Orientador: Rafael Zadorosny / Resumo: Este trabalho está estruturado em duas partes. A primeira, envolve o estudo do estado de vórtices e comportamento magnético de um supercondutor mesoscópico submetido a um gradiente térmico. Foi veri cado que a influência de um gradiente térmico altera o estado de vórtice gigante e pode levar a magnetização a ter um comportamento anômalo, o qual depende do parâmetro de Ginzburg-Landau e das temperaturas nas regiões "quente" e "fria". Na segunda parte, abordamos o estado resistivo durante o movimento de um vórtice e durante a dinâmica de par vórtice-antivórtice (V-AV) para os tipos de Abrikosov e Cinemáticos. Estudamos quais parâmetros afetam as propriedades do não equilíbrio e os mecanismos de dissipação presentes no processo de difusão térmica. Para os processos envolvendo vórtice ou V-AV de Abrikosov, os mecanismos de dissipação devido à processos de relaxação contribuem para a maior parte da potência total dissipada e, consequentemente, exercem uma grande contribuição para a variação de temperatura. Por outro lado, para o caso de V-AV cinemáticos, os elétrons normais aparecem como principal mecanismo de dissipação, todavia os processos de relaxação tem uma contribuição considerável para esse caso. Para realizar tais estudos, utilizamos o teorema da energia livre e as equações de Ginzburg- Landau generalizadas. / Abstract: This work is structured in two parts. In the rst one, the vortices states and the magnetic behavior were studied for a mesoscopic superconductor under a thermal gradient. It was veri ed that the thermal gradient changes the giant vortex state and could lead a exotic behavior of the magnetization curve, depending on the parameter and the temperature of the hot and the cold . In the second part, the focus was the resistive state during one vortex motion and during the annihilation process of a vortex-antivortex (V-AV) pair both with Abrikosov and Kinematic types. We de ned which parameters a ect the nonequilibrium proprieties and the dissipative mechanisms present in the thermal di usion processes. For the cases involving a vortex or an Abrikosov's V-AV pair, the dissipative mechanisms due to the relaxation process have the major contribution for the total power dissipated power and consequently, exert great contribuition on the temperature variation. On the other hand, for the case of kinematic V-AV, the normal eletrons rised as the main dissipative mechanism, however relaxation process has a considerable contribuition for this case. To realize such studies, we used the free energy theorem and the generalized Ginzburg-Landau equations. / Doutor

In this dissertation, we employ variational methods to obtain a new existence result for solutions of a Ginzburg-Landau type equation on a Riemannian manifold. We prove that if $N$ is a compact, orientable 3-dimensional Riemannian manifold without boundary and $\gamma$ is a simple, smooth, connected, closed geodesic in $N$ satisfying a natural nondegeneracy condition, then for every $\ep>0$ sufficiently small, $\exists$ a critical point $u^\ep\in H^1(N;\mathbb{C})$ of the Ginzburg-Landau functional \bd\ds E^\ep(u):=\frac{1}{2\pi |\ln\ep|}\int_N |\nabla u|^2+\frac{(|u|^2-1)^2}{2\ep^2}\ed and these critical points have the property that $E^\ep(u^\ep)\rightarrow\tx{length}(\gamma)$ as $\ep\rightarrow 0$. To accomplish this, we appeal to a recent general asymptotic minmax theorem which basically says that if $E^\ep$ $\Gamma$-converges to $E$ (not necessarily defined on the same Banach space as $E^\ep$), $v$ is a saddle point of $E$ and some additional mild hypotheses are met, then there exists $\ep_0>0$ such that for every $\ep\in(0,\ep_0),E^\ep$ possesses a critical point $u^\ep$ and $\lim_{\ep\rightarrow 0}E^\ep(u^\ep)=E(v)$. Typically, $E$ is only lower semicontinuous, therefore a suitable notion of saddle point is needed. Using known results on $\mathbb{R}^3$, we show the Ginzburg-Landau functional $E^\ep$ defined above $\Gamma$-converges to a functional $E$ which can be thought of as measuring the arclength of a limiting singular set. Also, we verify using regularity theory for almost-minimal currents that $\gamma$ is a saddle point of $E$ in an appropriate sense.

