PHYSICAL REVIEW B 72, 064410 共2005兲
Nonlinear magnetization dynamics in a ferromagnetic nanowire with spin current
Peng-Bin He and W. M. Liu
Joint Laboratory of Advanced Technology in Measurements, Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China
共Received 16 November 2004; revised manuscript received 22 June 2005; published 4 August 2005兲
We investigate the effect of spin transport on nonlinear excitations in a ferromagnetic nanowire with nonuniform magnetizations. In the presence of spin-polarized current, the exact soliton solutions propagating along
the wire’s axis are obtained in two cases of high or low quality factors 共Q兲. The current can change the
velocities of magnetic solitons and affect the solitons’ collision with phase shift. Also, ac current induces
vibration of the center of the solitons.
DOI: 10.1103/PhysRevB.72.064410
PACS number共s兲: 75.75.⫹a, 05.45.Yv, 72.25.Ba, 73.21.Hb
I. INTRODUCTION
In magnetic multilayers, the spin-polarized current causes
many unique phenomena, such as spin-wave excitation,1
magnetization switching and reversal,2–6 and enhanced Gilbert damping.7 These phenomena attribute to the spintransfer effect, proposed by Slonczewski8 and Berger.9 Macroscopically, the magnetization dynamics in the presence of
spin-polarized current can be described by a generalized
Landau-Lifshitz-Gilbert 共LLG兲 equation, including terms related to spin-polarized current, spin accumulation, and spintransfer torque. Bazaliy et al. considered spin-polarized current in half-metal and reduced the current effects as a
topological term in the LLG equation.10 The spin torque has
different forms for uniform magnetization and nonuniform
ones. In the uniform case, such as spin-valve structure, the
spin torque is ␶a = −共aJ / M s兲M ⫻ 共M ⫻ M̂ p兲 , where M is the
magnetization of the free layer, M̂ p is the unit vector along
the direction of the magnetization of the pinned layer, M s is
the saturation magnetization, and aJ is a model-dependent
parameter and is proportional to the current density.8 For the
nonuniform magnetization, Li and Zhang gave a spin torque
taking form ␶b = bJ ⳵ M/ ⳵ x, where bJ depends on the materials parameters and current.11 The spin torque is very different
from precession and damping terms in the LLG equation. It
can induce not only precession but also damping. Due to the
unique property of the spin torque, the magnetization dynamics has been extensively studied in spin-valve pillar
structures,12,13 magnetic nanowires geometry,14 and pointcontact geometry.15
Nonlinear excitations are general phenomena in magnetic
ordered materials. Extensive investigations have been made
in insulated thin films for easy excitation and detection, such
as YIG.16 In confined magnetic metals, nonlinear excitations
are also interesting due to the interaction between spinpolarized current and local magnetizations. The solitary excitations of isotropic ferromagnets in the present of spinpolarized current is well studied using the inverse scattering
method.17
In this paper, we will discuss the linear and nonlinear
magnetization dynamics in a uniaxial ferromagnetic nanowire injected with current. By stereographic projection of the
unit sphere of magnetization onto a complex plane,18 we
1098-0121/2005/72共6兲/064410共5兲/$23.00
transform the LLG equation into a nonlinear equation of
complex function and obtain two kinds of soliton solutions
propagating along the wire’s axis for a high-Q model and a
low-Q one 关Q = Hk / 共4␲ M s兲 is the quality factor兴. From these
solutions, we study the effect of a spin-polarized current on
the nonlinear excitation of the magnetization in the ferromagnetic metal nanowire.
II. HIGH-Q CASE
We consider a very long ferromagnetic nanowire with a
uniform cross section, as shown in Fig. 1. To excellent approximation, this nanowire can be viewed as infinite in
length. The electronic current flows along the long length of
the wire, defined as the x direction. The z axis is taken as the
directions of the uniaxial anisotropy field and the external
field. Also, we assume the magnetization is nonuniform only
in the direction of current. The current injected in the nanowire can be polarized and produces a spin torque acting on
the local magnetization, which is written as ␶b = bJ ⳵ M/ ⳵ x,
where bJ = Pje␮B / 共eM s兲, P is the spin polarization of the
current, je is the electric current density and flows along the
x direction, ␮B is the Bohr magneton, and e is the magnitude
of electron charge.11 So, the modified LLG equation with this
spin torque is
⳵M
⳵M
␣
= − ␥M ⫻ Heff +
+ ␶b ,
M⫻
⳵t
Ms
⳵t
共1兲
where ␥ is the gyromagnetic ratio, M s is the saturation magnetization, ␣ is the Gilbert damping parameter, and Heff is
the effective magnetic field including the external field, the
064410-1
FIG. 1. Geometry of the nanowire.
