is plotted in (d) (black curve) along with the PL spectra of QD2 taken in σ co polarization. (e) PLE map between 2046 meV and 2013 meV presenting several excited states, detecting in π cross polarization. (f ) Zoom on a particular excited state presenting a splitting inversion, taken in π cross detection.

Figure

  Figure IV.7: (a) Low temperature (T = 5 K) PL spectra of the exciton in QD2 (X-Cr) for co-circularly (blue) and cross-circularly (red) polarized excitation/detection taken for an excitation around 2120 meV. (b) -(c) PLE map at T = 5 K between 2046 meV and 2013 meV presenting several excited states, detected in σ co (b) and σ cross (c).

  Fig. IV.8). This anti-crossing arises from a mixing of the bright and dark excitons

  Fig. IV.1(a)), while two linearly polarized peaks appear in QD3 spectra (Fig. IV.1(c)). Investigating both the biexciton and the exciton in the same Cr-doped QD, we can also analyze the impact of the carrier-Cr interaction on the ne structure of the Cr spin. The magnetic eld dependence of X 2 -Cr emission in QD3 is presented along with the X-Cr emission as a contour plot in Fig. IV.9 (a) and (b) respectively.

Figure

  Figure IV.8: (a) Evolution of the circularly polarized X-Cr PL of QD1 under magnetic eld applied along the growth axis (B z ) at T = 5 K. See text for the identication of the anti-crossing (1), (2), (3) and (4). (b) Corresponding PL spectra taken at 0 and 10T for both circular polarization.

Figure IV. 9 :

 9 Figure IV.9: Linear polarization intensity map (a) X 2 -Cr and (c) X-Cr in QD3. (b) -(d) Respective intensity map of the longitudinal magnetic eld dependence of the emission (bottom panel) of

Figure IV. 10 :

 10 Figure IV.10: (a) Evolution in magnetic eld of QD4 X-Cr circularly polarized PL. (b) QD3 X-Cr PL at B z = 0 T and B z = 7 T in both polarization. A schema presenting the spin conguration for the most intense outer peak under magnetic eld is joined on the side.

Figure

  Figure IV.11: (a) Evolution in magnetic eld of the PL of a single Manganese atom coupled to an exciton in a II-VI QD. (b) PL spectra of the X-Mn system taken at B z = 0 T and B z = 11 T in both circular polarizations. These experimental results are taken from Yoan Léger PhD thesis [51].

  Figure IV.12: (a) Calculated linear polarization PL intensity map of X-Cr at zero eld. The 0 • polarization angle correspond to an emission polarized along the [100] axis. (b) Right: Calculated magnetic eld dependency of X-Cr circularly polarized PL. See text for details of the model. Parameters are listed in Tab. IV.1. Corresponding anti-crossing are highlighted in the same way as on Fig. IV.8 and IV.10. Left: PL spectra calculated for B z = 0 T and B z = 11 T in both circular polarizations. The peaks intensity was calculated using only a thermal distribution. (c) Schema of the magnetic eld dependency of the energy levels of the low energy Cr spin states S z = 0 and S z = ±1, and corresponding bright (| + 1 blue, | -1 red) and dark (| ± 2 green) X-Cr energy levels.

  applied on the X-Cr levels. The results of the model obtained with T ef f = 20 K, D 0 = 2.2 meV and an electron-Cr (hole-Cr) exchange interaction I eCr = -30 µeV (I hCr = 250 µeV) are reported in Fig. IV.12. The exchange parameters cannot be extracted independently, and are chosen to t the data to the closest. ∆ lh D 0 is needed to see no eect of the light-holes on the magneto-optics maps. ∆ lh = 25 meV was chosen. The parameters not specic to Cr-doped QDs are listed in Tab. IV.1. Such parameters do not aim to precisely t the data and are only reasonable order of magnitude to qualitatively reproduce the experimental results of the PL of X-Cr at zero eld and its evolution in magnetic eld. The splitting of the central line at zero eld (anti-crossing (1)) and the anti-crossings under magnetic eld (anti-crossings (2) and (3) around B z = 6 T for the Cr spin states S z = | + 1 and anti-crossings (4) with the dark exciton around B z = 2 T) are well reproduced by the model.

