Curvilinear magnonics

New routes to affect the characteristics of materials with ferromagnetic order are to bend the thin film membranes. Bending the membrane can lead to internal strains and to a break of the local inversion symmetry [1], resulting for example in an unambiguous distinction between the outer and inner surfaces in case of curved geometries like nanotubes. The internal energies are also affected, especially when the curvature radius reaches intrinsic length scales. As shown in [2], in strongly curved systems the off-diagonal elements of the exchange interaction are not negligible, leading to chiral ordering. The dipolar fields are also influenced by the break of the inversion symmetry. Due to the modified energies the magnetic ordering and the magnetization dynamics differs from those known for thin-films [3-5]. Therefore the curvature can be accounted as an extra degree of freedom for controlling the characteristics of ferromagnetic materials.

In Magnonics, spin waves (SWs) or magnons are proposed to be used to carry, transport and process information analogous to the electron currents in electronics. Engineering magnon properties to control the SW excitation and propagation is an unavoidable task. The membrane curvature can be used to extend the toolbox of operations for controlling SWs, required in communication and logic devices [6–8]. Geometries like Möbius rings, helices, grooves stripes, and nanotubes can be accounted as few sets of layouts wherein the system curvature has an impact on the SW dynamics. Such examples will be referred in this talk, being the later (magnetic nanotubes) the one of our main focus. The tunable non-reciprocal SW properties induced by the nanotubular curvature [6–8] will be highlighted. In particular, that the dispersion relation is asymmetric regarding the sign of the propagation vector. Therefore the counter propagating magnons have different wave vectors and characteristic length for the transport. Figure 1(a) sketches a Permalloy nanotube in circular magnetic state wherein the SWs are excited by an rf-field applied at the center of the tube. Quasi-monocromatic magnons of different orders (n = 0, ±1, ±2) excited at 4.7 GHz are shown in Figure 1(b, c). The radial component of the excited magnon modes is represented by the color code in Figure 1(b). Note that the wavelength and transport length of counter-propagating magnons differ. Figure1(c) shows the magnon field distribution along the nanotube perimeter. The case of non-reciprocal SW dispersion and intrinsic linewidth are presented in Figure 2 (a) and (b), respectively. Aspects like the optimization of the curvature-induced non-reciprocity as function of the system size and magnetic ground state, some means to control the magnons mode profile and the tuning of non-reciprocity via weak DC external magnetic fields, will be also discussed.
We believe that three dimensional curvilinear magnetic membranes, in particular nanotubes, can be exploited as a novel layouts for non-reciprocal conduits, for magnons transport along curved paths, and as one-dimensional magnonic crystals.