©2005 The American Physical Society
PHYSICAL REVIEW B 72, 064410 共2005兲
P.-B. HE AND W. M. LIU
anisotropy field, the demagnetization field, and the exchange
field. This effective field can be written as Heff
= 共2A / M s2兲⳵2M/ ⳵ x2 + 关共HK / M s − 4␲兲M z + Hext兴ez, where A is
the exchange constant, HK is the anisotropy field, Hext is the
applied field, and ez is the unit vector along the z direction.
When the uniaxial anisotropy field is larger than the demagnetization field, namely, Q ⬎ 1 共This is the case for some
metallic magnetic films19 and multilayers20兲, we can get dark
magnetic solitons for the mz component. After rescaling the
time coordinate and space coordinate by characteristic time
t0 = 1 / 关␥共HK − 4␲ M s兲兴 and length l0 = 冑2A / 关共HK − 4␲ M s兲M s兴
and taking M = M sm , the LLG equation can be simplified as
冉
冊冉
冊
⳵m
⳵ 2m
Hext
= − m ⫻ 2 − mz +
共m ⫻ ez兲
⳵t
⳵x
HK − 4␲ M s
⳵ m b Jt 0 ⳵ m
+
.
+ ␣m ⫻
⳵t
l0 ⳵ x
冉 冊
⳵ ␺ ⳵ 2␺
⳵␺
␺*
= 2 −2
⳵t ⳵x
1 + ␺␺* ⳵ x
共2兲
−
2
−
i
共3兲
We will get linear and nonlinear dynamics of magnetization
from ␺.
As shown by Li and Zhang,11 the spin-wave solution of
the LLG equation is obtained in the presence of spinpolarized current. In the high-magnetic field, the deviation of
magnetization from the the direction of the field is small. So,
we only consider low-energy excitation. We take the spinwave ansatz
m = 共␦mxex + ␦myey兲ei共k·r−␻t兲 + ez ,
共4兲
corresponding to ␺ = ␺0ei共k·r−␻t兲, where ␺0 = 共␦mx + i␦my兲 / 2,
and ␦mx and ␦my represent the rescaled amplitude of the spin
waves in the x and y directions. Inserting ␺ into Eq. 共3兲 and
keeping the linear terms in ␺0, we obtain the dispersion relation of the spin wave
␻ = 共1 + i␣兲␻0关1 + l0⬘2共kx − bJM s/␥A兲2兴,
共5兲
where
␻0 = 关␥共HK − 4␲ M s + Hext兲 − b2J M s/2␥A兴/共1 + ␣2兲,
l0⬘2 = 16␥2A2/关8␥2M sA共HK − 4␲ M s + Hext兲 − M s2b2J 兴.
冊
By the Hirota bilinear method,22 the one- and two-soliton
solutions of Eq. 共7兲 are easily obtained. The one-soliton solution is ␺ = kRei关␩I+共1+Hext/共HK−4␲M s兲兲t兴 / cosh关␩R + C / 2兴, where
␩ = k关x + 共bJt0 / l0 − ik兲t兴, eC = e2兩␭兩 / 共4kR2 兲, the subscripts R and I
denote the real and imaginary parts, and k is an arbitrary
complex parameter and can be determined by the initial soliton location and amplitude. From ␺, we obtain the solution
of magnetization,
Hext
␺
HK − 4␲ M s
1 − ␺␺*
b Jt 0 ⳵ ␺
␺+i
.
l0 ⳵ x
1 + ␺␺*
冉
⳵ ␺ ⳵ 2␺
t0 ⳵ ␺
Hext
=
− 1+
␺ + 2兩␺兩2␺ + ibJ
.
⳵ t ⳵ x2
l0 ⳵ x
HK − 4␲ M s
共7兲
Considering the fact that the magnitude of the magnetization
is a constant at temperatures well below the Curie temperature, namely, m2 = 共M/M s兲2 = 1, it is reasonable to take stereographic transformation,18 ␺ = 共mx + imy兲 / 共1 + mz兲. Then,
the equation about the complex function ␺ is obtained,
i共1 − i␣兲
Now we investigate the nonlinear excitation of LLG equations. Considering small deviations of magnetization from
the equilibrium direction 共along the anisotropy axis兲 corresponding to m2x + m2y Ⰶ mz2, or, 兩␺兩2 Ⰶ 1, and taking the longwavelength approximation,21 where l0k Ⰶ 1, and k is the
wave vector of spin wave that constitutes nonlinear excitations, Eq. 共3兲 can be simplified, by keeping only the nonlinear terms of the order of the magnitude of 兩␺兩2␺. Without
damping, we obtain the following nonlinear Schrödinger
equation
共6兲
It is easy to see that the spin-polarized current changes the
energy
gap.
For
large
current
density,
je
艌 e / 共បP兲冑2AM s共HK − 4␲ M s + Hext兲, the gap vanishes. This
means that without other stimulation, such as thermal effects,
the spin-polarized current excites spin waves when its density increases beyond a critical value. The Gilbert damping
gives the energy gap a rescaling and damps the spin wave.
