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Elongated Skyrmion as Spin Torque Nano-Oscillator and Magnonic Waveguide

Spin Torque Nano-Oscillator has been extensively studied both theoretically and experimentally in recent decades due to its potential applications in future microwave communication technology and neuromorphic computing. In this work, we present a skyrmion-based spin torque nano-oscillator driven by a spatially uniform direct current, where the skyrmion is confined by two pinning sites. Different from other skyrmion-based oscillators that arise from the circular motion or the breathing mode of a skyrmion, the steady-state oscillatory motions are produced by the periodic deformation of an elongated skyrmion. Through micromagnetic simulations, we find that the oscillation frequency depends on the driving current, the damping constant as well as the characteristics of pinning sites. This nonlinear response to direct current turns out to be universal and can also appear in the case of antiskyrmions, skyrmioniums and domain walls. Furthermore, the elongated skyrmion possesses a rectangle-like domain wall, which could also serve as a magnonic waveguide. Utilizing the propagation of spin waves in this waveguide, we propose a device design of logic gate and demonstrate its performance.

Compared with the circular motion and breathing mode of skyrmions, the oscillation produced by the third mechanism is much less studied. In this work, we present a nanotrack with two pinning sites, in which an elongated skyrmion can be confined. Within a certain range of the driving current density, the magnetic potential well induced by the pinning regions as well as the competition between the spin torques and inherent damping will yield a sustained oscillation of magnetization, where the frequency can reach a few GHz. The oscillation is also caused by the deformation of a skyrmion, but the driven current is spatially uniform, which is different from the method reported in Ref. [26]. On the other hand, both theoretical and experimental results have shown that a domain wall naturally acts as a magnonic waveguide, where spin waves can be excited easily and propagate with low dissipation [60][61][62][63]. In our model, the elongated skyrmion involves a rectanglelike domain wall so that the spin wave can propagate in this channel. Furthermore, we design a logic gate based on this concept, which realizes the basic operations of logic AND, OR and XOR functions.

II. MODEL AND METHODS
Here, we consider a ferromagnetic nanotrack grown on a heavy metal that provides sufficiently strong Dzyaloshinskii-Moriya interaction (DMI) [64,65] to stabilize chiral skyrmion-like structures and allows the spin Hall effect when an electric current flows through it due to the strong spin-orbit arXiv:2203.02969v1 [cond-mat.mes-hall] 6 Mar 2022 interaction. To obtain an elongated skyrmion, two pinning sties are placed in the nanotrack, which is constructed by the local perpendicular magnetic anisotropy (PMA) reduction and indicated by the pink region of width w in Fig. 1(a). The PMA constant in pinning sites is K p = 0.55 MJ·m −3 and the width is w = 10 nm in the following simulation, except for special indication. Besides, the distance d between the pinning site and the end of the nanotrack is fixed to be 30 nm. In general, this type of pinning sites can be created by locally applying an external voltage because of the voltagecontrolled-magnetic-anisotropy (VCMA) effect [66] or by using ion irradiation in experiment [67]. Recent work has also shown the static properties and current-induced dynamics of square-shaped skyrmions in a magnetic thin film with orthogonal defect lines formed by the reduction of PMA [68].
In this work, we use the public code project objectoriented micromagnetic framework (OOMMF) [69] based on the Landau-Lifshitz-Gilbert (LLG) equation to study the dynamics of magnetization, The three terms on the right side of the above equation describe the gyromagnetic precession, the dissipation originating from the nonzero Gilbert damping, and the damping-like spin torque induced by the driving current, respectively. It should be noted that the last term will be removed when discussing the propagation of spin waves in the elongated skyrmion. Here, m denotes the local normalized magnetization, γ is the gyromagnetic ratio and α is the damping constant. H eff represents the effective magnetic field associated with the total energy density, where A is the Heisenberg exchange stiffness, K is the PMA constant and ε d denotes the demagnetization energy density. The last term arises from the asymmetric exchange interaction DMI, which determines the configuration and chirality of the spin structures. For example, the Bloch-type skyrmion is stabilized by the bulk DMI ε DMI = Dm · (∇ × m), the Néeltype skyrmion is formed in the thin film with the interfacial , while for the magnetic system with crystallographic class D 2d that is used to stabilize an antiskyrmion, the DMI energy density is given by ε DMI = Dm · (∂ x m ×x − ∂ y m ×ŷ) [70]. Meanwhile, the sign of DMI constant D indicates the chirality of skyrmionlike structures, i.e., the left-handedness or right-handedness.
