APPLIED PHYSICS LETTERS 98, 012514 共2011兲
Magnetic vortex wall motion driven by spin waves Soo-Man Seo,1 Hyun-Woo Lee,2 Hiroshi Kohno,3 and Kyung-Jin Lee1,a兲 1
Department of Materials Science and Engineering, Korea University, Seoul 136-701, Republic of Korea Department of Physics, Pohang University of Science and Technology, Kyungbuk 790-784, Republic of Korea 3 Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan 2
共Received 16 September 2010; accepted 19 December 2010; published online 7 January 2011兲 A magnetic vortex wall motion driven by propagating spin waves in a nanostrip is investigated by means of micromagnetic simulation. Propagating spin waves can drive a vortex wall into a stream motion in spite of its complex internal spin structure. Compared to the transverse wall, the vortex wall moves faster and its velocity is less sensitive to the spin wave frequency. The amplitude of spin waves changes when passing through the domain wall, closely related to the domain wall velocity. This domain-wall-type-specific study provides important information for developing the theory of the interaction between domain wall and spin waves. © 2011 American Institute of Physics. 关doi:10.1063/1.3541651兴 Domain wall 共DW兲 motion in a patterned magnetic nanostrip has attracted considerable interest in the research field of spintronics due to its potential application to data storage1 and logic devices.2 The DW motion driven by magnetic fields or electric currents has been intensively explored. Recently, Han et al. numerically demonstrated an alternative means to drive the DW motion by propagating spin waves 共SWs兲 共Ref. 3兲 and Jamali et al. reported numerical results on current-induced DW motion assisted by SWs.4 However, the understanding about the interaction between SWs and DW is still incomplete, although SWs are ubiquitous in magnetic systems. The type of DW can be classified depending on the internal spin structure, i.e., a transverse wall 共TW兲 and a vortex wall 共VW兲. Due to the competition between the exchange and the magnetostatic energies, the DW-type is determined by the cross-sectional dimension of nanostrip.5,6 The DW motion driven by magnetic fields or electric currents significantly depends on the DW-type, such as the microscopic deformation of DW during its translational motion.7,8 In addition, for the electrical measurement, the anisotropic magnetoresistance signal from a DW is altered by the DW-type.9 Therefore, it is important to understand SW-induced DW motion for a different type of DW. In this work, we performed micromagnetic simulation to investigate VW motion driven by propagating SWs. In contrast with TW, VW has more complex spin structure such as in-plane spiral magnetization and out-of-plane magnetization at the center of DW. Nevertheless, we observed that propagating SWs can drive VW motion. Moreover, for the same strength of the magnetic field to generate SWs, the VW velocity is remarkably larger than the TW velocity. Also, the SW frequency dependence of the DW velocity shows a clear difference between the two types of DW. We found that the VW velocity is quantitatively correlated with the amplitude change of SWs. For the micromagnetic simulation, we used the Landau– Lifshitz–Gilbert 共LLG兲 equation describing the timedependent magnetization dynamics, a兲
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M M ␣ = − ␥M ⫻ Heff + . M⫻ t MS t
共1兲
where M is the vector of local magnetization, ␥ is the gyromagnetic ratio 共=1.76⫻ 107 s−1 Oe−1兲, ␣ is the Gilbert damping constant, and M S is the saturation magnetization. Heff is the effective field consisting of the magnetostatic, exchange, anisotropy, and external field. The Permalloy 共Ni80Fe20 , Py兲 nanostrip is 4 m long, 160 nm wide, and 20 nm thick as a model system depicted in Fig. 1. We used the rectangular unit cells of 4 ⫻ 4 ⫻ 20 nm3 and the standard material parameters of Py as follows: 800 emu/ cm3 for M S, 0.01 for ␣, 1.3⫻ 10−6 erg/ cm for Aex 共exchange constant兲. The crystalline anisotropy and the temperature were assumed to be zero. For a ground state, a VW was placed at the center of the nanostrip 共x = 2 m兲. By applying ac magnetic field Hext = H0 sin共2 f 0t兲yˆ at x = 200– 220 nm, SWs were generated and propagated along the long axis of nanostrip 共x-axis兲. The absorbing boundary condition was considered to prevent reflected SWs at the long edges of nanostrip.10,11 Also, the DW can be affected by the stray field caused by the dipole-dipole interaction as it gets closer to the end of the nanostrip. To exclude those effects, we assumed that artificial Py nanostrips of 20 m long are attached at both left and right edges. In Fig. 2共a兲, the position of VW as a function of the time is shown at various frequencies 共f 0兲 and amplitudes of ac field 共H0兲. In the early time stage, VW does not move since SWs do not arrive at the location of VW. For t ⬍ 10 ns, there are transient behaviors, indicating that the VW position does not linearly increases with time. These transient behaviors occur for both field- and current-driven DW motions. Figure
FIG. 1. 共Color online兲 Schematic view of model system. The VW is initially placed at the center of nanostrip. The thick arrow represents the externally applied ac field.
