Spin waves of a current-injected thin ferromagnetic stripe R. P. Erickson* and D. P. Pappas National Institute of Standards and Technology, Boulder, Colorado 80305, USA Received 4 August 2008; revised manuscript received 24 September 2008; published 27 October 2008 Within a micromagnetic model we present the theory of linearized spin waves of a current-carrying rectan- gular ferromagnetic stripe treated as a slab of infinite extent. After determining the nonuniform scissorlike magnetic ground state that results when a dc electric current is applied along an in-plane easy axis, we calculate both ferromagnetic resonances and spin-wave dispersion as a function of slab thickness. For Permalloy stripes less than 1 m in thickness, increasing current stiffens the response of bulk spin waves, and their dispersion becomes increasingly asymmetric with respect to the easy axis—shifting to lower higher frequencies with opposite the direction of current. Also, the frequency and direction of propagation of the Damon-Eshbach surface mode are substantially modified by the current, with changed surface-mode behavior exhibited. Above 1 m in thickness the lowest-lying resonance frequency of the Permalloy stripe softens to zero with increasing current and a gap opens up to finite wavelengths along the direction of current, indicative of a ground-state instability. We discuss the implication of our results to the characterization of the magnetic state of these rectangular structures. DOI: 10.1103/PhysRevB.78.144421 PACS numbers: 75.30.Ds, 75.25.+z, 76.50.+g I. INTRODUCTION Fabrication of nanoscale materials that couple electron charge to spin quanta, so-called spintronic or magnetoelec- tronic devices, has created a burgeoning field of study in the last couple of decades, 1 elicited by the discovery of giant magnetoresistance GMR in Fe/Cr/Fe multilayers. 2 Most notably, the spin-valve device, which exploits the GMR ef- fect, has gained widespread use in read-write heads within the magnetic recording industry. More recently, experiments involving the injection of spin-polarized current into mag- netic nanopillars 3,4 have led to observations of coupled non- linear magnetic oscillations of significant size within these structures. 5,6 In particular, Pufall et al., 4 in their elegant ex- periment, suggested that the interaction of their nano- oscillators is mediated by spin waves. Recent theoretical studies of spin waves in the presence of a transport current have focused on the disk shape. 7,8 Other theoretical studies of spin waves in cylindrical ferromagnetic nanowires have also been discussed in the presence of both Zeeman 9 and microwave 10 fields. As demonstrated in Ref. 4, pulses of electric current ap- plied at frequencies in the gigahertz range can induce spin waves in ferromagnetic structures. In the present paper we consider unpolarized current pulses of frequencies orders of magnitude lower, where the temporal width of the pulse is of sufficient duration that one may model the current by its dc component and corresponding Oersted field. In this limit a well-defined magnetic ground state can be sustained for the duration of the pulse, and excitations of this ground state are spin waves generated by means other than the current itself. While there is a long history of theoretical descriptions of micromagnetic order and spin waves in rectangular, cylindri- cal, and spherical geometries, as far as we know a thorough account of spin-wave excitations of the ferromagnetic slab in the presence of a transport current, as we envisage here, has not been presented in the literature. Some years ago, Smith et al. 11 presented a micromagnetic theory of ferromagnetic order in current-carrying thin films, with application to Per- malloy stripes possessing an in-plane easy axis of magneti- zation, associated with volume anisotropy of strain-induced origin. In this situation, the application of current along the easy axis causes the magnetization to spread out in either direction, away from the easy axis, in an arrangement that can be described as a scissor configuration. This fanlike ar- rangement is a consequence of the fact that the applied cur- rent induces an Oersted field in plane and perpendicular to the easy axis but in opposite directions with respect to the top and bottom of the slab. The Oersted field increases lin- early in magnitude from zero, at the center of the slab, to a maximum value at the slab surfaces. The success of Ref. 11 was to show how the model exchange constant could be accurately quantified by fitting to magnetoresistive measure- ments obtained while current was applied along the easy axis. In the present paper we expand on the work in Ref. 11 by providing an account of the linearized spin-wave excitation spectrum of the nonuniform classical magnetic ground state that arises in this rectangular geometry—with an unpolarized dc electric current applied along the in-plane easy axis. As in Ref. 11, we have in mind rectangular Permalloy stripes where the width perpendicular to the easy axis is much greater than the thickness of the stripe, such that we can approximate the stripe as a ferromagnetic slab with top and bottom surfaces of infinite extent. A classical theory of spin waves established within a micromagnetic model of these structures permits their characterization through such experi- mental techniques as ferromagnetic resonance FMR and Brillouin light scattering BLS. 12 Additionally, it allows us to investigate the stability of the scissorlike ground state un- der such forms of magnetic excitation as these experimental techniques induce. Our model assumes a spatial continuum wherein ex- change enters through the gradient of magnetization. Thus, PHYSICAL REVIEW B 78, 144421 2008 1098-0121/2008/7814/14442114 144421-1