Band Diagram of Spin Waves in a Two-Dimensional Magnonic Crystal S. Tacchi, 1 F. Montoncello, 2 M. Madami, 1 G. Gubbiotti, 1,3 G. Carlotti, 1,4 L. Giovannini, 2 R. Zivieri, 2 F. Nizzoli, 2 S. Jain, 5 A. O. Adeyeye, 5 and N. Singh 5 1 CNISM-Unita ` di Perugia, Dipartimento di Fisica, Via A. Pascoli, I-06123, Perugia, Italy 2 Dipartimento di Fisica and CNISM, Universita ` di Ferrara, via G. Saragat 1, I-44100 Ferrara, Italy 3 Istituto Officina dei Materiali (CNR-IOM), Sede di Perugia, c/o Dipartimento di Fisica, Via A. Pascoli, I-06123 Perugia, Italy 4 Dipartimento di Fisica, Universita ` di Perugia, Via A. Pascoli, 06123 Perugia, Italy 5 Department of Electrical and Computer Engineering, National University, Singapore, Singapore (Received 28 March 2011; published 16 September 2011) The dispersion curves of collective spin-wave excitations in a magnonic crystal consisting of a square array of interacting saturated nanodisks have been measured by Brillouin light scattering along the four principal directions of the first Brillouin zone. The experimental data are successfully compared to calculations of the band diagram and of the Brillouin light scattering cross section, performed through the dynamical matrix method extended to include the dipolar interaction between the disks. We found that the fourfold symmetry of the geometrical lattice is reduced by the application of the external field and therefore equivalent directions of the first Brillouin zone are characterized by different dispersion relations of collective spin waves. The dispersion relations are explained through the introduction of a bidimen- sional effective wave vector that characterizes each mode in this magnonic metamaterial. DOI: 10.1103/PhysRevLett.107.127204 PACS numbers: 75.30.Ds, 75.75.c, 75.78.Cd, 78.35.+c The concept of electronic band structure in crystalline solids is one of the most important achievements of con- densed matter physics [1]. Similarly, the propagation of electromagnetic waves in artificial materials with periodi- cally modulated dielectric constant (photonic crystals) is subject to the existence of allowed frequency ranges, alter- nated with forbidden band gaps [2]. The same also occurs if one considers collective spin waves (SWs) propagating in magnonic crystals (MCs), a new class of metamaterials with periodically modulated magnetic properties [3–7]. Since the wavelengths of these excitations are shorter than those of electromagnetic radiation in the GHz fre- quency range, MCs offer better prospects for miniaturiza- tion at these frequencies with the advantage that frequency position and width of the band gap are tunable by the applied magnetic field [8,9]. This suggests an unprece- dented opportunity to design and exploit a new generation of spin logic devices, filters, and waveguides operating in the GHz frequency range [10,11]. However, a knowledge of the magnonic band structure of a specific MC is pre- liminary to any desired application. Up to now, experimental studies have been concentrated on one-dimensional (1D) MCs consisting of arrays of longitudinally magnetized nanostripes, with the wave vec- tor directed along the array periodicity [12,13]. These studies demonstrated the existence of dispersive Bloch waves whose periodicity in the reciprocal space matches the width of the Brillouin zone. On the contrary, there are only a few reports on either the imaging [14] or the magnonic band structure [15] of collective excitations in two-dimensional MCs. In particular a complete mapping of the Brillouin zone (BZ) and a detailed discussion of the physics underpinning the dispersion curves of magnonic modes is still lacking in the literature. In this work, we exploit the Brillouin light scattering technique (BLS) to achieve a complete mapping of the spin-wave dispersion curves, along the principal symmetry directions of the first BZ, for a two-dimensional (2D) MC consisting of a square array of Ni 80 Fe 20 disks. The experi- mental data are successfully compared to the spin-wave band diagram and the BLS cross section calculated through the dynamical matrix method (DMM) [16–18], which was extended to the case of 2D periodic structures [19]. The magnonic band structure within the first Brillouin zone is explained in terms of a bidimensional effective wave vec- tor for each mode. The sample, fabricated using deep ultraviolet lithogra- phy, consists of a large array of 50 nm thick Ni 80 Fe 20 disks having a diameter d ¼ 600 nm and arranged in a square matrix with interdot separation of Á ¼ 55 nm (period a ¼ 655 nm)[20]. This corresponds to a square first BZ of side 2=a ¼ 2  4:8  10 4 rad=cm. Brillouin light scattering experiments were performed in the backscatter- ing geometry by using a 200 mW solid state laser operating at wavelength  ¼ 532 nm. The sample was mounted on a two-axis goniometer which allows us to choose a specified angle of incidence of light () as well as to rotate the sample around the surface normal (azimuthal rotation, ), with an accuracy of 1  . By changing , it is possible to select the magnitude of the in-plane component q of the SW wave vector [q ¼ 4ðsinÞ=] entering into the scattering process, while variation of the azimuthal angle  corresponds to change the in-plane direction of q. The SW frequency dispersion was studied along the principal PRL 107, 127204 (2011) PHYSICAL REVIEW LETTERS week ending 16 SEPTEMBER 2011 0031-9007= 11=107(12)=127204(5) 127204-1 Ó 2011 American Physical Society