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An equation for the density of chirality in a circular and cylindrical ferromagnet exhibiting a vortex-state configuration of the magnetization is found. This equation can be considered the equivalent of the vorticity equation of a fluid. The derivation is performed following a semi-classical approach and starting from the Landau-Lifshitz equation with no losses which describes spin precession around the equilibrium direction of the magnetization. First, the equation is derived in the simplest case, that of a circular ferromagnetic ring exhibiting a vortex-state and neglecting damping. For a classical ferromagnet in the vortex flux-closure configuration it is useful to write down, in analogy with the classical fluid, a circulation in the form of a line integral of the magnetization field over a contour line of the surface S filled by the vortex magnetization. By applying the Stokes theorem, the density of chirality or chirality per unit area is defined as the curl of the magnetization field. The density of chirality equation is obtained according to the two following steps: 1) Applying the curl operator on both members of the purely precessional Landau-Lifshitz equation 2) Linearizing on the second member, namely decomposing both the total magnetization and the effective field into a static and a small dynamic part. As a result, the time derivative of the density of chirality turns out to be proportional to terms depending either on the coupling between the magnetization and the spatial derivative of the effective field or on the coupling of the static effective field and the spatial derivative of the magnetization. The physical meaning of each term is discussed. An extension of the equation to the case of a circular and cylindrical dot in the presence of a vortex core region with out-of-plane magnetization is performed. The equation for the density of chirality is further generalized by including the Gilbert intrinsic damping and the spin-transfer torque damping and the additional physical effects are discussed

Publisher: Politecnico di Milano

Year: 2013

OAI identifier:
oai:iris.unife.it:11392/1857909

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