We first investigated the size effect of the nanostructures on magnetic structures. The simulations were carried out on a L ×   L square lattice with free boundary conditions. The zero-temperature equilibrium states for the lattices with different sizes were obtained, and some of the typical configurations were presented in Figure 1. Here a magnetic field B z = − 0.008   J F M (corresponding to ∼ −0.1 Tesla, as estimated with the magnetic moment ∼3.0   μ B obtained from density functional theory for Pd/Fe/Ir(111) in Ref. [40]) is applied along the −z-axis, which is sufficient for creating the skyrmion states here. In the following, we used an external magnetic field to generate the exotic magnetic kink textures in the nanostructures, and then we demonstrated an alternative way for creating kink by applying spin-polarized current.

As one may see in Figure 1A, a bubble domain arises in the square nanostructure at small lateral size L = 20 (∼10 nm). As the size increases to L = 24 (∼12 nm), a kink-like structure forms in the corner of the square nanostructure, surrounded by the ferromagnetic domain, as will be discussed below. For L = 50 (∼25 nm), it was observed that some half-skyrmion-like (meron-like) structures with topological charge | Q | ≈ 1/2 and elongated stripes (fractional skyrmion) with topological charge 0 < | Q | < 1 emerges at the edge of the lattice, together with a kink-like structure that appears in the corner. Further increasing the size, the magnetic states evolve into multi-domains composed of spiral domains, edge-merons, and skyrmion as seen in Figure 1D for the relatively large size L = 80 (i.e., ∼40 nm). It is interesting that the kink-like, meron-like, and skyrmion states tend to nucleate in the corner, the edge, and the inner region of the nanostructure, respectively. These are reasonable likely due to the different boundary constrictions on the formation of magnetic structures, which manifests the strong boundaries and geometric confinement effects in the nanostructures. It is noted that some similar edge states are also found experimentally and theoretically in diverse magnetic materials that host skyrmions with constricted geometry, as a consequence of the effects of geometric boundaries and confinements [37, 45].

Now we focus on the intriguing features of kink-like structures and analyzing their current-induced dynamics in this paper. For this, we adopted a small square shaped nanomagnet consisting of 24 × 24 square lattices (∼12 × 12 nm) for studying the kink-like structure, in the rest part of this work. We first tested the stability of the kink-like structure in Figure 1B once the magnetic field was removed. For this, the magnetic structure was relaxed for the equilibrium state by solving the LLG equation, in which we used the kink-like state shown in Figure 1B as the initial state, and set B z = 0 in the calculation. It was found that the kink-like structure remains stable without the assistance of an external magnetic field, as presented in Figure 2A. Note that the magnetic structure enclosed by the dash lines in Figure 2A is called Néel-type magnetic kink in this work, in which the magnetic moments undergo 180° rotation from the upward direction at its center to the downward direction in the periphery. The schematic magnetic configuration for a Néel-type kink is displayed in Figure 2D. For simplicity, we call the kink-like state as the kink state hereafter.

To explore the possible ground state for the nanomagnet of 24 × 24 square lattices at B z = 0, the system was initialized from a the paramagnetic state at sufficiently high temperature T, and cooled down gradually until it reaches a very low T under B z = 0, using Monte Carlo simulations with simulated annealing technique [17, 18, 44]. Then the system was further relaxed for the zero-temperature equilibrium state by solving LLG equation. As a result, a spiral structure forms as shown in Figure 2B. The numerical calculations indicate that the total energy of the spiral structure is a little smaller than that of the Néel-type kink structure. These results suggest the spiral structure to be the possible zero-field ground state, while the Néel-type kink structure to be a metastable state at B z = 0, other than the minimum-energy state.

To further estimate the stability of the metastable Néel-type kink state under B z = 0, we calculated the reduced local effective field b i ≡ B i e f f / | B i e f f | acting on the i-th magnetic moment m i . Interestingly, it was found in Figure 2C that the b i distribution and the magnetic kink state in Figure 2A are similar in spatial profile, and almost all the magnetic moments in the kink structure are oriented in parallel to the local field b i , making the metastable kink structure self-sustaining. In addition, it was noted that the anisotropy energy ℋ a n i = − A ∑ i ( m i ⋅ z ^ ) 2 contributes to the effective field in terms of − ( 1 / ℏ γ ) ( ∂ ℋ a n i / ∂ m i ) = ( 2 / ℏ γ ) A m i z e z . This suggests that the anisotropy plays an important role in guaranteeing the robustness of the metastable kink state, since the effective field ( 2 / ℏ γ ) A m i z e z   makes the stabilization of the out-of-plane magnetization component m i z [46, 47]. In this regard, we reckon that the kink structure is a robust state after the withdrawal of the magnetic field, due to the kink state is trapped in an energy valley in the configurational energy landscape [47].

