Ultrafast lattice dynamics studied by photoelectron spectroscopy

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Time-resolved optical measurements

The ultrafast laser heating can also modify the optical properties of materials. Therefore, the analysis of an optical probe pulse reflected on the sample can provide interesting direct or indirect information about the electron and ion dynamics.
The principle of these techniques relies on the measurement of the optical modifications of the probe beam, such as phase, amplitude and polarization. Frequency Domain Interferometry (FDI) and time-resolved ellipsometry are examples of these techniques. FDI uses two optical probe pulses that are reflected upon the surface of the target [51]. The first probe pulse arrives before the pump pulse and the second one after. The probe pulses interfere before being detected, therefore producing an interference pattern for both P and S polarization which can be detected separately. Time-resolved ellipsometry analyzes changes in the polarization state from the reflected light at different angles.
The goal of these optical techniques is to reconstruct the dielectric function, that depends upon the state of the sample (solid, liquid, plasma), the electronic density ne and the Te and Ti [13]. The optical techniques need ultrafast optical pulses to attain the desired temporal resolution. At the same time, the surface quality of the sample must be sufficiently good to allow a proper reflection of the probe beams.
The information obtained with these techniques is mainly related to the electronic population of the sample. The reflectivity of metals depends directly on the conduction band electrons through its complex dielectric constant. Also, with the FDI technique it is possible to obtain, through the measured optical phase changes, direct information about the sample surface, whether if its expanding or losing its sharpness (in the case of an expanding plasma generated at the surface) [12].

Time-resolved photoelectron spectroscopy

We have selected photoelectron spectroscopy (PES) as our experimental technique. It consists in the detection of photoelectrons using what is called an electron energy analyzer that is capable of detecting electrons as a function of their kinetic energies forming a spectrum. The generated photoelectron spectrum carries important information about the material. As it will be discussed in this section, PES probes the electronic states through the photoelectron spectrum that are sensitive to the conditions of the lattice such as temperature and ordering. In this way, PES retrieves indirect information about the structure of the atomic lattice.
Historically, P. Lenard was the first to correctly describe, in 1902 the light-induced emission of electrons from metals including the so-called work function that couldn’t be described by the classical wave theory of light [55]. In 1905, Einstein proposed the quantum perspective of photoemission stating clearly that the photoemitted electrons contain information about the material [56]. In 1914, a team led by Rutherford realized that the kinetic energy of the photoelectron is equal to the difference between the photon energy and the binding energy of the electrons [57].
The utility of x-ray PES (XPS) in chemical analysis was recognized in the 20-30’s when line shifts caused by the chemical bonding were observed [58]. In the 50’s the first XPS apparatus with high resolution was built allowing to record photoelectron spectra where the definition of the lines was limited by their natural linewidth [59]. This motivated the development of the technique and more robust equipments were built to the point that nowadays XPS is a routine technique in many fields of research such as chemistry, surface analysis and material science.
The PES technique in a time-resolved configuration (Tr-PES) is suitable for studying ultrafast phenomena. Tr-PES has become more common thanks to the development of femtosecond table top lasers with pulse energies in the mJ range, meeting the conditions needed for the ultraviolet (UV) and extreme-ultraviolet (XUV) light generation [60, 61, 62] (using up-conversion or High-Order Harmonic Generation, for example). This makes the table-top, laser-based Tr-PES setups quite convenient, suitable and more available than Tr-PES experiments at large-scale facilities such as synchrotrons and x-ray free electron lasers (XFELs).
As all other mentioned techniques, Tr-PES setups are also ideal for studying the ultrafast laser-matter interaction because the picosecond and femtosecond required resolution can be achieved. These reasons lead us to choose the Tr-PES technique based on a high-order harmonic generation beamline, as it will be detailed in chapter II. This technique is versatile as it can also be coupled with different detectors for angle resolved PES [63], spin-resolved PES suitable for magnetic materials [64] and finally, this technique can be used with semiconducting and metallic targets.

