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Localized domain-wall excitations in patterned magnetic dots probed by broadband ferromagnetic resonance

We investigate the magnetization dynamics in circular Permalloy dots with spatially separated magnetic vortices interconnected by domain walls (double vortex state). We identify a novel type of quasi one-dimensional (1D) localised spin wave modes confined along domain walls, connecting each of two vortex cores with two edge half-antivortices. Variation of the mode eigenfrequencies with the dot size is in quantitative agreement with the developed model, which considers a dipolar origin of the localized 1D spin waves or so-called Winter\'s magnons [J.M. Winter, Phys.Rev. 124, 452 (1961)]. These spin waves are analogous to the displacement waves of strings, and could be excited in a wide class of patterned magnetic nanostructures possessing domain walls, namely in triangular, square, circular or elliptic magnetic dots.

Vortex forms are found in a variety of physical systems, ranging from superfluids and superconductors to tornadoes. As for the single magnetic vortex (SV) state in confined geometry, its first observation already revealed the coexistence of a SV and domain walls (DW), which connect the SV with the vertices of a triangular nanomagnet [1]. Special interest in the confined magnetic vortices is inspired by the possibility of excitation or switching of the vortex core [2,3] which has been suggested as a potential new road to the creation of nanoscale memory and spin torque vortex oscillators [4]. It is now well established that coexisting vortex-DW configurations are formed in the ground states of magnetic rectangles, triangles or ellipses [1,[5][6][7], before the single-domain state appears upon decreasing the particle size [5]. Although excitations of the confined DW or SV states are well understood [8][9][10][11][12], the nature of the spin waves excited in nanomagnets involving both vortices and domain walls remains unclear. Circular dots are an example of nanomagnets where eigenmodes in the SV ground state can simply be described analytically [11,13]. Decreasing the external magnetic field from saturation, these nanomagnets are known to have two intermediate metastable states: the double magnetic vortex (DMV) and the so-called S-state, which is formed by a DW only [14]. Recent reports show that circular magnetic dots with reduced thickness, and therefore enhanced pinning, may accommodate the DMV as a long-living metastable state, with two magnetic vortices connected via a DW to edge localized antivortices (AV) [14][15][16][17].
Circular dots provide a unique opportunity to investigate the spin wave dynamics of both the vortex-DW (DMV) and SV configurations by reducing or increasing the external magnetic field below or above the vortex nucleation field, H n . In this system, the vortex cores can be pinned by microstructure related defects, intergrain boundaries, etc. [18], thereby trapping the system in the metastable DMV state. Provided the perturbation is small enough, the dot can stay in its metastable state for at least the duration of the measurements. We show here that for low fields the magnetization dynamics of circular Py dots with diameter of 1000 nm and a thickness below 25 nm corresponds to DMV state excitations. The first few lowest frequency spin wave modes present the first example of confined (standing) quasi-1D spin waves localised along domain walls pinned between the vortex and half-antivortex cores.
For the experiments, a vector network analyzer (VNA-FMR) was used to conduct broadband measurements of the magnetization dynamics in the square arrays of Permalloy dots with spacing between dot edges exceeding diameter in order to avoid influence of dipolar interaction. We excited the dots by applying small (0.1 Oe) in-plane driving fields created by a co-planar waveguide (CPW), parallel to the magnetic bias field. This technique ensures small precession amplitudes of the magnetization around the effective field. The data was analyzed on the basis of the transmission model, under the assumption that the dominant CPW mode is TEM mode and by neglecting the effect of reflection [19]. In the experiments we investigated the evolution of the broadband dynamic response of the dot as a function of a slowly changing inplane magnetic field. We first saturate the sample by applying a positive field that is two times larger than the positive vortex annihilation field (H a ). Subsequently we slowly sweep the field downwards, lowering it past the vortex nucleation field (H n ) and after changing the field polarity we end at the negative vortex annihilation field. The microwave field drive, parallel to the external bias field, with an amplitude a thousand times smaller than H a , mainly couples to DW where the exchange energy is the highest. This permits precise detection of possible localized excitations in the DW [11]. We investigated circular dots with the thicknesses L= 50, 25, 20 and 15 nm with diameter 2R = 1035 nm and 20 nm thick dots with 2R=570 nm. The micromagnetic simulations were carried out using OOMMF code [20] for circular Py dots with a diameter of 1035 nm and thickness of 20 nm, with simulated cell size 5 x 5 x 20 nm 3 , γ/2π = 2.96 MHz/Oe, the exchange stiffness constant A = 1.4 x 10 -11 J/m, and the Gilbert damping constant α = 0.01.
