A spin-torque nano-oscillator based on interlayer-coupled meron-skyrmion pairs with a fixed orbit

In recent years, magnetic skyrmion-based spin-torque nano-oscillators (STNOs) attract considerable interest for their prospect in future-generation communication and spintronic technologies. However, some critical issues, which hamper their practical applications, e.g., the long start-up time and variable skyrmion gyration orbit, remain to be resolved. Here, we numerically demonstrate a realization of a fixed-orbit STNO, which is based on an interlayer-coupled meron-skyrmion (MS) pair other than a magnetic skyrmion. In this STNO, the MS pair possesses a structurally defined, fixed orbit within a broad range of driving current, even in the presence of random defects. The output frequency range of the STNO based on an MS pair far exceeds that of the STNO typically based on a single skyrmion. Moreover, the output frequency of this STNO can be further elevated if more MS pairs are incorporated. Our results reveal the nontrivial dynamics of the interlayer-coupled MS pair, opening perspectives for the design and optimization of fundamental spintronic devices.


I. INTRODUCTION
2][3][4][5] The topological charge  ∬    d d of a ferromagnetic (FM) skyrmion is equal to ±1, where m = m(x, y) is the normalized magnetization vector. 5Triggered by a sufficiently large driving force, a skyrmionwhen touching the sample edge-loses the topological protection and converts into an edgemeron with a fractional topological charge, 6 which risks being expelled from the sample by an outward attractive force from the edge. 7,8[11] Spin-torque nano-oscillators (STNOs) function as microwave generators by converting an input direct current into a microwave voltage. 12Benefiting from the nanoscale size and compatibility with existing CMOS technology, they are considered key ingredients of nextgeneration spintronic devices.As compared to other types of STNOs, such as the ones featuring a single domain [13][14][15] or magnetic vortex, [16][17][18] skyrmion-based STNOs offer several advantages, including the reduced driving current, smaller frequency linewidth, wider tunable frequency range, and synchronization of multiple skyrmions in a single STNO. 19,20However, there exist a few innate shortcomings for skyrmion-based STNOs: 19,20 specifically, the frequency of skyrmion-based STNOs comprising a single FM nanodisk is typically restricted to a few gigahertz, the initiation time for the current-driven skyrmion to enter a steady orbit is undesirably long (typically, 20 ns), and the orbit of skyrmion gyration depends on the applied current complicating the design of detection units based on the magnetoresistance effect.
[26][27][28] Nevertheless, it remains challenging to accurately modify a unique ring area in a nanodisk using even state-of-the-art experimental techniques.Until now, a solution that can simultaneously address all the stated problems is still lacking.
By subtly combining distinct spin textures and exploiting their interactions, emerging functionalities and applications unattainable with individual magnetic entities might be opened.
In this research, we devise an STNO based on an SAF nanodisk and exploit the gyrational dynamics of an interlayer-coupled meron-skyrmion (MS) pair stabilized in the SAF structure.
We employ micromagnetic simulations to examine the spectral characteristics of the STNO dependent on the geometric and material properties.It is seen that the frequency range of the MS pair-based STNO is significantly extended relative to that of a typical skyrmion-based STNO.Increasing the number of MS pairs in the STNO can further elevate its output frequency.
Moreover, weak disorder in the sample does not remarkably modify the main characteristics of the STNO.

