The effective properties of two-dimensional (2D) and three-dimensional (3D) magnonic metamaterials are presented according to micromagnetic and analytical calculations. Micromagnetic calculations were performed by using the Hamiltonian-based Dynamical Matrix Method (HDMM) extended to periodic systems. The 2D systems are composed by periodic square arrays of circular antidots (holes) embedded into a permalloy ferromagnetic film. In the calculations both the diameters d of the holes and the array periodicities are in the nanometric range. Two geometries have been investigated: 1) The external magnetic field H is applied along the y direction and perpendicularly to the Bloch wave vector K placed along the x direction in the sample plane. 2) H (K) forms an angle of 45o degrees with respect to the y (x) axis. Magnonic modes dispersion is calculated as a function of K for both extended and localized modes and opening of band gaps at Brillouin zone boundaries is explained in terms of inhomogeneity of the internal field [1,2]. Band gaps are discussed as examples of metamaterial properties of 2D magnonic crystals. In both geometries it is possible to identify, for a given collective mode, a characteristic wavelength which is commensurable with the periodicity of the system [3]. Since collective modes are mainly affected by the finite size of the holes rather than the periodicity and since the characteristic wavelength is much larger than d, the dynamics is described in terms of effective properties and an effective medium approximation is used to model the metamaterial wave in the propagative regime. These properties can be regarded as metamaterial properties [4,5]. In this way, the characteristic wavelength can be regarded as an effective wavelength λeff related to the scattering from the holes of the given collective mode. Interestingly, the effective wavelength, which can be defined for each mode of the spectrum, is not necessarily equal to the Bloch wavelength. In the cases studied the ratio d /λeff <<1 for the whole range of Bloch wave vectors investigated. Correspondingly, also a small wave vector is introduced and important effective rules that do not contradict the Bloch’s theorem, but complete it, are derived. A description of scattering from antidots in terms of momentum conservation is given and the differences with the well-known Bragg diffraction law are outlined [6,7]. The 3D systems are composed by 2D periodic arrangements of circular nanodots of cobalts partially or totally embedded into a permalloy film, but subdivided into a stack of layers along the z-direction accounting for the nonuniform magnetization of the two materials along z [8,9]. Band structure of the most representative collective modes, namely a mode mainly localized in the cobalt dots and a mode prevantly concentrated in the permalloy film is studied. The dependences of band gap amplitudes on the cobalt volume and on the cobalt position with respect to the permalloy film are also discussed. Effective “surface magnetic charges” are introduced to explain the demagnetizing field behaviour associated to the two materials and effective quantities, like an effective magnetization and an effective exchange stiffness constant, are introduced and the dispersion of the corresponding metamaterial wave in the propagative regime is calculated. It is also shown that the interchange between the two materials in the system leads to different band structure of the two most representative collective modes. Finally, a concentration factor is introduced to quantitatively express the localization of the most relevant collective modes [9] in analogy with the corresponding one defined for electromagnetic waves in 2D photonic crystals. [1] R. Zivieri et al., Physical Review B 85 012403-1-6 (2012). [2] R. Zivieri, Solid State Physics 63, 151-216 (2012). [3] R. Zivieri, Proceedings of Metamaterials ’2012, 6th International Congress on Advanced Electromagnetic Materials in Microwave and Optics, 624-626 (2012). [4] R. Zivieri and L. Giovannini, Metamaterials 6, e127-138 (2012). [5] R. Zivieri and L. Giovannini, “Effective quantities and effective rules in 2D ferromagnetic antidot lattices” Photonics and Nanostructures - Fundamentals and Applications 11, 191-202 (2013). [6] R. Zivieri, “Metamaterial description of magnonic modes along ΓM direction in a 2D antidot lattice” in press in Proceedings of the 7th International Congress on Advanced Electromagnetic Materials in Microwaves and Optics – Metamaterials 2013, Bordeaux, France, 16-21 September 2013. [7] R. Zivieri, “Effective scattering of collective modes in two-dimensional magnonic crystals” submitted to New Journal of Physics. [8] R. Zivieri and P. Malagò, “Effective properties of a three-dimensional permalloy/cobalt binary system”, in press in Proceedings of the 7th International Congress on Advanced Electromagnetic Materials in Microwaves and Optics – Metamaterials 2013, Bordeaux, France, 16-21 September 2013. [9] R. Zivieri, P. Malagò and L. Giovannini, “Band structure of magnonic modes in three-dimensional permalloy-cobalt binary systems: a micromagnetic study”, submitted to Physical Review B.

