Unidirectional sub-100 ps magnetic vortex core reversal

The magnetic vortex structure, an important ground state configuration in micron and sub-micron sized ferromagnetic thin film platelets, is characterized by a curling in-plane magnetization and, in the center, a minuscule region with out-of-plane magnetization, the vortex core, which points either up or down. It has already been demonstrated that the vortex core polarity can be reversed with external AC magnetic fields, frequency-tuned to the (sub-GHz) gyrotropic eigenmode or to (multi-GHz) azimuthal spin wave modes, where reversal times in the sub-ns regime can be realized. This fast vortex core switching may also be of technological interest as the vortex core polarity can be regarded as one data bit. Here we experimentally demonstrate that unidirectional vortex core reversal by excitation with sub-100 ps long orthogonal monopolar magnetic pulse sequences is possible in a wide range of pulse lengths and amplitudes. The application of such short digital pulses is the favourable excitation scheme for technological applications. Measured phase diagrams of this unidirectional, spin wave mediated vortex core reversal are in good qualitative agreement with phase diagrams obtained from micromagnetic simulations. The time dependence of the reversal process, observed by time-resolved scanning transmission X-ray microscopy indicates a switching time of 100 ps and fits well with our simulations. The origin of the asymmetric response to clockwise and counter clockwise excitation which is a prerequisite for reliable unidirectional switching is discussed, based on the gyromode - spin wave coupling.


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In thin soft magnetic layers with thicknesses of a few tens of nm and lateral dimensions ranging from 100 nm to 10 µm the vortex structure is the ground state showing an in-plane curling magnetization and a perpendicularly magnetized core at the center with a size of about 10 to 20 nm [1][2][3]. In spite of its high stability with respect to static external magnetic fields the vortex core can be switched dynamically with low-power sine or pulsed magnetic fields or spin polarized currents [4][5][6][7]. This was achieved by resonantly exciting the gyrotropic mode with its eigenfrequency typically in the range from 100 MHz to 1 GHz. By applying rotating in-plane fields this reversal occurs in a unidirectional manner [8][9][10][11][12] since the exciting field couples only to the vortex gyrotropic mode if the sense of rotation of the excitation corresponds to the sense of the vortex gyrotropic mode which is determined by the core polarity. Due to these phenomena the vortex core has been discussed as an extremely stable bit element in digital magnetic random access storage media providing a low-power [7] and selectively addressable switching mechanism with a speed in the range of ns [9]. At much higher (GHz) frequencies vortex structures possess spin wave eigenmodes arising from the magnetostatic interaction. Spin wave mediated vortex core reversal was demonstrated experimentally with excitation of multi-GHz rotating field bursts [13]. Hereby unidirectional vortex core reversal was achieved and corresponding micromagnetic simulations indicated a switching time slightly above 200 ps [14,15] when using one period bursts.
Here we explore, by systematic experimental studies and micromagnetic simulations, how to speed up unidirectional vortex core reversal by excitation with a sequence of two orthogonal monopolar magnetic pulses, less than 100 ps in total duration. The fact that digital pulses can be used for fast spin wave mediated switching instead of rotating field bursts is an attractive aspect for potential technological applications. The phase diagram of vortex core reversal is measured, i. e., the dependence of switching on pulse amplitudes and their durations. A region of unidirectionality is found to be rather large and thus robust against sample dimension variations. The finding that the region of unidirectionality is larger than the corresponding region for rotating field bursts is a further important result. Time-resolved x-ray microscopy movies image the dynamics of the switching process, in excellent agreement with simulations. The measurements indicate a switching time below 100 ps which is in accordance with our simulations. Furthermore, the switching time of unidirectional vortex core reversal by pulsed excitation is simulated in samples of different sizes and the influence of material parameters is studied indicating a lower universal limit of about 70 ps for the switching time. Coupling between spinwaves and the vortex gyromode is identified as the origin of the asymmetric response of the magnetization to clockwise (CW) or counter clockwise (CCW) excitation which is responsible for the unidirectionality in the vortex core reversal by this broad band orthogonal short pulse excitation.