Photoelectron spectroscopy in the context of lattice heating

Both core-level and valence band PES spectra are sources of valuable information regarding the material chemical composition, oxidation state and even crystalline structure.
The study of core-level peaks in steady-state PES for quantitative analysis is very well established and developed [57], whereas the valence band (VB) spectrum is much more complex and is still not routinely used as a standard practice. Core-level and VB-level can both be studied by Tr-PES in order to obtain insight about the evolution of Te and Ti. Core-level peaks can broaden upon heating of the lattice, giving an indirect measurement of the Ti [63]. On the other hand, the Fermi edge can be fitted to an exponential function distribution to deduce the Te [76]. The works from Refs. [63, 76] performed these type of measurements on superconducting and topological insulators optically pumped at very low fluences (1-10 J/m2). In order to perform these kind of measurements a very fine spectral resolution is needed (< 200 meV) which is not the case of our experimental setup due to the spectral width of our photon source ( 1 eV). And, as it will be discussed later, performing Tr-PES with high pump energies (enough to induce a phase transition in metals, as in our case), imposes some constraints that entail a loss of resolution due to multiphoton emission of electrons that disturb the photoelectron spectrum [77].
In our case of interest, which is the ultrafast heating of metals, we will focus on the study of the VB changes. It is expected that after heating of the sample, the VB changes can be unveiled using the Tr-PES technique. This is the goal of this work.
The study of the VB spectrum evolution upon heating of the sample is justified if it is sensitive to the temperature and structural changes. We justify this approach based on different published works that measure statically the VB at different temperatures, assess the modification of the DOS performing theoretical calculations and study the VB in a time resolved manner.

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XUV source: a 100 eV beamline based on high order harmonic generation

The probe arm is constituted by a beamline based on High Order Harmonic Generation (HHG) in Ne gas. First, a brief description of the theoretical fundamentals of HHG will be presented. Then, the main components of the XUV beamline will be described along with its characterization and performance.

XUV beamline design and characterization

Our XUV beamline, depicted in Fig. II.5, needs 6 mJ laser pulses in order to operate. The pulses are delivered by the Aurore laser facility into the experimental setup. They arrive with p-polarization. The beam is divided by a beam splitter (BS). The probe beam goes through an energy regulator (HWP and PC) and directed towards a grating compressor (C1, working in p-polarization). A half-wave plate (HWP) rotates the polarization to s needed for the operation of the beamline (detailed in section b)). Then, the beam is steered inside the XUV vacuum line through a dispersion-less window. The HHG beamline can be divided into three main parts : (1) generation, (2) selection and focusing and (3) characterization. Each of these stages will be presented followed by the beamline performance.
In Figure II.6, a photo of the installation is presented highlighting the already mentioned beamline stages.

HH generation stage

The laser beamsize is regulated with a diaphragm (D1 in Fig.II.5) right after compression, and then steered into the vacuum setup. It arrives in the HH Generation stage where it is directed towards a spherical mirror (SM) of 1 m focal length that focuses the beam in the gas cell (GC). This focal length ensures a mild focusing (compared to shorter focal distances) and a larger Rayleigh waist, reducing the phase mismatch from propagation.
The gas cell was designed at the INSP and built to be installed in the beamline. As shown in Fig. II.7 , the specificity of this gas cell is that the gas chamber length can be varied while under vacuum from 1 mm to 20 mm in order to optimize the signal output. Moreover, the gas pressure can be precisely controlled with a manometer and is remotely stabilized. In other words, the number of gas atoms of the generating medium are precisely optimized. The gas cell is completely adjustable under vacuum with respect to the laser axis since it is mounted on:
• a (x, y) micrometric manual stages.
• a z pico-motorized stage.
• a rotation and tilt angle stages with micrometric screws.

Pump beamline focusing and fluence calculations

As seen on Figure II.5, after the beam splitter (BS) that forms the pump and probe arms of the experimental setup, the pump beam goes through an energy regulator consisting of a half wave plate (HWP) plus a polarizing cube (PC) and then to its dedicated compressor (C2). It is steered through the room towards a motorized delay line (DL) and then directed onto the sample with a 1 inch, 45 mirror placed under vacuum.
The Tr-PES pump/probe experiment has certain requirements for the pump beam focusing:
1. an homogeneous focal spot, ideally with a top-hat or Gaussian profile to ensure an homogeneous transverse heating of the sample.
2. a focal spot that is larger than the probe spot to ensure that the probed region is heated uniformly.
3. a sufficiently high fluence to trigger the desired dynamics in the samples (in the range of 5000-9000 J/m2).