In fact, one may take account of the kink structures from the prototypical Néel skyrmion, whose configuration can be viewed as a coplanar spiral with magnetic moments lying in a plane perpendicular to the xy plane [1, 38]: { m ( r ) = sin ( q ⋅ r ) e q + cos ( q ⋅ r ) e z | c o r e − u p ,   p = 1 m ( r ) = sin ( q ⋅ r + π ) e q + cos ( q ⋅ r + π ) e z | c o r e − d o w n ,   p = − 1 , (5) where the spatial variables of magnetization are defined in polar coordinates r = ( r ⁡ cos ⁡ φ , r ⁡ sin ⁡ φ ) , and the magnitude of spiral wave vector is | q | = π / R , with the direction denoted by unit vector e q = cos ⁡ φ e x + sin ⁡ φ e y . For the isolated Néel skyrmion configuration, the core magnetization points upward or downward, and smoothly changes to the opposite direction in the peripheral circle with radius R . We considered skyrmion structure with core-up (core-down) magnetization carries core polarity p = 1 ( p = −1), as the skyrmion with p = 1 depicted in Figure 2E. Here, R is used to define the radius of skyrmion, with 0 ≤ r ≤ R and the azimuthal angle 0 ≤ φ ≤   2 π for a single skyrmion.

To proceed, one may divide an isolated skyrmion configuration into four quadrants along the dash lines, as illustrated in Figure 2E. The magnetic structures in quadrant I, II, III, and IV can be described by Eq. 5, with the azimuthal angle 0   ≤ φ I ≤ π/2, π/2 ≤ φ II ≤ π, π ≤ φ III ≤ 3π/2, 3π/2 ≤ φ IV ≤ 2π, respectively. The kink may appear in one of the four possible corners, i.e., the upper left (UL), upper right (UR), lower left (LL) or lower right (LR) corner of the nanostructure in the simulations, as shown in Figure 2F. The LL, LR, UR, and UL kink structures correspond to the well-defined kinks residing in the quadrant I, II, III, and IV, respectively. In addition, we define the polarity of the kink with core-up (core-down) as p = 1 ( p = −1). Therefore, the kinks in the nanostructure can carry the core polarity p = ± 1, and can reside in one of the four possible corners, constituting 8 degenerate kink states. By comparison, the single skyrmion often appears in circular shaped nanodisks, and possesses only two degenerate states with core polarity p = ±1 [8, 48–50]. In this sense, the kinks in the square shaped nanostructure have the special feature of multiple degenerate states.

In this section, we studied the generation of kinks by injecting into the nanostructure an in-plane spin-polarized current pulse. Here, the direction of the CIP injection is defined by the angle φ with respect to the x-axis (see in Figure 3B), and φ is tunable. Note that in the following study, no external magnetic field was applied.

The simulations start from the initial ferromagnetic phase, with all magnetic moments aligning along z-axis at t = 0.0 ps (see Figure 3A). We first tested the effect of CIP on the kink dynamics, using a moderate current density j CIP = 0.1 κ CIP ∼ 1.4 × 10¹² Am⁻² with current duration t d u r = 250   τ ∼ 50 ps. Figure 3A shows the formation process of the UR kink for a typical case of φ = 10°. At the beginning ( t = 20 ps) of the current duration, the embryo of kink pattern emerges in the UR corner of the nanostructures. This pattern gradually enlarges, and becomes a rough kink at t = 50 ps. Subsequently, the current is turned off and the system evolves into an equilibrated kink state at t = 120 ps.

Further simulations demonstrated that the generation of kinks is sensitive to injection angle φ, with the simulated results summarized in Figure 3C. It was found that the UR, UL, LL, and LR quarters can also be created respectively for −42° ≤ φ ≤ 14°, 48° ≤ φ ≤ 104°, 138° ≤ φ ≤ 194°, and 228° ≤ φ ≤ 284°, while some bubble domain states form beyond these angles φ (see the while region in the Figure 3C). Note that the kinks with p = −1 presented here are generated from the initial ferromagnetic phase with magnetization along the z-axis. If the initial ferromagnetic phase is magnetized along the −z-axis, the lattices will evolve to the kinks with p = 1. Therefore, all the eight degenerate kink states can be created by tuning the direction of CIP injection with two different initial ferromagnetic orientations.

