We present a theoretical model to obtain analytically the spin modes frequencies of cylindrical magnetic dots in the vortex state as a function of an external applied field. In the calculations both the core and the out-of-core ex- change and dipolar fields induced by surface and volume charges are taken into account. Because of the radial sym- metry the linearized integro-differential Landau-Lifshitz equation of motion may be put into a differential form. The analytical solution of the differential equation is a linear combination of Bessel functions of the first and of the second kind of integer order times their correspond- ing angular functions. The wave vector quantization is determined by applying general radial boundary condi- tion at the dot lateral surface. The linearized equation of motion is put into a matrix form and an analytic expres- sion for the spin modes frequencies is obtained from the vanishing of the coe±cient determinant. The spin exci- tations with higher frequencies correspond to the axially symmetric zero order Bessel solutions, whereas the modes at lower frequencies correspond to the Bessel functions of higher orders. In this formalism also the translational vortex mode at lowest frequency is included. Numerical frequency calculations are performed in the vortex state for different dot thicknesses and radii as a function of an external field. Due to the demagnetizing field arising from the vortex displacement, the modes frequencies decrease with increasing the external field. -- Presentazione poster by R. Zivieri - Conferenza internazionale
Spin modes in vortex-state ferromagnetic cylindrical dots -- Presentazione poster by R. Zivieri - Conferenza internazionale
ZIVIERI, Roberto;NIZZOLI, Fabrizio
2004
Abstract
We present a theoretical model to obtain analytically the spin modes frequencies of cylindrical magnetic dots in the vortex state as a function of an external applied field. In the calculations both the core and the out-of-core ex- change and dipolar fields induced by surface and volume charges are taken into account. Because of the radial sym- metry the linearized integro-differential Landau-Lifshitz equation of motion may be put into a differential form. The analytical solution of the differential equation is a linear combination of Bessel functions of the first and of the second kind of integer order times their correspond- ing angular functions. The wave vector quantization is determined by applying general radial boundary condi- tion at the dot lateral surface. The linearized equation of motion is put into a matrix form and an analytic expres- sion for the spin modes frequencies is obtained from the vanishing of the coe±cient determinant. The spin exci- tations with higher frequencies correspond to the axially symmetric zero order Bessel solutions, whereas the modes at lower frequencies correspond to the Bessel functions of higher orders. In this formalism also the translational vortex mode at lowest frequency is included. Numerical frequency calculations are performed in the vortex state for different dot thicknesses and radii as a function of an external field. Due to the demagnetizing field arising from the vortex displacement, the modes frequencies decrease with increasing the external field. -- Presentazione poster by R. Zivieri - Conferenza internazionaleI documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
