Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript presents a numerical investigation, via micromagnetic simulations, of a skyrmion-pair-driven racetrack memory mechanism utilizing Hall motion. The authors propose a driving method based on a non-contacting wire, which generates a spatial Oersted (Oe) field gradient along the x-axis (as illustrated in Figure 1(a)). This Oe field causes the skyrmions to move away from the track center to minimize system energy, due to SkHE propels them along the nanotrack in the y-direction. This mechanism forms the core finding of the manuscript.

While the concept appears promising, the presentation of work is oversimplified and leaves several key physical aspects underexplored. A more comprehensive analysis of skyrmion dynamics under this configuration is necessary to meet the publication standards of this journal.

I have the following major question.

1. A primary concern is the long-range dynamics of the skyrmions along extended nanotracks. For exapmle, At t=0, the skyrmion moves from the center of nanotrack along width due to the Oe field, and SkHE initiates motion along the length of nanotrack. However, it remains unclear what happens once the skyrmion reaches the width edge of the nanotrack and becomes pinned. How is sustained motion ensured over longer tracks? This aspect needs clarification.

2. Can author tell, about how much maximum and minimum current is applied in non- contacting wire so that desired condition will be obtained (meaning only two skyrmion and a Domain wall) and start propagating along nanotrack, for this may author need do phase analysis of magnetization of nanotrack and Oe field generated by non- contacting wire.

3. The authors report that skyrmion pairs propagate faster than a single skyrmion, despite the system not being configured as a synthetic antiferromagnet (SAF). What is the physical mechanism behind this behavior and Is there any skyrmion-skyrmion interaction considered? Additionally, in the single-skyrmion case, the relative placement of the non-contacting wire with respect to the nanotrack is unclear, and the origin of the Oe field gradient in this scenario should be explicitly described.

4. Missing LLG Equation and Effective Field Description, LLG equation is not included in the manuscript. For completeness and reproducibility, the authors should present the governing equation and describe how the effective field or energy terms are modeled in the simulation.

5. It would strengthen the manuscript to compare the proposed Oe-field-based mechanism with alternative driving gradients, such as Dzyaloshinskii–Moriya interaction (DMI) or anisotropy gradients. Such a comparison could help highlight the advantages and trade-offs of the current approach.

6. Can author modify equation (3) in term of x as variable, since g is derivative of x

 

Author Response

Comments 1: 

A primary concern is the long-range dynamics of the skyrmions along extended nanotracks. For exapmle, At t=0, the skyrmion moves from the center of nanotrack along width due to the Oe field, and SkHE initiates motion along the length of nanotrack. However, it remains unclear what happens once the skyrmion reaches the width edge of the nanotrack and becomes pinned. How is sustained motion ensured over longer tracks? This aspect needs clarification.

Response 1:

We thank this referee for raising this important piece of suggestion. We have added the explanation for the consideration of ultra-long tracks in the main text: 

However, the skyrmions only have the Hall motion induced by SkHE along the x direction due to the boundary effect. Furthermore, when the track length reaches 5 μm, snapshots of the skyrmion motion process (Supplementary Figure S1) show the skyrmion Hall motion induced by the transverse Oersted field can be fully utilized for directional transport along the x direction, without reaching the width edge of the track and being pinned.

We have also added a detailed explanation in the supplementary materials: 

For instance, under the 128 nm nanotrack width and the transverse Oersted field gradient (g=0.0234 mT/nm), the maximum offset of the transverse displacement is 10 nm.  Due to the boundary effect, the skyrmion will no longer undergo the transverse offset and only move along the x direction.

When the track length reaches 5 μm, snapshots of the skyrmion's motion process are shown in Supplementary Figure S1. It shows that the skyrmion Hall motion induced by the transverse Oersted field can be fully utilized for directional transport along the x direction, without reaching the width edge of the track and being pinned.

Figure S1 the motion of skyrmion pair in the ultra-long strip (5 μm).

Comments 2: 

Can author tell, about how much maximum and minimum current is applied in non- contacting wire so that desired condition will be obtained (meaning only two skyrmion and a Domain wall) and start propagating along nanotrack, for this may author need do phase analysis of magnetization of nanotrack and Oe field generated by non- contacting wire.

