We study the magnetization dynamics of coupled vortices in arrays of Permalloy disks via analytical calculations and scanning transmission x-ray microscopy. The Thiele approach is used to derive linear equations of motion of the vortices. Thereby, vortex motions following a nanosecond field pulse are described by a superposition of eigenmodes that depend on the vortex polarizations. Eigenmodes are calculated for a specific polarization pattern of a 3 × 3 vortex array. With magnetic field pulses distinct oscillations are excited and imaged in space and time. The calculated eigenmodes precisely describe the measured oscillations.

Complexity created by periodic arrangement of well-understood building blocks plays an important role in biochemistry, photonics, engineering and nanoelectrics. The periodic arrangement of atoms or molecules as basis determines the physical properties of crystals. With the flexibility of nanometer precise electron-beam lithography here magnetic interactions are engineered yielding two-dimensional magnonic crystals that benefit from the magnetic vortex core as crystal basis. Using scanning transmission X-ray microscopy at the MAXYMUS beamline at BESSY II in Berlin, Germany the magnonic crystal dynamics are imaged with time resolution in the sub-nanosecond regime and simultaneous spatial resolution on the nanometer scale. Self-organized vortex core state formation by adiabatic reduction of a high frequency magnetic field excitation is observed. The emerging polarization states are shown to depend on the frequency of excitation and the strength of the dipolar interaction between the elements. In spite of the complexity of the investigated system, global order caused by local interactions creates polarization states with a high degree of symmetry. An analytical dipole model and numerically solved coupled equations of motion are adopted to analytically describe the experimental results. The emerging states can be predicted by a fundamental stability criterion based on the excitability of eigenmodes in the crystal. Further experiments with ferromagnetic absorption spectroscopy are carried out that give insight into the characteristic frequencies of the vortex dynamics that are crucially influenced by the self-organized state formation. This is emphasized with experiments on benzene-like magnetic vortex molecules whose motions show strong similarities to the vibrational modes of the actual benzene molecule (C 6 H 6 ). The symmetry of both systems determines the motions of the oscillators, i.e., the carbon atoms or the magnetic vortices. This allows to simplify the derivation of the fundamentally different dispersion relations depending on the previously tuned polarization state. The experiments confirm the calculations and prove that the magnetic vortex molecule features a reprogrammable band structure or dispersion relation. Consequently, this work allows further research studies to tailor the characteristic properties of various magnetic vortex arrangements by tuning the polarization state. I KurzzusammenfassungDas Zusammenspiel einer Vielzahl gleichartiger, gut verstandener Bausteine spielt eine entscheidende Rolle in vielen komplexen Systemen verschiedenster wissenschaftlicher Disziplinen wie Biochemie, Photonik, Ingenieurswissenschaften oder Nanoelektronik. Zum Beispiel bestimmen periodische Anordnungen von Atomen oder Molekülen die physikalischen Eigenschaften von Kristallen. In dieser Arbeit werden magnetisch wechselwirkende Wirbel, sogenannte magnetische Vortizes, zu magnonischen Kristallen angeordnet. Die verwendete Elektronenstrahl-Lithographie erlaubt es, die Vortizes mit Nanometer-Präzision herzustellen und ...

Magnonic vortex crystals are studied via scanning transmission x-ray microscopy and ferromagnetic-resonance spectroscopy. We investigate a two-dimensional vortex crystal by imprinting waves with tunable wave vectors. The dispersion relation ω(k) is determined via ferromagnetic-resonance spectroscopy with a tunable frequency and wave vector for two vortex core polarization patterns that are adjusted by self-organized state formation prior to the measurement. We demonstrate that the band structure of the crystal is reprogrammed by tuning the vortex polarizations.

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The dynamics of emergent magnetic quasiparticles, such as vortices, domain walls and bubbles are studied by scanning transmission X-ray microscopy (STXM), combining magnetic (XMCD) contrast with about 25 nm lateral resolution as well as 70 ps time resolution. Essential progress in the understanding of magnetic vortex dynamics is achieved by vortex core reversal observed by sub-GHz excitation of the vortex gyromode, either by ac magnetic fields or spin transfer torque. The basic switching scheme for this vortex core reversal is the generation of a vortex-antivortex pair. Much faster vortex core reversal is obtained by exciting azimuthal spin wave modes with (multi-GHz) rotating magnetic fields or orthogonal monopolar field pulses in the x and y direction, down to 45 ps in duration. In that way unidirectional vortex core reversal to the vortex core "down" or "up" state only can be achieved with switching times well below 100 ps. Coupled modes of interacting vortices mimic crystal properties. The individual vortex oscillators determine the properties of the ensemble, where the gyrotropic mode represents the fundamental excitation. By self-organized state formation we investigate distinct vortex core polarization configurations and understand these eigenmodes in an extended Thiele model. Analogies with photonic crystals are drawn. Oersted fields and spin-polarized currents are used to excite the dynamics of domain walls and magnetic bubble skyrmions. From the measured phase and amplitude of the displacement of domain walls we deduce the size of the non-adiabatic spin-transfer torque. For sensing applications, the displacement of domain walls is studied and a direct correlation between domain wall velocity and spin structure is found. Finally the synchronous displacement of multiple domain walls using perpendicular field pulses is demonstrated as a possible paradigm shift for magnetic memory and logic applications.

Collective modes in three-dimensional crystals of stacked permalloy disks with magnetic vortices are investigated by ferromagnetic resonance spectroscopy and scanning transmission X-ray microscopy. The size of the arrangements is increased step by step to identify the different contributions to the interaction between the vortices. These contributions are the key requirement to understand complex dynamics of three dimensional vortex crystals. Both vertical and horizontal coupling determine the collective modes. In-plane dipoles strongly influence the interaction between the disks in the stacks and lead to polarity-dependent resonance frequencies. Weaker contributions discern arrangements with different polarities and circularities that result from the lateral coupling of the stacks and the interaction of the core regions inside a stack. All three contributions are identified in the experiments and are explained in a rigid particle model.

A novel approach to investigate geometrical frustration is introduced using two-dimensional magnonic vortex crystals. The frustration of the crystal can be manipulated and turned on and off dynamically on the timescale of milliseconds. The vortices are studied using scanning transmission x-ray microscopy and ferromagnetic resonance spectroscopy. They are arranged analogous to the nanomagnets in artificial spin-ice systems. The polarization state of the vortices is tuned in a way that geometrical frustration arises. We demonstrate that frustrated polarization states and non-frustrated states can be tuned to the crystal by changing the frequency of the state formation process.

Spin waves with wavelengths in the nanometre range could serve as data carriers in future magnonic logic or signal processing devices. We investigate the interference of spin waves emitted from magnetic vortices in two exchange-coupled vortex stacks. The spin-wave dynamics are studied using scanning transmission X-ray microscopy and micromagnetic simulations. Stacks of vortices provide an excellent controllability of spin-wave properties including a tunable wavelength in the 100 nm regime and manipulation of their propagation direction via the magnetisation configuration. Furthermore, interference gives rise to amplified or reduced spin-wave amplitudes in distinct areas of the structure providing controlled confinement crucial for future applications of spin waves.