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FIG. 2. 共Color online兲 共a兲 VW position as a function of the time at various frequencies 共f 0兲 and amplitudes of ac field 共H0兲. 共b兲 VW velocity 共VW兲 and TW velocity 共TW兲 as a function of the frequency 共f 0兲 at H0 = 0.3 kOe.
2共b兲 shows the VW velocity 共VW兲 and the TW velocity 共TW兲 as a function of f 0 for H0 = 0.3 kOe. VW was evaluated from the averaged velocity in the time interval of 10–20 ns. Note that the cross-sectional dimension of nanostrip for the TW study is 40⫻ 5 nm2 and thus the aspect ratio 共=width/ thickness兲 is the same to the nanostrip for the VW study. For f 0 ⬍ 8 GHz, both types of DW do not move because SW propagation is not allowed due to the geometric confinement.10,12 At f 0 = 10.5 GHz, VW is maximum 共=5 m / s兲. Then, VW gradually decreases with increasing f 0. However, for a TW there are three distinct peaks of DW velocity, i.e., TW = 0.3, 0.37, and 0.1 m/s at f 0 = 17, 28, and 33.5 GHz, respectively, consistent with the previous report for a TW 共Ref. 3兲. Interestingly, we observed negative velocity corresponding to the backward motion of TW for f 0 = 10– 17 GHz. Note that the geometry of our nanowire is different from that used in Ref. 3. When we use the same geometry to Ref. 3, the negative velocity was not observed. For f 0 = 10– 17 GHz, we found that the SW amplitude does not change when SWs pass through TW, indicating no interaction between SW and TW 共see discussion related to Fig. 3兲. We attribute this negative velocity to the symmetry breaking of dipolar field due to SW excitations on the left side of TW. Note that the effective magnetization is reduced at the position of SW source. It breaks the symmetry of dipolar fields and causes the TW motion toward SW source. From Fig. 2共b兲, one finds that SWs more effectively drive VW into a stream motion in spite of more complex spin structure and wider width of DW compared to TW. In addition, the frequency dependence of VW velocity is qualitatively different from that of TW velocity. For TW, a resonant frequency for the TW motion is pronounced due to the exis-
FIG. 3. 共Color online兲 Normalized M y共=M y / M S兲 as a function of the time for 共a兲 f 0 = 20 GHz and x = 0.8 m, 共b兲 f 0 = 10 GHz and x = 0.8 m, 共c兲 f 0 = 20 GHz and x = 3.2 m, and 共d兲 f 0 = 10 GHz and x = 3.2 m. For all cases, H0 = 0.3 kOe. The black and red lines correspond to the cases in the absence of the VW 共“without VW”兲 and in the presence of the VW 共“with VW”兲, respectively.
Appl. Phys. Lett. 98, 012514 共2011兲
FIG. 4. 共Color online兲 Log-scaled FFT amplitude of M z vs position 共x兲 at 共a兲 f 0 = 10 GHz and 共b兲 f 0 = 20 GHz. 共c兲 The amplitude change, ⌬AFFT = AFFT 共without VW兲—AFFT 共with VW兲, as a function of the position 共x兲 for f 0 = 10, 20 GHz. 共d兲 VW velocity 共VW兲 and averaged ⌬AFFT as a function of the frequency 共f 0兲. For all cases, H0 = 0.3 kOe.