II. MODEL AND METHODS
As depicted in Fig. 1(a), the STNO consists of three main parts: the fixed layer, the spacer layer, and the free layer.Unlike in a conventional STNO, an SAF multilayer structure serves as the free layer in our STNO.The lower nanodisk with a bigger radius rl (layer-l) and the upper nanodisk with a smaller radius ru (layer-u), separated by a ruthenium (Ru) layer, form an asymmetric SAF structure, so that only the center of layer-l is subject to the Ruderman-Kittel-Kasuya-Yosida (RKKY) interlayer exchange coupling with layer-u.Both layer-l and layer-u assume an FM/HM (heavy-metal) bilayer structure to enable the presence of a Dzyaloshinskii-Moriya interaction (DMI) alongside the perpendicular magnetocrystalline anisotropy (PMA).
[23][24]26,27 An out-ofplane electric current is fed into the STNO through a top electrode with a radius re.
The public-domain micromagnetic solvers MuMax3 37 and OOMMF 38 (modified with the interfacial DMI extension module 39 ) are employed to find the static magnetization configuration and subsequently to trace the current-induced dynamics of magnetization, which is governed by the Landau-Lifshitz-Gilbert-Slonczewski equation, 19,20 ∂ ∂ where γ is the gyromagnetic ratio, α is the Gilbert damping constant, t is time, and Heff is the effective magnetic field, which includes the Heisenberg exchange field, magneto-dipolar field, perpendicular anisotropic field, DMI effective field, and interlayer exchange field.     is the spin-transfer torque, where the spin-polarization vector has a radial vortex-like distribution,  cos , sin , 0 , with  arctan  20 and  ℏ is the efficiency factor with ħ representing the reduced Planck constant, P the spin polarization ratio, J the current density, μ0 the vacuum permeability, e the elementary charge, Ms the saturation magnetization, and tf the thickness of each FM layer in the SAF nanodisk.
The material parameters for both free layers (layer-u and layer-l) feature the Pt/Co multilayer film: [40][41][42] the saturation magnetization Ms = 580 kA/m, exchange stiffness A = 15 pJ/m, PMA constant Ku = 0.8 MJ/m 3 , DMI strength D = 3.5 mJ/m 2 , and Gibert damping constant α = 0.02.The bilinear interfacial exchange coefficient quantifying the strength of interlayer antiferromagnetic exchange interaction is σ = -2 mJ/m 2 unless explicitly stated otherwise. 43It is assumed that P = 0.4 and the spin-polarization attenuation between layer-u and layer-l is negligible, so that the spin-transfer torque manifests an identical efficiency in both free layers.This supposition is reasonable considering the small thickness of each layer in the SAF nanodisk. 44The unit cell size 1×1×1 nm 3 is adopted regardless of the sample size.
Open boundary conditions are assumed.

III. RESULTS AND DISCUSSION
To create an MS pair, a spin-polarized current can be applied locally to a desired region at the boundary of layer-l, where the magnetization is switched under the action of spin-transfer torque.Subsequently, a mirroring nucleation site for the meron is formed in layer-u as a result of the strong interlayer exchange coupling. 36After the nucleation of the spin textures, the applied current is turned off allowing the system to relax freely.During the relaxation process, the meron experiences an outward attractive force  from the edge of layer-u and an inward dragging force  from the skyrmion.Meanwhile, the skyrmion undergoes an inward repulsive force  from the edge of layer-l and an outward dragging force  from the meron.
In a certain range of ru/rl [highlighted in Fig. the injected skyrmion is remote from the border of layer-l, whereby the inward force  on the skyrmion from the edge of layer-l cannot equilibrate the outward force  on the meron from the edge of layer-u.Consequently, the meron is expelled from layer-u and meanwhile the skyrmion is dragged toward the edge of layer-l.Ultimately, the skyrmion resides between the borders of layer-u and layer-l, because it would encounter a potential barrier when approaching either border. 36,45For this configuration, the topological charge of the SAF structure is Q ~ 1 A/m 2 , and the corresponding mode pattern shown in the inset identifies the trajectory of the MS pair.Note that the multiple, equidistantly spaced high harmonics in the frequency spectra arise from the truncated waveform of mz(t). 46r simulations of the spectral characteristics of the STNO suggest that the frequency of the STNO not only varies with the applied current density, but also depends on the geometric and material properties of the SAF nanodisk.As shown in Fig. 3 For comparison, we conduct additional simulations on an STNO based on an SS pair, which is stabilized in a symmetric SAF nanodisk, 23,40 where the two free layers separated by the Ru layer are identical in size, namely, ru = rl = 60 nm.As shown in Fig. 3(d), the output frequency of the STNO based on an SS pair predominates at small current densities, but drops below the frequency of the STNO based on an MS pair when the current density becomes high enough.We also examined the dynamics of an STNO based on a single-layer skyrmion 20 (in this case, the free layer-u and the Ru layer are absent).Under a negative driving current, the single skyrmion in the STNO can move counterclockwise in a circular orbit, manifesting a frequency about hundreds of megahertz.Its spectral characteristics is plotted in the inset of Fig.

