Nonlinear magnetization dynamics in a ferromagnetic nanowire with spin current

Peng-Bin He and W. M. LiuJoint Laboratory of Advanced Technology in Measurements, Beijing National Laboratory for Condensed Matter Physics,

Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China�Received 16 November 2004; revised manuscript received 22 June 2005; published 4 August 2005�

We investigate the effect of spin transport on nonlinear excitations in a ferromagnetic nanowire with non-uniform magnetizations. In the presence of spin-polarized current, the exact soliton solutions propagating alongthe wire’s axis are obtained in two cases of high or low quality factors �Q�. The current can change thevelocities of magnetic solitons and affect the solitons’ collision with phase shift. Also, ac current inducesvibration of the center of the solitons.

DOI: 10.1103/PhysRevB.72.064410 PACS number�s�: 75.75.�a, 05.45.Yv, 72.25.Ba, 73.21.Hb

I. INTRODUCTION

In magnetic multilayers, the spin-polarized current causesmany unique phenomena, such as spin-wave excitation,1

magnetization switching and reversal,2–6 and enhanced Gil-bert damping.7 These phenomena attribute to the spin-transfer effect, proposed by Slonczewski8 and Berger.9 Mac-roscopically, the magnetization dynamics in the presence ofspin-polarized current can be described by a generalizedLandau-Lifshitz-Gilbert �LLG� equation, including terms re-lated to spin-polarized current, spin accumulation, and spin-transfer torque. Bazaliy et al. considered spin-polarized cur-rent in half-metal and reduced the current effects as atopological term in the LLG equation.10 The spin torque hasdifferent forms for uniform magnetization and nonuniformones. In the uniform case, such as spin-valve structure, the

spin torque is �a=−�aJ /Ms�M� �M�M̂p� , where M is the

magnetization of the free layer, M̂p is the unit vector alongthe direction of the magnetization of the pinned layer, Ms isthe saturation magnetization, and aJ is a model-dependentparameter and is proportional to the current density.8 For thenonuniform magnetization, Li and Zhang gave a spin torquetaking form �b=bJ�M/�x, where bJ depends on the materi-als parameters and current.11 The spin torque is very differentfrom precession and damping terms in the LLG equation. Itcan induce not only precession but also damping. Due to theunique property of the spin torque, the magnetization dynam-ics has been extensively studied in spin-valve pillarstructures,12,13 magnetic nanowires geometry,14 and point-contact geometry.15

Nonlinear excitations are general phenomena in magneticordered materials. Extensive investigations have been madein insulated thin films for easy excitation and detection, suchas YIG.16 In confined magnetic metals, nonlinear excitationsare also interesting due to the interaction between spin-polarized current and local magnetizations. The solitary ex-citations of isotropic ferromagnets in the present of spin-polarized current is well studied using the inverse scatteringmethod.17

In this paper, we will discuss the linear and nonlinearmagnetization dynamics in a uniaxial ferromagnetic nano-wire injected with current. By stereographic projection of theunit sphere of magnetization onto a complex plane,18 we

transform the LLG equation into a nonlinear equation ofcomplex function and obtain two kinds of soliton solutionspropagating along the wire’s axis for a high-Q model and alow-Q one �Q=Hk / �4�Ms� is the quality factor�. From thesesolutions, we study the effect of a spin-polarized current onthe nonlinear excitation of the magnetization in the ferro-magnetic metal nanowire.

II. HIGH-Q CASE

We consider a very long ferromagnetic nanowire with auniform cross section, as shown in Fig. 1. To excellent ap-proximation, this nanowire can be viewed as infinite inlength. The electronic current flows along the long length ofthe wire, defined as the x direction. The z axis is taken as thedirections of the uniaxial anisotropy field and the externalfield. Also, we assume the magnetization is nonuniform onlyin the direction of current. The current injected in the nano-wire can be polarized and produces a spin torque acting onthe local magnetization, which is written as �b=bJ�M/�x,where bJ= Pje�B / �eMs�, P is the spin polarization of thecurrent, je is the electric current density and flows along thex direction, �B is the Bohr magneton, and e is the magnitudeof electron charge.11 So, the modified LLG equation with thisspin torque is

�M

�t= − �M � Heff +

�

MsM �

�M

�t+ �b, �1�

where � is the gyromagnetic ratio, Ms is the saturation mag-netization, � is the Gilbert damping parameter, and Heff isthe effective magnetic field including the external field, the

FIG. 1. Geometry of the nanowire.

