Modelling spin waves in noncollinear antiferromagnets: spin-flop states, spin spirals, skyrmions and antiskyrmions

Spin waves in antiferromagnetic materials have great potential for next-generation magnonic technologies. However, their properties and their dependence on the type of ground-state antiferromagnetic structure are still open questions. Here, we investigate theoretically spin waves in one- and two-dimensional model systems with a focus on noncollinear antiferromagnetic textures such as spin spirals and skyrmions of opposite topological charges. We address in particular the nonreciprocal spin excitations recently measured in bulk antiferromagnet $\alpha$--$\text{Cu}_2\text{V}_2\text{O}_7$ utilizing inelastic neutron scattering experiments [Phys.\ Rev.\ Lett.\ \textbf{119}, 047201 (2017)], where we help to characterize the nature of the detected spin-wave modes. Furthermore, we discuss how the Dzyaloshinskii-Moriya interaction can lift the degeneracy of the spin-wave modes in antiferromagnets, resembling the electronic Rashba splitting. We consider the spin-wave excitations in antiferromagnetic spin-spiral and skyrmion systems and discuss the features of their inelastic scattering spectra. We demonstrate that antiskyrmions can be obtained with an isotropic Dzyaloshinskii-Moriya interaction in certain antiferromagnets.Comment: 26 pages, 9 figure

First-principles investigation of spin wave dispersions in surface-reconstructed Co thin films on W(110)

We computed spin wave dispersions of surface-reconstructed Co films on the W(110) surface in the adiabatic approximation. The magnetic exchange interactions are obtained via first-principles electronic structure calculations using the Korringa-Kohn-Rostoker Green function method. We analyze the strength and oscillatory behavior of the intralayer and interlayer magnetic interactions and investigate the resulting spin wave dispersions as a function of the thickness of Co films. In particular, we highlight and explain the strong impact of hybridization of the electronic states at the Co-W interface on the magnetic exchange interactions and on the spin wave dispersions. We compare our results to recent measurements based on electron energy loss spectroscopy [E. Michel, H. Ibach, and C.M. Schneider, Phys. Rev. B 92, 024407 (2015)]. Good overall agreement with experimental findings can be obtained by considering the possible overestimation of the spin splitting, stemming from the local spin density approximation, and adopting an appropriate correction.Comment: 14 pages, 14 figure

Aprendiendo funciones complejas con GeoGebra

In this paper we describe a teaching experiment targeting with students of Complex Analysis attending an undergraduate course of a Portuguese university. Our main goal is the understanding of the GeoGebra role with respect to visualization and as a technological mediator, according to Vygotsky theory, in the teaching and learning process of complex functions. The first step of our study was the conception of a sequence of didactical tasks and the development of GeoGebra tools related to the target didactical objectives. Here we will describe the procedure related to the tasks implementation in a classroom environment and the achieved results based on the collected data composed by written assignments produced by students, video recording the student performance during the experiment, and the student constructions with GeoGebra. All the collected data were analysed from a qualitative and interpretative paradigm.publishe

Nonreciprocity of spin waves in noncollinear magnets due to the Dzyaloshinskii-Moriya interaction

Broken inversion symmetry in combination with the spin-orbit interaction generates a finite Dzyaloshinskii-Moriya interaction (DMI), which can induce noncollinear spin textures of chiral nature. The DMI is characterized by an interaction vector whose magnitude, direction and symmetries are crucial to determine the stability of various spin textures, such as skyrmions and spin spirals. The DMI can be measured from the nonreciprocity of spin waves in ferromagnets, which can be probed via inelastic scattering experiments. In a ferromagnet, the DMI can modify the spin-wave dispersion, moving its minimum away from the $\Gamma$ point. Spin waves propagating with opposite wavevectors are then characterized by different group velocities, energies and lifetimes, defining their nonreciprocity. Here, we address the case of complex spin textures, where the manifestation of DMI-induced chiral asymmetries remains to be explored. We discuss such nonreciprocal effects and propose ways of accessing the magnitude and direction of the DMI vectors in the context of spin-polarized or spin-resolved inelastic scattering experiments. We show that only when a periodic magnetic system has finite net magnetization, that is, when the vector sum of all magnetic moments is nonzero, can it present a total nonreciprocal spin-wave spectrum. However, even zero-net-magnetization systems, such as collinear antiferromagnets and cycloidal spin spirals, can have spin-wave modes that are individually nonreciprocal, while the total spectrum remains reciprocal.Comment: 12 figure

