We prove some discrete space and time symmetries of linearized equations of motion in vortex-state ferromagnetic dots. 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. First, we discuss the space symmetries. By placing the vortex parallel to the reflection plane (xy plane) and by introducing the space reflection operator σxy, the following symmetry transformation holds: σxy c= -c. The c switching leads to a change of sign of the in-plane static magnetization component (M →-M) and of the in-plane dynamic one, namely δmu→-δmu which leads to a change of sign of the corresponding dynamic effective field component δhu→-δhu. Moreover, there is also the switching δ mvC → −δ mvC of the core (C) component δ mvC = −δ mz cosθ with θ the polar angle. These transformations lead to the invariance of the equation of motion. By placing the vortex perpendicularly to the reflection plane (xy plane), we get σxy p= - p. The p switching (vortex core switching) leads to δmu→-δmu and to δmz→-δmz leaving again the equation of motion invariant. A remarkable physical effect on spin dynamics is the interchange kv→-kv of the phases of azimuthal vortex modes characterized by the azimuthal number kv = ±1, ±2, .. obtained by using a reflection operator xy σ in a quantum mechanical scheme We now 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 moment of electrons, leaving Eq.(1) invariant like in the case of space symmetries. Within a quantum mechanical scheme, according to Wigner anti-unitary operator [2] the effect is equivalent to that of xy σ (kv→-kv) and leads to a change of the sense of rotation of azimuthal vortex modes including the gyrotropic. Instead, 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.
Symmetry properties and invariance of vortex-state linearized equations of motion in ferromagnetic dots -- Presentazione poster by R. Zivieri - Conferenza internazionale
ZIVIERI, Roberto;
2013
Abstract
We prove some discrete space and time symmetries of linearized equations of motion in vortex-state ferromagnetic dots. 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. First, we discuss the space symmetries. By placing the vortex parallel to the reflection plane (xy plane) and by introducing the space reflection operator σxy, the following symmetry transformation holds: σxy c= -c. The c switching leads to a change of sign of the in-plane static magnetization component (M →-M) and of the in-plane dynamic one, namely δmu→-δmu which leads to a change of sign of the corresponding dynamic effective field component δhu→-δhu. Moreover, there is also the switching δ mvC → −δ mvC of the core (C) component δ mvC = −δ mz cosθ with θ the polar angle. These transformations lead to the invariance of the equation of motion. By placing the vortex perpendicularly to the reflection plane (xy plane), we get σxy p= - p. The p switching (vortex core switching) leads to δmu→-δmu and to δmz→-δmz leaving again the equation of motion invariant. A remarkable physical effect on spin dynamics is the interchange kv→-kv of the phases of azimuthal vortex modes characterized by the azimuthal number kv = ±1, ±2, .. obtained by using a reflection operator xy σ in a quantum mechanical scheme We now 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 moment of electrons, leaving Eq.(1) invariant like in the case of space symmetries. Within a quantum mechanical scheme, according to Wigner anti-unitary operator [2] the effect is equivalent to that of xy σ (kv→-kv) and leads to a change of the sense of rotation of azimuthal vortex modes including the gyrotropic. Instead, 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.
