Statistical thermodynamics of chiral skyrmions in a ferromagnetic material

Solitons are a challenging topic in condensed matter physics and materials science because of the interplay between their topological and physical properties and for the crucial role they play in topological phase transitions. Among them, chiral skyrmions hosted in ferromagnetic systems are axisymmetric solitonic states attracting a lot of attention for their dazzling physical properties and technological applications. In this paper, the equilibrium statistical thermodynamics of chiral magnetic skyrmions developing in a ferromagnetic material having the shape of an ultrathin cylindrical dot is investigated. This is accomplished by determining via analytical calculations for both Néel and Bloch skyrmions: (1) the internal energy of a single chiral skyrmion; (2) the partition function; (3) the free energy; (4) the pressure; and (5) the equation of state of a skyrmion diameters population. To calculate the thermodynamic functions for points (2)-(5), the derivation of the average internal energy and of the configurational entropy is crucial. Numerical calculations of the thermodynamic functions for points (1)-(5) are applied to Néel skyrmions. These results could advance the field of materials science with special regard to low-dimensional magnetic systems

Nonlinear dispersion relation in anharmonic periodic mass-spring and mass-in-mass systems

The study of wave propagation in chains of anharmonic periodic systems is of fundamental importance to understand the response of dynamical absorbers of vibrations and acoustic metamaterials working in nonlinear regime. Here, we derive an analytical nonlinear dispersion relation for periodic chains of anharmonic mass-spring and mass-in-mass systems resulting from considering the hypothesis of weak anharmonic energy and a periodic distribution function as ansatz of a general solution of the nonlinear equations of motion. Numerical simulations show that this expression is valid for anharmonic potential energy up to 50% of the harmonic one. This work provides a simple tool to design and study nonlinear dynamics for a class of seismic metamaterials.Comment: 18 pages, 5 figure

A strategy for the design of skyrmion racetrack memories

Magnetic storage based on racetrack memory is very promising for the design of ultra-dense, low-cost and low-power storage technology. Information can be coded in a magnetic region between two domain walls or, as predicted recently, in topological magnetic objects known as skyrmions. Here, we show the technological advantages and limitations of using Bloch and Neel skyrmions manipulated by spin current generated within the ferromagnet or via the spin-Hall effect arising from a non-magnetic heavy metal underlayer. We found that the Neel skyrmion moved by the spin-Hall effect is a very promising strategy for technological implementation of the next generation of skyrmion racetrack memories (zero field, high thermal stability, and ultra-dense storage). We employed micromagnetics reinforced with an analytical formulation of skyrmion dynamics that we developed from the Thiele equation. We identified that the excitation, at high currents, of a breathing mode of the skyrmion limits the maximal velocity of the memory

Critical phenomena in ferromagnetic antidot lattices

In this paper a quantitative theoretical formulation of the critical behavior of soft mode frequencies as a function of an applied magnetic field in two-dimensional Permalloy square antidot lattices in the nanometric range is given according to micromagnetic simulations and simple analytical calculations. The degree of softening of the two lowest-frequency modes, namely the edge mode and the fundamental mode, corresponding to the field interval around the critical magnetic field, can be expressed via numerical exponents. For the antidot lattices studied we have found that: a) the ratio between the critical magnetic field and the in-plane geometric aspect ratio and (b) the ratio between the numerical exponents of the frequency power laws of the fundamental mode and of the edge mode do not depend on the geometry. The above definitions could be extended to other types of in-plane magnetized periodic magnetic systems exhibiting soft-mode dynamics and a fourfold anisotropy

Combined frequency-amplitude nonlinear modulation: theory and applications

In this work we formulate a generalized theoretical model to describe the nonlinear dynamics observed in combined frequency-amplitude modulators whose characteristic parameters exhibit a nonlinear dependence on the input modulating signal. The derived analytical solution may give a satisfactory explanation of recent laboratory observations on magnetic spin-transfer oscillators and fully agrees with results of micromagnetic calculations. Since the theory has been developed independently of the mechanism causing the nonlinearities, it may encompass the description of modulation processes of any physical nature, a promising feature for potential applications in the field of communication systems.Comment: 8 pages, 4 figures, to be published on IEEE Transactions on Magnetic

