Magnetic vortex gyration affected by Dzyaloshinskii

Magnetic vortex gyration affected by Dzyaloshinskii–Moriya interaction Y. M. Luo, C. Zhou, C. Won, and Y. Z. Wu Citation: Journal of Applied Physics 117, 163916 (2015); doi: 10.1063/1.4919423 View online: http://dx.doi.org/10.1063/1.4919423 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/117/16?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of Dzyaloshinskii-Moriya interaction on the magnetic vortex oscillator driven by spin-polarized current J. Appl. Phys. 117, 17B720 (2015); 10.1063/1.4915476 The effects of Dzyaloshinskii-Moriya interactions on the ferromagnetic resonance response in nanosized devices J. Appl. Phys. 115, 17C902 (2014); 10.1063/1.4870138 Effect of Dzyaloshinskii–Moriya interaction on magnetic vortex AIP Advances 4, 047136 (2014); 10.1063/1.4874135 Chiral magnetization textures stabilized by the Dzyaloshinskii-Moriya interaction during spin-orbit torque switching Appl. Phys. Lett. 104, 092403 (2014); 10.1063/1.4867199 Low-amplitude magnetic vortex core reversal by non-linear interaction between azimuthal spin waves and the vortex gyromode Appl. Phys. Lett. 104, 012409 (2014); 10.1063/1.4861779

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JOURNAL OF APPLIED PHYSICS 117, 163916 (2015)

Magnetic vortex gyration affected by Dzyaloshinskii–Moriya interaction Y. M. Luo,1 C. Zhou,1 C. Won,2 and Y. Z. Wu1,a) 1

Department of Physics, State Key Laboratory of Surface Physics and Collaborative Innovation Center of Advanced Microstructures, Fudan University, Shanghai 200433, People’s Republic of China 2 Department of Physics, Kyung Hee University, Seoul 130-701, South Korea

(Received 6 March 2015; accepted 18 April 2015; published online 29 April 2015) The effect of the Dzyaloshinskii–Moriya interaction (DMI) on magnetic vortex gyration is investigated systematically through micromagnetic simulations. Our results show that the DMI can lift the degeneracy of vortex gyration eigenfrequencies for vortices with left- and right-handedness. For vortex gyration excited by an in-plane AC resonant field, the DMI can strongly influence the gyration amplitude and the critical field for core switching, depending on the sign of the DMI and the vortex handedness. The DMI-induced edge state has a strong effect on the vortex core gyration C 2015 AIP Publishing LLC. as the core approaches the disk edge. V [http://dx.doi.org/10.1063/1.4919423]

I. INTRODUCTION

The recent discovery of novel chiral spin structures, such as skyrmion lattice and chiral domain wall, which are induced by the Dzyaloshinskii–Moriya interaction (DMI),1,2 has attracted considerable interest because these novel chiral spin structures can exhibit exotic dynamic properties3 and unconventional transport phenomena.4–6 The DMI arises from inversion asymmetry and large spin orbit coupling, and thus was found in bulk materials with a lack of space inversion symmetry,7,8 and at the interface between a magnetic layer and a strong spin-orbit coupling adjacent layer.9–12 In order to realize new spintronic devices, the influence of DMI on spin structures and spin dynamics in confined structures has been explored. The skyrmions that are induced by DMI show striking dynamic properties, opening a path towards new concepts for magnetic memories based on skyrmion motion in nanotracks,13,14 and the chiral Neel-type wall induced by DMI in nanowires was shown to be of significance for high speed wall motion in a racetrack memory device.15,16 In micromagnetic disks made from soft ferromagnetic materials, the magnetic vortex often exists as a stable state that can be characterized by an in-plane curling magnetization (chirality) and a nanometer-scale central region with an out-of-plane magnetization (polarity). Magnetic vortices have generated considerable interest because of their chiral nature and unique gyration dynamics,17,18 and their potential applications in information storage and nano-oscillators.19 Normally, vortex chirality and polarization are not coupled and can be switched independently, but one recent experiment indicated that DMI can break this symmetry,20 although the DMI should be very weak in normal metallic materials.21 Several theoretical investigations were carried out to explore the influence of the DMI on magnetic vortices. Through an analytical analysis, Butenko et al.22 found that the DMI can modify the size of the vortex cores, but without considering its effect on the magnetization at the disk edge. a)

