Dr Solignac

Jean-Baptiste Moussy

Danylo Babich

Rostyslav Danylo

Pavel Leshuk

Anton Pylypenko

Irina Andriy

Irina Alexandr

Ludmila Anatoliy

In

Keywords: electron Drude and restoring-force Lorentz models, sampling detectivity enhancement, Laser pump fluence dependence, metal stack amplitude absorption, MgO amplitude absorption, optical properties, 35

Furthermore, I would like to thank all members of the Institut des Molécules et Matériaux du Mans, and especially the past and present NoVA team researchers, for their help and assistance. I can not factor out the vital theoretical contribution of Prof. Dr. Brice Arnaud

This long-short four-year journey was not possible without the help of many people. This brief letter is to express my most profound gratitude for their contributions to this work.

First and foremost, I sincerely thank my supervisors, Dr. Vincent Juvé and Prof. Dr. Pascal Ruello, for giving me the possibility to carry out this work. Furthermore, I sincerely appreciate their constant scientific, experimental, and personal guidance, which made the completion of this Ph.D. project possible. Moreover, I am immensely thankful for their availability, patience and encouragement, especially during the final steps of the manuscript and defense preparation. I could not have hoped for better supervisors and mentors. I sincerely thank Prof. Dr. Steven Johnson, Prof. Dr. Luca Perfetti, Dr. Arnaud Arbouet, and Dr. Christine Boeglin for being the jury members. I highly appreciate their effort to come to Le Mans, time spent reading the manuscript, profound questions, and valuable suggestions about the work.

In addition, I want to acknowledge the enormous technical help and support provided by Dr. Gwenaëlle Vaudel. She was the one who has kept the main laser system running in a stable condition, allowing me to carry out the experiments on a daily basis. In addition, her reactivity and responsivity have simplified and accelerated the resolution of numerous problems.

Essentially, the experimental work carried out in the past four years was impossible without the high-quality samples. I thank Dr. Mathieu Edely for the in-lab growth of the thin metallic films of Aluminum and Chromium. In addition, I want to acknowledge our colleagues from the University of Silesia in Katowice for supplying us with beautiful nanofilms of Bismuth Telluride: Dr. Bartosz Wilk, Dr. Mateusz Weis, Dr. Katarzyna Balin, and Pof. Dr. Jacek Szade.

Ultimately, our fruitful collaboration with CEA SPEC in the frame of the ANR SANTA project has allowed us to investigate and understand the complex physics in a multitude of thin magnetic heterostructures. Given this opportunity, I want to acknowledge Dr. Michel Viret, Dr. Jean-Yves Chauleau, Dr. Aurélie

Introduction

Solid-state physics at the nanoscale has attracted considerable scientific and industrial interest over the past century. Today, we see an exponential increase in the amount of data generated, stored, and transmitted both locally and all around the web. However, in addition to the everincreasing computational complexity, the question of the most powerful, robust, and energy-efficient core for future electronics remains unanswered. As chipmakers relentlessly pursue Moore's Law, advances in pulsed femtosecond lasers have allowed scientists to study physical processes six times faster than the clock speed of the best processors on the market. Ultimately, the rapid development in ultrashort lasers has defined new research directions in nanophotonics, solid-state physics, and spintronics. Therefore, understanding the ultrafast coupling between electrons, phonons, and spins in nanostructures is vital for new promising solutions to the numerous technological challenges. Today, we routinely use the microwave part of the spectrum (up to 300 GHz) for communication, remote control, and radar technology. In addition, we have mastered the production of electromagnetic waves in the infrared, visible, ultraviolet, and radiographic range, widely used for security, optical imaging, memory, and cancer treatment, to name a few. However, until recently, it was challenging to generate light in the so-called terahertz (THz) range (300 GHz-40 THz), which is very promising for the fundamental research and industrial applications. In particular, the THz frequency range is fascinating for scientific applications, as these frequencies resonate with collective modes of matter, such as the vibrations of molecules or crystals, collective spin excitation (magnons), or the motion of the free electrons [START_REF] Kampfrath | Resonant and nonresonant control over matter and light by intense terahertz transients[END_REF]. For instance, the large amplitude THz transients generated by intense mid-infrared femtosecond pulses can be tuned in resonance with the infrared active vibrational modes inaccessible by thermal, mechanical, or optical excitation [START_REF] Först | Mode-Selective Control of the Crystal Lattice[END_REF]. To explore INTRODUCTION such intricate THz-matter interaction, developing the efficient, broadband, and tunable source of the pulsed THz radiation is a priority task. For example, it has been shown that femtosecond laser illumination of nm-thick magnetic heterostructures with strong spin-orbit interaction can lead to broadband THz emission with a tunable polarization state [START_REF] Kampfrath | Terahertz spin current pulses controlled by magnetic heterostructures[END_REF]. Furthermore, the selective control over subtle modulations of the matter's vibrational, electric, and magnetic properties, possibly achievable with the THz, is of particular interest for high-performance and low-power electronics. This thesis will contribute to both aspects of THz physics: the excitation of vibrational modes of a solid with THz pulses and the characterization of the promising spintronic THz emitters.

The first part of this thesis will be devoted to the experimental study of the coherent generation of acoustic phonons with picosecond pulsed THz radiation. Using ultrafast nonlinear spectroscopy in a pump-probe geometry, we will study the transient response of metal films (Cr, Al) and a narrow bandgap, n-doped topological insulator (Bi 2 Te 3 ) at the picosecond time scale. Through a quantitative comparison between terahertz (4 meV) and near-infrared (1.5 eV) excitation, we will show that the acoustic phonon generation process in thin metal films with THz is mainly driven by thermoelastic stress caused by the ultrafast Joule heating of the lattice. To the best of our knowledge, it is the first demonstration of the THz-driven coherent acoustic phonons in metals and narrow-band semiconductors.

The following discussion will focus on the rich lattice dynamics when the terahertz pulse excites the Bi 2 Te 3 nanofilm. We will demonstrate the non-resonant generation of coherent Raman active phonons, with frequencies up to five times higher than the center frequency of the driving THz pulse. Thus, this work contributes to the description of emerging nonlinear solid-state phononics and discusses different physical mechanisms of the selective control of the lattice vibrations with THz light.

The third cornerstone of this thesis aims to detect the ultrafast signature of the spin-charge interconversion due to the spin transport in the bulk of CoFeB combined with the injectionbased spin-to-charge conversion mechanisms. In particular, the study will be focused on the subpicosecond dynamics of the recently discovered Rashba-split band structure at the CoFeB/MgO interface [4]. Therefore, this thesis will try to probe the THz emission due to the ultrafast inverse Rashba-Edelstein conversion within the CoFeB/MgO bilayers. Among the results, we will show that the inverse Rashba Edelstein effect is sensitive to the pump photon energy, which opens exciting perspectives for ultrafast spintronic emitters.

The manuscript is constructed as follows. Chapter 1 will touch on the fundamentals of solidstate physics, including electron, phonon, and spin properties, basics of electromagnetism, and nonlinear optics. Chapter 2 is dedicated to the presentation of the experimental methods and techniques. Finally, the Chapters 3 and 4 present the results about coherent phonon excitation with pulsed THz radiation and ultrafast spin-to-charge conversion, respectively. We will then conclude this manuscript with a general summary of this thesis's main achievements and prospects for future work.

Introduction

Back in 1912, M. Laue proposed a rather simple experiment (Fig. 1.1a) and observed the X-Ray interference due to the crystalline nature of a sample, for which he was awarded the Nobel Prize in 1914. One hundred years later, thanks to the progressively increasing knowledge and scientific interest in the condensed matter physics, we developed some remarkable solid-state devices such as transistors, semiconductor lasers, optical fibers, lithium batteries, liquid crystal display, and many others.

Here, we introduce the baseline theory necessary for discussion of the following chapters. This chapter will start with the notions about the arrangements and properties of atoms, electrons, and spins in solid matter. Next, the basics of the light-matter are explained with the definition of the electromagnetic wave, optical properties of the material, and second-order nonlinear processes in crystals. At the end, a brief description of the temporal coupling of light with the electrons, atoms, and spins will be given. First, we will define what the ideal crystal is. The ideal three-dimensional (3D) crystal can be imagined as a periodic array of elementary building blocks, i.e., atoms or a group of atoms, that is called the basis (red circles on Fig. 1.2a). The set of mathematical points (coordinates) to which the basis is attached is called the lattice [START_REF] Kittel | Introduction to Solid State Physics[END_REF].

3D lattices, due to the point symmetry, form 14 different lattice types in total: in addition to general triclinic, there are 13 other lattices named by the corresponding unit cell type. These 14 lattices are called Bravais or direct lattices. For example, 3 possible configurations of the cubic lattice are shown in Fig. 1.2 (a).

CHAPTER 1. THEORETICAL BACKGROUND

The next important notation is the crystal plane orientation. The set of parallel planes that intercept the crystal unit-cell can be noted as (hkl), where Miller's indices h, k, and l define the point of intercept on the given crystal axes a, b, c at position 1/h, 1/k and 1/l respectively. The example of the crystal plane notation using Miller indices is shown in the Fig. 1.2b. Usually, the reciprocal values of h, k, l are reduced to the set of integers with the least common multiplier. If plane cuts the axis on the negative side, the corresponding negative index is then noted as h, k, l, as seen from Fig. 1.2c. The type of Bravais lattice combined with the crystal plane orientation can affect the combination of the solid matter's elastic, electronic, magnetic, and optical properties. The 3D crystal is built with an ever-repeating pattern of atoms. For simplicity, within the harmonic potential approximation let us consider an infinite 2D diatomic chain, where two atoms with masses M 1 and M 2 lie on the plane, tied together with the elastic force constant C. This two-atom basis has the repeat distance a as shown in Fig. 1.3a. We assume that each atom interacts only with its nearest neighbors. Therefore, the simple equations of motion for the ions in the chain reads: and we will look for the nontrivial solutions in the form of the wave with respective amplitudes v and u:

Ionic subsystem

Phonon dispersion relation

       M 1 d 2 u n dt 2 = C(v n + v n-1 -2u n ); M 2 d 2 v n dt 2 = C(u n+1 + u n -2v n ) , ( 1
u n = ue inka e -iωt ; v n = ve inka e -iωt . (1.2)
Given the periodicity of this classical diatomic chain, we can apply the periodic boundary condition (Born-von Karman):

e ikN a = 1, (1.3)
where N is the number of lattice repetitions, that defines the total chain length as L = N a.

Solution of this boundary condition for k returns the value for the edges of the first Brillouin zone, containing all unique values of atomic vibrational frequency ω. If we substitute Eq.1.2 in Eq.1.1 we find solutions for ω 2 :

ω 2 ac = C M 1 M 2 M 1 + M 2 -M 2 1 + M 2 2 + 2M 1 M 2 cos (ka) ; ω 2 opt = C M 1 M 2 M 1 + M 2 + M 2 1 + M 2 2 + 2M 1 M 2 cos (ka) .
(1.4)

These solutions tell us about the frequency range of the permitted oscillations of the crystal within the first Brillouin zone, called phonon dispersion curves are referred to as an acoustical and optical branch, and are shown in the Fig. 1.3b. Within the acoustical branches, two atoms oscillate in the same phase, following the similar pattern of the sound wave. For the optical branch, two ions oscillate out of phase. Such vibration can be imagined as the action of electromagnetic (EM) waves on the two dipoles with opposite charges [START_REF] Kittel | Introduction to Solid State Physics[END_REF], hence the optical branch name. Optical and acoustical branches are split by the amount of 2C M 1 -2C M 2 at the edge of the Brillouin zone, defining the range of forbidden vibrational frequencies within the crystal, named phononic bandgap.

CHAPTER 1. THEORETICAL BACKGROUND

The example of the phonon dispersion curve of a ZnTe crystal is shown in Fig. 1.4a. ZnTe has fcc lattice and is from F 43m [START_REF] Liu | Spin-torque ferromagnetic resonance induced by the spin Hall effect[END_REF] space group (Hermann-Mauguin notation), also known as Zincblende. The 3 degrees of freedom (x, y, z) of each atom allow for 3p vibrational modes in total, where p is the number of atoms per unit cell. ZnTe phonon dispersion curves have 3 (1 Longitudinal (LA) and 2 Transverse (TA)) acoustical modes, where the remaining 3p -3 will be reserved for Longitudinal (LO) and Transverse (TO) optical phonon branches. Along the strong symmetry axes (ΓX, ΓL, see Fig. 1.4b), due to the degeneracy of transverse branches, we can observe only 4 modes: TA, TO, LA and LO. If we move perpendicularly to the (110) crystallographic plane, i.e., along [START_REF] Weis | Ultrafast lightinduced coherent optical and acoustic phonons in few quintuple layers of the topological insulator Bi2Te3[END_REF] direction, TO and two TA branches lose that degeneracy, resulting in 3 × 2 = 6 predicted vibrational modes in total.

Lattice heat capacity

The quanta of lattice vibrational energy is called a phonon. Energy of a given elastic mode, occupied by n s phonons (n phonons within the branch s) with frequency ω s and wave vector k defined as:

ϵ s = (n s + 1/2)ℏω s (k). (1.5)
Therefore, the total vibrational energy of the crystal could be simply expressed as the sum over all discrete energy values for all existing phonon normal modes:

E ph = s,k n s + 1 2 ℏω s (k). (1.6)
Phonons are bosons. The mean number of phonons that occupy n s (k) energy state at given temperature T can be approximated with Planck distribution:

⟨n s (k)⟩ = 1 exp ℏωs(k) k B T -1 , ( 1.7)
where k B is the Boltzmann constant. This distribution tells us that high-k states are more likely to be occupied at higher temperatures. Knowing the thermal distribution of the phonon modes, we can write the generalized expression for the crystal vibrational energy density [START_REF] Ashcroft | Solid State Physics[END_REF]:

U ph = U eq + 1 2V k,s ℏω s (k) + 1 V k,s
⟨n s (k)⟩ ℏω s (k), (1.8) where U eq is the lattice energy at equilibrium (T = 0), and the second term corresponds to the zero-point vibration of the lattice mode s. The third term introduces the temperature dependence of U ph . Finally, we can define the lattice heat capacity c L , at constant volume as:

c L = ∂U ph ∂T V = 1 V k,s ∂ ∂T ⟨n s (k)⟩ ℏω s (k), (1.9)
CHAPTER 1. THEORETICAL BACKGROUND

showing the temperature dependence of c L . For given temperature range, there are different approximations for c L . For the majority of crystals at T > 300 K, one can use the law of Dulong and Petit (c L ∝ 3k B n s /V ). At low temperatures Debye approximation (c L ∝ T 3 ) assumes that only low-ω acoustical modes are excited. This makes phonon frequency ω s directly proportional to the wavevector k, resulting in a constant sound velocity. While Debye approximation works for acoustical branches, Einstein approximation is usually chosen as a first approximation to describe the energy contribution of optical phonons to c L . Both Debye and Einstein laws are designed to work within specific temperature range, and still, they both converge to the classic law of Dulong and Petit at high temperatures [START_REF] Kittel | Introduction to Solid State Physics[END_REF]. For the considerable number of the phonon modes, one can replace the sum over k in the Eq.1.9 by an integral, saying that crystal has D s (ω)dω modes in the frequency interval [ω, ω + dω] and in the branch s:

c L = 1 V s D s (ω)dω ∂ ∂T ⟨n s (k)⟩ ℏω s ((k), (1.10)
where D s (ω) is the phonon density of states. In its general form, D s (ω) ∝ 1 v g where v g is the phonon group velocity, also known as the speed of sound.

Electronic subsystem

When considering the electronic Hamiltonian with a constant potential (even zero by selecting a specific energy scale), the electron wave function becomes that of a free electrons' Sommerfeld model with parabolic electron dispersion relation: .11) where m is the mass of the electron in vacuum. Adding some periodic boundary Born von Karman condition, we obtain the quantization of the wave vector and the associated energy. The surface at a constant energy, plotted in the k space, defines the Fermi sphere.

ϵ = ℏ 2 k 2 2m , ( 1
In reality, the periodic arrangement of the lattice and its translational symmetry invariance highly impact how the electron behaves inside the crystal. The electrons are indeed affected by the periodic potential generated by neighboring electrons and lattice ions that leads to specific electron-ions interaction, creating some "gaps" in the electronic band structure. The eigenfunction of the electron, or Bloch wave, is written as the plane wave having periodicity defined by the Bravais vectors R [START_REF] Ashcroft | Solid State Physics[END_REF]. The electron wave functions become:

ψ n b ,k (r) = e ik•r u n b ,k (r), (1.12) where k is the electron wave-vector, and n b is the so-called band index. The lattice periodicity is represented by the term u n b ,k (r) with the symmetry invariance: and the periodic potential (1.14) This impinges the same periodicity on the electron wave function with ψ n b ,k (r + R) = e ikR ψ n b ,k (r).

u n b ,k (r + R) = u n b ,k (r), (1.13)
U n b ,k (r) = U n b ,k (r + R).
The time-independent Schrodinger equation for the electron in a periodic potential reads [START_REF] Ashcroft | Solid State Physics[END_REF]:

Hψ n b ,k (r) = - ℏ 2 2m ∇ 2 + U n b ,k (r) ψ n b ,k (r) = ϵ n,k ψ n b ,k (r), (1.15)
where the set of eigenenergies ϵ n,k associated with the basis of the Bloch wave functions establish the electronic band structure of the crystal (see Fig. 1.5a). In the case of metal at T = 0 K, some electron bands are partially filled up to a specific energy level, so-called Fermi energy E F .

In another possible configuration, electrons completely fill a certain number of bands (valence bands), leaving others empty (conduction bands), with E F positioned in between. Material with such an electronic configuration is called an insulator. The difference between highest occupied and lowest unoccupied electron levels is defined as band-gap energy E g . At some specific points of the Brillouin zone (called valleys), the approximation of parabolic dispersion also usually works for semiconductors [START_REF] Ashcroft | Solid State Physics[END_REF], with:

ϵ(k) = ℏ 2 k 2 2m * e , ( 1.16)
where m * e is the effective mass that is defined by the curvature of the electron band. m * e is related to the change of electron group velocity along the wave vector k with m * e = 1/ ∂ 2 ϵ(k) ∂k 2 and can be positive (electron) or negative (holes) at the Brillouin zone center of GaAs for instance. In CHAPTER 1. THEORETICAL BACKGROUND a) Fermi-Dirac distribution of metal for the different temperatures. Here, the chemical potential µ is equaled to EF , which is usually true for T = 0 K. For an insulator, µ is chosen to lie equidistantly from the bottom of the valence and top of the conduction band. For an intrinsic semiconductor, if ϵ = µ, f (ϵ) = 1 2 [START_REF] Kittel | Introduction to Solid State Physics[END_REF]. At kBT2 > Eg, the electron is promoted from the valence to conduction band, creating an electron-hole pair, resulting in conducting behavior. b) Electron and hole density of states (orange), Fermi-Dirac distribution fe(ϵ) for the electrons (blue). Hole distribution f h (ϵ) is inverse to the one of electrons. The product of the electron-hole density of states and Fermi-Dirac distribution for an insulator, in the parabolic band approximation, is shown in red. Adapted from Wikipedia. that case the free-electron "picture" is then useful. However, such parabolic approximation does not hold for the whole Brillouin zone [START_REF] Ashcroft | Solid State Physics[END_REF]. The electron dispersion relation, accounting for the lattice periodicity, generally leads to a rather complicated electronic band structure, with Fig. 1.5b showing the electron dispersion relation of a ZnTe crystal.

Statistics. Fermi-Dirac distribution

Electrons are Fermions, so they obey the Pauli exclusion principle. At T = 0 K electrons will fill the energy bands up to the E F . For T > 0 K, the distribution of thermally activated electron can be described by the Bose statistics [START_REF] Ashcroft | Solid State Physics[END_REF]:

f (ϵ, µ, T ) = 1 1 + exp[(ϵ(k) -µ)/k B T ] , (1.17)
where µ is the chemical potential, and at T = 0 K, µ approaches the value of the Fermi energy E F . This distribution show that thermally activated electrons will gradually occupy higher energy states with a discrete interval of ≈ k B T , as shown in Fig. 1.6a.

Electron density of states. Heat Capacity

If we assume the parabolic shape of the electron dispersion relation (Sommerfeld model), we can write a rather simple expression for the electronic density of states (DOS). For the metal, it will take form:

ρ met (ϵ) = dn dϵ = 1 π 2 m * e ℏ 2 3 2 √ 2ϵ, (1.18)
CHAPTER 1. THEORETICAL BACKGROUND when for the semiconductor it will be:

ρ sc e (ϵ) = dn dϵ = 1 2π 2 2m * e ℏ 2 3 2 ϵ -E c ; ρ sc h (ϵ) = dn dϵ = 1 2π 2 2m * h ℏ 2 3 2 E v -ϵ; , (1.19)
where m * e and m * h are the effective mass of electron and hole, respectively; E c is the energy of the top of conduction band, and E v corresponds to energy at the top of the valence band. In metals, knowing the electron density of states, one can express the total electronic energy per unit of volume:

U e = ∞ 0 ϵρ met (ϵ)f (ϵ)dϵ.
(1.20)

To get the electron energy density for an insulator, one needs to account for the electron and hole density of states and adjust the integration boundaries for these two contributions. The temperature dependent Fermi-Dirac distribution implies the temperature-dependent electron heat capacity, defined as:

c e = ∂U e ∂T V . ( 1

Electrical conductivity in the semi-classical approach

Let us picture the Fermi surface of a crystal centered at the origin of k space, as shown in Fig1.7a. In the ground state, the net electron momentum (m e v = ℏk) is zero, since for every electron orbital at k x,y there is an occupied opposite electron state at -k x,y . In the presence of an external electric force F = -eE acting on these electrons, the second law of motion reads:

-

eE x = ℏ dk x dt = m e dv x dt . (1.22)
Over time t, each electron will experience the equivalent displacement in k-space, as shown in Fig. 1.7b, and the total shift of the Fermi surface could be written as:

δk x = -eE x t ℏ . (1.23)
After the electric field is lifted, the electrons will return to equilibrium. The electron momentum, gained in the presence of E x can be dissipated via collisions with phonons, lattice imperfections and impurities. We can define the characteristic time t = τ e during which the electrons will maintain

CHAPTER 1. THEORETICAL BACKGROUND k x k y k x k y E �k x a) b)
Circular Fermi surface of a 2D metal, with dots being the occupied free-electron states. a) Without applied field E the net momentum zero, since for each electron state (kx, ky) there is an electron at (-kx, -ky). b)With -eEx force applied on the electrons, Fermi surface will be shifted (red circle) from its initial position (dotted circle), gaining the increase of electron momentum δpx = δkx the δk x shift of the Fermi surface. Therefore, the density of electric current generated under the action of E x is [START_REF] Kittel | Introduction to Solid State Physics[END_REF]:

j = -env x = e 2 nτ e m e E x = σE x , (1.24)
which is the definition of microscopic Ohm's law, with σ being the conductivity of the metal and τ e is electron scattering time. For the conductivity of the semiconductor, one needs to account for the contribution of both electrons and holes:

σ sc = e 2 nτ e m e + e 2 hτ h m h , ( 1.25)
where τ e and τ h are the electron and hole collision time, respectively.

We then can explore the total amount of kinetic energy gained by the conduction electrons.

With change of electron momentum ℏδk x , change in kinetic energy per unit volume is:

δU k = n (ℏδk x ) 2 2m e = 1 2
m e e 2 n j 2 = 1 2 τ e j 2 σ . (1.26)
The total density of absorbed energy by the electron subsystem will then read as:

W k = ∂U k ∂t = 1 2 j 2 σ -1 = 1 2 j 2 ρ, ( 1.27)
with resistivity ρ = σ -1 . This is a microscopic analogy of the well-known Joule heating process, saying that the electrical current generated by the motion of the charge, and it will contribute to the increase of the internal energy of the crystal. Another important value related to the electron scattering time τ e is the distance electrons traveled between the collisions, namely the electron mean-free path:

Λ = v x τ e .
(1.28)

Usually, the electron velocity v x is defined in terms of Fermi velocity v F , which accounts for the curvature of the electron dispersion band. However, in the case of free-electron gas approximation, the average thermal electron velocity v 2 th is proportional to the increase of system kinetic energy by amount 3 2 k B T /m e .

Spin subsystem

The electron state in an atom can be described with 4 quantum numbers: principal quantum number n, orbital quantum number l, magnetic quantum number m l , and spin quantum number given axis are referred to as spin-up and spin-down. In quantum mechanics, components of spin angular momentum operator ŝ obey the following commutation relation [START_REF] Coey | Magnetism and Magnetic Materials[END_REF]:

[ŝ i , ŝj ] = iℏϵ ijk ŝk , (1.29)
with ϵ ijk being Levi-Civita permutation symbol. This permutation relation holds for orbital angular momentum operator l as well. The eigenvalues of ŝ are ±ℏ/2. Two commuting operators ŝz and ŝ2 have respective eigenvalues of ±ℏs and ℏ 2 s(s + 1). Electrons' orbital motion and spin associated with respective orbital μorb and spin μs magnetic moment operators as:

μorb = - µ B ℏ l; μs = - µ B g s ℏ ŝ, (1.30)
where µ B = |e|ℏ/2m e is Bohr magneton with e being elementary charge and g s is the Landé's gfactor. The electron total angular momentum operator ĵ (Fig. 1.8b2), and hence the total magnetic moment operator are defined as:

ĵ = ŝ + l; μe = μs + μl . (1.31)
The first existence of the electron spin was confirmed experimentally by Stern and Gerlach [START_REF] Gerlach | Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld[END_REF] in 1922, although the result of their experiment was misinterpreted. Researchers theoretically introduced the intrinsic magnetic moment of the electron 6 years after Stern's experiment as the consequence of the relativistic corrections [START_REF] Robert | Quantum Theory of Magnetism : Magnetic Properties of Materials[END_REF].

Spin-orbit interaction

In 1928 Dirac published the paper on the quantum mechanics, where he derived the relativistic version of the Schrödinger equation, and introduced the spin-operators [START_REF] Adrien | The quantum theory of the electron[END_REF]. The relativistic singleelectron Dirac Hamiltonian in the symmetric potential, under the static magnetic field Btot , was derived as [START_REF] Robert | Quantum Theory of Magnetism : Magnetic Properties of Materials[END_REF]:

Ĥ = T + Trel + ĤC + ĤD + ĤSO + ĤZ , ( 1.32)
The first two terms in Eq.1.32 are non-relativistic T and relativistic Trel corrections to the electron's kinetic energy. The next two terms are the potential energy of the electron in the nucleus potential, i.e., Coulomb force ĤC , and the so-called Darwin term ĤD , arising from the relativistic fluctuations of the electron's potential energy. Interestingly, in the Dirac representation, the wave functions associated with Ĥ are called spinors, and compared to vectors, they transform differently under rotation in 3D space [START_REF] Robert | Quantum Theory of Magnetism : Magnetic Properties of Materials[END_REF].

The fifth term, ĤSO , describes the electron spin-orbit interaction, which helped explain the doublets in the hydrogen spectrum observed by Michelson back in 1892 [START_REF] Michelson | XXX. On the application of interference methods to spectroscopic measurements. -II[END_REF]. One can approach the spin-orbit coupling term from the rest frame of the moving electron. The orbital motion of the nucleus around the stationary electron is equivalent to the current loop that will generate magnetic field B SO ∝ E × v e /c 2 (Fig. 1.8b1). This magnetic field will affect the electron spin CHAPTER 1. THEORETICAL BACKGROUND magnetic moment μs , introducing the coupling between the orbital and spin angular momenta of a single electron. Generally, the spin-orbit Hamiltonian can be written as:

ĤSO = eℏ 2mc ŝ • (E(r) × ve c ) = λ SO l • ŝ, (1.33)
with the speed of light c, electron band velocity ve , and total electric field E experienced by the electron. Here, λ SO = eℏE(r)/(2mc 2 r) is the spin-orbit parameter (in classical notation), and it decreases rapidly with increasing nuclear distance [START_REF] Nolting | Quantum theory of magnetism[END_REF].

There are two edge cases of angular and spin-orbital-momentum interactions. First one is known as Russell-Saunders (LS) coupling, saying that the total spin angular momentum Ŝ = i ŝi will be coupled to the total orbital angular momentum L = i li , as shown in Fig. 1.8b2. LS coupling is true for the atoms with a small nuclear charge Z, resulting in a weak spin-orbit interaction.

For heavy atoms, the individual coupling between the electron total angular momentum (JJ) Ĵ = i ĵi = i (s i + l i ) is stronger than the electrostatic (LS) interaction between them. Pure JJ and LS interactions are rare, and spin-orbit coupling in atoms is regarded as the intermediate case between the two.

The last term in the Eq.1.32 is the Zeeman interaction operator ĤZ :

ĤZ = -μ e • Btot . (1.34)
It describes the interaction between the total electron magnetic moment and the static magnetic field. All magnetic moments in a solid will align parallel to the external magnetic field only when Zeeman interaction will compensate for the spin-orbit interaction. Therefore, the magnitude of the external magnetic field used to align the magnetic moments may vary depending on the crystallographic plane direction it is applied along. For example, to saturate the magnetization of bcc Fe, the magnitude of the magnetic field applied along the ⟨111⟩ direction (hard axis) will be significantly higher than the one applied along the ⟨100⟩ direction (easy axis).

Since the Zeeman and spin operators do not commute ([ ĤZ , ŝ] ̸ = 0), electron dispersion bands in solids may exhibit spin-dependent behavior in the presence of intrinsic or external magnetic field [START_REF] Stöhr | Magnetism: From fundamentals to nanoscale dynamics[END_REF]. The latter will lead to the two derivatives of the Zeeman effect, known as Dresselhaus and Rashba effects, that describe the creation of the spin-polarized electron dispersion curves in solids [START_REF] Yehuda | Electronic Properties of Solids[END_REF]. The Rashba Hamiltonian, and its relation to the spin-to-charge conversion is discussed in the Chapter 4.

Direct exchange interaction

Let us now consider the system with two electrons, namely a and b, interacting with positively charged nucleus Ze. For simplicity, we will not account for the relativistic corrections. The eigenenergies associated with single electron Hamiltonian Ĥa and Ĥb are assumed to be known,

and are E a and E b respectively. The total Hamiltonian of a two-electron system can be expressed CHAPTER 1. THEORETICAL BACKGROUND in a simple form:

Ĥab = Ĥa (r 1 ) + Ĥb (r 2 ) + e 2 4πϵ 0 |r 1 -r2 | , (1.35)
where the third term is the perturbation due to electron-electron interaction. Since we neglected spin-orbit interaction, the electron wave function can be factorized in orbital Φ i (r j ) and spin Ω i (ŝ j ) parts [START_REF] Stöhr | Magnetism: From fundamentals to nanoscale dynamics[END_REF]. For such a system of two fermions, due to the Pauli exclusion principle, each particle can be either spin up or down, resulting a total combination of four basis states of the spin-part wave function Ω i (ŝ j ): three Symmetric triplets:

|↑, ↑⟩ (|↑, ↓⟩ + |↑, ↓⟩)/ √ 2 |↓, ↓⟩        S = 1ℏ ; (1.36)
and one Asymmetric singlet:

(|↑, ↓⟩ -|↑, ↓⟩)/ √ 2 ; S = 0, (1.37) where S denotes the total spin of the basis state. When the electron-electron interaction is small compared to the E a and E b , it is assumed that there is no coupling of the orbital states. If a set of orbital-part electron wave functions form an orthonormal basis, it is possible to derive the eigenenergies of a singlet E s and triplet E t states [START_REF] Robert | Quantum Theory of Magnetism : Magnetic Properties of Materials[END_REF]

E s = E a + E b + K ab + J ab ; E t = E a + E b + K ab -J ab , (1.38)
that are split by the amount of 2J ab . Here, the

K ab = dr 1 dr 2 |Φ a (r 1 )| 2 |Φ b (r 2 )| 2 (1.39)
is the quantitative measure of the overlap between two electron orbitals and is called the overlap integral. J ab is an analogy of two-site Coulomb interaction and is called the interaction integral:

J ab = dr 1 dr 2 Φ * a (r 1 )Φ * b (r 2 ) e 2 4πϵ 0 r 12 Φ b (r 1 )Φ a (r 2 ). (1.40)
Paul Dirac derived the eigenenergies from Eq.1.38 by introducing the spin-dependent term in the Hamiltonian instead of two-electron Coulomb exchange, as demonstrated above. The so-called Dirac exchange Hamiltonian Ĥex for the two-electron system is:

Ĥex = 1 4 (E s + E t ) - 1 4 (E s -E t )ŝ 1 • ŝ2 /ℏ 2 , (1.41) CHAPTER 1. THEORETICAL BACKGROUND
where the terms in parenthesis correspond to the energy of bonding 2(E s -E t ) and antibonding (E s + E t ) orbitals. This equation says that the exchange interaction is a purely electrostatic effect that arises from the spin-spin interaction and Pauli exclusion principle. For N -electron system, the exchange interaction Hamiltonian is known as the Heisenberg Hamiltonian:

Ĥex = -2 N i<j J ex ij ŝi • ŝj /ℏ 2 , (1.42)
where the exchange integral J ex ij has the same form as J ab . When the exchange integral J ex ij is positive, according to Eq.1.38, adjacent spins point in the same direction, i.e., aligned in parallel, which is the sign of ferromagnetic coupling. Inversely, if J ex ij is negative, consecutive spins point in the opposite direction, resulting in antiferromagnetic interaction. If more than one magnetic substrate with an antiparallel spin orientation do not cancel each other, ferrimagnetic ordering can occur [START_REF] Nolting | Quantum theory of magnetism[END_REF].

The magnitude of the exchange integral J ex ij is determined by the overlap integral K ij (see K ab in Eq.1.39). If the wave functions of the participating electrons do not overlap, there will be no spontaneous spin ordering. When the distance between two magnetic ions is large, meaning that the direct exchange not possible, the spin-spin exchange mechanism can be mediated by superexchange (via oxygen orbitals in oxides, for instance) [START_REF] Stöhr | Magnetism: From fundamentals to nanoscale dynamics[END_REF].

Ferromagnetism. Stoner model

The direct exchange interaction in solids can result in spontaneous spin ordering, giving rise to the macroscopic magnetization M defined as:

M = 1 V i ⟨ μi ⟩, (1.43)
where ⟨ μi ⟩ is the expected value of the total magnetic moment operator (see Eq.1.31).

Measurement of the spontaneous magnetization in Fe, Co, and Ni revealed that the magnetic moment per atom is not indeed an integer multiple of Bohr magneton, contradicting the existing at that time theory. This problem was solved by Mott [START_REF] Mott | A discussion of the transition metals on the basis of quantum mechanics[END_REF], Slater [START_REF] Slater | The Ferromagnetism of Nickel[END_REF], and Stoner [START_REF] Stoner | Collective electron ferronmagnetism[END_REF] around the 1940s. As an example, the Stoner band model was developed to explain broken µ B numbers as a result of ferromagnetism .

The classical Stoner model is used to predict spontaneous ferromagnetic ordering in solids, and it assumes separate band structures for spin-up and spin-down electron states. Since it is always hard to keep track of the true sample magnetization, we can apply a static external magnetic field (B ext ) to align the magnetization of the sample parallel to the applied field, as shown in Fig. 1.9a.

Since spin magnetic moment and electron spin have opposite directions (see Eq.1.30), the states with larger electron population (majority band) will be aligned in the opposite direction to B ext , while ones having a smaller population (minority band) will be parallel to the applied magnetic field. Hence, the magnetization and the spin minority will always point in the direction of B ext . We can define the difference in the number of electrons in the minority and majority bands as [START_REF] Stöhr | Magnetism: From fundamentals to nanoscale dynamics[END_REF]:

N maj e -N min e = N ↓ e -N ↑ e = E F -∞ ρ ↓ (ϵ)dϵ - E F -∞ ρ ↑ (ϵ)dϵ, (1.44)
with ρ ↑↓ (ϵ) being minority or majority electron DOS. The magnetic moment of a solid (m in classical notation) will be proportional to the difference of electrons (or holes) between these two bands:

|m| = µ B (N maj e -N min e ) = µ B (N maj h -N min h ). (1.45)
The difference in the electron density of states at the Fermi level E F and exchange interaction field B ex (generated by exchange interaction Hamiltonian (see Eq.1.42)) will result in the majority and minority bands split by the amount of E ex [START_REF] Stöhr | Magnetism: From fundamentals to nanoscale dynamics[END_REF]:

E ex ∝ 2m • B ex . (1.46)
The Stoner model, assumes the parabolic shape of electron DOS, while "real" electron DOS has far more complex shape. The spin-polarized density of states of bcc-Fe is shown in Fig. 1.9b, with E ex ≈ 2.2 eV. Generally, the exchange interaction competes with other energies in solid [START_REF] Coey | Magnetism and Magnetic Materials[END_REF]. As an example , one can expect that the thermal atomic disorder may overcome the exchange interaction contribution, resulting in a random alignment of magnetic moments. Indeed, the magnetization CHAPTER 1. THEORETICAL BACKGROUND M is proportional to [START_REF] Stöhr | Magnetism: From fundamentals to nanoscale dynamics[END_REF]:

M ∝ (T crit -T ) α , (1.47)
where T crit is the critical temperature (Curie or Neel temperature for ferro-and antiferromagnets, respectively) above which the magnetic order vanishes, and α is the critical exponent that differs with the material.

Magnons

Ferromagnetic metal in the ground state at T = 0 K, according to Eq.1.38, prefers to have its spins aligned in parallel, which leads to the lowest possible energy state. If we treat spins as classical vectors, the total exchange energy in this ground state will be equal to -2N J ex s 2 with N being the number of neighbors. Thus, we can estimate the thermal energy (Eq.1.47) required to eliminate the spontaneous ground-state magnetization as [START_REF] Coey | Magnetism and Magnetic Materials[END_REF]:

k B T C = -2N J ex ŝ2 ℏ 2 . ( 1.48)
Let us imagine an electron spin-chain with two nearest neighbors, i.e., N = 2. The magnetic order will vanish at critical temperature T C , with corresponding thermal energy of k B T C = J ex . In the mean time, the energy required to flip the spin of a single electron is equal to 2J ex (-4J ex s 2 → 4J ex s 2 ), i.e., two times the k B T C . Clearly, the single-electron spin-flip is not the best "energyefficient" solution, and can not occur at low temperatures. Instead, all adjacent electrons will share the spin-reversal, which will form an excitation with a much lower energy cost per electron [START_REF] Kittel | Introduction to Solid State Physics[END_REF].

Such oscillation of the relative orientation of spins on the lattice, depicted in Fig. 1.10, is called magnons, or spin-waves.

The exchange interaction Hamiltonian for the two nearest neighbours in the 1D electron chain can be rewritten in terms of:

ĤN=2 ex = ŝj Bex j , (1.49)
with the exchange field Bex j = -2J ex (ŝ j-1 + ŝj+1 )/ℏ 2 generated by the neighbours at lattice sites j ± 1. According to the Ehrenfest equation (in the absence of damping), this exchange field Bex j will act as a source term of the spin precession [START_REF] Coey | Magnetism and Magnetic Materials[END_REF]:

d dt ⟨ ŝj ⟩ = g s µ B ℏ ŝj × Bex j . (1.50)
In Cartesian coordinates, the precession of the ⟨ŝ j ⟩ components along the given axis could be expressed as the system of three coupled differential equations. We can neglect the products of x CHAPTER 1. THEORETICAL BACKGROUND

Perspective and an aerial view of spins on a lattice with lattice constant a. Spin wave in aerial view is drawn through the end of the spin vectors. The image was redrawn from [START_REF] Kittel | Introduction to Solid State Physics[END_REF] Figure 1.10 -Spin wave on the 1D chain of spins.

and y components, since the expected values of s x,y j ≪ s z j [START_REF] Nolting | Quantum theory of magnetism[END_REF]:

             d dt s x j = - J ex gµ B ℏ 2s y j -s y j-1 -s y j+1 d dt s y j = - J ex gµ B ℏ 2s x j -s x j-1 -s x j+1 d dt s z j = 0 (1.51)
We will look for the solutions in the form of traveling wave ∝ exp[ik j a-ω m t], with lattice constant a. The solution for the precession with a frequency ω m defines the magnon dispersion relation [START_REF] Kittel | Introduction to Solid State Physics[END_REF]:

ℏω m = -4J ex S(1 -cos(ka)). (1.52)
The magnon dispersion relation for ka ≪ 1 has the form ℏω m = D SW k 2 , with stiffness parameter

D SW = 2J ex Sa 2 .
Magnons are bosons, and the quantized energy of a magnon is defined as:

ϵ m = n m + 1 2 ℏω m , (1.53)
where n m is the number of magnons in the mode with frequency ω m at the zone center [START_REF] Coey | Magnetism and Magnetic Materials[END_REF]. For T < T crit magnons behave like bosons, so they follow the Bose statistics. Interestingly, the magnons are in some way similar to phonons. Instead of the relative precession of an atomic position on a lattice, spin waves are oscillations of the relative orientation of the lattice spin magnetic moments.