3(d).
As expected, if a positive current is applied, the skyrmion will return to the disk center in a clockwise sense. 20,21[20][21][22][23][24][25][26][27][28][29]31,42,45 As illustrated in Fig. 4(a), for the meron, and for the skyrmion, where the first to fifth terms represent the Magnus force, the edge-induced force, the internal force between the meron and the skyrmion, the dissipative force, and the driving force, By adding Eq. ( 2) to Eq. ( 3), one obtains the Thiele equation for an MS pair in steady motion,     0, where  as an internal force is absent.As illustrated in Fig. 4(a), these forces can be projected onto the radial and tangential directions.In the circular motion, the radial velocity of the MS pair is zero, and the net velocity of the MS pair can be derived from the Thiele equation in the tangential direction,   0 , i.e.,    0. To simply the calculation, we consider an instant when the MS pair is passing through the rightmost end of the circular orbit, where the velocity of the MS pair is oriented along -y.In this scenario,  dd  •  and  dd   •  .
One will be able to calculate  and  if an explicit form of m(x, y) is known.In fact, more than one ansatz of m(x, y) has been proposed for a skyrmion, 42,47,48  upon the current density can be found in Fig. 5.
As revealed by the data for the MS pair, the value of  for the meron is far smaller than that of the skyrmion, whereas the values of S for both the meron and the skyrmion are almost equal.In all cases, S is more sensitive to the current density than .In particular, for the MS pair, S rises with the increased current density, whereas for the SS pair and the single-layer skyrmion, S diminishes with the increased current density.These features result in the consequence that, at a sufficiently high current density, the MS pair moves faster and possesses a higher frequency than the SS pair.Another important feature is that, in the relevant range of current density, the value of S for the single-layer skyrmion is several times smaller than those of the MS and SS pairs, which explains why an STNO based on a single skyrmion has very small frequencies. 19,20 identify the role of interlayer exchange interaction in the synchronized motion of the MS pair, we carry out a set of control simulations, where the spin-polarized current only flows into a single free layer, either layer-u or layer-l, at a time. 22In this scenario, both frequencies of the MS pair and the SS pair are seen to drop [Figs.3(d) and 4(b)], implying that a doublelayer current is preferable in order to synchronize the meron and the skyrmion. 22,49ne distinctive feature of the skyrmion-based STNO lies in the synchronized motion of multiple skyrmions in a single nanodisk, which can elevate the output frequency remarkably. 19,23Following this strategy, we arrange multiple MS pairs (N = 2, 3, 4, 5) in the SAF nanodisk, as illustrated in Fig. 6(a), and investigate their dynamics under a spin-polarized current.As anticipated, the frequency of the STNO with multiple MS pairs increases with the increasing number N, as shown in Fig. 6(b).It is worthy to note that, the symmetric distribution of the MS pairs would be destroyed if an exceedingly large current is utilized, leading to the emergence of additional frequencies.
Finally, we examine the influence of disorder on the dynamic characteristics of an MS pair.Typically, we concentrate on a kind of disorder closely related to the polycrystalline ultrathin films fabricated by magnetron sputtering. 31,50To capture the key property of a granular film, we introduce the random variation ΔKu to the PMA constant Ku and employ Voronoi tessellation to generate two grain patterns: i) the mean grain size is 10 nm and ΔKu from grain to grain is within 5%, as depicted in Fig. 7(a), and ii) the average grain size is 25 nm and ΔKu is within 2%, as indicated in Fig. 7(b).The frequency spectra and mode patterns of an MS pair in the granular and clean samples, plotted in Fig. 7(c), suggest that the weak disorder only leads to a fluctuating velocity of the MS pair but has no appreciable effect on its gyro-orbit.

