Dependence of Transverse Domain Wall Dynamics on ... - IEEE Xplore

IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 5, MAY 2010

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Dependence of Transverse Domain Wall Dynamics on Permalloy Nanowire Dimensions Matthew T. Bryan1 , Thomas Schrefl1;2 , and Dan A. Allwood1 Department of Engineering Materials, University of Sheffield, Sheffield S1 3JD, U.K. St. Poelten University of Applied Sciences, 3100 St. Poelten, Austria Micromagnetic modeling was used to investigate transverse domain wall propagation through Permalloy nanowires of width, , between 10 and 120 nm and thickness, , between 2 and 20 nm. The transverse wall mobility below Walker breakdown, , and Walker breakdown field, , depended on the exact wire dimensions, rather than simply a function of , as suggested by 1-D analytical modeling. Empirical equations describing the and dependence of and were found from the micromagnetic data. As the model for transverse domakes no assumptions about magnetization uniformity, these equations provide a generic description of and main walls in Ni80 Fe20 nanowires.

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Index Terms—Mobility, thickness, transverse domain wall, Walker breakdown, width.

I. INTRODUCTION

ECENT interest in magnetic nanowire domain wall applications [1]–[3] has led to several studies of domain wall motion due to applied magnetic fields and spin-polarized currents [4]–[10]. The general field-dependence of the domain wall velocity has been identified [6] and is well understood theoretically [11], [12]. At low fields, the domain wall velocity, , increases almost linearly with the applied axial magnetic , until reaching a maximum at the Walker breakdown field, . Above the Walker breakdown field, wall motion befield, comes nonuniform and the wall structure regularly changes [9], [12]. These characteristics of Walker breakdown are periodic in smooth-sided wires. When edge roughness is present, the periodicity is degraded [10]. Just above the Walker breakdown field, wall motion against the drive field causes negative differential domain wall mobility [6]. At higher fields, the retrograde motion is less significant, leading to a recovery of positive wall mobility, although the high-field mobility is lower than the [6]. mobility below Experimental measurements of the field-driven domain wall velocity in nanowires can vary dramatically. For example, two different studies of Ni Fe (Permalloy) nanowires reported domain wall velocities of 100 m/s and 1500 m/s at the same drive field [5], [6]. The difference between the measured velocities may be due to the Walker breakdown field in the wires [6], which had different dimensions. In addition, it is likely that the structure of the domain walls in two studies were different [13], [14], which may also have affected the wall velocity [13]. Whereas some micromagnetic studies have indicated that both the wall velocity and the Walker breakdown field are dependent

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Manuscript received July 15, 2009; revised November 12, 2009; accepted January 08, 2010. First published March 08, 2010; current version published April 21, 2010. Corresponding author: M. T. Bryan (e-mail: [email protected]. uk). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2010.2040622

on the wire width and thickness [13], [15]–[17], the precise relationship has not been studied. Indeed, current understanding of the effect of the wire dimensions on the domain wall dynamics is largely based on one-dimensional (1-D) analytical models. In the 1-D models, magnetization is assumed to be uniform and the wall mobility throughout the wire cross-section and below Walker breakdown, , are described by [16] (1) (2) where is the Gilbert damping constant, is the saturation magnetization, and are demagnetization factors along is the gyromagnetic ratio and the wire width and thickness, is the wall width. The wall width is dependent on and , as well as the applied field; and depend only on the width-to-thickness ratio [16]. Therefore, in the 1-D model wires with the same width-to-thickness ratio behave identically, regardless of their absolute dimensions. Although the 1-D models predict the general characteristics of domain wall motion, they do not adequately describe motion above the Walker breakdown field, where the domain wall periodically alternates between a transverse and a vortex or anti-vortex structure [12]. Here, the relationship between the wire dimensions and the domain wall dynamics is studied in detail using micromagnetic simulations. Equations are fitted to the data, in order to provide insight into dynamic effects present in wires. Edge roughness, which is understood to increase the Walker breakdown field, suppress negative differential domain wall mobility and cause variable Walker breakdown periodicity [10], [18], is not included in the model presented here, in order to concentrate on the effects of the wire dimensions. II. MODEL SETUP Smooth-sided wires of length , width – nm, and thickness – nm were modeled using a finite-element method to solve the Landau-Lifshitz-Gilbert equation [19]. Wire dimensions

