Experiments shows that the conductance of a quantum point contact is quantized in steps of  2e²/h, where e is the charge of the electron and h is Planck’s constant, and thereby Ohm’s law is not valid for nanostructures. By using the approximation method finite difference, the transmission for one-dimensional contacts and one- and two-dimensional potentials are investigated. In the case of two-dimensional contacts and a two-dimensional potential the Green’s function method is used. It turns out that if electrons are treated as waves, the transmission and the conductance just differ by the constant 2e²/h, which in this thesis is interpreted numerically in Matlab by using the Green’s function method.

We discuss a relativistic transport theory of condensed matter based on a microscopic system containing bosonic and fermionic degrees of freedom interacting via 3 - and 4-point interactions. We use the Dyson hierarchy as a solution to the underlying field theory and truncate this hierarchy by parametrizing the 2-particle-irreducible kernels of the 4- and 5-point vertex functions. We then perform a complete crossing-symmetric reduction of the 2-particle intermediate states of the theory. We obtain a reduction hierarchy and show how to explore the quality of our truncation scheme using this reduction hierarchy. Finally we discuss the problem of regularization of the theory in the case of hadronic matter by either putting form factors directly in the action or by using dispersion relations to introduce causal form factors into singular diagrams. / Graduation date: 1992

Thesis (Ph. D.)--University of Washington, 2001. / Vita. Includes bibliographical references (leaves 150-155).

Thesis (Ph. D.)--Duke University, 2007. / Includes bibliographical references.

Thesis (Ph. D.)--Michigan State University. Applied Mathematics, 2008. / Title from PDF t.p. (viewed on July 8, 2009) Includes bibliographical references (p. 123-131). Also issued in print.

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 186-201).

Thesis (Ph. D. in Interdisciplinary Materials Science)--Vanderbilt University, Aug. 2007. / Title from title screen. Includes bibliographical references.

Orientador: Guillermo Gerardo Cabrera Oyarzún / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-08-27T21:31:38Z (GMT). No. of bitstreams: 1 Fernandes_ImaraLima_D.pdf: 2572980 bytes, checksum: 17e1019655bf7dc13c5d5180196e89db (MD5) Previous issue date: 2015 / Resumo: Esta tese de doutorado abordou principalmente o estudo teórico das propriedades de transporte dependente do spin em nanoestruturas magnéticas. As principais estruturas estudadas foram junções magnéticas de tunelamento e sistemas compostos por um arranjo de pontos quânticos acoplados a eletrodos ferromagnéticos. Com o intuito de obter propriedades físicas do sistema como, a corrente elétrica, a corrente de spin, a densidade local de estados, a ocupação média nos pontos quânticos e a corrente induzida por spin transfer-torque utilizamos o formalismo de funções de Green de não-equilíbrio. Na primeira parte deste trabalho, estudamos os efeitos da inversão do spin nas propriedades de transporte em junções de tunelamento. Para estes sistemas, o fenômeno da magnetorresistência tem origem na densidade de estados dos elétrons de condução dos eletrodos e mostramos que ela é fortemente afetada pela inversão do spin no tunelamento. Além disso, foi possível observar que a inversão do spin induz um spin-torqueadicional ao sistema. Na segunda parte deste trabalho, investigamos as propriedades de transporte em um sistema composto por dois pontos quânticos em forma de T acoplados a eletrodos ferromagnéticos. Com a mesma metodologia empregada anteriormente, encontramos que o aparecimento da ressonância de Fano e a formação de estados ligados dependem fortemente dos parâmetros do sistema. Explorando em detalhes o sistema, foi observado que é possível controlar a condutância elétrica do sistema através de um potencial de gate. Em particular, ao variar a posição do nível de energia do ponto quântico central é possível inverter a ressonância de Fano. Na última parte, apresentamos os resultados numéricos para o sistema considerando um ponto quântico com dois níveis de energia acoplado a dois eletrodos ferromagnéticos, e no interior do ponto quântico levamos em conta a interação e-e e a interação spin-órbita de Rashba. A interação de Rashba introduz a transição entre os níveis de energia com a inversão do spin, o que originou interessantes propriedades no transporte dependente do spin. Em particular, para eletrodos não magnéticos obtivemos que o acoplamento spin-órbita resultou na criação de corrente de spin polarizada / Abstract: In this work, we have studied the spin-dependent quantum transport in magnetic nanostructures. The main structures studied are magnetic tunneling junctions and systems composed of an arrangement of quantum dots coupled to ferromagnetic electrodes. Using the nonequilibrium Green's function techinique, we were able to calculate selected properties of the systems such as the electric current, the spin current, local density of states and the current-induced spin-transfer torque. In the first system, we have observed the effects of the spin-flip scattering in the transport properties considering tunneling junctions composed by an insulating layer between two ferromagnetic electrodes. The results obtained for this system showed that the magnetoresistance is related to density of state effects at the ferromagnetic electrodes, and we have found that it is strongly affected by the spin-flip scattering. Besides, we also observed that the spin-flip scattering gives rise to an additional spin-torque to the system. For the system composed of two quantum dots T-shaped electrodes coupled to ferromagnetic electrodes, we investigated the spin-dependent properties. We have observed that the appearance of the Fano resonance and the formation of bound states rely strongly on system parameters. Another interesting finding is the possibility to control the electrical conductance via a gate voltage. We figured out that changing the energy level of the central dot affected directly the Fano resonance peak intensities. Lastly, we have showed that for the system composed by a quantum-dot with two energy levels coupled to two ferromagnetic electrodes must be modified when the \textit{e-e} and the Rashba spin-orbit interaction are taken into account. The Rashba interaction induces level transitions with spin-flip resulting in interesting properties in the spin-dependent transport. We have found out that the spin-orbit interaction strongly contributes to the spin current even for non-magnetic electrodes / Doutorado / Física / Doutora em Ciências