IV. CONCLUSION
In summary, we propose a spin-torque nano-oscillator (STNO) that is based on an interlayercoupled meron-skyrmion (MS) pair.The output characteristics of the STNO has been examined via micromagnetic simulations and the Thiele model.Comparative simulations reveal that this STNO surpasses, in several aspects, the STNO based on either interlayercoupled skyrmion-skyrmion pair or conventional single-layer skyrmion.The synchronized motion of multiple MS pairs in a single STNO can raise the output frequency of the STNO further by several times.Furthermore, the presence of weak disorder appears not to markedly modify the key characteristics of the STNO.This type of STNO holds great potential to overcome some critical bottlenecks, such as the ultralong initiation time, experienced by skyrmion and vortex-based STNOs in practical applications.Our research suggests that the interplay of distinct features 29,31,51 provides a viable route toward settling some critical technical issues.For all magnetic entities,  is insensitive to the current density, but the meron has a much smaller  than the skyrmion.While the values of S for the MS pair rise with the current density, the values of S for the SS pair and the single skyrmion decrease.
1(b) by the white box], the resultant force on the MS pair can become zero (specifically,   0 and   0 ), so that the interlayer-coupled MS pair is stabilized near the edge of layer-u.As revealed by the simulation results, the resulting equilibrium spin configuration carries a topological charge Q ~ 0.5 [as illustrated in Fig. 1(c), middle panel], which is consistent with the expected value of the topological charge (Q = Qem + Qsk) of a pair of edge-meron (Qem = -0.5)and skyrmion (Qsk = +1).At a smaller ru/rl than required [e.g., the region in Fig. 1(b) marked by the yellow box],

[
Fig. 1(c), left panel].When ru/rl is much higher [e.g., the region in Fig. 1(b) indicated by the magenta box], both the meron and the skyrmion are expelled out, resulting in two antiferromagnetically coupled single domains with a topological charge Q ~ 0 [Fig.1(c), right panel].Note that no SS pair (also known as bilayer-skyrmion 40 ) appears in the phase diagram for the considered values of [ru, rl], since the formation of an SS pair involves conversion of the meron to a skyrmion-a process associated with a high potential barrier.This study concentrates on the dynamics of an MS pair driven by a perpendicular current.Unless otherwise specified, all results and discussion are based on the STNO with [ru, rl] = [40 nm, 60 nm] and re = 40 nm.The snapshots of the MS pair under a given current density are depicted in Fig. 2(a), indicating that the meron and the skyrmion are tightly bound by the interlayer exchange interaction and move synchronously along the edge of layer-u.Fig. 2(b) shows oscillations of the position (Rx, Ry) and average magnetization component mz of the meron.It is evident that, immediately after the application of the current, the MS pair moves steadily in a fixed circular orbit, resulting in a very short start-up time.The Fourier spectra of Rx and mz, plotted in Fig 2(c), displays the frequency (f = 8.53 GHz) of the STNO at J = 1.5×10 11 (a), for various values of the antiferromagnetic coupling coefficient σ, each curve of f(J) terminates at a unique critical current density, above which the meron and the skyrmion cannot be bound together, unless the interlayer exchange coupling is further enhanced.That is, the critical current density scales with the interlayer exchange coupling strength.For sufficiently strong interlayer exchange coupling, the output frequency at a given current density appears insensitive to the magnitude of σ.However, the polarizer angle φ can dramatically modify the characteristics of the STNO, as shown in Fig.3(b).Specifically, the threshold current density of the STNO changes nonmonotonically with φ.Once φ attains π/3, the threshold current density drops rapidly.Fig 3(c) shows the output frequency versus driving current density at φ = 0 for different values of re.It is obvious that a smaller electrode contributes to a reduced output frequency and an elevated threshold current density.
respectively.The Magnus force   , where v denotes the tangential velocity of the MS pair,  4z is the gyrovector with ẑ denoting the unit vector along +z, and     /.The edge-induced force  ∇U, where U is the potential energy.The dissipative force   ⃖ ⃗ •  , where  ⃖ ⃗  0 0  in which   ≡  dd  •  with i = (x, y).The current-induced driving force      can be decomposed into a radial force and a tangential force, where F   in which  dd   •  with j = (r, τ) denoting the radial or tangential direction.The internal force      between the paired meron and skyrmion originates from the interlayer exchange coupling.

.
yet no proper ansatz exists for an edge-meron.Despite the lack of an available ansatz for the meron, numerical values of  and  can still be found from the simulation results by rewriting  in the discrete form, Then, after some procedures, we obtain the numerical values of  and S for the meron and the skyrmion.Considering specifically an MS pair at J = 5×10 10 A/m 2 , for the meron, m = 95.0 and Sm = 93.9nm, while for the skyrmion, s = 172.3and Ss = 96.7 nm.Having these numerical values, one acquires the velocity of the MS pair,    e , where p m + s and Sp = Sm + Ss.The dependencies of  and S