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anisotropy field, the demagnetization field, and the exchangefield. This effective field can be written as Heff= �2A /Ms

2��2M/�x2+ ��HK /Ms−4��Mz+Hext�ez, where A isthe exchange constant, HK is the anisotropy field, Hext is theapplied field, and ez is the unit vector along the z direction.

When the uniaxial anisotropy field is larger than the de-magnetization field, namely, Q�1 �This is the case for somemetallic magnetic films19 and multilayers20�, we can get darkmagnetic solitons for the mz component. After rescaling thetime coordinate and space coordinate by characteristic timet0=1/ ���HK−4�Ms�� and length l0=�2A / ��HK−4�Ms�Ms�and taking M=Msm , the LLG equation can be simplified as

�m

�t= − �m �

�2m

�x2 � − �mz +Hext

HK − 4�Ms��m � ez�

+ �m ��m

�t+

bJt0

l0

�m

�x. �2�

Considering the fact that the magnitude of the magnetizationis a constant at temperatures well below the Curie tempera-ture, namely, m2= �M/Ms�2=1, it is reasonable to take ste-reographic transformation,18 �= �mx+ imy� / �1+mz�. Then,the equation about the complex function � is obtained,

i�1 − i����

�t=

�2�

�x2 − 2�*

1 + ��*� ��

�x�2

−Hext

HK − 4�Ms�

−1 − ��*

1 + ��*� + ibJt0

l0

��

�x. �3�

We will get linear and nonlinear dynamics of magnetizationfrom �.

As shown by Li and Zhang,11 the spin-wave solution ofthe LLG equation is obtained in the presence of spin-polarized current. In the high-magnetic field, the deviation ofmagnetization from the the direction of the field is small. So,we only consider low-energy excitation. We take the spin-wave ansatz

m = �mxex + myey�ei�k·r−t� + ez, �4�

corresponding to �=�0ei�k·r−t�, where �0= �mx+ imy� /2,and mx and my represent the rescaled amplitude of the spinwaves in the x and y directions. Inserting � into Eq. �3� andkeeping the linear terms in �0, we obtain the dispersion re-lation of the spin wave

= �1 + i��0�1 + l0�2�kx − bJMs/�A�2� , �5�

where

0 = ���HK − 4�Ms + Hext� − bJ2Ms/2�A�/�1 + �2� ,

�6�l0�

2 = 16�2A2/�8�2MsA�HK − 4�Ms + Hext� − Ms2bJ

2� .

It is easy to see that the spin-polarized current changes theenergy gap. For large current density, je

�e / ��P��2AMs�HK−4�Ms+Hext�, the gap vanishes. Thismeans that without other stimulation, such as thermal effects,the spin-polarized current excites spin waves when its den-sity increases beyond a critical value. The Gilbert dampinggives the energy gap a rescaling and damps the spin wave.

Now we investigate the nonlinear excitation of LLG equa-tions. Considering small deviations of magnetization fromthe equilibrium direction �along the anisotropy axis� corre-sponding to mx

2+my2 mz

2, or, ���2 1, and taking the long-wavelength approximation,21 where l0k 1, and k is thewave vector of spin wave that constitutes nonlinear excita-tions, Eq. �3� can be simplified, by keeping only the nonlin-ear terms of the order of the magnitude of ���2�. Withoutdamping, we obtain the following nonlinear Schrödingerequation

i��

�t=

�2�

�x2 − �1 +Hext

HK − 4�Ms�� + 2���2� + ibJ

t0

l0

��

�x.