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Fig. 1. The change in the time-averaged domain wall velocity with applied field in Permalloy wires of different width (w ) and thickness (T ). The arrows indicate the Walker breakdown fields.

were defined such that in all wires studied. This allowed us to systematically study the effect of the wire , cross section on domain wall motion under a drive field, applied along the wire axis. The wires were discretized into a tetrahedral mesh with a maximum cell size of 5 nm and the material parameters of Permalloy were used (exchange stiffness constant, Jm ; magneto-crystalline Jm ; kAm ; ). anisotropy, Head-to-head transverse domain walls were initialized 400 nm from one end of the wire and were driven to the other end of remained constant during wall propagation. The the wire. time-averaged domain wall velocity was calculated from the rate of change of the average magnetization along the axis of the wire [16], excluding the transient time required for the wall to accelerate to its maximum instantaneous velocity from its initial (stationary) state. III. RESULTS AND DISCUSSION The field-dependence of the average domain wall velocity is illustrated for a few wires of different widths and thicknesses in Fig. 1. Individually, each wire displays the general trends of field-driven domain wall motion seen previously [6], [13]. However, the effect of the wire dimensions on and is more complicated than that suggested by (1) and (2), as wires with the same width-to-thickness ratio do not behave identically. For exis significantly larger in the 25 nm wide, 5 nm thick ample, wires than the 50 nm wide, 10 nm thick wires, even though the two wires have similar . Fig. 1 shows that decreases and increases for wires of similar thickness but increasing width – nm, nm). However, when the wire width ( and remains constant and the thickness is increased, both decrease (Fig. 1, nm, – nm). As and have different dependencies on the wire dimensions, no two wires behave identically. Similar trends can be seen in previous studies that were performed using either a constant [13] or [16]. By systematically varying both and , the effects of the wire shape on the domain wall dynamics can be examined in detail. Fig. 2(a) shows the dependence of the wall mobility on the width-to-thickness ratio, calculated using the micromag-

Fig. 2. The domain wall mobility, , at H = 10 Oe (below H ) for Permalloy wires of various widths, w , and thicknesses, T , as a function of (a) w=T and (b) w =T . The solid line shows wall mobility calculated using a 1-D model. The dotted line, described by (3), is a linear fit of the entire data set.

netic and 1-D models. The wall mobility is calculated at , ensuring that the walls were propagated below the Walker breakdown field in all wires. In general, the wall mobility increases as the width-to-thickness ratio increases. Agreement between the 1-D and micromagnetic models is strongest in the narrowest wires, suggesting that the wire may nm. be considered truly “one-dimensional” when However, in wider wires, the micromagnetically calculated wall structure gradually becomes nonuniform across the wire width and becomes increasingly divergent from that predicted by the 1-D model. In addition, the dependence of on the width-to-thickness ratio changes as the wire width increases, suggesting that the micromagnetic wall mobility is determined by the absolute, rather than relative, values of the wire width and thickness. For example, at , increases by 30% as increases from 25 nm to 75 nm. Despite the inaccuracy of the 1-D model, the analytical expressions it provides are useful for understanding the physical processes governing domain wall motion. Further insight can be achieved by analyzing the micromagnetically calculated data to find relations between the dynamic properties of domain walls and the wire width at constant thickness (and similarly for the wire thickness at constant width). As these are empirical relations describing transverse wall motion, they may not hold for , where vortex domain walls or for square cross sections the configuration of the domain wall changes such that the magnetization lies along a diagonal [20]. Fig. 2(b) shows as a func. For all of the wires studied, the wall tion of the ratio Oe is described by mobility at

(3)