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Micromagnetic simulations23 showed that the DMI could induce tilting magnetization at the disk edge. This therefore leaves an open question whether or not the analytical model can correctly describe the edge state of the vortex. In addition, apart from examination of the influence of the DMI on the static spin configuration in a magnetic vortex, few studies have examined how the DMI influences the gyration dynamics of the magnetic vortex, which is crucial for practical application of the magnetic vortex. In this paper, we studied the effects of DMI on vortex dynamics using micromagnetic simulations. The edge magnetization tilting of the vortex could be quantitatively described through analytical analysis. The DMI lifted the gyration eigenfrequency degeneration for vortices with opposite handedness. We also studied vortex core gyration when excited by an in-plane AC resonant field and found that the DMI can strongly affect both the gyration amplitude and the critical excitation field to switch the vortex core. The edge state induced by the DMI also has a significant effect on the vortex dynamics. II. METHODS

In our simulation, the spin system of the disk is described by a two-dimensional (2D) square lattice with a local magnetic moment j~ m j ¼ MS at each site. The Hamiltonian can be written as X * * X* * * r ij mi mj E¼ J mi mj þ D X* * 1 X* * mi l0 (1) mi H d l0 H 2 which includes the ferromagnetic exchange interaction, the magnetic dipole interaction, the Zeeman coupling, and the DMI, but does not*consider the magnetocrystalline anisot* * ropy. J, D, r ij , l0 , H , and H d denote the exchange constant, the DM constant, the distance vector between spin sites i and j, the magnetic permeability, the external magnetic field, and * the demagnetization field, respectively. In the simulation, H d can be calculated exactly using either the magnetostatic

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equation24,25 or the local dipole approximation.26 In this study, we only considered DMI as it occurs in bulk materials with B20 structures, and this bulk DMI can induce helical spiral stripes.7 The spiral period is generally determined using the ratio J=D, and the helical direction is determined by the sign of D: a negative D produces a left-handed helical structure, while a positive D produces a right-handed helical structure.7 It is well known that the spin configuration and spin dynamics can be simulated via numerical solution of the Landau–Lifshitz–Gilbert equation *

*

* dmi a * dmi * ¼ cmi H ef f ; mi dt dt Ms

(2)

*

with total effective field H ef f , Gilbert gyromagnetic ratio c, *

and damping constant a. H ef f can be calculated using the *

equation H ef f ¼ l1

0

@E *, @mi

where E is the total energy of the

system from Eq. (1). Typically, the effective field induced by the DMI, which is termed the DM field, can be expressed as23 *

H DM ¼

2D ð * r mÞ: l0 Ms

(3)

We performed the simulation by adding a DMI module to the standard micromagnetic simulation software OOMMF.27 In the simulation, the nanodisk diameter is 340 nm, and the disk thickness is 20 nm. To compare the results of the micromagnetic simulation with a real material system, we selected typical parameters for the simulation, including Ms ¼ 8 exchange stiffness J ¼ 1:3 1011 J=m, 105 A=m, a ¼ 0:01, and unit cell size 2 2 20 nm3 , but the DM constant was regarded as the tuning parameter. We also did the testing simulation with the three-dimensional square lattice with unit cell 2 2 4 nm3 , and the resulted effect of DMI on the spin configuration and the vortex gyration eigenfrequency is same as that calculated with the 2D square lattice. We found that the DMI can lift the degeneracy of vortices with opposite handedness, and thus, we only focused on the effects of DMI with a positive D value on vortices with positive polarity but different values of circularity c, i.e., c ¼ þ1 for right-handed circularity and c ¼ 1 for left-handed circularity. For a vortex with negative core polarity and a negative D value, the effect of the DMI could be understood through symmetry analysis. III. RESULTS AND DISCUSSION

First, we must discuss the effects of DMI on the static spin configuration in the magnetic vortex, because the vortex dynamics are usually related to the vortex spin configuration. The micromagnetic simulations already showed that the DMI could change the vortex core size, and could also induce tilting of the magnetization at the disk edge. Through analytical calculations, Butenko et al.22 found that the DMI can considerably change the sizes of vortices, but they ignored the effect of DMI on the magnetization at the disk edge, which exists in a real magnetic vortex system.23 Here,