The Riemann sphere in GeoGebra

The stereographic projection is a bijective smooth map which allows us to think the sphere as the extended complex plane. Among its properties it should be emphasized the remarkable property of being angle conformal that is, it is an angle measure preserving map. Unfortunately, this projection map does not preserve areas. Besides being conformal it has also the property of projecting spherical circles in either circles or straight lines in the plane This type of projection maps seems to have been known since ancient times by Hipparchus (150 BC), being Ptolemy (AD 140) who, in his work entitled "The Planisphaerium", provided a detailed description of such a map. Nonetheless, it is worthwhile to mention that the property of the invariance of angle measure has only been established much later, in the seventeenth century, by Thomas Harriot. In fact, it was exactly in that century that the Jesuit François d’Aguilon introduced the terminology "stereographic projection" for this type of maps, which remained up to our days. Here, we shall show how we create in GeoGebra, the PRiemannz tool and its potential concerning the visualization and analysis of the properties of the stereographic projection, in addition to the viewing of the amazing relations between Möbius Transformations and stereographic projections

Complex functions with Geogebra

Complex functions, generally feature some interesting peculiarities, seen as extensions real functions, complementing the study of real analysis. However, the visualization of some complex functions properties requires the simultaneous visualization of two-dimensional spaces. The multiple Windows of GeoGebra, combined with its ability of algebraic computation with complex numbers, allow the study of the functions defined from ℂ to ℂ through traditional techniques and by the use of Domain Colouring. Here, we will show how we can use GeoGebra for the study of complex functions, using several representations and creating tools which complement the tools already provided by the software. Our proposals designed for students of the first year of engineering and science courses can and should be used as an educational tool in collaborative learning environments. The main advantage in its use in individual terms is the promotion of the deductive reasoning (conjecture / proof). In performed the literature review few references were found involving this educational topic and by the use of a single software

Using GeoGebra to study complex functions

The aim of this workshop to present some of the strategies studied to use GeoGebra in the analysis of complex functions. The proposed tasks focus on complex analysis topics target for students of the 1st year of higher education, which can be easily adapted to pre-university students. In the first part of this workshop we will illustrate how to use the two graphical windows of GeoGebra to represent complex functions of complex variable. The second part will present the use of the dynamic color Geogebra in order to obtain Coloring domains that correspond to the graphic representation of complex functions. Finally, we will use the threedimensional graphics window in GeoGebra to study the component functions of a complex function. During the workshop will be provided scripts orientation of the different tasks proposed to be held on computers with Geogebra version 5.0 or high

GeoGebra as a learning mathematical environment

GeoGebra, a software system for dynamic geometry and algebra in the plane, since its inception in 2001, has gone from a dynamic geometry software (DGS), to a powerful computational tool in several areas of mathematics. Powerful algebraic capabilities have joined GeoGebra, an efficient spreadsheet that can deal with many kinds of objects, an algebraic and symbolic calculation system and several graphical views that expand the possibility of multidimensional representations, namely, by using colouring domain techniques, expanded to representations in the Riemann sphere, making this DGS a powerful research tool in mathematics. On the other hand, GeoGebra can create applications easily and export to HTML, and the possibility to quickly integrating these applets in several web platforms provides this DGS with an excellent way to create strong collaborative environments to teach and learn mathematics. Recently was added to GeoGebra powerful capabilities that transform this software a real Learning Mathematical Environment, using the GeoGebraBooks and GeoGebraGroups, plain of collaborative functionality between students and teachers

