We prove some discrete space and time symmetries of linearized equations of motion in vortex-state ferromagnetic dots in the absence of an external magnetic field. The space symmetries are strictly related to the vortex-state configuration of the magnetization characterized by a chirality c= ±1 (counterclockwise or clockwise rotation of the magnetization) and a polarity p = ±1 (outward or inward core magnetization) and to its nature of axial vector. For a purely conservative dynamics, the linearized equation of motion in vortex-state ferromagnetic dots is written in the approximated form [1]. The approximated linearized equation can be applied to ferromagnetic dots of any shape exhibiting a curling configuration and its solution expresses the deviation from the ground-state of vortex-state excitations. First, we discuss the space symmetries. By placing the vortex parallel to the reflection plane (xy plane) the switching of the vortex chirality takes place. The chirality switching leads to a change of sign of the in-plane static magnetization component and of the in-plane dynamic one which, in turn, leads to a change of sign of the corresponding dynamic effective field component Moreover, there is also the switching of the core azimuthal component. These transformations lead to the invariance of the linearized equation of motion. By placing the vortex perpendicularly to the reflection plane (xy plane), the switching of the vortex polarity takes place leading to a change of sign of both dynamical magnetization components leaving again the equation of motion invariant. A remarkable physical effect on spin dynamics is the interchange of the phases of azimuthal vortex modes characterized by the azimuthal number kv = ±1, ±2, .. obtained by using a reflection operator in a quantum mechanical scheme. This corresponds to a change of the sense of rotation of azimuthal vortex modes including the gyrotropic. We also discuss the time-reversal T symmetry, namely t -t. Under this symmetry transformation both the static and dynamic quantities change sign, due to the reversal of angular momentum of electrons leaving the linearized equation of motion invariant like in the case of space symmetries. Within a quantum mechanical scheme, according to Wigner anti-unitary operator [2], the effect of time reversal is equivalent to that of the reflection operator in a quantum mechanical scheme. The dynamics of radial modes (kv = 0) is not affected by the T symmetry. [1] R. Zivieri and F. Nizzoli, Phys. Rev. B 75 (2005), 014411-1-5; ibidem Phys. Rev. B 74 (2006), 219901(E). [2] E.P. Wigner, Group theory and its application to the quantum mechanics, New York, Academic Press (1959), p.325.
Symmetries of vortex-state linearized equations of motion in ferromagnetic dots -- Presentazione poster by R. Zivieri -- Conferenza nazionale
ZIVIERI, Roberto
2013
Abstract
We prove some discrete space and time symmetries of linearized equations of motion in vortex-state ferromagnetic dots in the absence of an external magnetic field. The space symmetries are strictly related to the vortex-state configuration of the magnetization characterized by a chirality c= ±1 (counterclockwise or clockwise rotation of the magnetization) and a polarity p = ±1 (outward or inward core magnetization) and to its nature of axial vector. For a purely conservative dynamics, the linearized equation of motion in vortex-state ferromagnetic dots is written in the approximated form [1]. The approximated linearized equation can be applied to ferromagnetic dots of any shape exhibiting a curling configuration and its solution expresses the deviation from the ground-state of vortex-state excitations. First, we discuss the space symmetries. By placing the vortex parallel to the reflection plane (xy plane) the switching of the vortex chirality takes place. The chirality switching leads to a change of sign of the in-plane static magnetization component and of the in-plane dynamic one which, in turn, leads to a change of sign of the corresponding dynamic effective field component Moreover, there is also the switching of the core azimuthal component. These transformations lead to the invariance of the linearized equation of motion. By placing the vortex perpendicularly to the reflection plane (xy plane), the switching of the vortex polarity takes place leading to a change of sign of both dynamical magnetization components leaving again the equation of motion invariant. A remarkable physical effect on spin dynamics is the interchange of the phases of azimuthal vortex modes characterized by the azimuthal number kv = ±1, ±2, .. obtained by using a reflection operator in a quantum mechanical scheme. This corresponds to a change of the sense of rotation of azimuthal vortex modes including the gyrotropic. We also discuss the time-reversal T symmetry, namely t -t. Under this symmetry transformation both the static and dynamic quantities change sign, due to the reversal of angular momentum of electrons leaving the linearized equation of motion invariant like in the case of space symmetries. Within a quantum mechanical scheme, according to Wigner anti-unitary operator [2], the effect of time reversal is equivalent to that of the reflection operator in a quantum mechanical scheme. The dynamics of radial modes (kv = 0) is not affected by the T symmetry. [1] R. Zivieri and F. Nizzoli, Phys. Rev. B 75 (2005), 014411-1-5; ibidem Phys. Rev. B 74 (2006), 219901(E). [2] E.P. Wigner, Group theory and its application to the quantum mechanics, New York, Academic Press (1959), p.325.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