In this work, we focus on the Néel skyrmion, skyrmionium and antiskyrmion, which are realized by considering different DMIs, either the crystallographic class or the strength.
The primary magnetic parameters are adopted as in Ref. [52]: saturation magnetization M s = 580 kA·m −1 , exchange stiffness A = 15 pJ·m −1 , PMA constant K = 0.8 MJ·m −3 in the clean magnetic area without pinning sites. The DMI constant is set to be 3.5 mJ·m −2 , at which both the skyrmion and the skyrmionium are the metastable states corresponding to the minimal values in the energy profile [71]. To study the skyrmion oscillation, the rectangular mesh of 1.0 × 1.0 × 0.4 nm 3 is used to discrete a ferromagnetic film with the dimension of 200 × 100 × 0.4 nm 3 and the Gilbert damping coefficient is 0.1. However, for the discussion about the propagation of spin waves in an elongated skyrmion, the model size is expanded to 400 × 200 × 1 nm 3 and the mesh size becomes 1×1×1 nm 3 . The Gilbert damping constant is reduced to 0.05 to suppress the dissipation of spin waves in the simulation.

III. RESULTS AND DISCUSSION
The initial spin configuration of an elongated skyrmion is generated by perpendicularly injecting a spin-polarized current pulse into the central region indicated by the dashed box in Fig. 1(a), and then relaxed for a sufficiently long time [72]. Fig. 1(b) and (c) respectively demonstrate the magnetization structure of an elongated skyrmion and the corresponding spatial distribution of the topological charge density defined as ρ = m · (∂ x m × ∂ y m). When the driving current is applied to the heavy metal along x axis, the elongated skyrmion will deform periodically and generate a stable spin oscillation that can be detected by the magnetoresistance effect. To quantify this dynamical behavior, we plot the temporal evolution of m z as shown in Fig. 1(d), where · · · represents the spatial average of a component of magnetization over the entire nanotrack, and calculate the oscillation frequency by using the fast Fourier transform (FFT). From the frequency spectra in Fig. 1(e), it is noted that some higher-order harmonics are also excited, in addition to the fundamental frequency. Therefore, the profile of m z is not a pure sine-like curve, as illustrated in Fig. 1(d).
To understand the skyrmion oscillation, eight snapshot images of the time-dependent spin configuration and the corresponding spatial distribution of the topological charge density within one period are shown in Fig. 2. It is seen that, driven by the spin-polarized current, the top domain wall (i.e., the red stripe) periodically oscillates, similar to an ordinary rope, while the bottom one just has a slight drift. This phenomenon is strongly dependent on the orientation of the local magnetization and the polarization direction of the driving current. Fig. 2(c) shows the distribution of the in-plane magnetization component and the schematic diagram of the current-induced damping-like torque τ d ∝ m × (p × m) acting on the local magnetization, where p is the unit polarization vector. Considering the fact that the local magnetization tends to align in the direction of polarization in the presence of spin torque and that the angle between p and m in the upper domain wall is greater than that in the bottom one, the magnetic moments in the former case are more fragile to reverse. Therefore, the top domain wall severs as a free end and the bottom one is a pinned section during the oscillation. Altering the chirality of the skyrmion or the polarization direction of the driving current can switch the free end of this oscillation [72]. Here, the method in Ref. [73] is used to describe the magnetization dynamics, where the overall deformation of a spin configuration is reflected in the motion of half skyrmions. In Fig. 2(b), the element marked by the yellow dashed circle is formed by the accumulation of negative topological charges, which can be regarded as a half skyrmion with negative fractional topological number Q = (1/4π) ρdxdy. In the left pinning site, this element moves upward along the border of the pinning region, and gradually separates from the pinned part at the bottom. Due to the repulsive force from the upper boundary of the nanotrack, it cannot continue to move upwards, but moves to the right pinning site under the action of a spin-polarized current. Now, we turn to tracking the upper element marked by the green dashed circle in the right pin- ning site. Since the force provided by the driving current is not enough to overcome the potential barrier of the pinning region, the topological element is confined in this area and moves downward to merge with the lower part of negative topological charge. Afterwards, it moves up analogous to the left marked element. However, due to the approach of the element marked by yellow circle, positive topological charges are induced between the two negative topological charge accumulation regions, corresponding to the bend of the domain wall string between two protrusions as shown in Fig. 2(a). Consequently, the upward motion of the element marked by the green circle will be suppressed, and it tends to compensate with the positive charges as indicated by the purple dashed circle. The element from the left side finally reaches the right pinning site and becomes the element marked by the green circle for the next