Next, we investigated the switching between these kinks using in-plane current pulse, which is fundamental to understand their dynamics properties. The current pulse ( j CIP = 1.4 × 10¹² Am⁻², current duration t d u r = 50 ps) with varying angle φ is exerted on the initial UR kink state with p = −1. Simulated results reveal that the kink can transfer from UR corner to the LR corner in a clockwise (CW) direction, driven by current with 116° ≤ φ ≤ 178°. However, it was noted that the UR kink cannot move to LL corner in a counterclockwise (CCW) direction or to diagonal UL corner with the adopted simulation parameters. Figures 4A–H show the dynamic process for the typical case of φ = 135°. We can see clearly that the upper part of the UR kink is pushed towards the right-edge of the nanostructures, and the lower part is simultaneously dragged to the bottom of the nanostructures (see Figures 4A–D). Although the entire kink pattern deforms largely in this process, it gradually turns in a CW direction and eventually moves to the LR corner at t = 50 ps (see Figures 4E,F). After that the current is switched off to zero for reaching the equilibrated kink state in the relaxation procedure. Moreover, simulations for the kinks with p = −1 indicate that their switching sequences can be summarized as follows: UR kink → LR kink → LL kink → UL kink with suitable angle φ , as shown schematically in Figure 4I. In this procedure, the switching sequences is in a CW direction, and the kink polarity has not been reversed by the CIP injection.

In this section, we presented an effective way to annihilate and create the kink by applying two successive CIP pulses, as the results shown in Figure 5. Our simulations start from the initial UR kink state with p = –1 in Figure 5A. We first found that the kink can be annihilated by applying a CIP current pulse, with a larger current density   j CIP = 2.8 × 10¹² Am⁻² and an appropriate injection angle φ 1 = 135°. Here two successive current pulses are used to annihilate and create the kink states respectively, with injection angle φ 1   = 135° for the first pulse, and tunable injection angle φ 2   for the second one. The current duration t d u r = 200  ps is considered for both two pulses, as shown in Figure 5C. Figure 5A displays the annihilation of the kink driven by the first current pulse with angle φ 1 , whose pattern rotates along a CW direction and disappears at the LR corner of the nanostructure, forming the ferromagnetic-like state at t = 40 ps. The ferromagnetic-like state remains at 40 ps ≤ t ≤ 200 ps, and we may see that most magnetic moments orientate along the z-axis in the ferromagnetic-like state, though some remaining magnetic moments align in the xy-plane at the LR and LL corners of the nanostructure.

More interestingly, the simulated results reveal that some other kink states can be created from the ferromagnetic-like state, by applying the second current pulse with varying injection angle φ 2 . The typical case for φ 2 = 230° is presented in Figure 5B, in which the domains first appear and gradually grow in the LL, and LR corners during 200 ps ≤ t ≤ 240 ps. The new embryonic kink forms at t = 280 ps and remains during 280 ps ≤ t ≤ 400 ps, and it finally evolves into a core-up kink in the UL corners after the relaxation with j CIP = 0. In this process, the kink has been annihilated and created by two successive current pulses, in which the switching between the kink and ferromagnetic-like states is reversible. This indicates the capability of reversible writing and deleting the kink states.

Further simulations generate the phase diagram for the equilibrated states at t = 520 ps as a function of φ 2 , as summarized in Figure 5D. It was noted that current pulses with 37° ≤ φ 2 ≤ 55°, 143° ≤ φ 2 ≤ 145°, and 320° ≤ φ 2 ≤ 325° may create the kinks with p =   −1 in the UL, LL, and UR corners of the nanostructure, while current pulses with 226° ≤ φ 2 ≤ 238°, 252° ≤ φ 2 ≤ 253° ∪ 314° ≤ φ 2 ≤ 315°, and φ 2 = 268° generate the kinks with p = 1 in the UL, LL, and UR corners, respectively. In addition, some bubble or spiral states form beyond these angles φ (see the striped-pattern and grid-pattern region in Figure 5D), with some typical configurations shown at the bottom of Figure 5D. Regarding early investigations on magnetic vortex, it was demonstrated that the vortex polarity reversal may be triggered by a CIP pulse through the formation of a vortex-antivortex pair [51], or by an alternating CIP through the resonant dynamics [52]. However, for the skyrmion in chiral magnets, CIP is usually used for displacing the skyrmion, while cannot change the skyrmion polarity in the dynamics [8, 41, 42]. In this sense, the core polarity reversal of kink can be achieved by CIP pulse, which is analogous to that of vortex system [51, 52]. From the simulation results and analyses in Generation and Switching of Kinks by In-Plane Current Pulse and Annihilation and Creation of Kinks by In-Plane Current Pulses, one may create, annihilate, displace, and reverse the kink on demand by adjusting the CIP pulse injections.