Results
We conduct our experiments on Permalloy (Ni 80 Fe 20 ) discs with a thickness of 50 nm and diameters of 490-500 nm. The discs are prepared on top of cross-like copper striplines (cf. top of Fig. 1). By sending current pulses through these crossed striplines [10], magnetic field pulses are generated which excit the Permalloy sample. The excitation consists of two orthogonal monopolar magnetic inplane field pulses with pulse amplitude B 0 , pulse length T and a delay of ½ T between the two pulses.
The resulting total magnetic field (cf. top of Fig. 1) is B(t)=(B 0 p(t-½T, T), B 0 p(t, T), 0) corresponding to CW excitation and B(t)=(B 0 p(t, T), B 0 p(t-½ T, T), 0) for CCW excitation, where p(t, T) describes the pulse shape: erf is the error function and k defines the rise time of the pulses. We define t=0 ps when the first pulse reaches 50 % of its full amplitude as the start of the excitation and t=3/2 T when the second pulse falls below 50 % of its full amplitude as the end of the excitation. Correspondingly the total excitation time is 3/2 T.
To determine the switching phase diagram the core polarity of the vortex structures is measured before and after the pulse excitation using scanning transmission x-ray microscopy at the MAXYMUS endstation at BESSY II, Berlin. For details of the experimental procedure, see Methods section. The experimentally monitored switching behaviour for both vortex core polarities is presented in Fig. 1a-d. The pulse length T is varied from 45 ps to 90 ps and the delay between the two pulses is always tuned to ½ T. This results in a total excitation time ranging from 67 ps to 135 ps. For a CCW sense of rotation (Fig. 1a, b) the pulse B x in x-direction starts before the pulse B y in y-direction. For this sense of rotation, it is found that the switching threshold for an initial vortex core 'down' is nearly twice as high as for an initial vortex core 'up'. This asymmetry in switching threshold is nearly independent of the pulse length and proves the undirectionality of the process for pulse amplitudes B 0 ranging from 15 mT to 30 mT. Due to symmetry reasons, a vortex core 'up' ('down') excited with a CW sequence is expected to have the same switching behaviour as a vortex core 'down' ('up') excited by a CCW sequence. This is checked and confirmed for the shortest pulse lengths by inverting the timing of the pulse sequence so that B y started before B x corresponding to a clockwise excitation.
Within the accuracy of the measurements the switching threshold found for core `up` at CW excitation ( Fig. 1c) is the same as for core `down` at CCW excitation ( Fig. 1b) and the threshold for core `down` at CW excitation ( Fig. 1d) is the same as for core `up` at CCW excitation (  It has to be pointed out that the switching thresholds in our experiments are found to be systematically lower by about 33 % compared to the simulations. This tendency, which has been observed before in the case of core reversal by static out of plane fields [17], seems to be a systematic phenomenon. It may be partly be explained by sample defects acting as nucleation centers for the reversal process and by thermal excitations, which are not accounted for in the simulations. Additionally, in micromagnetic simulations the treatment of a Bloch point which is 6 present during the reversal process is problematic and can increase the switching threshold as discussed by Thiaville et al. in detail in [17]. It might be of considerable technological interest that the experimental switching thresholds are systemically lower than expected from the simulations as the required fields can be realized easier.
Spin wave mediated vortex core reversal. To get more insight into the dynamics of the reversal process the out-of-plane component of the magnetization during the switching process is imaged by stroboscopic time-resolved scanning transmission x-ray microscopy and is compared with the result from a corresponding micromagnetic simulation (Fig. 2). Therefore the switching times as a function of pulse amplitude and pulse length (T=40 ps, T=60 ps, T=90 ps) are deduced from the micromagnetic simulations shown in Fig. 1 (for a diameter d 1 =500 nm) and for two additional Permalloy platelets with smaller diameter (d 2 =250 nm,  [17,18].