cycle. This process occurs repeatedly and produces a stable spin oscillation without any decay. Figure 3(a) demonstrates the relationship between the oscillation frequency and the driving current density. It is found that there are three distinct regions in the whole working window, as illustrated by different color backgrounds. When the driving current density j is smaller than the first critical value j 1 , the elongated skyrmion is completely pinned, where the initial m z oscillation gradually decays to 0, indicating that the driving force is insufficient to overcome the intrinsic force due to the topological protection and the damping force. However, if the current density is larger than the second critical value j 2 , the elongated skyrmion will depin from the pinning sites and be annihilated around the upper boundary of the nanotrack. This motion behavior is similar to that of a skyrmion in a homogeneous nanotrack [see Fig. 3(d)].
In the range of j 1 < j < j 2 , the steady skyrmion oscillation occurs, and the frequency increases with the increasing j. The dependence of oscillation on the damping coefficient α is also shown in Fig. 3(b). There are still three phases, but, on the contrary, the pinned and annihilation states occur in the high-damping and low-damping areas, respectively, and the frequency is inversely proportional to α in the oscillation region. For a rigid topological object driven by a spin-polarized current, the Thiele equation is given by [74] where the first term represents the Magnus force with G = (0, 0, G) and G = 4πQ, D is the dissipative tensor with D ij = δ ij D, F j denotes the current-induced driving force proportional to the current density j and F r is the repulsive force imposed by the boundary. If this driving current is applied along x axis with p = −ŷ and F r = F rŷ , we obtain the following velocity relationship: When the topological body moves along the upper boundary of the nanotrack, the Magnus force is balanced by the repulsive force from the edge, v y = 0 and v x = F j /(αD). It should be noted that the velocity is proportional to the driving current density and increases as the damping constant decreases. Therefore, a fast motion of the marked element in Fig. 2(b) requires a large current density and low damping, which can reduce the motion period and hence increase the oscillation frequency. Here, we continue to investigate the skyrmion oscillation driven by a spin-polarized current as a function of the PMA value K p and the width w of the pinning sites. As shown in Fig. 4, for each of driving current density j, the increase in K p yields a reduction in the oscillation frequency. It is also noted that the working window of j is wider in low-PMA region, which is attributed to the strong pinning force caused by the large PMA difference between the pinning site and the clean region. On the other hand, the response of this oscillation to the width of pinning sites is non-monotonic, and there is an optimal size that corresponds to the maximum frequency. For a narrow pinning region, the strength of the pinning force is not strong enough to protect the skyrmion from annihilation. However, if the width w is too large, most part of the skyrmion will be pinned, thus suppressing its deformation and oscillation. Consequently, to obtain a high performance of the proposed skyrmion oscillator, the pinning site with low PMA and an appropriate optimal width w should be chosen. In addition, it should be mentioned that this oscillation can also occur in the case of antiskyrmions, skyrmioniums and domain walls (see supplemental movies). Furthermore, such an elongated skyrmion is also an ideal magnonic waveguide, in which the spin wave travels along the rectangle-like domain wall. To excite spin waves, one microwave antenna is placed on the center of the nanotrack along y axis, which is realized by locally applying an alternating magnetic field in the simulation. We calculate the fluctuation in x component of the magnetization δm x = m x (t) − m x (0) to demonstrate the spectra of spin waves in real-space-time domain intuitively, where m x (t) and m x (0) are the instant and the initial magnetization, respectively. First, the excitation field is applied to an area of 4 × 50 nm 2 , i.e., the top half of the nanotrack indicated by the dashed box in Fig. 5(a). It is seen that the excited spin waves propagating in two directions are nonreciprocal due to the presence of DMI [60,61] and spread successfully from the upper branch to the lower branch along the domain wall. Extending the antenna to cover the nanotrack width [see Fig. 5(b)], we find that the spin waves generated on the two branches of the elongated skyrmion are asymmetrical. Figure 6(a) shows the distribution of the energy density, where the white channel corresponds to the rectangle-like domain wall of an elongated skyrmion with the maximum energy density. Excited by a low-frequency microwave stimuli, the magnetic moments in this region is easier to precess than that outside of the domain wall, thus confining spin waves in this narrow channel and forming a waveguide. In addition, considering that the directions of the magnetization or the effective field in the upper and lower branches are different [see Fig. 2(c) and Fig. 6(b)], one can understand why the propagation of excited spin waves is asymmetrical in Fig. 5(b).
