Soutenance mémoire
The experimental starting point
The aim of this thesis is to develop an efficient method for the description of observables related to the electronic structure of matter. To validate the approach, we will benchmark it with the existing state–of–the–art theories. Therefore, we will not directly face the comparison with experiments. However, the connection with the experimental world is still of primary importance, as it can guide us on choosing which quantities are important to reproduce theoretically. All the chapters that follow will focus on one particular measurable quantity, the spectral function. This is a key quantity for the interpretation of different crucial experiments, which are based on phenomena that have marked the development of quantum theory itself: the photoelectric effect and quantum tunnelling. Indeed, the diagonal of the spectral function in real space is the fundamental observable for describing scanning tunnelling spectroscopy. Its diagonal in reciprocal space is the cornerstone of angle resolved photoemission spectroscopy. Finally its trace, which is basis independent, is the many–body quantity that is needed for reproducing photoemission and inverse photoemission experiments. Tunnelling spectroscopy is essentially a surface sensitive technique. Also photoemission and its variants, according to the photon energy, are sensitive to the surface. However, they are extremely useful for investigating also the bulk properties of a system. In this thesis we concentrate on the bulk, and photoemission will therefore be our primary reference experiment.
Photoemission spectroscopy
Photoemission experiments are the modern times development of some famous investigations performed by Hertz in 1887, who observed what became known as photoelectric effect: under particular circumstances, a beam of monochromatic light is able to knock out electrons, thus called photoelectrons, from a solid. The quantum theoretical explanation [13] of this effect earned Einstein his Nobel prize: in an independent–particle picture , a photon of frequency ω/2π is absorbed by one of the electrons of the material, with initial energy −εB −φ, where φ is the work function (energy needed to eject an electron from the highest occupied level, namely, in a metal, the vacuum minus the Fermi energy) and εB > 0 is the binding energy of the electron, measured with respect to the Fermi energy µ. If the energy gain ħω is sufficiently large, that is if ħω > εB + φ, the electron escapes the material into the vacuum, with positive kinetic energy εk = ħω− φ − εB and momentum ħk, and is then collected by an analyser.
ħω + φ, and since εk is measured by a detector, ħω is chosen by the experimentalist and φ can be known, the electron binding energy can be determined from the kinetic energy of the photoelectron released in the vacuum.
Plotting the intensity of the signal, which is proportional to the number of photoelectrons, as a function of the binding energy, one obtains an extremely rich spectrum, where a series of distinct features reflects the different electronic energy states allowed in the material. To make the discussion concrete, I will refer to a particular experiment in which I have taken part. It is an angle resolved photoemission experiment (ARPES, see below) on bulk aluminum, performed at the PEARL beamline of the Swiss Light Source (SLS ). The aim was collecting experimental data on the electronic structure of aluminum, and use a theoretical approach (the cumulant expansion) to describe them.
The photoemission process
The standard approach to deal with the photoemission process from a many–body point of view simplifies the single photoemission event into the succession of three independent steps (three–step model [18]): an electron is excited by a photon in the solid, it propagates to the surface and it finally leaves the surface into the vacuum. Each of these steps contributes to the final photocurrent (detected electrons per unit time): an effective mean free path and a transmission probability through the surface take into account the last two steps [19], while the intrinsic photoexcitation of the electron can be described by the transition rate wf i from the ground state of the N electron system |Ψ (N)0〉 to an excited state |Ψ(N)f〉 – driven by a perturbing Hamiltonian Hˆ int .
Angle Resolved Photo Emission Spectroscopy (ARPES)
In the last decades, Angle Resolved Photo Emission Spectroscopy (ARPES) has become feasible: besides energy levels, also their dependence on the wavevector k is experimentally accessible. ARPES is an extremely useful technique for investigating the dispersion of valence states which, as already mentioned, present an important itinerant nature; in particular, for Fermi liquids, the position of the main quasiparticle peak as a function of k is the measured band structure. Also the Fermi surface, as a mapping of the k–points with energy µ, is directly accessible by ARPES.