In this dissertation, we employ variational methods to obtain a new existence result for solutions of a Ginzburg-Landau type equation on a Riemannian manifold. We prove that if $N$ is a compact, orientable 3-dimensional Riemannian manifold without boundary and $\gamma$ is a simple, smooth, connected, closed geodesic in $N$ satisfying a natural nondegeneracy condition, then for every $\ep>0$ sufficiently small, $\exists$ a critical point $u^\ep\in H^1(N;\mathbb{C})$ of the Ginzburg-Landau functional \bd\ds E^\ep(u):=\frac{1}{2\pi |\ln\ep|}\int_N |\nabla u|^2+\frac{(|u|^2-1)^2}{2\ep^2}\ed and these critical points have the property that $E^\ep(u^\ep)\rightarrow\tx{length}(\gamma)$ as $\ep\rightarrow 0$. To accomplish this, we appeal to a recent general asymptotic minmax theorem which basically says that if $E^\ep$ $\Gamma$-converges to $E$ (not necessarily defined on the same Banach space as $E^\ep$), $v$ is a saddle point of $E$ and some additional mild hypotheses are met, then there exists $\ep_0>0$ such that for every $\ep\in(0,\ep_0),E^\ep$ possesses a critical point $u^\ep$ and $\lim_{\ep\rightarrow 0}E^\ep(u^\ep)=E(v)$. Typically, $E$ is only lower semicontinuous, therefore a suitable notion of saddle point is needed. Using known results on $\mathbb{R}^3$, we show the Ginzburg-Landau functional $E^\ep$ defined above $\Gamma$-converges to a functional $E$ which can be thought of as measuring the arclength of a limiting singular set. Also, we verify using regularity theory for almost-minimal currents that $\gamma$ is a saddle point of $E$ in an appropriate sense.

In part I, a numerical study of the mixed states in a mesoscopic type-II superconducting cylinder is described. Steady-state configurations and transient behavior of the magnetic vortices for various values of the applied magnetic field H are presented. Transitions between different multi-vortex states as H is changed is demonstrated by abrupt changes in vortex configurations and jumps in the B vs H plot. An efficient scheme to determine the equilibrium vortex configuration in a mesoscopic system at any given applied field, not limited to the symmetry of the system, is devised and demonstrated. In part II, a superconducting thin film is subject to a non-uniform magnetic field from a vertical magnetic dipole, consisting of two magnetic monopoles of opposite charges. For a film with constant thickness and with no pins, it has been found that the film carries two pairs of vortex-antivortex in the steady state in the imposed flux range of 2.15 < (Phi)+ < 2.90 (in units of flux quantum) and no vortex at all for (Phi)+ <= 2.15. Transitions from a superconducting state with 3 pairs of vortex-antivortex to one with 2 pairs, where a pair of vortex-antivortex annihilates, have been observed in the pseudo-time sequence. With a perturbation with antidots (holes), vortexantivortex pair has been created for lower magnetic fluxes down to (Phi)+ = 1.3. In the sample of size 16(Xi) x 16(Xi), the attraction force between the vortex and antivortex always dominates over the pinning force, so that they eventually come out of pins, move toward each other, and annihilate each other. The annihilation rate, measured with time taken for the annihilation, is reduced noticeably by the increase of the distance between pins, or the increase in the pin size. A simulation of the magnetic vortex pinning in the sample of size 32(Xi) x 32(Xi) suggests we are likely to achieve pinning of the vortex-antivortex pair with the sample size around this and vortex-antivortex separation of 22(Xi). Using this sample as a template, the maximum density of pinned vortices achievable is calculates to be about 7.6 x 10^14 vortices/m2 for (Xi) =~ 1.6A°.