In this section, we investigated the core polarity reversal of the kinks with out-of-plane current, which is also a fundamental issue for understanding their dynamics properties. One may build a magnetic tunnel junction (MTJ) or spin valve to locally address the kinks [5, 53]. As schematically shown in Figure 6A, a typical MTJ structure consists of the top and bottom two layers of ferromagnet, which is separated by an ultrathin spacer of insulator. The top ferromagnetic layer is a free layer which presents a kink state, while the bottom one is a fixed layer with the magnetization M fix fixed along the z direction. When a spin current is injected into the bottom fixed layer along the z direction, the spin is polarized along z direction. The polarized spin current then flows through the insulating layer to the free layer, acting on its magnetizations with CPP-type STT, described by Eq. 4. One may reverse the polarization direction of the spin current, by injecting the spin current into the top free layer along the −z direction. Here we defined the current density   j CPP > 0 and   j CPP < 0 in Eq. 4 corresponding to the injection of spin current flowing along z and −z direction, respectively. Our simulations start from an initial UR kink with polarity p = −1 as a general representative. Note that the CPP-type STT induced by current j cpp < 0 tends to align the magnetic moments in the free layer along the −z direction, thus the kink may be reversed to its image structure with polarity p = 1. Contrarily, the kink with polarity p = 1 can be reversed, once the spin current is injected along the z direction.

We first studied the dynamics of the kink driven by various current densities j CPP and duration time t d u r , and the phase diagram for the equilibrated states obtained after the current pulses was summarized in Figure 6B. The results show that the kink is immovable under low current densities | j CPP | < 0.04 κ CPP (∼3.4 × 10¹² Am⁻²), due to the weak STT strength. It was seen that the kink may be pulled at current densities larger than the critical value | j CPP | = 0.05 κ CPP (∼4.3 × 10¹² Am⁻²). At an intermediate current density j CPP   ∼ −4.3 × 10¹² Am⁻² and duration time   t d u r = 160 ps − 180 ps, the kink moves in a CW direction to near LR corner of the nanostructure, accompanying with the kink polarity reverse. For the current pulse with a relatively large current density 0.06κ_CPP ≤ | j CPP | ≤ 0.11 κ CPP (i.e., 5.1 × 10¹² Am⁻² ≤ | j CPP | ≤ 9.4 × 10¹² Am⁻²) and duration ∼20 ps ≤ t d u r ≤ ∼ 60 ps, it was found that the kink polarity is also reserved, while the UR kink shifts diagonally to the LL corner of the nanostructure. It was also seen in the phase diagram that the large   j CPP and large t d u r usually lead to the bubble domains, and some spiral states appear at 0.06κ_CPP ≤ | j CPP | ≤ 0.07κ_CPP with a short pulse duration.

To further investigate the repeatability of polarity reversals of kink driven by CPP pulses, we took an examination on injecting some discontinuous current pulses to the system, as presented in Figure 7A. The insert shows the kink states at some typical time before and after the pulse injections in the evolutions. It was observed that all the four CPP pulses reverse the kink polarity, and the pulse with spin current j CPP < 0 ( j CPP > 0) is used to reverse the kink with polarity p = −1 ( p = 1). The first and second CPP pulses with j CPP   = ± 4.3 × 10¹² Am⁻² and t d u r = 160 ps rotate the kink in CW and CCW directions respectively (see state transitions ①   →   ②   →   ③ ), which is a reversible switching. Note that the CW and CCW rotational directions here depend on the core polarity of the pre-evolutionary kink. The third and fourth CPP pulses with j CPP = ± 0.07 κ CPP ∼ ± 6.0 × 10¹² Am⁻² and t d u r   = 40 ps move the kink to the diagonal corner of the nanostructure (see state transitions ③ →   ④   →   ⑤ ), which is also repeatable. For tracking the variations of the magnetic structures, the out-of-plane magnetization M z = ∑ i m   i z / N is used to characterize the reversal of kink, where N is the total number of magnetic moments. Although some fluctuant variations appear in the M z − t curve in Figure 7B, it is clear that M z flips to the opposite direction after every current pulse, in response to the core polarity reversal of the kinks. Therefore, we may use the CPP current to reverse the kink polarity and displace their position.