Table of contents :

Introduction
I Matter under extreme conditions 
I.1 Ultrafast lattice dynamics and solid-liquid phase transitions
I.1.1 Phenomenological description
a) Energy absorption, heating and equilibration
b) Thermal and non-thermal phase transitions
I.1.2 Theory: modelling and limitations
a) Energy absorption
b) The two-temperature model
c) Hydrodynamic simulations
d) Molecular dynamic simulations
e) Quantum molecular dynamic simulations
I.2 Experimental state of the art
I.2.1 Time-resolved x-ray absorption spectroscopy
I.2.2 Time-resolved optical measurements
I.2.3 Time-resolved x-ray and electron diffraction
I.3 Time-resolved photoelectron spectroscopy
I.3.1 Principle
I.3.2 Interpretation
I.3.3 Characteristics
a) Probing depth
b) Photoionization cross section
I.3.4 Photoelectron spectroscopy in the context of lattice heating
a) State of the art
b) Experimental scheme
c) Challenges
II Experimental method 
II.1 Infrared femtosecond laser source: the Aurore facility
II.2 XUV source: a 100 eV beamline based on high order harmonic generation
II.2.1 Theoretical principles of high order harmonic generation .
a) Microscopic aspects of the HHG
b) Macroscopic aspects of the HHG
II.2.2 XUV beamline design and characterization
a) HH generation stage
b) HH selection and focusing
c) HH characterization
d) Performance
II.3 Pump/Probe experiment
II.3.1 Pump beamline focusing and fluence calculations
II.3.2 Pump pulse temporal characterization
II.3.3 Pump/Probe synchronization and spatial overlap
II.4 Interaction chamber for photoelectron spectroscopy
II.4.1 Vacuum requirements
II.4.2 Photoelectron spectrometer
II.4.3 Sample manipulation, cleaning and XPS characterization .
II.4.4 Valence band photoelectron spectrum measurements
III Space-charge effect study 
III.1 Introduction and motivation
III.2 Experimental Results
III.3 Numerical models for the pump-probe space-charge effect
III.3.1 Pump initial spectrum calculations: the jellium-Volkov approximation
III.3.2 Simulated Matter Irradiated by Light at Extreme Intensities (SMILEI) simulations
III.3.3 A Space-Charge Tracking Algorithm (ASTRA) simulations
III.4 Numerical simulations: results and comparison to experimental measurements
III.4.1 Jellium-Volkov Pump initial emission
III.4.2 SMILEI laser-plasma simulations
III.4.3 Pump/probe Jellium-Volkov-ASTRA simulations
a) Standard simulation
b) Comparison of the (x, y, z) distribution with an (x, y, t) cathode emission distribution
c) Number of electrons as the key parameter
d) Transversal size of the electron bunches
e) Angular distribution
f) Sorting of particles
g) Delay on the mirror charges
h) Analyzing shift and broadening
III.4.4 Pump/probe SMILEI-ASTRA simulations
a) Varying the low temperature (TLow)
b) Varying the high T temperature (THigh)
c) Varying proportion between THigh and TLow populations
d) Varying the cutoff of the initial pump spectrum
III.4.5 Pump/probe experimental pump-ASTRA simulations
a) Gaussian fits from the experimental spectrum .
b) Experimental spectrum
III.4.6 Summary and discussion of the pump induced space-charge effect study
III.5 Experimental reduction of the pump-probe space-charge effect
IV Ultrafast lattice dynamics studied by photoelectron spectroscopy 
IV.1 Sample
IV.1.1 Selection of the samples
IV.1.2 Sample preparation and characterization
IV.2 Tr-PES experimental measurements
IV.2.1 Acquisition conditions
IV.2.2 Data analysis procedure
IV.2.3 Measurements of the different pump fluences
IV.3 Interpretation
IV.3.1 Disentangling the space-charge effect
IV.3.2 Interpretation of the photoelectron spectrum
a) ABINIT density functional theory calculations of the density
of occupied states
b) Hydrodynamic calculations and the two temperature model:
The ESTHER Code
IV.4 Summary and conclusion
Conclusions
Context and contributions
Bibliography 

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