BRYAN et al.: DEPENDENCE OF TRANSVERSE DOMAIN WALL DYNAMICS ON PERMALLOY NANOWIRE DIMENSIONS

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This predicts transverse wall velocities that are approximately Oe) [13], contwice of those calculated previously (at sistent with the fact that the damping constant in that study was twice that used here [17]. The mobility calculated from (3) for a 200 nm wide, 5 nm thick wire is approximately twice the mobility seen experimentally [21]. However, the experimental result shows evidence of domain wall motion via thermal activation over multiple energy barriers, which may account for the reduced mobility. Equation (3) indicates that the wall mobility is more sensitive to the wire width than thickness, which may account for the increasing divergence between the micromagnetic and 1-D models as increases. The power law dependence of the mobility on the thickness predicted here is substantially different from that predicted for thin films containing Néel walls . This suggests that the wall type [22], where depends on strongly influences the response of the domain wall. The effect of the wire dimensions on the Walker breakdown field is shown in Fig. 3. It shows that as either wire width decreases. [Fig. 3(a)] or thickness [Fig. 3(b)] is increased, This is observed, but not commented on in previous studies [13]. However, due to the systematic variation of both and , and the it is possible to quantify the relationship between decreases exponenwire dimensions. Fig. 3(c) shows that tially with and , approximating to the relation (4) at all wire dimensions studied (within 3 Oe). This approximation is in reasonable agreement with other models that use [18], but agrees less the same damping constant well with models by the same authors with different damping constants [13] (even assuming is proportional to [17]). The form of (4) suggests that the Walker breakdown field is an order of magnitude more sensitive to the wire thickness than the width. These results have particular implications for the miniaturization of magnetic devices. If both and are scaled down, reduces, indicating that will also reduce, while will increase. However, previous from (4) it is clear that discussions of miniaturization [3] have suggested that to maintain thermal stability below a certain wire width will require to increase. This change in shape anisotropy will reduce fur. These factors could mean that improved ther and reduce performance with miniaturized nanowire devices is limited by the magnetization dynamics, over and above overcoming fabrication and engineering obstacles.

Fig. 3. The effect of (a) the wire width, w , and (b) the wire thickness, T , on the Walker breakdown field, H , for several wire dimensions. (c) H as an exponential function of w and T . The dashed line, described by (4), is a fit of all the data.

IV. CONCLUSION

ACKNOWLEDGMENT

We have used micromagnetic modeling to show that the dynamics properties of domain walls in Permalloy nanowires are dependent on the absolute values of the wire width and thickness. Despite this complexity, the effect of the wire dimensions on the Walker breakdown field and mobility below the Walker breakdown field could be summarized into reasonably simple functions of the width and thickness. As the micromagnetic model includes the effect of a nonuniform wall

This work was supported by EPSRC under Grant GR/T02942/01.

magnetization across the wire width, these equations provide a more realistic description of the wall dynamics than the 1-D analytical models that have previously been used to analyze wall motion. Although 1-D equations are useful aids to understanding, their predictions become increasingly divergent from those of micromagnetic models in wires of width larger than 10 nm. Using the equations presented here allows detailed comparisons between studies using different wire dimensions. We are currently working on experiments for comparison with these results. Further modeling is required to understand the influence of wire dimensions on field-drive vortex wall propagation and on current-induced wall motion. This work will guide the design and miniaturization of devices based on field-induced movement of domain walls.

REFERENCES [1] R. D. McMichael, J. Eicke, M. J. Donahue, and D. G. Porter, “Domain wall traps for low-field switching of submicron elements,” J. Appl. Phys., vol. 87, p. 7058, 2000. [2] M. Diegel, R. Mattheis, and E. Halder, “360 degree domain wall investigation for sensor applications,” IEEE Trans. Magn., vol. 40, no. 4, pp. 2655–2657, Jul. 2004.