Response 2:

We thank this referee for raising this important piece of suggestion. We have added the description for the phase analysis of magnetization and the Oersted field in the main text:

Under the combined action of the Oersted fields and the DMI, a chiral DW is formed under the wire, forming a skyrmion-DW-skyrmion magnetic moment structure in the track as shown in Figure 1 (c). To investigate the correlation between the current and the skyrmion-DW-skyrmion structure, the phase diagram as a function of the Oersted field, the corresponding current, and PMA is presented in Figure 1 (d). When the Oersted field exceeds the maximum tolerable threshold, it will provide sufficient driving force for skyrmions to surmount the energy barrier, thereby inducing their gradual annihilation and a subsequent phase transition to the ferromagnetic (FM) state. Notably, as the PMA increases, the magnetic system exhibits an enhanced tendency to transition to the FM state. Consequently, the maximum Oersted field that the system can tolerate decreases accordingly.

Figure 1. The skyrmion pair racetrack device. (a) Device diagram. (b) FEM calculation of the current-generated Oersted fields. (c) The magnetic moment distribution of the skyrmion-DW-skyrmion structure. (d) The phase diagram as a function of the Oersted field, the corresponding current, and PMA.

Comments 3: 

The authors report that skyrmion pairs propagate faster than a single skyrmion, despite the system not being configured as a synthetic antiferromagnet (SAF). What is the physical mechanism behind this behavior and Is there any skyrmion-skyrmion interaction considered? Additionally, in the single-skyrmion case, the relative placement of the non-contacting wire with respect to the nanotrack is unclear, and the origin of the Oe field gradient in this scenario should be explicitly described.

Response 3:

We thank this referee for raising this important piece of suggestion. As for the first comment, we have added the explanation of skyrmion-DW-skyrmion interaction in the main text:

Due to the presence of the skyrmion-DW-skyrmion coupling structure, a “push” effect emerges between the intermediate DW and the skyrmions on both sides under the influence of the Oersted field gradient. Specifically, the “push” effect serves to enhance the y direction driving force acting on them, resulting in a stronger SkHE and thus further leading to a higher motion velocity in the x direction.

As for the second comment, we have added the device diagram of the single skyrmion racetrack device in the main text:

In contrast, in the case of single skyrmion, the current wire is fabricated on the right side of the entire nanostrip (its structure is shown in Supplementary Figure S2).

Additionally, we have added the explanation of the single skyrmion racetrack device diagram in the supplementary materials:

In contrast, in the case of single skyrmion, the current wire is fabricated on the right side of the entire nanostrip, which is shown in Supplementary Figure S2. DW is not formed on the right side of the nanostrip, and the skyrmion is only laterally driven by the Oersted field gradient. Due to the absence of repulsive force between the skyrmion and DW, the speed of the single skyrmion is slower than that of the skyrmion pair.

Figure S2 The device diagram of the single skyrmion racetrack device.

Comments 4: 

Missing LLG Equation and Effective Field Description, LLG equation is not included in the manuscript. For completeness and reproducibility, the authors should present the governing equation and describe how the effective field or energy terms are modeled in the simulation.

Response 4:

We thank this referee for raising this important piece of suggestion. We have provided an description for the LLG with the effective field in the main text: 

The Oersted field gradient has been shown to effectively manipulate the movement of skyrmions [42-45]. All micromagnetic simulations were performed using Mumax3 [46], based on the Landau-Lifshitz-Gilbert (LLG) equation, expressed as [47]:

                                                                                                                          (S1)   

where γ is the gyromagnetic ratio,  α is the damping coefficient, m is the normalized magnetic vector. H_eff is the effective field where the total energy E in our studied system is E=E_DMI+E_ex+E_de+E_an+E_Oe . We consider the Neumann boundary conditions, and the interfacial Dzyaloshinskii-Moriya interaction (i-DMI) energy as:

                                                                                                                               (S2)

where D is the i-DMI strength. The exchange energy is given by:

                                                                                                                                                      (S3)

where A  is the exchange constant. The demagnetization energy is:

                                                                                                                                              (S4)

 where H_de is the demagnetization field, μ₀ is the vacuum permeability, and  M_s is the saturation magnetization. The uniaxial anisotropy energy E_an is

                                                                                                                                             (S5)

where K_u is perpendicular magnetization anisotropy. The effect of Oersted fields is given by a Zeeman energy E_Oe as: 

                                                                                                                                            (S6)

where  B is the Oersted field. We study the case that the skyrmion pair move under a spatially inhomogeneous magnetic field of the form B=[0, 0, B_z(x)], with a gradient along the x-direction (∂B_z/∂x).