To further obtain the whole picture of the characteristics of wave propagation, the dispersion relations along two branches are also calculated, as shown in Fig. 6(c)  Ref. [60].
One application of the magnonic waveguide based on an elongated skyrmion is logic computing. As shown in Fig. 7(a), the proposed device consists of two inputs, where sinusoidal fields are applied to excite spin waves, and one output to detect the arriving spin waves. Note that the digital data 0 (or 1) at the input is distinguished by turning on (or off) the excitation field, while for the output this information is encoded in the net amplitude of spin waves by setting a threshold value A c . Taking the damping into account, spin waves gradually dissipate during their propagation. Therefore, the strength of excitation field at the input I 1 is stronger than that at the input I 2 , so that the corresponding spin waves have nearly same amplitude when they arrive at the output. Besides, the excitation frequencies of the two incident spin waves are identical and the distance between two inputs must be integer multiple of wavelength. As a result, the signal intensity measured at the output is determined by the phase difference ∆ϕ between the two incident waves.
The basic operations of logic AND, OR and XOR gates can be implemented by this proposed device. In the first two cases, the incident spin waves must be in phase (∆ϕ = 0), while for the last one they are completely out of phase (i.e., the phase difference is π). Firstly, the OR gate is the simplest one. As shown in Fig. 7(b) and (c), as long as there is one incident spin wave, the signal at the output will be detected, which corresponds to the fact that the output is 1 if at least one input is 1. Therefore, the threshold amplitude is 0 < A c < A 1 , where A 1 is the amplitude of a single input wave propagating to the output. Since the two spin waves are set to be in phase, they interfere constructively and the net amplitude produced by two incident waves A 11 is significantly larger than A 1 . The criterion for the AND gate thus satisfies A 1 < A c < A 11 , at which the output is 1 only when both inputs are 1. However, for the XOR gate, two input spin waves must be out of phase as mentioned so that they interfere destructively. In this case, A 11 < A c < A 1 ; when the two input signals are the same (i.e., 00 or 11), the output is 0, otherwise it is 1 [see Fig. 7(b) and (d)]. It is important to note that, in our proposed model of the logic computing gate, the XOR function is implemented with simple structures and circuits, rather than complex combinations of standard logic gates in most situations [75].

IV. CONCLUSION
In conclusion, we demonstrate the current-induced oscillation of an elongated skyrmion on a nanotrack with two pinning sites. Combining the Thiele equation and the motion of topological elements, the stable oscillation caused by the periodic deformation of a skyrmion is described phenomenologically. We show the effect of the driving current density, the damping coefficient and two primary parameters of the pinning region on the performance of the skyrmion oscillator. The working window of this oscillation is also determined. Furthermore, the waveguide properties of such an elongated skyrmion are corroborated through micromagnetic simulations and the logic gate based on the propagation of spin waves in this model is also proposed. Our results are crucial for future digital information technology based on topological spin structures and spin waves, such as the generation and the processing of microwave signals.