�7�

By the Hirota bilinear method,22 the one- and two-solitonsolutions of Eq. �7� are easily obtained. The one-soliton so-lution is �=kRei��I+�1+Hext/�HK−4�Ms��t� / cosh��R+C /2�, where�=k�x+ �bJt0 / l0− ik�t�, eC=e2��� / �4kR

2�, the subscripts R and Idenote the real and imaginary parts, and k is an arbitrarycomplex parameter and can be determined by the initial soli-ton location and amplitude. From �, we obtain the solutionof magnetization,

mx =2kR

�cos�I + �1 +

Hext

HK − 4�Ms�tcosh�R +

C

2 ,

my =2kR

�sin�I + �1 +

Hext

HK − 4�Ms�tcosh�R +

C

2 ,

mz =1

�cosh2��R +

C

2� − kR

2 , �8�

where �=cosh2��R+C /2�+kR2 .

This one-soliton solution represents a 2� domain wallwhich is the bound state of a double � wall.23 The height ofthe soliton is the rescaled magnetization. The velocity the ofwall is V=−bJ−2kIl0 / t0. As we see it, the spin-polarized cur-rent alters the soliton velocity as −bJ, proportional to theelectron current density. The external fields induce the mxand my components of the soliton to oscillate periodically.If applying alternating currents, bJ= Pje�B / �eMs�cos�t�,and the only change to the solutions is �=k�x+bJ / �l0�sin�t0t�− ikt�. The center of the mass of solitonsvibrate with the same frequency as ac currents. Inversely, weconsider the effect of nonlinear excitations on the spin-polarized current. In ferromagnetic material with nonuniformmagnetizations and adiabatic approximation, the spin-currentdensity is jx= ��B /e�Pjem11. Thus, the one-soliton excitationin nanowire gives spin current a magnetic pulse like theoptical pulse in nonlinear media. In the same way, we canget the two-soliton solutions �= �g1+g3� / �1+ f2

+ f4�ei�1+Hext/�HK−4�Ms��t, and nonlinear dynamics of magneti-zation,

mx =� + �*

1 + ��* , my = − i� − �*

1 + ��* , mz =1 − ��*

1 + ��* , �9�

where

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g1 = �1e�1 + �2e�2,

g3 = e�1+�1*+�2+�1 + e�1+�2

*+�2+�2,

f2 = e�1+�1*+R1 + e�2+�2

*+R2 + e�1+�2*+ + e�1

*+�2+*,

f4 = e�1+�1*+�2

*+�2+R3, �10�

with

e�1 =��1�2�2�k1 − k2�2

�k1 + k1*�2�k1

* + k2�2 ,

e�2 =��2�2�1�k1 − k2�2

�k2 + k2*�2�k2

* + k1�2 ,

eR1 =��1�2

�k1 + k1*�2 ,

eR2 =��2�2

�k2 + k2*�2 ,

e =�1�2

*

�k1 + k2*�2 ,

eR3 =��1�2��2�2�k1

* − k2*�4

�k1 + k1*�2�k2 + k2

*�2�k1 + k2*�4

, �10�

and �i=ki�x+ �bJt0 / l0− iki�t�, k1, k2, �1, and �2 are four com-plex parameters and determined by the initial conditions. Inthe limits t→ ±�, the two-soliton solution is reduced to twosingle solitons similar to Eq. �8�. That is to say, the two-soliton solution is combined with two one-solitons asymp-totically. Asymptotical analysis indicates that the two soli-tons collide without amplitude switching, but the associatephase shift is �= �R3−R2−R1� /2, namely, the center of thesolitons displace with �. We show this two-soliton solutionin Fig. 2, where we use the materials parameters of CoPt3:HK=33778 Oe, A=1.0�10−6 erg/cm, Ms=1125 G, �=1.75�107 Oe−1 s−1, P=0.35, and take Hext=2000 Oe, je=106 A/cm2. So, we get t0=2.91�10−12 s, l0=3.01�10−7 cm, bJ=18 cm/s. The amplitudes and widths of mxand my vary periodically with time because of the oscillating

factor induced by the external and anisotropic fields. Thetwo-soliton excitations bring the spin current two magneticpulses with different velocities.

III. LOW-Q CASE

When Q�1, such as in permalloy, the nonlinear solutionsof Eq. �1� are different. In this case, the characteristic timeand length are t0=1/ ���4�Ms−HK�� and l0

=�2A / ��4�Ms−HK�Ms�. By the same method, the LLGequation �1� is transformed into a nonlinear Schrödingerequation

i��

�t=

�2�

�x2 + �1 −Hext

4�Ms − HK�� − 2���2� + ibJ

t0

l0

��

�x.

