Electric-Field Coupling to Spin Waves in a Centrosymmetric Ferrite

A systematic control of spin waves via external electric fields has been a long standing issue for the design of magnonic devices, and is of fundamental interest. One way to attain such control is to use multiferroics [1], whose electric and magnetic polarizations are inherently coupled. The lack of electric polarization in a centrosymmetric ferrite, however, makes direct coupling of its magnetization to external electric fields a challenge. Indirect electric control of spin waves has been accomplished by hybridizing yttrium iron garnet (YIG), a centrosymmetric ferrite, with a piezoelectric material [2]. Here, we predict direct control of spin waves in YIG by {\it a flexoelectric interaction}, which couples an electric field to the spatial gradient of the magnetization, and thus the spin waves [3]. Based on a superexchange model, which describes the antiferromagnetic coupling between two nearest neighbor iron ions through an oxygen ion, including spin-orbit coupling, we estimate the coupling constant and predict a phase shift linear in the applied electric fields [4]. The theory is then confirmed by experimental measurement of the electric-field-induced phase shift in a YIG waveguide [5]. In addition to the flexoelectric effect, another electric effect is observed, which couples the electric field directly with the magnetization of YIG. We call this a magnetoelectric effect. By adjusting the direction of the electric field, the two effects can be well separated. Experimental results agree quantitatively with the theoretical prediction. A phenomenological coupling constant for the magnetoelectric effect is also obtained. Our findings point to an important avenue for manipulating spin waves and developing electrically tunable magnonic devices. [1] P. Rovillain \textit{et al}., Nat. Mater. {\bf 9}, 975 (2010). [2] M. Bao \textit{et al}., Appl. Phys. Lett. {\bf 101}, 022409 (2012). [3] T. Liu and G. Vignale, J. Appl. Phys. {\bf 111}, 083907 (2012). [4] T. Liu and G. Vignale, Phys. Rev. Lett. {\bf 106}, 247203 (2011). [5] X. Zhang \textit{et al}., Phys. Rev. Lett. {\bf 113}, 037202 (2014). [6]The author gratefully acknowledges collaborations with G. Vignale, M.E. Flatte', X. Zhang and H. X. Tang. This work is supported by DARPA MESO and an ARO MURI.