The magnon contribution to the heat capacity of the solid can be expressed in terms of the spin-correlation function ⟨ŝ i • ŝj ⟩ (see Eq.1.42) [START_REF] Coey | Magnetism and Magnetic Materials[END_REF]:

c m = -2N ZJ ex ∂⟨ŝ i • ŝj ⟩ ∂T (1.54)

Light-matter interaction 1.3.1 Optical properties and dielectric function

Most optical phenomena can be explained with the classical electromagnetic (EM) approach, from the perspective of the physical interaction between charged particles. The light propagation CHAPTER 1. THEORETICAL BACKGROUND inside the material can be described by Maxwell's equations:

∇ • D = ρ, ∇ • B = 0, ∇ × E = - ∂B ∂t , ∇ × H = j + ∂D ∂t , (1.55)
where D is electric flux density, E is electric field strength, B and H are magnetic flux and magnetic field respectively, ρ is charge density and j is electric current density (free carriers). In the linear regime, when the material is not charged, homogeneous, and isotropic the following set of equations can be established:

ρ = 0, j = σE, D = ε 0 εE, B = µ 0 H, (1.56)
where σ is conductivity, ε 0 is electric constant, µ 0 is magnetic constant and ε is the medium permittivity. Therefore, using Eq.1.55 one can derive the generalized wave equation for the dielectric system:

∇ 2 E - ε µ 0 ε 0 ∂ 2 E ∂t 2 = µ 0 ∂j ∂t
, which gives:

k 2 E = ω 2 µ 0 εE + iµ 0 ωσE. (1.57)
By noting that speed of light is c =1 √ ε0µ0 and dielectric constant of the material is ε = ε 0 ε b , the wave vector expressed in terms of light frequency ω as:

k = ω c ε b + iσ ε 0 ω = k ′ + ik ′′ , (1.58)
with ε b and iσ/ε 0 ω are the bound and free electron contributions to the relative dielectric constant, with k ′ and k ′′ being the real and imaginary parts of the complex wavevector, respectively. We then can rewrite k as:

k = ω c ñ(ω) (1.59)
where ñ(ω) = n(ω)+iK(ω) is complex index of refraction, with refractive index n(ω) and extinction coefficient K(ω). With the extinction coefficient K(ω) one can estimate the EM wave intensity 1 attenuation in medium along the z-direction. From the Beer-Lambert law we obtain:

I(z)
I(z) = I 0 exp(-αz), (1.60)
with α = 2ωK/c = 4πK/λ is the absorption coefficient (λ is the light wavelength). The absorption of EM has an exponential profile for K ̸ = 0, while K = 0 defines lossless transmission.

CHAPTER 1. THEORETICAL BACKGROUND

The well-known relation between ñ(ω), electric susceptibility χ(ω) and complex permittivity ε(ω) of the material written as:

ε(ω) = 1 + χ(ω) = ñ2 (ω) = ε 1 (ω) + iε 2 (ω), (1.61)
where ε 1 (ω) is real and ε 2 (ω) is the imaginary part of the frequency dependent dielectric constant.

In a crystal, light-matter interaction depends on the crystal symmetry and light propagation direction with respect to the crystallographic plane orientation, resulting ε, χ(ω) and ñ being tensors.

In this section, for simplicity, we will assume that our solids are isotropic, lifting the tensor notation for all the optical constants discussed below.

Drude and Lorentz Model: semi-classical approach

The electron degrees of freedom within partially filled electron bands and interband transitions play an important role in the optical properties of both metals and semiconductors. In this subsection, we address the intraband Drude and interband Lorentz contributions to the dielectric function ε.

In the Section 1. We can express the single-electron dipolar moment as p = er. Then, the macroscopic polarization P created by the displacement of the electron cloud reads [START_REF] Boyd | Nonlinear Optics[END_REF]:

P = nr = ε 0 χ(ω)E(t). (1.63)
Combining the two previous equations, we can deduce the free-electron contribution to the electric susceptibility of the crystal. The so-called Drude susceptibility is:

χ D (ω) = - ne 2 ε 0 m e 1 ω 2 + iω/τ e = - ω 2 p ω 2 + iω/τ e , ( 1.64)
with

ω p = ne 2 ε 0 m e (1.
m e d 2 r dt 2 + 1 τ dr dt + ω 2 0,n r = -eE(t), (1.67)
that differs from the Drude model only by an additional restoring force term, which corresponds to the interband transition. This model does not lead to well-defined energy gaps, but it is widely used to describe non-conducting materials [START_REF] Dressel | Electrodynamics of Solids[END_REF]. The derived Lorentz complex dielectric function, for multiple electron transitions reads:

εL (ω) = 1 + n ω 2 p,n ω 2 0,n -ω 2 + iω/τ n , ( 1.68)
where ω p,n denotes the characteristic frequency associated to the electronic population of the level n, τ n is the phenomenological damping time, defining the width of the Lorentz oscillator and ω 0,n is the resonant frequency of the n-th type electron interband transition. Interestingly, the classical Lorentz model is really close to its quantum counterpart [START_REF] Yu | Fundamentals of Semiconductors[END_REF]:

εL (ω) = 1 + n f n ω 2 p,n (ω 2 j -ω 2 ) + iω/τ n , ( 1.69)
with oscillator strength defined as:

f j = 2m * e ℏ 2 ℏω n |r n | 2 , (1.70)
where r n is the displacement of the n-th electron. The simple Lorentz model, with only one oscillator (see Eq.1.68) has 4 distinct spectral regimes:

1. ω < (ω 0 -1/τ ): Low frequency range, where ε L 2 is infinitely small, resulting little to no absorption (ℏω < E g ); 2. ω ≈ ω 0 : Absorption range, where electrons can be excited across the bandgap;

Metals. Interband contribution

The Drude model shows good agreement with the experimental data in the low-frequency range, rapidly diverging from the experiment at higher frequencies. The reason is that the Drude model does not take into account the band structure of the metal. For the real 3D metal, the absorption of a photon with sufficiently high energy can provoke an interband transition of the electron from the valence band to the conduction band above the E F . For example, we can separate the bound and free-electron contributions and account for possible interband transitions by introducing a Lorentz term into the Drude model. The combined Drude-Lorentz model gives:

εDL (ω) = 1 - ω 2 p ω 2 + iω/τ e + n=k n=1 ω 2 p,n (ω 2 0,n -ω 2 ) + iω/τ n , (1.71)
for k is the number for all allowed interband transitions. As seen from Fig. 1.12a, the composite Drude-Lorentz model (with n = 1) slightly extends the prediction range of ε(ω). Even when accounting only for one Lorentz term, we still can estimate the energy of an interband transition threshold of Au (ℏω IB ≈ 2.7 eV), which is not so far from the real value of 2.4 eV. In a more general form, the dielectric constant of the metal is written as a function of both interband and intraband contributions as:

ε(ω) = 1 + χ D (ω) + χ IB (ω), (1.72)
where χ IB (ω) is interband electric susceptibility that accounts for all possible interband transitions.

Interband transitions in metals can not be neglected in the high-frequency range and play an important role in the optical properties of metals. The optical frequencies above the interband transition threshold ℏω IB will be highly absorbed. For example in Au, light with wavelengths below 520 nm will be strongly absorbed, translating to its "yellowish" color.

The excitation of an electron above the Fermi level will leave a vacancy at the electrons' initial position, known as a hole. Electron-hole pair excitation (due to intraband or interband absorption) must satisfy the law of momentum conservation:

k = k ′ + K, (1.73)
where k and k ′ are the electron and hole wave vectors, and K is reciprocal lattice vector. The momentum conservation can be fulfilled with two-or three-particle interactions.

The electron can absorb the photon if the photon energy corresponds to the energy difference between the hole and electron state: which defines the direct optical (or vertical) electron transition. In addition, the momentum conservation can be fulfilled by the emission/absorption of a third particle, for example, a phonon.

ℏω = ϵ(k) -ϵ(k ′ ), (1.74)
In this case, the momentum conservation law reads:

ℏω = ϵ(k) -ϵ(k ′ ) ± ℏω ph , (1.75)
where ω ph is the frequency of emitted phonon. Normally, the direct optical transition requires the lowest amount of energy, and most of the time, it will be dominant during photon absorption.

Calculation of an interband contribution

The interband contribution to the dielectric constant can be calculated using the so-called Rosei model [START_REF] Lynch | Thermomodulation Spectra of Al, Au, and Cu[END_REF][START_REF] Rosei | Temperature modulation of the optical transitions involving the fermi surface in Ag: Theory[END_REF][START_REF] Guerrisi | Splitting of the interband absorption edge in Au[END_REF]. This model takes into account the effective masses of the electrons in the parabolic band structure of noble metals in different directions relative to the Γ point. If we consider the photon with energy ℏω that stimulates the transition from initial i to final f band, we can write:

ϵ f (k) = ℏω + ϵ i (k), (1.76)
with energies ϵ f , ϵ i in final and initial electron bands, respectively. With respect to [START_REF] Rosei | Temperature modulation of the optical transitions involving the fermi surface in Ag: Theory[END_REF][START_REF] Otomalo | Ultrafast optical response of complex plasmonic nanoparticles[END_REF], we can write the joint density of states J i→j as:

J i→j (ℏω) = D i→j (ϵ, ℏω)[1 -f (ϵ + ℏω, µ, T )]dϵ, (1.77)
CHAPTER 1. THEORETICAL BACKGROUND a) Coupling between photons and transverse optical phonon, called phonon-polariton. Redrawn from [START_REF] James | Solid-State Physics[END_REF] and Wikipedia. b) High-frequency interband and low-frequency polariton contributions to ZnTe dielectric function. Both real and imaginary parts of the dielectric constant are flat in 8 -400 THz range.

High-frequency properties are redrawn from [START_REF] Sato | Optical properties of ZnTe[END_REF]. where D i→j (ϵ, ℏω) is the energy-dependent density of states, which depends on the type of the optical transition (d → s or p → s, for example). Then, the imaginary part of the dielectric function can be calculated using the Lindhard theory [START_REF] Lindhard | On the properites of a gas of charged particles[END_REF]:

ϵ IB 2 (ω) ∝ 1 ω 2 M i→f 2 J i→j (ℏω), (1.78)
where M i→f is the averaged electron momentum matrix element between the initial and final bands. The 1/ω 2 term comes from the electric dipole approximation. Interestingly, the Eq.1.78 can be used to calculate the dielectric constant for the semiconductor [START_REF] Yu | Fundamentals of Semiconductors[END_REF], which does not include the spin-orbit interaction. The real part of the dielectric function can be obtained by applying the Kramers-Kronig relationship to Eq.1.78.

Semiconductors. Ion contribution

As discussed in the Section 1.2.1, crystals with two (or more) atoms per unit cell can support optical vibrational modes. Furthermore, some polar optical phonons (also known as infrared active) can be directly coupled to the incident EM wave [START_REF] James | Solid-State Physics[END_REF]. This happens due to the momentum conservation between the photons and phonons, resulting in the excitation of a quantized particle called phonon-polariton (see Fig. 1.13a).

The coupling between photons and phonons can be modeled with slightly-modified Lorentz model [START_REF] Barker | Transverse and Longitudinal Optic Mode Study in Mg F2[END_REF]:

ε(ω) = ε ∞ + j S j ω 2 j ω 2 j -ω 2 -iω/τ j , (1.79) CHAPTER 1. THEORETICAL BACKGROUND
where ε ∞ is the dielectric constant of the ion background (i.e that of the conduction electrons, see the Eq.1.71). Here, we account for the contribution of all lattice oscillations that can interact with EM fields, with eigenfrequency ω j , damping time τ j , and dimensionless oscillator strength S j . For most semiconductors, this equation can be simplified by restricting the sum to the first low-frequency TO mode:

ε(ω) = ε ∞ + S T O ω 2 T O ω 2 T O -ω 2 -iω/τ T O . (1.80)
The oscillator strength S T O can be deduced from the DC dielectric constant [START_REF] Barker | Transverse and Longitudinal Optic Mode Study in Mg F2[END_REF] as:

ε(0) = ε ion + S T O . (1.81)
From the Lyddane-Sachs-Teller relation [START_REF] Lyddane | On the Polar Vibrations of Alkali Halides[END_REF], Kurosawa [START_REF] Kurosawa | Polarization Waves in Solids[END_REF] showed the direct connection between S T O and the frequency difference between non-excited ω LO and resonant ω T O modes:

S T O ω 2 T O = ω 2 LO -ω 2 T O . (1.82)
Due to the phonon-polariton coupling in ionic materials, this equation suggests that the lattice will strongly absorb photons with energies from ℏω T O to ℏω LO . This frequency range is called the Reststrahlen band and ω T O sometimes referred to as Restrahl frequency.

To give a real example of phonon-polariton, the complex dielectric function of ZnTe from DC to 1400 THz is displayed in Fig. 1.13b. The high-frequency contribution was redrawn from [START_REF] Sato | Optical properties of ZnTe[END_REF],

showing well-pronounced interband contribution2 to the dielectric constant around ≈ 800 THz (375 nm wavelength). Low-frequency phonon-polariton contribution calculated with parameters from [START_REF] Leitenstorfer | Detectors and sources for ultrabroadband electro-optic sampling: Experiment and theory[END_REF], showing strong photon absorption around 5.3 THz. The presence of the Reststrahlen band is the main limiting factor for the semiconductor optics in the THz frequency range.

Nonlinear Regime

Light-matter interaction can modify the macroscopic polarization of the crystal (see Eq. 1.63) proportional to its electric susceptibility. The total electronic displacement under the applied electric field is defined as:

D = ε 0 ε(ω)E(t) = ε 0 [1 + χ(ω)]E(t) = ε 0 E(t) + P (ω, t). (1.83)
For the electric fields with amplitudes comparable to the strength of the atomic field, i.e.:

E at = e 4πε 0 a 2 0 = 5.14 × 10 11 , V m (1.84)
where a 0 is the Bohr radius of the Hydrogen atom, can induce the nonresonant precession of the macroscopic polarization P [START_REF] Boyd | Nonlinear Optics[END_REF]. This nonlinearity is accounted for by expressing the polarization Material χ (2) (pm/V) r (pm/V) P as a power series in the field strength E:

P (t) = ε 0 [χ (1) E(t) + χ (2) E 2 (t) + χ (3) E 3 (t) + .
..] ≡ P (1) (t) + P (2) (t) + P (3) (t) + ..., (1.85) with χ (1) known as linear susceptibility, χ (2) and χ (3) are 2 nd -and 3 rd -order nonlinear optical susceptibilities, and so on. By isolating the linear contribution P L , Eq.1.85 becomes:

P = P L + P N L , ( 1.86)
with nonlinear contribution P N L containing all power terms of E. In this subsection, the discussion will be limited to the 2 nd order nonlinear processes, when relevant to this work higher-order nonlinearities will be mentioned in the experimental section.

Second Order Nonlinear Processes

Let us first estimate the value of second-order nonlinear susceptibility in condensed matter. In the nonlinear regime, at some peak value of E peak one can expect electrons to "explore" the non harmonic part of ionic potential, resulting in equal contributions of linear P (1) (t) and nonlinear terms P (2) (t) to the total polarization P (t). In most solids, χ (1) normally is in the order of unity.

By setting E peak = E at , the Eq.1.85 gives:

χ (2) = 1 E at = 1.94 × 10 -12 . m V (1.87)
This estimation is quite accurate compared to experimentally measured values displayed in Table 1.1. The value of χ (2) can be interpreted as the efficiency of nonlinear interaction in a given solid.

In the case of centrosymmetric medium, the shape of electron potential obeys V (r) = V (-r) [START_REF] Boyd | Nonlinear Optics[END_REF]. Simple estimation of the second-order nonlinear polarization induced by the EM wave with frequency ω:

-P (2) = χ (2) (ω)[-E(ω)][-E(ω)] = χ (2) (ω)E 2 (ω) = P (2) .

(1.88)

Obtained equality -P (2) = P (2) holds only when χ (2) = 0. The group theory predicts that each element of χ (2) vanishes in the centrosymmetric crystals [START_REF] Boyd | Nonlinear Optics[END_REF], allowing only nonlinear interactions CHAPTER 1. THEORETICAL BACKGROUND proportional to the odd powers of the electric field. However, nonlinear interactions of both odd and even power of E are allowed in noncentrosymmetric crystals, such as ZnTe, GaP, Beta Barium Borate (BBO), and more.

In the case of a noncentrosymmetric, isotropic medium with no dispersion, by taking into account linear and nonlinear polarization, the driven Maxwell wave-equation becomes [START_REF] Boyd | Nonlinear Optics[END_REF]:

∇ 2 E - ε (1) c 2 ∂ 2 E ∂t 2 = 1 ε 0 c 2 ∂ 2 P N L ∂t 2 , ( 1.89)
with ε (1) = 1 + χ (1) , where the second order time derivative of nonlinear polarization acts as the source term. We now examine the set of possible frequencies of electron polarization precession in the nonlinear media. Let us apply two-color EM field:

E = E 1 e -iω1t + E 2 e -iω2t + c.c., (1.90)
where E 1 = A 1 e ik1z and E 2 = A 1 e ik2z . With Euler's formula cos(x) = (e ix + e -ix )/2, the complete second-order nonlinear polarization induced by the two-color EM field:

P (2) (t) = ε 0 χ (2) E(t) 2 = = 1 2 ε 0 χ (2)     E 2 1 [cos(2ω 1 t) + cos((ω 1 -ω 1 )t)]+ +E 2 2 [cos(2ω 2 t) + cos((ω 2 -ω 2 )t)]+ +2E 1 E 2 cos[(ω 1 + ω 2 )t] + 2E 1 E 2 cos[(ω 1 -ω 2 )t]     . (1.91)
With four different permutations between two frequencies ω 1,2 the grouped contributions to the second order nonlinear polarization are:

P (2) (2ω 1,2 ) = 1 2 ε 0 χ (2) E 2 1,2 cos(2ω 1,2 t)
Second harmonic generation (SHG);

P (2) (0ω) = 1 2 ε 0 χ (2) E 2 1,2
Optical rectification (OR);

P (2) (ω 1 + ω 2 ) = ε 0 χ (2) E 1 E 2 cos[(ω 1 + ω 2 )t]
Sum frequency generation (SFG);

P (2) (ω 1 -ω 2 ) = ε 0 χ (2) E 1 E 2 cos[(ω 1 -ω 2 )t] Difference frequency generation (DFG), (1.92)
with SHG and OR being the special cases of SFG and DFG. The nonresonant movement of dipole moments P N L will create the nonlinear current density J N L , equal to:

dP N L dt = J N L . (1.93)
Such transient precession of the J N L can lead to the emission of light at different frequencies, as shown in Fig. 1.14: Bottom: SHG and OR mediated by the χ (2) interaction in the frequency domain. However, it adds an important constraint: the driven wave equation (see Eq.1.89) must hold for all frequencies, including the newly created ones.

E N L (t) ∝ ∂J N L ∂t ∝ ∂ 2 P N L ∂t 2 . ( 1
Let us consider only the SFG created by the two-color field. Under slowly varying field amplitude approximation [START_REF] Boyd | Nonlinear Optics[END_REF], Eq.1.89 can be reduced to the spatial variation of ω 3 = ω 1 + ω 2 :

dA 3 dz = iχ (2) (ω 1 + ω 2 , ω 1 , ω 2 )ω 3 n 3 c A 1 A 2 e i[k1+k2-k3]z , (1.95)
where n 3 is crystal refractive index at ω 3 . This relation shows that the amplitude of the EM wave A 3 at ω 3 depends on the coupling between A 1 and A 2 . We can introduce the so-called wavevector-, momentum-or phase mismatch as ∆k:

∆k = k 1 + k 2 -k 3 . (1.96)
By integration of Eq.1.95 over the distance d, the intensity of SFG radiation will be ∝ sinc 2 (∆kz/2).

This means that the most efficient coupling between ω 1 and ω 2 is when

k 3 = k 1 + k 2 .
Usually, the increase in light frequency ω is not compensated by the decrease in ñ, meaning that the perfect phase matching is rarely achieved. Multiple techniques were developed to minimize the ∆k, with birefringent phase-matching being the most used one. In addition, wavevector mismatch is sensitive to the crystal imperfections, laser beam quality, laser polarization state, and more. The discussion about the phase-matching importance for the second-order nonlinear processes will continue in the experimental section.

The remaining second-order nonlinear process of interest is the linear electro-optic effect, known as the Pockels' effect. Pockels' effect explains the modification of refractive indices of the crystal with an applied electric field. The contracted notation of the electrooptic effect is given by [START_REF] Boyd | Nonlinear Optics[END_REF]:

∆ 1 n 2 i = j r ij E j , (1.97)
where r ij is the electrooptic tensor. Electrooptic effect can be pictured as follows: external DC electric field will tilt the crystals' optical axes by an amount ∝ ± 1 2 n 3 0 r ij E z , introducing the phase retardation between two orthogonal light polarization components [START_REF] Boyd | Nonlinear Optics[END_REF]:

Γ = (n i -n j ) ωL c = n 3 0 r ij E j ωL c , ( 1.98)
with the speed of light in vacuum c, crystal thickness L and refractive index n 0 in the absence of electric field. Electrooptic effect is used for the high-frequency light modulation and detection of the THz radiation, for example. Interestingly, the response time of a Pockels' effect is in the order of femtoseconds [START_REF] Hu | Breaking the energy-bandwidth limit of electro-optic modulators: theory and a device proposal[END_REF], allowing ultrafast switching of the light polarization with picosecond electromagnetic pulses [START_REF] Juvé | Ultrafast tunable modulation of light polarization at terahertz frequencies[END_REF].

Polarization, reflection, and transmission of an electromagnetic wave

Polarization

Light can be described as the periodic oscillation of the EM field in the xy plane travelling along the z axis in Cartesian space. Mathematically, we can write equation of the complex EM field as:

E = E 0 exp [i(kz -ωt + ϕ)], (1.99)
where E is the electric field with the field-wave vector k frequency ω and phase ϕ. At the given time t, the direction of electric field can be expressed through the unit vectors x and y:

E = E 0x exp [i(kz -ωt + ϕ x )]x + E 0y exp [i(kz -ωt + ϕ y )]y. (1.100)
The direction of the electric field vector E along x and y axes is known as polarization of EM wave, and it is completely determined by the amplitudes and phase difference of the two orthogonal electric field components. It can be written in the form of the two-element matrix, called Jones vector:

E 0 = E 0x e iϕx
E 0y e iϕy .

(1.101)

With the Jones matrix notation, it is easy to represent the linearly, elliptically, and circularly polarized light, including the polarization handedness direction. Randomly polarized or nonpolarized light does not require the use of the matrix notation. Linear polarization of the light in the xy plane, according to the Jones vector notation defined as:

E 0 = E 0x E 0y , ( 1.102)
with both E 0x and E 0y being real, non-zero field amplitudes. The periodically varying amplitude of the electric field is inclined relative to the origin by the angle α = tan -1 (E 0y /E 0x ) [START_REF] Frank | Introduction to Optics[END_REF].

E 0x =
1, E 0y = 0 correspond to the linear light polarization along the x axis.

The general expression for the elliptically polarized light given as:

E 0 = E 0x E 0y e ±iϕy , ( 1.103)
where the sign of the ϕ y defines the direction of the polarization rotation. If ϕ y is positive, electric field direction will rotate in counterclockwise direction, when negative ϕ y corresponds to the clockwise handedness. The inclination angle of a polarization ellipse equals to the absolute value of ϕ y . If E 0x = E 0y and ϕ y = π/2, such EM wave has left-handed circular polarization.

Reflection and Transmission. Transfer matrix formalism

The transmission and reflection of an EM radiation at the singular plane surface depends on the following set of variables: the angle of incidence θ 1 and refraction θ 2 , the refractive indices of incident n 1 and refractive media n 2 , and the light polarization direction. The ratio between the total, reflected and transmitted EM amplitudes can be derived from the Maxwell equations at the boundary condition as shown elsewhere [START_REF] Frank | Introduction to Optics[END_REF]. Obtained equations, namely, the polarization Optical stack (two layers between the two half-spaces) with N=4. vi, wi are the amplitudes of the forward and backward propagating waves. Boundary conditions defined as vN-1 = t, wN-1 = 0 and v0 = 1, w0 = r. The image adapted from [START_REF] Steven | Multilayer optical calculations[END_REF] Figure 1.15 -Beam transmission and reflection from a multi layer film stack.

dependent Fresnel reflection r and transmission t coefficients read:

r s = n 1 cos θ 1 -n 2 cos θ 2 n 1 cos θ 1 + n 2 cos θ 2 , r p = n 2 cos θ 1 -n 1 cos θ 2 n 2 cos θ 1 + n 1 cos θ 2 t s = 2n 1 cos θ 1 n 1 cos θ 1 + n 2 cos θ 2 , t p = 2n 1 cos θ 1 n 2 cos θ 1 + n 1 cos θ 2 (1.104)
where subscripts p and s correspond to the parallel and perpendicular alignment of the electric field polarization relative to the lights' plane of incidence.

The transmission and reflection of light in a complex multilayer system can be approached from the view of the superposition of EM waves within the given layer. For example, the arbitrary N layer configuration has n = 0, 1, ..., N -1 optically different interfaces, that contribute to the total transmission and reflection. An example of the N=4 stack system is shown in the Fig. 1.15, defining the amplitudes of the forward and backward propagating waves. The amplitudes of transmitted and reflected light arriving at the interface n = 1, 2, . . . , N -2 can be expressed in the matrix notation as:

v n w n = M n v n+1 w n+1 , ( 1.105)
where the n th layer transfer matrix M n is equal:

M n ≡ e -iδn 0 0 e iδn 1 r n,n+1 r n,n+1 1 1 t n,n+1 , (1.106)
CHAPTER 1. THEORETICAL BACKGROUND with the n th layer phasor δ n = d n k n , where d n is the thickness of the layer. Here, the M n is the product of so called phase/absorption matrix:

P n = e -iδn 0 0 e iδn , ( 1.107)
and refraction/transmission matrix

D n D -1 n-1 = 1 r n,n+1 r n,n+1 1 1 t n,n+1 , (1.108)
with Fresnel reflection r n,n+1 and transmission t n,n+1 coefficients for the light directed from the interface n towards the interface n + 1. The repeated application of phase/absorption and refraction/transmission matrices over N-2 layers and N-1 interfaces will result in the following expression [START_REF] Charalambos | General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference[END_REF]:

1 r = D -1 0 N -2 n=1 D n P n D -1 n D N -1 t 0 = T 11 T 12 T 21 T 22 t 0 .
(1.109)

This 2 × 2 matrix accounts for the coherent interaction of the reflected and transmitted beams at each interface of the multilayer optical media. Usually, T11 T12 T21 T22 is referred to as system transfer matrix T = T 0/(N -1) .

Temporal evolution of optical properties

Ultrafast processes induced by the absorption of an ultrashort laser pulse in metals. Image is reprinted from the thesis of T.Seifert [START_REF] Sebastian | Spintronics with Terahertz Radiation : Probing and driving spins at highest frequencies[END_REF] Figure 1.16 -Evolution of the electronic distribution after an ultrashort laser action.

The development of ultrashort laser pulses allowed to study the ultrafast dynamics of solids, molecules, and chemical compounds on the timescales comparable to the carrier relaxation times [START_REF] Gamaly | Physics of ultra-short laser interaction with matter: From phonon excitation to ultimate transformations[END_REF]. For example, let us consider interaction of the fs laser pulse with the matter. Ultrafast laser action and the carrier-photon absorption will initially create highly-nonlinear photoexcited electronic distribution, which will then try to reach new equilibrium state between electron, lattice, and spin subsystems.

CHAPTER 1. THEORETICAL BACKGROUND

In metals, immediately after laser irradiation, the light-induced dipole polarization will create the superposition of the electron states above and below E F . Simultaneously, light absorption will lead to an athermal, nonlinear distribution of the electron-hole pairs. The following energy transfer from the photoexcited carriers involves the multiple-particle interactions (electron-electron, electron-phonon, and electron-magnon), in order to bring the excited system to the thermal equilibrium.

The fastest many-particle relaxation process in the electron-electron interaction, which is happening at the sup-picosecond time scale. Electron subsystem will thermalize via carrier multiplication, reaching the Boltzmann distribution at elevated temperature [START_REF] Gamaly | Physics of ultra-short laser interaction with matter: From phonon excitation to ultimate transformations[END_REF]. Subsequently, the thermalization process between electronic, ionic, and spin baths occurs at the timescales of 0.1-10 picoseconds. After electrons, magnons, and phonons have thermalized, heat dissipation to the environment will occur, lasting more than 10 picoseconds. The sketch of the electron thermalization process is displayed in the Fig. 1.16.

Phenomenological Two and Three-Temperature Models

The phenomenological macroscopic model was established to explain the interaction between electron, phonon, and spin subsystems of a solid at the ultrashort timescale with the evolution of experimental techniques. The first description of electron-phonon interaction after the absorption of the laser pulse was proposed by the introduction of the energy balance equation [START_REF] Si Anisimov | Electron emission from metal surfaces exposed to ultrashort laser pulses[END_REF]:

     c e (T e ) dT e dt = -G e-ph (T e -T l ) + I(t); c L (T l ) dT l dt = -G e-ph (T l -T e ), (1.110)
where G e-ph is the electron-phonon coupling constant and I(t) is the laser source term. The carriers and phonon (heat) diffusion terms are neglected to simplify the model. I(t) applied only to the electron subsystem since the initial energy absorption is mediated only by the electrons.

These equations were then denoted as the phenomenological two-temperature model (TTM) [START_REF] Qiu | Heat Transfer Mechanisms During Short-Pulse Laser Heating of Metals[END_REF].

In the work of Beaurepaire et al. studying the ultrafast response of a ferromagnetic Ni, an extension of the TTM, which includes the electron-phonon-spin interaction was proposed [START_REF] Beaurepaire | Ultrafast Spin Dynamics in Ferromagnetic Nickel[END_REF].

Beaurepaire's system of equation, named as the three-temperature model (3TM), reads:

             c e (T e ) dT e dt = -G e-ph (T e -T l ) -G e-m (T e -T s ) + I(t); c s (T s ) dT s dt = -G e-m (T s -T e ) -G m-ph (T s -T l ); c L (T l ) dT l dt = -G e-ph (T l -T e ) -G m-ph (T l -T s ), (1.111)
with additional electron-magnon G e-m and magnon-phonon G m-ph coupling constants. Similar to the TTM, this model suggests that the heat exchange between electron, phonon, and spin subsystems is proportional to the temperature difference between the two given baths. The respective coupling constant then defines the efficiency of the heat transfer (see Fig. of the magnon, phonon, and electron dispersion curves, Boltzmann statistics, and with knowledge about the respective particles' scattering times [START_REF] Kittel | Introduction to Solid State Physics[END_REF][START_REF] Ashcroft | Solid State Physics[END_REF][START_REF] Yu | Fundamentals of Semiconductors[END_REF]. However, the magnitudes of coupling constants and characteristic population decay times can be extracted with specific time-resolved measurements. For example, some coupling constants can be derived in the frame of 3TM-based molecular dynamics simulations, where experimental data used as the reference [START_REF] Ma | Spin-lattice-electron dynamics simulations of magnetic materials[END_REF].

With the knowledge of CoFeB coupling constants from the literature [START_REF] Navabi | Efficient Excitation of High-Frequency Exchange-Dominated Spin Waves in Periodic Ferromagnetic Structures[END_REF][START_REF] Liu | Ferromagnetic resonance and damping properties of CoFeB thin films as free layers in MgO-based magnetic tunnel junctions[END_REF][START_REF] Conca | Lack of correlation between the spin-mixing conductance and the inverse spin Hall effect generated voltages in CoFeB/Pt and CoFeB/Ta bilayers[END_REF][START_REF] Heinrich | Spin pumping at the magnetic insulator (YIG)/normal metal (Au) interfaces[END_REF][START_REF] Lin | Electron-phonon coupling and electron heat capacity of metals under conditions of strong electron-phonon nonequilibrium[END_REF][START_REF] Chatterji | Enhancement of spintransfer torque switching via resonant tunneling[END_REF], we can use 3TM, neglecting weak magnon-phonon energy exchange, to display the temporal evolution of electron, spin, and magnon temperatures upon laser absorption (see Fig.

Introduction

This chapter will focus on the description of the experimental techniques. First, we address the principle of the stroboscopic measurements, allowing us to record the transient dynamics of the material at the femtosecond and picosecond timescales. Second, we discuss the THz generation via second-order nonlinear processes, and its particular configuration, the tilted-pulse front method. The next part is devoted to the field-resolved detection of the pulsed THz radiation.

After describing the electrooptic sampling technique, we calculate the detector response function for the crystals used in this work. In addition, we introduce and characterize the method of the electrooptic detection improvement, which made possible the detection of the THz emission from the spintronic emitters discussed in the Chapter 4. The last subsection is about the basics of the THz Time domain spectroscopy (THz-TDS), where we present measured properties of silicon, silica, and x-cut Quartz in the THz range.

Pump-probe spectroscopy

Since its first implementation, the stroboscopic approach has dramatically improved and now it is widely used in science, art, and the entertainment industry. For example, the fastest highspeed camera can record up to 1 Million frames per second (Phantom v2512) and is used in the field of microfluidics and particle velocimetry [START_REF] Cui | Magnetothermochromic coupling Janus sphere for dual response display[END_REF]. In addition, the ultrafast photodiodes with the rise-time of about 15 ps are now commercially available. However, since the temporal resolution The subsequent exponential decay of the relative transmittance corresponds to electron-electron, electron-phonon coupling, and heat dissipation. In addition, one can reveal some coherent dynamics (high and low-frequency modulation of the ∆T /T ) with the pump-probe spectroscopy. is limited by the shutter speed or carrier velocities in electronic detectors, it is impossible to use these gating devices at the sub-picosecond timescales.

However, one can achieve sub-picosecond temporal resolution with pulsed laser radiation, where the width of the sampling pulse envelope can be as short as a couple of tens of attoseconds (10 × 10 -18 s) [START_REF] Paul | Observation of a train of attosecond pulses from high harmonic generation[END_REF][START_REF] Cavalieri | Attosecond spectroscopy in condensed matter[END_REF][START_REF] Orfanos | Attosecond pulse metrology[END_REF]. In this thesis, ultrafast nonlinear spectroscopy was performed in pumpprobe geometry. For this, the output laser beam is spatially split into two parts. The first part, called the pump, is used to excite the electronic subsystem of the sample, triggering the change of the dielectric, structural or magnetic properties. The second part of the laser beam, is used to monitor or probe the change in reflectance (∆R(τ )/R) or transmittance (∆T (τ )/T ) of the probe beam as the function of temporal delay τ between the pump and the probe pulses. The τ is adjusted by changing the optical path length of one of two beams (τ = ∆l/c), as shown in Fig. 2.1a.

Today, various pump-probe based experimental techniques can measure the dynamics of a solid at the sub-picosecond timescales, using different parts of the EM spectrum of light. For example, experiments can be done with different configurations, such as all-optical pump-probe, time-resolved magneto-optical Kerr effect [START_REF] Beaurepaire | Ultrafast Spin Dynamics in Ferromagnetic Nickel[END_REF], time-resolved X-ray diffraction [START_REF] Lindenberg | Time-resolved X-ray diffraction from coherent phonons during a laser-induced phase transition[END_REF], and many more.

All those pump-probe "derivatives" are designed to probe and/or excite a specific degree of freedom of matter. For example, during the all-optical pump-probe experiment, one measures the relative change in the intensity of probing light with a central frequency of ω pr over the time τ , linked to the temporal change in the dielectric function of the excited material. Let us write the total electric field inside the studied media as [START_REF] Joffre | Optique non-linéaire en régimes continu et femtoseconde[END_REF]:

E(τ ) = 1 2 E pr (k pr , τ ) + E pump (k pump , τ ) + E * pr (k pr , τ ) + E * pump (k pump , τ ) . (2.1)
From the perturbation theory, the first order interaction describes the absorption of either the pump and the probe beams, governed by χ (1) susceptibility. Next, the second order field-interaction includes the second harmonic generation (SHG), optical rectification (OR) of pump and probe beam, and the frequency mixing between the pump and the probe (ω pr ±ω pump ) pulses, as discussed in the Section 1.3. In our case, we are only interested in the ω pr field components able to reach the detector (i.e. propagating along the k pr ). Here, the pump-probe interaction inside the media, that does not affect the change in k pr could be described in terms of third-order modification of the nonlinear polarization as [START_REF] Joffre | Optique non-linéaire en régimes continu et femtoseconde[END_REF]:

P (3) pump-pr (τ ) = 3ϵ 0 2 dω pr 2π dω pump 2π dω pump 2π χ (3) (ω pr , ω pump , -ω pump )× E pr (ω pr )E pump (ω pump )E * pump (ω pump ) exp[-i(ω pr + ω pump -ω pump )τ ], (2.2)
where χ (3) (ω pr , ω pump , -ω pump ) is the third order susceptibility, governing the particular interaction when the two pump pulses arrive simultaneously:

k = k pr + k pump -k pump = k pr , (2.3)
where k pr is conserved. In other words, for the short time-delays τ , the pump-probe experiment allows us to access the population relaxation of the irradiated matter. In the low-perturbation regime (for small ∆ε), we can express the time-dependent modification of the probe beam transmittance or reflectivity as [START_REF] Yu | Fundamentals of Semiconductors[END_REF]:

       ∆T T (ω pr , τ ) = ∂ ln T ∂ε 1 (ω pr )∆ε 1 (ω pr , τ ) + ∂ ln T ∂ε 2 (ω pr )∆ε 2 (ω pr , τ ); ∆R R (ω pr , τ ) = ∂ ln R ∂ε 1 (ω pr )∆ε 1 (ω pr , τ ) + ∂ ln R ∂ε 2 (ω pr )∆ε 2 (ω pr , τ ), (2.4)
where ∆ε 1 (ω pr , τ ) and ∆ε 2 (ω pr , τ ) are time-dependent change of the real and imaginary part of complex dielectric function ε(ω pr ), caused by the pump-pump population change within the material of interest.

In this work, we use a Ti:Al 2 O 3 laser system based on the chirped-pulse amplification technique [START_REF] Strickland | Compression of amplified chirped optical pulses[END_REF]. Our laser delivers 3 mJ pulses, centered around 800 nm wavelength, at 1 kHz repetition rate (3 W of mean power). The pulse duration was measured by a second-order intensity autocorrelator and was found to be ≈ 165 fs (at FWHM).

The sketch of the experimental setup is shown in Fig. 2.2. The beam is split with the ratio of 90/10, where 90 % of power is used for the pump or THz pulse generation, and remaining 10 % of intensity is used as the probe. The 0.5 mm Beta Barium Borate (BBO) crystal, introduced in the probe beam path, via SHG process converts the 800 nm into 400 nm light. The probe beam is a) 800 nm pump and 400 nm probe setup. The Beta Barium Borate (BBO) crystal is used for the frequency doubling of the probe beam. b)1 THz pump and 400 nm probe setup. The THz pulse is generated via the tilted-pulse front setup discussed in the manuscript. For both experimental configurations, the pump beam is incident close to the normal incidence regarding the sample surface.

For the probe, the angle of incidence is about 10 to 15°. For the balanced detection, 50 % of optical power is directed towards the reference photodiode.

The signal part of the probe beam is focused onto the sample (with the spot size of ≈ 150 µm at FWHM) and is recorded by the signal photodiode of the NewFocus 2307 detector. The difference between the signal and reference inputs of NewFocus 2307 is fed to the SRS-830 Lock-in amplifier.

The lock-in extracts the signal at the modulation frequency of the mechanical chopper (500 Hz) introduced in the pump beam path.

The conventional 800 nm pump/400 nm probe configuration is shown in the Fig. 2.2a. Here, the pump is focused down to ≈ 200 µm spot diameter at normal incidence in respect to the sample surface. Visible probe and near-IR pump beam sizes were measured with a CCD camera.

The main experimental configuration in this work is the THz pump/400 nm probe, shown in the

Terahertz generation and detection

The part of the electromagnetic spectrum with corresponding photon energies in the range of 0. Here, SC stands for semiconductor, FE is ferroelectric, and OC is an abbreviation of organic crystal.

From [START_REF] Hebling | Tunable THz pulse generation by optical rectification of ultrashort laser pulses with tilted pulse fronts[END_REF] Table 2.1 -Properties of the materials suitable for the optical rectification in laboratory conditions via optical rectification in semiconductors [START_REF] Leitenstorfer | Detectors and sources for ultrabroadband electro-optic sampling: Experiment and theory[END_REF][START_REF] Wu | 7 terahertz broadband GaP electro-optic sensor[END_REF], ferroelectrics [START_REF] Hirori | Single-cycle terahertz pulses with amplitudes exceeding 1 MV/cm generated by optical rectification in LiNbO3[END_REF], organic crystals [START_REF] Schneider | Generation of terahertz pulses through optical rectification in organic DAST crystals: theory and experiment[END_REF], in air plasma [START_REF] Hamster | Subpicosecond, electromagnetic pulses from intense laser-plasma interaction[END_REF], using synchrotron radiation [START_REF] Evain | Stable coherent terahertz synchrotron radiation from controlled relativistic electron bunches[END_REF], free-electron lasers [START_REF] Carr | High-power terahertz radiation from relativistic electrons[END_REF],

photoconductive switches [START_REF] Bacon | Photoconductive emitters for pulsed terahertz generation[END_REF] and spintronic emitters [START_REF] Kampfrath | Resonant and nonresonant control over matter and light by intense terahertz transients[END_REF]. Furthermore, the THz radiation can be detected and measured with multiple techniques, such as pyroelectric detection, electrooptic sampling (EOS) [START_REF] Auston | Electrooptic Generation and Detection of Femtosecond Electrical Transients[END_REF], using photoconductive antennas [START_REF] Grischkowsky | Far-infrared timedomain spectroscopy with terahertz beams of dielectrics and semiconductors[END_REF][START_REF] Uhd Jepsen | Generation and detection of terahertz pulses from biased semiconductor antennas[END_REF], bolometers, air biased coherent detection [START_REF] Dai | Detection of broadband terahertz waves with a laser-induced plasma in gases[END_REF], and THz radiation enhanced emission of fluorescence [START_REF] Liu | Terahertz-radiation-enhanced emission of fluorescence from gas plasma[END_REF]. This section will discuss the THz generation via optical rectification and EOS detection principles.