The topic of this thesis is vortex-physics in multi component Ginzburg- Landau models. These models describe a newly discovered class of super- conductors with multiple superconducting gaps, and possess many properties that set them apart from single component models. The work presented here relies on large scale computer simulations using various numerical techniques, but also on some analytical methods. In Paper I, Type-1.5 Superconducting State from an Intrinsic Proximity Effect in Two-Band Superconductors, we show that in multiband supercon- ductors, even an extremely small interband proximity effect can lead to a qualitative change in the interaction potential between superconducting vor- tices, by producing long-range intervortex attraction. This type of vortex interaction results in an unusual response to low magnetic fields, leading to phase separation into domains of two-component Meissner states and vortex droplets. In paper II, Type-1.5 superconductivity in two-band systems, we discuss the influence of Josephson coupling and show that non-monotonic intervortex interaction can also arise in two-band superconductors where one of the bands is proximity induced by Josephson interband coupling. In paper III, Type-1.5 superconductivity in multiband systems: Effects of interband couplings, we investigate the appearance of Type-1.5 superconduc- tivity in the case with two active bands and substantial inter-band couplings such as intrinsic Josephson coupling, mixed gradient coupling, and density- density interactions. We show that in the presence of these interactions, the system supports type-1.5 superconductivity with fundamental length scales being associated with the mass of the gauge field and two masses of normal modes represented by linear combinations of the density fields. In paper IV, Semi-Meissner state and nonpairwise intervortex interactions in type-1.5 superconductors, we demonstrate the existence of nonpairwise in- tervortex forces in multicomponent and layered superconducting systems. We also consider the properties of vortex clusters in a semi-Meissner state of type- 1.5 two-component superconductors. We show that under certain conditions nonpairwise forces can contribute to the formation of complex vortex states in type-1.5 regimes. In paper V, Length scales, collective modes, and type-1.5 regimes in three- band superconductors, we consider systems where frustration in phase dif- ferences occur due to competing Josephson inter-band coupling terms. We show that gradients of densities and phase differences can be inextricably intertwined in vortex excitations in three-band models. This can lead to long-range attractive intervortex interactions and the appearance of type-1.5 regimes even when the intercomponent Josephson coupling is large. We also show that field-induced vortices can lead to a change of broken symmetry from U (1) to U (1) ⇥ Z2 in the system. In the type-1.5 regime, it results in a semi-Meissner state where the system has a macroscopic phase separation in domainswithbrokenU(1)andU(1)⇥Z2 symmetries. In paper VI, Topological Solitons in Three-Band Superconductors with Broken Time Reversal Symmetry, we show that three-band superconductors with broken time reversal symmetry allow magnetic flux-carrying stable topo- logical solitons. They can be induced by fluctuations or quenching the system through a phase transition. It can provide an experimental signature of the time reversal symmetry breakdown. In paper VII, Type-1.5 superconductivity in multiband systems: Magnetic response, broken symmetries and microscopic theory – A brief overview, we give an overview of vortex physics and magnetic response in multi component Ginzburg-Landau theory. We also examine Type-1.5 superconductivity in the context of microscopic theory. In paper VIII, Chiral CP2 skyrmions in three-band superconductors, we show that under certain conditions, three-component superconductors (and, in particular, three-band systems) allow stable topological defects different from vortices. We demonstrate the existence of these excitations, charac- terised by a CP2 topological invariant, in models for three-component super- conductors with broken time-reversal symmetry. We term these topological defects “chiral GL(3) skyrmions,” where “chiral” refers to the fact that due to broken time-reversal symmetry, these defects come in inequivalent left- and right-handed versions. In certain cases, these objects are energetically cheaper than vortices and should be induced by an applied magnetic field. In other situations, these skyrmions are metastable states, which can be produced by a quench. Observation of these defects can signal broken time-reversal sym- metry in three-band superconductors or in Josephson-coupled bilayers of s± and s-wave superconductors. In paper IX, Phase transition in multi-component superconductors, we ex- amine the thermodynamics of frustrated multi-components superconductors and show that their highly complex energy landscape can give rise new types of phase transitions not present in single component superconductors. / <p>QC 20131205</p>

Freiburg (Breisgau), Universiẗat, Diss., 1999.

