Theory of Dipole-Exchange Spin Waves in a Ferromagnetic Nanotube. Consideration of Volume and Surface Modes
Abstract
The paper extends study of dipole-exchange spin waves in a ferromagnetic nanotube with a circular crosssection started by the author in the previous paper. The proposed model considers the magnetic dipole-dipole interaction, the exchange interaction, the anisotropy effects, the damping effects, the general boundary conditions and the existence of both volume and surface modes for the considered spin waves. Therefore, a new method of obtaining the values’ spectrum of the orthogonal (to the translation axis) wavenumbers for the investigated spin waves is proposed in addition to the previously obtained dispersion law. The method is based on the application of general boundary conditions for the magnetic field on a superposition of the above-mentioned modes. The obtained spectrum is shown to be a quasi-one-dimensional one – similar to that in a thin ferromagnetic field – for typical ferromagnetic nanotubes. Exploitation of the above-mentioned method essentially extends the area of application of the obtained results compared to the previous paper.
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Introduction
At the present time, a variety of actual and prospective technologies are based on the applications of spin waves in nanosystems. In particular, such waves are promising for application in information technologies – for creating new data storage [1], transfer [1,2] and processing [3] devices. One of the key problems for developing such technologies is theoretical modeling of spin-wave processes in these nanosystems. Such modeling is required not only for direct applications of spin waves, but also for synthesizing materials with preset magnetic properties because these properties are often influenced by spin-wave processes. This modeling, in turn, requires deeper understanding of the corresponding processes in magnetic nanosystems. In the proposed paper, one of the problems of the above-described type is solved.
It has been shown by numerous studies that the properties of nanosystems – in particular, spin-wave properties – depend essentially on their size and shape. Unfortunately, a general theory of spin waves in magnetic nanosystems has not been created at the moment. Therefore, spin waves in nanosystems of different geometries are studied separately. Among the variety of magnetic nanosystems of different configurations, a special class is represented by shell-type ferromagnetic nanosystems (nanoshells, nanotubes and others). These nanosystems exhibit unique – not inherent to traditional continuous nanosystems – magnetic properties that are prospective for numerous technical applications. For instance, magnetic properties of such nanosystems can be regulated more flexibly than properties of corresponding continuous nanosystems. However, such nanosystems remain poorly researched at the moment. In particular, study of spin waves in synthesized recently magnetic nanotubes [4] represents an actual topic of research.
The paper continues the study of dipole-exchange spin waves in a ferromagnetic nanotube with a circular cross-section started by the author in the papers [5,6]. In the study, the magnetic dipole-dipole interaction, the exchange interaction, the anisotropy effects and the damping effects are considered. In the previous papers of the author [5,6], a dispersion relation for the above-described spin waves has been obtained. However, for complete description of the considered waves, this relation should be complemented by either a relation between the wavenumber components or values’ spectrum of the orthogonal (to the tube axis) wavenumber component. For the most nanosystems, that represents more challenging task than just finding the dispersion relation. Moreover, for thin films and nanotubes one should consider existence of both volume and surface modes that are, generally speaking, hybridized in the considered cases. In the papers [5,6], only a volume spin wave mode has been considered and the above-mentioned spectrum has been obtained only for a very specific particular case (the material outside the nanotube has been assumed to be a high-conductivity metal) thus essentially limiting the area of application of the entire obtained result. The proposed paper overcomes this limitation by considering both modes’ types and applying a different – essentially more general – method of obtaining the above-mentioned spectrum. As a result, the obtained values’ spectrum of the orthogonal wavenumber component has an essentially wider range of applications. The obtained spectrum of wavenumbers is shown to have a quasi-one-dimensional form for the considered (thin) nanotubes.
Conclusion
Thus, dipole-exchange spin waves in a ferromagnetic nanotube (easy-axis ferromagnet) have been studied in the paper. The magnetic dipole-dipole interaction, the exchange interaction, the anisotropy effects and the damping effects have been considered. It has been shown that – similarly to the known case of a thin ferromagnetic film – in the considered nanotube, both surface and volume spin wave modes can be observed as well as hybrid modes. The dispersion law for such waves – obtained in the previous paper of the author – has been complemented with the spectrum of orthogonal wavenumber values. Unlike the previous papers of the author [5,6], the above-mentioned spectrum have been obtained using general boundary conditions for the magnetic field. This essentially extends the area of application of the obtained results compared to the papers [5,6] in which the spectrum has been found for a very specific particular case. It has been shown that for both modes’ types the orthogonal wavenumber (for the surface modes – its real part) values’ spectrum is quasi-one-dimensional on the most part of orthogonal and longitudinal wavenumbers’ ranges. Therefore, the obtained results can be used for any round ferromagnetic nanotube of the studied configuration as long as the general model used in the paper (thin nanotube, linear waves, constant absolute value of the magnetization vector etc.) can be applied – and the mentioned model is applicable for typical round ferromagnetic nanotubes synthesized nowadays.
A graphical representation of the resulting dispersion relation with account for the obtained orthogonal wavenumber values’ spectrum has been given. Numerical estimations of the spin waves’ frequency for typical nanoshell parameters have been performed. The estimations have shown that the resulting frequency, really, lies within the frequencies interval for typical observed spin waves.
The method proposed in the paper can be applied to nanotubes of more complex configurations – in particular, to synthesized recently ferromagnetic nanotubes with an elliptic cross-section – as well as for more complex configurations of shell-type nanosystems in general. For some of them boundary conditions can be applied separately on volume and surface modes.