tence of the inherent mode between SWs and TW 共Ref. 3兲. In contrast, VW is less sensitive to f 0. In order to understand the frequency dependence of VW motion, we investigated the dynamics of local magnetizations. Figure 3 shows the normalized M y共=M y / M S兲 as a function of the time. In all graphs, the black and red lines correspond to the cases in the absence of VW 共“without VW”兲 and in the presence of VW 共“with VW”兲, respectively. It is observed that the time-dependent dynamics depends on the presence of VW and the position 共x兲. At x = 0.8 m, the amplitude is almost the same for both cases of without VW and with VW. On the other hand, at x = 3.2 m, the amplitude is evidently smaller for the case of with VW. This means that the amplitude of SWs is reduced after passing through the VW. The amplitude reduction at f 0 = 10 GHz 共VW = 2.67 m / s兲 is larger than that at f 0 = 20 GHz 共VW = 0.64 m / s兲. Therefore, the amount of the amplitude change would be correlated with the velocity of VW. For a quantitative analysis of the relationship between the amplitude change of SWs and the VW velocity, we conducted FFT 共fast Fourier transformation兲 for the timedependent dynamics of the magnetizations. Figures 4共a兲 and 4共b兲 show log-scaled spatially resolved FFT amplitude of M z calculated from the f 0-spectral power 共examples are shown in the right side images of Fig. 5兲 averaged over the nanostrip width 共y-axis兲, for the time window of 0–20 ns. Note that the FFT amplitude 共AFFT兲 means the SW amplitude 共ASW兲. For the case of without VW 关black lines in Figs. 4共a兲 and 4共b兲兴, the log-scaled AFFT linearly decreases with respect to the position 共x兲. This is the intrinsic property of SW propagation, where ASW exponentially attenuates due to the intrinsic damping. For the case of with VW 关red lines in Figs. 4共a兲 and 4共b兲兴, the peak of AFFT at x = 2 m is caused by the presence of VW. At f 0 = 10 GHz 关Fig. 4共a兲兴, small oscillations are observed on the left side of VW 共i.e., x ⬍ 2 m兲 关blue dotted circle in Fig. 4共a兲兴, implying the reflected SWs from the VW. The difference of AFFT between “before-DW 共x ⬍ 2 m兲” and “after-DW 共x ⬎ 2 m兲” is clearly observed. For x ⬍ 2 m, the AFFT of two cases 共with VW and without VW兲 almost coincide. In contrast, for x ⬎ 2 m, AFFT is smaller when a VW present. The value of the reduced amplitude, ⌬AFFT = AFFT 共without VW兲—AFFT 共with VW兲, as a function of the position 共x兲 is plotted in Fig. 4共c兲. ⌬AFFT at f 0 = 10 GHz is evidently larger than that at f 0
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FIG. 5. 共Color online兲 Top-view of spatial distribution of normalized M z at t = 20 ns and corresponding f 0-spectral images with the time window of 20 ns for 共a兲 without VW 共f 0 = 10 GHz兲, 共b兲 with VW 共f 0 = 10 GHz兲, 共c兲 without VW 共f 0 = 20 GHz兲, and 共d兲 with VW 共f 0 = 20 GHz兲.
