Theory of Dipole-Exchange Spin Waves in a Ferromagnetic Nanotube in the Presence of a Thermoelectrically Induced Spin-Transfer Torque. Elliptic Nanotube Case

A theoretical model of linear dipole-exchange spin waves in a conducting ferromagnetic (easy-axis ferromagnet) nanotube with an elliptic cross-section subjected to an axial temperature gradient is developed. The temperature gradient generates a thermoelectric spin-polarized current, whose action on magnetization dynamics is described by Zhang–Li spin-transfer torque terms. Exchange interaction, dipole-dipole interaction, uniaxial anisotropy and Gilbert damping are taken into account within the linearized Landau–Lifshitz–Gilbert equation and the magnetostatic approximation. The corresponding dispersion relation is derived. It is shown that the thermally induced spin current produces a Doppler-type shift of the real part of the spin-wave frequency and modifies the effective damping through the nonadiabatic spin-transfer contribution. The condition of spin-wave excitation is obtained. The ellipticity of the cross-section affects the spectrum through transverse quantization: the fundamental mode remains identical to that of a circular nanotube, whereas nonzero transverse modes are described by Mathieu functions in elliptic cylindrical co-ordinates. Implicit expression for the transverse wavenumber has been found and (for a nanotube close to a circular one) simplified into an explicit asymptotic expression for the ellipticity-induced correction to the transverse wavenumber. The leading effect of ellipticity is shown to be splitting of the transverse doublet for the first mode, while higher modes acquire no first-order wavenumber shift in this approximation. The results demonstrate that the temperature gradient controls longitudinal propagation and damping, whereas ellipticity provides an additional geometrical mechanism for tuning transverse spin-wave modes in nanoscale magnonic and spin-caloritronic devices.