we prove that the edge tilting induced by the DMI can also be understood through analytical analysis. Static magnetic configurations can be derived by minimizing the energy of Eq. (1). Here, we consider the axial symmetric distributions of magnetization in the vortex, and * express the magnetization vector m in spherical coordinates. * The magnetization vector m at the position ðr cos u; * r sin u; 0Þ can be expressed as m ¼ ðsin h cos w; sin h sin w; cos hÞ, as shown in Fig. 1(a). By considering the rotational symmetry of the vortex, we find w ¼ u þ c p2, and h ¼ hðrÞ. With the local dipole approximation, a variational calculation of Eq. (1) leads to the equation for the equilibrium vortex profile hðrÞ22 d2 h 1 dh 1 D sinh cos h sin2 h þ K sin h cos h ¼ 0; J þ dr 2 r dr r 2 r (4) where K ¼ 12 l0 Ms2 denotes the shape anisotropy in the local dipole approximation and r denotes the distance to the disk center. The correct boundary conditions are required for full solution of this equation. However, Butenko et al.22 solved the equation by only considering hjr¼0 ¼ 0 in the core center and the free boundary condition of dh dr jr¼R ¼ 0. This boundary condition at r ¼ R, therefore, prohibits discovery of the edge state of the tilted magnetization, thus making the analytical analysis deviate with spin configuration in the micromagnetic simulation.23 Recently, Rohart et al.28 studied the effects of the DMI on the boundary conditions in a nanodisk structure, and for bulk-like DMI, the magnetization at the boundary should meet the condition *

dm *

dn

¼

D ð * *Þ mn ; 2J

(5)

*

where n denotes the vector normal to the edge. For a nanodisk with perpendicular anisotropy, Rohart et al.28 found that the interface DMI could induce tilting magnetization at the edge of the disk with an in-plane magnetization component. Alternatively, based on the boundary condition shown in the Appendix of Ref. 28, we found that the bulk DMI in a magnetic vortex tilts the spins towards the out-of-plane direction D at the disk edge under the boundary condition dh dr jr¼R ¼ 2J . Figure 1(b) shows typical vortex spin profiles with D ¼ 0.8 mJ/m2 simulated under explicit calculation conditions and using a local approximation of the dipole interaction, and these profiles show a clear canted magnetization at the disk edge. Using the boundary conditions hjr¼0 ¼ 0 and dh D dr jr¼R ¼ 2J , we can numerically solve Eq. (4) and obtain a stable vortex spin profile, as indicated by the solid red lines in Fig. 1(b), which shows excellent agreement with the simulated results under the local approximation of the dipole interaction. Figure 1(c) shows the value of dh dr at the edge that was extracted from the simulations, which is linear relative to the D value with a slope of 2J1 , and again shows excellent agreement with the results of the analytical model. In reality, dipolar coupling can slightly modify the static spin configuration. As the magnetization turns outward from


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FIG. 1. (a) Geometry and definition of variables. (b) Representative mz profiles across the core for a right-handed vortex with D ¼ þ0.8 mJ/m2 simulated under explicit calculation conditions and the local approximation of the dipole interactions. The black curve is offset for ease of comparison. The red lines indicate the analytical results from solution of Eq. (4). (c) dh dr at the disk edge as a function of D, simulated under explicit calculation conditions and the local approximation of dipole interactions. The red line in (c) is the theoretical result, with a slope of 2J1 .

the surface, the local dipolar coupling can reduce the effective demagnetization field. Through numerical simulations, we calculated the spin profile based on a full dipolar coupling calculation, as shown in Fig. 1(b). In comparison with the results calculated using the local approximation of the dipole interaction, the overall magnetization profile shape is very similar, but the center core is broader and the edge tilt is higher. In Fig. 1(c), the calculated value of dh dr at the edge from a full dipolar coupling calculation is always slightly D . larger than 2J Next, we numerically studied the effect of the DMI on the vortex gyration eigenfrequency. The simulation can only be limited to DMI strengths of less than 1.8 mJ/m2 for the right-handed vortex and less than 1.0 mJ/m2 for the lefthanded vortex. For D > 1.0 mJ/m2, only the right-handed vortex can exist, and for D > 1.8 mJ/m2, the vortex state is no longer stable and subsequently transforms into the stripe phase.23 To obtain the vortex gyration eigenfrequency, we first applied a 50 Oe in-plane field along the y direction to translate the vortex core from the disk center, and then removed the applied field to relax the vortex core to the ground state through gyration motion. During the relaxation process, the x-component (mx ) of the magnetization exhibited damped oscillation, and the gyration frequency can be

obtained through fast Fourier transform (FFT) analysis. Figure 2(a) shows representative mx oscillation curves for the left- and right-handed vortices with D ¼ 0.5 mJ/m2, and the inset shows the corresponding FFT curves, which demonstrate that the DMI can lift the gyration eigenfrequency degeneracy of vortices with opposite handedness. As shown in Fig. 2(b), the gyration frequency increases with DMI strength for the right-handed vortex, but decreases with DMI strength for the left-handed vortex. For a DMI with a negative sign, i.e., D < 0, the situation should be reversed. The frequency difference between the vortices with opposite handedness is linearly dependent on the DMI strength. Therefore, if one can produce a magnetic vortex that contains DMI, it is possible to evaluate the DMI strength by simply measuring the eigenfrequency difference with opposite handedness. The change in the gyration frequency can be attributed to the DM field that is induced by DMI. The z-component of the DM field in the vortex can be expressed as *

