Direct imaging of a zero-field target skyrmion and its polarity switch in a chiral magnetic nanodisk

A target skyrmion is a flux-closed spin texture that has two-fold degeneracy and is promising as a binary state in next generation universal memories. Although its formation in nanopatterned chiral magnets has been predicted, its observation has remained challenging. Here, we use off-axis electron holography to record images of target skyrmions in a 160-nm-diameter nanodisk of the chiral magnet FeGe. We compare experimental measurements with numerical simulations, demonstrate switching between two stable degenerate target skyrmion ground states that have opposite polarities and rotation senses and discuss the observed switching mechanism.


Introduction
Recent studies of nanoscale topological spin textures have changed the landscape of magnetism [1]. Magnetic skyrmions, which are characterized by topological indices, exhibit novel dynamics and are of interest as carriers of binary digits or logic elements in future spintronic devices [2][3][4][5]. The engineering difficulty and energy consumption of such devices would be reduced by the realization of zero-field skyrmion states [6,7]. Unfortunately, the stabilization of skyrmions in chiral magnets usually results from a combination of ferromagnetic exchange, anti-symmetric chiral Dzyaloshinskii-Moriya (DM) interactions and the presence of an externally applied magnetic field, with the first two coupling terms giving rise to a spin helix ground state [8]. However, a zero-field skyrmion is predicted to be possible in a nanopatterned structure because the magnetostatic energy then prefers a flux-closed state such as a magnetic skyrmion [9]. Such a spin configuration would offer the prospect of a purely electrical skyrmion-based device. As a result, there is great interest in the study of skyrmion physics in confined geometries [4,10,11]. In particular, theoretical work on nanodisks has suggested the possibility of forming an exotic topological texture termed a target skyrmion [10][11][12][13][14][15][16]. However, the target skyrmion has not yet been observed experimentally.
A major obstacle that hinders the imaging of the magnetic texture of a target skyrmion is the fact that the size of the nanodisk that supports it must be comparable to the size sk of the skyrmion itself, which is usually on the deep-submicron scale [10][11][12][13][14]. In a previous report [17], Lorentz transmission electron microscopy (TEM) was used to study chiral magnetic FeGe nanodisks with diameters d > 180 nm, that were approximately twice as large as sk ~ 80 nm for FeGe. The ground state in these nanodisks was a spin helix [17]. The study of smaller samples has been hindered both by sample preparation and by limitations in the spatial resolution of magnetic imaging techniques. For example, it is difficult to measure spin textures in nanostructures using Lorentz TEM because of the presence of Fresnel fringes, which are affected by local changes in sample thickness and composition [18,19]. Such unwanted contributions to the recorded contrast can be eliminated more easily when using the TEM technique of off-axis electron holography (EH), which has been used to provide direct access to electron-optical phase images of FeGe nanostructures with sizes of below 0.1 µm with nm spatial resolution [20,21]. Here, we use EH to study spin configurations in an FeGe nanodisk of diameter d ~ 160 nm. We identify zero-field target skyrmions and use a perpendicular applied magnetic field to reverse their polarities.

Magnetic configuration of a target skyrmion
A target skyrmion consists of a central skyrmion surrounded by one or more concentric helical stripes (Fig. 1) and can be regarded as a curved spin helical state [10,22]. From the center of the disk to its boundary, the out-of-plane component of the magnetic moment rotates by an angle that is larger than the value of for a typical skyrmion. The parameter / can then be used to characterize the number of half helical periods along any radial direction. As a result of the boundary confinement [10,[12][13][14][15], the outermost spin helix, termed as an edge twist, may correspond to an irrational fraction of a period. The value of is therefore not necessarily an integer multiple of . The topological charge of a target skyrmion is also not integer-valued [23], and is given by the expression Following definitions that are widely used to describe magnetic vortices in soft magnetic disks [24], we make use of two parameters -the polarity p and the    using EH after zero-field cooling (ZFC), which avoids possible metastable magnetic states [25]. When compared with the calculated in-plane magnetization shown in Fig. 1c, this image provides unequivocal evidence for a target skyrmion ground state.
The central region with circularity c = -1 is a complete skyrmion and is surrounded by an edge twist for which c = +1. The size of the central skyrmion is ~ 85 nm, which is close to the skyrmion lattice constant in thin film FeGe [26]. The polarity of the skyrmion cannot be measured directly, but can be inferred from its field-driven evolution, as discussed below. We label this target skyrmion Type 1. As mentioned above, target skyrmions with opposite polarity have the same energy in zero magnetic field. The same ZFC annealing procedure was used to obtain a magnetic state in which the central skyrmion corresponded to c = +1 (Fig. 2d). This target skyrmion is labeled Type 2. The size of the central skyrmion in the Type 2 is almost identical to that in the Type 1 configuration. However, the position of the central skyrmion deviates slightly from the center of the nanodisk. These differences are thought to result from pinning effects due to sample imperfections.