= 20 GHz. For a direct comparison between VW and ⌬AFFT, ⌬AFFT is averaged for the range of x = 2.5– 3.5 m. The averaged ⌬AFFT and VW as a function of f 0 are plotted in Fig. 4共d兲. It is found that the f 0-dependence of ⌬AFFT and VW are in good agreement. This means that the VW motion via coupling with SWs can be quantified by the change of ASW. However, we found that the agreement between ⌬AFFT and VW becomes worse for f 0 ⬎ 18 GHz. We attribute this unsatisfactory agreement to the SW quantization along the nanostrip width. Figure 5 shows spatial distributions of normalized M z at t = 20 ns and corresponding f 0-spectral images calculated by FFT with the time window of 20 ns. At f 0 = 10 GHz, the excited mode of SWs is monochromatic over the whole sample, and this feature is maintained after coupling with the VW 关Figs. 5共a兲 and 5共b兲兴. At f 0 = 20 GHz, in contrast, ASW are periodically reduced along the edge as well as in the middle of the nanostrip, as indicated by oblique arrows in Fig. 5共c兲. It is also observed from the spatially resolved FFT 关black arrows in Fig. 4共b兲兴. The periodicity of these localized modes is much larger than the wavelength of the propagating mode of the SWs. This behavior was experimentally confirmed and well described in Ref. 13, which is caused by the SW quantization along the nanostrip width. The quantization modes can be described by the wavevector along the y-axis, ky = n / w, where n is an odd integer and w is the nanostrip width. Thus, when the SW frequency 共=energy兲 is large enough to excite the first excited mode corresponding to n = 3, the lowest mode 共n = 1兲 and the first excited mode 共n = 3兲 interfere with each other. Note that in the nanostrip used for VW study, the first excited mode appears at 18 GHz. It is clearly observed that the dynamics of the SWs becomes more complex after coupling with the VW 关denoted by the dotted square in Fig. 5共d兲兴, which results in the reduction of the coupling efficiency and is the reason of less satisfactory agreement between ⌬AFFT and VW at above 18 GHz. We also note that in the nanostrip for TW study, the SW quantization is observed at much higher SW frequency compared to the case of VW case since the nanostrip width is smaller and thus a higher energy is required to develop the first excited mode. We also examined the VW motion depending on its polarity 共p兲 = +1 or ⫺1 共core pointing out of the plane or into
the plane兲 and chirality 共q兲 = +1 or ⫺1 共clockwise or counterclockwise of the rotating sense of the magnetization inside the VW兲. We found that the velocity is independent of p and q because there is no vertical movement of the vortex core during its translational motion along the x-axis. To conclude, the VW motion can be driven by the propagating SWs. For the same amplitude of ac field generating the SWs, the VW moves faster than the TW. From the FFT analysis of SWs, we found that the VW velocity is quantitatively correlated with the change of ASW. However, the quantized SW modes are developed for higher SW frequencies, which makes the coupling efficiency between SWs and DW be reduced. As a result, the quantitative correlation between ⌬AFFT and VW becomes worse at higher frequencies. This work was supported by the Korea 共NRFK兲-Japan 共JSPS兲 Joint Research Project, the Fundamental R&D Program for Core Technology of Materials funded by the Ministry of Commerce, Industry and Energy, Republic of Korea, the DRC Program funded by KRCF, and the National Research Foundation of Korea 共NRF兲 grant funded by the Korea government 共MEST兲 共Grant No. 2010-0023798兲. S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 共2008兲. D. A. Allwood, G. Xiong, M. D. Cooke, C. C. Faulkner, D. Atkinson, N. Vernier, and R. P. Cowburn, Science 296, 2003 共2002兲. 3 D.-S. Han, S.-K. Kim, J.-Y. Lee, S. Hermsdoerfer, H. Schiltheiss, B. Levan, and B. Hillebrands, Appl. Phys. Lett. 94, 112502 共2009兲. 4 M. Jamali, H. Yang, and K.-J. Lee, Appl. Phys. Lett. 96, 242501 共2010兲. 5 R. D. McMichael and M. J. Donahue, IEEE Trans. Magn. 33, 4167 共1997兲. 6 Y. Nakatani, A. Thiaville, and J. Miltat, J. Magn. Magn. Mater. 290–291, 750 共2005兲. 7 J.-Y. Lee, K.-S. Lee, S. Choi, K. Y. Guslienko, and S.-K. Kim, Phys. Rev. B 76, 184408 共2007兲. 8 S.-M. Seo, K.-J. Lee, W. Kim, and T.-D. Lee, Appl. Phys. Lett. 90, 252508 共2007兲. 9 M. Hayashi, L. Thomas, C. Rettner, R. Moriya, X. Jiang, and S. S. P. Parkin, Phys. Rev. Lett. 97, 207205 共2006兲. 10 S.-M. Seo, K.-J. Lee, H. Yang, and T. Ono, Phys. Rev. Lett. 102, 147202 共2009兲. 11 D. V. Berkov and N. L. Gorn, J. Appl. Phys. 99, 08Q701 共2006兲. 12 K. Y. Guslienko, R. W. Chantrell, and A. N. Slavin, Phys. Rev. B 68, 024422 共2003兲. 13 V. E. Demidov, S. O. Demokritov, K. Rott, P. Krzysteczko, and G. Reiss, Phys. Rev. B 77, 064406 共2008兲. 1 2
