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Concept# Nearly free electron model

Summary

In solid-state physics, the nearly free electron model (or NFE model and quasi-free electron model) is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model is closely related to the more conceptual empty lattice approximation. The model enables understanding and calculation of the electronic band structures, especially of metals.
This model is an immediate improvement of the free electron model, in which the metal was considered as a non-interacting electron gas and the ions were neglected completely.
Mathematical formulation
The nearly free electron model is a modification of the free-electron gas model which includes a weak periodic perturbation meant to model the interaction between the conduction electrons and the ions in a crystalline solid. This model, like the free-electron model, does not take into account electron–electron interactions; that is, the independent electron appro

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Julien Stanislas Pierre Dominski

In tokamak fusion plasmas, micro-turbulence transport is known to be the cause of large losses of heat and particles. The present work deals with the study of electrostatic micro-turbulence transport driven by instabilities of essentially two types: the ion temperature gradient (ITG) modes and the trapped electron modes (TEM). The plasma is described within the gyrokinetic framework, which permits to save computational resources compared to the classical Vlasov kinetic description. In gyrokinetic simulations of fusion plasmas, the passing electrons are often assumed fast enough so that they respond instantaneously to the electrostatic perturbations. In this case, their response is computed adiabatically instead of kinetically. The main advantage is that this simplified model for the electron response is less demanding in computational resources. This assumption is nonetheless incorrect, in particular near mode rational surfaces where the non-adiabatic response of passing electrons cannot be neglected. This thesis work focuses on the study of this passing electron non-adiabatic response, whose influence on microturbulence is studied by means of numerical simulations carried out with the gyrokinetic codes GENE and ORB5. In the first part of this thesis work, the response of passing electrons in ITG and TEM microturbulence regimes is studied by making use of the flux-tube version of the GENE code. Results are obtained using two different electron models, fully kinetic and hybrid. In the hybrid model, passing particles are forced to respond adiabatically while trapped are handled kinetically. Comparing linear eigenmodes obtained with these two models enables one to systematically isolate fine radial structures located at corresponding mode rational surfaces, clearly resulting from the non-adiabatic passing-electron response. Nonlinear simulations show that these fine structures on the non-axisymmetric modes survive in the turbulent phase. Furthermore, through nonlinear coupling to axisymmetric modes, they induce radial modulations in the effective profiles of density, ion and electron temperature and zonal flows $E \times B$ shearing rate. Finally, the passing-electron channel is shown to significantly contribute to the transport levels, at least in our ITG case. Also shown is that the passing electrons significantly influence the $E \times B$ saturation mechanism of turbulent fluxes. Following this study in flux tube geometry, the influence of the non-adiabatic passing electron response near mode rational surfaces is further studied in global geometry with the global gyrokinetic code ORB5, in which a new field solver is implemented for the gyrokinetic quasi-neutrality equation valid at arbitrary wavelength, overcoming the former long wavelength approximation made in the original version of the code. A benchmark is conducted against the global version of the gyrokinetic code GENE, showing very good agreement. Nonlinear simulations are carried out with the new solver in conditions relevant to the TCV tokamak, with the physical deuterium to electron mass ratio ($m_i/m_e=3672$) and are compared to simulations carried out with heavy electrons ($m_i/m_e=400$). The particular spectral organization of the passing electron turbulent flux and its dependence on the radial profile of the safety factor are revealed. In particular, the formation of short-scale transport barriers is studied near low-order mode rational surfaces. Results show that quantitatively correct nonlinear fully-kinetic simulations of tokamak transport must be carried out in a full torus and with the physical mass ratio.

In my thesis work I have concentrated on the growth and the in-depth analysis of high temperature superconducting thin films with the central aim to elucidate their electronic properties, predominantly by in-situ angle resolved photoemission spectroscopy (ARPES). I have used two somewhat complementary approaches and two laser ablation set-ups. The first one, developed previously in Wisconsin, was used mainly for studies of strained La2-xSrxCuO4 (LSCO) with a transfer to the Scienta analyzer via an appropriate suitcase. The second one, at the EPFL, where I have built a new pulsed laser deposition (PLD) system, was used to optimize the growth of Bi2Sr2-xLaxCuO4 (Bi-2201) and study in-situ ARPES. In-situ ARPES is the most direct tool to probe the electronic structure. We performed it at the Synchrotron Radiation Center (SRC, University of Wisconsin), where we used the aforementioned experimental set-up consisting in a dedicated PLD system coupled with the SCIENTA beamline. The sample transfer procedure assures that the surface quality is preserved on the way to the SCIENTA analyzer. There we studied in detail the effect of strain in LSCO thin films. In a previous work the in-plane compressive strain was studied and the main result was that the Fermi surface (FS) topology changed from hole-like to electron-like. The tensile strained films showed completely different results. ARPES analysis show evidence for a 3-dimensional (3D) electronic dispersion relation in contrast to the strictly 2-dimensional (2D) dispersion observed in all other studied LSCO films. In this thesis this result has been confirmed mapping the FS at different photon energies. We found that the strain related to the thickness of the films, is playing an important role in inducing a 3D dispersion. Furthermore, the 3D parameters, evolve according to the level of strain. Moreover, we observe a staircase structure for different photon energies, revealing both the 3D nature of the electronic dispersion and the quantization of the electron wave vector along the direction normal to the film surface. Taking advantage of the wavevector quantization we were able to determine directly the band parameters and map the FS without using the nearly-free-electron approximation (NFEA). Moreover, introducing an effective anisotropic photoelectron effective mass, related to the local structure of the excited band, improves the use of the NFEA for single photon energy measurements. In parallel, I have built an improved PLD system at the EPFL which can be connected to the SCIENTA analyzer, and which enables us to perform in-situ ARPES measurements at any time rather than only during allocated beamtimes at the synchrotron. We also produced our own targets for the laser ablation and all the films were fully characterized at the EPFL performing X-ray diffraction (XRD), resistivity and magnetic measurements. I analyzed in detail the growth mechanism of Bi-2201 and I investigated the presence of random intergrowths. We developed a model to explain the presence of these polytypes and studied their presence as a function of the deposition parameters and the annealing treatment. The model predicts a very particular spatial distribution of defects: a Markovian-like sequence of displacements along the grow direction, as well as a two-component in-plane correlation function, characteristic of self-organized intercalates. We varied the growth conditions in order to study the presence of intergrowths and to produce single-phase samples. Subsequently, we performed in-situ photoemission experiments on thin films of Bi-2201 films free from intergrowths and we analyzed their FS. This method is successful and can be extended to other related oxide films.

Claudia Cancellieri, Ping-Hui Lin, Davor Pavuna

We have observed the wavevector quantization in LaSrCuO films thinner than 12 unit cells grown on SrTiO3 substrates. Low energy dispersions were determined in situ for different photon energies by angle resolved photoemission spectroscopy. From the wavevector quantization, we extract three dimensional dispersions within a tight-binding model and obtain the Fermi surface topology, without resorting to the nearly free-electron approximation. Such method can be extended to similar confined electron nanostructures.

2008