Introductory Chapter

This book deals with the recent advancements in two topical subjects of condensed mater physics, superluidity, and superconductivity. In principle, the two phenomena are very similar because they occur as a function of temperature and in the presence of the vanishing of a physical quantity marking a phase transition below a critical temperature. A superluid is a luid having zero viscosity while a superconductor is a conductor with zero resistance. Superluidity occurs in liquid helium and in ultracold atomic gases while superconductivity is typical of elements like niobium and lead, of some niobium alloys, or compounds like ytrium barium and copper oxide and compounds containing iron. Regarding the later, since the irst discoveries, the interplay between superconductivity and magnetism has also been investigated inding that the magnetic state of superconductors can be described as ideal diamagnetism. The behaviour toward the external magnetic ield allows to distinguish between irst- and second-type superconductors. Instead, the critical temperature in correspondence of which superconductivity arises allows to distinguish between low- and high-critical temperature superconductors. After their initial discovery, superluidity was explained as a quantum mechanical phenomenon, while superconductivity was described irst according to a phenomenological and classical theory and only in a second moment in terms of a microscopic quantum mechanical theory

Metamaterial Properties of 2D Ferromagnetic Nanostructures: From Continuous Ferromagnetic Films to Magnonic Crystals

In recent years the study of low-dimensional magnetic systems has become topical not only for its several technological applications but also for achieving a deep understanding of the underlying physics of magnetic nanostructures. These efforts have considerably advanced the field of magnetism both theoretically and from an experimental point of view. Very recently, for their challenging features, great attention has been given to the investigation of the static and dynamical properties of magnetic nanostructures with special regard to magnonic crystals, a class of periodic magnetic systems. As shown by micromagnetic and analytical methods, the ferromagnetic materials composing magnonic crystals can be regarded as metamaterials since they exhibit effective properties directly linked, for instance, to the definition of an effective magnetization, an effective permeability, and an effective wavelength. Hence, the aim of this chapter is to give an overview of the recent results obtained on the study of metamaterial properties of two-dimensional ferromagnetic nanostructures ranging from those of thin films to the ones of two-dimensional magnonic crystals. Some possible applications based on the effective properties for tailoring new magnetic devices are suggested

Hamiltonian and Lagrangian Dynamical Matrix Approaches Applied to Magnetic Nanostructures

Two micromagnetic tools to study the spin dynamics are reviewed. Both approaches are based upon the so-called dynamical matrix method, a hybrid micromagnetic framework used to investigate the spin-wave normal modes of confined magnetic systems. The approach which was formulated first is the Hamiltonian-based dynamical matrix method. This method, used to investigate dynamic magnetic properties of conservative systems, was originally developed for studying spin excitations in isolated magnetic nanoparticles and it has been recently generalized to study the dynamics of periodic magnetic nanoparticles. The other one, the Lagrangian-based dynamical matrix method, was formulated as an extension of the previous one in order to include also dissipative effects. Such dissipative phenomena are associated not only to intrinsic but also to extrinsic damping caused by injection of a spin current in the form of spin-transfer torque. This method is very accurate in identifying spin modes that become unstable under the action of a spin current. The analytical development of the system of the linearized equations of motion leads to a complex generalized Hermitian eigenvalue problem in the Hamiltonian dynamical matrix method and to a non-Hermitian one in the Lagrangian approach. In both cases, such systems have to be solved numerically

Entropy for the Brain and Applied Computation

An understanding of the interplay between the brain region and the state of consciousness and the evolution of brain complexity, regarded as an entropy-enhancing process, has recently been proposed, leading to an increase in the space of states that can be visited, and to the accessibility of new channels. Nowadays, nonlinear and complex system theories are considered promising candidates for analyzing the principles of operation of neural networks and their entropic content. The concept of computational entropy, for example, has been utilized in cryptographic primitives, namely, in cryptographic algorithms that are used to create cryptographic protocols in computer security systems

Topological, non topological and instanton droplets driven by spin-transfer torque in materials with perpendicular magnetic anisotropy and Dzyaloshinskii-Moriya Interaction

The interfacial Dzyaloshinskii-Moriya Interaction can modify the topology of droplets excited by a localized spin-polarized current. Here, we show that, in addition to the stationary droplet excitations with skyrmion number either one (topological) or zero (non-topological), there exists, for a fixed current, an excited mode with a non-stationary time behavior. We call this mode "instanton droplet", which is characterized by time domain transitions of the skyrmion number. These transitions are coupled to an emission of incoherent spin-waves that can be observed in the frequency domain as a source of noise. Our results are interesting from a fundamental point of view to study spin-wave emissions due to a topological transition in current-driven systems, and could open the route for experiments based on magnetoresistance effect for the design of a further generation of nanoscale microwave oscillators