We first saturate the dot, using an in-plane field of 1000 Oe. After releasing the field in one step to zero, in agreement with [14], the dot relaxes by means of two clearly defined steps that have a slope of energy vs. time of almost zero (not shown). These steps are two metastable magnetic states which we identify as first the S-state and afterwards the DMV state. For our simulations these states remain stable for about 10 ns only. This can be attributed to the fact that we neglected the magnetic anisotropy and all the imperfections that the real samples usually have.
To excite magnetization dynamics, a variable driving field (Gaussian field pulse) with the amplitude of 1 Oe and FWHM of 1 ps was applied. We perform local Fourier transforms over all simulation cells and average these spectra to obtain the eigenfrequencies. bias fields below vortex nucleation field |H n | correspond to the azimuthal modes [19] and are symmetric with respect to the field direction, as expected for the SV state (Fig.1a,c). However, this low field response drastically changes for the Py dots with smaller L. Starting from 25 nm the magnetization dynamics becomes asymmetric at low fields (Fig.1d). Besides, we find for a parallel excitation field and for bias fields just below H n , additional peaks with frequencies between 2 and 5 GHz, which are lower than the first azimuthal mode frequency of the SV state, but much higher than the vortex gyrotropic frequency [6]. We claim that these surprising spectra originate from another, intermediate state. Namely, we consider that the Py dots in the array are in a metastable DMV state in the field interval indicated by vertical dashed lines in Fig.1. A reduction of the vortex core size and a decrease in the dot's thickness [21], accompanied by the enhancement of the effective damping, may be an origin of the stabilization of the metastable DMV state for the thinner dots, in agreement with previous measurements [14][15][16][17].
In order to investigate the possible metastable states of the dots, we performed micromagnetic simulations. We identified the measured asymmetric eigenmodes (Fig.1d) by comparison of the calculated excitation spectra for different metastable (S or DMV) states with the experimental results. Comparison of the dynamic simulations for the S-type and DMV states with experimental data reveals that the spin wave modes excited between the positive H n and small negative bias fields, are close to those expected for the DMV state (Fig.2). The robustness of the dynamic response (not shown) further confirms that the detected spin waves are due to the DMV state. Figure 2a shows the three strongest peaks (above the gyrotropic frequency) of the simulated spectra compared to the experiments. Clearly, there is a good match between the experiments and simulations.
In order to get more insight into the spin dynamics in the DMV state we investigate the spatial distribution of the main eigenmodes. Knowing the local distribution of the phases and amplitudes for every cell for selected a eigenfrequency, we can reconstruct the eigenmode profiles [22]. We note that in the case of the SV excitations in square dots [7] the VC is a crossing of the four 90-degrees DWs connecting the core and the square corners. The VC position oscillations then correspond to change of the DW shape (DW bulging) and length. The VC mode and DW bulging modes are naturally coupled and have the same oscillation eigenfrequency (230 MHz in [7]). These bulging DW modes are essentially different from the here considered DW flexural modes, having eigenfrequencies about 2 GHz and higher.
There are two branches of the spin excitations in the presence of a DW: ¨free spin waves¨ with excitations resembling those in an infinite uniform ferromagnet, and wall-bounded excitations or Winter´s magnons [24], representing oscillations of the DW shape about the static equilibrium position which can be described as a plane. The DW displacement waves are analogous to the displacements that can propagate along elastic strings (e.g., violin strings) and to the capillary waves on the surface of liquids. The Winter´s magnons were considered for the Bloch type of the DW [24] in an infinite ferromagnet. But in our case of flat soft magnetic dots, the spins lay in-plane due to strong magnetostatic energy and a Néel DW should be considered as a ground state. To understand 1D spin waves confined along the Néel DW that is pinned by the V(AV) cores, we determine spin excitations of the wall applying the approach by Slonczewski , where δ is the DW width. Then, we consider oscillations of the DW shape writing coupled equations of motion for the DW displacement from its rest position ( ) t y q , and the magnetization deviation angle ϑ from the dot plane xOy ( ϑ π + = Θ 2 / ). The variables q and ϑ describe flexural modes of the DW or Winter´s magnons [24]. We introduce a strong pinning near the VC position in the DW (y=0) and no pinning or a strong pinning near the AV ). This allows using the eigenfunctions