In this thesis, we discuss some of the results that were proven by Fabes and Stroock in 1984. Our main purpose is to give a self-contained presentation of the proof of this results. The first result is on the existence of a “reverse H ̈older inequality” for the Green’s function. We utilize the work of Muckenhoupt on the reverse Ho ̈lder inequality and its connection to the A∞ class to establish a comparability property for the Green’s functions. Additionally, we discuss some of the underlying preliminaries. In that, we prove the Alexandrov-Bakelman-Pucci estimate, give a treatment to the Ap and A∞ classes of Muckenhoupt, and establish two intrinsic lemmas on the behavior of Green’s function.

The Liouville resolvent method is an unconventional technique used for finding a Green function for a Hamiltonian. Implementation of the method entails the calculation of commutators of a second-quantized Hamiltonian operator with particular generalized stepping operators that are elements of a Hilbert space and that represent transitions between many-particle states. These commutators produce linear combinations of stepping operators, so the results can be arrayed as matrix elements of the Liouville operator L̂ in the Hilbert space of stepping operators. The resulting L̂ matrix is usually of infinite order, and in principle its eigenvalues and eigenvectors can be used to construct the Green function from the L̂ resolvent matrix. Approximations are usually necessary, at least in the form of truncation of the L̂ matrix, and if one produces a sequence of such matrices of increasing order and calculates the eigenvalues and eigenvectors of these matrices, a sequence of approximations for the L̂ resolvent matrix can be produced. This sequence is mathematically guaranteed to converge to the exact result for the L̂ resolvent matrix (except at its singularities). The accuracy of an approximation depends on the order of the matrix at which the sequence is truncated. Application of the method to a Hamiltonian representing interactions between fermions and bosons involves complications arising from the large number of terms generated by the commutation properties of boson operators. This dissertation describes the method and its use in the study of fermion-boson couplings. Approximations to second order in stepping operators are calculated for simplified Froehlich and Lee models. Limited thermodynamic results are obtained from the Lee model. Exact energy eigenvalues are obtained by operator algebra for simplified Froehlich, Lee and Dirac models. These exact solutions comprise the main contribution of this research and will prove to be valuable starting points for further research. Suggestions are made for further research. / Ph. D. / incomplete_metadata