Made available in DSpace on 2014-11-10T11:09:42Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-05-21Bitstream added on 2014-11-10T11:58:44Z : No. of bitstreams: 1 000793304.pdf: 3817236 bytes, checksum: c88f0b4b23997905abcf1aeddcf17d72 (MD5) / O estudo de amostras supercondutoras em escala mesoscópica é de grande interesse científico e tecnológico pelos efeitos de confinamento que dela advém, por isso da intensificação dos trabalhos voltados a essa classe de materiais nas últimas décadas. Estas escalas estão próximas aos comprimentos característicos dos supercondutores, i.e., do comprimento de coerência (T) e da profundidade de penetração (T). Neste trabalho, usamos a Teoria de Ginzburg-Landau dependente do tempo, TDGL, para descrever o parâmetro de ordem e o campo magnético no interior de amostras supercondutoras de seção reta quadrática e comprimento infinito, ao longo do eixo z, sob um campo externo constante. Nestas amostras foi inserido um defeito extenso na forma de uma fenda e, então, os estudos foram focados na análise da influência desta fenda em diversas posições sobre algumas propriedades do material. É importante frisar que a TDGL permite descrever os estados metaestáveis da configuração de vórtices e o equilíbrio é obtido deixando-se o sistema relaxar para o estado estacionário. A discretização das equações TDGL foi embasada no método das variáveis de ligação e, então, as simulações computacionais foram realizadas. Assim, analisamos a influência da fenda, em diferentes posições, na primeira penetração de vórtices e nos campos críticos Hc1 e Hc3. Simulamos amostras de supercondutores do tipo II, k = 5, com dimensões mesoscópicas, 10 (0) x 10 (0) e o campo magnético externo foi variado com passos de H= 10-3 para diversos valores de temperatura. Também foram analisadas as condições nas quais a primeira penetração de vórtice ocorre pela fenda, assim, novas simulações foram realizadas para amostras com tamanhos laterais de 20 (0) x 20 (0) e (30) (0) / The study of superconducting samples in a mesoscopic scale is of scientific and technological great interest due to their confinement effects. As a consequence, this fact led to an intensification of works focused in this category of materials in the last decades. Such scale lies in the fundamental superconducting lengths, i.e., the coherence length k = 5,and the penetration depth (T). In this work, we use Time Dependent Ginzburg-Landau theory, TDGL, to describe the order parameter and the magnet field inside superconducting samples of square cross section and infinite length, in the z axis, under a constant external magnetic field. In such samples was inserted a long slit-like defect, and thus, we focused our studies on the influence of this slit, positioned in several positions of the border of the sample, in the properties of the material. It is important to emphasize that the TDGL model allows us to describe the metastable states of the vortex configuration and the equilibrium state is obtained by leaving the system rich the stationary state. The discretization of the TDGL equations was obtained by the variable link method. Thus, the computational simulations were carried out and we analyzed the influence of a slit, in several positions, in the first vortex penetrations as well as in the and .critical fields. It were simulated type II superconducting samples, , with mesoscopic dimensions of . The external applied magnet field was changed in steps of = 10-3, for several values of temperature. It was also analyzed the conditions for which the first penetration of the vortex occur by the slit. In this interim, we carried out simulations for samples with lateral sizes of 20 (0) x 20 (0) e (30) (0)

