Spin wave mediated unidirectional Vortex Core Reversal by Two Orthogonal Monopolar Field Pulses: The Essential Role of Three-dimensional Magnetization Dynamics

Scanning transmission x-ray microscopy is employed to investigate experimentally the reversal of the magnetic vortex core polarity in cylindrical Ni81Fe19 nanodisks triggered by two orthogonal monopolar magnetic field pulses with peak amplitude $B_0$, pulse length ${\tau}$=60 ps and delay time ${\Delta}$t in the range from -400 ps to +400 ps between the two pulses. The two pulses are oriented in-plane in the x- and y-direction. We have experimentally studied vortex core reversal as function of $B_0$ and ${\Delta}$t. The resulting phase diagram shows large regions of unidirectional vortex core switching where the switching threshold is modulated due to resonant amplification of azimuthal spin waves. The switching behavior changes dramatically depending on whether the first pulse is applied in the x- or the y-direction. This asymmetry can be reproduced by three-dimensional micromagnetic simulations but not by two-dimensional simulations. This behavior demonstrates that in contrast to previous experiments on vortex core reversal the three-dimensionality in the dynamics is essential here.

Scanning transmission x-ray microscopy is employed to investigate experimentally the reversal of the magnetic vortex core polarity in cylindrical Ni81Fe19 nanodisks triggered by two orthogonal monopolar magnetic field pulses with peak amplitude B0, pulse length 60 ps and delay time ∆ in the range from 400 ps to 400 ps between the two pulses. The two pulses are oriented in-plane in the x-and y-direction. We have experimentally studied vortex core reversal as function of B0 and ∆ . The resulting phase diagramshows large regions of unidirectional vortex core switching where the switching threshold is modulated due to resonant amplification of azimuthal spin waves. The switching behavior changes dramatically depending on whether the first pulse is applied in the x-or the y-direction. This asymmetry can be reproduced by three-dimensional micromagnetic simulations but not by two-dimensional simulations. This behavior demonstrates that in contrast to previous experiments on vortex core reversal the three-dimensionality in the dynamics is essential here.
The zero field magnetic ground state of a flat cylindrical, soft ferromagnetic nanodisk (thickness h of a few tens of nm and diameter 2 of several hundred nm) is the vortex state. In the plane of the disk the magnetization curls in a clockwise (CW) or counter clockwise (CCW) circulation ( 1 or 1). The exchange energy is minimized by the formation of a region with a typical diameter of 10 to 20 nm at the center of the disk where the magnetization turns out of the plane. This vortex core points up or down corresponding to the two polarities 1 and 1. The polarity can be considered as binary data bit, e.g., 1 for 1 and 0 for 1. Since the vortex polarity is very stable and well controllable, such structures may be used in the future for information storage and data processing devices. It is well established that the vortex polarity can be switched by applying dynamical external magnetic in-plane fields [1][2][3][4][5][6] or dynamical spin-polarized currents [7]. For excitation, short sinusoidal bursts [1] or unipolar pulses [2,8] can be used. Unidirectional switching which is a prerequisite for data storage applications is possible with rotating fields in resonance either with the gyrotropic eigenmode of the vortex core [3] with frequencies in the range of 100 MHz to 1 GHz (depending on the disk dimensions and the material of the disk), or with azimuthal spin waves in the higher GHz frequency range [4]. Alternatively, the excitation can consist of short multi-GHz rotating field bursts [5], or can have the shape of two orthogonal monopolar pulses [6] with pulse amplitude B, pulse length  and a delay ∆ /2 between the pulses. For the latter type of excitation, switching times shorter than 100 ps were found [6] for pulse amplitudes of a few tens of mT. The fact that digital pulses can be used for fast switching instead of rotating field bursts is an attractive aspect for potential technological applications.
For all these types of excitations the dynamic switching mechanism is in general the same (see ref. [9] and references therein). Due to the excitation a local 'dip' region [10] is formed close to the vortex core where the out-of-plane components of the magnetization are antiparallel to the out-of-plane components of the core. If the excitation is strong enough, this dip splits into a vortex-antivortex pair with polarities opposite to the one of the original vortex, and the switching is performed via annihilation of the antivortex with the original vortex. This behavior has been shown by micromagnetic simulations [2]. In previous investigations on vortex core reversal the general results were very similar for two-dimensional [1,[3][4][5] and for three-dimensional [2,6] simulations, showing that the three-dimensionality of the magnetization dynamics has an effect on the details of the reversal mechanism [9] but has only minor influence on whether the core is switched or not.
In the present paper we report on experimental results for the switching of the vortex core by two orthogonal monopolar in-plane magnetic field pulses in x-and y-direction with delay times ∆ between the two pulses which are considerably longer than the half width of the pulses. We explore experimentally the vortex core switching behavior to investigate unidirectional vortex core reversal.
Therefore the phase diagrams for vortex core reversal as a function of pulse amplitude B0 and ∆ are are studied. We compare two-dimensional and three-dimensional simulations with the experimental findings to demonstrate the importance of a three-dimensional treatment of the problem. The x-ray magnetic circular dichroism effect at the Ni L3-edge is used as a contrast mechanism.