Fig. IV. 13 .

 13 Fig. IV.13. Two other anti-crossings appear when increasing the value of E. They appear on the low and high energy peaks, around B = 4 T and are labelled (7) and (8) in Fig. IV.13. They occur when the states |S z = +1, X z = -1 and |S z = -1, X z = -1 are brought together by the Zeeman eect. These states are composed of two Cr spin states separated by two units of spin coupled to the same exciton state. Their coupling by E causes the anti-crossing by splitting the initial state of the recombination.Fig. IV.13 (b) shows the evolution in linear polarization and the circularly

Fig. IV. 15

 15 Fig. IV.15 (a) presents such a PL map versus bias voltage. The rst visible feature is the strong electric eld dependency of the emission energy, more pronounced for X-Cr than for X + -Cr and X --Cr. The maximum shift of X-Cr is of about 2.9 meV. Another remarkable point is evidenced in Fig. IV.15(b): the splitting between each peak of X + -Cr changes with the applied electric eld. The total splitting, measured between the high and low energy peaks, decreases from about 0.7 meV at 8 V down to 0 meV at -4.5 V.

Figure IV. 16 :

 16 Figure IV.16: PL of X-Cr in QD7 (dot390) at T = 5 K. (a) Linear polarization dependence of the PL intensity, taken at -2.5 V bias voltage. (b) Circular PL intensity evolution as a function of the applied bias voltage. (c)-(d) Circular PL for an applied bias voltage of, respectively, 0 V and -2.5 V.

Fig. IV. 16

 16 Fig. IV.16 shows that, using bias voltage, one can manipulate the splitting of any given charged state of the QD. For all positive bias voltages between 0V and 7V, X-Cr presents a broad emission containing six peaks in linear emission,

  photon correlation techniques. Auto-correlation of the photons emitted by the QD under continuous optical excitation reveals uctuations of the localized spin with a timescale in the 10 ns range. Cross-correlation experiments allow to extract quantitative transfer time between Cr spin states.In the second part, we demonstrate the possibility to prepare the Cr spin using resonant optical pumping. Monitoring the time dependence of the intensity of the resonant uorescence of the QD during the pumping permits to probe the dynamics of the optical initialization of the Cr spin. Using resonant excitation, we

Figure V. 1 :

 1 Figure V.1: Auto-correlation of the PL intensity of QD2, collected in circular polarization on the X-Cr lines (1), (4) and (5) and compared with the autocorrelation of the exciton in a nonmagnetic QD (black line). The curves are shifted for clarity. For line (1), the auto-correlation is also recorded under a transverse magnetic eld B x = 0.42 T (blue line). Inset: Circularly polarized PL spectra of QD2. The lines corresponding number are indicated above them.

  .2 Resonant optical pumping of a Cr spin An ecient way to probe the dynamics of a Cr spin under excitation or in the dark is to perform resonant optical pumping. To do that, we tuned a laser in resonance with one of the transition of the QD: an exciton can only be injected if the Cr spin is in the resonantly excited state. If the intraband relaxation time of the X-Cr complex is smaller than the one of the Cr alone (τ Cr > τ X-Cr ), the Cr may undergo spin-ips after an absorption, progressively decreasing the population of the resonantly excited state. Once it is empty, the resonant laser cannot inject exciton in the QD anymore.Resonantly exciting the high energy transition, a signature of the pumping can be detected by looking at the cross-polarized PL of the low energy line of the dot, after a spin-ip of the exciton. In this conguration, both the excitation and the detection are done on the same Cr spin state. It supposes that the spin ip time of the exciton in presence of Cr is smaller than the relaxation time of the Cr in the excited state τ X-Cr . This process is illustrated on the inset of Fig. V.4 for an excitation of |S z = -1, X z = -1 state.