*

ðHDM Þz ¼ l2DMs ðr mÞz . The DM field direction is deter0 mined by the vortex chirality and is independent of the polarization, so the DMI can generate a bias field for core polarization switching that could be as large as 214 mT when D ¼ 0.5 mJ/m2.23 Using the spin configuration in the


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FIG. 2. (a) Time dependent mx oscillation during core gyration for the leftand right-handed vortices with D ¼ 0.5 mJ/m2. The inset shows results of the FFT analysis of the corresponding mx oscillations. (b) Vortex gyration frequency f as a function of D for the left- and right-handed vortices.

magnetic vortex, the DM field on the core can be calculated. For D > 0, the DM field is parallel to the core polarization for the right-handed vortex, while the opposite applies for the left-handed vortex. It has been reported that the out-ofplane field can increase the vortex gyration eigenfrequency when the field is parallel to the core polarization, and can also reduce the gyration eigenfrequency when the field is opposite to the polarization,29 and this field effect is similar to the effect of the DM field shown in Fig. 2(b). The DM field is highly localized around the core center because hðrÞ changes dramatically across the radius in the core. The spins outside the core lie almost in the film plane, and this results in a negligible DM field. Because of the localized nonuniform DM field, it remains difficult to derive a quantitative analytical relationship between the gyration eigenfrequency and the DMI strength. It is known that the in-plane oscillatory AC field can excite static gyration motion, and the gyration radius increases with increasing excitation field.18 When the AC field is sufficiently strong, the core gyration reaches its maximum radius rmax, and the core will then switch its polarization.18 Because DMI can lift the degeneration between vortices with opposite handedness, it is expected that both gyration motion excitation and core switching can also be affected by DMI. Here, we resonantly excited the vortex gyration using an in-plane AC field HAC at the vortex gyration eigenfrequency, as shown in Fig. 3(a). As HAC gradually increases, the core will be continuously driven to a

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larger gyration radius with higher velocity until HAC reaches the critical point HSW , then the vortex core becomes unstable and switches polarization. As shown in Fig. 3(b), without the DMI, the gyration radius increases gradually with HAC, and the vortex core switches its polarization at HSW 5 mT with a maximum gyration radius rgmax of 100 nm. However, for D ¼ 0.8 mJ/m2, HSW can reach 30 mT with rgmax 155 nm. The simulation results clearly demonstrate the strong effect of the DMI on both vortex gyration motion and core switching. Figures 3(c) and (d) summarize Hsw and rgmax for the vortex gyration motion as a function of the positive D value. Hsw decreases with increasing D for the left-handed vortex, but increases for the right-handed vortex. Notably, for the right-handed vortex, Hsw shows a rapid change for D values between 0.2 mJ/m2 and 0.5 mJ/m2. However, rgmax only increases continuously from 100 nm to 155 nm with increasing D, but then saturates at 155 nm. For D > 0.5 mJ/m2, the vortex core gyrates out of the disk and switches polarization. Because rgmax represents the maximum radius of static gyration motion, it therefore cannot be larger than 155 nm when considering a core size of 15 nm. The rapid change in Hsw for D values between 0.2 mJ/m2 and 0.5 mJ/m2 is likely to be related to the fact that the vortex core is gradually approaching the disk edge. The relationship between the DMI strength and the critical excitation field can also be attributed to the DM field. Figure 4(a) shows a typical snapshot image of the mz distribution during gyration motion and the mz profile across the vortex core for a moving right-handed vortex without DMI. Because of the gyrofield induced by the gyration motion,30 there is a negative dip beside the vortex core. This gyrofield increases with increasing gyration velocity, and thus the core polarity is reversed when the excitation is strong enough. Based on the definition of the gyrofield given in Ref. 30, we can calculate the distribution of the z-component of the gyrofield, which is localized around the vortex core. For comparison, Fig. 4(b) shows a snapshot image of the mz distribution and the mz profile across the core for a right-handed vortex with D ¼ 0.8 mJ/m2. In addition to the gyrofield, the DM field is also calculated using Eq. (3). It is clear that for a right-handed vortex, the DM field is parallel to the core polarization, and thus the DM field can stabilize the vortex core and prevent the formation of the negative dip beside the core. In this case, core switching for the disk with DMI requires a higher core gyration velocity to produce a stronger gyrofield,30 and then a stronger excitation field is needed, so both Hsw and rgmax increase with increasing D. However, different with the rapid change of Hsw for D values between 0.2 mJ/m2 and 0.5 mJ/m2, rgmax increases continuously with the D value. For a right-handed vortex, to realize vortex core switching, the gyrofield has to compete with the DM field, as shown in Fig. 4. According to Eq. (3), the DM field is proportional to D, thus both the critic gryofield and the critic vortex core gyration velocity for core switching should increase with D, since the gyrofield is proportional to the vortex core gyration velocity.30 Fig. 2(b) shows that the vortex gyration eigenfrequency increases almost linearly with D; thus, it is reasonable that rgmax also increases continuously