Job quality of Portuguese health professionals: 2011 vs 2015

The present research uses data from European Union Labour Force Survey to analyse job quality of Portuguese health professionals, comparing data from the years 2011 and 2015. The data from both years were processed separately through Fuzzy Clustering Analysis, which enabled the identification of employment profiles within each year. The empirical findings identified sets of three and four profiles to 2011 and 2015, respectively. In these sets, the duality of bad and good jobs is distinguishable. The bad jobs are characterized by affecting mainly young professionals and are identified by the existence of precarious situations such as limited duration contracts, part-time jobs and having low income. Good jobs have the opposite characteristics. The good jobs are mainly occupied by the older professionals. These professionals have permanent and full-time jobs and are taking home wages placed within the highest deciles of income in the Portuguese society. Medical Doctors is the professional category with better contractual and income indicators, though is the category to be clearly identified as having more weekly working hours. The category Other Health Professionals is the one more affected by the precariousness in the early years of practice, which indicates the need of a deeper research on who these other professionals are, and if there are other relevant differences within the category. Apart from the found duality, this research also found a duality within the duality, which is the existence of atypical work within the profiles with better jobs, creating a distinction between professionals.A presente pesquisa utiliza dados do "European Union Labour Force Survey" para analisar a qualidade do emprego dos profissionais de saúde portugueses, comparando dados dos anos de 2011 e 2015. Os dados de cada ano foram tratados através de "Fuzzy Clustering Analysis", que permitiu a identificação de conjuntos de perfis de emprego dentro de cada ano. Os resultados empíricos identificaram conjuntos de três e quatro perfis de emprego para 2011 e 2015, respetivamente. Nestes conjuntos observa-se a existência da dualidade dos maus e bons empregos. Os maus empregos caracterizam-se por afetar jovens profissionais e são identificados pela existência de situações precárias, como contratos de tempo limitado, empregos a tempo parcial e salários de baixo valor. Os bons empregos são ocupados principalmente pelos profissionais de saúde mais antigos. Estes profissionais têm empregos com contratos sem termo e a tempo inteiro, levando para casa salários colocados nos níveis mais altos de rendimento em Portugal. Os médicos são a categoria profissional com melhores indicadores contratuais e de rendimento mensal, embora seja a categoria que é claramente identificada como tendo mais horas semanais de trabalho. A categoria dos outros profissionais da saúde é a mais afetada pela precariedade nos primeiros anos de prática, o que indicia a necessidade de uma pesquisa aprofundada sobre quem são esses outros profissionais, e se existem outras diferenças relevantes mesmo dentro da categoria. Encontrou-se, ainda, uma dualidade dentro da dualidade, que é a existência de trabalho atípico em alguns dos perfis dos bons empregos, criando separação entre os profissionais

Quais as soluções da equação (x^2-5x+5)^(x^2-9x+20)

Este trabajo presenta los procesos de resolución de una ecuación por estudiantes del último año de educación secundaria, de un curso profesional, en Portugal. Con base en el constructivismo social, como marco teórico para el aprendizaje, y el uso de los mediadores como sea el grupo de alumnos y la tecnología, en particular el GeoGebra 3.0, se eligieron las tareas propuestas a los estudiantes y se gestionó el trabajo en el aula. Los datos fueron analizados e interpretados a través de un paradigma cualitativo interpretativo. La tarea se sitúa en el contexto de la utilización de la metodología de Aprendizaje Basado en Problemas. El profesor-investigador estaba interesado en aprender cómo los estudiantes respondieron a estas tareas, así como entender las estrategias desarrolladas y el tipo de tecnología por ellos utilizada para resolver el problema. A partir de estas premisas se analizan e interpretan los resultados obtenidos