Fig. V. 5

 5 Fig. V.5 shows the experimental set-up. The intensity of both lasers can be switched ON and OFF with accousto-optic modulators with a time response of about 10 ns. This permits to create trains of pulses with dierent durations and delays. The two lasers are focused on the sample with a microscope objective. A high index (n ≈ 2.5) Solid Immersion Lens (SIL) is also mounted on the surface of the sample, to increase the spatial resolution and the collection of photons emitted by the QD. The emitted light is dispersed and ltered by a double monochromator

  Figure V.6: (a) PL transients recorded in circular polarization on line (3) and on line (4) (as dened in the inset) under the resonant (pump on (1)) and quasiresonant optical excitation sequences displayed at the bottom. Inset: PL of X-Cr and conguration of the resonant excitation and detection. (b) and (c): Energy detuning dependence of resonant PL intensity (I 1 , at the beginning and I 2 , at the end of the pump pulse) and of the corresponding normalized amplitude of pumping transient ∆I/I 2 = (I 1 -I 2 )/I 2 .

Figure V. 7 :

 7 Figure V.7: Optical pumping under a CW probe at E = 2004 meV. (a) Pumping on line (1) and detecting on line (4). (b) Pumping on line (4) and detecting on line (1).

Figure V. 8 :

 8 Figure V.8: PL transients measured on line (4) while pumping on line (1) at P pump = 250 µW (red) or without pumping (dark). (a) P probe = 125 µW. (b) P probe = 250 µW. (c) P probe = 500 µW.

Fig. V. 10

 10 Fig. V.10 (a) shows the PL detected during the probe and pump pulse on the low energy line in red and in the central peak in black. The detection on the low energy line is similar to the one presented in Fig. V.6. It is reported here for comparison with the evolution of the state S z = 0.

Figure V. 10 :

 10 Figure V.10: (a) Pumping experiment for a circular polarized excitation on line (1) and cross-circularly polarized detection either on line (3) (black) or on line (4) (red). The inset presents the PL of the dot with a reminder of the line numbers. (b) Cross-linearly polarized pumping experiment, exciting on line (1) and detecting on line (3).

Figure V. 11 :

 11 Figure V.11: Relaxation paths for the X-Cr system for a resonant excitation on the high energy line (|S z = +1, X z = +1 in σ+ polarization, or |S z = -1, X z = -1 in σpolzarization). The relaxation involves a h spin-ip (τ h ) and a hole-Cr spins ip-op (τ hCr ).

  Pikus hamiltonian H BP (Eq. I.24). The phonon induced deformation comes into play through the o-diagonal term of this hamiltonian. The phonon-mediated coupling between |0 | ⇑ h , ↓ e and | + 1 | ⇓ h , ↓ e can then be found through the hamiltonian term: ⇓ h , ↓ e | +1|H BP |0 | ⇑ h , ↓ e = 2 ×with r * as dened in Eq. I.53.

Figure V. 12 :

 12 Figure V.12: Relaxation time τ hCr between the states |S z = ±1, X z = ∓2 and |S z = 0, X z = ±1 as a function of the energy splitting between those states, calculated at a temperature T = 7 K with the gaussian hole wave function parameters l z = 1.2 nm and l ⊥ = 3 nm, and the parameters I hCr = 0.25 meV and ∆ lh = 25 meV. CdTe parameters can be found in Tab. III.2.

  Fig. V.12 shows τ hCr as a function of the splitting between the two states. The splitting between |S z = ±1, X z = ∓2 and |S z = 0, X s = ±1 is expected to be in the meV range in the studied Cr-doped QDs. It gives a τ hCr of a few ns. This short ip-op time, consistent with the time measured in optical pumping experiments (see Fig. V.9), controls the exciton-Cr dynamics under resonant excitation.

Figure V. 13 :

 13 Figure V.13: (a) Evolution of the pumping transient intensity in a pump-probe experiment as a function of the distance between the probe laser and the dot. (b) PL transients recorded for a probe laser at d = 4 µm from the dot. Red: probe on; black: probe o. (c) Evolution of the pumping transient intensity as a function of the probe power, for a probe laser at d = 4 µm from the dot. (d) PL transient recorded for a probe laser at E probe = 2014.5 meV, close to the resonant laser (E pump = 2001.2 meV). Red: probe on; black: probe o. (e) Evolution of the pumping transient intensity as a function of the probe power.