Light propagating inside a nonlinear optical medium can create second-order nonlinear polarization P (2) . As discussed in the Section 1.3.1.1, the rapidly varying femtosecond pulse envelope may lead to transient precession of P (2) in the far-or mid-infrared frequency range due to the difference-frequency mixing of a spectrally-broad light pulse. The second-order nonlinear susceptibility (χ (2) (Ω, ω 1 , ω 2 )) describes the efficiency of the optical rectification process. As an example, for F 43m space group (ZnTe, GaP, GaAs) the χ (2) (Ω, ω) tensor has only one non-vanishing diagonal component and is linked to the nonlinear optic tensor d ij (ω i , ω j ) and electrooptic coefficient r ij as [START_REF] Boyd | Nonlinear Optics[END_REF][START_REF] Auston | Electrooptic Generation and Detection of Femtosecond Electrical Transients[END_REF]:

χ (2) ij (Ω, ω i , ω j ) = 2d ij = - n 2 i n 2 j r ij 4 . ( 2.5)
For simplicity, we will drop the tensor notation, and will speak in terms of the effective nonlin-

ear susceptibilities χ (2)
eff (Ω, ω) and electrooptic coefficients r eff (Ω). The list of some THz emitters suitable for optical rectification is shown in the Table 2.1.

Terahertz generation: optical rectification

Since OR is a second-order nonlinear effect, the efficiency of the THz generation depends on the so-called velocity matching along the volume of the nonlinear crystal. For the in-phase precession of P (2) inside the nonlinear crystal, the following condition needs to be satisfied [START_REF] Nahata | A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling[END_REF]: i.e., the group velocity of the femtosecond pump should match the phase velocity of the emitted THz pulse. From the Table 2.1, we see that for some crystals (ZnTe and GaP), this condition is somewhat fulfilled, leading to the broadband THz emission (within 3-7 THz frequency range) [START_REF] Wu | 7 terahertz broadband GaP electro-optic sensor[END_REF][START_REF] Gallot | Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation[END_REF]. However, the phonon-polariton coupling can severely reduce the optical rectification (OR) bandwidth for the photons with frequencies close to the lattice optical phonon modes (see Section 1.3.1).

v 800 nm

The titled pulse front method:

From the table 2.1, we see that lithium niobate (LiNbO 3 ) ferroic crystal has high values of r eff , almost one order of magnitude higher than the ones of ZnTe and GaP. But, due to the high-velocity mismatch between the optical pulse and THz wave, complemented with increased THz absorption, these crystals are not efficient THz emitters in the colinear geometry, with conversion efficiency less than 0.001 %. However, as proposed by Janos Hebling et al. [START_REF] Hebling | Tunable THz pulse generation by optical rectification of ultrashort laser pulses with tilted pulse fronts[END_REF][START_REF] Hebling | Velocity matching by pulse front tilting for large area THz-pulse generation[END_REF], by tilting the pump-pulse Fig. 2.3a shows the geometry of the tilted-pulse front THz generation method. The diffraction grating, in pair with the two lenses, is used to tilt the intensity front of the pump by an angle γ tilt :

v 800 nm gr cos(γ tilt ) ≈ v THz . (2.7)
For the THz emission perpendicular to the crystal surface, the latter should be cut at the specific angle θ LN , equal to the front-tilt angle γ tilt . Then, the experimental parameters, such as diffraction angle θ d , crystal angle θ LN , and diffraction grating groove density p are chosen to satisfy the two following relations [START_REF] Hirori | Single-cycle terahertz pulses with amplitudes exceeding 1 MV/cm generated by optical rectification in LiNbO3[END_REF]:

tan γ tilt = mλ 0 p n 800 nm gr β 1 cos θ d and tan θ LN = n 800 nm β 2 tan θ d , ( 2.8)
with diffraction order m and pump pulse carrier wavelength λ 0 , where β 1 and β 2 are the horizontal magnification factors of the 4-f system for the pump pulse front and the grating image, respectively.

In this work, we use MgO doped LiNbO 3 cut at θ LN = 63°. The 4-f imaging system is built with two spherical lenses with

f 1 = f 2 =

Electrooptic terahertz detection

Back in the 1980s, Auston [START_REF] Auston | Electrooptic Generation and Detection of Femtosecond Electrical Transients[END_REF] introduced a method of time-resolved gating of the THz electric field with the stroboscopic approach, displayed in Fig. 2.4a. Such EOS detection technique is based on the Pockels effect (or linear Kerr-effect) in noncentrosymmetric crystals describing the electric field induced birefringence. For the EOS detection, Pockels effect is used to measure the change in the sampling pulse ellipticity inside the electrooptic crystal in the presence of the THz field.

Let us use the Jones Matrix formalism to describe the final polarization state of the sampling pulse, initially polarized along the x-axis [START_REF] Frank | Introduction to Optics[END_REF]: According to Eq.1.98, the electrooptic crystal will act as a THz field amplitude dependent phase retarder, where the phase shift Γ equal to:

p pr = 1 0 . ( 2
Γ = 2πn 3 0 r eff E THz L λc , (2.10)
where L is the crystal thickness and λ is the wavelength of the probe beam. Then, the total phase shift experienced by the probe is equal to [START_REF] Frank | Introduction to Optics[END_REF]:

C ZnTe = 1 2 e iΓ + e -iΓ e iΓ -e -iΓ
e iΓ -e -iΓ e iΓ + e -iΓ .

(2.11)

We then introduce Quarter wave plate (QWP) after the EOS crystal, with its fast axis oriented at +45°to probe beam polarization:

Q +45 = 1 √ 2 1 i i 1 . (2.12)
Next, the resulting polarization state of the probe beam will write: with a Wollaston prism (WP). Then, we can measure the difference between x and y intensities using the balanced detection (See Fig. 2.4a). In the absence of THz electric field inside the crystal,

p = Q +45 • C ZnT e • p pr = 1 √ 2 cos Γ -sin Γ i(cos Γ + sin Γ) . ( 2
since |p x | 2 = |p y | 2
, measured signal will be zero. Then, in the presence of E THz (E THz ̸ = 0), the normalized power seen by the balanced photodiodes is:

S = |p y | 2 -|p x | 2 |p x | 2 + |p y | 2 = sin(2Γ). (2.14)
For Γ ≪ 1, we can assume that sin(2Γ) ≈ 2Γ. Then, with Eq.2.10 and Eq.2.14 we can write:

Γ = 2πn 3 0 r eff E THz L λ ⇒ E T Hz = Sλ πn 3 0 r eff L (2.15)
By changing the temporal overlap between the short sampling pulse and the THz waveform, one can point-by-point record S(τ ) (see Eq.2.14), and thus measure the shape of the THz electric field.

Moreover, the efficiency of the THz detection strongly depends on the orientation of the THz polarization (angle ψ) and the probe beam polarization with respect to the crystal z-axis (angle ϕ), depicted in the Fig. 2.4b. Planken et al. [START_REF] Paul | Measurement and calculation of the orientation dependence of terahertz pulse detection in ZnTe[END_REF] have deduced the dependence of the electrooptic signal strength on the angles ψ and ϕ:

S(τ ) ∝ (cos ψ sin 2ϕ + 2 sin ψ cos 2ϕ). (2.16)
We remind that ψ and ϕ are the azimutal angles between the E THz and E p in respect to electrooptic detection crystal z-axis. In this thesis, we have set ϕ = ψ (meaning that E p ∥ E THz ), and we rotated the crystal to obtain maximum possible signal.

THz detection sensitivity improvement:

In Chapter 4, we sample low-amplitude E THz (τ ) fields, which requires a lot of averaging in the conventional electrooptic detection scheme (with Q +45 ). To improve the sensitivity of the EOS, we use the simple experimental technique first described by J. A. Johnson et al. [START_REF] Johnson | Distortion-free enhancement of terahertz signals measured by electro-optic sampling II Experiment[END_REF] and then further simplified by P. Krauspe et al. [START_REF] Krauspe | Effective detection of weak terahertz pulses in electro-optic sampling at kilohertz repetition rate[END_REF], with a theory established by J. Brunner et. al. [START_REF] Fabian | Distortion-free enhancement of terahertz signals measured by electro-optic sampling I Theory[END_REF].

Experimentally, this approach could be regarded as the decrease in the ratio between the optical bias and the effective electrooptic signal. In a general way, the measured EOS signal (see Eq.2.14) can be defined as the function of Γ and Quarter Wave Plate (QWP) angle α (see Fig. 2.5a) as [START_REF] Krauspe | Effective detection of weak terahertz pulses in electro-optic sampling at kilohertz repetition rate[END_REF]:

arcsin(S(α, Γ)) = 2Γ sin(2α) + 2cot(2α) 2 • Γ 2 , (2.17)
where the first term is the linear part of the signal, directly proportional to E THz . The second term is the quadratic contribution, proportional to |E THz | 2 . One can retrieve the conventional EOS equation (Eq.2.14) for α = ±45°, since the quadratic term and the α-dependence can be CHAPTER 2. EXPERIMENTAL METHODS neglected. However, for small E THz -induced phase (Γ ≪ 1, sin(Γ) ≈ Γ), the subtraction of signals measured for ±α yields:

S(α, Γ) -S(-α, Γ) = 4Γ sin(2α) , ( 2.18)
returning only the linear contribution to the S(τ ), but improved (or enhanced) by the factor of 1/ sin 2α. In analogy, the addition of the two signals recorded for ±α:

S(α, Γ) + S(-α, Γ) = 4 cot(2α) 2 • Γ 2 (2.19)
will contain only information about the quadratic contribution to the measured EOS signal.

To check this method, we first use the 500 µm ZnTe emitter and detector, and measure THz waveform in the function of the QWP angle α. For α ̸ = 45°, a variable attenuator (VA, shown in Fig. 2.5a) needs to be used to re-balance the photodiodes. Fig. 2.5b depicts the change in S(τ

)
with the decrease of QWP angle α. The data for α > 1.7°is not shown, since the waveform is indistinguishable from the one recorded for α ≈ 45°. For α < 1.7°, the nonlinear contribution starts to dominate, resulting in the noticeable distortion of the measured THz field [START_REF] Johnson | Distortion-free enhancement of terahertz signals measured by electro-optic sampling II Experiment[END_REF].

Since we can not control QWP angle α with high precision, manual rotation of the mechanical components can lead to questionable eperimental reproducibility of the undistorted THz waveform retrieval, especially for small values of α. Due to this mechanical limitation, the study of the THz emission from spintronic emitters was done for the fixed value of α. QWP angle α was selected so that the quadratic contribution to S(τ ) is negligible (i.e. 2Γ/sin(2α) ≪ 2cot(2α) 2 •Γ 2 , see Eq.2.17).

In this case, the measurement for ±α can be avoided.

In order to select the appropriate value of α, we replaced the 500 µm ZnTe crystal with spintronic THz emitter CoFeB(20 nm)/MgO(6 nm) 1 . All other experimental parameters were kept the same as before. The recorded THz waveforms for α ≈ 0.25°and α ≈ 45°are shown in the Fig. 2.5c.

When two measurements normalized by the peak-to-peak amplitude of the respective signal, perfect overlap of the normalized THz waveforms indicates the absence of nonlinear distortion (see inset of Fig. 2.5c). This is expected, since THz emitted from CoFeB(20)/MgO( 6) is way weaker than the one generated with 500 µm ZnTe crystal, roughly by the factor of 800. In addition, the fieldinduced phase retardation Γ of CoFeB(20 nm)/MgO(6 nm) at α = 0.25°is still smaller than the signal obtained 0.5 mm ZnTe at α ≈ 45°.

From this, in the absence of the signal distortion, one can directly estimate the detectivity enhancement factor Λ EOS as: which, for CoFeB(20 nm)/MgO(6 nm) is equal to:

Λ EOS = S(α, Γ) S(45°, Γ) , ( 2
Λ EOS = S(0.25°, Γ) S(45°, Γ) ≈ 50. (2.21)
All the THz signals discussed in the Chapter 4, were measured with this technique. However, at first all signals were recorded without the EOS detectivity enhancement. Then, the experiment was repeated for the fixed value of α ≈ 0.7°, for which THz detected from all spintronic emitters showed no distortion. Figure 2.18d displays the "raw" (α = 45°) and enhanced signal (α ≈ 0.7°) of the most efficient spintronic emitter in our disposal, CoFeB(5)/Pt(3). For the enhancement factor Λ EOS of about 8.5, both the THz waveform and its spectrum remains unchanged, indicating the presence of only linear contribution to S(τ ). In addition, the dependence of emitted E THz amplitude on the NIR pump power was measured at conventional α ≈ 45°, in order to be free from the possible detection nonlinearities present for the higher-amplitude THz fields.

In theory, for the low electric field amplitudes, the improvement factor Λ EOS can be increased even further, but mainly limited by the damage threshold of the EOS crystal and the reasonable amount of optical power required for the balanced detection. For example, one can increase the sampling beam size in order to reduce the optical fluence Φ inside the detection crystal. In addition, one can replace a conventional balanced detector by the pair of avalanche photodiodes for low α < 0.1°. Ultimately, this technique can be applied for any polarization-sensitive detection.

In principle, this THz detectivity enhancement technique based on the QWP-detuning can be used for both kHz and MHz oscillators, but it requires an excessive sampling pulse optical power density, and signal-to-noise ratio may be limited by the electronic noise in the case of MHz oscillators.

Electrooptic response function:

Since EOS is governed by the crystals' effective electrooptic coefficient r eff , the detection efficiency and bandwidth strongly depend on the ionic and the optical properties of the detector. For example, when THz phase -sampling pulse group velocity mismatch is negligible, and with THz pulse having frequency components out of the range of detection crystal lattice optical phonon resonances, measured signal will be the direct replica of the THz electric field, i.e., S(τ ) ∝ E THz . In reality, however, detected signal S(τ ) is a result of the convolution between the frequency-dependent detector response and THz electric field [START_REF] Leitenstorfer | Detectors and sources for ultrabroadband electro-optic sampling: Experiment and theory[END_REF][START_REF] Wu | 7 terahertz broadband GaP electro-optic sensor[END_REF][START_REF] Gallot | Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation[END_REF]:

S(τ ) = E THz (τ ) ⊛ h EOS (τ ), (2.22)
where h EOS (τ ) is the EOS response function of the nonlinear medium, and ⊛ is the convolution operator. In the frequency domain, the latter equation can be rewritten as:

S(Ω) = E THz (Ω) × h EOS (ω, Ω), (2.23)
where ω and Ω are the angular frequencies of the Near-IR sampling pulse and THz waveform, respectively.

From the physics perspective, h EOS (ω, Ω) contains the information about the phase-matching, sampling pulse-THz wave interaction, and EOS detection bandwidth. According to Kampfrath et.al. [START_REF] Kampfrath | Sampling of broadband terahertz pulses with thick electro-optic crystals[END_REF], h EOS (ω, Ω) is expressed as:

h EOS (ω, Ω) = 4it 12 (Ω) c 2 ω>Ω dωE * s (ω)E s (ω -Ω)C(ω, Ω)T (ω, Ω)P (ω, Ω), (2.24)
where t 12 is the Fresnel transmission coefficient for the air-crystal interface, E s (ω) is the spectrum of a sampling pulse. The first coefficient C(ω, Ω) describes the nonlinear terahertz -sampling pulse interaction:

C(ω, Ω) = ω 2 × χ (2) ef f (ω -Ω, Ω)/k(ω), (2.25)
with k(ω) = n(ω)ω/c being the sampling pulse wave-vector. Next, T (ω, Ω) is the "composite" sampling pulse transmission coefficient:

T (ω, Ω) = t * 12 (ω) × t 12 (ω -Ω) × exp[-2d tot Im{k(ω)}] × |t 21 | 2 , (2.26)
with t 21 being the crystal-air Fresnel coefficient, and d tot is the total thickness of the EOS crystal (electrooptically active layer plus the substrate). The last term, P (ω, Ω) is the phase-matching coefficient, describing the frequency-dependent velocity mismatch between the THz and sampling pulses:

P (ω, Ω) = exp(iΩd[v -1 ph (Ω) -v -1 g (ω)]) -1 iΩ[v -1 ph (Ω) -v -1 g (ω)] , ( 2.27)
with d being the thickness of the electrooptic crystal (only of the active layer).

Here, we use ⟨110⟩ 500 µm thick ZnTe and ⟨110⟩ 200 µm thick GaP for EOS detection. For ZnTe, refs [START_REF] Marple | Refractive Index of ZnSe, ZnTe, and CdTe[END_REF] and [START_REF] Gallot | Electro-optic detection of terahertz radiation.pdf[END_REF] were used to calculate the NIR and THz refractive indices, respectively, with refs [START_REF] Bi | The dispersion of the refractive index of III-V semiconductors[END_REF] and [START_REF] Faust | Dispersion in the nonlinear susceptibility of GaP near the reststrahl band[END_REF] containing information about GaP detector. The χ

(2)
ef f (ω -Ω, Ω) was calculated from [START_REF] Leitenstorfer | Detectors and sources for ultrabroadband electro-optic sampling: Experiment and theory[END_REF] and [START_REF] Faust | Mixing of visible and near-resonance infrared light in GaP[END_REF] for ZnTe and GaP, respectively. For the NIR pulse parameters, we use the central wavelength of 800 nm and pulse duration of 165 fs. We see roughly 10 times less signal for the GaP detector, which is expected due to the 4 times smaller r eff (see Table 2.1), 2.5 times thickness difference and difference in the refractive indices (see Table 2.1). However, after accounting for h EOS (ω, Ω) of a given detector, we recover nearly identical shape of the E THz , as shown in Fig. 2.6d.

CHAPTER 2. EXPERIMENTAL METHODS

One should note that Fig. 2.6d gives us the approximate shape of E THz (τ ) in front of the detection crystal. In Chapter 3, we apply such h EOS (ω, Ω)-deconvolution to the THz emitted from LiNbO 3 crystal, since the information about the E THz waveform just in front of the sample is essential for the measurements interpretation and first-principle calculations. However, due to the fact that the THz emission experiments presented in Chapter 4 were done for the same detection crystal, it allows us to directly compare all the measured signals. In addition, one can estimate the electric field waveform at the emitter position, taking into account the spatio-temporal transformation focusing distortions of the THz pulse [START_REF] Gürtler | Terahertz pulse propagation in the near field and the far field[END_REF][START_REF] Kužel | Spatiotemporal transformations of ultrashort terahertz pulses[END_REF].

THz time-domain spectroscopy

The THz time-domain spectroscopy (THz-TDS) allows studying the static or transient material properties in the far-IR/THz range. The review of the different sources used for the THz-TDS, potential noise sources, the influence of the experimental settings were described by J. Neu and C.

A. Schmuttenmaer [START_REF] Neu | Tutorial: An introduction to terahertz time domain spectroscopy (THz-TDS)[END_REF].

Free-standing film or bare substrate: The dielectric function of the free-standing film can be measured as follows. First, the waveform of the reference pulse S ref (τ ) with no sample in the path of THz, in pair with the noise S noise (τ ) 3 , propagating through the free space is recorded.

Second, the sample with the thickness d is introduced in the THz beam path, and sample signal S sam (τ ) is recorded. Next, one can write the ratio between the reference and sample signals in the frequency domain as:

H(Ω) = S(Ω) sam S(Ω) ref = E sam (Ω) × h EOS (ω, Ω) E ref (Ω) × h EOS (ω, Ω) = E sam (Ω) E ref (Ω) , ( 2.28)
where the response function of the detector is mathematically irrelevant. In this type of experiment, the signal-to-noise ratio is really important, especially for frequencies close to the detector lattice resonance modes, for example.

Next, the amplitude A(Ω) and the phase ϕ(Ω) of the THz-TDS ratio H(Ω) are calculated. By knowing the thickness of the sample d, the refractive index of the free-standing film n(Ω) is given as [START_REF] Peter | Dynamic range in terahertz time-domain transmission and reflection spectroscopy[END_REF]:

n(Ω) = 1 + c Ωd ϕ(Ω). (2.29)
From the calculated refractive index, the absorption coefficient α(Ω) can be estimated as:

α(Ω) = - 2 d ln A(Ω) [n(Ω) + 1] 2 4n(Ω) = 2ωIm(k(Ω)) c . (2.30)
However, it is important to know the sensitivity of the experiment. The ratio between S ref (Ω)/S noise (Ω)

3. When the THz source is blocked. is used to estimate the experimental dynamical reserve (DR) 4 of the experiment [START_REF] Peter | Dynamic range in terahertz time-domain transmission and reflection spectroscopy[END_REF]:

α max (Ω) = 2 ln DR(Ω) 4n(Ω) ([n(Ω) + 1] 2 . (2.31)
It should be noted, that α(Ω) < α max (Ω) is the criterion of the reliable measurement of both n and α. In addition, we manually limit the THz-TDS resolution up to 3 THz, close to the high-distortion of the h EOS (see Fig. 2.6a).

Next, we have measured refractive indices and absorption coefficients of the High-Resistivity

Float-Zone Silicon (HRFZS) wafer, High-Density Polyethylene (HDPE), and Float-Zone Glass substrates (see Fig. 2.7). We observe a flat refractive index and absence of absorption in HRFZS and HDPE, in good agreement with the literature [START_REF] Dai | Terahertz time-domain spectroscopy characterization of the far-infrared absorption and index of refraction of highresistivity, float-zone silicon[END_REF][START_REF] Sommer | THz-Spectroscopy on High Density Polyethylene with Different Crystallinity[END_REF]. On the other hand, the glass shows strong n(Ω) and α(Ω) frequency dependence, which leads to the strong distortion of the transmitted THz pulse [START_REF] Naftaly | Terahertz time-domain spectroscopy of silicate glasses and the relationship to material properties[END_REF].

The primary purpose of the HDPE sheet (with band-gap energy in the range of 4.6 eV or 270 nm [START_REF] Majeed | Effect of High Density Polyethylene (HDPE) on Structural and Optical Properties of (PP/PMMA) Blends[END_REF]) is to diffuse and filter out the pump beam, used for the THz generation in the collinear configuration (ZnTe, GaP, spintronic emitters). In addition, we use HRFZS wafer to completely remove any remaining pump residue. The HRFZS THz transmission, placed after the single HDPE sheet, was estimated to T HRFZS = 67 %, in good agreement with multiple reports [START_REF] Hirori | Single-cycle terahertz pulses with amplitudes exceeding 1 MV/cm generated by optical rectification in LiNbO3[END_REF][START_REF] Dai | Terahertz time-domain spectroscopy characterization of the far-infrared absorption and index of refraction of highresistivity, float-zone silicon[END_REF][START_REF] Ro/Nne | Investigation of the temperature dependence of dielectric relaxation in BIBLIOGRAPHY liquid water by THz reflection spectroscopy and molecular dynamics simulation[END_REF].

However, the removal of HDPE sheet before the HRFZS leads to the photoexcitation and pumppower dependent THz absorption of the latter, as discussed in the Appendix F.

Thin metallic films:

For the thin film with the thickness d film deposited onto the thick substrate, the THz amplitude transmission H(Ω) (see Eq.2.28) will be the product of the THz pulse multiple reflections at the film interfaces, paired with THz absorption in the metal film and the substrate [START_REF] Laman | Terahertz conductivity of thin metal films[END_REF]:

H(Ω) = t 12 t 23 t 31 • exp[ik 2 (Ω)d film ] • exp[ik 3 (Ω)d sub ] 1 + r 12 r 23 exp[2ik 2 (Ω)d film ] , ( 2.32)
where k 1 , k 2 and k 3 are the complex THz wave vectors in air, metallic film and substrate, respectively. However, one can simplify the latter equation,if the THz-TDS signal of the bare substrate S sub (Ω) was used as the reference [START_REF] Seifert | Terahertz spectroscopy for all-optical spintronic characterization of the spin-Hall-effect metals Pt, W and Cu80Ir20[END_REF]:

H(Ω) = S sam (Ω) S sub (Ω) • exp[i(n 3 (Ω) -1)∆d sub Ω/c], (2.33)
where we account for the reference and sample substrate thickness mismatch with ∆d sub . If the metallic film is smaller than the THz wavelength and penetration depth, the Eq.2.32 becomes:

H(Ω) = n 2 (Ω) + n 3 (Ω) n 2 (Ω) + n 3 (Ω) + Z 0 d film 0 dzσ(Ω) , ( 2.34)
with the free space impedance Z 0 = 377 Ω. Here, the frequency dependent conductivity σ(Ω) can be calculated with the Drude model: where τ e is the electron scattering rate. It is relatively straightforward to measure the optical properties of a thin metallic film on the substrate, using THz transmitted though the bare substrate as the reference. In that case, the use of sophisticated THz-propagation models is not necessary [START_REF] Peretti | THz-TDS Time-Trace Analysis for the Extraction of Material and Metamaterial Parameters[END_REF].

σ(Ω) = σ DC 1 -iΩ/τ e , ( 2
THz-TDS is a powerful technique that allows directly measuring and extracting the material properties in the THz range, but it has its limitations. For example, too strong (α = α max ) or negligible absorption in the THz range define the sensitivity of the experiment. On the other hand, the physical properties of the detector and the environment the experiment is performed in (vacuum or air) ultimately define the detection bandwidth.

THz pulse shaping

Here, we present a method to modify the spectrum of the THz pulse after the propagation through the birefringent Quartz crystal. We then study the influence of 3 mm thick x-cut Quartz 57 CHAPTER 2. EXPERIMENTAL METHODS crystal on the THz pulse polarization and spectrum. In this experiment, we use a 500 µm ZnTe crystal as both THz source and detector.

First, we have recorded the reference THz spectrum in the absence of the x-cut Quartz, displayed as a black curve in Fig. 2.8c. Next, we placed the x-cut Quartz crystal between the Wire Grid WG polarizer analyzer, with its fast axis oriented at θ Quartz ≈ 45°relative to E THz polarization, and measured the THz spectrum in the function of the WG analyzer angle (see Fig. 2.8b).

The rotation of the WG analyzer will select a specific polarization component of the E THz , which is then directed towards the EOS detector. To be in an optimal EOS detection condition, we have kept the angles between z-axis of the detection crystal, E THz and E p field vectors fixed through the experiment (as described in the Section 2.3.2).

THz spectra, measured for θ = 0°, 50°, and 90°are shown in the Fig. 2.8c. The observed modification of the THz spectrum amplitude, can be explained with the Eq.1.98. For this we can calculate the frequency-dependent phase shift within 0 to 4 THz range, experienced by the sampling pulse: We then can calculate the spectral centroid in the function of the WG analyzer angle, i.e. to see how spectrum center of mass evolves for selected E THz projection:

Γ(Ω) = (n e -n o ) ΩL c , ( 2
Spectral Centroid = N -1 n=0 nA[n] N -1 n=0 A[n] , ( 2.37)
where A[n] is the FFT amplitude corresponding to the n-th point of the spectrum. One should note that the most direct way to shape the spectrum of the THz pulse is to change the thickness of the electrooptic crystal (see Eq.2.27). As an alternative, the tilted pulse front setup (TPFS) can be realigned to satisfy the phase-matching condition within 0.5-3 THz range [START_REF] Hebling | Tunable THz pulse generation by optical rectification of ultrashort laser pulses with tilted pulse fronts[END_REF][START_REF] Morita | Terahertz pulse generation by the tilted pulse front technique using an M-shaped optical system[END_REF].

However, due to the high realignment complexity of the TPFS setup, the 3 mm x-cut Quartz crystal could be a simple alternative for the post-emission THz pulse-shaping. Since we directly measure the waveform of the THz pulse, we have access to both the real and imaginary part of the h Quartz (Ω, θ). We then can use the measured h Quartz (Ω, θ) to estimate the shape of the LiNbO 3 THz waveform as a function of WG analyzer angle θ:

E THz (Ω, θ) = E THz (Ω) × h Quartz (Ω, θ), ( 2

Introduction

The works of H.J. Maris [START_REF] Thomsen | Coherent Phonon Generation and Detection by Picosecond Light Pulses[END_REF], J.M. Wiesenfeld [START_REF] Wiesenfeld | Acoustic phonon generation in the picosecond dynamics of dense electronhole plasmas in InGaAsP films[END_REF], G. Eesley [START_REF] Eesley | Generation of nonequilibrium electron and lattice temperatures in copper by picosecond laser pulses[END_REF] and Y. Yan [START_REF] Yong Xin Yan | Impulsive stimulated scattering: General importance in femtosecond laser pulse interactions with matter, and spectroscopic applications[END_REF] have kickstarted the field of picosecond laser acoustics, describing photogeneration and optical detection of picosecond acoustic pulses in metals and semiconductors. At nearly the same moment, ultrashort laser pulses allowed to explore the optical phonon dynamics [START_REF] Zeiger | Theory for displacive excitation of coherent phonons[END_REF][START_REF] Merlin | Generating coherent THz phonons with light pulses[END_REF]. The detection of coherent acoustic and optical phonons is possible by analyzing the variation of the reflected or transmitted probe beam intensity or polarization, which is caused by change in material properties in the presence of the phonon strain field [START_REF] Thomsen | Picosecond interferometric technique for study of phonons in the brillouin frequency range[END_REF]. Most coherent phonons have been generated by nearinfrared, visible [START_REF] Weis | Ultrafast lightinduced coherent optical and acoustic phonons in few quintuple layers of the topological insulator Bi2Te3[END_REF][START_REF] Lejman | Ultrafast acousto-optic mode conversion in optically birefringent ferroelectrics[END_REF], or near-ultraviolet pulsed radiation [START_REF] Maznev | Generation of coherent phonons by coherent extreme ultraviolet radiation in a transient grating experiment[END_REF][START_REF] Maznev | Generation and detection of 50 GHz surface acoustic waves by extreme ultraviolet pulses[END_REF], i.e., a light pulse close to the resonant electronic excitation. In these cases, phonon generation is driven by different electronphonon collisions processes (deformation potential or thermoelasticity, for example) that we will briefly describe in the first part of this chapter. Moreover, in some cases, stimulated Raman and Brillouin scattering can lead to coherent lattice motion [START_REF] Merlin | Generating coherent THz phonons with light pulses[END_REF].

CHAPTER 3. ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ

Over the last decades, thanks to the recent developments in nonlinear optics, the tabletop amplified laser systems can be used to routinely generate high peak-amplitude THz pulses [START_REF] András Fülöp | Laser-Driven Strong-Field Terahertz Sources[END_REF].

Such intense electric field transients (with peak electric field > 100 kV cm -1 ) having characteristic energy in meV range, can be used to study the collective particle motion in a solid. In particular, resonant and nonresonant control over matter [START_REF] Kampfrath | Resonant and nonresonant control over matter and light by intense terahertz transients[END_REF][START_REF] Nicoletti | Nonlinear light-matter interaction at terahertz frequencies[END_REF] is possible through electron acceleration in the presence of an intense THz field [START_REF] Kuehn | Terahertz-induced interband tunneling of electrons in GaAs[END_REF][START_REF] Schubert | Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations[END_REF] up to impact ionization [START_REF] Hirori | Extraordinary carrier multiplication gated by a picosecond electric field pulse[END_REF]. In addition, resonant antiferromagnon excitation with THz has also been reported [START_REF] Kampfrath | Coherent terahertz control of antiferromagnetic spin waves[END_REF]. THz radiation can also be directly coupled to phonons, and intense THz pulses are proclaimed as the ultimate tool of coherent lattice vibration control [START_REF] Först | Mode-Selective Control of the Crystal Lattice[END_REF]. Once said, only a few reports of coherent acoustic phonon generation with THz exist at the moment, demonstrating that this area of research is almost unexplored. In the past few years, observation of acoustic phonon signatures in AlGaAs intrinsic semiconductor has been discussed but without establishing the nature of the microscopic mechanism at play [START_REF] Manceau | Direct acoustic phonon excitation by intense and ultrashort terahertz pulses[END_REF].

Some reports of strain induced by microwave radiation (electromagnetic pulse in the GHz regime) also exist [START_REF] Grill | Excitation of Coherent and Incoherent Terahertz Phonon Pulses in Quartz Using Infrared Laser Radiation[END_REF] without a clear conclusion at the moment. This situation deeply contrasts with the generation of coherent optical phonons by a THz pulse, for which there is much more experimental evidence. One can cite the optical phonon generation in Tellurium [START_REF] Huber | Coherent phonon spectroscopy of non-fully symmetric modes using resonant terahertz excitation[END_REF], diamond [START_REF] Maehrlein | Terahertz Sum-Frequency Excitation of a Raman-Active Phonon[END_REF] and Bi 2 Se 3 nanofilms [START_REF] Melnikov | Coherent phonons in a Bi2Se3 film generated by an intense single-cycle THz pulse[END_REF][START_REF] Melnikov | Phonon-driven ultrafast symmetry lowering in a Bi2Se3 crystal[END_REF], for example. Despite numerous reports, there are still some debates about the microscopic origin of the THz-driven optical phonon dynamics, as a result of non-linear ionic

Raman scattering [START_REF] Först | Mode-Selective Control of the Crystal Lattice[END_REF][START_REF] Mills | Ionic contributions to the Raman tensor of insulators[END_REF] or THz sum frequency excitation [START_REF] Dominik | Sum-frequency ionic Raman scattering[END_REF]. This Chapter will be split into two parts, where we are going to discuss the experimental

Acoustic phonons generation by pulsed THz radiation

Theoretical Background

Following the Chapter 1 discussion, carriers generated by the absorption of the laser pulse undergo scattering and energy exchange through couplings between electron, phonon, and spin baths. Even though NIR-VIS-NUV light was mainly used as the pump pulse, the following model is relevant for any kind of initial electron excitation. The resulting conversion of optical into internal mechanical energy may lead to stress generation both at the macroscopic scale and at the unit-cell level. From the thermodynamic point of view, the energy increase of an adiabatic system will lead to the change in the internal pressure of a solid, known as the photoinduced stress σ PI [START_REF] Vitalyi | Laser Optoacoustics[END_REF]. For the isotropic, 1D medium, we can write the classical equation of elastic motion, with σ PI acting as the source term [START_REF] Ruello | Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action[END_REF]:

∂ 2 u(t, z) ∂t 2 -V 2 ph ∂ 2 u(t, z) ∂z 2 = 1 ρ i ∂σ PI,i (t, z) ∂z , ( 3.1)
where u(t, z) is the particle displacement, V ph is the strain pulse velocity, ρ is the density of the medium, and σ PI,i (t, z) is the spatio-temporal profile of the specific contribution to the photoinduced stress. In complex systems (for example, in multiferroics) σ PI may have various simultaneous contributions of different physical origins [START_REF] Ruello | Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action[END_REF]:

σ PI = σ TE + σ DP + σ PE + σ ES + σ MS , (3.2)
where σ TE , σ DP , σ PE , σ MS , and σ ES are the thermoelastic (TE), deformation potential (DP), inverse piezoeletric (PE), magnetostriction (MS) and electrostriction (ES) stresses. In this thesis, however, we will discuss only the most relevant contributions for a simple metal and a semiconductor, namely the σ TE and σ DP .

Thermoelastic stress:

The most familiar mechanism of coherent GHz-THz acoustic phonon generation in metals is the thermoelastic stress σ TE (z, t). The initial distribution of optical energy deposited per unit of volume can be estimated by [START_REF] Thomsen | Surface generation and detection of phonons by picosecond light pulses[END_REF]:

W opt (z) = (1 -R) Q A • ξ e -z/ξ for d≫ ξ, ( 3.3)
where A is the irradiated area, Q is the pulse energy, ξ is the optical penetration depth, R is the laser intensity reflection coefficient and d is the film thickness. As one can notice, W opt (z) is initially defined over a finite distance, equal to ξ (see Fig. 3.1a).

Following the very fast electron-electron scattering (see Fig3.1b, electron-electron thermalization), the hot carriers will interact with the phonons to decay to the lowest energy state (electronphonon thermalization). This intraband process will lead to thermal (or incoherent) phonon emission. Such increase in incoherent phonon population of the phonon band m (ρ p m (k)) will contribute to the thermoelastic stress as [START_REF] Ashcroft | Solid State Physics[END_REF][START_REF] Ruello | Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action[END_REF]:

σ TE (z, t) = - m,k δρ p m (k)ℏω m,k γ p m,k = -γ L c L δT L (z, t), ( 3.4)
where γ L is the averaged Grüneisen coefficient for all k-dependent Grüneisen factors over m phonon bands, c L is the lattice heat capacity per unit of volume and δT L is the increase in the lattice temperature. From the macroscopic point of view, the latter equation is equivalent to the one observed with the thermodynamic relation:

β = 1 3B ∂P ∂T V , ( 3.5)
where β is the linear expansion coefficient, P is the pressure, and B is the bulk modulus. The lattice expansion is related to the increase in the internal pressure of a solid (σ = -∆P by convention).

With this, we arrive to the macroscopic formulation of the thermoelastic stress:

σ TE (z, t) = -3BβδT L (z, t). (3.6)
One should note that temperature-driven lattice expansion is possible only for an anharmonic interatomic potential [START_REF] Ashcroft | Solid State Physics[END_REF][START_REF] Ruello | Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action[END_REF]. For metals, if we assume the complete transfer of the energy from electron to phonon system, we can write [START_REF] Ruello | Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action[END_REF]: where N is the photoexcited carrier concentration and hν is the pump photon energy. In addition to the stress generation through the lattice heating, one needs to account for the photoinduced change in the electronic distribution, i.e., deformation potential stress, resulting from the light-induced modification of the electron-cation interaction.

δT L (z, t) = N (z, t)hν c L , ( 3

Deformation potential stress:

The deformation potential stress (σ DP ) can be described as interatomic potential modification due to the change in the electronic distribution. For example, the change in electronic energy in the band n by an amount δE n,k (see Fig. 3.2a), may lead to the volume change of a solid [START_REF] Yu | Fundamentals of Semiconductors[END_REF][START_REF] Ruello | Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action[END_REF]:

δV V n,k = δE n,k d eh n,k , ( 3.8)
where d eh n,k is the deformation potential coefficient of the energy level E n,k . The change in electronhole distribution described by δE n,k will affect the interatomic potential. This will result in the ion displacement and creation or annihilation of a phonon, as shown in the Fig. 3.2b. The deformation potential stress is usually expressed in terms of the light-induced modification of carrier population δρ e n (k) at different energy levels E n,k as [START_REF] Ruello | Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action[END_REF]:

σ DP = - n,k δρ e n (k)E n,k γ e nk = n,k δρ e n (k) ∂E n,k ∂η n,k , ( 3.9)
where η n,k is the strain generated by the change in electronic distribution, and γ e k is the electron Grüneisen coefficient.

General case:

In reality, both σ DP and σ TE can simultaneously contribute to the total photoinduced stress in both metals and semiconductors. In metals, the non-negligible σ DP can exist for high electron temperatures. Therefore, the total photoinduced stress, as a sum of thermoelastic CHAPTER 3. ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ and electronic equivalent of the phononic pressure is defined as [START_REF] Ruello | Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action[END_REF][START_REF] Perner | Observation of Hot-Electron Pressure in the Vibration Dynamics of Metal Nanoparticles[END_REF]:

σ = σ TE + σ DP = -3BβδT L -γ e c e δT e (Metal) . (3.10)
with γ e and c e the electronic Grüneisen coefficient and the electronic heat capacity.

In the semiconductor, upon the electron and hole intraband thermalization (i.e. when the electrons and holes are localized at the bottom or top of the conduction or valence bands, respectively), the deformation potential can be written as:

σ DP = N ∂E g ∂ξ = -N B ∂E g ∂P = -N d eh , ( 3.11)
where d eh is the electron-hole deformation potential parameter for this particular electron-hole energy distribution. Finally, when the incident photon energy is larger than the bandgap (ℏω > E g , see Figure 3.2c), one should account for both the deformation potential and thermal phonon contributions [START_REF] Ruello | Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action[END_REF][START_REF] Thomsen | Surface generation and detection of phonons by picosecond light pulses[END_REF]:

σ = σ DP + σ TE = -N d eh -3BβN (ℏω -E g ) c L (Semiconductor) , (3.12)
where

δT L = N (ℏω-Eg) c L
is the contribution to the heating of the lattice due to the intraband relaxation process. Usually, linear expansion coefficient β is positive for most solids, so incoherent phonons produce negative stress. On the other hand, the sign of d eh may be positive or negative (case of photoexcited silicon [START_REF] Wright | Acoustic generation in crystalline silicon with femtosecond optical pulses[END_REF]), so the total photoinduced strain may be the sum or difference of DP and TE contributions.

According to Eq.3.1, the non-uniform change in the internal pressure will lead to the generation of the strain pulse in the material. For the semi-infinite solids, Thomsen et al. have established the expression of the bipolar strain pulse emission from a free surface [START_REF] Thomsen | Coherent Phonon Generation and Detection by Picosecond Light Pulses[END_REF][START_REF] Thomsen | Surface generation and detection of phonons by picosecond light pulses[END_REF]. As a first approximation, if there is no fast carrier or heat diffusion process, the spatial distribution of the strain pulse is limited by the optical penetration depth of the pump [START_REF] Vitalyi | Laser Optoacoustics[END_REF]. In that case, the characteristic frequency of the emitted acoustic pulse is f = V ac /2ξ.

In addition, for a thin film with a thickness comparable with an optical penetration depth, as in the samples presented later, the photoinduced stress is defined over the entire film thickness H. The thin-film geometry restricts the spatial distribution of the photoinduced stress, which results in filtering the phonon spectrum (Fabry-Perot cavity effect) to specific values given by the equation:

f = lV ac 2H , ( 3.13)
with l, V ac , and H the mode number, the longitudinal speed of sound, and the film thickness, respectively. Apart from these breathing modes of the thin film, the other frequencies interfere destructively.