Figures captionsFigure 1 .
Figures captionsFigure1.Device structure and possible spin configurations.(a) Schematic diagram of a spintorque nano-oscillator (STNO).The STNO includes three functional layers: the fixed layer, the spacer layer, and the free layer.The free layer comprises a synthetic antiferromagnetic (SAF) nanodisk, in which the upper free layer (layer-u) and the lower free layer (layer-l) are separated by an ultrathin ruthenium (Ru) layer to ensure the occurrence of interlayer exchange coupling.Both layer-u and layer-l are supposed to adopt a ferromagnet/heave-metal (FM/HM) structure to stabilize chiral spin textures by introducing a Dyzaloshinskii-Moriya interaction.The radii of layer-u and layer-l are denoted as ru and rl, respectively.Layer-l is set to be wider than layeru, i.e., rl > ru, to facilitate the nucleation of a meron-skyrmion (MS) pair.The fixed layer assumes a radial vortex-like magnetization distribution.The current flows perpendicular to the fixed layer via a top electrode with a radius re (not shown for brevity).Without external stimuli, the outward attractive force  on the meron from the edge of layer-u counteracts the inward repulsive force  on the skyrmion from the edge of layer-l, via the mediation of the internal forces  and  between the meron and the skyrmion, resulting in stabilization of the MS pair at the border of layer-u.The origin of coordinate is at the lower-left rim of layer-l.(b) Phase diagram for the spin configurations in the SAF nanodisk.This diagram, rendered as dependency of the total topological charge in the SAF nanodisk on the radii ru and rl, contains three regions, typically, the yellow, white, and magenta dashed boxes highlighting Q = 0.93 for [ru, rl] = [24 nm, 60 nm], Q = 0.52 for [ru, rl] = [40 nm, 60 nm], and Q = -0.04 for [ru, rl] = [56 nm, 60 nm], respectively.(c) The spin configurations in the SAF nanodisk corresponding to Q = 0.93, Q = 0.52, and Q = -0.04.The spin configuration with Q = 0.52 represents an MS pair.

Figure 2 .
Figure 2. Frequency characteristics of an MS pair.(a) Temporal sequences of the MS pair under a current, illustrating clockwise gyration.The current density J = 1.5×10 11 A/m 2 .The orange grid box in layer-u highlights the detection region of the output signal.(b) Oscillations of the positions (Rx and Ry) and average magnetization component mz of the meron with time.(c) Fourier-transform spectra of Rx and mz.The inset shows the mode pattern of the gyrating MS pair with a frequency of f = 8.53 GHz.In these illustrations, [ru, rl] = [40 nm, 60 nm].The output signal mz is detected, at the region marked by the orange grid box in each panel, via the magnetoresistance effect.

Figure 3 .
Figure 3. Spectral characteristics of an MS pair.Panels (a)-(d) plot the frequency versus current density for various parameters: (a) strengths σ of the interlayer exchange coupling, (b) polarizer angle φ, (c) radius re of the electrode, and (d) radius ru of layer-u.

Figure 4 .
Figure 4. Force diagram and coupling-assisted motion of an MS pair.(a) Thiele forces on the

Figure 6 .
Figure 6.Spectral characteristics of an STNO carrying multiple MS pairs.(a) Centrosymmetrically distributed MS pairs in a single STNO.The number of the MS pairs, denoted as N, is indicated in each panel.In these illustrations, [ru, rl] = [56 nm, 76 nm].(b)Comparison of the frequency versus current density for different numbers of MS pairs.Inset: zoom-in view of the curves at small current densities.

Figure 7 .
Figure 7. Effect of disorder on the characteristics of an MS pair.Panels (a) and (b) depict two realizations of the disorder in the SAF nanodisk modeled as a granular sample, in which both the grain size and the anisotropy constant Ku are randomly distributed around their respective mean values.In these illustrations, [ru, rl] = [40 nm, 60 nm].(a) The average grain size is 10 nm and Ku in each grain varies randomly within 5% of the mean value  = 0.8 MJ/m 3 , namely, ∆ /   / ≤ 5%.(b) The average grain size is 25 nm and Ku in each grain varies randomly within 2%, i.e., ∆ / ≤ 2%.(c) Comparison of the frequency characteristics of an MS pair in clean and granular nanodisks under J = 1.5×10 11 A/m 2 .Insets: mode patterns of the STNO for the two granular samples.Apparently, the weak disorder in a