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FIG. 3. (a) Picture of the gyration motion of a vortex with radius rg excited by an in-plane AC resonant field HAC with gyration frequency f . The blue circle shows the vortex core trajectory. (b) Vortex core gyration radius rg as a function of HAC for vortices with D ¼ 0 mJ/m2 and 0.8 mJ/m2. The blue arrows indicate Hsw , at which core reversal occurs, and the horizontal dashed line indicates the location of the disk edge. (c) Critical field Hsw for core reversal as a function of D for the left- and right-handed vortices. The blue dashed lines are intended as guidelines. (d) Maximum gyration radius rmax as a function of D for the left- and right-handed vortices.

without a rapid change. For a left-handed vortex, the DM field direction is opposite to the core, which destabilizes the g core, and thus, as shown in Figs. 3(c) and 3(d), Hsw and rmax decrease with increasing DM strength until D ¼ 1.0 mJ/m2, above which the left-handed vortex is unstable, even in the static state. As shown in Fig. 1, the DMI can induce tilting magnetization at the disk edge, and this type of edge state can form an energy barrier for the core as it approaches the edge. In an earlier study of skyrmion dynamics in constricted geometries, Iwasaki et al.14 also found that the interface DMI could induce a potential barrier at the edge. For D < 0.2 mJ/m2, the core reversal happens far away from the disk edge, and Hsw increases with increasing D because of the DM field. When

D > 0.2 mJ/m2, the core can be driven to be close to the disk edge before switching, and then the DMI-induced edge state will play an additional role in the core gyration motion, which responds to the rapid change in Hsw shown in Fig. 3(c). For a right-handed vortex with D > 0.5 mJ/m2, the core can reach the boundary before reversing its polarization, and can then gyrate out of the disk to form a new core with opposite polarization at the edge.31 In this case, the vortex core reversal must overcome the edge state with an out-of-plane magnetization component that is opposite to the core polarization, and which is proportional to the strength of the DMI. IV. CONCLUSION

In summary, we studied the effects of the bulk-like DMI on magnetic vortex spin configuration and gyration dynamics. The DMI-induced edge magnetization tilting can be quantitatively described through analytical analysis. Micromagnetic simulations show that the DMI can lift the degeneracy of the vortex gyration frequency for both leftand right-handed vortices. Vortex core gyration excited by the in-plane AC resonant field was studied, and the DMI strongly influenced both the vortex gyration amplitude and the critical excitation field for core polarization reversal. The edge state induced by the DMI also influenced the dynamic properties. With DMI, a vortex with high stability and core gyration with large amplitude can be realized, and this may be useful in the development of vortex-based nanooscillators. ACKNOWLEDGMENTS FIG. 4. Representative mz profiles across the core and the distribution of the z-component of a DM field and a gyrofield for a right-handed vortex with (a) D ¼ 0 mJ/m2 and (b) D ¼ 0.8 mJ/m2. The insets show snapshot images of the spatial mz distribution during vortex gyration motion.

The authors would like to acknowledge Dr. Shulei Zhang and Professor Shufeng Zhang for helpful discussions with regard to this work. This work was supported by the National Key Basic Research Program (Grant No.


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2015CB921401) and the National Science Foundation (Grant Nos. 11274074, 11434003, and 11474066) of China, and by a National Research Foundation of Korea Grant funded by the Korean Government (Grant No. 2012R1A1A2007524). 1

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