�12�

The one-soliton solution is �=−�ei���1+ei��+ �1−ei��tanh�� /2��, where �=kx+ �2�2+k2+kbJt0 / l0+1+Hext / �4�Ms−HK��t+�0, �=arctan�P�4�2− P2 / �P2

−2�2��, and �= P�x− ��4�2− P2−2k+bJt0 / l0�t�+�0, in whichk, �, P, �0, and �0 are real constants. So, the solutions ofmagnetization are

mx =�

��cos�� + �� − cos � + �cos�� + ��

+ cos ��tanh��/2�� ,

my =�

��sin�� + �� − sin � + �sin�� + ��

+ sin ��tanh��/2�� ,

mz =1

4��4�1 − �2� + P2 − P2 tanh2��/2�� , �13�

where �= �4�1+�2�− P2+ P2 tanh2�� /2�� /4. In this solutions,in addition to the change of velocity, spin-polarized currentinduces the oscillation of the mx and my components.

Also, it is easy to find the two-soliton solutions of Eq.�12� �=−�ei��1+g1+g2� / �1+ f1+ f2�, and the nonlinear evo-lution of magnetization like that of Eq. �9�. The functions inthe solution are as follows:

g1 = ei�1 exp �1 + ei�2 exp �2,

g2 = Cei��1+�2�exp��1 + �2� ,

f1 = exp �1 + exp �2,

f2 = A exp��1 + �2� ,

where

� j = Pj�x − ��4�2 − Pj2 − 2k + kbJt0/l0�t� + � j0,

� j = arctan�Pj�4�2 − Pj

2/�Pj2 − 2�2��, j = 1,2,

FIG. 2. The elastic collision of two dark solitons of mz compo-nent where k1=1+0.5i, k2=2−0.5i, �1=0.4+2i, �2=2i.

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C =�P1 − P2�2 + ��4�2 − P1

2 − �4�2 − P22�2

�P1 + P2�2 + ��4�2 − P12 − �4�2 − P2

2�2,

with P1, P2, �10, and �20 are real constants. Using asymptoti-cal analysis, we find that the two solitons collide with phaseshift �=ln C. After the collision, the centers of solitonsdisplace with ln A. In Fig. 3, we plot this two-soliton solutionwith material parameters of permalloy: HK=10 Oe, A=1.3�10−6erg/cm, Ms=800 G, P=0.5 and Hext=2000 Oe, je=106 A/cm2. The characteristic time and length is t0=4.55�10−11s, l0=1.14�10−6cm, and bJ=51 cm/s.

When the external magnetic field deviates from the z axis,in general, the equations cannot be exactly solved.24 How-ever, some special solutions can be found. For example, weconsider the case that Hext=Hxex+Hzez and Q�1. In thepresence of spin-polarized current, there exists a dynamicsoliton solution

mx =1 − D2 cosh2 �

1 + D2 cosh2 �cos��Hzt0t� ,

my =1 − D2 cosh2 �

1 + D2 cosh2 �sin��Hzt0t� ,

mz =2D cosh �

1 + D2 cosh2 �,

where �=�1−�t0Hx�x+bJt0 / l0t�, D=���Hxt0� / �1−�Hxt0�.This solution is plotted in Fig. 4 with the same material

parameters as above. The depth and width of the trough de-crease as Hx increases. The amplitudes of mx and my oscillatewith frequency �Hz. According to this solution, when theexternal field perpendicular to wire’s axis vanishes, the mag-netization rotates purely in a plane parallel to the wire’s axis.

IV. CONCLUSION

Using a stereographic projection, we transform the modi-fied LLG equation with a spin torque into a nonlinear equa-tion of complex function. By the Hirota bilinear method, weget one- and two-soliton solutions describing nonlinear mag-netization in ferromagnetic nanowire with spin-polarizedcurrent, and these solitons propagate along the wire’s axis.These results indicate that the soliton velocity varies due tothe spin torque, the change is proportional to current density,and the amplitudes are modulated by spin-polarized current.The centers of the solitons vibrate periodically with the fre-quency of alternating currents. In the case of multisolitonsolutions, magnetic solitons collide elastically with phaseshift.