Results and discussion

So far, the generation of coherent acoustic phonons has been studied only with laser pulses in the visible, near infrared and near ultraviolet range [START_REF] Maznev | Generation of coherent phonons by coherent extreme ultraviolet radiation in a transient grating experiment[END_REF][START_REF] Ruello | Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action[END_REF]. In the meantime, it has been shown that the THz pulse can be coupled to optical phonons [START_REF] Huber | Coherent phonon spectroscopy of non-fully symmetric modes using resonant terahertz excitation[END_REF][START_REF] Maehrlein | Terahertz Sum-Frequency Excitation of a Raman-Active Phonon[END_REF], while there is no definitive report on the coherent generation of acoustic phonons with THz. Through the comparison of NIR and THz photoexcitation (1.54 eV versus 4 meV), we will attempt to identify the main mechanisms of acoustic phonon generation in metal and narrow-band semiconductor thin films. This Section is based on the already published paper "Coherent acoustic phonons generated by ultrashort terahertz pulses in nanofilms of metals and topological insulators," prepared by A. Levchuk, B. Wilk, G.

Vaudel, F. Labbé, B. Arnaud, K. Balin, J. Szade, P. Ruello, and V. Juvé [START_REF] Levchuk | Coherent acoustic phonons generated by ultrashort terahertz pulses in nanofilms of metals and topological insulators[END_REF].

Experimental details

The experiments were done at room temperature, in the ultrafast pump-probe configuration in the transmission geometry, as shown in Fig. 3.3a. As the THz source, we used the LiNbO 3 crystal in the tilted pulse front configuration discussed in the Chapter 2. In addition, two free-standing Wire Grid (WG) polarizers were introduced in the THz beam path for pump-power-dependent measurements. THz beam and THz optics characterization are presented in the Chapter 2 and Appendix A. The peak THz electric field amplitude used in the experiment was estimated to be around 275 kV/cm1 , which corresponds to a pulse energy of roughly 1.2 µJ (according to a calibrated pyroelectric detector).

In this section, we have compared THz to NIR (1.54 eV, 800 nm) pump/probe measurements.

In both experiments (THz and NIR excitation), the probe wavelength has been kept to the same energy, 3.1 eV (400 nm), to avoid spurious effects due to the detection process.

CHAPTER 3. ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ

Samples

Chromium and Aluminum films : Thin polycrystalline films were prepared by pulsedvapor deposition on a transparent mica substrate with the thicknesses of H = 14 ± 0.5 nm (Cr) and H = 20 ± 0.5 nm (Al) measured by x-ray reflectivity. Both deposition and x-ray reflectivity experiments were done at the IMMM laboratory by Dr. Mathieu Edely.

Bi 2 Te 3 thin film: Bismuth Telluride, (Bi 2 Te 3 ), is a narrow-gap semiconductor (with E g ≈ 0.2 eV [START_REF] Zhang | Topological insulators in Bi 2 Se 3, Bi 2 Te 3 and Sb 2 Te 3 with a single Dirac cone on the surface[END_REF]), known for its thermoelectric properties, with high electrical conductivity and low lattice thermal conductivity [START_REF] Satterthwaite | Electrical and thermal properties of Bi2 Te3[END_REF][START_REF] Park | Thermal and Electrical Conduction of Single-crystal Bi2Te3 Nanostructures grown using a one step process[END_REF][START_REF] Witting | The Thermoelectric Properties of Bismuth Telluride[END_REF], making it appealing for industrial applications. In addition, Bi 2 Te 3 can support unique surface conduction bands, first measured by Angle-Resolved Photoemission Spectroscopy (ARPES) by Noh et al. [START_REF] Noh | Spin-orbit interaction effect in the electronic structure of Bi 2 Te 3 observed by angle-resolved photoemission spectroscopy[END_REF], making this material a topological insulator [START_REF] Hasan | Colloquium: Topological insulators[END_REF].

As depicted in the Fig. 3.3b, topological insulators have a bulk bandgap like an ordinary insulator and so-called surface conducting Dirac states. The topological insulator nature of Bi 2 Te 3 arises from both spin-orbit interaction (Z=83 for Bi and Z=52 for Te) and time-reversal symmetry breaking at the interface [START_REF] Zhang | Topological insulators in Bi 2 Se 3, Bi 2 Te 3 and Sb 2 Te 3 with a single Dirac cone on the surface[END_REF]. However, in this thesis, we do not probe the topological order of Bi 2 Te 3 . Firstly, due to the strong n-doping of the sample (see ARPES measurement of [START_REF] Weis | Hot-carrier and BIBLIOGRAPHY optical-phonon ultrafast dynamics in the topological insulator Bi 2 Te 3 upon iron deposition on its surface[END_REF]) the Fermi level is shifted closer to the conduction band, therefore making the manipulation of Dirac states more difficult. Secondly, the visible probe has an optical penetration depth of around 9 nm [START_REF] Greenaway | Band structure of bismuth telluride, bismuth selenide and their respective alloys[END_REF] and is not sensitive to the surface states. The samples were grown by molecular beam epitaxy in the Physics Institute of Katowice, the University of Silesia, by Dr. K. Balin and Prof. J. Szade. As described in previous references [START_REF] Weis | Ultrafast lightinduced coherent optical and acoustic phonons in few quintuple layers of the topological insulator Bi2Te3[END_REF][START_REF] Weis | Hot-carrier and BIBLIOGRAPHY optical-phonon ultrafast dynamics in the topological insulator Bi 2 Te 3 upon iron deposition on its surface[END_REF][START_REF] Weis | Quantum size effect on charges and phonons ultrafast dynamics in atomically controlled nanolayers of topological insulators Bi2Te3[END_REF][START_REF] Weis | Growth and spectroscopic studies ( continuous and time-resolved ) of ultrathin films of topological insulators[END_REF], the sample is of high quality with single-crystalline growth and trigonal axis oriented perpendicular to the surface. The Bi 2 Te 3 layer was deposited onto the mica substrate.

Experimental results

In this part, we start first with experimental results on coherent acoustic phonon generation with THz and NIR ultrashort pulses in prototypical metals (Chromium and Aluminum). The second part will be an extension of this study to more complex materials such as topological insulator Bi 2 Te 3 .

Cr and Al nanofilms:

The change in the 400 nm probe beam transmission (δT (τ )/T ) as the function of the pump-probe delay for the Cr and Al thin films are displayed in Figure 3.4a and b, with both samples showing similar transient behavior. The black line in Fig. 3.4a shows the chromium thin film δT (τ )/T signal measured upon the THz excitation with pulse energy of roughly 1.2 µJ, which corresponds to peak E THz ≈ 275 kV cm -1 . The signal consists of a sharp increase in the probe transmission followed by a slower exponential decay. This exponential decay is attributed to the lattice temperature elevation (δT L , incoherent phonons) and subsequent heat diffusion [START_REF] Paddock | Transient thermoreflectance from thin metal films[END_REF][START_REF] David | Nanoscale thermal transport[END_REF][START_REF] Saito | Picosecond acoustic phonon pulse generation in nickel and chromium[END_REF][START_REF] Watabe | Optical pump-and-probe test system for thermal characterization of thin metal and phase-change films[END_REF]. On top of this thermal background, a clear oscillatory feature can be observed. To isolate only the oscillatory component, we fit the thermal background with the following function: linearly with the THz pulse energy (quadratically with the E THz amplitude). The same linear dependence is found for the NIR pump, as depicted in Fig. 3.4d.

f (τ ) = a 0 • 1 + erf τ -τ 0 s • a 1 exp τ -τ 0 τ 1 , ( 3.14)
Bi 2 Te 3 nanofilms: As the following step, we extended our study to thin Te rich and consequently n-doped Bi 2 Te 3 films with various thicknesses ranging from 4 to 25 nm. Figure 3.5a shows the signal of Bi 2 Te 3 16 nm film excited with THz and NIR pump pulse energy fixed at 1.2 µJ and 51 nJ, respectively. For both pump wavelengths, the transient optical transmission δT /T shows a sharp increase after the initial excitation and a fast recovery followed by a slower decaying signal attributed to the thermal background. We extracted the oscillatory part of the signal with a similar fit (see Eq.3.14) having two exponential decays. Fast exponential decay was found to be equal to 1.8 ps, and it is related to the hot carrier relaxation [START_REF] Wang | Measurement of Intrinsic Dirac Fermion Cooling on the Surface of the Topological Insulator Bi2Se3 Using Time-Resolved and Angle-Resolved Photoemission Spe[END_REF][START_REF] Hsieh | Selective probing of photoinduced charge and spin dynamics in the bulk and surface of a topological insulator[END_REF][START_REF] Glinka | Ultrafast carrier dynamics in thin-films of the topological insulator Bi 2 Se 3[END_REF].

When the thermal background is subtracted (see inset in Fig. 3.5a), we observe the δT /T residue modulation at two distinct frequencies: a fast oscillatory component at around 1.8 THz, which is attributed to the Raman active A 1 1g optical phonon [START_REF] Misochko | Polarization dependence of coherent phonon generation and detection in the three-dimensional topological insulator B i2 T e3[END_REF] (will be discussed in the Section 3.3.2) and a slower, with a period of about 11.6 ps (f BT = 86 GHz frequency). We report that the amplitude of the slow oscillatory component scales linearly with the THz pulse energy (see Fig. To confirm the nature of this low-frequency δT /T modulation, we start by analysing the mechanical boundary conditions, in order to establish which acoustic modes can be excited. We first note that since pump light penetration depth ξ is bigger than the film thickness H, it means that the photoinduced stress will be confined within the acoustic layer. Next, we estimate the acoustic reflection coefficient at the metal/mica interface, defined as [START_REF] Vitalyi | Laser Optoacoustics[END_REF]:

R ac = Z mica -Z layer Z mica + Z layer , ( 3.15)
where Z is the acoustic impedance :

Z = ρV ac , ( 3.16)
with the material density ρ and the acoustic velocity V ac . With the values from the literature, the acoustic impedance of all the materials chosen as the layer here (Cr, Al, Bi 2 Te 3 ) is larger than the acoustic impedance of the substrate 2 . This leads to R ac < 0 for all interfaces. The R ac < 0 value means that the acoustic wave reflected at the interface will be phase-shifted by π [START_REF] Vitalyi | Laser Optoacoustics[END_REF] after a single trip, repeating each time it bounces back and forth within the acoustic layer. This leads to the resonator-like vibration of the nanometric film at the frequency f = lV ac /2H (see Eq.3.13).

Then, the straightforward estimation of the eigenmode frequencies for all layers gives:

f Cr = 1 • 6600 m s -1 2 • 14 nm = 235.7 GHz (Experiment: 238 GHz); f Al = 1 • 6320 m s -1 2 • 20 nm = 158
GHz (Experiment: 139 GHz);

f BT = 1 • 2460 m s -1 2 • 16 nm = 77 GHz (Experiment: 86 GHz), (3.17)
and they are in a rather good agreement with experimentally measured values. The about 12 % difference for the Al thin film could be related to a decrease in V Al due to polycrystalline nature of the film, as observed before for Mg-Zn alloy for instance [START_REF] Orzi | Photoacoustic characterization of phase transitions in amorphous metal alloys[END_REF]. For the Bi 2 Te 3 , the speed of sound 2. ρ mica = 2880 kg m -3 , V mica = 4300 m s -1 , Z mica = 1.2 × 10 7 kg m -2 s -1 .

CHAPTER 3. ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ

was taken from [START_REF] Weis | Ultrafast lightinduced coherent optical and acoustic phonons in few quintuple layers of the topological insulator Bi2Te3[END_REF][START_REF] Weis | Quantum size effect on charges and phonons ultrafast dynamics in atomically controlled nanolayers of topological insulators Bi2Te3[END_REF]. In our study, the fit of acoustic period measured with THz excitation as a function of the film thickness (red dashed line in Figure 3.5d), gives the V BT = 2700 m s -1 , which deviates by less than 10 % from previous reports [START_REF] Wang | Acoustic phonon scattering in Bi2 Te3 / Sb 2 Te3 superlattices[END_REF].

Acoustic phonon generation mechanism in metals:

We continue the discussion with the pump power dependence of coherent acoustic phonon amplitude, presented in Fig. 3.4c and d. Importantly, for both THz and NIR excitation, δT /T | acoustic scales linearly with the pump pulse energy. This means that for all samples in the study, linearity of δT /T | acoustic for the THz excitation excludes highly nonlinear processes of energy deposition such as impact ionization [START_REF] Hirori | Extraordinary carrier multiplication gated by a picosecond electric field pulse[END_REF] or Zener tunneling [START_REF] Kuehn | Terahertz-induced interband tunneling of electrons in GaAs[END_REF]. With this in mind, we can now estimate the absorbed energy density in the case of THz and NIR excitation within the usual dielectric response approach.

With the THz time-domain spectroscopy, we have first measured the complex dielectric function of the 14 nm thick Cr film in the 0.2-2.5 THz frequency range [START_REF] Laman | Terahertz conductivity of thin metal films[END_REF], shown in Figure 3.6a. By fitting the measured data with a simple Drude model, we extract Chromium plasma frequency ω p ≈ 350 THz and the electronic scattering rate τ e ≈ 14 THz. These parameters lead to a nearlyfrequency-independent 3 conductivity σ Cr ≈ 4.5 × 10 5 Ω m -1 . In addition, as the characteristic scattering frequency in metals is typically well above the frequency of our driving THz field [START_REF] Aleksandar | Optical properties of metallic films for vertical-cavity optoelectronic devices[END_REF], we limit our model to the stationary drift electron transport regime and neglect any ballistic transport effects. These two assumptions allow us to ignore the E THz retardation effects, and we can write the time-dependent current density associated to the electron acceleration in the presence of the local THz pulse as:

j(t ′ ) = σ t ′ -∞ E loc THz (t ′′ )dt ′′ . (3.18)
Here, E loc THz (t) is the local THz electric field acting on the electrons, and σ is the frequencyindependent metal conductivity. Then, we can estimate the temporal profile of the absorbed energy with the ultrafast analogy of the well-known Joule effect:

W Abs. THz (t) = t -∞ E loc THz (t ′ ) • ⃗ j(t ′ )dt ′ = σ t -∞ E loc THz (t ′ ) 2 dt ′ . (3.19)
In the case of THz excitation, since our samples are much thinner than the driving field's wavelength, the local electric field can be approximated by the electric field inside the sample E loc THz . Prof. Brice Arnaud (Theory group at IMMM) has implemented the finite difference time domain (FDTD) method to simulate the electric field value in the sample and the substrate:

E loc THz (t) = s • E THz (t), (3.20)
with E THz (t) being the vacuum THz electric field and s ≈ 0.45 is the electric field attenuation coefficient, estimated from FDTD. Using Eq.3.19 and the experimental parameters, for the highest experimental THz field value (roughly 275 kV cm -1 , pulse energy of about 1.2 µJ) we can now estimate the total absorbed energy density, which becomes W Abs. THz ≈ 52 J cm -3 . Using the value of Chromium heat capacity from the literature (c L ≈ 3 × 10 6 J m -3 ), we can finally calculate the lattice temperature increase via the ultrafast THz Joule effect:

δT L (THz) = W Abs THz c L ≈ 17 K. (3.21)
In contrast to the THz excitation, the absorbed energy induced by NIR excitation is mediated by the direct optical transitions, which gives:

W Abs NIR = AN ℏω NIR V , ( 3.22)
with A being the optical absorption coefficient, N is the number of incident photons of energy hν NIR , and the excited volume V . The NIR absorption coefficient A was modeled by the transfer matrix method using tabulated optical constants [START_REF] Aleksandar | Optical properties of metallic films for vertical-cavity optoelectronic devices[END_REF]. Remarkably, for the NIR pulse energy of 0.07 µJ (see Fig. 3.4a) we obtain an absorbed energy of W Abs NIR ≈ 53 J cm -3 , equivalent to the one obtained with Eq.3.21, for the THz pulse energy of 1.2 µJ. Consequently, for both scenarios (NIR and THz excitation), the maximum lattice temperature increase is estimated around δT L ≈ 17 K. Figure 3.6b shows that this equivalence is valid over the entire absorbed energy range under investigation: the acoustic phonon amplitude is driven solely by the total absorbed energy and is not dependent on the photon energy of the excitation pulse. This indicates that the amplitude of the acoustic phonon is not drastically dependent on the pathway of energy relaxation from the electron to the phonon subsystem (which is supposed to be different for a NIR and THz excitation) but only on the total absorbed energy by the lattice. In order to better understand this result, it is useful to study the spectral components of the photoinduced stress. For this purpose, Prof. Brice Arnaud has computed, using the ab initio DFT calculation, the lattice and electron heat capacities for the 20 nm thick Al film. Then, the phenomenological two-temperature model, with the computed heat capacities allows us to estimate the dynamics of the electron and lattice pressure. In the case of the NIR excitation, Figure 3.7a

shows that the lattice stress contribution (σ TE , blue line) dominates the electron stress contribution (σ DP , red line), meaning that acoustic waves are indeed mainly driven by thermoelastic stress (σ TE ). Assuming that the thermoelastic stress is one of the possible mechanisms at play for the THz excitation, we can estimate the approximate shape of the σ TE using the phenomenological TTM model. To do so, we calculate the transient shape of the Joule effect with Eq.3.21, which gives us the associated change in the thermoelastic stress, depicted as the blue curve in Fig. 3.7b.

Here, for the same amount of absorbed energy, i.e. W Abs NIR = W Abs THz , we arrive to the same maximum value of σ TE as shown in Fig. 3.7a and b. We immediately see that the time evolution of the lattice stress around the time-delay zero is very distinct for the THz and NIR excitation. This is somewhat expected, since the interaction time, defined by the NIR and THz pump pulse duration, differs by close to one order of magnitude. We thus perform a Fast Fourier transform of σ TE , CHAPTER 3. ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ and restrict ourselves to the relevant coherent acoustic phonon frequency range (up to 500 GHz), displayed in Fig. 3.7c. Interestingly, we observe that the relevant ratio of the NIR and THz phonon spectral components (see blue line in Fig. 3.7b) is close to 1 in the region of interest. This perfectly confirms what we discussed before: for this rather low acoustic frequency, the more rapid lattice heating with a NIR compared to THz pulse is not important as soon as the characteristic rise of the σ TE (either for NIR or THZ excitation) is much shorter that the acoustic phonon period of interest.

As a short summary, the analysis of the pump fluence dependence, the existence of the linear response, the presentation thermoelastic stress based on the TTM model, including the discussion of the spectrum of the photoinduced stress, strongly suggest that the lattice heating, as a result of ultrafast Joule effect, is the driving mechanism of the coherent acoustic phonon generation in thin metallic films with THz.

Acoustic phonon generation mechanism in Bi 2 Te 3 : Concerning Bi 2 Te 3 , the linear increase in both thermal background (not shown here) and δT /T | acoustic in the function of the THz pulse energy strongly suggests that we can rule out highly-nonlinear light-excitation processes such as Zener tunneling [START_REF] Kuehn | Terahertz-induced interband tunneling of electrons in GaAs[END_REF] and impact ionization [START_REF] Hirori | Extraordinary carrier multiplication gated by a picosecond electric field pulse[END_REF] similarly to the observations we made in the previous subsection. Since the Bi 2 Te 3 bandgap is in the order of 200 meV [START_REF] Rapacz | Spectroscopic characterization of high-purity polycrystalline Bi-Te films grown by thermal evaporation[END_REF], we can exclude the multiphoton absorption and excitation process as well (E g ≈ 50ℏΩ THz ). However, it is worth to note that the recent ARPES measurements of 15 nm Bi 2 Te 3 film revealed a non-zero signal at the Fermi level [START_REF] Weis | Hot-carrier and BIBLIOGRAPHY optical-phonon ultrafast dynamics in the topological insulator Bi 2 Te 3 upon iron deposition on its surface[END_REF] (see Fig. 3.8b), and despite our sample was studied in air (passivated surface [START_REF] Weis | Hot-carrier and BIBLIOGRAPHY optical-phonon ultrafast dynamics in the topological insulator Bi 2 Te 3 upon iron deposition on its surface[END_REF]), our Bi 2 Te 3 could be very close to the metallic state. Moreover, we measure the nearly identical long-term residual signal (see Fig. 3.5a and the inset) extracted with the same fit parameters for the THz and NIR excitation. This may indicate a similar coherent acoustic phonon generation mechanism for two different pump photon energies, either originating from thermoelastic, deformation potential stress, or a combination of two. Notably, the evolution of the phase of the coherent acoustic phonons excited in Bi 2 Te 3 as a function of the sample thicknesses (see Fig. 3.5c) might be related to differences in the driving excitation process [START_REF] Voisin | Time-resolved investigation of the vibrational dynamics of metal nanoparticles[END_REF] due to the change in the electronic band structure for the thin samples with, in particular, some confinement effects observed below around 6 nm [START_REF] Hirori | Extraordinary carrier multiplication gated by a picosecond electric field pulse[END_REF][START_REF] Weis | Quantum size effect on charges and phonons ultrafast dynamics in atomically controlled nanolayers of topological insulators Bi2Te3[END_REF]. Due to the complexity of electron-phonon coupling in this correlated material, these assumptions about coherent acoustic phonon generation in Bi 2 Te 3 need to be supported with first-principle calculations.

Outlook: As a final element of discussion, it worth to tell that we can expect some potential differences between NIR and THz excitation in metallic nanostructures through the thermoelastic process at least. As an illustration, Figure 3.8b reveals a dominating thermoelastic contribution driven by Thz excitation within 1-2 THz range. That larger contribution from THz excitation at around 1.5 THz is directly related to the spectrum of the THz pulse. Therefore, one can imagine that THz excitation of high-frequency acoustic modes may reveal the very different nature of heating based either on intraband relaxation (with NIR excitation) or by multiple electron scattering at the Fermi level (with THz excitation). This makes it interesting to study the THz excitation of metallic nanoparticles of various sizes, as one can tune their natural vibrations to THz frequencies [START_REF] Juvé | Probing Elasticity at the Nanoscale: Terahertz Acoustic Vibration of Small Metal Nanoparticles[END_REF].

Conclusions

To conclude, we have evidenced the generation of coherent acoustic phonons in metallic and topological insulator thin films with pulsed THz radiation. The coupling efficiency is quadratic in the THz electric field strength (linear with the pulse energy) within the THz-power range of investigation. Remarkably, for both NIR and THz excitations of thin films of metals, the similar amount of light absorbed energy results in the nearly-identical contributions of the coherent acoustic phonons and the incoherent phonons to the experimental signal, as seen from Figure 3.6b.

This direct comparison between the NIR and THz pulsed excitation, despite the well-distinguished microscopic lattice "heating" mechanisms, allows us to demonstrate that the thermoelastic process is the main driving mechanism of coherent acoustic phonon generation in metals of Aluminum and Chromium. While for NIR excitation, the ultrafast lattice heating is achieved by intraband hot electron-phonon thermalization, but when the THz pulse drives the system, multiple electron-phonon scattering processes in the vicinity of the Fermi level are responsible for the lattice temperature increase (ultrafast Joule effect). Although the relaxation channels are different, we demonstrate that they do not impact the coherent acoustic phonon generation for the lowfrequency interest range (<500 GHz).

Regarding the nanofilms of n-doped Bi 2 Te 3 topological insulator, at this moment, we cannot conclude about the physical origin of the light-induced coherent acoustic phonon generation. The linear pump fluence dependence might indicate at least a possible contribution of the thermoelastic (coming from free carriers in this n-doped sample) and/or deformation potential mechanisms. In contrary to metals, the existence of long-lived carriers due to the bandgap (even in the range of ≈ 200 meV) and the rather large deformation potential constant [START_REF] Weis | Ultrafast lightinduced coherent optical and acoustic phonons in few quintuple layers of the topological insulator Bi2Te3[END_REF][START_REF] Weis | Quantum size effect on charges and phonons ultrafast dynamics in atomically controlled nanolayers of topological insulators Bi2Te3[END_REF] could lead indeed to an important contribution of non-thermal deformation potential stress ∆σ DP to the total ∆σ PI . In the future, in parallel with the first principle calculations needed to clarify the physics in this complex material, we plan to perform more experiments. For example, studying Bi 2 Te 3 film with various doping and different Fermi level positions should help us disentangle possible acoustic phonon WITH THZ generation mechanisms at play. In addition, ultrafast THz experiments at low-temperature could also be beneficial, as they will reduce the amount of thermally activated carriers in the conduction band of Bi 2 Te 3 .

In the future, the generation of coherent acoustic phonons by an ultrashort THz pulse could open other fundamental opportunities for controlling the macroscopic pressure and even be used to probe the elasticity or viscoelasticity in the soft matter. Moreover, as the new research direction in our group, the understanding of the coupling between the THz and the large internal polarization in multiferroic materials may be crucial for the efficient control of the ferroic order.

Coherent optical phonon generation in Bi 2 Te 3 with pulsed

THz radiation

This section will focus on the discussion about the possible mechanisms of coherent A 1 1g optical phonon generation in 16 nm thick Bi 2 Te 3 film with intense THz pulses. First, we will briefly introduce different theoretical models of coherent optical phonon excitation with pulsed light radiation.

Then, after the description of optical phonon properties of Bi 2 Te 3 measured by Raman scattering in general, we will present the experimental results where we will show the A 1 1g mode generation efficiency dependence on E THz field amplitude and THz pump pulse central frequency, to finally discuss the possible scenarios which may lead to coherent A 1 1g phonon excitation.

Theoretical background

Phenomenologically, the motion of coherent optical phonon can be modeled as the lattice response on the specific driving force, usually shorter than the inverse of the coherent phonon period of interest [START_REF] Först | Coherent Phonons in Bulk and Low-Dimensional Semiconductors[END_REF]. For example, one can use the classical equation of motion of the force-driven damped harmonic oscillator to model the displacement of coherent phonons in the lattice [START_REF] Först | Coherent Phonons in Bulk and Low-Dimensional Semiconductors[END_REF][START_REF] Dekorsy | Coherent phonons in condensed media[END_REF][START_REF] Subedi | Theory of nonlinear phononics for coherent light control of solids[END_REF]:

∂ 2 Q α (t) ∂t 2 + 2γ α ∂Q α (t) ∂t + ∂V (Q) ∂Q α = F Q α (t), (3.23)
where V (Q) is the total potential energy of the lattice, Q α is the normal mode coordinate 4 , γ α and Ω α are the phenomenological damping constant and frequency of the phonon mode α.

The driving force F Q α (t) can originate either from phonon-EM wave interaction (IR absorption or Raman scattering) or have the displacive nature due to immediate modification of the electronic distribution [START_REF] Merlin | Generating coherent THz phonons with light pulses[END_REF][START_REF] Cheng | Mechanism for displacive excitation of coherent phonons in Sb, Bi, Te, and Ti2O3[END_REF]. For example, NIR-VIS-NUV excitation of opaque materials can lead to a Displacive Excitation of Coherent Phonons (DECP), mediated by the deformation potential stress [START_REF] Kuznetsov | Theory of coherent phonon oscillations in semiconductors[END_REF].

The internal force ("harmonic and anharmonic restoring forces") is defined as the gradient of the lattice (phonon) potential energy V (Q). The V (Q) can be expressed as the polynomial expansion of the all phonon modes that exist in a material, as [START_REF] Subedi | Theory of nonlinear phononics for coherent light control of solids[END_REF][START_REF] Dominik | Coherent Optical Phononics[END_REF]:

V (Q) = α Ω 2 α 2 Q 2 α + α,β,γ a αβγ Q α Q β Q γ + α,β,γ,δ b αβγδ Q α Q β Q γ Q δ + . . . , (3.24)
where the first term is the harmonic phonon potential, and other terms describe the higher-order phonon anharmonicities (i.e., the nonlinear phonon-phonon interaction), and a αβγ and b αβγδ are the corresponding anharmonic coefficients. The subscripts α, β, γ, etc., denote the different lattice eigenmodes. For example, when α = β = γ, c ααα Q 3 α is the anharmonic potential of the phonon mode α. For α = β ̸ = γ, c ααγ Q 2 α Q γ is the anharmonic coupling between phonon modes α and γ. However, some specific anharmonic interaction terms between IR and Raman phonon modes can be forbidden due to the crystal symmetry [START_REF] Först | Mode-Selective Control of the Crystal Lattice[END_REF][START_REF] Först | Nonlinear phononics as an ultrafast route to lattice control[END_REF].

"Conventional" phononics: stimulated IR and Raman active modes

The lowest-order allowed electric field-phonon interaction is the absorption of the EM radiation by the IR active WITH THZ phonon mode. In this case, the IR mode equation of motion in the linear regime will read [START_REF] Först | Nonlinear phononics as an ultrafast route to lattice control[END_REF]:

∂ 2 Q I R(t) ∂t 2 + 2γ IR ∂Q IR (t) ∂t + Ω 2 IR Q IR (t) = Z * IR E(t), (3.25)
where Z * IR is the effective charge of IR active mode, and E(t) is the incident electric field. A classical example of such coupling is the phonon-polariton interaction, described in the Section 1.3.1.

The lowest-order excitation of the Raman-active phonon, due to its zero effective charge (Z * R = 0), can be mediated via the two-photon Raman process [START_REF] Dekorsy | Coherent phonons in condensed media[END_REF]. In this case, the Raman active phonon driving force is given as the second-order interaction of the electromagnetic field E and the Raman tensor R jk [START_REF] Först | Coherent Phonons in Bulk and Low-Dimensional Semiconductors[END_REF]:

F Q R (t) = R jk E j E k , (3.26)
where E j , E k are the field components of the incident electromagnetic pulse. The latter equation is sometimes expressed in terms of Raman polarizability tensor [START_REF] Dekorsy | Coherent phonons in condensed media[END_REF]. The Raman tensor of a given phonon mode is defined as [START_REF] Dominik | Coherent Optical Phononics[END_REF][START_REF] Porezag | Infrared intensities and Raman-scattering activities within density-functional theory[END_REF]:

R jk = V c ∂χ jk ∂Q R , ( 3.27)
where χ jk is the linear electric susceptibility. In this case, coherent phonon excitation is driven by the impulsive stimulated Raman Scattering (ISRS), which is possible due to the energy and wavevector conservation rules discussed before in Chapter 1. The driven, damped equation of motion for the Raman active phonon will be equivalent to Eq.3.25, but with the following driving force [START_REF] Maehrlein | Terahertz Sum-Frequency Excitation of a Raman-Active Phonon[END_REF]:

F (Q R ) ∝ ∂χ ∂Q R E 2 (t). (3.28)
One should note that this force-term, proportional to E 2 (t), consists of both sum-and differencefrequency components of the incident electric field E(t). For example, the difference frequency components of sub-picosecond VIS-NIR pulses are linked to the ISRS excitation of coherent phonons in transparent materials (see Fig. 3.9a) [START_REF] Yong Xin Yan | Impulsive stimulated scattering: General importance in femtosecond laser pulse interactions with matter, and spectroscopic applications[END_REF][START_REF] Cheng | Mechanism for displacive excitation of coherent phonons in Sb, Bi, Te, and Ti2O3[END_REF][START_REF] Cheng | Impulsive excitation of coherent phonons observed in reflection in bismuth and antimony[END_REF]. In the case of an electric field in the THz frequency range, if the sum-frequency component is resonant with a Raman-active phonon, excitation can occur via photonic THz sum-frequency excitation process [START_REF] Maehrlein | Terahertz Sum-Frequency Excitation of a Raman-Active Phonon[END_REF], depicted in the Figure 3.9b.

Nonlinear phononics: coupled IR and Raman active modes Notably, a population decay of optical phonons is believed to be dominated by lattice anharmonicity and can lead to the generation of lower-energy acoustic and sometimes optical phonon modes [START_REF] Först | Coherent Phonons in Bulk and Low-Dimensional Semiconductors[END_REF]. About forty years ago, it was proposed that excitation of an infrared-active phonon could serve as the intermediate state for Raman scattering through a process that relies on lattice anharmonicities rather than second-order electron-phonon interactions [START_REF] Nicoletti | Nonlinear light-matter interaction at terahertz frequencies[END_REF][START_REF] Mills | Ionic contributions to the Raman tensor of insulators[END_REF][START_REF] Subedi | Theory of nonlinear phononics for coherent light control of solids[END_REF]. As an example, let us expand the Eq. anharmonic terms [START_REF] Först | Mode-Selective Control of the Crystal Lattice[END_REF]:

V (Q R , Q IR ) N L = 1 2 Ω 2 R Q 2 R + 1 2 Ω 2 IR Q 2 IR + a 12 Q IR Q 2 R + a 21 Q 2 IR Q R , ( 3.29)
where a 12 and a 21 are anharmonic coupling constants. It should be noted, that a 12 Q IR Q 2 R term is zero in centrosymmetric crystals, like in the sample of study, Bi 2 Te 3 . With this nonlinear phonon potential, the coupled equations of motion for the centrosymmetric medium [START_REF] Först | Mode-Selective Control of the Crystal Lattice[END_REF][START_REF] Nicoletti | Nonlinear light-matter interaction at terahertz frequencies[END_REF][START_REF] Dominik | Sum-frequency ionic Raman scattering[END_REF]:

∂ 2 Q IR ∂t 2 + 2γ IR ∂Q IR ∂t + Ω 2 IR Q IR = 2a 21 Q IR Q R + Z * IR E THz , ∂ 2 Q R ∂t 2 + 2γ R ∂Q R ∂t + Ω 2 R Q R = a 21 Q 2 IR , (3.30)
where the time dependence of Q R , Q IR and E THz was omitted for the clarity of the notation.

As a result, strong nonlinear coupling between IR and Raman modes may "shift" or displace the energy potential of a Raman mode Q R , as shown in Figure 3.10a. This will result in simultaneous lattice oscillations along the infrared coordinate Q IR (red curve in Fig. 3.10b), extorting a displacive force on the Raman active phonon coordinate Q R (blue curve in Fig. 3.10b). As a result, the forceterm driving the Q R coherent phonon is proportional to [START_REF] Dominik | Sum-frequency ionic Raman scattering[END_REF][START_REF] Först | Nonlinear phononics as an ultrafast route to lattice control[END_REF]:

F (Q R ) ∝ a 21 Q 2 IR ∝ E 2 . (3.31)
Importantly, for both Raman excitation [START_REF] Maehrlein | Terahertz Sum-Frequency Excitation of a Raman-Active Phonon[END_REF] and anharmonic IR-Raman coupling [START_REF] Melnikov | Coherent phonons in a Bi2Se3 film generated by an intense single-cycle THz pulse[END_REF][START_REF] Dominik | Sum-frequency ionic Raman scattering[END_REF] (see Figure 3.9b), the effective driving force should be proportional to the square of the incident EM field (E 2 ). Note also that according to Eq. 3.29, due to large coupling with a Raman active mode R, the IR mode frequency is re-normalized to Ω 2 IR -2a 21 Q R .

CHAPTER 3. ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ

Interestingly, since the majority of coherent optical modes exist within the 1-40 THz frequency range, pulsed, broadband THz radiation can resonantly or non-resonantly couple to the optical vibrational modes. For example, there have been several observations of THz-driven anharmonic coupling between optical phonons in manganite PrMnO 3 (PMO) [START_REF] Rini | Control of the electronic phase of a manganite by mode-selective vibrational excitation[END_REF] 5 , LaSrMnO 3 [START_REF] Först | Nonlinear phononics as an ultrafast route to lattice control[END_REF] and Bismuth Selenide (Bi 2 Se 3 ) [START_REF] Melnikov | Coherent phonons in a Bi2Se3 film generated by an intense single-cycle THz pulse[END_REF][START_REF] Melnikov | Phonon-driven ultrafast symmetry lowering in a Bi2Se3 crystal[END_REF]. In addition, there are experimental [START_REF] Maehrlein | Terahertz Sum-Frequency Excitation of a Raman-Active Phonon[END_REF] and theoretical [START_REF] Dominik | Sum-frequency ionic Raman scattering[END_REF] reports of the impulsive Raman THz sum-frequency excitation of coherent phonons in the diamond.

Before proceeding to the experimental section discussing the optical phonon generation with pulsed THz radiation, let us recall the phonon properties of the Bi 2 Te 3 .

Bi 2 Te 3 coherent optical phonon properties

Bi 2 Te 3 is a trigonal rhombohedral crystal with five atoms in a primitive unit cell, and it belongs to the D 5 3d (R 3m) space group. The Bi 2 Te 3 crystallographic structure is usually described within the hexagonal unit cell along the [START_REF] Lejman | Ultrafast acousto-optic mode conversion in optically birefringent ferroelectrics[END_REF] direction, consisting of 3 quintuple layers (QL) of Te 1 -Bi-Te 2 -Bi-Te 1 atoms 6 . The Bi 2 Te 3 quintuple layers are linked with Van der Waals bonds. According to the group theory, Bi 2 Te 3 can support 15 normal phonon modes at the zone center Γ [START_REF] Misochko | Polarization dependence of coherent phonon generation and detection in the three-dimensional topological insulator B i2 T e3[END_REF][START_REF] Kullmann | Effect of Hydrostatic and Uniaxial Pressure on Structural Properties and Raman Active Lattice Vibrations in Bi2Te3[END_REF]:

Γ = 2A 1g ⊕ 2E g ⊕ 3A u ⊕ 3E u , (3.32)
where two E u and one A u are acoustic transverse and longitudinal modes, respectively. This leaves a total of 12 optical modes, shown in Fig. 3.11a. Since Bi 2 Te 3 is a centrosymmetric crystal, two A 1g and two doubly-degenerate E g optical modes are Raman active [START_REF] Richter | A Raman and far-infrared investigation of phonons in the rhombohedral V2-VI3 compounds Bi2Te3, Bi2Se3, Sb2Te3 and Bi2(Te1-xSex)3, (Bi1-ySby)[END_REF], when two A 1u and two doubly-degenerate E u are the IR active modes [START_REF] Wang | Origin of coherent phonons in Bi2Te3 excited by ultrafast laser pulses[END_REF]. The Raman tensors of A 1g and E g modes are defined as [START_REF] Misochko | Polarization dependence of coherent phonon generation and detection in the three-dimensional topological insulator B i2 T e3[END_REF]:

A 1g =     a a b     , E g =     -c -d -c -d     , and     c -c d d     .
(3.33)

The Table 3 After [START_REF] Wang | Origin of coherent phonons in Bi2Te3 excited by ultrafast laser pulses[END_REF]. b) Raman spectra of Bi2Te3 in backscattering geometry in parallel (VV) and crossed (VH) polarization of incident and scattered electric field, respectively. After [START_REF] Misochko | Polarization dependence of coherent phonon generation and detection in the three-dimensional topological insulator B i2 T e3[END_REF]. Subscripts u and g correspond to the Raman (in-plane) and IR (out-of-plane) lattice vibration modes, respectively. After [START_REF] Kullmann | Effect of Hydrostatic and Uniaxial Pressure on Structural Properties and Raman Active Lattice Vibrations in Bi2Te3[END_REF][START_REF] Richter | A Raman and far-infrared investigation of phonons in the rhombohedral V2-VI3 compounds Bi2Te3, Bi2Se3, Sb2Te3 and Bi2(Te1-xSex)3, (Bi1-ySby)[END_REF][START_REF] Wang | Origin of coherent phonons in Bi2Te3 excited by ultrafast laser pulses[END_REF].

Results and discussion

Experimental results

Before discussing the experimental results on coherent optical phonon generation with THz pulses in 16 nm thick Bi 2 Te 3 film, let us first define the time delay zero of the experiment. For this, we first fit the electrooptic trace of E THz of the LiNbO 3 source 7 with the following equation:

E THz (τ ) = E 0 exp - 4 ln 2τ 2 σ 2 FWHM sin[Ω 0 τ + ϕ 0 ], (3.35)
where E 0 is the value of the peak electric field, σ FWHM is the pulse width at FWHM, Ω 0 is the angular THz central frequency, and ϕ 0 is the carrier-envelope phase (CEP). Figure 3.12a shows the THz pulse waveform E THz (τ ) (line-dot) and its fit (solid line) with the following parameters:

σ FWHM = 1.317 ps, E 0 ≈ 275 kV cm -1 , Ω 0 /2π = 0.64 THz and ϕ 0 = 0.12π. Next, with the Eq.3.19 we can calculate the approximate shape of W abs (τ ):

W abs (τ ) ∝ τ -∞ E 2 THz (t ′ )dt ′ . (3.36)
Both E 2 THz (τ ) and calculated W abs (τ ) are shown in the Fig. 3.12b (blue and red dash-dot lines, respectively). In this case, Bi 2 Te 3 conductivity in the THz range, electron-electron, and electronphonon scattering rates, that all may affect the shape of W abs (τ ) were not taken into account.

In particular, such calculation of W abs assumes an immediate dissipation of the electromagnetic energy. Finally, we manually superpose the measured signal δT /T with calculated W abs (τ ) profile as shown in the Fig. 3.12b, which gives us the time-delay zero position of the experiment.

