Formation of skyrmion and skyrmionium in confined nanodisk with perpendicular magnetic anisotropy

Magnetic skyrmions are nano-scale spin configurations with topological protection, which have vortex-like spin texture [1, 2]. Different from traditional ferromagnetic (FM) states, the nontrivial topological properties make skyrmions exhibit unique physical features, like potential small size of tens of nanometers, high stability and easy manipulation. Therefore, skyrmions are considered to have great potential in next-generation data storage with high-density, low-energy and non-volatile memory [3, 4]. Notably, racetrack memory is one of the most promising applications [5], which could realize three-dimensional information storage by correlating the presence or absence of skyrmions with the binary data bits '1' or '0' [6, 7]. In this assumption, the data encoding of '0' is based on the quantization of distances between adjacent skyrmions on a track. However, even though the topological structure of skyrmions is stable, they still have high mobility and will drift, as a whole, under the influence of thermal fluctuations, making it impossible to maintain their original positions. Additionally, there are significant challenges in fabricating artificial pinning centers at the nanoscale to solve this problem [8].

Previously, people have studied the topological structure of skyrmionium [9–14], which is generated by elastically coupling a skyrmion core and its surrounding domain wall under the action of magnetostatic repulsion [15]. Skyrmionium is also called 2π-state skyrmion and target skyrmion [16–18] with zero topological charge [11, 19]. Moreover, the skyrmion and skyrmionium have been able to be distinguished experimentally [17]. Thus, skyrmion and skyrmionium could be treated as data bit '1' and '0', instead of the aforementioned presence and absence of skyrmions. This assumption does not require a fixed distance between adjacent data carriers because '1' and '0' will correspond to the two spin configurations one-to-one. Therefore, the formation of skyrmion and skyrmionium and the transition between them are of great importance [20, 21]. Additionally, it has been reported that in finite-size magnetic systems, the transition promoted by the boundary has a much lower energy barrier than other energy paths [22]. Therefore, it is interesting to study their transition in confined geometry.

Here, with micromagnetic simulations, we investigate the formation of skyrmion (Sk) and skyrmionium (Skium) by varying the effective anisotropy constant (K) in nanodisks with different radius (R) and propose an anisotropy-induced method to realize their transition.

In order to control the size and dynamic properties of skyrmions, many studies have been carried out in magnetic thin film/heavy metal heterostructure and multilayer structures [23–26], where the Dzyaloshinskii–Moriya interaction (DMI) [27, 28] and the perpendicular magnetic anisotropy (PMA) could be controlled by adjusting the film thickness or composite structure. As shown in figure 1, our model is an ultrathin FM CoFeB nanodisk grown on a nonmagnetic heavy metal (HM) Pt layer. The thickness of FM and HM layer are both fixed at 2 nm.

Zoom In Zoom Out Reset image size

Interfacial DMI is usually induced by the broken inversion symmetry at the interface and the strong spin–orbit coupling (SOC) [29] in the HM layer. It is a kind of indirect exchange between the two adjacent magnetic atoms in HM layer. For the CoFeB/Pt bilayers, the value of interfacial DMI constant depends on the film thickness and is found to saturate at 0.45 mJ m⁻² when Pt thickness is larger than ~2 nm [30]. Thus, in this model, the DMI constant is chosen to be 0.4 mJ m⁻².

The effective anisotropy constant K can be expressed by equation (1) as follows [31]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle K={{K}_{{\rm bulk}}}+\frac{{{K}_{{\rm s}}}}{{{d}_{{\rm FM}}}}\nonumber \end{align} \tag{ 1 }$$

where K_bulk is the magneto-crystalline anisotropy, d_FM is the thickness of FM layer, and K_s is the surface anisotropy, which depends on the interface effect including crystalline lattice symmetry breaking, electronic bonding, and lattice mismatch [32]. In ultrathin films where d_FM is sufficiently small, the second term dominates and causes K value to be positive, so the system exhibits PMA. From previous studies, it has been reported that the PMA can be adjusted by external stimuli such as electric field [33] or strain field [34], which to some extent, provide practical possibilities for our theoretical simulations.

