Soutenance mémoire
Photoemission spectroscopy
The photoemission spectroscopy originates from the discovery of the photoelectric effect by Hertz [21], which was later explained by Einstein [22], who obtained the Nobel prize in 1921. After its discovery photoemission spectroscopy has become more and more used to understand the electronic properties of materials [4]. In a general spectroscopy experiment, one perturbs an electronic system by transferring some external energy (perturbation), and measures the response of the system after excitation. This allows one to access a great deal of information about the electronic properties of the system. Photoemission spectroscopy can be divided into two complementary processes, namely direct and inverse photoemission spectroscopy ((I)PES). In the direct photoemission the sample absorbs photons and the electrons in the sample are excited and emitted. Knowing the photon energy and by measuring the kinetic energy of the photoelectrons, one can infer information about the energy distribution of electronic states (the occupied states in an independent-particle picture) of the sample. The inverse photoemission (IPES) can be considered as the time-reversal of PES, where the electrons are absorbed and photons are emitted and measured in the detector. Consequently one can access the information of the possible empty (unoccupied) states (in an independent-particle picture) of the sample.
What happens during photoemission spectroscopy?
A schematic view of the PES and IPES processes are shown in Fig. 1.1, where Ekin and hν are the energies of electron and photon, respectively. In an (I)PES experiment, these two energies can be measured such that the energy difference of the sample before and after the measurement can be determined by energy conservation.
The quasi-particle excitation
In an independent-particle picture , the energy difference between the incoming photons and outgoing photoelectrons in (AR)PES gives directly the energy of the one-particle level that the one electron was occupying before being emitted from the system. Therefore under the independent-particle assumption, we will observe a series of δ-peaks corresponding to different independent-particle states in the sample . Since there is no interaction between particles, each δ-peak has a normalized weight of Z = 1 representing one particle. Moreover, there is no relaxation of remaining electrons in the independent-particle picture such that the excitation has an infinite lifetime (the excitation energy is real-valued as opposed to a complexed valued energy from the GW approximations that will be discussed later) and the occupation number of each state is either 1 (for states below the Fermi level) or 0 (for states above the Fermi level).
However, the spectrum one actually obtains from an (AR)PES experiment is most of the time different from the simple independent-particle picture. In fact, the emitted electron leaves a hole in the electronic system corresponding to a positive charge. The remaining electrons in the system can react to that hole, which screens the hole. Alternatively speaking, the hole left behind attracts the surrounding electrons due to the attractive interaction between positive and negative charges. The screened particles are called quasi-particles (QPs). The interaction between QPs is weaker than the interaction between bare electrons: instead of the bare Coulomb interaction, the effective interaction in the system is the dynamically screened Coulomb interaction W (see Fig. 1.4). Since the dynamical screening corresponds to excitations into which the QP can decay, the latter corresponds to an excited state of the system with a finite lifetime. As a consequence, in an ARPES experiment we observe a broadened peak with finite width, which gives a measure of the interactions due to the presence of a hole. The broadened QP peaks, which replace the sharp δ-peaks of the independent-particle picture .
Coupling to neutral excitations
Moreover, besides the broadening the observed QP weight is reduced compared to the independent-particle picture. This is because the hole induced by the photoemission excites all remaining electrons, which often leads to collective wave like oscillations in the system. This collective oscillation is a neutral excitation as opposed to the charged excitation, because after the photoemission, the remaining electrons move inside the system and the total number of electrons is conserved. The neutral excitations are of bosonic nature because they are constituted by pairs of fermions (e.g., excitons and plasmons) . Therefore besides the QPs, the photon energy has been used also to excite the additional excitations (e.g., the plasmon excitation will be discussed intensively throughout this thesis) yielding new structures in the spectrum. The renormalized QP weight is reduced by giving some weight to these additional structures, leading to a fractional renormalization factor Z < 1. In simple systems like simple metals or semiconductors, the weight of the plasmon excitations are smaller than the QP weight. The peaks due to the plasmon excitations are called satellites in the photoemission spectrum. As a collective oscillation, the plasmon excitation should be able to propagate as all other waves in nature (e.g., waves of water), the longer it propagates, the smaller its weight (possibility of excitation) indicating that we should be able to observe not only one plasmon satellite but its replica with decreasing weights .
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Table des matières
1 Introduction
1.1 Photoemission spectroscopy
1.1.1 What happens during photoemission spectroscopy?
1.1.2 Theoretical approach to photoemission spectroscopy
1.2 Green’s function theory
1.2.1 Definition of the one-body Green’s function
1.2.2 The Lehmann representation and the intrinsic spectral function
1.3 Summary
2 State-of-the-art theories in electron spectroscopy
2.1 Density functional theory
2.1.1 The many-body problem
2.1.2 Kohn-Sham DFT
2.2 Hedin’s equations and the GW approximation
2.2.1 The GW approximation in theory
2.2.2 The GW approximation in practice
2.3 The cumulant expansion approximation
2.3.1 The TOC96
2.3.2 The retarded cumulant
2.4 Summary
3 Unified derivations of the GWA and CEA
3.1 Linearization
3.2 The GW approximation from the linearized differential equation
3.3 The TOC11 from the linearized differential equation
3.3.1 Cumulant from Decoupling approximation
3.3.2 The GW quasi-particle correction
3.4 The retarded cumulant
3.4.1 Ansatz that gives the GWA
3.4.2 Ansatz that gives a retarded cumulant
3.5 Summary
4 Performance in Model Hamiltonians
4.1 The model Hamiltonians
4.1.1 One-level hole-plasmon coupling Hamiltonian
4.1.2 Two-level hole-plasmon coupling Hamiltonian
4.2 Performance of approximations to spectra in the model Hamiltonians
4.2.1 The one-level hole-plasmon coupling model
4.2.2 The GWA in HPC-2
4.2.3 The CEAs in HPC-2
4.2.4 The RC
4.3 Summary
5 The constrained retarded cumulant Green’s function
5.1 An approximate fully analytical Green’s function of the HPC-2 3
5.2 The constrained retarded cumulant in the HPC-2
5.3 Generalization and implementation of the CRC for the calculation of real systems
5.3.1 A four-level system with dispersionless plasmons
5.3.2 Implementation into a computer code
5.4 Summary
6 Cumulant expansion approximation in bulk sodium
6.1 Sodium photoemission spectrum from the time-ordered cumulant
6.1.1 The TOC11 in sodium core
6.1.2 The TOC11 in sodium valence
6.1.3 The TOC96 using RPA screening
6.2 Sodium valence photoemission spectrum from the constrained retarded cumulant
6.3 The screening beyond the random phase approximation
6.4 Computational details
6.5 Summary
7 Conclusion
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