ACKNOWLEDGMENTS

Peng-Bin He thanks Lu Li and Zai-Dong Li for helpfuldiscussions. This work was supported by the NSF of Chinaunder Grants Nos. 60490280, 90403034, and 90406017.

1 J. C. Slonczewski, J. Magn. Magn. Mater. 195, L261 �1999�.2 J. Z. Sun, J. Magn. Magn. Mater. 202, 157 �1999�.3 C. Heide, P. E. Zilberman, and R. J. Elliott, Phys. Rev. B 63,

064424 �2001�.4 S. Zhang, P. M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601

�2002�.5 M. Tsoi, V. Tsoi, J. Bass, A. G. M. Jansen, and P. Wyder, Phys.

Rev. Lett. 89, 246803 �2002�.6 T. Y. Chen, Y. Ji, C. L. Chien, and M. D. Stiles, Phys. Rev. Lett.

93, 026601 �2004�.7 Y. Tserkovnyak, A, Brataas, and G. E. W. Bauer, Phys. Rev. Lett.

88, 117601 �2002�.8 J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 �1996�.

9 L. Berger, Phys. Rev. B 54, 9353 �1996�.10 Ya. B. Bazaliy, B. A. Jones, and S. C. Zhang, Phys. Rev. B 57,

R3213 �1998�.11 Z. Li and S. Zhang, Phys. Rev. Lett. 92, 207203 �2004�; Phys.

Rev. B 70, 024417 �2004�.12 E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A.

Buhrman, Science 285, 867 �1999�.13 J. Z. Sun, D. J. Monsma, M. J. Rooks, and R. H. Koch, Appl.

Phys. Lett. 81, 2202 �2002�.14 J. E. Wegrowe, D. Kelly, Y. Jaccard, Ph. Guittienne, and J. Ph.

Ansermet, Europhys. Lett. 45, 626 �1999�; J. E. Wegrowe, X.Hoffer, Ph. Guittienne, A. Fabian, L. Gravier, T. Wade, and J.Ph. Ansermet, J. Appl. Phys. 91, 6806 �2002�.

FIG. 3. The elastic collision of two bright solitons of an mz

component where P1=1.2, P2=3.9, �=2, k=1, �0=2, �10=1,�20=3.

FIG. 4. Two neighbor bright solitons of mz component withHy =1000 Oe.

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15 M. Tsoi, A. G. M. Jansen, J. Bass, W. C. Chiang, M. Seck, V.Tsoi, and P. Wyder, Phys. Rev. Lett. 80, 4281 �1998�; M. Tsoi,A. G. M. Jansen, J. Bass, W. C. Chiang, V. Tsoi, and P. Wyder,Nature �London� 406, 46 �2000�.

16 Linear and Nonlinear Spin Waves in Magnetic Films and Super-lattices, edited by M. G. Cottam �World Scientific, Singapore,1994�, pp. 421.

17 Z. D. Li, J. Q. Liang, L. Li, and W. M. Liu, Phys. Rev. E 69,066611 �2004�.

18 M. Lakshmanan and K. Nakamura, Phys. Rev. Lett. 53, 2497�1984�.

19 Y. Yamada, W. P. Van Drent, E. N. Abarra, and T. Suzuki, J. Appl.Phys. 83, 6527�1998�.

20 L. Belliard, J. Miltat, V. Kottler, V. Mathet, C. Chappert, and T.Valet, J. Appl. Phys. 81, 5315 �1997�.

21 A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Phys. Rep.194, 117 �1990�.

22 R. Hirota, J. Math. Phys. 14, 805 �1973�.23 W. M. Liu, B. Wu, X. Zhou, D. K. Campbell, S. T. Chui, and Q.

Niu, Phys. Rev. B 65, 172416 �2002�.24 Dynamical Problems in Soliton Systems, edited by S. Takeno

�Springer-Verlag, Berlin, 1984�, pp. 210.

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