Additional reference:

[47]Liu J, Song C, Zhao L, Cai L, Feng H, Zhao B, et al. Manipulation of skyrmion by magnetic field gradients: a Stern–Gerlach-like experiment. Nano Letters, 23(11), 4931-4937 (2023).

Comments 5:

It would strengthen the manuscript to compare the proposed Oe-field-based mechanism with alternative driving gradients, such as Dzyaloshinskii–Moriya interaction (DMI) or anisotropy gradients. Such a comparison could help highlight the advantages and trade-offs of the current approach.

Response 5:

We thank this referee for raising this important piece of suggestion. We add the comparison with anisotropy gradients in the supplementary materials:

To verify the effectiveness of the proposed method, we compared it with the approach that drives skyrmion Hall motion along the x direction utilizing the PMA gradient, as illustrated in Supplementary Figure S3. Specifically, the driving force generated by the PMA gradient along the y direction can also propel skyrmions to undergo Hall motion along the x direction. However, skyrmions can only stably exist within a specific range of PMA due to their high sensitivity to PMA, which consequently restricts the feasible range of the PMA gradient. For the present material system, we conducted a comparative analysis of the two methods in a 256-nm width racetrack. Under the maximum tolerable PMA gradient (12.5 GJ/m⁴), the skyrmion velocity achieved by this approach was merely 0.17 m/s. However, under the maximum Oersted field gradient (0.146 mT/nm), the skyrmion velocity can reach 0.287 m/s. In addition, the implementation of the Oersted field gradient along the x direction is technically simpler than that of the PMA gradient.

Figure S3 the skyrmion pair velocity comparison between the methods of PMA gradient and Oersted field gradient.

Comments 6:

Can author modify equation (3) in term of x as variable, since g is derivative of x.

Responses 6:

We thank this referee for raising this important piece of suggestion. We have modified the equations (1-3) in term of x as variable in the main text.

                                                                 

Reviewer 2 Report

Comments and Suggestions for Authors

I attached the report

Comments for author File: Comments.pdf

 

Author Response

Comments1:

The authors use Ku = 470 kJ/m3, Ms = 7.7 × 10^5 A/m, A = 1 × 10^−11 J/m, and D = 1 mJ/m^2. Which material system are these values based on? Are they represent known chiral magnet with finite DMI? This should be explicitly stated.

Rsponses1:

We thank this referee for raising this important piece of suggestion. we have explicitly stated the explanation for the material system parameters in the main text:

In the micromagnetic simulation, the magnetic parameters are based on a known chiral magnet system, specifically the Ta/CoFeB/MgO material parameter system in reference [47], which exhibits finite DMI that can stabilize skyrmions:

[47]Liu J, Song C, Zhao L, Cai L, Feng H, Zhao B, et al. Manipulation of skyrmion by magnetic field gradients: a Stern–Gerlach-like experiment. Nano Letters, 23(11), 4931-4937, (2023).

Comments2:

The simulation mesh is 4 nm × 4 nm × 1 nm. The results could be sensitive to discretization. Was these mesh size smaller the exchange length? What happens to size of skyrmion when these parameters are changed?

Rsponses2:

We thank this referee for raising this important piece of suggestion. We have added the explanation for the mesh size in the supplementary materials:

The standard definition formula of the magnetic moment exchange length is:

                                                                                                                                                                 (S1)

As the exchange constant is A=1.0×10^-11 J/m, the saturation magnetization is M_s=7.7×10⁵ A/m, the permeability of free space is μ₀=4π×10^-7 T·m/A, and the exchange length is l_ex=5.17 nm according to equation (S1).