  will then be able to excite phonons in it and inject them in the semiconductors without injecting any carriers[130, 131].V.4 Cr spin relaxation in the darkUsing resonant optical pumping technique presented in Sec. V.2 to prepare and read-out the Cr spin, we performed pump-probe experiments to measure its relaxation time in the absence of carriers (Fig. V.14). A non-equilibrium distribution of the Cr spin population is prepared with a circularly polarized resonant pump pulse on the high energy X-Cr line (line (1)). The pump laser is then switched o, and switched on again after a dark time τ dark . The amplitude of the pumping transient observed on the resonant PL of the low energy line depends on the Cr spin relaxation time in the dark τ dark . As presented in Fig. V.14 (b), the amplitude of the transient seems to be fully restored after a dark time of about 10 µs, suggesting that after this delay the Cr spin is in equilibrium with the lattice temperature (T = 5 K). Let us note, however, that the initial amplitude of the pumping transient in this case is weaker than the one observed after a quasi-resonant probe pulse (Fig. V.6 (a)).

  mechanisms. However, Fig. V.14 shows only one time, suggesting that the second might occur at a longer time scale.

Figure V. 15 :

 15 Figure V.15: Comparison of the relaxation of the Cr spin after resonant pumping in the line (1) with (red) or without (blue) a probe pulse (E probe = 2070 meV). (a) Time resolved PL of line (4) for a cross-circular resonant pump on line (1) with a probe pulse. (b) ∆I/I 2 as a function of the dark time τ dark measured for the relaxation between the probe and the pump pulses (red) or between two pump pulses (blue). (c) Time resolved PL of line (4) for a cross-circular resonant pump on line (1) with no probe pulse.

  are anti-linear (noted |I, n in Fig. V.16) and linear (|I I, n ) combination of the unperturbed states. They are split proportionally to the Rabi opping frequency Ω r . This splitting is called Autler-Townes splitting [135] and is well described by the dressed-atom picture. It could be exploited to control the coherent dynamics of a Cr spin [27, 76].

Fig. V. 16

 16 Fig. V.16 for |G = |Cr and |E = |X -Cr , with S z = +1 and X z = +1.

  Figure V.16: Illustration of the energy level of a Cr-doped quantum dot and formation of the dressed-states.

  δ = 0), c = s = 1/2. The states |E, n -1 and |G, n corresponding to the upper and lower levels of the transitions have then equal contribution to the dressed levels |I, n and |II, n : the dressed states are entangled atom-eld states.To observe the inuence of a high power resonant laser on a QD doped with a single Cr atom, we used the optical conguration presented in Fig. V.17. A high intensity laser was put in resonance with the high energy transition (line (1)), in σ+ polarization, exciting the Cr spin state S z = +1 in order to induce Autler-Townes splitting of this state. The splitting can be observed on any state sharing the same ground state. A quasi-resonant laser, as presented in Sec. V.2, was used to generate PL on those states. It was detected in cross-circular polarization on the low energy line (4) (recombination from |S z = +1, X z = -1 ) and the dark exciton line (5) (recombination from |S z = +1, X z = -2 ).

Figure V. 17 :

 17 Figure V.17: (a) PL of QD2 X-Cr and conguration of excitation in the resonant optical control experiments. The inset illustrate the laser induced splittings in the ground and excited states for a σ+ excitation on |S z = +1 . (b) PL intensity maps of lines (5) and (4) for an excitation on (1) as a function of the detuning (top) and of the excitation intensity (bottom). The PL is produced by a second quasi-resonant laser.

Figure V. 19 :

 19 Figure V.19: PL of line (1) and (2) (high energy line) for a laser on resonance with the dark exciton state (5). Inset: PL intensity map of line (1) as a function of the laser detuning around (5).

Table 1 :

 1 List of dierent possible transition metals and their key properties in the context of our study.

	Inserted atom	V	2+	Cr 2+	Mn 2+	Fe	2+	Co	2+	Ni	2+	Cu 2+
	d-shell	d 3	d 4	d 5	d 6		d 7		d 8		d 9
	Electronic spin	3/2	2	5/2	2		3/2	1		1/2
	Nuclear spin	7/2	0	5/2	0		7/2	0		3/2

Table I .