To interpret ARPES experiments, one has to relate the measured momentum (in vacuum) to the wave vector k inside the solid: crossing the surface, the parallel (to the surface) component of the photoelectron momentum is conserved, while the perpendicular is not, and different approaches are used to determine it [19]. Besides the conservation of energy that we implemented above, also the momentum conservation in the solid must be taken into account: since the photon carries a negligible momentum in most cases, only vertical transitions are allowed, and the momentum of the electron in the material can be modified by reciprocal lattice vectors G only. These vertical transitions between bands are determined by selection rules in the matrix elements.
Scanning Tunnelling Spectroscopy (STS)
A completely different experimental technique is Scanning Tunnelling Spectroscopy (STS), a purely surface–sensitive method based on the tunnel effect. The principle is very simple: a conducting probing tip is moved closer and closer to the surface of the sample, till the many– body wavefunctions of sample and tip overlap: in such a situation, if a suitable bias V is applied between tip and sample, tunnelling of electrons through the vacuum between the two becomes possible; thus, a tunnelling current J (V ) can be measured. As the tip moves around, by measuring the tunnelling current one can achieve a complete reconstruction and visualization of the surface with atomic resolution (∼ 10⁻¹ ÷10⁰Å), producing a real atomic microscope (STM: scanning tunnelling microscope [24]). Furthermore, besides visualization ot single atoms [25], also the manipulation of them became feasible within this technique [26].
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Table des matières
I Background
1 The experimental starting point
1.1 Photoemission spectroscopy
1.1.1 The photoemission process
1.1.2 Angle Resolved Photo Emission Spectroscopy (ARPES)
1.2 Scanning Tunnelling Spectroscopy
2 The many–body problem
2.1 The system
2.1.1 The Born–Oppenheimer decoupling
2.1.2 The electronic system
2.2 The problem
2.3 Observables
2.4 Reformulations of the problem: functionals
2.4.1 The Rayleigh-Ritz principle
2.4.2 Hohenberg–Kohn Density Functional Theory (DFT)
2.4.3 One–body Reduced Density Matrix Functional Theory (RDMFT)
2.5 Green’s function
2.5.1 Lehmann representation
2.5.2 Analytic properties of the Green’s function: the spectral function
2.5.3 Standard route to the spectral function: the self energy
2.5.4 Dyson and Hedin equations
2.6 The Hubbard model
II Auxiliary systems
3 Auxiliary systems: an introduction
3.1 Reduced quantities
3.2 How to build auxiliary systems
3.2.1 The generalized Sham–Schlüter equation
3.2.2 What we have and what we have not
3.3 The Kohn–Sham system
3.3.1 The Sham–Schlüter equation
3.4 An auxiliary system for the density matrix
3.5 The spectral potential
3.6 DMFT and spectralDFT
4 Auxiliary systems: explicit examples
4.1 The Hubbard model on the Bethe lattice
4.1.1 The auxiliary system
4.2 The symmetric Hubbard dimer
4.2.1 The auxiliary system
4.2.2 Sham–Schlüter equation approach
4.3 The homogeneous electron gas
4.3.1 Real system viewpoint: HSE06 solution
4.3.2 The auxiliary system
III The connector and dynLCA
5 The connector
5.1 The connector idea: Local Density Approximation (LDA)
5.2 Dynamical Mean Field Theory (DMFT)
5.3 A generalization
5.4 The dynamical local connector approximation (dynLCA)
5.5 The connector
5.5.1 General structure of the connector
5.5.2 Perturbation expansion
5.6 Model systems without auxiliary systems
5.6.1 Sham–Kohn Quasi Particle LDA
6 The connector for the Hubbard dimer
6.1 The real system: one–fourth filling solution
6.2 Approximations to the self energy
6.2.1 Hartree approximation
6.2.2 The GW approximation
6.3 Dynamical Connector Approach (dynCA)
6.3.1 The auxiliary system
6.3.2 The model system
6.3.3 Different quantities to import
7 Dynamical Local Connector Approximation in practice: real systems
7.1 Implementation
7.2 Fermi energy alignment
7.3 A shortcut: local alignment
7.4 Band structure correction
7.5 External correction
7.6 The dynLCA band structure
Conclusion
Acknowledgments
Appendices
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