Field-driven polarity reversal of a target skyrmion
In order to switch between two degenerate target skyrmions, we applied a magnetic field H perpendicular to the disk. We define positive H as pointing upwards (marked with the symbol ⊙ in Fig. 3). For the Type 1 target skyrmion (Fig. 2c), an increase in the magnetic field shrinks the central skyrmion, while both c and p are unchanged during the magnetization process until a saturated magnetic state is achieved at large H (Fig. 3a). This observation is in agreement with the conventional field-driven evolution of a skyrmion in a two-dimensional thin film [3,8,26], in which a skyrmion is stable when an external magnetic field is applied antiparallel to the magnetization direction of its core. We conclude that the polarity of this target skyrmion is p = -1, as shown in Fig. 2c. Interestingly, the size of the central skyrmion exhibits a non-monotonic dependence on applied magnetic field and has a maximum value of 110 nm at H ~ 134 mT, which is almost 1.5 times sk ~ 81 nm (Fig. S4).
This behavior is associated in part with the flexibility of the edge twist, which results from its non-quantized topological charge [5,20]. A different process is observed for a Type 2 target skyrmion (Fig. 3b).  Fig. 3b).

Numerical simulation of magnetic-field-driven polarity reversal
Our experimental results demonstrate two-fold degenerate zero-field target skyrmions and their field-driven polarity reversal. However, they only measure the projected in-plane magnetic induction [21]. As the disk has a thickness of 90 nm, which is comparable to sk ~ 80 nm, a three-dimensional (3D) magnetization state may be supported [27][28][29]. We performed numerical simulations using a spin model for a 3D isotropic chiral magnet, in order to understand the spin arrangement and magnetic phase transitions in more detail. Parameters for the real sample were used in the simulation. Following a standard approach [12,13], we first determined the lowest energy magnetic state by comparing the energies of typical equilibrium states relaxed from different initial states. The results show that the target skyrmion always has a lower energy than any other magnetic state, indicating that the ground state is consistent with the experimental results (Fig. S5). As in the experiment results, the simulations followed the field-driven evolution of magnetic states starting from the two types of target skyrmion. For the Type 1 target skyrmion, the simulations show no switching during the entire magnetization process (Fig. S6). In contrast, a polarity switch is reproduced for the Type 2 target skyrmion. Fig 4a shows snapshots of the simulations during the switching process.
The applied magnetic field was extended further to the negative branch, allowing a complete hysteresis loop to be obtained, as shown in Fig. 4b. Just as in the experiment, a polarity reversal is identified at Hs. The transition field Hs in the simulation (~ 320 mT) is larger than that in the experiment (~ 220 mT). This difference is The agreement between experimental results and theoretical simulations allows us to obtain insight into the mechanism of stability of the target skyrmion. Previous theoretical investigations have proposed that both the demagnetization energy [29,30] and the magnetization variation normal to the disk plane [27][28][29] help to stabilize target skyrmions. These effects are confirmed by our analysis. The role of demagnetization energy is revealed in three ways. First, if the demagnetization energy is omitted, the target skyrmion cannot be achieved at equilibrium no matter what initial state is chosen (Fig. S5). Second, if we use the stabilized zero-field target skyrmion as the initial state and then turn off the demagnetization field, the target skyrmion is no longer the ground state (Fig. S5). In order to quantify the effect of the demagnetization field, we used different values of saturation magnetization to tune the demagnetization energy, while keeping other interactions unchanged. As before, the system was relaxed from different initial states and the energies of the final states were compared. Fig S8 shows that the helical state has lower energy than the target skyrmion when the demagnetization field is 0.4 times as small as the original value.
The variation in spin configuration along the z direction is identified directly in the simulation [27], as shown in Fig 4c. A slight twist of the magnetization at the top and bottom surfaces occurs because the presence of such a twist normal to the surface saves energy from the DM interaction along the z direction. In comparison, the lowest energy state for a 2D nanodisk is a complex helical state relaxed from a random initial magnetic configuration (Fig. S5).
Our numerical results confirm that two factors stabilize a zero-field target skyrmion, whose size is essentially determined by DM and FM couplings. Since the material parameters that are used here can be reduced to a small number of dimensionless variables, a scalar behavior is followed [31]. Our simulated results are then of universal significance for other helimagnetic materials. Moreover, skyrmion sizes in bulk are widely tunable from hundreds to tens of nm [32] and the temperature region of the skyrmion phase can be extended to room temperature [33]. The formation of a zero-field target skyrmion of much smaller size at room temperature is therefore anticipated.
The zero-field target skyrmion that we observe in nanodisks here is similar to the well-studied vortex in micron-sized elements of soft magnets [24]. Magnetic vortices have been investigated extensively, in the hope of building spintronic devices [34], because of the formation of four-fold-degenerate ground states with two polarities and two circularities. The present zero-field target skyrmion with a two-fold degeneracy can be an alternative to magnetic vortices in similar applications.

Conclusions
We have directly observed zero-field target skyrmions in a chiral magnetic material in a strongly confined nanodisk geometry using state-of-art off-axis electron holography. In the presence of an external magnetic field applied perpendicular to the nanodisk, the two types of target skyrmion can be switched. Our results are well reproduced and additional details are captured in numerical calculations. Our demonstration of the stability and switching of target skyrmions provides a solid physical basis for future skyrmion-based applications. The emergence of target skyrmions in other systems and their switching using different stimuli will be discussed in future works.