##### Scheda prodotto non validato

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

`http://hdl.handle.net/11392/1869727`

Titolo: | Effective description of 2D and 3D magnonic metamaterials -- Invited talk by R. Zivieri -- Conferenza internazionale |

Autori interni: | ZIVIERI, Roberto |

Data di pubblicazione: | 2013 |

Abstract: | The effective properties of two-dimensional (2D) and three-dimensional (3D) magnonic metamaterials are presented according to micromagnetic and analytical calculations. Micromagnetic calculations were performed by using the Hamiltonian-based Dynamical Matrix Method (HDMM) extended to periodic systems. The 2D systems are composed by periodic square arrays of circular antidots (holes) embedded into a permalloy ferromagnetic film. In the calculations both the diameters d of the holes and the array periodicities are in the nanometric range. Two geometries have been investigated: 1) The external magnetic field H is applied along the y direction and perpendicularly to the Bloch wave vector K placed along the x direction in the sample plane. 2) H (K) forms an angle of 45o degrees with respect to the y (x) axis. Magnonic modes dispersion is calculated as a function of K for both extended and localized modes and opening of band gaps at Brillouin zone boundaries is explained in terms of inhomogeneity of the internal field [1,2]. Band gaps are discussed as examples of metamaterial properties of 2D magnonic crystals. In both geometries it is possible to identify, for a given collective mode, a characteristic wavelength which is commensurable with the periodicity of the system [3]. Since collective modes are mainly affected by the finite size of the holes rather than the periodicity and since the characteristic wavelength is much larger than d, the dynamics is described in terms of effective properties and an effective medium approximation is used to model the metamaterial wave in the propagative regime. These properties can be regarded as metamaterial properties [4,5]. In this way, the characteristic wavelength can be regarded as an effective wavelength λeff related to the scattering from the holes of the given collective mode. Interestingly, the effective wavelength, which can be defined for each mode of the spectrum, is not necessarily equal to the Bloch wavelength. In the cases studied the ratio d /λeff <<1 for the whole range of Bloch wave vectors investigated. Correspondingly, also a small wave vector is introduced and important effective rules that do not contradict the Bloch’s theorem, but complete it, are derived. A description of scattering from antidots in terms of momentum conservation is given and the differences with the well-known Bragg diffraction law are outlined [6,7]. The 3D systems are composed by 2D periodic arrangements of circular nanodots of cobalts partially or totally embedded into a permalloy film, but subdivided into a stack of layers along the z-direction accounting for the nonuniform magnetization of the two materials along z [8,9]. Band structure of the most representative collective modes, namely a mode mainly localized in the cobalt dots and a mode prevantly concentrated in the permalloy film is studied. The dependences of band gap amplitudes on the cobalt volume and on the cobalt position with respect to the permalloy film are also discussed. Effective “surface magnetic charges” are introduced to explain the demagnetizing field behaviour associated to the two materials and effective quantities, like an effective magnetization and an effective exchange stiffness constant, are introduced and the dispersion of the corresponding metamaterial wave in the propagative regime is calculated. It is also shown that the interchange between the two materials in the system leads to different band structure of the two most representative collective modes. Finally, a concentration factor is introduced to quantitatively express the localization of the most relevant collective modes [9] in analogy with the corresponding one defined for electromagnetic waves in 2D photonic crystals. [1] R. Zivieri et al., Physical Review B 85 012403-1-6 (2012). [2] R. Zivieri, Solid State Physics 63, 151-216 (2012). [3] R. Zivieri, Proceedings of Metamaterials ’2012, 6th International Congress on Advanced Electromagnetic Materials in Microwave and Optics, 624-626 (2012). [4] R. Zivieri and L. Giovannini, Metamaterials 6, e127-138 (2012). [5] R. Zivieri and L. Giovannini, “Effective quantities and effective rules in 2D ferromagnetic antidot lattices” Photonics and Nanostructures - Fundamentals and Applications 11, 191-202 (2013). [6] R. Zivieri, “Metamaterial description of magnonic modes along ΓM direction in a 2D antidot lattice” in press in Proceedings of the 7th International Congress on Advanced Electromagnetic Materials in Microwaves and Optics – Metamaterials 2013, Bordeaux, France, 16-21 September 2013. [7] R. Zivieri, “Effective scattering of collective modes in two-dimensional magnonic crystals” submitted to New Journal of Physics. [8] R. Zivieri and P. Malagò, “Effective properties of a three-dimensional permalloy/cobalt binary system”, in press in Proceedings of the 7th International Congress on Advanced Electromagnetic Materials in Microwaves and Optics – Metamaterials 2013, Bordeaux, France, 16-21 September 2013. [9] R. Zivieri, P. Malagò and L. Giovannini, “Band structure of magnonic modes in three-dimensional permalloy-cobalt binary systems: a micromagnetic study”, submitted to Physical Review B. |

Handle: | http://hdl.handle.net/11392/1869727 |

Appare nelle tipologie: | 04.3 Abstract (Riassunto) in convegno in Rivista/Volume |