An analytical model able to calculate frequencies of surface spin modes in vortex-state ferromagnetic circular cylindrical dots at zero applied field is formulated. The calculation is based upon the tensorial Green's function formalism. In dots of nanometric radius, the radial eigenvectors of the nonaxially symmetric modes are Bessel functions of order one or greater than one, while the axially symmetric modes correspond to zero order Bessel functions. Instead, in dots of radius in the submicrometric and micrometric range the radial eigenvectors of the whole set of modes are Bessel functions of order one. The calculated frequencies of the most representative m = 0, n modes (m is the azimuthal number and n the radial number) are compared to available experimental data for Permalloy (Py) disks. We have found that the frequencies behaviour of both axially and nonaxially symmetric modes shows, in dots of moderate aspect ratio, a deviation from the linear dependence from (L/R)^1/2 where L is the cylinder thickness and R the dot radius as found in literature. The dipolar contribution is calculated exactly for each class of modes taking into account the 3-dimensional effect on the spin modes frequencies. The dynamics is studied in arrays of cylindrical dots with radius ranging from the nanometric to the micrometric size. We restrict ourselves to arrays of dots where the interdot distance is large, so that the interdot magnetostatic energy may be considered negligible. In this way the spin dynamics of a single dot may be studied and a comparison with frequencies measured by means of BLS technique in arrays of dots whose separation is equal to the dot diameter can be performed. In the numerical calculation we neglect any anisotropy contribution, because we consider here Permalloy (Py) dots where anisotropy effects are in general small. The calculated splitting of the m = + -1 doublet compares well with the measured splitting at different aspect ratios. We also show that the general effect of the static exchange field arising from the "curling" configuration causes a downshift of the spin mode frequencies in nanometric dots. This behaviour is similar to that of the demagnetizing field in the saturated state. A comparison of the results of the analytical model with a recent micromagnetic approach based on the dynamical matrix approach is performed for the most representative modes of the spectrum and the differences between the two approaches are shown. The study is extended to circular ferromagnetic rings in the vortex-state in the absence of an external field. Also for these systems an analytical expression of the quantized spectrum of spin modes is derived from the corresponding one of the standard disk. Radial boundary conditions are proposed on both the outer and the inner lateral surfaces. Analogies and differences with spin dynamics in disks are shown. In particular, it is demonstrated that modes localized in the core region exhibiting nodes along the thickness are not anymore present in circular magnetic rings due to the vortex core removal. The effect of the core removal on the spin modes frequencies of the most representative modes is discussed. The real effect of the core on the frequencies of spin modes evaluated by forcing the magnetization to lie in the dot plane in a standard disk is also studied. Finally, the numerical results of the analytical model are compared with those obtained by means of micromagnetic calculations in rings and the analogies and differences between the two approaches are discussed for these systems. Support by CNISM-CNR is gratefully acknowledged. -- Presentazione orale by R. Zivieri - Conferenza internazionale
Spin excitations in vortex-state magnetic cylindrical dots and rings: from nanometric to micrometric size -- Presentazione orale by R. Zivieri - Conferenza internazionale
ZIVIERI, Roberto;NIZZOLI, Fabrizio
2008
Abstract
An analytical model able to calculate frequencies of surface spin modes in vortex-state ferromagnetic circular cylindrical dots at zero applied field is formulated. The calculation is based upon the tensorial Green's function formalism. In dots of nanometric radius, the radial eigenvectors of the nonaxially symmetric modes are Bessel functions of order one or greater than one, while the axially symmetric modes correspond to zero order Bessel functions. Instead, in dots of radius in the submicrometric and micrometric range the radial eigenvectors of the whole set of modes are Bessel functions of order one. The calculated frequencies of the most representative m = 0, n modes (m is the azimuthal number and n the radial number) are compared to available experimental data for Permalloy (Py) disks. We have found that the frequencies behaviour of both axially and nonaxially symmetric modes shows, in dots of moderate aspect ratio, a deviation from the linear dependence from (L/R)^1/2 where L is the cylinder thickness and R the dot radius as found in literature. The dipolar contribution is calculated exactly for each class of modes taking into account the 3-dimensional effect on the spin modes frequencies. The dynamics is studied in arrays of cylindrical dots with radius ranging from the nanometric to the micrometric size. We restrict ourselves to arrays of dots where the interdot distance is large, so that the interdot magnetostatic energy may be considered negligible. In this way the spin dynamics of a single dot may be studied and a comparison with frequencies measured by means of BLS technique in arrays of dots whose separation is equal to the dot diameter can be performed. In the numerical calculation we neglect any anisotropy contribution, because we consider here Permalloy (Py) dots where anisotropy effects are in general small. The calculated splitting of the m = + -1 doublet compares well with the measured splitting at different aspect ratios. We also show that the general effect of the static exchange field arising from the "curling" configuration causes a downshift of the spin mode frequencies in nanometric dots. This behaviour is similar to that of the demagnetizing field in the saturated state. A comparison of the results of the analytical model with a recent micromagnetic approach based on the dynamical matrix approach is performed for the most representative modes of the spectrum and the differences between the two approaches are shown. The study is extended to circular ferromagnetic rings in the vortex-state in the absence of an external field. Also for these systems an analytical expression of the quantized spectrum of spin modes is derived from the corresponding one of the standard disk. Radial boundary conditions are proposed on both the outer and the inner lateral surfaces. Analogies and differences with spin dynamics in disks are shown. In particular, it is demonstrated that modes localized in the core region exhibiting nodes along the thickness are not anymore present in circular magnetic rings due to the vortex core removal. The effect of the core removal on the spin modes frequencies of the most representative modes is discussed. The real effect of the core on the frequencies of spin modes evaluated by forcing the magnetization to lie in the dot plane in a standard disk is also studied. Finally, the numerical results of the analytical model are compared with those obtained by means of micromagnetic calculations in rings and the analogies and differences between the two approaches are discussed for these systems. Support by CNISM-CNR is gratefully acknowledged. -- Presentazione orale by R. Zivieri - Conferenza internazionaleI documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