In the following, we will focus only on the optical phonon part of the signal, extracted by fitting the acoustic mode, electronic and thermal background with the following function: where τ 1 , τ 2 and τ ac are the characteristic decay times, with their respective amplitudes a 1 ,a 2 and a ac , and s is linked to the rising-time of the signal 8 . Ω ac and ϕ ac are the angular frequency and the phase of breathing acoustic phonon mode. The background fits for NIR and THz excitation are shown as the blue solid line in Figure 3.13a. In the following, we define δT /T | optic as the difference between "raw" measured signal and the background fit, as:

δT /T | bg =A • 1 + erf τ s • a 1 exp - τ τ 1 + a 2 exp - τ τ 2 + + a ac cos[Ω ac (τ ) + ϕ ac ] exp - τ τ ac , ( 3
δT /T | optic = δT /T -δT /T | bg . (3.38)
The Figure 3.13b displays δT /T | optic of the sample, excited with THz (black) and 800 nm (red) pulses. Interestingly, for both THz and NIR (800 nm) photoexcitation we see the signal modulation with approximately 540 fs period (1.85 THz frequency), attributed to the Raman active optical phonon mode A 1 1g (see Table 3.1). The observation of the A 1 1g mode with NIR pump is also consistent with previous measurements in the group [START_REF] Weis | Ultrafast lightinduced coherent optical and acoustic phonons in few quintuple layers of the topological insulator Bi2Te3[END_REF]. Moreover, the FFT spectrum of the signal recorded with the NIR pump (red curve in Figure 3.13c) displays a shallow signature of the A 2 1g mode around 4.1 THz, which was already observed before [START_REF] Misochko | Polarization dependence of coherent phonon generation and detection in the three-dimensional topological insulator B i2 T e3[END_REF][START_REF] Wang | Origin of coherent phonons in Bi2Te3 excited by ultrafast laser pulses[END_REF]. In our case, such low detected A 2 1g amplitude may be related to pump and probe pulse duration of about 160 fs at FWHM, insufficient to properly sample A 2 1g mode with a period of 240 fs [START_REF] Shannon | Commnunication theory in the presence of noise[END_REF]. In addition, since coherent phonon generation in Bi 2 Te 3 with pulsed NIR radiation using both isotropic and anisotropic detection techniques was already presented elsewhere [START_REF] Misochko | Polarization dependence of coherent phonon generation and detection in the three-dimensional topological insulator B i2 T e3[END_REF][START_REF] Wang | Origin of coherent phonons in Bi2Te3 excited by ultrafast laser pulses[END_REF], we will focus on the results of optical phonon generation with THz pulses only.

It is worth to underline that we can excite the 1.85 THz coherent motion with a THz pulse having a central frequency of 0.64 THz indicating the non-linear nature of the phonon generation process.

In addition, another noticeable feature arising from THz photoexcitation is 1.4 THz "sideband" (see black line in Fig. 3.13b). Such THz-pump characteristic "sideband" was also observed at the short time-delays by Melnikov et al. in Bi 2 Se 3 [START_REF] Melnikov | Coherent phonons in a Bi2Se3 film generated by an intense single-cycle THz pulse[END_REF], and by Huber et. al. in Te [START_REF] Huber | Coherent phonon spectroscopy of non-fully symmetric modes using resonant terahertz excitation[END_REF]. To investigate the physical mechanism of A 1 1g phonon generation and the origin of this "sideband", we studied the influence of the THz-field amplitude and THz-central frequency on the overall shape of δT /T | optic and amplitude of A 1 1g phonon.

First, we have measured the E THz field amplitude -A 1 1g phonon amplitude dependence. As seen from Figure 3.14b, two measurements overlap perfectly, meaning that A 1 1g amplitude does not depend on the THz pump polarization. This finding is in good agreement with previous reports [START_REF] Misochko | Polarization dependence of coherent phonon generation and detection in the three-dimensional topological insulator B i2 T e3[END_REF]. In addition, A 1 1g amplitude scales with the square of the THz field (δT /T | optic ∝ E 2 THz ), underlining the nonlinear mechanism of this Raman active excitation.

Next, we recorded the δT /T | optic for slightly different E THz pump spectrum. In order to tune the THz central frequency Ω 0 /2π from ≈ 0.52 and up to ≈ 0.73 THz, we have used the x-cut Quartz plate, as discussed in the Section 2.5. Figure 3.15a shows measured signals for Ω 0 /2π = 0.52, 0.63 and 0.73 THz. We see that both amplitude of A 1 1g phonon and signal around τ = 0 is strongly affected by the E THz change in Ω 0 /2π. The Fig. 3.15b displays the FFT δT /T | optic signals discussed above. We observe that both A 1 1g amplitude and characteristic to THz excitation "sideband" are strongly correlated to the incident E THz spectrum (violet dashed line in Figure 3.15b). This sideband seems to be related to the sumfrequency component of E THz and completely disappears when the FFT is performed on the signal for > 1 ps time delays (see dash-dot lines in Fig. 3.15b) [START_REF] Huber | Coherent phonon spectroscopy of non-fully symmetric modes using resonant terahertz excitation[END_REF]. Indeed, as presented in Appendix B, we find the correlation between the "sideband" and E THz SF component for all signals recorded 1 THz E 2 g phonon mode (see Table 3.1). Notably, close to the natural frequency of the E 2 g mode, we see again this asymmetric "sideband", which corresponds to δT /T | optic modulation at ≈4Ω 0 /2π, indicating the possible of higher-than-two order ion-electric field interaction (i.e., a fourth harmonic of the E THz component, see Eq.3.28). Figure 3.16d shows the FFT of the same signals, but for the time delays >1.5 ps. We see that neither E 1 g nor E 1 u phonons were detected. This absence of E 1 g mode in the anisotropic spectrum could be attributed to a non-optimal angle between the THz pump/probe polarization with respect to Bi 2 Te 3 azimuth angle, as was observed for the NIR excitation of Bi 2 Te 3 [START_REF] Misochko | Polarization dependence of coherent phonon generation and detection in the three-dimensional topological insulator B i2 T e3[END_REF] and a similar system, Bi 2 Se 3 [START_REF] Melnikov | Coherent phonons in a Bi2Se3 film generated by an intense single-cycle THz pulse[END_REF]. In principle, the signatures of the E u modes can be revealed with THz-TDS. However, since the sensitivity of the THz-TDS depends on the sample absorption and the volume of interaction, THz transmission of the 16 nm thick Bi 2 Te 3 film, compared to 490 nm thick one [START_REF] Kovalev | Terahertz signatures of ultrafast Dirac fermion relaxation at the surface of topological insulators[END_REF] gives no signature of IR absorption (not shown here). Finally, the study of the E g and E u modes are to be continued in the future. In the following, we concentrate on the discussion of the generation mechanisms of the largest detected optical phonon mode in Bi 2 Te 3 , i.e., A 1 1g .

Discussion

In the section above, employing isotropic detection scheme, we detected the A 1 1g coherent Raman phonon in 16 nm thick Bi 2 Te 3 , excited by THz pulses with central frequency Ω 0 /2π from ≈ 0.53 THz and up to ≈ 0.73 THz. For the fixed Ω 0 /2π ≈ 0.64 THz we have shown that the amplitude of the A 1 1g phonon seems to scale with E 2 THz , and is not affected by the polarization of the THz pulse (according to Fig. 3.14b). Next, Figure 3.15c shows that the amplitude of A 1 1g phonon is sensitive to the change of the THz pump central frequency and can be increased by a factor of 3, which is not fully explained by the increase in THz power (see Fig. 3.15b). The latter observation underlines the crucial role of the THz spectrum on the non-linear process efficiency. Moreover, preliminary anisotropic detection experiments revealed the presence of E 2 g phonon, detectable at >1.5 ps time delays. Both E 1 u and E 1 g phonons were not observed, most likely due to nonoptimal experimental conditions. Note, since the strongest A 1 1g mode is Raman active, the resonant excitation (Q A 1 1g ∝ E THz ) is forbidden by the means of symmetry.

We first briefly compare our work to previously published results on coherent phonon excitation with THz. First, T. Huber et al. reported the resonant THz excitation of non-symmetric TO phonons in Tellurium (Te) [START_REF] Huber | Coherent phonon spectroscopy of non-fully symmetric modes using resonant terahertz excitation[END_REF]. Then, A. Melnikov et al. proposed THz-ionic Raman scattering mechanism of E 2 g phonon excitation in Bi 2 Se 3 (THz-IRS, depicted in the Fig. 3.17a) [START_REF] Melnikov | Coherent phonons in a Bi2Se3 film generated by an intense single-cycle THz pulse[END_REF][START_REF] Melnikov | Phonon-driven ultrafast symmetry lowering in a Bi2Se3 crystal[END_REF]. In addition, the nonlinear phononics chapter in the L. Braun's Ph.D. thesis discusses E 2 g coherent phonon generation in Bi 2 Te 3 and Sb 2 Te 3 [START_REF] Braun | Electron and Phonon Dynamics in Topological Insulators at THz Frequencies[END_REF], suggesting the displacive THz-IRS mechanism of E 2 g phonon generation. Next, S. Maerhrlein et al. reported the generation of F 2g 40 THz Raman active phonon in Diamond via an impulsive THz-sum frequency excitation (THz-SFE) mechanism (see Fig. 3.17a) [START_REF] Maehrlein | Terahertz Sum-Frequency Excitation of a Raman-Active Phonon[END_REF]. They observe the maximum of F 2g phonon amplitude for THz pulse central frequency equal to the half of natural frequency of F 2g mode (i.e. for Ω 0 /2π = 2 × Ω F2g ). As a result, in the absorbing, semi-metallic Bi 2 Te 3 , the A 1 1g phonon generation may be governed by the THz-SFE and/or THz-IRS processes. Let us now discuss in more detail the experimental THz spectrum. The violet line in Figure 3.17b shows the FFT of the pristine E THz (for Ω 0 /2π = 0.64 THz). We can note that the highfrequency shoulder of the THz pulse overlaps with the IR-active E 1 u mode, allowing for the resonant excitation of the latter. As established in the Theory section of this Chapter, the driving force of E 1 u phonon was defined as

F (Q E 1 u ) = Z * E 1 u E THz . Assuming high-amplitude Q E 1 u displacement in the presence of E THz , it may result in A 1
1g phonon excitation through anharmonic IR-Raman interaction, where the force-term is

F (Q A 1 1g ) ∝ aQ E 1 u ∝ E 2
THz (see Eq.3.31 and Fig. 3.17a).

Next, let us examine the frequency components of the square of the THz electric field. In the Fourier domain, E 2 THz (Ω) will be given as:

F E 2 THz (τ ) = ∞ 0 ∞ 0 dΩ 1 dΩ 2 E(Ω 1 ) E(-Ω 2 ) F [E 2 THz (Ω)] DF + ∞ 0 ∞ 0 dΩ 1 dΩ 2 E(Ω 1 ) E(Ω 2 ) F [E 2 THz (Ω)] SF , (3.39)
where the first and the second terms are the difference-(DF) and sum-frequency (SF) contributions, respectively, that are shown in blue and red in Figure 3.12b. In our case, only the SF components of E THz have non-negligible spectral amplitude in the range of A 1 1g mode. Therefore, one can imagine A 1 1g phonon excitation through the photonic THz-SFE, in analogy to the similar effect observed in the diamond [START_REF] Maehrlein | Terahertz Sum-Frequency Excitation of a Raman-Active Phonon[END_REF]. Importantly, in the case of both THz-SFE and THz-IRS, coherent phonon amplitude is expected to increase quadratically with applied E THz field amplitude, making the distinction between these processes not straightforward. At this stage, we can only compare the phase of the experimentally measured A 1 1g phonon to the one expected from THz-SFE and THz-IRS mechanisms, as explained below.

For example, the displacive THz-IRS and impulsive THz-SFE of A 1 1g phonon generation should result in different phase of excited coherent phonon motion. To compare the expected A 1g 1 phonon 90 CHAPTER 3. ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ phase as a result of THz-IRS and THz-SFE mechanisms, we phenomenologically solve the two sets of the differential equations describing the two distinct pathways of A 1 1g excitation. However, since we do not have any information about the values of frequency-dependent Raman tensor R(Ω), the life-time of IR E 1 u mode, its effective charge Z * E 1 u , and strength of the anharmonic interaction between E 1 u and A 1 1g mode, the further discussion is based on simple numerical modeling. These crucial parameters and the likelihood of THz-SFE and THz-IRS processes are yet to be confirmed by the first-principle calculations.

Despite the absence of the theoretical parameters, we still able to numerically solve the set of differential equations for Q A 1 1g displacement. For the THz-IRS, based on Eq.3.29 we write:

QE 1 u + 2γ E 1 u QE 1 u + Ω 2 E 1 u Q E 1 u = 2aQ E 1 u Q A 1 1g + Z * E 1 u E THz , QA 1 1g + 2γ A 1 1g QA 1 1g + Ω 2 A 1 1g Q A 1 1g = aQ 2 E 1 u , (3.40)
where a is the anharmonic coupling coefficient, γ E 1 u , and γ A 1 1g are the IR and Raman damping constants, and Z * E 1 u is the effective charge of the E 1 u mode. For E THz , we used the experimental THz pulse shown in Fig. 3.12a. The γ A 1 1g was set to 0.285 ps -1 , and it corresponds to A 1 1g phonon damping time of 3.5 ps, extracted from fitting the experimental data. We set Z *

E 1 u = -1, a = -1, 10
and γ E 1 u = 1 ps -1 . The frequency of the E 1 u mode (Ω E 1 u = 1.5 THz) was taken from Table 3.1. The solution for Q A 1 1g , normalized to match the experimental data, and the driving force ∝ aQ 2

E 1 u are
shown in the Figure 3.18a (as red line and pink filled area, respectively).

In the case of THz-SFE, one can write the following equation of motion [START_REF] Dominik | Sum-frequency ionic Raman scattering[END_REF][START_REF] Meyer | Observation of the spin Nernst effect[END_REF]:

QA 1 1g + 2γ A 1 1g QA 1 1g + Ω 2 A 1 1g Q A 1 1g = RE 2 THz . ( 3.41)
Here, we disregard the frequency dependence of the Raman tensor and set it to unity (R = 1). We use the same shape of E THz as in the THz-IRS model. Solution for Q A 1 1g along with its driving force ∝ E 2 THz are shown in the Figure 3.18a (as orange line and filled red area, respectively). Now, let us compare the experimental data and Q A 1 1g displacement calculated with two models proposed above. Figure 3.18a compares the around time-delay zero experimental signal and solutions of the two models. Especially for the negative time delays, we see the big difference between the δT /T | optic (black line) and calculated Q A 1 1g displacement for both THz-IRS (red line) and THz-SFE (orange line). We believe this is because we model only part of the signal

∝ χ Q A 1 1g Q A 1
1g . In addition, "parasitic" E THz and E 2 THz contributions may be present in the "raw" δT /T experimental signal [START_REF] Maehrlein | Terahertz Sum-Frequency Excitation of a Raman-Active Phonon[END_REF][START_REF] Hoffmann | Terahertz Kerr effect[END_REF], that are not included in our simple model.

From Figure 3.18c, we see that the THz-SFE model (orange line) describes better the phase of the experimental signal (black line). However, it should be noted that both γ E 1 u damping constant and an anharmonic coefficient a used in the THz-IRS model heavily alter the phase of Q A 1 1g displacement, making this comparison inconclusive. As a result, there is always an open question about the time delay zero definition (see Fig. 3.12b), making the direct experiment-simulation phonon phase comparison quite a delicate task. As a final step, using the same numerical modeling, we can calculate the effect of the E THz pump central frequency on the A 1 1g phonon amplitude for both THz-IRS and THz-SFE mechanisms. Figure 3.19a displays the experimental data (blue circles, presented in Fig. 3.15c) superimposed with the results of the THz-IRS and THz-SFE models. Surprisingly, we have a good agreement between the experiment and each proposed pathway of A 1 1g phonon generation. In addition, we see that the maximum of Q A 1 1g displacement is expected for the THz central frequency of Ω 0 /2π = 0.925 THz (half of A 1 1g phonon mode natural frequency), which is out of reach in current experimental conditions. Finally, at this stage, without first-principle calculations, we can not assign the generation of A 1 1g phonon in 16 nm thick n-doped Bi 2 Te 3 to either THz-SFE or THz-IRS.

Conclusion

In this section, we have demonstrated the excitation of A 1 1g coherent optical phonon in Bi 2 Te 3 with THz pulsed radiation. We show that the generation efficiency of the A 1 1g phonon scales quadratically with THz field amplitude, and is affected by the change in the THz central frequency. We have compared the experimental results to the two recently-proposed models of A 1 1g phonon excitation, namely impulsive THz-SFE and displacive THz-IRS. Given our time-delay zero definition, defined as the temporal overlap between the "raw" δT /T signal and calculated W abs THz profile, we find that the THz-SFE model does a better job in matching the experimental phase of the A 1 1g . In addition, preliminary anisotropic detection results have not revealed the presence of either Raman E 1 g or IR E 1 u phonons. At the same time, we saw a clear, long-lived signature of the E 2 g mode, with a frequency roughly four times higher than the Ω 0 /2π of the driving THz electric field. At this moment, we can not conclude on the possible mechanism of E 2 g mode excitation, and more experimental studies are needed.

Finally, Bi 2 Te 3 is a perfect system to study the complex nonlinear and anharmonic interaction between various optical phonon modes. One can imagine a rather complex nonlinear pathways of

A 1
1g , E 1 g and E 2 g generation, which may be governed solely by the THz-SFE, THz-IRS, or combination of both, as sketched in Fig. 3.19b). To confirm or contradict this assumption, theoretical modeling and additional experimental developments are needed. From the theoretical point of view, our colleagues Brice Arnaud and Rémi Busselez are working on first-principle calculations (DFT) to evaluate the respective amplitudes of impulsive and displacive forces acting on the A 1 1g phonon. Concerning the experimental developments, the Bi 2 Te 3 sample orientation, E THz and E probe angular dependence of δT /T are to be measured, both in isotropic and anisotropic detection configurations. In addition, an automated tilted-pulse front realignment feedback-loop could be built, which will allow us to tune the THz pulse central frequency in the range of 0.4-3 THz [START_REF] Morita | Terahertz pulse generation by the tilted pulse front technique using an M-shaped optical system[END_REF].

Such tunable THz source could help us to better understand the coherent phonon generation in the highly off-resonance THz-pumping conditions. In addition, a two-dimensional time-resolved pumpprobe spectroscopy can be used to study and control the nonlinear interaction between selected phonon modes in Bi 2 Te 3 [START_REF] Kuehn | Terahertz-induced interband tunneling of electrons in GaAs[END_REF][START_REF] Kuehn | Strong correlation of electronic and lattice excitations in GaAs/AlGaAs semiconductor quantum wells revealed by two-dimensional terahertz spectroscopy[END_REF][START_REF] Houver | 2D THz spectroscopic investigation of ballistic conduction-band electron dynamics in InSb[END_REF][START_REF] Choi | Theory of Two-Dimensional Nonlinear Spectroscopy for the Kitaev Spin Liquid[END_REF]. Understanding the nonlinear generation and coupling between optical phonon modes is crucial for the selective control over the coherent states of the lattice vibrations.

Conclusions and perspectives

This chapter demonstrates the generation of coherent acoustic phonons with THz pulses in nanometric films of metals (Chromium and Aluminum) and narrow-band semiconductor (Bismuth Telluride) with pulsed THz radiation. We show that the coherent acoustic phonon amplitude in Cr, Al, and Bi 2 Te 3 nanofilms scale linearly with the THz pump pulse energy. In addition, the acoustic period is proportional to the thin film thickness, as demonstrated for the Bi 2 Te 3 nanofilms. We conclude that the THz excitation leads to the ultrafast change in the thermoelastic stress due to the ultrafast Joule effect for the Cr and Al nanofilms. In comparison, while the THz-induced thermoelastic stress is the primary driving acoustic phonon generation mechanism in metals, the Bi 2 Te 3 's linear acoustic response as a function of the THz-pulse energy could be a preliminary indication of a thermoelastic and/or a deformation potential generation process.

However, more controlled experiments and first-principle calculations are needed to determine the origin of coherent acoustic phonon generation with pulsed THz radiation in Bi 2 Te 3 .

In the second part of this chapter, we conduct the experimental study of coherent optical phonon generation in 16 nm thick Bi 2 Te 3 film with pulsed THz radiation. For the first time, we demonstrate that the A 1 1g Raman active optical phonon with 1.85 THz frequency can be excited with a THz pulse of a 0.64 THz central frequency. Furthermore, we show that the amplitude of the

A 1
1g mode scales quadratically with the amplitude of the applied THz electric field and does not depend on the THz pump polarization. In addition, we show that the change of the THz pump central frequency from 0.52 THz to 0.73 THz leads to the three times more efficient generation of the A 1 1g phonon. We then compare the experimental data with the two phenomenological models: impulsive Raman THz-Sum Frequency excitation and displacive anharmonic THz-Ionic Raman scattering. We find that the phase of the THz-Sum Frequency excitation model fits better the phase of experimental A 1 1g signal. However, first-principle calculations are necessary to estimate the relative amplitudes of displacive and impulsive forces that can contribute to the A 1 1g phonon generation in 16 nm thick Bi 2 Te 3 film.

In future work, the study of the acoustic phonon generation with THz could be extended to metallic nanoparticles that may be fabricated to support acoustic vibrations in the THz range. In addition, the experiments on the Bi 2 Te 3 film with various doping and tunable Fermi levels could help identify the primary microscopic mechanism of acoustic phonon generation in narrow-band semiconductor films. Furthermore, the experimental developments in the frequency tunable THz source combined with the 2D time-resolved spectroscopy may allow us to explore the different excitation conditions of optical phonon modes in Bi 2 Te 3 . Finally, by comparing the experiment and first-principle calculations, we should be able to identify the dominant mechanism of A 1 1g phonon generation in Bi 2 Te 3 .

Introduction

Over the last decade, electricity consumption by the information and communication technology sector (ICTS) has rapidly increased. It has been estimated that starting from 2020, ICTS will be responsible for more than 11% of Global use of electricity [START_REF] Andrae | On Global Electricity Usage of Communication Technology: Trends to 2030[END_REF]. With an exponential increase in the generated and stored data, there is a necessity for energy-efficient, fast, non-volatile memory and energy-harvesting devices. Spintronics is regarded today as a promising field of physics capable of proposing some possible solutions of this problem [START_REF] Puebla | Spintronic devices for energy-efficient data storage and energy harvesting[END_REF].

The main goal of spintronics is to develop an efficient way to excite, "flip," transport, and convert spins under external stimuli. The study of the spin-transport and spin-to-charge conversion phenomena in the Magnetic/Non-Magnetic (NM) heterostructures in the DC regime displays the possibility of efficient control of the magnetic degrees of freedom with a relatively small amount of applied current (within ≈ nV -µV range) [START_REF] Sangiao | Control of the spin to charge conversion using the inverse Rashba-Edelstein effect[END_REF][START_REF] Wang | Anomalous spin Hall and inverse spin Hall effects in magnetic systems[END_REF][START_REF] Miao | Inverse Spin Hall Effect in a Ferromagnetic Metal[END_REF]. However, the development of fast memory devices requires an understanding of the magnetic system response times upon rapidly-varying external stimuli, such as magnetic, electric and light pulses. Interestingly, the ultrafast response of a Ferromagnetic (FM)/NM heterostructure to an ultrashort laser pulse can be probed with THz emission spectroscopy, as discussed in the following Theoretical background section. The physical mechanisms discussed in this Chapter are based on further development of the theory, introduced back in the Section 1.2.3. This will serve as literature review of the recent achievements in THz-spintronics based on the manipulation of spin precession and spin currents. The experimental part of this chapter is inspired by the existence of the Rashba states in CoFeB/MgO bilayers, recently observed with quasi-static ferromagnetic resonance spin-pumping measurements, where the samples are excited by a microwave magnetic field at a fixed frequency [4]. Interestingly, they show an identical spin-to-charge conversion efficiencies between the best-performing inverse spin Hall emitter (CoFeB/Pt) and Rashba converter (CoFeB/MgO) (see Fig. 4.1)c). As a result, Rousseau et al. [4] claim that the deposition of a thin 6 nm MgO layer on top of the FM metal, such as Co, Fe, CoFe, or CoFeB, will lead to the creation of the Rashba states inside FM, in the vicinity to the MgO interface.

In the Experimental results section we show that ultrashort laser pulse illumination of some hybrid magnetic nanostructures leads to the emission of sub-ps THz pulses. In addition, we will demonstrate that the emitted THz is sensitive to different experimental parameters, allowing us to investigate different microscopic spin-to-charge conversion mechanisms. In particular, by comparing the ultrafast light-induced THz-emission in CoFeB/Pt and CoFeB/MgO systems we will try to identify and estimate the relative amplitudes of ultrafast Inverse Spin Hall effect (ISHE), bulk conversion inside the ferromagnetic CoFeB, and the Inverse Edelstein effect (IREE). At last, we demonstrate the striking difference between the ISHE and IREE effects as a result of the pump photon energy sensitive IREE spin-to-charge conversion. To the best of our knowledge, this effect was never observed before, and opens prospects for light-controlled spin-orbit THz emitters.

Spin transport: Theoretical background

The spin-orbit interaction (SOI) can affect and alter the propagation direction of the spin and charge current in the material. For example, some FM/NM interfaces with broken time-reversal symmetry may act as the source of the so-called spin-galvanic effects [START_REF] Ganichev | Spin-galvanic effect[END_REF]. In addition, particular spin and charge transport effects can arise from temperature gradient within the FM or at the FM/NM interface. These effects are often used for the spin-injection [START_REF] Dai Tian | Separation of spin Seebeck effect and anomalous Nernst effect in Co/Cu/YIG[END_REF] and spin-to-charge conversion [START_REF] Chuang | Enhancement of the anomalous Nernst effect in ferromagnetic thin films[END_REF]. In the following subsections, we will discuss the main spin-dependent transport phenomena and how one can probe them at the sub-ps time scale.

Spin precession

In the Section 1.2.3, we discussed the undamped precession of the spins on the lattice as the result of the near-neighbor exchange interaction field Bex j (see Eq.1.49). Here, let us look at the change of the expected value of the spin operator of a single electron subjected to the timedependent external magnetic field Bext (t). Given the Zeeman term Ĥz = -μe • Bext (t), the undamped spin-precession in the presence of an external magnetic field will read [START_REF] Lakshmanan | The fascinating world of the Landau-Lifshitz-Gilbert equation: an overview[END_REF]:

d dt ⟨ŝ⟩ = g s µ B ℏ ⟨ŝ⟩ × Bext (t), (4.1)
where the time-dependent change in ⟨ŝ⟩ will happen at the specific angular frequency ω Larmor = µ B ℏ -1 B ext , known as Larmor precession frequency [START_REF] Coey | Magnetism and Magnetic Materials[END_REF].

Moreover, the additional spin-dependent terms in the Hamiltonian for the many-electron system show that the spin-precession can be coupled to the magnetic anisotropy, lead to the demagnetization, modify the exchange interaction, and interact with various magneto-elastic contributions [START_REF] Robert | Quantum Theory of Magnetism : Magnetic Properties of Materials[END_REF].

In reality, due to the dissipation of the angular momentum, the perturbed system will return to equilibrium, resulting in the damped precession of a spin around the field felt by the magnetic system. Therefore, the spin-damping is accounted for by an additional term, and the initial action of B ef f on the spin-operator named the spin-torque.

In the frame of macroscopic magnetization (M , see Eq.1.43), the precessional damping of the magnetization can be described in terms of the Landau-Lifshitz-Gilbert (LLG) equation [START_REF] Gilbert | Classics in Magnetics A Phenomenological Theory of Damping in Ferromagnetic Materials[END_REF]:

∂M ∂t = γ G M × B ef f - α G M M × ∂M ∂t , ( 4.2)
where M is the magnitude of M , γ G = γ/(1 + α 2 G ) with gyromagnetic ratio γ and the Gilbert damping parameter α G . Importantly, while the direction of M is subject to change, the magneti-a) [START_REF] Kampfrath | Resonant and nonresonant control over matter and light by intense terahertz transients[END_REF] The flow of electrons with equal density and velocity of up and down spins along the same direction results in the pure charge current, where no transfer of angular momentum can exist. (2) The electron flow where the mobility and concentration of up and down carriers are not equivalent, the charge transport will carry the information about the angular momentum, leading to the spin-polarized electron current. (3) When the current density with up and down spin polarization flows in the opposite direction, only flow of the angular momentum can exist. This corresponds to the pure spin current.

Figure after [START_REF] Castro | Spin-to-charge current conversion in SrTiO3-based two-dimensional electron gases[END_REF]. b) Electron propagation in the presence of an external electric field is affected by the scattering and deflection of the impurities. The mean distance of the change in electron direction or energy is called mean free path Λ. The mean distance describing the flip of an electron's spin is denoted as spin diffusion length, λs. Figure adapted from [START_REF] Noel | Dynamical spin injection and spin to charge current conversion in oxide-based Rashba interfaces and topological insulators[END_REF]. zation magnitude M will not be affected [START_REF] Brataas | Spin pumping and spin transfer[END_REF]. The second term in Eq.4.2 describes the angular momentum loss via precessional damped-motion of magnetization through interaction with the conduction electrons, phonons, magnons, or lattice defects [START_REF] Gilbert | Classics in Magnetics A Phenomenological Theory of Damping in Ferromagnetic Materials[END_REF]. To summarize, this equation is valid for T ≪ T C while the phenomenological damping term corresponds to the multitude of complex energy transfer processes within the magnetic material.

The generalized forms of the LLG equation are used to model the temporal dynamics of the magnetization precession M in magnetic materials [START_REF] Silsbee | Coupling between ferromagnetic and conductionspin-resonance modes at a ferromagnetic-normal-metal interface[END_REF] subjected to external stimuli [START_REF] Xu | Handbook of Spintronics[END_REF]. For example, one can change the direction of M : by exciting the FM with an EM wave of a specific polarization and frequency close to ω Larmor , via spin accumulation in the adjacent NM layer or by thermal effects [START_REF] Brataas | Spin pumping and spin transfer[END_REF]. Notably, the opposite is true: dynamical magnetization will "pump" spins into the adjacent NM layer. We note that such pure spin current generation does not involve any contribution of electron transport.

Spin currents

Electron transport in magnetic materials can lead to the generation of the spin current. In addition to the charge current, usually defined as the flow of the electrons along the given direction (See Fig. 4.2a), magnetic materials allow for the transport of the intrinsic angular momentum.

Regarding the ongoing discussion about the definitive formulation of the spin-current [START_REF] Shi | Proper Definition of Spin Current in Spin-Orbit Coupled Systems[END_REF][START_REF] An | The universal definition of spin current[END_REF],

we will address it in the frame of a phenomenological two spin-channel model described elsewhere [START_REF] Castro | Spin-to-charge current conversion in SrTiO3-based two-dimensional electron gases[END_REF][START_REF] Dominik | Spin Currents in Metallic Nanostructures[END_REF].

In the frame of the Stoner model introduced in the Section 1.2.3, one can express the the electron charge j c and the angular momentum j s in the terms of spin-dependent carrier density 99 CHAPTER 4. ULTRAFAST SPIN-TO-CHARGE CONVERSION ρ ↑,↓ and velocity v ↑,↓ as:

j c = j ↑ + j ↓ = -e(ρ ↑ v ↑ + ρ ↓ v ↓ ); j s = - ℏ 2 (ρ ↑ v ↑ -ρ ↓ v ↓ ) = - ℏ 2e (j ↑ -j ↓ ). (4.3)
This equation summarizes three possible variations of the electron current depicted in Fig. 4.2.

Usually, in NM conductors, the equal density of the spin-up and spin-down electrons at the Fermi level renders j s = 0, establishing the pure charge current flow. However, in the FM both ρ ↑ ̸ = ρ ↓ and v ↑ ̸ = v ↓ are true, allowing for simultaneous charge and spin-current flow called spin-polarized current (see Fig. 4.2a2). However, when j ↑ = -j ↓ , as displayed in Fig. 4.2a3, only the transport of the electron spin can exist, denoted as pure spin current. Moreover, the generation of the pure spin current is quite challenging task, sometimes requiring extreme experimental conditions and use of specific magnetic materials [START_REF] Huang | Pure spin current phenomena[END_REF].

In analogy with the electron mean path Λ, the collision and scattering events may alter the particles' spin direction. The mean distance that thermalized electrons can travel without change in their angular momentum, or without so-called "spin-flip"(as shown in Fig. 4.2b), is called spin diffusion length λ s [START_REF] Bass | Spin-diffusion lengths in metals and alloys, and spinflipping at metal/metal interfaces: an experimentalist's critical review[END_REF]:

λ s = 1 √ 3 v F √ τ s τ e , (4.4)
where τ s is the spin-flip time or spin lifetime. λ s is directly linked to the electron velocity at the Fermi level v F and scattering time τ e . This equation, however, does not take into account the different spin-flip diffusion lengths for the majority and minority of spin-polarized carriers due to the asymmetric scattering [START_REF] Bass | Spin-diffusion lengths in metals and alloys, and spinflipping at metal/metal interfaces: an experimentalist's critical review[END_REF]. Most materials with strong SOI are known to have short λ s in the order of tens of nanometers. The spin-flip is known to occur via two competing mechanisms: Elliot-Yafet, [START_REF] Elliott | Theory of the Effect of Spin-Orbit Coupling on Magnetic Resonance in Some Semiconductors[END_REF] which dominates for centrosymmetric conductors, and D'yakonov-Perel' [START_REF] Glazov | D'yakonov-Perel' spin relaxation controlled by electronelectron scattering[END_REF], superior within the 2D systems and semiconductors (GaP, ZnTe, GaAs) lacking space-inversion symmetry.

Interestingly, the Elliot-Yafet mechanism predicts the spin-rotation during the collisions, while the D'yakonov-Perel model describes spin-flip in-between scattering events.

Field-driven spin transport effects in DC (quasi-static)

Hall and Anomalous Hall effects in terms of both Hall (ρ Hall ) and AHE (ρ AHE ) contributions:

ρ tot xy = ρ Hall B ext,z + ρ AHE M z . ( 4.6)
It was observed that in the absence of M , i.e., when working with NM conductor, ρ xy is increasing linearly with B ext , as expected from Eqs.4.5,4.6. However, for the FM Ni, the ρ xy linearly increases at low B ext [START_REF] Pugh | Hall Effect in Ferromagnetic Materials[END_REF], and then rapidly saturates due to the parallel alignment of all spins along the direction of the applied magnetic field [START_REF] Nagaosa | Anomalous Hall effect[END_REF].

The AHE describes the charge to spin-polarized current conversion. In general, the total AHE conductivity tensor (σ AH xy ) can be expressed as the sum of the following contributions [START_REF] Nagaosa | Anomalous Hall effect[END_REF]:

1 ρ AHE xy = σ AHE xy = σ AHE-int xy + σ AHE-sj xy + σ AHE-skew xy , ( 4

Intrinsic deflection:

The intrinsic contribution to the AHE (σ AHE-int xy ) depends on the band structure of the perfect crystal [START_REF] Karplus | Hall effect in ferromagnetics[END_REF]. The total velocity of a Bloch electron in the band n, with band momentum k in the real space, in the presence of the applied electric field E can be written as [START_REF] Nagaosa | Anomalous Hall effect[END_REF][START_REF] Ong | GEOMETRY AND THE ANOMALOUS HALL EFFECT IN FERROMAGNETS[END_REF]:

∂⟨r⟩ dt = 1 ℏ ∂ϵ n (k) ∂k normal velocity - ℏk ∂t × b n (k) anomalous velocity , ( 4.8)
where the first term corresponds to the ordinary electron velocity in the band n at the energy level structure, and it is the gauge-invariant representation of Bloch wave function geometric properties.

The anomalous contribution is similar to the classical Lorentz force (see Eq.4.5). Still, it originates from the SOI where b n (k) acts as a magnetic field in the momentum space [START_REF] Ong | GEOMETRY AND THE ANOMALOUS HALL EFFECT IN FERROMAGNETS[END_REF]. The intrinsic contribution usually dominates the side-jump and skew-scattering in pure crystals with large SOI and does not require electron scattering [START_REF] Sinova | Spin Hall effects[END_REF].

Side-jump scattering:

The side-jump contribution (σ AHE-sj xy ) to AHE is the scattering process where electron momentum and energy are not affected. The scattering of the electron wave packet from the impurity will lead to spin-dependent transverse displacement. The electrons with opposite spins will experience an equal shift in the opposite directions, as shown in Fig. 4.4b.

Interestingly, the numerical prediction of the σ AH-sj xy is highly challenging since it depends on the intricate details of the material disorder.

Skew-scattering: Skew-scattering alters the electron momentum after interaction with the impurity. Its efficiency is proportional to the electron scattering time τ e [START_REF] Nagaosa | Anomalous Hall effect[END_REF]. Skew-scattering can occur in the presence of a strong SOI, leading to the spin-dependent change in the electron momentum, as shown in Fig. 4.4. Additionally, the skew-scattering can happen in the absence of SOI, when the electrons are subject to motion in the spin-orbit-coupled band structure. The latter can be thought of as the impurity-analogy of the heavy-atoms Mott-scattering [START_REF] Sinova | Spin Hall effects[END_REF].

Spin Hall and Inverse Spin-Hall effects

The description of the Spin Hall effect (SHE) and Inverse Spin Hall Effect (ISHE) are directly related to the physics of the AHE. SHE and ISHE effects describe the conversion of the conserved charge quantity to the decaying and dephasing amount of spin [START_REF] Sinova | Spin Hall effects[END_REF]. This defines the SHE and ISHE as charge-to-spin and spin-to-charge conversion mechanisms respectively, first predicted by Dyakonov and Perel in 1971 [212]. Similarly to the AHE, the SHE conductivity tensor σ SH xy can be expressed as the sum of three mechanisms: The vital distinction between the AHE and the SHE is that AHE can happen only in magnetic materials, while the SHE and the ISHE are suitable for all materials with strong SOI. For example, injection of pure spin current in the NM will establish transverse charge current due to the asymmetric scattering arising from different concentration of the minority and majority spin populations at the Fermi level. Inversely, NM will convert pure spin current into pure charge current.

The amount of spin or charge generated via the SHE and ISHE can be described with the following equations [START_REF] Sinova | Spin Hall effects[END_REF]:

j SHE s = - ℏ 2e θ SHE [j c × σ s ], j ISHE c = - 2e ℏ θ SHE [j s × σ s ], (4.10)
where θ SHE is the so-called spin Hall angle that describes the conversion efficiency, and σ s is the unit vector of the spin-/spin-current polarization.

The well-studied field-driven effects, such as AHE, SHE, and ISHE, are routinely used to measure and characterize spin currents. The magnitude of the θ SHE can be measured by studying the efficiency of the spin-to-charge conversion with a variety of different techniques: FMR spinpumping [START_REF] Tserkovnyak | Spin pumping and magnetization dynamics in metallic multilayers[END_REF], thermal spin injection via the spin Seebeck effect [START_REF] Qu | Self-consistent determination of spin Hall angles in selected 5d metals by thermal spin injection[END_REF], spin Hall magneto-resistance [START_REF] Nakayama | Spin Hall Magnetoresistance Induced by a Nonequilibrium Proximity Effect[END_REF], and spin torque FMR [START_REF] Liu | Spin-torque ferromagnetic resonance induced by the spin Hall effect[END_REF].

Rashba-Edelstein and Inverse Rashba-Edelstein effects

As discussed in the Section 1.2.3, the Zeeman Hamiltonian can lift the spin degeneracy of the electronic band structure, leading to the Dresselhaus and Rashba splitting. However, special conditions have to be fulfilled regarding the time-reversal and space-inversion symmetries of the material.

Symmetry considerations:

When the time-reversal symmetry is preserved, i.e., T : t → -t, following the Kramer's degeneracy theorem, the energies of fermions with opposite momentum This equation leads to the parabolic electron dispersion relation. However, the effective SOI field can affect the trajectory of charged particles depending on their spin. Therefore, the SOI may lift the time-reversal symmetry. In the case of centrosymmetric medium with SOI, both spin-and k-dependent effects are allowed, establishing the well-known SHE and ISHE effects in Pt and Ta, for example.

Non-centrosymmetric medium, by definition, lacks the space-inversion symmetry, so that:

E k,↑(↓) ̸ = E -k,↑(↓) . (4.12)
Space and time-reversal symmetries will be lifted if we introduce the SOI in non-centrosymmetric crystals. In addition, broken space-inversion symmetry can indirectly create the intrinsic, stray electric field, which can be coupled to the electron spin via SOI. Such electric field (E) can also exist as a result of a junction effect when two layers are in contact (as in the case of the hybrid nanostructures studied in this Chapter). Thereby, the electrons with velocity v feel in their frame of refference a local magnetic field called the Rashba field (B * ) given by B * = c -2 • v × E [START_REF] Coey | Magnetism and Magnetic Materials[END_REF],

creating the Rashba-split band-structure ( an example is shown in Fig. 4.6).

CHAPTER 4. ULTRAFAST SPIN-TO-CHARGE CONVERSION

The Rashba Hamiltonian can be expressed as the interaction between the spin-vector s and the electron momentum vector k as [START_REF] Sangiao | Control of the spin to charge conversion using the inverse Rashba-Edelstein effect[END_REF]:

ĤR ∝ B * × s = α R (k × z) • s, (4.13)
with the coordinate normal to the interface z, and the Rashba coefficient α R that is proportional to the splitting-electric field. The Rashba Hamiltonian ĤR will lead to the following electron dispersion relation [4]:

E ↑,↓ (k) = ε 0 + ℏ 2 k 2 2m * e ± |α R ||k|, (4.14)
where ε 0 is the band-crossing energy at k = 0 (see Fig. 4.6a). Importantly, α R is not a constant, and can be expressed as:

α R = 2E R k R , ( 4.15)
with E R being the band-splitting energy at k = k R [4], displaying the relationship between the Rashba coefficient and the electron momentum.

The Rashba-split band structure allows for the charge-to-spin and spin-to-charge conversion. It was first theoretically predicted in 2D systems [START_REF] Aronov | Nuclear electric resonance and orientation of carrier spins by an electric field[END_REF] and was later called Edelstein or spin-galvanic effect [START_REF] Ganichev | Spin-galvanic effect[END_REF]. The spin-galvanic effect was then first measured at the Ag/Bi interface [START_REF] Rojas Sánchez | Spin-to-charge conversion using Rashba coupling at the interface between non-magnetic materials[END_REF].

