Walker’s modes in ferromagnetic finite hollow cylinder
Introduction
The properties of spin waves in non-simply connected domains are of great interest for the possibility to exploit interference effects or to induce additional phases by an electric field, like in the Ahronov–Casher effect [1], [2], [3], [4]. These effects are seen as the possible working principles of future magnonic devices working as logic ports in which the phase of the spin waves are expected to carry the information to be processed [5]. From the point of view of a theoretical description the geometry of a hollow cylinder represents a simple but meaningful example also because nanorings, nanotubes and other kinds non simply connected geometries are being currently being realized and studied [6], [7], [8], [9], [10]. By neglecting the effect of ferromagnetic exchange spin waves and modes are entirely due to the magnetostatic field and the Walker’s equation holds [11]. The problem is a classical one that has been largely investigated in the literature going from the spheroids [11] to ferromagnetic thin films [12], [13], [14] and to cylinders [15], [16]. Hollow cylinders have been considered by Das and Cottam who derived the dispersion relation and shown that the uniform modes along the cylinder axis are surface non reciprocal azimuthal modes concentrated at the inner or outer surfaces of the cylinder [16], however the analytic expression for the magnetic scalar potential and for the magnetization field were missing. These analytic expression are useful when the coupling with external radiofrequency antennae has to be designed [17]. In the present contribution we solve the Walker’s equation in cylindrical coordinates and we extend the results of [16] by deriving the analytical expression for the magnetic scalar potential and the magnetization components.
Section snippets
Walker’s equation
In this paper the magnetostatic modes are described by the Walker’s partial differential equation, whose solution describes the physical behavior of the magnetic scalar potential . The Walker’s equation (the detailed derivation can be found in the literature [13], [14]) is based on the linearization of the magnetization dynamics and on the magnetostatic approximation of Maxwell’s equations. By neglecting the damping the magnetization dynamics is given by the precession equation for the
Solutions for the finite hollow cylinder
In the following we apply the Walker’s theory to the geometry of a hollow cylinder with inner and outer radii and , respectively, and length , as shown in Fig. 1. The appropriate coordinate system to describe the problem is the cylindrical one defined by the versors (). By appropriately solving the time independent magnetostatic problem along each of the cylindrical axis one may formulate a set of equations for the magnetostatic modes for axial, azimuthal or radial
Surface azimuthal modes
With the solutions are simplified as and so that one easily obtains the following dispersion relation for the surface azimuthal modes (see also [16]) shown in the plot of Fig. 2 top. It should be noticed that the mode with exists with but is suppressed in the limit . The non reciprocal field displacement is clearly shown by deriving the analytic expression for the magnetic scalar potential in the ferromagnetic
Discussion and conclusions
While it was already clear from the work of Das and Cottam that the uniform modes along the cylinder axis are surface non reciprocal azimuthal modes [16], the explicit analytic expression for the magnetic scalar potential and for the magnetization field was missing in the literature. These analytic expression have been derived and shown in the present paper. The surface modes developing at the inner or outer surfaces of the cylinder. These modes have several similarities with the Damon–Eshbach
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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