 I 1: Physical parameters for CdTe and ZnTe.

		CdTe	ZnTe
	E g	1606 meV	2391 meV
	ε r	10.6	9.7
	a 0	6.48 Å	6.10 Å
	∆ SO	0.90 eV	0.91 eV
	γ 1	4.8	4.07
	γ 2	1.5	0.78
	γ 3	1.9	1.59
	m hh,z	0.556	0.398
	m hh,⊥	0.159	0.206
	m lh,z	0.128	0.178
	m lh,⊥	0.303	0.303
	m e	0.096	0.116

I.1.2 Lattice mismatch and the Bir-Pikus Hamiltonian

ZnTe crystal has a lattice parameter of a ZnT e = 6.10 Å, while CdTe one is of a CdT e = 6.48 Å. Since we grow CdTe on a ZnTe substrate epitaxially, this lattice mismatch results in strains in a CdTe layer grown on a ZnTe substrate: ε =

a ZnT e -a CdT e a CdT e = -5.8%

(I.

[START_REF] Gall | Optical stark eect and dressed exciton states in a Mn-doped CdTe quantum dot[END_REF]

  The conduction electron wave function can be written as |ψ k |σ; σ z ≡ |ψ k ; σ z , |ψ k being the Bloch function of the electron. First, we only consider electrons of the conduction band. On the other side, considering a magnetic atom at r = R d , we write the spatial component of the wave function Φ d (r -R d ). Its total electronic spin, sum of the electron spins on its d orbital, is noted |S; S z . The whole wave function of the magnetic atom is then |Φ d ; S Z .

	For simplication, let's begin with a single magnetic atom in the semiconductor
	lattice. Using Born-Oppenheimer approximation, separating the movement of the
	electrons from the one of the nuclei, we can write the hamiltonian of this material:

  Schrieer and Wol rewrote the Anderson hamiltonian H hyb in order to give a form closer to the Kondo hamiltonian [62]:

Table I .

 I 2: Values for spin to strain coupling coecients of Cr in bulk CdTe (in

Table II .

 II 1: Targeted ux in Beam Equivalent Pressure (BEP) for each cell during the growth of the strained QDs.

	Elements	Targeted BEP (Pa)	Targeted ux (atoms.cm -2 .s -1 )
	Zn	9.1 × 10 -5	1.3 × 10 14
	Te	6.0 × 10 -5	5.8 × 10 13
	Cd	6.0 × 10 -5	6.7 × 10 13
	in Tab. II.1 for each cell used during the growth. These uxes were measured via
	a Bayard-Alperd pressure gauge inside the MBE chamber, and are therefore given
	in Beam Equivalent Pressure (BEP). It was shown that the best quality of ZnTe

was achieved for a growth in excess of Zn

[87]

. Otherwise, vacancies appear in the bulk, optically visible, and the surface is rougher. The ZnTe barriers were therefore grown in excess of Zn.

Table II

 II

	Sample	Cleaning	# CdTe	Targeted Cr	# Cr-doped
	name	process	MLs	concentration (%)	dots found
	dot334	Br etching	6.5	0.09	3
	dot338	Br etching	6.5	0.05	2
	dot359	Br etching	6.5	0.11	1
	dot363	Br etching	6.5	0.21	2

.2: List of samples where single dots with single Cr was found.

  • C. Starting at 320 • C, we opened the Te cells in order to stabilize the surface. When the substrate temperature reached 360 • C, we opened the Cd cell and grew a 2.35 µm of CdTe on top of the 3 µm grown on the TableII.4: Strain-free QDs doped with Cr. This may be caused by higher steps than expected at the CdTe/Cd 0.7 Mg 0.3 Te interface.

	Sample	# CdTe MLs	Cr aimed concentration (%)	Probability of Cr-doped QD
	SFD4	4	0.35	None found
	SFD5	2	0.15	None found
	SFD6	2	0.54	None found
	SFD7	2	0.35	None found
	SFD8	2	0.75	None found
	grown in Grenoble.

Table IV .