Rashba Edelstein Effect: Let us consider the direct Rashba Edelstein effect (REE). The action of an external electric field E x leads to a shift of the Fermi sphere. As a consequence there is a total non-zero electron momentum in the x-direction with ∆k x ∝ -E x ∝ -j x c , as shown in Fig. 4.7a. The net spin current density generated under the action of E x can be expressed as:

δs = δs ↑ -δs ↓ ∝ k out F δk x -k in F δk x , (4.16)
where k out F and k in F are the radii of the outer and inner Fermi contours. Considering similar electron scattering times for both contours, the finite spin accumulation will happen under the action of E x or the injection of the charge current j x c .

Inverse Rashba Edelstein Effect: Like the ISHE, the Inverse Rashba Edelstein effect (IREE) describes the charge current generation upon injection of the spin current at the Rashba interface. As shown in Fig. 4.7b, the injection of the spin-current polarized along the y-direction will shift the outer and inner Fermi contours by an equal amount δk x . However, the total displacement δk total x will be proportional to:

δk total x ∝ ∆k out x -∆k in x ∝ k out F δk x -k in F δk x . (4.17)
a) Direct Rashba-Edelstein effect (REE). Charge current j x c applied along the -x direction will equally shift the Rashba-split Fermi contours by the amount ∆kx. This will create two unequal spin densities δs ↓ and δs ↑ (red and blue shaded regions, respectively) that will lead to the finite spin accumulation. b).

Inverse Rashba-Edelstein effect (IREE). The injection of the spin-current along z, polarized along the y-axis, will lead to the unequal displacement of the Rashba contours by the amount of ∆k out x and ∆k in x . Uneven carrier acceleration in the opposite directions will lead to the charge flow along the x-axis.

Redrawn from [START_REF] Castro | Spin-to-charge current conversion in SrTiO3-based two-dimensional electron gases[END_REF] The non-zero δk total x will result in the generation of the charge current perpendicular to the polarization of the injected spin-current (along k z ). One can notice that the REE and IREE efficiency depends on the splitting of inner and outer contours, which is related to the magnitude of the Rashba coefficient α R . The charge current generated by the IREE, sensitive to the spincurrent polarization direction σ s , can be written as [START_REF] Rojas Sánchez | Spin-to-charge conversion using Rashba coupling at the interface between non-magnetic materials[END_REF]:

j IREE c ∝ λ IREE • j s × σ s , ( 4.18)
where "effective Rashba length" λ IREE = α R τ s /ℏ, with τ s being effective eletron spin relaxation time.

Thermally-driven spin transport effects in DC (quasi-static)

Experimental and theoretical developments of the Hall-driven effects led to the discovery of thermally-driven spin-transport phenomena, such as the spin-Seebeck [START_REF] Uchida | Observation of the spin Seebeck effect[END_REF] and the Nernst [START_REF] Ettingshausen | Ueber das Auftreten electromotorischer Kräfte in Metallplatten, welche von einem Wärmestrome durchflossen werden und sich im magnetischen Felde befinden[END_REF] effects.

Seebeck effects

Seebeck effect: Let us take two metals with different transport properties. The thermal gradient ∇T applied along a given direction will lead to charge redistribution in the bulk of each metal. If we bring those two metals into electrical contact, we will establish the current flow along the junction due to the well-known Seebeck effect, as shown in Fig. 4.8a. The current flow will be proportional to:

j SE c ∝ ∇T (K 1 -K 2 ), (4.19)
where K 1 and K 2 are the Seebeck coefficients. The Seebeck effect is the operational principle of the thermocouples with K = ∆V /∆T the Seebeck constant and V is the Seebeck voltage.

Spin-dependent Seebeck Effect:

The spin-dependent Seebeck effect (SDSE) describes the generation of spin-polarized charge current (j SDSE s-c ) under heat gradient ∇T in the bulk of a FM metal, shown in Fig. 4.8. It is explained by the different transport properties of the minority-and majority-spin electrons that are related to the spin-dependent Seebeck coefficients S maj and S min [START_REF] Gerrit | Spin caloritronics[END_REF]:

j SDSE s-c
∝ ∇T (S maj -S min ).

(4.20)

However, the spin-diffusion length λ s is the determining factor concerning the efficiency of the SDSE in the FM metals. The strong spin-flip scattering in the bulk of the FM layer can drastically reduce the number of spins injected in the neighboring NM layers [START_REF] Qu | Self-consistent determination of spin Hall angles in selected 5d metals by thermal spin injection[END_REF][START_REF] Qu | Inverse spin Hall effect in Cr: Independence of antiferromagnetic ordering[END_REF].

Spin Seebeck Effect: Contrary to the SDSE, the spin Seebeck effect (SSE) is related to the generation of pure spin current (j SSE s ) in magnetic insulators [START_REF] Uchida | Observation of the spin Seebeck effect[END_REF]. Here, the magnons carry the pure spin current, which is usually detected by the ISHE in a NM layer. As shown in Fig. 4.8c, this spin-current in the NM material is generated through the temperature difference between the electron temperature T e in the NM metal and magnon temperature T M in the magnetic insulator [START_REF] Adachi | Theory of the spin Seebeck effect[END_REF]: where K SSE is the spin Seebeck coefficient. In ferromagnetic metals, however, both Spin-Seebeck and Spin-dependent Seebeck effects can simultaneously exist [START_REF] Gerrit | Spin caloritronics[END_REF][START_REF] Slachter | Thermally driven spin injection from a ferromagnet into a non-magnetic metal[END_REF].

j SSE s = K SSE • (T e -T m ), ( 4

Nernst effects

Nernst Effect: The Nernst effect, discovered by v.Ettingshausen and Nernst in 1886, is the temperature analogy of the Hall effect [START_REF] Ettingshausen | Ueber das Auftreten electromotorischer Kräfte in Metallplatten, welche von einem Wärmestrome durchflossen werden und sich im magnetischen Felde befinden[END_REF]. In the presence of an external magnetic field B ext applied perpendicularly to the thermal gradient ∇T , as the result of the SOI, the Nernst current will be generated perpendicular to both ∇T and B ext , as shown in Fig. 4.9a.

Anomalous Nernst Effect: When the thermal gradient is applied to a FM metal, the macroscopic magnetization M can result in the strong Nernst effect even in the absence of the external magnetic field, as displayed in Fig. 4.9b. Such thermal counterpart of AHE is called the Anomalous Nernst effect (ANE). One can express the transverse charge flow established by the ∇T in the FM metal as [START_REF] Sakuraba | Potential of thermoelectric power generation using anomalous Nernst effect in magnetic materials[END_REF]:

j ANE s-c ∝ θ AN E ∇T × σ s , ( 4.22)
with θ AN E being the anomalous Nernst coefficient, describing the longitudinal-to-transverse charge conversion efficiency. The study of the j ANE c dependence on the FM layer thickness d F M show the decrease of the θ AN E due to the increase in resistance of FM. Importantly, for FM with thickness d F M > λ s , ANE can compete or even completely dominate the SDSE [START_REF] Dai Tian | Separation of spin Seebeck effect and anomalous Nernst effect in Co/Cu/YIG[END_REF].

Spin Nernst Effect:

The spin Nernst effect (SNE) describes the pure transverse spin current generation arising from the longitudinal heat gradient [START_REF] Meyer | Observation of the spin Nernst effect[END_REF][START_REF] Shu | Spin Nernst effect and Nernst effect in two-dimensional electron systems[END_REF]. The SNE originates from the different transverse scattering rates for the above Fermi level ("hot") and below Fermi level ("cold") electrons flowing respectively along and against the temperature gradient direction, as shown in CHAPTER 4. ULTRAFAST SPIN-TO-CHARGE CONVERSION Fig. 4.9c. Thus, spin current originating from the SNE can be expressed in terms of [START_REF] Bose | Direct detection of spin Nernst effect in platinum[END_REF]:

j SN E s ∝ j Hot s -j Cold s ∝ θ SN E j h × σ s , (4.23)
where θ SN E is the Spin Nernst coefficient and j h = -κ∇T is the heat current with κ being the thermal conductivity. The spin Nernst effect was directly observed in the heavy metal (HM) like

Tungsten [START_REF] Sheng | The spin Nernst effect in tungsten[END_REF] or Platinum [START_REF] Meyer | Observation of the spin Nernst effect[END_REF][START_REF] Bose | Direct detection of spin Nernst effect in platinum[END_REF].

Terahertz emission as the signature of the ultrafast spin-charge conversion

As of today, researchers are mainly focused on the characterization of the spin-to-charge conversion in the DC and GHz regimes via SHE and ISHE [START_REF] Sinova | Spin Hall effects[END_REF], AHE [START_REF] Nagaosa | Anomalous Hall effect[END_REF], SSE and SDSE [START_REF] Uchida | Observation of the spin Seebeck effect[END_REF][START_REF] Uchida | Longitudinal spin-Seebeck effect in sintered polycrystalline (Mn,Zn) Fe2 O4[END_REF],

ANE [START_REF] Bose | Recent advances in the spin Nernst effect[END_REF], and IREE [START_REF] Sangiao | Control of the spin to charge conversion using the inverse Rashba-Edelstein effect[END_REF][START_REF] Shen | Spin-to-Charge Conversion in Ag/Bi Bilayer Revisited[END_REF]. Hypothetically, all of these mechanisms can lead to the spin-charge conversion not only at the GHz and DC frequencies but in the THz range as well.

The charge current generated from the spin-to-charge and charge-to-charge conversion within the FM/NM heterostructures is usually detected in the DC/GHz regime by placing physical contacts on top of the NM layer, used to measure the transverse current. On the other hand, in the THz regime the spin-to-charge conversion can be probed in a non-contact manner [START_REF] Kampfrath | Terahertz spin current pulses controlled by magnetic heterostructures[END_REF][START_REF] Feng | Spintronic terahertz emitter[END_REF]. If we recall the Maxwell wave-equation (see Eq.1.93), the transient charge current (j c ) generated in the heterostructure can act as the source term, responsible for the emission of an electromagnetic radiation:

E THz (t) ∝ ∂j c (t) ∂t . ( 4.24)
Let us consider what happens upon the excitation of the FM film by a femtosecond laser pulse propagating along the z-axis. By applying the external magnetic field H along the x-axis to fix the magnetization direction M of the FM layer (i.e. H x ∥ M x ∥ σ x s ), the total transverse spinpolarized current (j y s,c ) generated by the fs of laser pulse propagation inside the FM will be given as:

j y s,c (t) ∝ i σ x s (t) × j z c,s (t), (4.25)
and it could be the combination of all possible spin-to-charge conversion mechanisms (such as AHE, ISHE, and IREE, DMG), as displayed in the Fig. 4.10. As of today, for THz spintronic emitters, the ISHE remains the most efficient conversion mechanism to date, completely overpowering ultrafast AHE and IREE conversion mechanisms [START_REF] Jungfleisch | Control of Terahertz Emission by Ultrafast Spin-Charge Current Conversion at Rashba Interfaces[END_REF][START_REF] Zhang | Terahertz Emission from Anomalous Hall Effect in a Single-Layer Ferromagnet[END_REF]. The following section will provide a brief literature overview of the most relevant and impactful studies about the THz spin-charge conversion in magnetic nanometric heterostructures and ultrafast spin-injection mechanisms at the picosecond timescale.

Supperdiffusive spin current injection

The ultrafast quenching of the magnetization in the ferromagnets at the 100 fs time scale was addressed with the superdiffusive spin transport theoretical model that relies on the spin-dependent transport of the hot electrons [START_REF] Battiato | Superdiffusive spin transport as a mechanism of ultrafast demagnetization[END_REF][START_REF] Melnikov | Ultrafast transport of laser-excited spin-polarized carriers in Au/Fe/MgO(001)[END_REF]. This model describes the competition between the diffusive and ballistic transport of the spin-polarized carriers at the fs timescale. Recent publications on THz spintronic emitters use the superdiffusive spin model to compare the experimental data with Finite-difference time-domain simulations of the excited spin currents via femtosecond laser action [START_REF] Dang | Ultrafast spin-currents and charge conversion at 3 d-5 d interfaces probed by time-domain terahertz spectroscopy[END_REF][START_REF] Hawecker | Spin Injection BIBLIOGRAPHY Efficiency at Metallic Interfaces Probed by THz Emission Spectroscopy[END_REF]. As a result, the bipolar shape of the emitted THz pulse is explained by the superdiffusive transport and multiple carrier reflections within the magnetic heterostructure [START_REF] Dennis M Nenno | Simulation of Hot-Carrier Dynamics and Terahertz Emission in Laser-Excited Metallic Bilayers[END_REF][START_REF] Tian | Interface reflectivity of a superdiffusive spin current in ultrafast demagnetization and terahertz emission[END_REF].

ISHE-driven THz emission

In Furthermore, various groups reported THz emission from stacked STEs [START_REF] Yang | Powerful and Tunable THz Emitters Based on the Fe/Pt Magnetic Heterostructure[END_REF], address the influence of the sample annealing on the detected THz waveform [START_REF] Sasaki | Annealing effect on laser pulse-induced THz wave emission in Ta/CoFeB/MgO films[END_REF][START_REF] Sasaki | Laser-induced terahertz emission in Co 2 MnSi/Pt structure[END_REF], interface roughness [START_REF] Li | THz emission from Co/Pt bilayers with varied roughness, crystal structure, and interface intermixing[END_REF],

and temperature dependence on the ISHE conversion efficiency [START_REF] Matthiesen | Temperature dependent inverse spin Hall effect in Co/Pt spintronic emitters[END_REF]. In addition, one can controll the THz emission from FM/HM bilayers depending on the sample geometrical orientation [START_REF] Qiu | Layer thickness dependence of the terahertz emission based on spin current in ferromagnetic heterostructures[END_REF][START_REF] Garik Torosyan | Optimized Spintronic Terahertz Emitters Based on Epitaxial Grown Fe/Pt Layer Structures[END_REF],

and even by changing the type of the sample substrate [START_REF] Seifert | Efficient metallic spintronic emitters of ultrabroadband terahertz radiation[END_REF][START_REF] Li | Laser induced THz emission from femtosecond photocurrents in Co/ZnO/Pt and Co/Cu/Pt multilayers[END_REF][START_REF] Nenno | Modification of spintronic terahertz emitter performance through defect engineering[END_REF].

Ultrafast SSE spin current injection

Seifert et al. reported the injection of ultrafast spin current from the FM insulator into HM via the SSE [START_REF] Seifert | Femtosecond formation dynamics of the spin Seebeck effect revealed by terahertz spectroscopy[END_REF]. They note that the spin current injection is mediated by the thermal nonequilibrium between the electron and magnon baths in HM and FM insulators, respectively. The bandwidth of the THz-SSE was then attributed to the electron-phonon coupling speed, the few hundreds of femtoseconds after the laser excitation.

AHE-driven THz emission

Zhang et al. report that the THz emission can occur via AHE in the FM films with a thickness of less than 20 nm [START_REF] Zhang | Terahertz Emission from Anomalous Hall Effect in a Single-Layer Ferromagnet[END_REF][START_REF] Zhang | Terahertz Emission from an Exchange-Coupled Synthetic Antiferromagnet[END_REF]. In total, they measure the THz emission from four different FM/MgO(4 nm) bilayers on the Quartz substrate. They claim that the charge gradient established within the FM is governed by the back-flow charge current, created by non-equal hot-carrier reflectance at the interfaces: to the accumulation of Pt with a higher concentration close to the given interface or affect the distance between the scattering centers [START_REF] Baldasseroni | Effect of capping material on interfacial ferromagnetism in FeRh thin films[END_REF][START_REF] Han | Effects of overlayer capping and lattice strain on perpendicular magnetic anisotropy of TM|FePt|MgO heterostructures[END_REF]. Such interface-sensitive Pt buffer may amplify the charge current generation, and introduce the sample asymmetry, which could explain the THz polarity reversal in the thin samples observed by the authors [START_REF] Zhang | Terahertz Emission from Anomalous Hall Effect in a Single-Layer Ferromagnet[END_REF]. In addition, they do not address the possible existence of the Rashba interface created at the FM/insulator interface [4,[START_REF] Vetter | Observation of carrier concentration dependent spintronic terahertz emission from n -GaN/NiFe heterostructures[END_REF].

j c ∝ θ AHE [r 1 -r 2 ], (4.26

IREE-mediated THz emission

As of today, there are couple of papers speaking of an ultrafast spin-to-charge conversion via the IREE at the Ag/Bi interface [START_REF] Jungfleisch | Control of Terahertz Emission by Ultrafast Spin-Charge Current Conversion at Rashba Interfaces[END_REF][START_REF] Zhou | Broadband Terahertz Generation via the Interface Inverse Rashba-Edelstein Effect[END_REF]. The authors demonstrate the importance of the Ag/Bi stacking order and discuss the possible contribution of ISHE in Bi. Both works stress that they measure the superposition of the THz emitted via IREE and ISHE. However, the reported efficiency of the THz emission with the IREE is about 50 times smaller than the one achieved with Pt-based ISHE emitters.

Importantly, for the Ag/Bi interface where the IREE leads to the THz emission, the phase of the emitted THz pulse depends on the Ag/Bi layer ordering, which should be equivalent to changing of the sample geometrical orientation [START_REF] Zhou | Broadband Terahertz Generation via the Interface Inverse Rashba-Edelstein Effect[END_REF]. As seen from Fig. 4.12, the detected THz pulse is the sum or difference of the IREE contributions occurring at the Ag/Bi interface and Additionally, the Schottky barrier may explain the enhancement of the THz emission at the FM/Semiconductor interface due to the difference in the FM and semiconductor work functions [START_REF] Vetter | Observation of carrier concentration dependent spintronic terahertz emission from n -GaN/NiFe heterostructures[END_REF]. This may establish a "stray" electric field, leading to the Rashba splitting at the FM/semiconductor interface. As a result, they find that the THz generation efficiency heavily depends on semiconductor doping, opening the way to the IREE emitter optimization.

Demagnetization-driven THz emission (DMG)

Back in 2004, E. Beaurepaire et al. reported the emission of the THz radiation by the ultrafast demagnetization (DMG) of the Cr(3 nm)/Ni(4.2 nm)/Cr(7 nm) film [START_REF] Beaurepaire | Coherent terahertz emission from ferromagnetic films excited by femtosecond laser pulses[END_REF]. They suggested that the emitted THz electric field, polarized along the y axis and propagating along the z direction, is proportional to the second time derivative of the ultrafast demagnetization:

E y (t) ∝ µ 0 4π 2 r ∂ 2 M x ∂t 2 (t -r/c), (4.27)
where r is the distance to the magnetic dipole. However, they stress that this is only a phe-

Influence of the pump wavelength on the THz emission

As the last experiment, we study the influence of the pump wavelength on the THz emission from the CoFeB/MgO samples and the reference CoFeB(5)/Pt(3) emitter. For this experiment, we have set pump fluence to the fixed value of 159 µJ cm -2 for two different pump wavelengths (800 nm and 400 nm). The absorbed optical density of energy was estimated to differ only within ±10 % margin for the two pump wavelengths. The direction of the external magnetic field H and sample orientation (Front Face) was kept fixed through the experiment.

As the first benchmark, we compare 800 nm (solid line) and 400 nm wavelength (dash-dot line) excitation of the CoFeB(5)/Pt(3) emitter, shown in the Figure 4.16a. For this ISHE-based STE, we do not observe a significant change in the shape, spectrum, and amplitude of the emitted E THz . This observation is in good agreement with the literature [START_REF] Th Papaioannou | Efficient Terahertz Generation Using Fe/Pt Spintronic Emitters Pumped at Different Wavelengths[END_REF][START_REF] Beigang | Efficient terahertz generation using Fe/Pt spintronic emitters pumped at different wavelengths[END_REF].

As the next test, the same experiment was conducted for the CoFeB(5)/MgO(6) sample, and is displayed in the Fig. 4.16b. We note drastic change in the detected THz spectrum for this emitter (see inset of Fig. 4.16b) which results in more broadband THz emission, in contrast with CoFeB(5)/Pt(3). In addition, CoFeB(5)/MgO(6) excited at 400 nm wavelength leads to the complete reversal of THz polarity. We should note, that this behaviour is completely reversible, and sample is not modified in the process. Furthermore, the same experiment was performed for Finally, the pump wavelength dependence measured for the CoFeB(20)/MgO(6) is shown in Fig4.16d. Again, this sample behaves differently from the CoFeB(5)/MgO(6) bilayer. Figure 4.16d demonstrates that CoFeB(20)/MgO(6) excited at 400 nm wavelength does not show the modification in the polarity and spectral components of emitted THz radiation, but results in the three-fold increase of the E THz amplitude.

Discussion

First off all, we will speak about the "prototypical" ISHE-based emitters, i.e., the conventional CoFeB(5 and 20)/Pt(3) bilayers. We observe that the THz emission from CoFeB(5 and 20)/Pt [START_REF] Kampfrath | Terahertz spin current pulses controlled by magnetic heterostructures[END_REF] samples is affected by the external magnetic field orientation and is polarized perpendicular to the direction of external magnetic field H (see Figs. 4.13b and e). Next, Figures 4.14b andc demonstrate the THz waveform polarity reversal when bilayer is excited from the side of the substrate (Front Face) or from the side of the Pt capping layer (Back Face). Finally, we find that THz emission from CoFeB(5)/Pt(3) is not affected by the change of the pump wavelength, as seen from Figure 4.16a. All these observations are in perfect agreement with previously published results [START_REF] Kampfrath | Terahertz spin current pulses controlled by magnetic heterostructures[END_REF][START_REF] Seifert | Efficient metallic spintronic emitters of ultrabroadband terahertz radiation[END_REF][START_REF] Th Papaioannou | Efficient Terahertz Generation Using Fe/Pt Spintronic Emitters Pumped at Different Wavelengths[END_REF][START_REF] Beigang | Efficient terahertz generation using Fe/Pt spintronic emitters pumped at different wavelengths[END_REF].

For the novel samples in this study, i.e., CoFeB/MgO bilayers, we find that the polarity of the emitted THz is sensitive and perpendicular to H (see Fig. [START_REF] Sebastian | Spintronics with Terahertz Radiation : Probing and driving spins at highest frequencies[END_REF][START_REF] Seifert | Frequency-Independent Terahertz Anomalous Hall Effect in DyCo5, Co32Fe68, and Gd27Fe73 Thin Films from DC to 40 THz[END_REF]. One should note that we neglect the possible existence of the Rashba interface at the CoFeB/Pt interface [START_REF] Hawecker | Spin Injection BIBLIOGRAPHY Efficiency at Metallic Interfaces Probed by THz Emission Spectroscopy[END_REF]. With this phenomenological formulation, we will proceed with the discussion about the sample geometry, pump fluence Φ and wavelength dependence, in order to evaluate the relative bulk and IREE contributions of CoFeB/MgO-based samples.

External magnetic field dependence: We now address why the direction of the external magnetic field H affects the polarization of the emitted THz radiation. Following the discussion of Section 4.2 and Eq.4.25, the direction of the converted transverse charge current (j c ) is defined as the vector product between spin-polarized current flow (j z c,s ) and direction of the FM macroscopic magnetization (M ), defining the polarization of the spin current σ s as:

j c (t) ∝ i M (t) × j z c,s (t), (4.29)
where the sum over i represents all possible spin-to-charge (SCC) contributions to the transverse transient charge current j c (t). From the latter equation, we see that the direction of generated j c should be sensitive and perpendicular to M . As the sample is subjected to an external magnetic field H, its macroscopic magnetization M will be aligned parallel to H. Therefore, from the Eq.4.29 the polarization of the emitted E THz via SCC must be sensitive to H direction. As shown in the Figure 4.13b, the rotation of the H by 180°leads to the 180°THz polarity reversal for both CoFeB/Pt and CoFeB/MgO samples. This confirms the magnetic origin of the THz emission from CoFeB/MgO bilayers.

The following discussion will be split into two parts: first, we address the difference in the THz

Low-fluence regime (pristine samples):

This subsection deals only with the unchanged samples, i.e., that were not exposed to laser fluence above Φ crit ≈ 650 µJ cm -2 .

Sample geometry dependence:

As the first step, we address the possible origin of the THz polarity reversal due to the different sample geometry. Since the IREE, as ISHE, is an injectionbased SCC mechanism, the polarity of the emitted THz should depend on the j z c,s direction in respect to the Rashba interface. Consequently, the sample rotation around the y-axis will change the direction of j z c,s (j z c,s → j -z c,s , see Fig. 4.14a) and, according to Eq.4.29 it will lead to the polarity reversal of the emitted THz field. This effect is well known for the ISHE-based emitters [START_REF] Seifert | Efficient metallic spintronic emitters of ultrabroadband terahertz radiation[END_REF][START_REF] Seifert | Ultrabroadband single-cycle terahertz pulses with peak fields of 300 kV cm-1 from a metallic spintronic emitter[END_REF], and an identical experiment conducted by Vetter et al. [START_REF] Vetter | Observation of carrier concentration dependent spintronic terahertz emission from n -GaN/NiFe heterostructures[END_REF] (Front and Back Face pumping) on the n-GaN/NiFe bilayers IREE-based emitters. As seen from the Figure 4.14b and c, for the CoFeB(5)/MgO(6) STE we observe the complete E THz polarity reversal between the Front Face and Back Face sample orientations (see red and grey curves). Hence, as a first element of conclusion, we can attribute the observed THz polarity reversal in CoFeB(5)/MgO [START_REF] Richard | Understanding Solids[END_REF] to the presence of the injection-based IREE SCC mechanism. Now we discuss the symmetry of the bulk SCC contribution to the emitted THz. From the literature, it is well known that the efficiency of the injection-based ISHE emitters scales inversely with the FM layer thickness [START_REF] Yang | Powerful and Tunable THz Emitters Based on the Fe/Pt Magnetic Heterostructure[END_REF][START_REF] Sasaki | Annealing effect on laser pulse-induced THz wave emission in Ta/CoFeB/MgO films[END_REF][START_REF] Garik Torosyan | Optimized Spintronic Terahertz Emitters Based on Epitaxial Grown Fe/Pt Layer Structures[END_REF]. Moreover, for the Co films, Huang et al. demonstrated that the bulk SCC contribution increases with the FM layer thickness (i.e., ξ Co(30 nm) > ξ Co(5 nm) ) to the point that the THz amplitudes detected from 30 nm Co are not affected by the deposition of the HM Tantalum (ξ Co(30 nm) > γ Ta ) [START_REF] Huang | Direct observation of terahertz emission from ultrafast spin dynamics in thick ferromagnetic films[END_REF]. The combination of these reports indicates that with an increase of the FM layer thickness, one should simultaneously increase the bulk contribution (ξ CoFeB ), while decreasing the injection-based SCC contribution (γ P t and β MgO ) to the total emitted E tot THz . This peculiar effect is usually attributed to the short hot-electron spin diffusion length in the CoFeB (in the order of couple of nm) [START_REF] Seifert | Efficient metallic spintronic emitters of ultrabroadband terahertz radiation[END_REF][START_REF] Garik Torosyan | Optimized Spintronic Terahertz Emitters Based on Epitaxial Grown Fe/Pt Layer Structures[END_REF]. Since the THz emission from the bulk contribution (AHE, DMG, or combination of both) should be governed by the spin-polarized current flow inside the FM layer, its direction should not be altered upon the sample flip. As a result, the absence of the E THz polarity reversal between Back and Front Face configurations should be a strong indication of the bulk-dominant SCC. Indeed, as shown in the Low-fluence regime summary: As a result of the previous subsection discussion, we find that the ultrafast IREE-driven THz emission from the CoFeB(5)/MgO(6) bilayers is roughly 96 times less efficient compared to the reference ISHE-based CoFeB(5)/Pt(3) emitter. These results are in stark contrast with the nearly-equivalent conversion efficiency of the quasi-DC FMR experiments [4], which again stresses the importance of the carriers energy distribution for the IREE mediated spin-to-charge conversion. In fact, the band structure-dependent IREE mechanism can be detrimental for the efficiency of the ultrafast spin-to-charge conversion, since most Rashba interfaces favor the conversion close to the Fermi level [4,[START_REF] Holanda | Longitudinal spin Seebeck effect in permalloy separated from the anomalous Nernst effect: Theory and experiment[END_REF][START_REF] Dieny | Perpendicular magnetic anisotropy at transition metal/oxide interfaces and applications[END_REF]. Therefore, one can expect a reduced efficiency from the hot electrons, mainly responsible for the ultrafast superdiffusive spin current transport. Within the 5 nm thick layer with a nearly homogeneous gradient of the light absorption, the energy distribution of the injected electrons carrying angular momentum is indeed highly nonlinear and band structure-dependent, and can be relatively broad and even structured [START_REF] Melnikov | Ultrafast transport of laser-excited spin-polarized carriers in Au/Fe/MgO(001)[END_REF][START_REF] Rhie | Femtosecond Electron and Spin Dynamics in Ni/W(110) Films[END_REF]. Hot-electron distribution was reported to spread up to about 1.5 eV from the Fermi level in the early stage of the optical excitation, with a typical center weight around 0.2 eV for thermalized carriers at relevant timescales [START_REF] Rhie | Femtosecond Electron and Spin Dynamics in Ni/W(110) Films[END_REF]. In addition, several Rashba bands could simultaneously participate in the spin-to-charge conversion (as sketched in the Figure 4.17c), which may result in decreased efficiencies.

CHAPTER 4. ULTRAFAST SPIN-TO-CHARGE CONVERSION

To conclude this part of the discussion, we present the Table 4.1, where we compare the sensitivity of the ISHE, bulk, and IREE conversion mechanisms on the sample geometrical orientation and pump wavelength. Therefore, we deduce that the THz emission from CoFeB(5)/MgO(6) bilayer is mainly mediated by the IREE mechanism. In contrast, the THz generation in CoFeB(20)/MgO [START_REF] Richard | Understanding Solids[END_REF] emitter mainly occurs via spin-charge conversion in the bulk of the CoFeB layer, and wavelength sensitivity is explained by more important FM light absorption gradient at 400 nm. The additional data on the CoFeB/Pt samples and power-dependence measurement for CoFeB/Pt and CoFeB/MgO on the Si substrate are presented in Appendix D and Appendix F, respectively.

High-fluence regime:

In this last subsection, we demonstrate that the delicate Rashba-split bands in CoFeB(5)/MgO [START_REF] Eckert | Disputed discovery: the beginnings of X-ray diffraction in crystals in 1912 and its repercussions[END_REF] emitter could be modified or even destroyed at high laser fluence, leading to bulk-like THz emission. Here, we address the modification of the "test" CoFeB(5)/MgO(5) sample, when exposed to Φ > 650 µJ cm -2 . As shown in Figure 4.15a, the pristine (Serie 1) CoFeB(5)/MgO(5) bilayer emits a similar waveform and has nonlinear pump-fluence dependence in Φ < 650 µJ cm -2 range, just as CoFeB(5)/MgO [START_REF] Richard | Understanding Solids[END_REF]. However, when the CoFeB(5)/MgO(5) sample was exposed to Φ > 650 µJ cm -2 , we observed the "threshold" effect, which resulted in a drastic change in both THz waveform and E THz (Φ) dependence (see Figs 4.15a, Serie 2 and Serie 3). Furthermore, the measured linear E THz (Φ) dependence for Serie 2 and Serie 3 measurements (see Fig. for the CoFeB(20)/MgO(6) sample. Of course, at this moment, we can completely rely on the latter guesstimate of the relative amplitudes of IREE and bulk contributions using the permanent "modification" of the CoFeB(5)/MgO(5)* bilayer, since the absence of the Rashba states in the CoFeB(5)/MgO(5)* needs to be confirmed with ARPES or XPS measurements, if possible.

Conclusions and perspectives

In this chapter, we have discussed the THz emission as a result of the ultrafast spin-to- to-charge conversion mechanism. On the first hand, we find that our measurements of CoFeB/Pt emitters are in the perfect agreement with the literature. On the other hand, we observed the strong thickness dependence of THz emission from the CoFeB/MgO bilayers, which we attribute to the combination of the spin-to-charge conversion in the bulk of the CoFeB ferromagnet and at the CoFeB/MgO Rashba interface. For the CoFeB(20 nm)/MgO(6 nm) sample, it appears that the bulk contribution is the dominant spin-to-charge conversion mechanism. In particular, we report that the sample reversal does not affect the polarity of the emitted THz, as expected from the spin-to-charge conversion inside the bulk of CoFeB layer. This claim is supported by the linear E THz amplitude -pump fluence dependence, previously reported for the thick single-layer ferromagnetic films [START_REF] Huang | Direct observation of terahertz emission from ultrafast spin dynamics in thick ferromagnetic films[END_REF]. In contrast, in the CoFeB(5)/MgO(6) emitter we find that Inverse Rashba sample is directly related to the spin-to-charge conversion at the Rashba interface. To the best of our knowledge, this finding was never mentioned in the literature before. Finally, we show that at the high laser pump fluence (higher than 650 µJ cm -2 ) one may permanently modify or damage the delicate Rashba interface, which leads to the purely bulk spin-to-charge conversion.

As the future prospects, one can imagine tuning the Rashba bands to favor the conversion at high carrier-energies. This may result in a highly efficient, wavelength-sensitive spintronic device, allowing for an easy tunability of the generated THz waveform amplitude and spectrum. In addition, the estimation of the electrooptic response function, which includes the spatio-temporal transformation of the emitted THz pulse may allow us to access the shape of the transient charge current generated in the bulk of the CoFeB and at the Rashba interface. This information, complemented with a future in-lab time-resolved magneto-optical Kerr effect setup will certainly facilitate the first-principle calculations, anticipated in the near future.

General conclusion

In this thesis, we study how terahertz (THz) radiation could be used to excite or probe the ultrafast dynamics of electron, phonon, and spin in solids and nanostructures. In particular, we address the generation of coherent phonons in metals such as Chromium (Cr) and Aluminum (Al) and narrow-band topological insulator Bismuth Telluride (Bi 2 Te 3 ) with picosecond THz pulses.

In addition, we demonstrate how the specific aspects of the THz emission from optically-excited CoFeB/Pt and CoFeB/MgO magnetic nanostructures may provide insights into the relevant ultrafast spin-to-charge interconversion mechanisms.

The experiments in this manuscript were done using the ultrafast nonlinear spectroscopy in the pump-probe geometry with 1 kHz 800 nm 3 mJ amplified laser. For the ultrafast excitation of Cr, Al, and Bi 2 Te 3 films, we have used the tilted-pulse front method for Lithium Niobate crystal As a first achievement of this thesis, we show that one can generate coherent phonons with pulsed THz radiation in the thin films of metals and narrow-band semiconductors. First, we discussed the THz excitation of 14 nm thick Cr and 20 nm thick Al films, and compared it to well-known 800 nm pump excitation. We find that the general shape of the time-dependent change in the 400 nm probe beam transmittance is similar for the THz and near-IR (NIR) excitation. We detect the thin metallic film first harmonic acoustic breathing mode for the NIR and THz pump excitation. Finally, comparing the electron (deformation potential) and lattice (thermoelastic) contributions to the total photoinduced stress for both types of excitation allowed us to identify thermoelastic stress as the main driving mechanism of acoustic phonon generation with THz. We conclude that the lattice heating is linked to the scattering of the accelerated electrons at the Fermi level, which leads to an ultrafast analogy of the well-known Joule effect.

The second achievement of this work is the investigation of the coherent optical phonon generation in Bi 2 Te 3 16 nm thin film with THz. We demonstrate that 1.3 ps THz pulse with the central frequency of 0.64 THz is able to excite symmetric A 1 1g 1.85 THz Raman active phonon. We show that the A 1 1g phonon amplitude increases quadratically with the THz field amplitude, indicating the second-order nonlinear process required to excite such coherent motion. We find as well that 132 GENERAL CONCLUSION A 1 1g phonon amplitude does not depend on the THz pump polarization and is sensitive to the change of the central frequency of the THz pulse. We then compared the experimental findings with recently proposed THz ionic Raman scattering and THz sum-frequency excitation models of Raman phonon excitation. At this moment, we can only qualitatively compare the phase of experimentally measured A 1 1g phonon with these two models and not the amplitude. Therefore, the first principle calculations under progress should shed new light on these observed mechanisms. Among other opened questions, preliminary anisotropic detection revealed the presence of an antisymmetric E 2 g 3.1 THz Raman active phonon, which will enrich the discussion about the highly-nonlinear coherent lattice mode control with THz in this correlated material.

In the last part of this thesis, we investigate the THz emission from the magnetic bilayers as the signature of the spin-to-charge conversion. We show that the THz emission from CoFeB/MgO heterostructures may result from the combination of the spin-charge interconversion in the bulk of the CoFeB ferromagnet and at the Rashba interface created by the deposition of the MgO on top of the ferromagnetic layer. Furthermore, we illustrate that by probing the sample sensitivity to the given experimental factor, such as pump wavelength and geometrical orientation, one can determine the dominant spin-to-charge conversion mechanism in the CoFeB/MgO bilayer. In addition, we show that the Rashba-dominant CoFeB(5 nm/MgO(6 nm) emitter is sensitive to the pump pulse wavelength, which leads to complete change in the phase and spectrum of the emitted THz. This finding could be beneficial for the tunability and improved efficiency of the Rashba-based devices.

In the future, we will try to address acoustic phonon generation with pulsed THz radiation in nanoparticles and multiferroic materials from these perspectives. In addition, we are planning to study the THz emission from the bare ferromagnetic Cobalt and Cobalt/MgO films, which may allow us to properly isolate the bulk and Rashba contributions to the ultrafast spin-to-charge conversion. Finally, the control over the pump pulse duration may allow probing the characteristic bandwidth of the spin-to-charge conversion, which is believed to be inversely proportional to the temporal bandwidth of the excitation pulse. To conclude, we show that THz radiation can be used to trigger and probe the rich interaction between multiple degrees of freedom in a condensed matter. However, further developments and research are needed to selectively excite and control electronic and magnetic properties of the solid-state, essential for the upcoming high-performance and low-energy consumption electronics that today industries and end-users are waiting for. La première partie de cette thèse est l'étude expérimentale de la génération de phonons cohérents avec des impulsions ultrabrèves térahertz et proche infrarouge dans des nanofilms de Cr, Al et de Bi 2 Te 3 . Grâce à une comparaison quantitative de ces deux types d'excitation, nous montrons de manière inédite que le processus de génération de phonons acoustiques par térahertz est principalement induit par la contrainte thermoélas-tique (effet Joule ultra-rapide).

Dans un deuxième temps, nous montrons aussi que l'impulsion térahertz peut exciter des modes actifs Raman par des processus non-linéaires optiques et phononiques.

La troisième réalisation de cette thèse est le contrôle par excitation optique ultra-rapide de la conversion spin/charge dans des bicouches nanométriques magnétiques à fort couplage spin-orbite. En particulier, nous discutons de l'effet Edelstein inverse ultra-rapide dans les hétérostructures CoFeB/MgO. Nous montrons qu'il est sensible à l'énergie des photons de pompe, ce qui ouvre des perspectives excitantes pour les émetteurs spintroniques ultra-rapides.

Title: Probing and controlling coherent-phonon and spin-charge conversion in nanostructures with ultrafast visible and THz light pulses Keywords: Terahertz spectroscopy, pump-probe, phonons, spin/charge conversion Abstract: The study of ultrafast coupling between electrons, phonons, and spins remains a major challenge in condensed state physics. This research must ultimately respond to various challenges posed by very high-frequency technologies.

The first part of this thesis is the experimental investigation of the coherent phonon generation with terahertz and near-infrared ultra-short pulses in nanofilms of Cr, Al, and Bi 2 Te 3 . By comparing two distinct excitation processes, we unprecedentedly show that the mechanism of acoustic phonon generation with terahertz is mainly driven by thermoelas-tic stress (ultrafast Joule effect).

As the next step, we also show that the terahertz pulse can excite Raman active modes through non-linear photonic and phononic processes.

The third achievement of this thesis is the picosecond control of the spin-to-charge conversion driven by the strong spin-orbit coupling in magnetic nanometric bilayers. In particular, we discuss the ultrafast inverse Edelstein effect in CoFeB/MgO heterostructures. We show that it is sensitive to the pump photon energy, which opens exciting perspectives of ultrafast spintronics emitters.

vi LIST OF FIGURES 2 . 7

 27 THz refractive indices and absorbance of the free standing Si, HDPE and Glass substrates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Influence of the x-cut Quartz on the THz spectrum. . . . . . . . . . . . . . . . . . 2.9 Reconstruction of the LiNbO 3 THz waveform after the Quartz waveplate. . . . . . 2.10 Evolution of LiNbO 3 electric field spectrum in the presence of x-cut Quartz plate. 3.1 Strain generation via the thermoelastic process. . . . . . . . . . . . . . . . . . . . . 3.2 Band-structure picture of the deformation potential stress. . . . . . . . . . . . . . . 3.3 Band structure sketch of the metal and topological insulator and experimental setup.

3. 4

 4 Transient response and acoustic phonon amplitude of a thin Cr and Al film excited with 800 nm and THz pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 THz pump: Bi 2 Te 3 thickness dependence on the coherent phonon signal. . . . . . . 3.6 The dielectric function of Cr and acoustic phonon amplitude in the absorbed density of energy for THz and NIR excitation. . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Thin metallic films: Calculated dynamics of the photoinduced stress and associated phonon spectrum for NIR and THz excitation. . . . . . . . . . . . . . . . . . . . . 3.8 The ARPES measurement of Bi 2 Te 3 film and the extended frequency range of the thermoelastic stress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Photonic and ionic mechanisms of Raman phonon excitation. . . . . . . . . . . . . 3.10 Anharmonic coupling between IR and Raman modes. . . . . . . . . . . . . . . . . 3.11 Sketch and Raman spectra of the optical phonon modes in the Bi 2 Te 3 . . . . . . . . 3.12 Bi 2 Te 3 THz pump/400 nm probe: Time-delay zero definition. . . . . . . . . . . . . 3.13 Bi 2 Te 3 , NIR versus THz excitation: extraction of the optical phonon part of the signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 Bi 2 Te 3 A 1 1g generation efficiency -E THz field amplitude dependence. . . . . . . . . 3.15 Bi 2 Te 3 : Influence of the THz pump spectrum on the optical phonon part of the signal. 3.16 Bi 2 Te 3 response measured in polarization sensitive detection configurations. . . . . 3.17 Two proposed pathways of A 1 1g phonon excitation in Bi 2 Te 3 . . . . . . . . . . . . . . 3.18 Two proposed models of A 1 1g phonon excitation: experiment vs. phenomenological modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 -

 1 Figure 1 -Shades of the electromagnetic spectrum. From nict.go.jp website.

4 CHAPTER 1

 41 . THEORETICAL BACKGROUND a) Famous x-Ray diffraction experiment that was proposed by Max von Laue and performed by Walter Friedrich and Paul Knipping, exhibit at Deutsches Museum in Munich [5]. b) Rock salt crystal. The periodical arrangement of the Na and Cl atoms on the lattice is responsible for the electronic, phononic and magnetic properties of the NaCl. From Wikipedia.

