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.

Wave function based electronic structure methods have became a robust and reliable tool for the prediction and interpretation of the results of chemical experiments. However, they suffer from very steep scaling behavior with respect to an increase in the size of the system as well as very slow convergence of the correlation energy with respect to the basis set size. Thus these methods are limited to small systems of up to a dozen atoms. The first of these issues can be efficiently resolved by exploiting the local nature of electron correlation effects while the second problem is alleviated by the use of explicitly correlated R12/F12 methods. Since R12/F12 methods are central to this work, we start by reviewing their modern formulation. Next, we present the explicitly correlated second-order Mo ller-Plesset (MP2-F12) method in which all nontrivial post-mean-field steps are formulated with linear computational complexity in system size [Pavov{s}evi'c et al., {em J. Chem. Phys.} {bf 144}, 144109 (2016)]. The two key ideas are the use of pair-natural orbitals for compact representation of wave function amplitudes and the use of domain approximation to impose the block sparsity. This development utilizes the concepts for sparse representation of tensors described in the context of the DLPNO-MP2 method by Neese, Valeev and co-workers [Pinski et al., {em J. Chem. Phys.} {bf 143}, 034108 (2015)]. Novel developments reported here include the use of domains not only for the projected atomic orbitals, but also for the complementary auxiliary basis set (CABS) used to approximate the three- and four-electron integrals of the F12 theory, and a simplification of the standard B intermediate of the F12 theory that avoids computation of four-index two-electron integrals that involve two CABS indices. For quasi-1-dimensional systems (n-alkanes) the bigO{N} DLPNO-MP2-F12 method becomes less expensive than the conventional bigO{N^{5}} MP2-F12 for $n$ between 10 and 15, for double- and triple-zeta basis sets; for the largest alkane, C$_{200}$H$_{402}$, in def2-TZVP basis the observed computational complexity is $N^{sim1.6}$, largely due to the cubic cost of computing the mean-field operators. The method reproduces the canonical MP2-F12 energy with high precision: 99.9% of the canonical correlation energy is recovered with the default truncation parameters. Although its cost is significantly higher than that of DLPNO-MP2 method, the cost increase is compensated by the great reduction of the basis set error due to explicit correlation. We extend this formalism to develop a linear-scaling coupled-cluster singles and doubles with perturbative inclusion of triples and explicitly correlated geminals [Pavov{s}evi'c et al., {em J. Chem. Phys.} {bf 146}, 174108 (2017)]. Even for conservative truncation levels, the method rapidly reaches near-linear complexity in realistic basis sets; e.g., an effective scaling exponent of 1.49 was obtained for n-alkanes with up to 200 carbon atoms in a def2-TZVP basis set. The robustness of the method is benchmarked against the massively parallel implementation of the conventional explicitly correlated coupled-cluster for a 20-water cluster; the total dissociation energy of the cluster ($sim$186 kcal/mol) is affected by the reduced-scaling approximations by only $sim$0.4 kcal/mol. The reduced-scaling explicitly correlated CCSD(T) method is used to examine the binding energies of several systems in the L7 benchmark data set of noncovalent interactions. Additionally, we discuss a massively parallel implementation of the Laplace transform perturbative triple correction (T) to the DF-CCSD energy within density fitting framework. This work is closely related to the work by Scuseria and co-workers [Constans et al., {em J. Chem. Phys.} {bf 113}, 10451 (2000)]. The accuracy of quadrature with respect to the number of quadrature points has been investigated on systems of the 18-water cluster, uracil dimer and pentacene dimer. In the case of the 18-water cluster, the $mu text{E}_{text{h}}$ accuracy is achieved with only 3 quadrature points. For the uracil dimer and pentacene dimer, 6 or more quadrature points are required to achieve $mu text{E}_{text{h}}$ accuracy; however, binding energy of $<$1 kcal/mol is obtained with 4 quadrature points. We observe an excellent strong scaling behavior on distributed-memory commodity cluster for the 18-water cluster. Furthermore, the Laplace transform formulation of (T) performs faster than the canonical (T) in the case of studied systems. The efficiency of the method has been furthermore tested on a DNA base-pair, a system with more than one thousand basis functions. Lastly, we discuss an explicitly correlated formalism for the second-order single-particle Green's function method (GF2-F12) that does not assume the popular diagonal approximation, and describes the energy dependence of the explicitly correlated terms [Pavov{s}evi'c et al., {em J. Chem. Phys.} {bf 147}, 121101 (2017)]. For small and medium organic molecules the basis set errors of ionization potentials of GF2-F12 are radically improved relative to GF2: the performance of GF2-F12/aug-cc-pVDZ is better than that of GF2/aug-cc-pVQZ, at a significantly lower cost. / Ph. D.