Theory of Spin Waves in an Easy-Plane Ferromagnetic Nanotube with a Spin-Polarized Current

In the paper, dipole-exchange spin waves in a nanotube composed of an easyplane ferromagnet in the presence of a spin-polarized current are studied theoretically. The magnetic dipole-dipole interaction, the exchange interaction, the magnetic anisotropy, the dissipation effects, and influence of the spin-polarized current are considered. For such spin waves, an equation for the magnetic potential is obtained and (for the case of longitudinal-radial waves) solved. As a result, the dispersion law for such waves is found. This dispersion law is complemented with the relation between the wave-vector components, which is shown to degenerate into a quasi-one-dimensional values' spectrum of the orthogonal wave-vector component nearly everywhere. As shown, in most cases, influences of the spin-wave dissipation and spinpolarized current on the real part of its frequency are negligible. Branches (which correspond to different orthogonal modes) of both the real and imaginary parts of the dispersion law are shown to be close to parabolic ones; distance between branches increases with increase of the mode number. Presence of the spin-polarized current can strengthen or weaken the spin-wave damping, creating the 'effective dissipation' and, in some cases, leading to a spin-wave generation. The condition of such a generation is found as well as limitations on the transverse modes' number, for which the generation is possible. As shown, for typical values of nanosystem parameters, only first several modes can be excited via such generation (in most cases, only zero mode or none). The method proposed in the paper can be applied to nanotubes (and other nanosystems) of more complex configurations.