An open source micromagnetic simulation software, MuMax3 [35], is used to conduct the numerical simulations. The magnetization dynamics can be described by Landau–Lifshitz-Gilbert (LLG) equation [36]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{{\rm d}{\bf m}}{{\rm d}t}~=-\gamma {\bf m}\times {{{\bf H}}_{{\rm ef\,\!f}}}-\alpha {\bf m}\times \frac{{\rm d}{\bf m}}{{\rm d}t}\nonumber \end{align} \tag{ 2 }$$

where γ is the gyromagnetic ratio, α is the Gilbert damping constant, m is the unit vector of magnetization and H_eff is the effective magnetic field, which is determined from the variational derivative of system total energy E_total, which is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{E}_{{\rm total}}}=&\,\int{{\rm d}{\bf r}}\left[ ~{{A}_{{\rm ex}}}{{(\nabla \cdot {\bf m})}^{2}}+{{D}_{\operatorname{int}}}{{\left[ {\bf m}(\nabla \cdot {\bf m})-({\bf m}\cdot \nabla){\bf m} \right]}_{{\rm z}}}\right.\nonumber \\&\left.+K{{({\bf m}\cdot {{{\bf e}}_{{\rm z}}})}^{2}}-\frac{1}{2}{{\mu }_{0}}{{M}_{{\rm sat}}}{\bf m}\cdot {{{\bf H}}_{{\rm dm}}} \right]\nonumber \end{align} \tag{ 3 }$$

where the first term is the Heisenberg exchange energy, the second term is the DMI contribution, the third term is the PMA part, and the last term is the demagnetisation energy.

Here, our sample is discretized into cubes with a side length of 2 nm, which is less than the exchange length λ_ex of 3.24 nm [37]. Parameters are derived from a real experiment [30] and the key inputs are as follows: the saturation magnetization M_sat  =  1.23  ×  10⁶ A m⁻¹, the exchange constant A_ex  =  1  ×  10⁻¹¹ J m⁻¹, the interfacial DMI constant D_int  =  0.4  ×  10⁻³ J m⁻², and the Gilbert damping constant α  =  0.012. The effective anisotropy constant (K) and the nanodisk radius (R) are variables in this model.

Firstly, in order to explore different situations, the system is started as three fixed initial states (random, uniform, vortex) under some values of K in a nanodisk with R  =150 nm. And then by varying K, the three common initial states are observed to relax to different equilibrium states. The corresponding results are shown in supplementary figure S1 (stacks.iop.org/JPhysD/53/195001/mmedia) and K ranges from 0.1 to 1.1 MJ m⁻³ in steps of 0.2 MJ m⁻³. It is found that the equilibrium states are more abundant and diverse by using vortex as the initial state. And it is generally believed that the vortex and the skyrmionic state have many similarities in magnetic configuration, for example, the vortex core with perpendicular magnetization and its centrally symmetric structure are suitable origins for Sk and Skium nucleation. So, the vortex initial state is chosen for further in-depth study. Additionally, the formation of vortex states has been widely reported experimentally and theoretically and the relevant means are relatively mature [38, 39].

Given above results, K and R are set at 0.8 to 1.05 MJ m⁻³ and 50 to 300 nm, as shown in figure 2(a), aiming to capture Sk and Skium. When K  =  0.8 MJ m⁻³, the demagnetization effects which favor in-plane magnetization dominate, making the system present in-plane magnetization. And the larger the nanodisk radius, the stronger the demagnetization effects, so the smaller the vortex core size. As K continues to increase, the PMA compete with the demagnetization effects, thus starting to present out-of-plane magnetization.

Zoom In Zoom Out Reset image size

When R  =  50 nm and K is between 0.85 and 0.95 MJ m⁻³, Sk can directly form under the influence of PMA, as shown in figure 2(b). The size of Sk gradually decreases with increasing K, and eventually strong PMA destroys the Sk and leads to a uniform arrangement along  +z direction. It should be noted that the obtained skyrmion is a twisted skyrmion [40, 41] as shown in figure 1(a), which can be attributed to the fact that the initial vortex state is structurally similar to a Bloch-type skyrmion and the PMA is applied [42].

Furthermore, when R  =  250 nm and K  =  1.0 MJ m⁻³, Skium can be obtained, as shown in figure 2(c). When R  >  100 nm, and K  >  0.95 MJ m⁻³, a multi-domain state appears because of competition between PMA and exchange interaction. It can be predicted according to the trend of spin configuration changes that such multi-domain states will also gradually change into uniform states as K continues to increase because the strong PMA remains dominant. In addition, it is worth noting that all configurations shown in figure 2 are stable states with system energies minimum.

