|Ph.D Thesis||Department of Physics|
|Supervisors:||Prof. Akkermans Eric|
|Prof. Fishman Shmuel|
In this thesis the effects of interactions between electrons are studied in mesoscopic systems. One part is devoted to the study of mesoscopic super-conductors in which the size of the system is comparable to both coherence length and London penetration depth. In recent experiments performed on small aluminium disks the jumps of magnetisation as a function of external magnetic field were observed. These jumps correspond to the penetration of the magnetic field into the sample characteristic of type-II super-conductors. Relying on the Ginzburg-Landau theory of super-conductivity adopted to the restricted geometry (small cylinders or disks) we show that for certain values of the external magnetic field the system sustains a stable vortex-like solution and that the number of vortices is determined by minimization of the Ginzburg-Landau free energy functional. The equations of motion for the order parameter and magnetic induction in the super-conductor are analyzed and solved in some geometries analytically using either special dual point solutions for an infinite systems or the London regime calculations valid for the extreme type-II super-conductor. In both cases the resulting expressions for the magnetisation and the matching fields at which the number of vortices in the sample changes by one are in good qualitative agreement with the experimental results. Magnetisation curves at the dual point are in good quantitative agreement with experiments, since this regime is relevant for the most experimental situations. The solutions can be interpolated between the dual point and the London regime using a novel scaling expression for the free energy. The Ginzburg-Landau equation in the London regime can be used to study the topology of magnetic field profile. Each solution can be classified by the set of topological numbers (index), the number of vortices being one of them. The topological study of the solutions of the Ginzburg-Landau equations is not specific to the London regime and can be extended to other experimentally relevant regimes as well as to the more complicated geometries of the sample.
Another part of this work deals with an interacting electrons in mesoscopic conductors. In parallel to the recent studies of such systems we study the localization in the Hilbert space of Tomonaga-Luttinger model. For the standard version of this model, the states are found to be extended in the basis of Slater determinants, representing the eigenstates of the non-interacting system. The linear dispersion which leads to the fact that these eigenstates are extended is replaced by one with random level spacings modeling the complicated one-particle spectra of realistic models. The localization properties of the eigenstates are studied. The interactions are simplified and an effective one-dimensional Lloyd model is obtained. The effects of many-body energy correlations are studied numerically. The eigenstates of the system are found to be localized in Fock space for any strength of the interactions, but the localization is not exponential.