Orientador: Rafael Zadorosny / Banca: Edson Sardella / Banca: Claudio Antonio Cardoso / Resumo: O estudo de amostras supercondutoras em escala mesoscópica é de grande interesse científico e tecnológico pelos efeitos de confinamento que dela advém, por isso da intensificação dos trabalhos voltados a essa classe de materiais nas últimas décadas. Estas escalas estão próximas aos comprimentos característicos dos supercondutores, i.e., do comprimento de coerência (T) e da profundidade de penetração (T). Neste trabalho, usamos a Teoria de Ginzburg-Landau dependente do tempo, TDGL, para descrever o parâmetro de ordem e o campo magnético no interior de amostras supercondutoras de seção reta quadrática e comprimento infinito, ao longo do eixo z, sob um campo externo constante. Nestas amostras foi inserido um defeito extenso na forma de uma fenda e, então, os estudos foram focados na análise da influência desta fenda em diversas posições sobre algumas propriedades do material. É importante frisar que a TDGL permite descrever os estados metaestáveis da configuração de vórtices e o equilíbrio é obtido deixando-se o sistema relaxar para o estado estacionário. A discretização das equações TDGL foi embasada no método das variáveis de ligação e, então, as simulações computacionais foram realizadas. Assim, analisamos a influência da fenda, em diferentes posições, na primeira penetração de vórtices e nos campos críticos Hc1 e Hc3. Simulamos amostras de supercondutores do tipo II, k = 5, com dimensões mesoscópicas, 10 (0) x 10 (0) e o campo magnético externo foi variado com passos de H= 10-3 para diversos valores de temperatura. Também foram analisadas as condições nas quais a primeira penetração de vórtice ocorre pela fenda, assim, novas simulações foram realizadas para amostras com tamanhos laterais de 20 (0) x 20 (0) e (30) (0) / Abstract: The study of superconducting samples in a mesoscopic scale is of scientific and technological great interest due to their confinement effects. As a consequence, this fact led to an intensification of works focused in this category of materials in the last decades. Such scale lies in the fundamental superconducting lengths, i.e., the coherence length k = 5,and the penetration depth (T). In this work, we use Time Dependent Ginzburg-Landau theory, TDGL, to describe the order parameter and the magnet field inside superconducting samples of square cross section and infinite length, in the z axis, under a constant external magnetic field. In such samples was inserted a long slit-like defect, and thus, we focused our studies on the influence of this slit, positioned in several positions of the border of the sample, in the properties of the material. It is important to emphasize that the TDGL model allows us to describe the metastable states of the vortex configuration and the equilibrium state is obtained by leaving the system rich the stationary state. The discretization of the TDGL equations was obtained by the variable link method. Thus, the computational simulations were carried out and we analyzed the influence of a slit, in several positions, in the first vortex penetrations as well as in the and .critical fields. It were simulated type II superconducting samples, , with mesoscopic dimensions of . The external applied magnet field was changed in steps of = 10-3, for several values of temperature. It was also analyzed the conditions for which the first penetration of the vortex occur by the slit. In this interim, we carried out simulations for samples with lateral sizes of 20 (0) x 20 (0) e (30) (0) / Mestre

Made available in DSpace on 2014-06-12T18:03:15Z (GMT). No. of bitstreams: 2 arquivo7708_1.pdf: 2115875 bytes, checksum: e92e0b690885458dfc4d49ad5563e512 (MD5) license.txt: 1748 bytes, checksum: 8a4605be74aa9ea9d79846c1fba20a33 (MD5) Previous issue date: 2007 / Conselho Nacional de Desenvolvimento Científico e Tecnológico / Nesta tese estudamos teoricamente as configurações de vórtices em algumas nanoestruturas supercondutoras, como quadrados mesoscópicos com dois defeitos as- simétricos ou com um defeito preenchido com material metálico, anéis mesoscópicos e filmes finos submersos em diferentes tipos de materiais (metais, outros supercon- dutores, isolantes) ou com uma alta densidade de defeitos em sua interface. O estudo foi realizado mediante resolução numérica das equações de Ginzburg-Landau dependentes do tempo (TDGL) para os sistemas citados utilizando-se o método de campos de calibre com variáveis de ligação. Encontramos as configurações de vórtices para estas geometrias e calculamos as curvas de magnetização em função do campo magnético aplicado. Constatamos que as configurações de vórtices, os campos críticos e a própria maneira como os vórtices adentram o supercondutor, dependem fortemente do arranjo dos centros de aprisionamento e da geometria do sistema. No estudo de filmes finos com interfaces diversas, observamos que, devido ao contato com o material, existe um aumento ou uma supress~ao (dependendo do tipo de interface) da supercondutividade nas bordas da amostra, como revelam os resultados das curvas de magnetização, densidade de elétrons supercondutores e número de vórtices em função do campo magnético e do comprimento de extra- polação de De Gennes b. Para determinados valores de b o sistema tem uma resposta paramagnética na curva de magnetização, relacionada na captura do luxo magnético devido nas fronteiras da amostra

The motivation for this paper is a better understanding of the basic building blocks of FJRW theory. The basics of FJRW theory will be briefly outlined, but the majority of the paper will deal with certain multivariate polynomials which are the most fundamental building blocks in FJRW theory. We will first describe what is already known about these polynomials and then discuss several properties we proved as well as conjectures we disproved. We also introduce a new conjecture suggested by computer calculations performed as part of our investigation.