Figure 1 . 1 -

 11 Figure 1.1 -Experimental setup proposed by Max von Laue and NaCl crystal (rock salt).

1. 2

 2 Atoms, electrons, and spins in condensed matter Bravais lattices with cubic symmetry: simple cubic (SC), body centered cubic (BCC) and face centered cubic (FCC). Redrawn from "Physics in a nutshell". b) Miller indices for different crystal planes. c) Projection down the c-axis, showing all four equivalent 110 planes. Adapted from [6]. d) Bravais lattice of the Zinc-Blende ZnTe crystal. e) (110) oriented ZnTe crystal plane.

Figure 1 . 2 -

 12 Figure 1.2 -Three cubic Bravais lattices, Miller indices for different crystal planes and Bravais lattice of ZnTe crystal.

  a) Diatomic crystal chain and transverse optical and acoustical waves. Vectors vn, un denote the atomic displacement along the acoustic wavevector k of ions with masses M1 and M2 respectively. C is the spring constant, a is the lattice repetition rate. b) Phonon dispersion relation for the diatomic crystal chain. So called phononic bandgap, defined as 2C M1 -2C M2 , defines the region with no solutions to the atomic equation of motion.

Figure 1 . 3 -

 13 Figure 1.3 -Diatomic crystal chain with respective transverse acoustical and optical waves and its dispersion relation.

. 1 )

 1 a) Phonon dispersion relation of a ZnTe crystal. Hollow symbols are experimental points, and lines are theoretical calculations. From [8]. b) First Brillouin zone of the FCC lattice, with high-symmetry points.From Wikipedia.

Figure 1 . 4 -

 14 Figure 1.4 -ZnTe phonon dispersion curve and first Brillouin zone of the FCC lattice.

Figure 1 . 5 -

 15 Figure 1.5 -Schematic electron band structure of Metal and Insulator and ZnTe electron dispersion curve.

Figure 1 . 6 -

 16 Figure 1.6 -Fermi-Dirac distribution of the metal and insulator, and density of states for an insulator.

. 21 )

 21 The electron heat capacity c e dominates the lattice heat capacity c L for metals at low temperatures only when E F ≫ k B T . A derivative of Eq.1.21 leads to the linear relation c e (T e ) = λ e T with electron thermal conductivity λ e = 96.6 J.mol -1 .K -2 for copper for instance. For semiconductors, c e contribution to the heat capacity of the crystal dominates c L only in the high-temperature regime, when E F ≪ k B T , although not many materials can support that.

Figure 1 . 7 -

 17 Figure 1.7 -Shift of the circular Fermi surface under the presence of the electric field.

CHAPTER 1 . 1 )

 11 THEORETICAL BACKGROUND a) Depiction of atomic orbitals in molecules and solids. The principal quantum number n identifies electron shell when electrons with the identical orbital quantum number l from the subshell. The way electrons fill in shell and subshell levels is indicated by the arrows above. For example, because of the lower energy state, the 4s orbital fills before the 3d. Image is taken from Encyclopedia Britannica. b) (Classical view of the spin-orbit interaction. E generated by the nucleus in the rest frame of moving electron will transform into magnetic field Bso, interacting with electron spin. (2) LS coupling of orbital and spin angular momenta for atoms with Z < 10.

Figure 1 . 8 -

 18 Figure 1.8 -Sketch of electronic states in an atom and electron spin-orbit interaction.

m

  s . Electrons with the same n form an electron shell, and the principal quantum number can take integer values from 1 to 7. Orbital quantum number l defines an electron subshell within electron shell n, having n possible values from 0 to (n -1), as shown in Fig.1.8a. Hund's rules describe the population of electron shells and subshells. l defines electron orbital within a subshell, and m l gives a projection of the orbital angular momentum l on the given axis. Values of m l are integers in the range [-l; l]. m s gives the axis projection of intrinsic electron angular momentum, called spin s. For an electron, m s can take the fixed values of ±1/2, when two orientations of s along the CHAPTER 1. THEORETICAL BACKGROUND

CHAPTER 1 .Figure 1 . 9 -

 119 Figure 1.9 -Stoner model of ferromagnet and calculated density of states of bcc-Fe.

2 . 2

 22 we defined the electrical conductivity σ of a metal and introduced the average phenomenological electron and hole relaxation time τ e . Now, we want to explore the action of EM field E(t) = E 0 (exp[-iωt]) on nearly-free conduction-band electrons. The classical damped equation of motion, describing the change in electron position r in time is written as:

CHAPTER 1 .

 1 THEORETICAL BACKGROUND a) Complex Drude dielectric function of the metal, showing three different regimes of the light-matter interaction. Low-frequency Hagen-Rubens regime, with frequency-independent ε1. Relaxation regime with R = const, and Transparent regime, where the metal becomes transparent. b) Complex dielectric function of a narrow-band semiconductor, calculated from the Lorentz model. The Low-frequency regime is an analogy to the Hagen-Rubens regime in metals. Around resonance frequency ω0, photon absorption can excite the electrons over the bandgap, resulting in strong light absorption.

Figure 1 . 11 -

 111 Figure 1.11 -Free-electron Drude and restoring-force Lorentz models.

CHAPTER 1 .

 1 THEORETICAL BACKGROUND 3. (ω + 0 -1/τ ) < ω < ω n : Reflection range, analogy of Drude relaxation regime; 4. ω > ω n : Transparent regime, where reflectivity drops to zero, rendering material transparent.

CHAPTER 1 .

 1 THEORETICAL BACKGROUND a) Comparison between the intraband Drude and composite Drude-Lorentz contributions to the real ε1(ω) and imaginary ε2(ω) parts of the experimental dielectric function of gold. Inset: Interband contribution, defined as the difference between experimental data and intraband Drude model to the real and imaginary parts of the Au's dielectric function. Experimental data are taken from [26]. b) Representation of the three possible electron transitions in metals and insulators. Direct interband absorption, mediated by the electron-photon interaction. Indirect intraband or interband transitions, followed by the phonon emission, are required due to the momentum conservation law.

Figure 1 . 12 -

 112 Figure 1.12 -Intraband and interband contributions to complex dielectric function of gold and schema of allowed transitions in metals and semiconductors.

Figure 1 . 13 -

 113 Figure 1.13 -Polariton dispersion curve and low-and-high frequency contributions to ZnTe dielectric constant ε(ω).

Figure 1 . 14 -

 114 Figure 1.14 -SHG and OR for Continuous-Wave and Modulated laser pulses.

Figure 1 . 17 -

 117 Figure 1.17 -The three-temperature model sketch and temporal evolution of electron, phonon and spin temperatures of CoFeB film.

2 . 1 39 2. 2 39 2. 3 42 2. 3 . 1 43 2. 3 . 2 45 2. 4 53 2. 5

 2139239342314332454535 1.17b). The slow spin thermalization in CoFeB is explained with G e-m being ≈ 60 times smaller compared to G e-ph . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pump-probe spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . Terahertz generation and detection . . . . . . . . . . . . . . . . . . . Terahertz generation: optical rectification . . . . . . . . . . . . . . . . . Electrooptic terahertz detection . . . . . . . . . . . . . . . . . . . . . . . THz time-domain spectroscopy . . . . . . . . . . . . . . . . . . . . . . THz pulse shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

40 CHAPTER 2

 402 . EXPERIMENTAL METHODS a) Sketch of the pump-probe experiment in transmission. The probe pulse is delayed in time with respect to the pump beam. We record the relative change in the probe's transmission for the different temporal overlaps between the pump and the probe. b) Example signal, obtained with the Bi2Te3 15 nm film, while excited with 800 nm pump and probed with 400 nm pulse. The sharp increase in the ∆T /T corresponds to the temporal overlap between the pump and the probe and consequent electron subsystem excitation.

Figure 2 . 1 -

 21 Figure 2.1 -Two-color pump-probe: Setup geometry and a typical recorded ∆T /T variation of 15 nm Bi2Te3.

Figure 2 . 2 -

 22 Figure 2.2 -Schema: The two pump-probe setups used in this work.

Figure 2 .

 2 Figure 2.2b. Here, THz pulse with a central frequency of 0.64 THz is generated with the tilted-pulse front method, discussed later in the text. THz emitted from the lithium niobate crystal (LiNbO 3 )is collected and focused with a one-inch off-axis parabolic mirror (OAPM), having a two-inch focal length. The THz beam size was measured with a knife-edge method and was estimated to be equal to roughly 615 µm in diameter. The maximum THz pulse energy used in the experiment is around 1.2 µJ. This gives us the maximum value of a peak electric field in the order of 275 kV cm -1 . Details about the tilted pulse front method are discussed in the Section 2.3.1, and THz beam size characterization is shown in the Appendix A.

gr=v

  THz or n 800 nm gr = n THz , (2.6) a) The 4f-lens tilted-pump-pulse-front generation scheme. The diffraction angle θ d was set to 57.15°, with θ LN = 63°. Two lenses with f1 = f2 = 250 mm and f3 = f4 = 150 mm were used, with a magnification factor β = 0.6. The pump pulse front is tilted by the angle γ tilt , so that v 800 nm gr cos(γ tilt ) ≈ v THz , leading to the efficient THz generation. Figure adapted from [66]. b) The temporal shape and spectrum (inset) of the THz radiation emitted from the MgO:LiNbO3 source in the tilted pulse front experimental setup, measured with ⟨110⟩200 µm GaP. The E THz at the entry surface of the crystal was extracted by accounting for the response function of the detection crystal, discussed in the Section2.3.2. The THz field value was cross-verified with the calibrated pyroelectric detector. b) The linear polarization of the MgO:LiNbO3 THz source, measured with pyroelectric detector and Wire Grid polarizer.

Figure 2 . 3 -

 23 Figure 2.3 -Tilted pulse front generation layout and THz spectrum.

CHAPTER 2 .

 2 EXPERIMENTAL METHODS front, one can overcome the velocity matching restrictions inside the LiNbO 3 crystal. Using this technique, the group of K.Tanaka has achieved record generation efficiency of about 0.1 % with LiNbO 3 crystals, reaching the peak THz field values of 1.2 MV cm -1[START_REF] Hirori | Single-cycle terahertz pulses with amplitudes exceeding 1 MV/cm generated by optical rectification in LiNbO3[END_REF].

250 mm and f 3 =

 3 f 4 = 150 mm, leading to the magnification factors of β 1 = β 2 = 0.6. The deviation angle of θ = θ d -θ i ≈ 17.3°was achieved with grating period p = 1800 mm -1 and m = -1 diffraction order. The THz emitted from the LiNbO 3 crystal was collected with a one-inch off-axis parabolic mirror (OAPM) having a two-inch focal length. The THz temporal waveform, shown in Fig.2.3b, was recorded with a 200 µm ⟨110⟩ GaP crystal, in pair with two Wire Grid (WG) polarizers used as field attenuators. The estimation of peak THz power was done with the EOS response function (see Section 2.3.2) and with the help of calibrated pyroelectric detector, accounting for the focused THz beam profile (see Appendix A). In addition, the THz power measured as a function of the WG polarizer angle confirms the linear polarization of the LiNbO 3 source (see Fig. 2.3c), which is perpendicular to the polarization of the pump pulse.

  .9) a) Both 800 nm probe and THz pulses are incident onto the electrooptic sampling crystal (⟨110⟩ ZnTe).With the controllable temporal overlap between the probe and THz pulses (via mechanical delay line, for example), one can point-by-point measure the ellipticity of the probe beam proportional to the E THz .The ellipticity is then recorded by measuring the difference between the two orthogonal probe polarization components, equal to zero in the absence of the THz. b) THz detection geometry. In this thesis, we are always in E THz ∥ Ep, ϕ = ψ experimental configuration, which is kept constant for all THz detection experiments. Image is redrawn from[START_REF] Paul | Measurement and calculation of the orientation dependence of terahertz pulse detection in ZnTe[END_REF].

Figure 2 . 4 -

 24 Figure 2.4 -Principle of the field-resolved electrooptic detection.

  photodiodes.The different QWP angles α define the ratio between the effective EOS signal and the total laser intensity seen by the balanced photodiode. b) THz fields recorded for the ⟨110⟩ 0.5 mm ZnTe crystal. The distortion of the waveform for the low α is related to the dominance of the nonlinear contribution to the detected signal (see Eq.2.17). c) THz detectivity enhancement for the low-amplitude THz emitter CoFeB(20 nm)/MgO(6 nm). Inset: Normalized to the peak-to-peak field THz waveforms, showing the absence of distortion even at the low QWP angles α. d) THz emitted from the CoFeB(5 nm)/Pt(3 nm) before (α ≈ 45°) and after (α ≈ 0.7°) detectivity enhancement and its spectrum e), showing the x8.5 increase of the signal with negligible nonlinear distortion.

Figure 2 . 5 -

 25 Figure 2.5 -Electrooptic sampling detectivity enhancement.

  The calculated response function h EOS (ω, Ω) of the 500 µm thick ZnTe is shown in the Fig.2.6a. The rapid phase loss around 3 THz corresponds to the TO and LO phonon-polariton coupling with THz light, where f T O ≈ 5.3 THz [37]. For the ⟨110⟩ 200 µm thick GaP the response function is mainly limited by the sampling pulse bandwidth 2 , since the f T O ≈ 11 THz. In order to check the correctness of the calculated h EOS (ω, Ω), we measured THz emitted from the same ⟨110⟩ 500 µm thick ZnTe emitter using two distinct GaP and ZnTe detectors. The "raw" signals recorded with 500 µm thick ZnTe and 200 µm thick GaP crystals are shown in the Fig.2.6c.

CHAPTER 2 .

 2 EXPERIMENTAL METHODS Electrooptic response functions h EOS (ω, Ω) calculated for the ⟨110⟩ a) 500 µm thick ZnTe and b) 200 µm thick GaP crystals. c) The S(τ ) measured for the ZnTe and GaP detectors. While using the GaP crystal, we observe the roughly 10 times less signal compared to the ZnTe detector. d) Retrieved waveforms, obtained with the Eq.2.23.

Figure 2 . 6 -

 26 Figure 2.6 -ZnTe and GaP response functions and retrieval of the E THz shape.

CHAPTER 2 .

 2 EXPERIMENTAL METHODSRefractive index n and absorbance α of a) High-resistivity float-zone Si wafer, b) High-Density Polyethylene (HDPE) and c) Float-Zone Glass substrate. Shaded regions represent the unreliable frequency range, i.e. where neither n nor α could be estimated with the THz-TDS experiment. The values were calculated with Eq.2.29 and Eq.2.30. Dashed blue line corresponds to the THz-TDS resolution limit, defined by the response function of ZnTe.

Figure 2 . 7 -

 27 Figure 2.7 -THz refractive indices and absorbance of the free standing Si, HDPE and Glass substrates.

  .35) a) Schema of the THz beam-shaping experiment. The ZnTe crystal is used as the THz radiation and EOS THz detector source. THz filter is used remove the 800 nm pump residue. Both THz Wire Grid (WG) polarizer and WG analyzer are initially set for the maximum THz transmission. b) Sketch of the experimental geometry. x-cut Quartz, with its fast axis set at roughly 45°relative to E THz polarization direction. x-cut Quartz is then placed between the fixed WG polarizer and the rotating WG Analyzer. c) Recorded FFT amplitude in the absence of 3 mm Quartz crystal (black line), and for the given angle of the WG analyzer. d) Plot of the THz spectral centroid (see Eq.2.37) as a function of the WG analyzer angle.

Figure 2 . 8 -

 28 Figure 2.8 -Influence of the x-cut Quartz on the THz spectrum.

  .36) with x-cut Quartz thickness L = 3 mm. We took n o = 2.113 and n e = 2.156 values from the manufacturer, measured at 1 THz 5 , and do not account for the possible frequency dependent n e-o (Ω). The result of the calculation is shown in the Fig.2.8c as orange dash-dot line. For example, we see that the x-cut Quartz should be analogous to the half-waveplate at frequency of 1 THz, or act as quarter-plate at 0.5 THz. This is why we observe close to zero FFT amplitude at 1 THz for WG analyzer fixed at θ = 0°(behaves as λ/2 plate), and no change at 0.5 THz frequency (behaves as λ/4 plate).

  Figure 2.8d shows the Spectral Centroid as a function of the WG analyzer angle θ. As a result, due to the strong Γ(Ω) frequency dependence in Quartz, the THz spectral centroid of the ZnTe crystal can be tuned in 1.2-1.4 THz frequency range.

5 .

 5 See THz materials section on the Tydex website. a) The black line is the pristine THz waveform of the ZnTe 0.5 mm emitter. Red, blue, and green lines are the signals measured with the x-cut Quartz introduced between fixed WG Polarizer and rotating WG Analyzer. b) The FFT amplitude ratios between the reference waveform and signal sampled for the specified WG Analyzer angle. c) Pristine, or reference spectrum of the LiNbO3 THz source (black), and retrieved spectra of the LiNbO3 source for the given WG analyzer angle. d) Retrieved THz transients obtained by performing the Inverse Fast Fourier Transform (IFFT) to spectra presented in c). Signals shown in a) were multiplied by the Gaussian window of the constant width to eliminate ringing due to the water vapor absorption and THz reflection at the ZnTe-air interface.

Figure 2 . 9 -

 29 Figure 2.9 -Reconstruction of the LiNbO3 THz waveform after the Quartz waveplate.

Figure 2 . 10 -

 210 Figure 2.10 -Evolution of LiNbO3 electric field spectrum in the presence of x-cut Quartz plate.

. 39 )

 39 where E THz (Ω) is the pristine LiNbO 3 THz electric field, shown as the black line in Fig.2.9c. The reconstructed LiNbO 3 spectra for θ = 0°, 50°and 90°are shown in the Figure2.9c. By performing the inverse Fourier transform, we have access to the LiNbO 3 waveform, depicted in the Figure2.9d.Since we measure the transfer function of the complete WG Polarizer-Quartz-WG analyzer, this approach indirectly accounts for the WG analyzer frequency dependent THz attenuation factor, discussed in the Appendix A.

Figure 2 . 2 63 3. 2 . 1 63 3. 2 . 2 76 3. 3 77 3. 3 . 1 77 3. 3 . 2 91 3. 4

 226321632276377317732914 Figure 2.10a shows the FFT of the reconstructed E THz waveform for the 0-90°WG analyzer angles. The blue line shows the blue shift of the THz central frequency.Figure 2.10b links WG

(Bi 2 63 CHAPTER 3 .

 2633 evidence of acoustic and optical phonon generation with intense pulsed THz radiation. The first part starts by summarizing the theoretical background of the picosecond acoustic strain generation in metals and semiconductors through thermoelastic and deformation potential stresses. After this theoretical introduction, we will present our results of coherent acoustic phonon generation with THz. Then, to address the microscopic mechanism at play, we compare the experimental results of Near-IR (NIR) and THz excitation in metals such as nanometric films of Chromium (Cr) and Aluminum (Al) where no optical phonon exist. Finally, based on this comparison, we present the experimental evidence that thermoelastic stress in the studied metallic systems is the driving mechanism of coherent acoustic phonon generation with THz. In this work, we evidence an ultrafast Joule effect (carrier acceleration and electron-phonon scattering processes at the Fermi level in the presence of a THz pulse) that can lead to similar photoinduced strain amplitudes obtained by a lattice heating with interband NIR excitation. At the end of this part, to open new avenues, we show how it is possible to drive acoustic phonons in more complex material as the Bismuth Telluride Te 3 ) nanofilms. Such a system is a rich platform for studying phonon generation since we find that THz pulse can simultaneously generate both acoustic and optical phonons. Furthermore, this constitutes the second important part and achievement of this Chapter.The second part of this Chapter will introduce phenomenological models describing coherent optical phonon excitation in transparent and opaque solids, including recently proposed THz ionic Raman scattering (THz-IRS) and THz-sum frequency excitation mechanisms (THz-SFE)[START_REF] Först | Mode-Selective Control of the Crystal Lattice[END_REF][START_REF] Mills | Ionic contributions to the Raman tensor of insulators[END_REF][START_REF] Dominik | Sum-frequency ionic Raman scattering[END_REF]. Next, based on the discussion of our experimental results, we will suggest possible scenarios regarding the driving mechanisms of Raman active A 1 1g coherent optical phonon excitation in a 16 nm thick Bi 2 Te 3 film with high-peak field THz electric pulses. ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ

CHAPTER 3 .

 3 ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ a) Simplified schematic of picosecond acoustic pulse generation in metals. The metal film absorbs the laser pulse within a finite volume, given by the optical penetration depth ξ, leading to the non-homogeneous deposited energy profile. The rapid lattice thermal expansion or contraction leads to the emission of the strain pulse. b) From left to right: The laser pulse is locally absorbed by the electron subsystem, reaching the temperature Te. After the electron-electron thermalization, electrons transfer their excess energy to the lattice via electron-phonon coupling (G e-ph ). The lattice temperature increase will result in the contraction/dilatation of the lattice, leading to the emission of a strain pulse from the excited surface area. After[START_REF] Czerniuk | Picosecond Ultrasonics for the Modulation and Nanoscopy of Semiconductor Lasers[END_REF].

Figure 3 . 1 -

 31 Figure 3.1 -Strain generation via the thermoelastic process.

. 7 )a)

 7 The absorption of the laser pulse leads to the electron-hole pair creation, and the change in the electronic distribution δE. b) In turn, this shift in the electronic distribution δE modifies the interatomic interaction which leads to the strain generation (δV /V ). c) Over bandgap excitation (ℏω > Eg) leads to both deformation potential stress (δE) and thermal phonon emission. Adapted from[START_REF] Ruello | Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action[END_REF][START_REF] Mante | Electron-acoustic phonon coupling in single crystal CH3NH3PbI3 perovskites revealed by coherent acoustic phonons[END_REF].

Figure 3 . 2 -

 32 Figure 3.2 -Band-structure picture of the deformation potential stress.

CHAPTER 3 .

 3 ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ a) Sketch of the femtosecond time-resolved experiment in transmission geometry. b) Sketch of the electronic band structures of the metal and the degenerated n-doped Bi2Te3. Simple vertical arrows show the interband optical transition induced by the NIR pulse, and double black arrows represent carrier intraband acceleration by the THz electric field. The blue dashed lines represent the spin-polarized Dirac surface states.

Figure 3 . 3 -

 33 Figure 3.3 -Band structure sketch of the metal and topological insulator and experimental setup.

a)

  Transient optical transmission obtained with a THz or NIR pump excitation for a thin Cr film. Dashed lines correspond to the extracted oscillatory part of the signals. The pulse energy of the NIR is ⋍ 0.07 µJ and of the THz is ⋍ 1.2 µJ. b) Same as a) but for a thin 20 nm Al film. The pulse energy of the NIR pulse is ⋍ 0.04 µJ and the THz is ⋍ 1.05 µJ. c) THz pulse energy dependence of the acoustic phonon signal for the Cr (black) and Al (grey) thin films. d) NIR pulse energy dependence of the acoustic phonon signal for the Cr (red) and Al (dark red) thin films. Dash-dot lines are the respective linear fits.

Figure 3 . 4 -

 34 Figure 3.4 -Transient response and acoustic phonon amplitude of a thin Cr and Al film excited with 800 nm and THz pulses.

Figure 3 . 5 -

 35 Figure 3.5 -THz pump: Bi2Te3 thickness dependence on the coherent phonon signal.

  Fig.3.4c and d). In addition, we show that the period of the low-frequency oscillatory component present in the Bi 2 Te 3 transient response is directly proportional to the film thickness H (see Fig.3.5d) suggesting these oscillations are the mechanical resonances of the layers.

3 .

 3 Within the experimental THz frequency range. a) Dielectric function of Chromium in the THz frequency range. Experimental dielectric function of the Cr thin film measured by THz-TDS (dots and squares). The lines are fitted with the Drude model. b) Acoustic phonon amplitude as a function of the absorbed density energy for the Cr sample, calculated with Eq.3.19.

Figure 3 . 6 -

 36 Figure 3.6 -The dielectric function of Cr and acoustic phonon amplitude in the absorbed density of energy for THz and NIR excitation.

CHAPTER 3 .

 3 ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ Dynamics of the photoinduced stresses originating from the electrons (σ DP ) and from the non-coherent phonons (σ TE ) calculated for the prototypical 20 nm thick Aluminum film. The total absorbed energy density equal to ≈ 10 J cm -3 in the case of a) the NIR light pulse with duration of 165 fs, and b) Experimentally measured THz pulse. c) Calculated phonon spectrum (FFT of σ TE ) for the NIR excitation (red) and THz excitation (black) in the relevant acoustic phonon frequency range. The ratio between THz and NIR is displayed as blue line, where the two eigenmode frequencies for the Aluminum and Chromium nanometric films are marked with the arrows.

Figure 3 . 7 -

 37 Figure 3.7 -Thin metallic films: Calculated dynamics of the photoinduced stress and associated phonon spectrum for NIR and THz excitation.

CHAPTER 3 .

 3 ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ a) ARPES band structure for a thin film ARPES signal of a Bi2Te3 film (15 nm) measured at 85 K. The horizontal axis is the momentum along the ΓK direction of the Brillouin zone. From [142]. b) Calculated phonon spectrum after NIR excitation (red) and THz excitation (black) in the frequency range up to 3 THz. The ratio between NIR and THz is displayed in blue.

Figure 3 . 8 -

 38 Figure 3.8 -The ARPES measurement of Bi2Te3 film and the extended frequency range of the thermoelastic stress.

4 .

 4 With already included correction for the effective mass.

CHAPTER 3 .

 3 ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ a) The difference-frequency excitation mechanisms of a coherent Raman phonon: Impulsive stimulated Raman scattering, where ω1 -ω2 is resonant with Raman active phonon ΩR, and conventional ionic Raman scattering, leading to the shift of the Raman phonon potential. b) The sum-frequency coherent phonon excitation mechanisms: sum-frequency excitation (THz-SFE) and sum-frequency counterpart of ionic Raman scattering (THz-IRS). Adapted from [127].

Figure 3 . 9 -

 39 Figure 3.9 -Photonic and ionic mechanisms of Raman phonon excitation.

  3.24 including only one Raman Q R and one IR Q IR mode, taking into account only the lowest-order a) Parabolic energy potential of a QR phonon mode (dashed curve), displaced under the cubic coupling with QIR phonon mode (solid curve). From [2]. b) Dynamic response of the coupled IR QIR and Raman QR modes. The ion oscillation along QIR mode (red) leads to a directional displacement of QR mode (blue). This displacement is proportional to Q 2 IR .

Figure 3 . 10 -

 310 Figure 3.10 -Anharmonic coupling between IR and Raman modes.

CHAPTER 3 .

 3 ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ a) Optical phonon modes in Bi2Te3. The red and blue circles denote Te and Bi atoms, respectively.

Figure 3 . 11 -

 311 Figure 3.11 -Sketch and Raman spectra of the optical phonon modes in the Bi2Te3.

Figure 3 .

 3 Figure 3.11b displays the Raman spectra of Bi 2 Te 3 crystal, showing that one can probe specific modes selectively via polarization analysis of the probe pulse[START_REF] Först | Coherent Phonons in Bulk and Low-Dimensional Semiconductors[END_REF]. For example, modes containing only diagonal terms (A 1g ) will only cause intensity modulation of the transmitted or scattered light and therefore they are absent in depolarized spectrum in Figure3.11b. However, modes containing off-diagonal tensor terms (E g ) will only affect the polarization state of the scattered or transmitted light, which, ideally, can be probed only by the polarization sensitive anisotropic detection.

CHAPTER 3 .

 3 ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ a) Electrooptic trace of the THz pulse, and its fit by Eq.3.35. b) Experimental signal in THz pump-400 nm probe of the Bi2Te3 16 nm thin film at τ < 2 ps time scale (black curve). The signal is overlapped with squared THz electric field (dash dot red) and the THz absorbed energy (dash dot blue).

Figure 3 . 12 -

 312 Figure 3.12 -Bi2Te3 THz pump/400 nm probe: Time-delay zero definition.

.37) 7 .

 7 Deconvoluted from the detection crystal response.CHAPTER 3. ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ a) Measured change in transmittance of Bi2Te3 16 nm thick film for THz (black) and NIR (red) excitation. The blue lines are the background fit, defined by Eq.3.37. b) Extracted transient optical phonon part of the signal for the THz (black) and NIR (red) excitation. c) Fourier transform of the transients shown in b). Black dotted line corresponds to the spectrum with subtracted 1.85 THz oscillatory component. Compared to the NIR, THz excitation has a noticeable sideband, around 1.5 THz in the frequency domain. Plots are offset for clarity.

Figure 3 . 13 -

 313 Figure 3.13 -Bi2Te3, NIR versus THz excitation: extraction of the optical phonon part of the signal.

8 .

 8 Defined as convolution of the pump and the probe pulses. a) Experimental configuration: Two Wire Grid (WG) polarizers were introduced in the THz beam path. The rotation of the WG analyzer changed THz field amplitude or/and polarization. b) Amplitude of the optical phonon part of the signal plotted as a function of the E THz field amplitude for static WG analyzer with rotating WG polarizer (blue dots) and fixed WG polarizer, with rotating WG analyzer (red dots). Black dot and dash-dot lines are the respective linear and square with E 2 THz fits.

Figure 3 . 14 -

 314 Figure 3.14 -Bi2Te3 A 1 1g generation efficiency -E THz field amplitude dependence.

Figure 3 .

 3 14a shows the experimental configuration, where the THz pulse emitted from LiNbO 3 crystal propagates through a set of Wire-Grid (WG) polarizers. Two WG polarizers allow us to selectively change only E THz amplitude (fixed WG analyzer, rotating WG polarizer), or E THz amplitude and polarization (fixed WG polarizer, rotating WG analyzer).

Figure 3 .

 3 Figure 3.14b shows the Bi 2 Te 3 A 1 1g optical phonon generation efficiency as the function of E THz field strength when we change only E THz field amplitude (blue circles) or E THz field amplitude and

Figure 3 . 15 -

 315 Figure 3.15 -Bi2Te3: Influence of the THz pump spectrum on the optical phonon part of the signal.

9 .

 9 Related to the discrepancy in the values measured with calibrated pyrodetector and THz-TDS. a) Three different detection schemes: isotropic (blue box), anisotropic rotation (orange) and elipticity (red) sensitive detection. b) Transient optical phonon response of the Bi2Te3 16 nm thick film pumped with Ω0/2π ≈ 0.64 THz pulse, recorded in the isotropic (blue), anisotropic rotation (orange) and elipticity (red) detection configurations and c) their respective spectra. The arrows indicate the frequencies of the different coherent optical phonon modes, with values extracted from [174] and Table 3.1 d) The FFT transform of the phonon signal part for the >1.5 ps time delays.

Figure 3 . 16 -

 316 Figure 3.16 -Bi2Te3 response measured in polarization sensitive detection configurations.

Figure 3 .

 3 Figure 3.15c displays normalized to maximum A 1 1g amplitude as the function of THz central frequency Ω 0 /2π. We see approximately three times increase in the A 1 1g generation efficiency between central frequency of Ω 0 /2π=0.52 THz and Ω 0 /2π=0.73 THz. Remarkably, the measured change of the THz power, associated with the change of THz central frequency Ω 0 /2π increases only the factor of roughly 1.4, as shown in Figure 3.15d.

Figure 3 .

 3 Figure 3.16b depicts isotropic signal (blue line), in comparison to anisotropic change of δT /T | optic (orange and red lines). Interestingly, within the 0 to 1.5 ps time delay, we observe high-frequency

CHAPTER 3 .

 3 ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ a) Two proposed scenarios of the A 1 1g mode excitation: THz ionic Raman scattering (THz-IRS) and THz sum-frequency excitation (THz-SFE). Figure adapted from [127]. b) Spectrum of the E THz (violet) and E 2 THz (dash dot red), showing difference-frequency (DF) and sum-frequency(SF) components. Arrows indicate optical phonon modes in Bi2Te3.

Figure 3 . 17 -

 317 Figure 3.17 -Two proposed pathways of A 1 1g phonon excitation in Bi2Te3.

10 .

 10 The minus was chosen to better match the phase of experimentally measured A 1 1g phonon.

91 CHAPTER 3 . 2 E 1 u with E 2

 913212 ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ a) Comparison between the experimentally measured signal (black ) curve, THz-IRS (red curve) and THz-SFE (orange) models, with their respective driving forces shown around the time-delay zero (shaded color lines). b) Temporal overlap between the experimental E THz , and modeled Q THz . c) δT /T |optic versus Q A 1 1g displacement calculated with THz-IRS and THz-SFE models at the > 2 ps time delays.

Figure 3 . 18 -

 318 Figure 3.18 -Two proposed models of A 1 1g phonon excitation: experiment vs. phenomenological modeling.

CHAPTER 3 .

 3 ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ a) A 1 1g phonon amplitude, normalized by the THz pump power as a function of the central frequency of the THz pulse (blue), overlapped with Q A 1 1g calculated from THz-IRS and THz-SFE models (dashed and dotted lines). b) Different possible pathways of Raman phonon excitation in Bi2Te3 with pulsed THz radiation.

Figure 3 . 19 -

 319 Figure 3.19 -THz pump central frequency -A 1 1g phonon amplitude dependence and the sketch of possible scenarios of Raman active phonons generation in Bi2Te3.

CHAPTER 3 .

 3 ULTRAFAST GENERATION OF ACOUSTIC AND OPTICAL PHONONS WITH THZ

96 CHAPTER 4 .

 964 ULTRAFAST SPIN-TO-CHARGE CONVERSION a) The deposition of the MgO on top of the CoFeB will lead to the creation of the Rashba states at the CoFeB/MgO interface. The left column represents Fermi surfaces of opposite magnetization orientations for CoFe/MgO interface and given sign of the electron effective mass m * . b) The experimental configuration of the Ferro-magnetic resonance experiment. c) Detected spin-charge currents for CoFeB(5 nm)/Pt(3 nm) (blue closed dots) and CoFeB(5 nm)/MgO(6 nm) (red closed dots) with their respective fits. Inset: FMR line-widths as function of the RF exciting field frequency. Effective Gilbert damping of 5e-3 and 10e-3 can be extracted for CoFeB(5 nm)/Pt(3 nm) and CoFeB(5 nm)/MgO(6 nm)respectively. From[4].

Figure 4 . 1 -

 41 Figure 4.1 -The Rashba splitting (αR) calculated from the band structure of Co-Fe terminated CoFe/MgO interface and FMR spin-pumping experiments.

Figure 4 . 2 -

 42 Figure 4.2 -Different types of the electron current and spin-diffusion length.

E

  .H. Hall in 1879 [204] discovered the transverse deflection of the charged particles in metals subjected to the presence of an external magnetic field B ext . One can explain this observation in the means of the classical Lorentz force, which acts on moving charged particles with speed v: F = -e(E + v × B ext ). (4.5) About two years after, Hall discovered the same effect in ferromagnetic materials, which is of one order of magnitude stronger [205]. Newly discovered Anomalous Hall effect (AHE) was then combined with the Hall effect by Pugh and Lippert [206] to describe the total Hall resistivity (ρ tot xy ) CHAPTER 4. ULTRAFAST SPIN-TO-CHARGE CONVERSION a) The ordinary Hall effect. Lorentz force is acting on the driving current jc in the presence of the external field Bext, resulting in the transverse Hall current j HE c . b) The extraordinary or Anomalous Hall effect. The jc flow will generate the transverse spin-polarized current in the magnetic material with non-zero macroscopic magnetization M ̸ = 0 via the Anomalous Hall Effect. Figure adapted from [44].

Figure 4 . 3 -

 43 Figure 4.3 -The geometry of the Hall and Anomalous Hall effects.

. 7 )

 7 where σ AHE-int xy , σ AHE-sj xy and σ AHE-skew xy are the intrinsic, side-jump and skew-scattering contributions shown in Fig.4.4.

  ϵ n (k). The second term introduces the transverse (anomalous) electron velocity perpendicular to both applied E and Berry phase-curvature b n (k). b n (k) strongly depends on the electron band a) Intrinsic deflection. Electrons gain anomalous velocity perpendicular to the direction of the external electric field E that heavily depends on the electron band structure. b) Side-jump scattering. Electrons with opposite spins are deflected in the opposite directions due to the scattering of an impurity. c) Skew-scattering. Asymmetric, spin-dependent electron deflection as a result of the electron or impurity SOI. Adapted from[START_REF] Nagaosa | Anomalous Hall effect[END_REF]

Figure 4 . 4 -

 44 Figure 4.4 -Schema of Hall Effect contributions.

Figure 4 . 5 -

 45 Figure 4.5 -The geometry of the Spin Hall and Inverse Spin Hall effects.

a)

  The band structure of electrons in 2D system in the absence of the SOI (grey). In the presence of SOI, blue and red curves denote spin-split electron dispersion curves. Region I is the energy interval between the bottom of the bands, defined as ER and the binding energy ϵ0, with similar spin-dependent dynamics of the inner and outer branches. At high energy, inner and outer branches show opposite spin-direction dynamics. b) Angle-resolved photoemission spectroscopy (ARPES) intensity from the Au(111) surface state, showcasing the existence of the inner and outer electron branches. Adapted from[START_REF] Nechaev | Hole dynamics in a two-dimensional spin-orbit coupled electron system: Theoretical and experimental study of the Au(111) surface state[END_REF]

Figure 4 . 6 -

 46 Figure 4.6 -Schema of the parabolic spin-split band structure in pair with Rashba splitting observed in Au ⟨111⟩.

Figure 4 . 7 -

 47 Figure 4.7 -Schema of the spin-to-charge conversion at the Rashba interface.

a)

  Seebeck effect. Temperature gradient ∇T applied to the two metals in contact, with different transport properties and Seebeck coefficients K1,2, will lead to the charge flow along the junction ∝ ∇T (K1 -K2). b) Spin-dependent Seebeck effect. The different mobility of the majority and the minority electrons in the magnetic metal will result in the different spin-dependent Seebeck coefficientsSmaj and Smin respectively. The temperature gradient applied to the metal with the net magnetization will lead to the generation of the spin-polarized current ∝ ∇T (Smaj -Smin). c) Spin-Seebeck effect. The magnons excited by the temperature gradient ∇T will carry the pure spin current along the direction of the gradient. Dominant in the magnetic insulators. From[START_REF] Sebastian | Spintronics with Terahertz Radiation : Probing and driving spins at highest frequencies[END_REF].

Figure 4 . 8 -

 48 Figure 4.8 -Schemes of the multiple Seebeck effects.

  .21) a) Nernst effect. When thermal gradient ∇T is applied normal to the direction of external field Bext, the Nernst charge current j N E c is generated normal to both. b) Anomalous Nernst effect. In the materials with a spontaneous magnetization M , the Anomalous Nernst spin-polarized current j ANE s-c ∝ ∇T × M will be generated. c) Spin Nernst Effect. Different scattering rates for the electrons below and above Fermi level flowing in the different direction will generate the pure spin current inside the material.Adapted from[START_REF] Bose | Recent advances in the spin Nernst effect[END_REF]

Figure 4 . 9 -

 49 Figure 4.9 -Geometry of an Anomalous Nernst and Spin Nernst Effect.

a)

  photoexcitation of the Ferromagnetic (FM) layer gets injected into the Non-Magnetic (NM) layer with strong SOI, where it gets converted into the transverse charge current j ISHE c

Figure 4 . 10 -

 410 Figure 4.10 -Possible contributions to the THz emission, originating from the different ultrafast spinto-charge conversion mechanisms.

a)Figure 4 . 11 -

 411 Figure 4.11 -ISHE based emitters: Efficiency and comparison to conventional THz sources.

  2013, Kampfrath et al. reported the THz emission from FM metal/Metal heterostructures, such as Fe/Au and Fe/Ru[START_REF] Kampfrath | Terahertz spin current pulses controlled by magnetic heterostructures[END_REF]. They demonstrated that the polarity of the emitted E THz (t) depends on the sign of the Spin-Hall angle θ SHE and on the H direction. Furthermore, they show the influence of carrier mobility in Au and Ru on the amplitude and bandwidth of the detected THz radiation. Moreover, T.H. Dang et al. combined both FMR and THz-TDS to address the importance of the material-dependent electron transport properties and spin-dynamics on the spin-to-charge conversion[START_REF] Dang | Ultrafast spin-currents and charge conversion at 3 d-5 d interfaces probed by time-domain terahertz spectroscopy[END_REF].Later, Seifert et al. studied the ISHE-driven emission from more than 70 FM/NM bilayer configurations[START_REF] Seifert | Efficient metallic spintronic emitters of ultrabroadband terahertz radiation[END_REF]. They compared the bandwidth of the spintronic THz emitters (STE) to conventional OR-and photoconductive-based THz sources, as shown in Fig.4.11a and b. In the follow-up paper, they show the generation of the THz pulses with a peak amplitude of about 300 kV cm -1 , achieved by the upsizing of the magnetic trilayer, made from the FM "sandwiched" between two HM layers with opposite θ SHE . In addition, combining THz-emission and THz-TDS experiments, the THz conductivity and spin-diffusion length λ s in Pt, W, Cu in pair with the magnitude and sign of the spin Hall angle θ SHE were estimated[START_REF] Seifert | Terahertz spectroscopy for all-optical spintronic characterization of the spin-Hall-effect metals Pt, W and Cu80Ir20[END_REF][START_REF] Li | Laser induced THz emission from femtosecond photocurrents in Co/ZnO/Pt and Co/Cu/Pt multilayers[END_REF][START_REF] Zhang | Laser pulse induced efficient terahertz emission from Co/Al heterostructures[END_REF].