Details of the measurement technique are given in ref. [6]. The switching phase diagram is determined by measuring the core polarity of the vortex structure before and after the pulse excitation.
The corresponding micromagnetic simulations are based on the Landau-Lifshitz-Gilbert equation [11], using the object-orientated micromagnetic framework [12] (OOMMF). Thereby the simulation is discrete in space and time. For three-dimensional simulations we use cubic simulation cells with a lattice constant of 3.125 nm well below the exchange length 6.6 nm, for the two-dimensional simulations cells with a volume of 3.125 nm ² • are used. Standard Permalloy material parameters are used for the exchange constant 13 • 10 J/m, gyromagnetic ratio 2.21 • 10 m/As, and anisotropy constants are set to zero. The damping parameter 0.007 and the saturation magnetization 690 kA/m were determined as in ref. [6].  [13]. A further investigation of the time-resolved simulations reveals double-switching events (i.e. a succession of switching and back-switching) in the 2D case while no double switching events are found in the 3D case. Since the core polarity is the same as prior to the excitation, double switching events cannot be detected in the experiment.
In previous works [3,4,6,14] a sense of rotation of the external excitation field was defined. For pulsed excitation a CW (CCW) sense of rotation corresponded to positive (negative) delay times [6,14]. However, for delay times larger than half a period of the excited modes, it is no longer possible to attribute a specific sense of rotation to positive or negative delay times, as will be discussed in more detail below. We will discuss first the total energy pumped into the annihilation of the antivortex with the original vortex, is in general a three-dimensional process [9]. As was discussed in the introduction, so far only minor differences between three-and twodimensional simulations were found. In the following we show that here in contrast to previous investigations the situations in the 2D and 3D cases fundamentally differ even before the creation of the vortex-/antivortex pair, i.e. before the actual switching process begins. From the micromagnetic simulations we find that a strong z-dependent vortex core trajectory can be present before the start of the reversal process (see Fig. 4). This three-dimensionality changes the situation qualitatively and quantitatively and probably leads to differences in the vortex core reversal process: A moving vortex generates a gyrofield [15] that is responsible for the formation of the dip discussed in the introduction which may lead to a vortex core reversal. It is understandable that a z-dependent vortex core trajectory leads to a different gyrofield than a two-dimensional trajectory. This can be seen for example in Figs. The concept of a critical velocity [15] cannot be applied here as velocities of approx. 600 m/s can be found for positive and negative delay times alike. We think that it is not possible to describe the switching behavior by a simple dynamical mechanism since the large amplitude excitation of multiple modes leads to non-linear coupling effects even before the switching process starts. Nevertheless the origin of the observed three-dimensional vortex core trajectories can be explained by the character of the vortex eigenmodes. To the knowledge of the authors, in the context of vortex core reversal so far the eigenmodes involved could always be treated in a two dimensional fashion, e.g., the gyrotropic mode and dipolar spin-wave modes. However, a two dimensional approach is no longer valid for the eigenmodes in vortex structures with film thicknesses above approximately 40 nm. In contrast, the dynamic profiles of the eigenmodes become three-dimensional as they can have nodes in the oscillation amplitude along the disk thickness. In particular the presence of higher-order gyrotropic flexure modes [16][17][18][19] leads to three-dimensional dynamics of the vortex core. Specifically, dynamic hybridization effects introduce a three-dimensionality (z-dependence) to the dynamics of the vortex core also for azimuthal spin-wave modes [20], which are mainly excited in the present experiments (see Fig. 3).
To conclude we have shown experimentally that unidirectional vortex core reversal is possible using two orthogonal monopolar magnetic field pulses of duration 60 ps over a wide range of amplitudes and delay times between the pulses. This flexibility might be useful in future data storage applications, for example to compensate for different signal propagation times. For this type of excitation, energy is mostly coupled to spin-wave modes, which modulates the observed switching threshold. Despite the fact that the vortex gyrotropic mode is excited only very little, it probably has a significant influence on the switching experiment. The experimental phase diagram for the reversal of the vortex core polarity can be reproduced only by three-dimensional micromagnetic simulations.
This shows that the three-dimensionality of the magnetization dynamics is essential for this process and for this problem a two-dimensional simplification fails since the z-dependent vortex core dynamics are not captured.

FIG. 2.
Phase diagrams of vortex core reversal by two orthogonal in-plane magnetic field pulses of amplitude B0, length 60 ps and delay time ∆ (see Fig. 1) between the pulses.

FIG. 3.
The energy coupled into the vortex structure depends on the delay time between the two pulses, as can be seen by the maximum value of the total energy max[E(t)], which contains the demagnetization, exchange and Zeeman energy. The injected energy is largest for delay times where the CW and CCW spin waves are amplified by the second pulse as sketched at the top. Switching only occurs in regions of large energy: The white lines correspond to the switching threshold obtained from Fig. 2d.