Now that we have captured Sk and Skium with given parameters in figures 2(b) and (c), next, we focus on the ranges for their existence. Topological charge Q [43, 44] is used to plot a phase diagram of different topological structures. The topological charge is given as follows

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle Q=-\frac{1}{4\pi }\iint{{\bf m}}\cdot (\frac{\partial {\bf m}}{\partial x}\times \frac{\partial {\bf m}}{\partial y}){\rm d}x{\rm d}y\nonumber \end{align} \tag{ 4 }$$

where x and y  are the in-plane coordinates. Generally, Q of vortex is  ±0.5, while Q of Sk and Skium are  ±1 and 0, respectively. Thus, we calculate the topological charge and plot a diagram of R–K phase, as shown in figure 3.

Zoom In Zoom Out Reset image size

The phase diagram can be divided into three regions (I–III). In region I, PMA dominates and induces uniform magnetic moments along z direction, thus Q is approximately zero. In region II, the system exhibits a vortex state with Q  =  −0.5 initially. With increasing K, alternating positive and negative out-of-plane magnetic moments gradually form on the edge, meanwhile, Q increases slightly and finally reaches about  −0.55. In addition to these two regions with distinct topological structures, the rest is defined as region III with varied Q. From region III, it can be seen that Sk exists in region A, region B and region C, while Skium only exists in region D. In particular, although the polarities of Sk in region B and region C is opposite, the K values are approximately the same. Therefore, PMA favors the formation of Sk when K is 1.00–1.05 MJ m⁻³. In addition, for Sk in region C and Skium in region D, it can be observed that the nanodisk have same R (about 240–290 nm), which is favorable for the transition between Sk and Skium. Different from previous studies involving two variables, here we can obtain Sk or Skium by adjusting only one variable K.

In order to investigate the relationship between skyrmionic configurations and magnetic anisotropy concretely, two different fixed R at 50 nm and 260 nm are chosen, corresponding to two vertical blue dashed lines in figure 3. In this part, the system is still allowed to start as a vortex state characterized with a given K value, and by further varying this value, it can be observed that the relaxed configuration switch from one state to another different state. The phase diagram of out-of-plane magnetization component m_z(x) along nanodisk diameter at different K and corresponding magnetization configurations are shown in figure 4.

Zoom In Zoom Out Reset image size

For R  =  50 nm, the system presents an incomplete skyriom (iSk) state when K  <  0.86 MJ m⁻³. With increasing K, magnetic moment along  +z direction gradually expands toward inside and replaces the in-plane magnetic moment, leading to a decrease in the thickness of domain wall. Subsequently, iSk converts to Sk. The Sk core size increases gradually at the beginning and then decreases, finally, the core disappears when K reaches 0.96 MJ m⁻³. With further expansion of edge areas which contain out-of-plane magnetic moments, the system exhibits uniform magnetization.

For R  =  260 nm, Sk can be easily distinguished from Skium around the K ~ 1.02 MJ m⁻³. The Skium size does not change obviously, but the Skium core size first increases with increasing K, and then decreases when K approaches the critical value, and finally results in a transition to Sk. Further increases of K causes sharp decreases in Sk size. Furthermore, we notice a big difference in m_z when K is 1.05 or 0.99 MJ m⁻³, indicating that Sk or Skium has developed into an irregular configuration.

The transition between Sk and Skium can be observed in the nanodisk with R  =  240–290 nm, because in this range both Sk and Skium can be formed by varying K. However, when R is relatively small (such as R  =  50 nm), the edge effect limits the formation of Skium so that only skyrmion can be formed, thus making the transition impossible. It is noteworthy that this anisotropy-induced transition from Sk to Skium is similar to that induced by a magnetic field [45]. However, compared with the large value of applied field exceeding 100 mT, K causes different stable states with much smaller changes. This provides theoretical possibility to control the transition between Sk and Skium by tuning the magnetic anisotropy.

With micromagnetic simulations for an ultrathin confined nanodisk, we investigate the influences of effective anisotropy constant K and nanodisk radius R on the formation of Sk and Skium. It is demonstrated that both K and R play important roles; when K is about 1.00–1.05 MJ m⁻³, PMA favors the formation of Sk with two polarities most; Sk and Skium mainly appear when R is about 240–290 nm due to the competition between exchange interactions and anisotropy contributions. Besides, transition between Sk and Skium can be achieved by fine tuning the magnetic anisotropy. This finding not only provides a fundamental understanding of the formation and transition of skyrmionic equilibrium states in magnetic thin films but enlightens skyrmionics studies for other materials.

The authors gratefully acknowledge the Natural Science Foundation of Zhejiang Province (LR18E010001), the National Natural Science Foundation of China (U1704253, U1908220), and the Fundamental Research Funds for the Central Universities (N160208001).