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[3] D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P. Cowburn, “Magnetic domain-wall logic,” Science, vol. 309, pp. 1688–1692, 2005. [4] T. Ono, H. Miyajima, K. Shigeto, K. Mibu, N. Hosoito, and T. Shinjo, “Propagation of a magnetic domain wall in a submicrometer magnetic wire,” Science, vol. 284, p. 468, 1999. [5] D. Atkinson, D. A. Allwood, G. Xiong, M. D. Cooke, C. C. Faulkner, and R. P. Cowburn, “Magnetic domain-wall dynamics in a submicrometre ferromagnetic structure,” Nature Mater., vol. 2, p. 85, 2003. [6] G. S. D. Beach, C. Nistor, C. Knutson, M. Tsoi, and J. L. Erskine, “Dynamics of field-driven domain-wall propagation in ferromagnetic nanowires,” Nature Mater., vol. 4, p. 741, 2005. [7] G. S. D. Beach, C. Knutson, C. Nistor, M. Tsoi, and J. L. Erskine, “Nonlinear domain-wall velocity enhancement by spin-polarized electric current,” Phys. Rev. Lett., vol. 97, p. 057203, 2006. [8] M. Hayashi, L. Thomas, Y. B. Bazaliy, C. Rettner, R. Moriya, X. Jiang, and S. S. P. Parkin, “Influence of current on field-driven domain wall motion in permalloy nanowires from time resolved measurements of anisotropic magnetoresistance,” Phys. Rev. Lett., vol. 96, p. 197207, 2006. [9] M. Hayashi, L. Thomas, C. Rettner, R. Moriya, and S. S. P. Parkin, “Direct observation of the coherent precession of magnetic domain walls propagating along permalloy nanowires,” Nature Phys., vol. 3, p. 21, 2007. [10] S. Glathe, R. Mattheis, and D. V. Berkov, “Direct observation and control of the Walker breakdown process during a field driven domain wall motion,” Appl. Phys. Lett., vol. 93, p. 072508, 2008. [11] N. L. Schryer and L. R. Walker, “Motion of 180 degrees domain-walls in uniform Dc magnetic-fields,” J. Appl. Phys., vol. 45, p. 5406, 1974. [12] J.-Y. Lee, K.-S. Lee, S. Choi, K. Y. Guslienko, and S.-K. Kim, “Dynamic transformations of the internal structure of a moving domain wall in magnetic nanostripes,” Phys. Rev. B, vol. 76, p. 184408, 2007.


[13] Y. Nakatani, A. Thiaville, and J. Miltat, “Head-to-head domain walls in soft nano-strips: A refined phase diagram,” J. Magn. Magn. Mater., vol. 290, p. 750, 2005. [14] M. Laufenberg, D. Backes, W. Buhrer, D. Bedau, M. Klaui, U. Rudiger, C. A. F. Vaz, J. A. C. Bland, L. J. Heyderman, F. Nolting, S. Cherifi, A. Locatelli, R. Belkhou, S. Heun, and E. Bauer, “Observation of thermally activated domain wall transformations,” Appl. Phys. Lett., vol. 88, p. 052507, 2006. [15] Y. Nakatani, N. Hayashi, T. Ono, and H. Miyajima, “Computer simulation of domain wall motion in a magnetic strip line with submicron width,” IEEE Trans. Magn., vol. 37, no. 4, pp. 2129–2131, Jul. 2001. [16] D. G. Porter and M. J. Donahue, “Velocity of transverse domain wall motion along thin, narrow strips,” J. Appl. Phys., vol. 95, p. 6729, 2004. [17] A. Kunz, “Simulating the maximum domain wall speed in a magnetic nanowire,” IEEE Trans. Magn., vol. 42, no. 10, p. 3219, Oct. 2006. [18] Y. Nakatani, A. Thiaville, and J. Miltat, “Faster magnetic walls in rough wires,” Nature Mater., vol. 2, p. 521, 2003. [19] T. Schrefl, M. E. Schabes, D. Suess, and M. Stehno, “Dynamic micromagnetic write head fields during magnetic recording in granular media,” IEEE Trans. Magn., vol. 40, no. 4, pp. 2341–2343, Jul. 2004. [20] A. Thiaville, J. M. Garcia, and J. Miltat, “Domain wall dynamics in nanowires,” J. Magn. Magn. Mater., vol. 242, p. 1061, 2002. [21] D. Atkinson, D. A. Allwood, C. C. Faulkner, G. Xiong, M. D. Cooke, and R. P. Cowburn, “Magnetic domain wall dynamics in a permalloy nanowire,” IEEE Trans. Magn., vol. 39, no. 5, pp. 2663–2665, Sep. 2003. [22] I. Ruiz-Feal, T. A. Moore, L. Lopez-Diaz, and J. A. C. Bland, “Model for reversal dynamics of ultrathin ferromagnetic films,” Phys. Rev. B, vol. 65, p. 054409, 2002.