Driven Dynamic Mode Splitting of the Magnetic Vortex Translational Resonance K. S. Buchanan, 1, * M. Grimsditch, 2 F. Y. Fradin, 2 S. D. Bader, 1,2 and V. Novosad 2 1 Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439, USA 2 Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA (Received 10 April 2007; revised manuscript received 16 August 2007; published 27 December 2007) A magnetic vortex in a restricted geometry possesses a nondegenerate translational excitation that corresponds to circular motion of its core at a characteristic frequency. For 40-nm thick, micron-sized permalloy elements, we find that the translational-mode microwave absorption peak splits into two peaks that differ in frequency by up to 25% as the driving field is increased. An analysis of micromagnetic equations shows that for large driving fields two stable solutions emerge. DOI: 10.1103/PhysRevLett.99.267201 PACS numbers: 75.40.Gb, 75.30.Ds, 75.75.+a Nonlinear phenomena are ubiquitous in nature, existing in systems ranging from leaky faucets to atmospheric circulation to optics [1– 3]. Magnetic systems can be mod- els for improving our understanding of nonlinear phe- nomena since they are experimentally accessible and the equations of motion are generally tractable. In magnetism, nonlinear effects were first observed in high-power ferro- magnetic resonance experiments in the 1950s [4,5] where the premature saturation of the main absorption peak and the emergence of subsidiary peaks were attributed to spin- wave generation [6]. These initial nonlinear dynamic stud- ies focused on a saturated magnetic state. More recently, magnetic systems have been shown to exhibit a wealth of other interesting phenomena, such as spin-wave self- focusing [7], symmetry breaking spin-wave Mo ¨bius soli- tons [8], and foldover and bistability effects [9]. The spin vortex state, an in-plane flux-closure magneti- zation distribution with a small central core, is often ob- served in magnetically soft microstructures [10 –12]. Spin- polarized scanning tunneling microscopy shows that the core radius is 10 nm, comparable to the material’s ex- change length [13]. Because of the Magnus-type force (gyroforce) that acts on the core, magnetic vortices exhibit unique dynamic excitations, including the translational or gyrotropic mode that is characterized by sub-GHz, spiral- like core motion [14 – 20], distinct from the higher fre- quency (GHz range) quantized spin waves observed in restricted geometries [21– 23]. The core polarization p  1 determines the handedness of the spiral motion, as demonstrated by time-resolved magneto-optical Kerr [24 – 26] and x-ray experiments [27]; the restoring force is provided mainly by the magnetostatic energy [16,28]. Here we explore frequency- and amplitude-dependent dynamics of magnetic vortices in circular and elliptical microdisks excited by a radio frequency (rf) driving field. We find that as the magnitude of the rf field h ac is in- creased, the resonance peak corresponding to the vortex translational mode splits into two well-defined peaks whose separation increases with h ac . The appearance of mode splitting is unusual since for a single dot this is a nondegenerate mode. (There is a degeneracy in the sense that the ensemble contains random chiralities and polari- ties with the same eigenfrequency but we find no indication in simulations that their high-field response should differ.) However, comparing the results with micromagnetic cal- culations and a phenomenological analytical model reveals that this system is similar to a driven anharmonic oscillator where a nonlinear energy potential can lead to two reso- nance states, thereby resolving the apparent contradiction of a split, nondegenerate mode. Nevertheless, the prevail- ing theory for magnetic vortex dynamics does not fully explain our observations. We use a microwave reflection technique to investigate the excitations of vortices in magnetic microstructures [29]. Elliptical and circular permalloy (Fe 20 Ni 80 alloy) microdisks were patterned on the central strip of coplanar waveguides (CPW), 2000 per waveguide separated by >1 m to minimize dipolar interactions, using e-beam lithography and liftoff. A rf current in the waveguide generates an in-plane rf magnetic field that is preferentially absorbed when the frequency coincides with a resonance. The derivatives of the CPW impedance are recorded with respect to a small modulation field applied parallel to the static, in-plane magnetic field H, and h ac is calculated from the current in the CPW. We examine three samples: 2:2  1:1 m ellipses, 3:1  1:7 m ellipses, and circles of diameter 2:2 m, all 40-nm thick, referred to as samples A, B, and C, respectively. All are in the single-vortex ground state with a mixture of chiralities and polarities. Figure 1(a) shows microwave impedance spectra as a function of h ac for sample A with H  60 Oe along the ellipse minor axis, orthogonal to h ac . Magneto-optical Kerr effect measurements (not shown) indicate that this is below the vortex annihilation field of 2.5 kOe along the minor (hard) axis. A single, symmetric peak is found at 130 MHz for low h ac . As h ac is increased, it broadens and develops a shoulder (h ac  11 Oe) and then splits in two, where one branch increases in frequency and the other decreases, reaching a separation of 40 MHz for h ac  24 Oe. Figure 1(b) shows how the microwave spectra PRL 99, 267201 (2007) PHYSICAL REVIEW LETTERS week ending 31 DECEMBER 2007 0031-9007= 07=99(26)=267201(4) 267201-1  2007 The American Physical Society