We set the simulation mesh as 2 nm×2 nm×1 nm , 4 nm×4 nm×1 nm, and 8 nm×8 nm×1 nm, the simulation results are shown in Supplementary Figure S4. As the mesh size exceeds the exchange length, consequently, considerable inaccuracies arise in the simulation results.

Figure S4. the motion of skyrmion pair under different mesh size in the simulation.

Comments 3:

The claim that the domain wall suppresses transverse motion is interesting, but no quantitative Hall angle or displacement data are shown for comparison with single skyrmions.

Responses 3:

We thank this referee for raising this important piece of suggestion. According to the suggestion, we have added the Hall angle comparison between the skyrmion pair and the single skyrmion in the main text:

Due to the absence of repulsive force between the skyrmion and the DW, the speed of the single skyrmion is slower than that of the skyrmion pair, and the Hall angle of the single skyrmion is slightly smaller than that of the skyrmion with the same topological charge in the skyrmion pair. The Hall angle comparison between the skyrmion pair (Q=±1) and the single skyrmion is shown in Supplementary Figure S5.

Additionally, we have added the explanation of the Hall angle comparison between the skyrmion pair (Q=±1) and the single skyrmion in the supplementary materials:

In contrast, in the case of single skyrmion, the current wire is fabricated on the right side of the entire nanostrip (its structure is shown in Supplementary Figure S2). DW is not formed on the right side of the nanostrip, and the skyrmion is only laterally driven by the Oersted field gradient. Due to the absence of repulsive force between the skyrmion and DW, the speed of the single skyrmion is slower than that of the skyrmion pair, and the transverse displacement is smaller than that of the skyrmion pair. Furthermore, the y-direction driving forces acting on the two skyrmions with opposite topological charges (Q=±1) of the skyrmion pair, their displacement in the y direction exhibit an opposite characteristic, and their Hall angles correspondingly show a relationship of equal magnitude but opposite signs. The Hall angle is defined as Φ_Sk=tan^-1(v_y/v_x), where v_y and v_x  are the velocity in the y direction and in the x direction within the 10 ms. The Hall angle comparison of the skyrmion pair (Q=1 and Q=-1) and the single skyrmion is shown in the Supplementary Figure S5.

Figure S5 The Hall angle comparison of the skyrmion pair (Q=1 and Q=-1) and the single skyrmion under different nanostrip width.

Comments 4:

The FEM-calculated Oersted field is introduced into MuMax3 micromagnetic simulations. Was the full non-uniform field profile used, or was a uniform gradient approximation used?

Responses 4:

We thank this referee for raising this important piece of suggestion. In the MuMax3 micromagnetic simulations, a uniform gradient approximation was used. We have explained this situation in the main text:

The Oersted fields obtained by the FEM are introduced into the simulations through external magnetic fields, and their magnetic field gradient is approximated as a uniform value.

Comments 5:

Were periodic or open boundary conditions applied along width and length during simulations? Please clarify.

Responses 5:

We thank this referee for raising this important piece of suggestion. We have added the explanation of open boundary conditions in the main text:

In the simulation, the periodic boundary condition was applied only along the x direction of the nanostrip. 

Comments 6:

In the Introduction, where the authors state:“Magnetic skyrmions are a type of particle-like spin textures with a unique topological charge [1–7]” I recommend adding a citation to:

Ullah, A.; Balasubramanian, B.; Tiwari, B.; et al. Topological spin textures and topological Hall effect in centrosymmetric magnetic nanoparticles. Phys. Rev. B 108, 184432 (2023).

Responses 6:

We thank this referee for raising this important piece of suggestion. I have added some references into the main text:

Magnetic skyrmions are a type of particle-like spin texture with a unique topological charge [1-9].

Additional references

[8] Ullah A, Balasubramanian, B, Tiwari, B, Giri, B, Sellmyer, DJ, Skomski, R, Xu, XS. Topological spin textures and topological Hall effect in centrosymmetric magnetic nanoparticles. Physical Review B, 108(18),184432 (2023).

[9] Ullah A,Balamurugan B, Zhang W, Valloppilly S, Li XZ, Pahari R, Yue LP, Sokolov A, Sellmyer DJ, Skomski R. Crystal Structure and Dzyaloshinski-Moriya Micromagnetics. IEEE Transactions on Magnetics, 55(7),7100305 (2019).