Low-Dimensional Magnetic Systems

The interest in the nanoscale properties of low-dimensional magnetic systems has grown exponentially during the last decades and has attracted the attention of both experimentalists and theorists. The state of the art of these investigations has indeed allowed to give valuable insights into the underlying physics of complex magnetization dynamics driven by magnetic fields, electric currents and thermal effects. At the same time, such studies have found, in relatively short times, several applications at industrial level in the field of spintronics and magnonics as magnetic memories, microwave oscillators, modulators, sensors, logic gates, diodes and transistors. The goal of this special issue is to offer a variety of recent developments on this topic by gathering contributions arising from several specialists in the field of nanomagnetism. The strength of this issue lies indeed on its "variety": the properties of these systems are, in fact, investigated from the viewpoint of physicists, engineers and mathematicians. Also, the issue encloses studies carried out at both mesoscopic and atomic scales, as well as results of both theoretical approaches (analytical, numerical and, in some cases, even "hybrid") and experimental observations. The covered topics range from the micromagnetic modeling of domain wall motion, dynamics of vortex structures, phase-locking phenomena in spintronic oscillators, experimental techniques for realizing heterostructures based on magnon-induced spin transfer torque, band structure and exchange field in the Landau-Lifshitz equation for magnonic crystals, gap and gapless structures in fractional quantum Hall effect, semiclassical description of anisotropic magnets and classical critical behaviour of Heisenberg ferromagnets. More specifically, within the subject dealing with domain walls, for example, the structure of complex cross-tie/vortex wall structures in soft films has been studied in detail by using micromagnetic simulations whereas the influence of the Rashba spin-orbit coupling on the current-induced dynamics has been investigated analytically. Regarding the exchange interaction governing the dynamics in magnonic crystals, a full analytical calculation of the exchange field acting on spin-wave dynamics from themicroscopic Heisenbergmodel has been performed. Attention has been also devoted to the study of thermodynamics in the case of classical planar ferromagnets close to the zero-temperature critical point. Two reviews are also included in this special issue. The first one deals with two hybrid micromagnetic tools, based on Hamiltonian and Lagrangian approaches, to model the spin-dynamics in laterally confined magnetic systems. The second one is mostly devoted to the micromagnetic analysis of static and dynamic properties of magnetic domain walls in materials exhibiting perpendicular anisotropy

Micromagnetic understanding of the skyrmion Hall angle current dependence in perpendicular magnetized ferromagnets

The understanding of the dynamical properties of skyrmion is a fundamental aspect for the realization of a competitive skyrmion based technology beyond CMOS. Most of the theoretical approaches are based on the approximation of a rigid skyrmion. However, thermal fluctuations can drive a continuous change of the skyrmion size via the excitation of thermal modes. Here, by taking advantage of the Hilbert-Huang transform, we demonstrate that at least two thermal modes can be excited which are non-stationary in time. In addition, one limit of the rigid skyrmion approximation is that this hypothesis does not allow for correctly describing the recent experimental evidence of skyrmion Hall angle dependence on the amplitude of the driving force, which is proportional to the injected current. In this work, we show that, in an ideal sample, the combined effect of field-like and damping-like torques on a breathing skyrmion can indeed give rise to such a current dependent skyrmion Hall angle. While here we design and control the breathing mode of the skyrmion, our results can be linked to the experiments by considering that the thermal fluctuations and/or disorder can excite the breathing mode. We also propose an experiment to validate our findings

Topological skyrmion dynamics in magnetic materials in the presence of a spin-polarized current - Invited talk

It is shown the effect of the interfacial Dzyaloshinskii–Moriya Interaction (i-DMI) on the topology of droplets excited by a localized perpendicular spin-polarized current. According to micromagnetic simulations it is demonstrated that the phase diagram i-DMI as a function of the polarized current at zero magnetic field exhibits a complex scenario with regions characterized by static and dynamic states. In the dynamical part, it is possible to identify topological stable and unstable regions. The topological stable regions are linked to the excitation of droplets with skyrmion number either equal to one (topological) or zero (non-topological). The zero skyrmion number droplets are characterized by the non-stationary time domain excitation of both topological and non-topological droplet modes. The transition between these two modes is coupled to an emission of incoherent spin-waves. It is also developed an analytical model demonstrating that the topological droplet mode can be seen as a linear radial mode of a static skyrmion state stabilized by a perpendicular spin-polarized current. The results obtained by means of the analytical model confirm the red-shift behaviour of the topological mode frequencies as a function of the current density predicted by micromagnetic simulations. The interplay between topology and dynamics is discussed by introducing the notion of topological degeneracy and the non-reciprocal role of the spin polarized current on the topological mode dynamics is highlighted

Dynamic Permeability in a Dissipative Ferromagnetic Medium - Presentazione poster by R. Zivieri - Conferenza internazionale

It is calculated the dynamic relative permeability in a ferromagnetic film with in-plane magnetization in the presence of intrinsic dissipation. The behaviour of the dynamic permeability is studied for microwave frequencies belonging to those of backward volume waves where both the real part and the imaginary part exhibit negative values