 IV 1: Values of the parameters used in the model of Cr-doped CdTe/ZnTe quantum dot presented in Fig. IV.12. The value of the parameters not listed inthe table is 0. The chosen values are typical for CdTe/ZnTe quantum dots and can be compared with parameters extracted from Mn-doped quantum dots[START_REF] Varghese | Dynamics of a Mn spin coupled to a single hole conned in a quantum dot[END_REF] 56]. These values are reasonable to reproduce the emission of the QDs presented

	in this thesis.
	I eCr	I hCr	δ 0	δ 1	δ 12	δ 11	|s| ∆ lh	|r| ∆ lh
	µeV	µeV	meV µeV	µeV	µeV
	-30	250	-1	250	150	50	0.05	0.05
	arg(r) D 0	g Cr	g e	g h	γ	η	T ef f
	-	π 2	meV 2.2	2	-1	0.4	µeV /T 2 µeV 1.5 25	K 20

  The reason a Schottky gate deposited on the MBE grown sample as presented in Sec. II.3.1.

;⇑ h (0) = 1): a weak coherent transfer between the two ground states of the Λ system still exists. However, with this weak transfer only, the calculated PL intensity is always more than 50 times smaller than with τ f f . The contribution of this weak process to the calculated auto-correlation signal (Fig.III.12 (a)) can therefore be safely neglected, and we can consider the phonon

,↑e )

at the PL of single dots and see their evolution under the application of a bias voltage. Fig. II.8 (a) presents the results of such an experiment for a dot with no Cr embedded. hybrid substrate, in order to have a QW structure far from the surface that might have been slightly damaged during the conservation or the Te evaporation. We then raised the substrate temperature up to 360 • C, under a Te ux, in order to proceed to the growth of the upper barrier, acting also as a protective layer. The opening time was there calculated to grow 90 nm of Cd 0.7 Mg 0.3 Te.

Optical characterization

Four samples of strain-free dots doped with Cr were produced (see Tab. II.4). The tested Cr concentrations were taken high because it was observed in the strain-free dots grown in Grenoble that a higher concentration of Mn was needed in strainfree dots than in SK dots. It was supposed to be the same with Cr. The Cr concentration was later raised after no dot doped with a single Cr was found.

The sample presented sharp and intense peaks, as shown in The structure found at B z = 0 T is expected for QDs with a large in-plane anisotropy term E, as presented in Sec. IV.2. However, the behaviour of these QDs under magnetic eld and application of bias voltage is not consistent with high E QDs.

We propose that the PL behaviour presented in and in the absence of optical injection of carriers, this spin relaxation is likely to be controlled by the spin-lattice interaction. Despite the large spin-phonon coupling expected for this magnetic atom with an orbital momentum and a strain induced spin splitting in the meV range [START_REF] Lafuente-Sampietro | Individual Cr atom in a semiconductor quantum dot: optical addressability and spin-strain coupling[END_REF], the Cr spin relaxation time remains in the µs range. This spin memory is long enough for a practical use of Cr in an hybrid spin nano-mechanical system and could even be improved in dierent QD structures with weaker biaxial strain [START_REF] Besombes | Resonant optical pumping of a Mn spin in a strain-free quantum dot[END_REF], lower magnetic anisotropy splitting and consequently less coupling with acoustic phonons [107].

The Cr spin-ip time found for a relaxation in the dark (µs range) is a lot longer than the one found under optical excitation (tens of ns range, see Sec. V.2). As shown in Sec. V.3, the fast relaxation of the Cr under excitation is caused by the h-Cr ecient relaxation path and the interaction with non-equilibrium acoustic phonon. For the Cr alone, the picture is dierent.

After the pump pulse, the Cr spin state S z = -1 is partially empty. During the dark time, it is repopulated from the states S z = 0 and S z = +1. The transfer is possible from the state S z = 0 by a Cr spin ip mediated by the absorption of a phonon. This one phonon mechanism, called direct mechanism, depends on ∆ 3 i,j , with ∆ i,j the splitting between the Cr spin states i and j.

The direct mechanism is ecient for the transfer from S z = 0, with ∆ 0,-1 ≈ 2