Discussion
In the final part we discuss the physical origin of the unidirectionality of vortex core reversal with very short orthogonal magnetic pulses. It differs fundamentally from the excitation with a rotating magnetic field with a well-defined frequency [13]. There the frequency can be tuned to a specific CW or CCW spin wave mode eigenfrequency and excitation only occurs, if both the frequencies and the senses of rotation of the external field and this specific spin wave mode are the same. Only if this is the case, energy is resonantly coupled into the system and vortex core reversal occurs. In contrast, the very short pulses used in the present paper show a several GHz broad frequency spectrum and thus frequency tuning to a specific CW or CCW vortex core eigenmode fails. Nevertheless, a unidirectional vortex core reversal is found for such a short and broadband excitation (cf. Fig. 1), in spite of the fact that our micromagnetic simulations reveal that the energy coupled into the system is nearly the same for CW and CCW excitation.
We suggest assigning the physical origin for this CW/CCW asymmetry for short pulse excitation to a coupling of the spin waves with the gyrotropic mode. Such coupling is present as otherwise the wellknown frequency splitting of CW and CCW rotating spin wave modes would not exist [21,22]. For vortex polarity 'up' (p=+1) the vortex gyromode only shows a CCW sense of rotation whereas for vortex core polarity 'down' (p=-1) only a CW sense of rotation is observed. This is also well-known and can be calculated analytically using the Thiele equation [19,20]. As the spin wave modes are coupled to the gyromode the response of the magnetization to CW or CCW excitation is different depending on whether the sense of rotation of the excited spin wave mode is the same or opposite to the sense of rotation of the gyromode. to t=100ps. In the CCW case this dip region is closer to the original vortex core and more localized than in the CW case.
Also, in the CCW case the core moves towards this dip region, while it moves away from it in the CW case leading to vortex core reversal for CCW excitation only. Fig. 4 reveal the differences in vortex dynamics for CW and CCW excitation (pulse length T = 60 ps) which lead to the observed asymmetric switching thresholds. The vortex has a core pointing 'up' (polarity p=+1) and a clockwise in-plane magnetization (circulation c=-1). Other combinations of c, p and sense of the exciting fields are not discussed since they can be derived from the two cases shown here by simple symmetry considerations (see notes [23]). For a CW pulse sequence, mainly the n=1/m=-1 mode (eigenfrequency f=7.1 GHz) is excited while for CCW excitation mainly the n=1/m=+1 mode (eigenfrequency f=9.5GHz) is excited. Due to the coupling with the vortex gyromode, excitation of the spin waves also leads to a movement of the vortex core. Both the simulated magnetization profile (Fig.4a, b) and the trajectory of the vortex core (Fig. 4c, d) are

Micromagnetic simulations in
clearly different leading to core reversal for CCW excitation only. The difference in switching threshold can be explained by these non-symmetric vortex core trajectories. Guslienko et al. [24] explained the formation of the out-of-plane magnetized dip region preceding the vortex core inversion by the introduction of an effective gyrofield caused by the moving vortex core. In our case, the different core trajectories (Fig. 4c, d) lead to different gyrofields and therefore different dip formations. For both CW and CCW excitation the formation of the dip is located left to the vortex core movement (Fig. 4c). But in the CCW case the dip forms closer to the core (Fig. 4d), is more localized and partly enclosed by the core's trajectory. This results in core reversal where the dip is transformed into a vortex core pointing down while the original core is annihilated. In the CW case, the average distance between the created dip and the original vortex is larger and the dip is not confined by the core's trajectory. The dip partly splits up into a vortex-antivortex (VA) pair followed by the annihilation of the original vortex core. However, this process does not occur across the whole thickness of the disc. In contrast, the splitting into VA pair starts at the bottom layer of the threedimensional simulation, and progresses only to approximately one third of the disc thickness.
Subsequently, the remaining dip in the upper simulation layers dissolves, starting at the top layer.
Thus the reversal process is not completed and the original vortex core 'up' remains. This is in contrast to previous investigations [4,5,7,8,13,14,18] where the formation of a VA pair led to core reversal.
In conclusion, we have experimentally demonstrated high-speed unidirectional vortex core reversal by orthogonal monopolar short pulses with pulse lengths between 90 ps and 45 ps. We observe a unidirectional switching behavior in a broad range of field amplitudes and pulse lengths, in qualitative agreement with simulation. The asymmetry in the response to clockwise and counter clockwise excitation can be explained by the coupling between the gyrotropic mode and the spin waves resulting in a different evolution of the magnetization for opposite rotation senses. We  Fig. 2 consists of the CCW pulse sequence shown in the figure and an additional CW pulse sequence at t=6.5 ns that resets the vortex core its initial 'up' polarity (not shown in Fig. 2). The stroboscopic experiment is repeated with a frequency of 75 MHz over more than 10 11 cycles to obtain the required signal to noise ratio.

Micromagnetic simulations.
Three-dimensional micromagnetic simulations are performed using the object-oriented micromagnetic framework (OOMMF) [16]. Unless otherwise stated, disc shaped platelets with a diameter of 500 nm and a thickness of 50 nm with cubic simulation cells of