) where r 1

 1 and r 2 are the different hot-electron reflection coefficients at the Substrate/FM and FM/NM interfaces, respectively. For the case when r 1 ̸ = r 2 , in the thin FM layer with homogeneous light-absorption profile, the change in the geometrical orientation of the sample should lead to the THz polarity reversal. Their model explains the THz polarity flip when STE is excited from the side of the substrate or MgO layer but does not explain the absence of the sign-reversal in the thick samples. In addition, they performed an experiment where they vary the Pt concentration along with the FM layer, which resulted in the amplification of the THz emission or even to the reversal of the polarity of the emitted THz. The two-current model proposed by Zhang et.al. does not include the effect of the substrate (Quartz) and the capping layer (MgO) on the Pt distribution in the FM layer, which may lead

  The time-domain THz signals at different Bi thickness for a) Fe(2)/Bi(wedge), b) Fe(2)/Ag(2)/Bi(wedge), and b) Fe(2)/Bi(wedge)/Ag(2) respectively. The number inside each pair of brackets indicates the corresponding film thickness. The difference in the phase of the emitted THz and non-zero signal from the bare Bi wedge sample shows the importance of the interface stacking ordering and the composite nature of the detected THz. Redrawn from [258].

Figure 4 . 12 -

 412 Figure 4.12 -THz emission from the Inverse Rashba Edelstein Effect (IREE) at the Bi/Ag interface.

  nomenological model, and it is far too simple and does not describe all the mechanisms related to the ultrafast spin dynamics.A recent paper of W.Zhang et al. shows that one can access the ultrafast demagnetization dynamics from the THz emission experiments[START_REF] Zhang | Ultrafast terahertz magnetometry[END_REF]. Furthermore, they stress the phenomenological nature of the DMG and conclude that the THz emission from the MgO/Fe(10 nm)/MgO comes a). Schema of the geometry-dependent THz emission experiment. Front Face configuration denotes when the sample is excited from the side of the substrate and vice-versa for the Back Face. The sample rotation by 180°around the y-axis (or x) should only affect the direction of the injected spin-polarized current, j N M s-c . b) THz signal measured in the Front (color) and Back (gray) face configurations for all samples deposited on the Glass substrate. c) The geometry-dependent THz emission in the Front (color, glass substrate) and Back (gray, Si substrate) configuration for all samples in the study. The number is the scaling factor used to normalize the signal to the CoFeB(5)/Pt(3) emitter peak-to-peak amplitude.

Figure 4 . 14 -

 414 Figure 4.14 -Spintronic emitters: Sample orientation-dependent THz emission.

Figure 4 . 15 -

 415 Figure 4.15 -CoFeB/MgO: Laser pump fluence dependence.

CHAPTER 4 .

 4 ULTRAFAST SPIN-TO-CHARGE CONVERSION THz waveform emitted from the a) CoFeB(5)Pt(3), b) CoFeB(5)MgO(6), c) CoFeB(5)MgO(5)*, and d) CoFeB(20)MgO(6) samples. All samples were excited at the fixed pump fluence at 800 nm (solid line) and at 400 nm (dash-dot line). * star indicates that the sample was exposed to the high laser fluence Φ > Φcrit.

Figure 4 . 16 -

 416 Figure 4.16 -Influence of the pumping wavelength on the THz emission.

CHAPTER 4 .

 4 ULTRAFAST SPIN-TO-CHARGE CONVERSION CoFeB(5)/MgO(5)* 2 and is shown in Figure 4.16c. Interestingly, CoFeB(5)/MgO(5)* does not seem to be affected by the change of the pump wavelength, similarly to CoFeB(5)/Pt(3).

  4.13e), similarly to CoFeB/Pt emitters. However, for the geometry, pump fluence Φ, and pump wavelength measurements, the CoFeB/MgO emitters behave differently depending on the CoFeB thickness. For example, Figures 4.14b and c demonstrate that THz polarity reversal between the Back and Front Face bilayer orientation was observed only for CoFeB(5)/MgO(6), while we report no change for CoFeB(20)/MgO(6) sample. In addition, the excitation of the CoFeB(5)/MgO(6) emitter at 400 nm, compared to 800 nm pump wavelength leads to the THz polarity reversal and broadening of the THz spectrum (see 4.16b). The excitation of the CoFeB(20)/MgO(6) results in the three times more important E THz amplitudes, as depicted in Fig.4.16d. No E THz polarity/spectrum modification was observed for this sample. At last, pump Φ dependence has revealed the highly nonlinear increase in the emitted THz field amplitudes for the CoFeB(5)/MgO(6) bilayer (see Fig.4.15b), but clear linear dependence for CoFeB(20)/MgO(6) (Fig.4.15c). Moreover, Figure 4.15a reveals that the power dependence and E THz waveform of the "control" CoFeB(5)/MgO(5) sample 3 could be altered when exposed to pump fluence Φ > 650 µJ cm -2 . In addition, the E THz emitted from "modified" CoFeB(5)/MgO[START_REF] Eckert | Disputed discovery: the beginnings of X-ray diffraction in crystals in 1912 and its repercussions[END_REF] sample is no longer sensitive to the change of the pump wavelength, as shown in the Figure4.16c.Based on the Section 4.2.5 discussion and Figure4.10, multiple SCC mechanisms can contribute to the total detected E THz , and can be split into two "categories": the one that originates from the SCC within the bulk of the FM metal (AHE and/or DMG, for example), and another emerging from the addition of the capping layer like Pt or MgO (ISHE and IREE, respectively

emission behavior from CoFeB( 5 )

 5 /MgO[START_REF] Richard | Understanding Solids[END_REF] and CoFeB(20)/MgO(6) emitters in the low-fluence regime, and in the second part we focus on the consequences of the permanent CoFeB(5)/MgO[START_REF] Eckert | Disputed discovery: the beginnings of X-ray diffraction in crystals in 1912 and its repercussions[END_REF] sample modification.

Figure 4 . 123 CHAPTER 4 .

 41234 14 (orange vs grey curves), we observe the absence of the E THz polarity reversal between Back and Front Face configurations for the thick CoFeB(20)/MgO(6) sample. This means that the THz emission from CoFeB(20)/MgO(6) emitter is mainly governed by the bulk SCC (i.e. ξ CoFeB(20) > β MgO ). It is worth to note that this argument does not hold when comparing CoFeB(5)/Pt(3) and CoFeB(20)/Pt(3) since ξ CoFeB ≪ γ Pt , even for 20 nm thick CoFeB [251]. Laser pump fluence dependence: Let us now address the pump fluence Φ-E THz amplitude dependence. From the CoFeB(5)/MgO(6) measurement, shown in Figure 4.15b, we see the clear nonlinear increase in the emitted THz field amplitudes as a function of Φ. This could be explained by the competition of bulk ξ CoFeB and IREE β MgO SCC mechanisms (see Eq.4.28) having opposite signs, and possibly different SCC pump fluence efficiencies. At the same time, for the CoFeB(20)/MgO(6) emitter, we measure linear E THz (Φ) dependence. Such linear behaviour, previously reported for the bare Co films[START_REF] Huang | Direct observation of terahertz emission from ultrafast spin dynamics in thick ferromagnetic films[END_REF] again confirms our suggestion that the THz emission from CoFeB(20)/MgO(6) is dominated by the SCC in the bulk of the CoFeB layer (i.e., ξ CoFeB(20) > β MgO ). We do not expect the nonlinear behaviour from the bulk contribution since the pump-fluence dependent magnitude of the ultrafast demagnetization in the thin ferromagnetic layers is known to be linear for the fluence below 4 mJ cm -2[START_REF] Takahashi | Ultrafast demagnetization in NiCo2O4 thin films probed by time-resolved microscopy[END_REF][START_REF] He | Ultrafast demagnetization enhancement in CoFeB/MgO/CoFeB magnetic tunneling junction driven by spin tunneling current[END_REF][START_REF] Eschenlohr | Spin currents during ultrafast demagnetization of ferromagnetic bilayers[END_REF]. Regardless, time-resolved magneto-optical measurements for the set of CoFeB bilayers are to be measured in the future.Laser pump wavelength dependence: From the Figure4.16a we see that the change of the pump wavelength from 800 nm to 400 nm has no effect on the ISHE-based CoFeB(5)/Pt(3) emitter, as expected from the literature[START_REF] Th Papaioannou | Efficient Terahertz Generation Using Fe/Pt Spintronic Emitters Pumped at Different Wavelengths[END_REF][START_REF] Beigang | Efficient terahertz generation using Fe/Pt spintronic emitters pumped at different wavelengths[END_REF]. The pump wavelength-independent THz emission from CoFeB(5)/Pt(3) can be attributed to the fact that spin-dependent electron deflection and scatter-ULTRAFAST SPIN-TO-CHARGE CONVERSION a) Sketch of the excitation with 800 nm and 400 nm pump wavelength. Supposedly, for the same amount of photoexcited carriers, the energy distribution of spin-polarized charge current will be higher for 400 nm pump. b) Since Inverse Spin-Hall efficiency in the heavy metal is mainly defined by the Spin-Hall angle (θ SH ), it should not be sensitive to the hot-carrier energy distribution, resulting in equivalent E THz emission for 400 nm and 800 nm pump. c) For the Inverse Rashba Edelstein effect, the spin-polarized current with different energy hot-carrier energy distribution may couple to different Rashba bands, which may result in different spin-to-charge conversion efficiency or even change in the direction of the generated charge current jc. Please note the change of axes between a), b) and c).

Figure 4 . 17 -

 417 Figure 4.17 -Sketch of the pump-wavelength dependent excitation: ISHE vs IREE.

115 ;

 115 /MgO(6) = -ξ CoFeB(5) + β MgO (ℏω p ) = 1 CoFeB(20)/MgO(6) = -4.625 • ξ CoFeB(5) + 0.2 • β MgO (ℏω p ) = -1 170 . (4.32) Here, the minus sign corresponds to the opposite THz polarity of the bulk-dominant CoFeB(20)/MgO(6) sample compared to the CoFeB(5)/MgO(6) bilayer (see Fig.4.14c). Thus, solving this system of equations gives us the factors of roughly 96 and 580 for IREE and bulk SCC contributions, respectively. These factors, shown in Figure 4.19, correspond to the ratio between the peak-to-peak amplitude of the THz emitted from the refference CoFeB(5)/Pt(3) and the relative contribution to the THz measured from CoFeB/MgO bilayers. Please note that such rather "rough" subtraction can be performed only for the same experimental configuration, and it does not take into account any additional CoFeB thickness-related effects, apart from the change in ∇A z .

( 4 . 33 )

 433 4.15a) suggests the irreversible sample modification. We now compare the E THz waveform emitted from CoFeB(5)/MgO(5) (black solid) and CoFeB(5)/MgO(5)* (black dash-dot), shown in the Figure 4.15a. For the convenience, the same data is displayed in the Figure 4.20b, and complemented with THz transients recorded for the Back and Front Face sample geometrical orientations. From the Figure 4.20b we see that the THz emission from CoFeB(5)/MgO(5)* bilayer is somewhat symmetric, as in the case of bulk-dominant CoFeB(20)/MgO(6) emitter. Moreover, as shown in the Figure 4.16c, CoFeB(5)/MgO(5)* sample 800 nm or 400 nm wavelength excitation shows no significant effect on the E Thz amplitude and spectrum. This, compared to the reference ISHE-based emitter (CoFeB(5)/Pt(3), see Fig.4.16a) and bulk-dominant STE (CoFeB(20)/MgO(6), Fig.4.16d), strongly suggests the "disappearance"of the Rashba states, with preserved bulk contribution, such as ξ CoFeB(5) ≫ β MgO . Since this irreversible modification happened to the same sample, we could re-write the Eq.4.28 as.   E tot THz (CoFeB(5)/MgO(5)) = ξ CoFeB(5) • E bulk THz + β MgO • E IREE THz ; E tot THz (CoFeB(5)/MgO(5)*) = ξ CoFeB(5)* • E bulk THz .Of course, the latter equation is valid under the assumption that Rashba bands do not affect the SCC within the CoFeB layer, i.e. ξ CoFeB(5)* = ξ CoFeB(5) . In this case, the difference between the THz transients emitted by the CoFeB(5)/MgO(5) and CoFeB(5)/MgO(5)* should give us only the IREE contribution, shown as the red line in Figure 4.20c. For the CoFeB(5)/MgO(5) bilayer, we estimate the 1:2.75 ratio between the bulk and IREE contribution. This is a relatively good agreement with the estimation of IREE-to-bulk ratio of 1:6 made at the end of the low-fluence section. The factor of two difference between the low-fluence and high-fluence regime is most likely related to the fact that we used CoFeB/Pt to approximate the decrease in IREE contribution a). Schema of the geometry-dependent THz emission experiment. Front Face configuration denotes when the sample is excited from the side of the substrate and vice-versa for the Back Face. j IREE s-c is the spin-polarized current directed towards the Rashba interface. b) THz signal emitted from pristine CoFeB(5)/MgO(5) (solid) and CoFeB(5)/MgO(5)* (dash-dot) in the Front (color) and Back (gray) face configurations at 159 µJ cm -2 fluence and 800 nm pump wavelength. Temporal delay and waveform broadening for the Back Face orientation are due to the frequency-dependent THz absorption in the glass substrate. c) Extracted IREE contribution (red), obtained as the difference between the Front-Face signal, measured before and after sample modification. Bulk contribution is the CoFeB(5)/MgO(5)* waveform. The scaling factors indicate the ratio between the reference CoFeB(5)/Pt(3) emitter and the given signal. d) Normalized to 1 Fast Fourier transform of the c).

Figure 4 . 20 -

 420 Figure 4.20 -CoFeB(5)/MgO(5): Sample orientation-dependent THz emission.

  charge conversion from CoFeB/Pt and CoFeB/MgO magnetic bilayers. While CoFeB/Pt samples were chosen as the most characterized and best-performing bilayers to date, the choice of the CoFeB/MgO structures was stimulated by the recent discovery of the Rashba states created by the deposition of insulating MgO nanofilm on top of the ferromagnetic CoFeB layer. We show that by analyzing the effect of the sample geometrical orientation, laser pump fluence and wavelength dependence on the polarity and the shape of the emitted THz, one can identify the dominant spin-CHAPTER 4. ULTRAFAST SPIN-TO-CHARGE CONVERSION

  Edelstein effect contributes the most to the total detected E THz , despite the presence of the bulk spin-to-charge conversion. This conclusion is supported by the sample-geometry dependent THz emission characteristic to the injection-based THz emitters, and highly-nonlinear emitted THz field amplitude -pump fluence dependence that indicates the competition of the bulk and Rashba contributions.Consequently, we have estimated the 1:6 ratio between the amplitudes of the emitted THz fields driven by the bulk and Inverse Rashba Edelstein conversion in CoFeB(5)/MgO(6) emitter. This estimation makes the ultrafast Inverse Rashba Edelstein conversion in CoFeB(5)/MgO[START_REF] Richard | Understanding Solids[END_REF] nearly 100 less efficient than the Inverse Spin Hall effect in CoFeB(5)/Pt(3). At the same time, the Inverse Spin Hall and Inverse Rashba Edelstein effects are found to be equivalently efficient for the thermalized spin-polarized carriers[4]. Such difference in the Inverse Rashba Edelstein and Inverse spin Hall conversion rates at the ultrafast and quasi-DC timescales may indicate the reduced Rashba conversion efficiency when dealing with non-thermal distribution of injected spinpolarized carriers, since most Rashba bands favor spin-to-charge conversion close to the Fermi level[4,[START_REF] Holanda | Longitudinal spin Seebeck effect in permalloy separated from the anomalous Nernst effect: Theory and experiment[END_REF][START_REF] Dieny | Perpendicular magnetic anisotropy at transition metal/oxide interfaces and applications[END_REF]. Moreover, we evidenced pump wavelength sensitivity of the Inverse-Rashba Edelstein CoFeB/MgO emitters. While for CoFeB(20)/MgO(6) bilayer such change might be attributed to the more important light absorption gradient inside the CoFeB layer, the difference in THz emission (polarity of the emitted THz field, spectrum) observed for CoFeB(5)/MgO[START_REF] Richard | Understanding Solids[END_REF]

(LiNbO 3 )

 3 to generate the THz pulses with a peak electric field of about 275 kV cm -1 with a central frequency of 0.64 THz, that corresponds to a 1.2 µJ pulse energy. In the experimental section, we show how birefringent x-cut Quartz crystal could be used to shape the THz pulse, we calculate the electro-optic response function for the GaP and ZnTe electro-optic crystals and discuss the technique of THz time-domain spectroscopy sensitivity enhancement used for the detection of the weak THz fields in the Chapter 4.

134

  APPENDIX A. PYROELECTRIC CHARACTERIZATION OF THE LNO THZ SOURCE: a) THz power transmission measured in the function of the single Wire Grid (WG) polarizer angle θ. b) THz power transmission measured for the single (black line) and two (red line) Wire Grid (WG) polarizers in the THz beam path in the function of the first WG polarizer angle θ.

Figure A. 2 -

 2 Figure A.2 -Wire Grid polarizer characterization with the LiNbO3 THz source.

  Measurement with a) THz pyrocam i2S Vision, and b) the pyroelectric detector using knife-edge method.

Figure A. 3 -

 3 Figure A.3 -LiNbO3 THz beam size measurement.

Figure E. 1 -

 1 Figure E.1 -Pt based samples: THz metal stack amplitude absorption.

Figure E. 2 -

 2 Figure E.2 -MgO based samples: THz CoFeB and MgO amplitude absorption.

Table 1 .1 -Values

 1 of second order susceptibilities for ZnTe, GaP, BBO, LiNbO3 and GaAs.

	ZnTe	137	4.04
	GaP	49.6	1.94
	BBO	3.6	2.1
	LiNbO 3	336	30.9
	GaAs	131.1	1.43
		From [38]

  THz α THz cm -1

	CdTe (SC)	4.5	2.92	3.73	3.23	4.8
	GaAs (SC)	1.43	3.68	4.18	3.61	0.5
	GaP (SC)	0.97	3.18	3.57	3.34	1.9
	ZnTe (SC)	4.04	2.87	3.31	3.17	1.3
	GaSe (SC)	1.7	2.85	3.13	3.72	0.07
	LiTaO 3 (FE) 30.5	2.145	2.22	6.42	46
	LiNbO 3 (FE) 30.9	2.159	2.23	5.16	16
	DAST (OC)	77	2.38	3.31	2.4	150

1 eV-120 meV is called the THz range. Nowadays, terahertz pulses can be routinely generated CHAPTER 2. EXPERIMENTAL METHODS Material r eff (pm/V) n 800 nm 0 n 800 nm gr n

  .1 gives the frequency of the optical Bi 2 Te 3 modes in THz, measured with Raman, IR, and femtosecond spectroscopy, in pair with molecular dynamics (MD) simulations. As one can notice, all the optical modes are within 1 to 4 THz frequency range.

	The coherent optical phonon detection, in analogy to the expression established in the Chapter
	2, can be explained as the transmittance δT (or reflectance δR) modulation by the first-order
	Raman tensor (∂χ)/(∂Q) [161]:
	δT =	∂T ∂n	∆n ≈	∂T ∂χ	∂χ ∂Q	Q.	(3.34)

5.

With the theory provided by A.Subedi et.al.

[START_REF] Subedi | Theory of nonlinear phononics for coherent light control of solids[END_REF]

. 6. Superscripts denote different positions of the Te atoms.

Table 3 .1 -Bi2Te3

 3

optical phonon frequencies in THz.

  ). With this in mind, for two separate systems (CoFeB/Pt and CoFeB/MgO) this concept of combined THz CoFeB • E bulk THz + β MgO • E IREE THz , E tot THz (CoFeB/Pt) = ξ CoFeB • E bulk THz + γ Pt • E ISHE CoFeB , β MgO and γ Pt are the phenomenological coefficients describing the SCC efficiency inside the bulk of the CoFeB (ξ CoFeB ), at the Rashba interface (β MgO ), and due to the ISHE within the Pt layer (γ Pt ). ξ CoFeB is present in both CoFeB/Pt and CoFeB/MgO systems and can originate from DMG [259,266-268] and/or Anomalous Hall Effect (AHE)

		CHAPTER 4. ULTRAFAST SPIN-TO-CHARGE CONVERSION
	emission can be phenomenologically formulated as:
	  	E tot THz (CoFeB/MgO) = ξ THz ;	(4.28)
	where ξ

Table 4 . 1 -

 41 ISHE, Bulk, and IREE spin-to-charge conversion mechanisms sensitivity to the sample geometry and pump wavelength.

		Sample geometry Pump wavelength
	CoFeB(5)/Pt(3)	Sensitive	Not sensitive
	CoFeB(20)/Pt(3)	Sensitive	Not sensitive
	CoFeB(5)/MgO(6)	Sensitive	Sensitive
	CoFeB(20)/MgO(6)	Not sensitive	Sensitive
	ISHE	Sensitive	Not sensitive
	IREE	Sensitive	Sensitive
	CoFeB bulk interconversion	Not sensitive	Not sensitive

I(z) ∝ |E 0 exp(Kz)| 2

related to the excitation of excitons, which is not discussed here

Discussed more in the Chapter 4

From Shannon-Nyquist sampling theorem, the detection bandwidth of the 165 fs pulse is in the range of ≈ 3 THz.

Or signal-to-noise ratio.

1/e 2 radius of ≈ 615 µm measured by a knife-edge method, see Appendix A.

Reminder: star means the sample state after high-Φ exposure.

Which in its pristine condition was equivalent to CoFeB(5)/MgO[START_REF] Richard | Understanding Solids[END_REF].

i2S vision data-sheet can be downloaded here: link.

WG polarizer data-sheet can be downloaded here: link.

Acknowledgments

APPENDIX G. PYTHON DATA ACQUISITION SOFTWARE Bibliography

CHAPTER 4. ULTRAFAST SPIN-TO-CHARGE CONVERSION from ultrafast demagnetization in Fe. Importantly, they say that THz emission from single-layer ferromagnet could have a contribution of the coherent-phonon driven magneto-elastic effect.

From this brief literature overview, we see that there are many reports on different spin-tocharge (SCC) mechanisms at the ultrafast timescale. Therefore, the dominant mechanism of the SCC may vary depending on the sample orientation, the choice of FM metal/insulator and/or the capping layer. As a result, it might not be so straightforward to address and study the SCC conversion mechanism of interest. Therefore, in the next section, we will to propose a way how to experimentally identify the different contributions to the ultrafast SCC in CoFeB/MgO magnetic bilayers. At the end, we will compare the THz emission from the CoFeB/MgO samples and conventional CoFeB/Pt STEs.

Experimental results

Sample choice:

In this section, we measure the THz emission from the MgO-based spintronic emitters and compare the results to the well-known CoFeB/Pt samples. Here, we are interested in the efficiencies of the injection-based IREE (CoFeB/MgO), ISHE (CoFeB/Pt), and bulk (CoFeB) SCC mechanisms at the sub-picosecond timescales. Furthermore, we will try to identify the different driving mechanisms of the SCC based on the sample experimental geometry, pump-fluence dependence, and the influence of the pump wavelength on the THz emission. Consequently, we discuss THz emission from four different STEs: CoFeB(5)/Pt [START_REF] Kampfrath | Terahertz spin current pulses controlled by magnetic heterostructures[END_REF], CoFeB [START_REF] Slater | The Ferromagnetism of Nickel[END_REF]/Pt(3), CoFeB(5)/MgO [START_REF] Richard | Understanding Solids[END_REF] and CoFeB [START_REF] Slater | The Ferromagnetism of Nickel[END_REF]/MgO [START_REF] Richard | Understanding Solids[END_REF] on the SiO 2 and Si substrates, where number in the parentheses is the thickness of the preceding layer in nm.

We chose the CoFeB layer with an equivalence ratio of Co 40 /Fe 40 /B 20 due to the highest reported efficiency of the THz emission [START_REF] Seifert | Efficient metallic spintronic emitters of ultrabroadband terahertz radiation[END_REF] in pair with already measured THz conductivity [START_REF] Seifert | Terahertz spectroscopy for all-optical spintronic characterization of the spin-Hall-effect metals Pt, W and Cu80Ir20[END_REF].The capping layers, namely 3 nm of Pt and 6 nm of MgO were grown on top of the CoFeB in the same sputtering chamber. Pt layer was chosen as the "reference" since it is the most efficient spin-to-charge converting HM, and it is well characterized both in DC and at the picosecond timescale. As mentioned in the introduction, the deposition of MgO on top of the CoFeB layer creates the Rashba states inside the FM metal close to CoFeB/MgO interface [4]. All the samples were grown at the Condensed Matter Physics laboratory of IRAMIS (Saclay Institute for Radiation and Matter) by J.B. Moussy, A. Solignac, J.Y. Chauleau, and M Viret.

Experimental details:

The experiments were performed with the 1 kHz 165 fs amplified laser, operating at the central wavelength of 800 nm. The output laser beam was split 90/10, where 90% of the laser power was used to pump the STE, while the remaining 10% used for electrooptic detection.

STE with the size of 0.5 mm × 0.5 mm was fixed in the sample holder with a hole of ≈ 0.45 mm in diameter. Unfocused pump beam with the ≈ 0.48 mm beam diameter at FWHM was incident normal to the surface of the STE. The THz pulse, emitted along with the pump direction, was separated with the combination of two two-millimeter thick HDPE sheets followed by the halfmillimeter thick silicon wafer. The THz beam was then focused with one-inch gold, unprotected off-axis parabolic mirror (OAPM) with a six-inch focal length. The lateral position of the sample relative to the OAPM remained fixed through the experiment. The emitted THz was recorded with the EOS technique, discussed in the Section 2.4. The probe beam intensity was kept below the two-photon absorption limit of ZnTe, in pair with the constant optical power seen by the balanced photodiodes. In our experiment, the direction of the external magnetic field H was controlled by the pair of magnets introduced into the rotating sample holder, to insure the homogeneous magnetic field distribution across the sample. The applied magnetic field strength is in the order of 0.2 T, enough to saturate the CoFeB magnetization.

External magnetic field dependence and polarization of the emitted THz radiation

As the first experiment, we check the magnetic origin of the THz emission. For this, we install the samples (CoFeB covered either with Pt or MgO) in the sample holder with the two magnets, such as the pump beam first hits the substrate, as shown in Fig. 4.13a. The emitted THz was then recorded for two opposite orientations of the external magnetic field ±H, shown in Fig. 4.13b (solid and dotted lines). Figure 4.13b shows that all samples are sensitive to the direction of H, and 180°rotation of H in the xy-plane leads to the complete polarity reversal of the emitted THz.

We should note the slight difference in the emitted THz amplitude (up to 5 %) between waveforms recorded for ±H that can be attributed to the manual rotation of the magnets.

In addition, we have measured the polarization of the THz emitted from the CoFeB(5)/Pt(3) sample, using the pyroelectric detector and a Wire Grid (WG) polarizer, as shown in Fig. 4.13c.

For two orientations of the external magnetic field H (displayed in Fig. 4.13d), we find that the THz polarization is linear and perpendicular to the direction of the external magnetic field (see Fig. 4.13e), in perfect agreement with previously published results [START_REF] Kampfrath | Terahertz spin current pulses controlled by magnetic heterostructures[END_REF]. Furthermore, the THz polarization of the CoFeB(20)/Pt(3) and CoFeB(5 or 20)/MgO(6) bilayer is linear and follows the same tendency.

Sample geometry-dependent THz emission

Next, we study the THz emission from the CoFeB(Pt or MgO) heterostructures within two different geometrical sample orientations: we define the Front Face sample orientation when the pump beam is incident from the side of the substrate, while the excitation from the nonmagnetic layer will be called the Back Face configuration, with the two displayed in Figure 4.14a. For the fixed laser pump fluence Φ, Figure 4.14b shows E THz waveform, emitted in the Front Face (color) and Back Face (grey) experimental configurations. The Front Face signals were normalized to the peak-to-peak amplitude of the CoFeB(5)/Pt(3) emitter (blue line), with a scaling factor mentioned above respective signal in Fig. 4.14b. First, we observe the additional temporal delay for the THz emitted in the Back Face related to the emitted THz pulse propagation through glass. In addition, the broadening of the THz waveform in the Back configuration occurs due to the frequency-dependent glass THz absorption (see Section 2.4). The same effect was already observed before [START_REF] Li | Laser induced THz emission from femtosecond photocurrents in Co/ZnO/Pt and Co/Cu/Pt multilayers[END_REF][START_REF] Huisman | Femtosecond control of electric currents in metallic ferromagnetic heterostructures[END_REF][START_REF] Huisman | Spin-photo-currents generated by femtosecond laser pulses in a ferrimagnetic GdFeCo/Pt bilayer[END_REF]. Nevertheless, we report THz polarity reversal as a function of the sample orientation (color vs. grey line) for all the samples presented in Figure 4.14b, except for the CoFeB(20)/MgO [START_REF] Richard | Understanding Solids[END_REF].

To measure the undistorted THz waveform in the Back Face geometrical orientation, the samples were grown on the high-resistivity, float-zone silicone (Si) substrate. The pump fluence was then adjusted to equalize the absorbed density of energy in the CoFeB layer to the Front Face excitation 1 . All signals were normalized to the peak-to-peak amplitude of the CoFeB(5)/Pt(3) bilayer in the Front Face orientation. The Figure 4.14c shows a complete reversal of the CoFeB(5)/Pt [START_REF] Kampfrath | Terahertz spin current pulses controlled by magnetic heterostructures[END_REF] and CoFeB(20)/Pt(3) THz waveforms upon the sample flip around the y-axis. Such change in the THz polarity upon the flip of the ISHE emitter has been reported numerous times [START_REF] Seifert | Efficient metallic spintronic emitters of ultrabroadband terahertz radiation[END_REF][START_REF] Yang | Powerful and Tunable THz Emitters Based on the Fe/Pt Magnetic Heterostructure[END_REF][START_REF] Huisman | Spin-photo-currents generated by femtosecond laser pulses in a ferrimagnetic GdFeCo/Pt bilayer[END_REF][START_REF] Liao | Pulsed-laser epitaxy of topological insulator Bi 2 Te 3 thin films[END_REF].

CoFeB/MgO pump fluence dependence

In this subsection, we present the pump fluence dependence of the CoFeB/MgO samples, when the data for the CoFeB/Pt emitters is presented in the Appendix D. Here, we introduce the "control" samples CoFeB(5)/MgO [START_REF] Eckert | Disputed discovery: the beginnings of X-ray diffraction in crystals in 1912 and its repercussions[END_REF] and CoFeB(20)/MgO [START_REF] Eckert | Disputed discovery: the beginnings of X-ray diffraction in crystals in 1912 and its repercussions[END_REF], that were grown in the different batch and behave the same as their CoFeB/MgO(6) counterpart. The direction of the external magnetic field H and sample orientation (Front Face) was kept fixed through the experiment.

The first experiment was conducted for the "control" CoFeB(5)/MgO(5) bilayer grown on the glass substrate. All the experiments are performed as independent sets of measurements, denoted "Series". The Figure 4.15a shows that within Serie 1 measurement, the E THz amplitude -laser pump fluence (Φ) dependence of CoFeB(5)/MgO(5) sample evolves in a highly nonlinear manner.

In addition, from From the previous discussion, we concluded that the THz emission from CoFeB(20)/MgO [START_REF] Richard | Understanding Solids[END_REF] is mainly mediated by the bulk contribution (ξ CoFeB(20) > β MgO ). However, for the same amount of the absorbed density of energy Figure 4.16d demonstrates no change in the THz polarity, but a three-fold increase in the E THz amplitude when the sample is excited at 400 nm compared to 800 nm. We propose to explain this observation with the instantaneous hot-carrier flux/gradient inside the CoFeB, which should be important for the bulk spin-to-charge conversion, especially in thick single-layer FM films [START_REF] Beaurepaire | Coherent terahertz emission from ferromagnetic films excited by femtosecond laser pulses[END_REF][START_REF] Huang | Direct observation of terahertz emission from ultrafast spin dynamics in thick ferromagnetic films[END_REF]. Figure 4.18 shows the light absorption gradient calculated for both CoFeB(20)/MgO(6) (orange) and CoFeB(5)/MgO(6) (red) samples using the Transfer Matrix method 4 . The position 0 indicates the initial point of the laser impact. The average profile of the light absorption ∇A z was then estimated using the following equation:

where A abs (0) and A abs (d) are the fractions of the absorbed optical power at the beginning (z = 0) and the end of the FM layer (z = d). Importantly, for the CoFeB(20)/MgO(6) bilayer, the absorbed density of energy between 400 nm and 800 nm differs only by less than 10 %, while the ∇A z has increased by the factor of 2.5. In this case, the three times increase in the E THz observed for 400 nm excitation (see Fig. 4.16d) may be directly linked to the 2.5 times more important light absorption gradient (see Fig. 4.18) in the CoFeB layer. Even though such explanation does not contradict previous hypothesis of the bulk-dominant ultrafast SCC, it is yet to be confirmed with single-layer FM STEs with various thicknesses.

Estimate of the bulk and IREE contributions:

As the final step, we will make an attempt to evaluate the relative amplitudes of the IREE and bulk contributions to SCC from CoFeB/MgO emitters. To do so, we start with an estimate of the decrease in injection-based (ISHE, IREE) SCC related to the increase of the CoFeB thickness. From Fig. 4.14b and c, we see that for the constant thickness of the Pt layer, the THz emission from the CoFeB(20)/Pt(3) bilayer is reduced by the factor of five when compared to CoFeB(5)/Pt(3) emitter (blue vs. green ratios). Such decrease in the total emitted THz can be attributed to the short spin-diffusion length in CoFeB [START_REF] Seifert | Efficient metallic spintronic emitters of ultrabroadband terahertz radiation[END_REF][START_REF] Garik Torosyan | Optimized Spintronic Terahertz Emitters Based on Epitaxial Grown Fe/Pt Layer Structures[END_REF] and possible Fabry-Pérot cavity effect, that may enhance the THz emission due to the multiple pump and THz reflections at the CoFeB/Pt and substrate/CoFeB interfaces [START_REF] Seifert | Efficient metallic spintronic emitters of ultrabroadband terahertz radiation[END_REF]. In this case, since γ Pt ≫ ξ CoFeB , we can assign this five-fold decrease in THz emission to the reduced amount of the spin current injected in the HM Pt layer and diminished Fabry-Pérot for the thick CoFeB(20)/Pt [START_REF] Kampfrath | Terahertz spin current pulses controlled by magnetic heterostructures[END_REF] sample [START_REF] Garik Torosyan | Optimized Spintronic Terahertz Emitters Based on Epitaxial Grown Fe/Pt Layer Structures[END_REF].

We then can use this factor to estimate the relative amplitudes of competing bulk and IREE From the main text, we concluded that the "sideband" observed close to the natural frequency of the A 1 1g coherent phonon mode could be attributed to the "electronic" background of the signal, and is not related to excitation of a coherent phonon. To support this claim we present here the short time scale signal of 14 nm Chromium film discussed in the first part of a Chapter 3. We remind that Chromium does not have optical phonon, so we should not expect any additional high-frequency contributions to the time-delay zero signal. The δT /T signal reveals the "double bump" around the time delay zero, shown as the black line in THz (blue dash-dot) and E 2 THz (red dash-dot). This effect could be attributed to the THz-induced optical birefringence observed in liquids [START_REF] Hoffmann | Terahertz Kerr effect[END_REF], transparent media [START_REF] Maehrlein | Terahertz Sum-Frequency Excitation of a Raman-Active Phonon[END_REF] and metals [START_REF] Vivek Unikandanunni | Ultrafast electron dynamics in platinum and gold thin films driven by optical and terahertz fields[END_REF].

where a ph is the amplitude of the A 1 1g optical phonons, Ω ph is the frequency of the A 1 1g mode, fixed to 1.85 THz, φ ph is the phonon phase and τ ph is the A 1 1g mode damping time, set to 3.5 ps ±3 %. The s parameter was set to the same value we have fitted the "raw" signal δT /T . The best fits As the next step, we will calculate the sum-frequency component of the E THz , i.e., E 2 THz | SF . In the Fourier space, the square of the pump field E 2 THz (τ ) will be equal to [START_REF] Braun | Electron and Phonon Dynamics in Topological Insulators at THz Frequencies[END_REF]: THz | SF (θ). The red shaded area corresponds to the region of the signal, that gets highly distorted by the slope of the error-function, used to fit the "raw" δT /T .

Appendix C

STEs Light absorption

In this appendix, we present the numerical simulations of the light absorption profiles in the set of the spintronic emitters CoFeB(5 and 20)/Pt(3) and CoFeB(5 and 20)/MgO [START_REF] Richard | Understanding Solids[END_REF], where number in brackets is the thickness of the layer in nm. Calculations are done using the Transfer Matrix Method approach, discussed in the Section 1.3.2. For the CoFeB nanolayers, the optical constant were extracted from the work of X.Liang et.al [START_REF] Liang | Optical constant of CoFeB thin film measured with the interference enhancement method[END_REF]. For Pt, we used the data from Rakic et.al. [START_REF] Aleksandar | Optical properties of metallic films for vertical-cavity optoelectronic devices[END_REF].

The plots presented in this appendix are shown in terms of the fraction of absorbed optical power per unit of the distance (nm). The amount of the absorbed optical power A abs within the layer i of the thickness d was calculated as following:

where n is the discrete value of the absorbed optical power per unit of distance within the range d i-1 ≤d n ≤d i+1 , and P tot inc is total incident optical power. The data for the Back and Front Face excitations are shown in the Table C.1

The light absorption gradient ∇A z was estimated using this equation:

where A abs is the absorbed fraction of the incident pump power, θ NM ISHE is the Spin-Hall angle of the NM layer (Pt in our case), d 0 is the critical thickness of the FM layer [START_REF] Garik Torosyan | Optimized Spintronic Terahertz Emitters Based on Epitaxial Grown Fe/Pt Layer Structures[END_REF][START_REF] Wu | High-Performance THz Emitters Based on Ferromagnetic/Nonmagnetic Heterostructures[END_REF], λ rel is the hot electron ballistic-transport relaxation length. Here, third term describes the generation of the spin-current in the FM, its ballistic transport and possible Fabry-Pérot cavity effect, that may enhance the THz emission due to the multiple pump and THz reflections at the FM/NM, NM/FM, NM/air interfaces. Last term, Z(ω) is the impedance of the STE given by:

Appendix E

THz-TDS of the Pt-and

MgO-based spintronic emitters

From the discussion of the THz emission in the Back Face configuration, in the absence of the substrate THz absorption, Torosyan et.al. [START_REF] Garik Torosyan | Optimized Spintronic Terahertz Emitters Based on Epitaxial Grown Fe/Pt Layer Structures[END_REF] proposed to introduce an additional term into Eq.D.1, which accounts for the THz attenuation due to the propagation through the metal layer stack :

where ξ THz is the effective inverse attenuation coefficient of the THz radiation, proposed by the authors. We can experimentally check this model, since this term can be effectively measured by the means of THz-TDS.

In order to check the validity the latter equation, we performed simple THz-TDS spectroscopy on the set of the spintronic emitters. For this experiment, we recorded the THz pulse that propagates through the bare 500 µm silicone substrate S ref (t), and via FFT we extract the information about its spectral components, S ref (ω). After, the 500 µm silicone substrate was replaced by the CoFeB(5 and 20)/Pt(3) grown on the 300 µm silicone substrate, and we recorded S sam (t) with S sam (ω).

In the THz transmission for all the samples show nearly frequency independent behaviour, well in the agreement with previous reports [START_REF] Seifert | Terahertz spectroscopy for all-optical spintronic characterization of the spin-Hall-effect metals Pt, W and Cu80Ir20[END_REF].

With Eq.E.1 with constant ξ THz it is not possible to explain the measured THz attenuation factor, underlying the thickness and layer-dependent THz conductivity of the whole metal stack.

We then change the the Eq.E.1 by accounting for the thickness change of the specific layer, which to both saturation of the magnetization quenching [START_REF] Vincent Cardin | Wavelength scaling of ultrafast demagnetization in Co/Pt multilayers[END_REF] and spin-trapping inside the HM layer [START_REF] Kampfrath | Terahertz spin current pulses controlled by magnetic heterostructures[END_REF].

At this stage, however, we cannot separate these two effects. Nevertheless, we show rather good consistency between the two batches, with E THz peak-to-peak difference of about 20 %, that can be attributed to the CoFeB or Pt layer thickness incertitude, which highly affects spin-to-charge conversion efficiency [START_REF] Garik Torosyan | Optimized Spintronic Terahertz Emitters Based on Epitaxial Grown Fe/Pt Layer Structures[END_REF]. is the "single-page" window with inputs for the time-resolved scan parameters, Lock-in Amplifier (LIA) controls, alignment tools, and plots with measured data sets. It was made with an intent of user-friendly software configuration, hardware initialization and adjustment of the GUI itself. No prior programming knowledge required [START_REF] Astley | The one last peculiar reference-link for particularly attentive readers: https://youtu.be/dQw4w9WgXcQ[END_REF].

More information is available on the GitHub, at https://github.com/artie-l/pump-probe-experimentcontrol.