Intensity of Brillouin light scattering from spin waves in magnetic multilayers with noncollinear spin configurations: Theory and experiment

The scattering of photons from spin waves (Brillouin light scattering -- BLS) is a well-established technique for the study of layered magnetic systems. The information about the magnetic state and properties of the sample is contained in the frequency position, width, and intensity of the BLS peaks. Previously [Phys. Rev. B 67, 184404 (2003)], we have shown that spin wave frequencies can be conveniently calculated within the ultrathin film approach, treating the intralayer exchange as an effective bilinear interlayer coupling between thin virtual sheets of the ferromagnetic layers. Here we give the consequent extension of this approach to the calculation of the Brillouin light scattering (BLS) peak intensities. Given the very close relation of the BLS cross-section to the magneto-optic Kerr effect (MOKE), the depth-resolved longitudinal and polar MOKE coefficients calculated numerically via the usual magneto-optic formalism can be employed in combination with the spin wave precessional amplitudes to calculate full BLS spectra for a given magnetic system. This approach allows an easy calculation of BLS intensities even for noncollinear spin configurations including the exchange modes. The formalism is applied to a Fe/Cr/Fe/Ag/Fe trilayer system with one antiferromagnetically coupling spacer (Cr). Good agreement with the experimental spectra is found for a wide variety of spin configurations.Comment: 19 pages, 5 figure

Configurational entropy of magnetic skyrmions as an ideal gas

The study of thermodynamics of topological defects is an important challenge to understand their underlying physics. Among them, magnetic skyrmions have a leading role for their physical properties and potential applications in storage and neuromorphic computing. In this paper, the thermodynamic statistics of magnetic skyrmions is derived. It is shown that the skyrmion free energy can be modelled via a parabolic function and the diameters statistics obeys the Maxwell-Boltzmann distribution. This allows for making an analogy between the behavior of the distribution of skyrmion diameters statistics and the diluted gas Maxwell-Boltzmann molecules distribution at thermodynamical equilibrium. The calculation of the skyrmion configurational entropy, due to thermally-induced changes of size and shape of the skyrmion, is essential for the determination of thermal fluctuations of the skyrmion energy around its average value. These results can be employed to advance the field of skyrmionics.Comment: Main text 26 pages and 6 figures. Supplementary Information 4 page

Theory of nonreciprocal spin waves excitation in spin-Hall oscillators with Dzyaloshinkii-Moriya interaction

A two-dimensional analytical model for the description of the excitation of nonreciprocal spin waves by spin current in spin-Hall oscillators in the presence of the interfacial Dzyaloshinskii-Moriya interaction (i-DMI) is developed. The theory allows one to calculate the threshold current for the excitation of spin waves, as well as the frequencies and spatial profiles of the excited spin wave modes. It is found, that the frequency of the excited spin waves exhibits a quadratic red shift with the i-DMI strength. At the same time, in the range of small and moderate values of the i-DMI constant, the averaged wave number of the excited spin waves is almost independent of the i-DMI, which results in a rather weak dependence on the i-DMI of the threshold current of the spin wave excitation. The obtained analytical results are confirmed by the results of micromagnetic simulations.Comment: 23 pages,5 figure

Simultaneous existence of two spin-wave modes in ultrathin Fe/GaAs(001) films studied by Brillouin Light Scattering: experiment and theory

A double-peaked structure was observed in the {\it in-situ} Brillouin Light Scattering (BLS) spectra of a 6 \AA$$ thick epitaxial Fe/GaAs(001) film for values of an external magnetic field $H$, applied along the hard in plane direction, lower than a critical value $H_c\simeq 0.9$ kOe. This experimental finding is theoretically interpreted in terms of a model which assumes a non-homogeneous magnetic ground state characterized by the presence of perperpendicular up/down stripe domains. For such a ground state, two spin-wave modes, namely an acoustic and an optic mode, can exist. Upon increasing the field the magnetization tilts in the film plane, and for $H \ge H_{c}$ the ground state is homogeneous, thus allowing the existence of just a single spin-wave mode. The frequencies of the two spin-wave modes were calculated and successfully compared with the experimental data. The field dependence of the intensities of the corresponding two peaks that are present in the BLS spectra was also estimated, providing further support to the above-mentioned interpretation.Comment: Shortened version (7 pages). Accepted for publication in Physical Review