冊册 冋 册
冋 冉
冊册 冋 册
冋 冉
冋 冉 冊 册
mx =
2kR
Hext
C
cos ␩I + 1 +
t cosh ␩R +
,
⌬
HK − 4␲ M s
2
my =
2kR
C
Hext
sin ␩I + 1 +
t cosh ␩R +
,
⌬
HK − 4␲ M s
2
mz =
1
C
cosh2 ␩R +
− kR2 ,
⌬
2
共8兲
where ⌬ = cosh2共␩R + C / 2兲 + kR2 .
This one-soliton solution represents a 2␲ domain wall
which is the bound state of a double ␲ wall.23 The height of
the soliton is the rescaled magnetization. The velocity the of
wall is V = −bJ − 2kIl0 / t0. As we see it, the spin-polarized current alters the soliton velocity as −bJ, proportional to the
electron current density. The external fields induce the mx
and my components of the soliton to oscillate periodically.
If applying alternating currents, bJ = Pje␮B / 共eM s兲cos共␻t兲,
and the only change to the solutions is ␩ = k关x
+ bJ / 共␻l0兲sin共␻t0t兲 − ikt兴. The center of the mass of solitons
vibrate with the same frequency as ac currents. Inversely, we
consider the effect of nonlinear excitations on the spinpolarized current. In ferromagnetic material with nonuniform
magnetizations and adiabatic approximation, the spin-current
density is jx = 共␮B / e兲Pjem11. Thus, the one-soliton excitation
in nanowire gives spin current a magnetic pulse like the
optical pulse in nonlinear media. In the same way, we can
␺ = 共g1 + g3兲 / 共1 + f 2
get
the
two-soliton
solutions
+ f 4兲ei关1+Hext/共HK−4␲M s兲兴t, and nonlinear dynamics of magnetization,
mx =
where
064410-2
␺ + ␺*
,
1 + ␺␺*
my = − i
␺ − ␺*
,
1 + ␺␺*
mz =
1 − ␺␺*
,
1 + ␺␺*
共9兲
PHYSICAL REVIEW B 72, 064410 共2005兲
NONLINEAR MAGNETIZATION DYNAMICS IN A…
factor induced by the external and anisotropic fields. The
two-soliton excitations bring the spin current two magnetic
pulses with different velocities.
III. LOW-Q CASE
FIG. 2. The elastic collision of two dark solitons of mz component where k1 = 1 + 0.5i, k2 = 2 − 0.5i, ␭1 = 0.4+ 2i, ␭2 = 2i.
When Q ⬍ 1, such as in permalloy, the nonlinear solutions
of Eq. 共1兲 are different. In this case, the characteristic time
and
l0
and
length
are
t0 = 1 / 关␥共4␲ M s − HK兲兴
= 冑2A / 关共4␲ M s − HK兲M s兴. By the same method, the LLG
equation 共1兲 is transformed into a nonlinear Schrödinger
equation
g 1 = ␭ 1e ␩1 + ␭ 2e ␩2 ,
*
i
f2 = e
+e
共12兲
␩2+␩*2+R2
*
+e
␩1+␩*2+␦
+e
␩*1+␩2+␦*
*
,
共10兲
with
e ␪2 =
兩␭1兩2␭2共k1 − k2兲2
共k1 + k*1兲2共k*1 + k2兲2
兩␭2兩2␭1共k1 − k2兲2
共k2 + k*2兲2共k*2 + k1兲
e R1 =
e R2 =
␦
e =
e R3 =
兩␭1兩
␺ = −␭e 关共1 + ei⌽兲
The
one-soliton
solution
is
i⌽
+ 共1 − e 兲tanh共␨ / 2兲兴, where ⌿ = kx + 关2␭2 + k2 + kbJt0 / l0 + 1
⌽ = arctan关P冑4␭2 − P2 / 共P2
+ Hext / 共4␲ M s − HK兲兴t + ⌿0,
2
2
2
− 2␭ 兲兴, and ␨ = P关x − 共冑4␭ − P − 2k + bJt0 / l0兲t兴 + ␨0, in which
k, ␭, P, ⌿0, and ␨0 are real constants. So, the solutions of
magnetization are
i⌿
f 4 = e␩1+␩1+␩2+␩2+R3 ,
e ␪1 =
冊
*
g 3 = e ␩1+␩1+␩2+␪1 + e ␩1+␩2+␩2+␪2 ,
␩1+␩*1+R1
冉
⳵ ␺ ⳵ 2␺
t0 ⳵ ␺
Hext
= 2 + 1−
.
␺ − 2兩␺兩2␺ + ibJ
⳵t ⳵x
l0 ⳵ x
4␲ M s − HK
,
+ cos ⌿兴tanh共␨/2兲兴,
,
2
,
+ sin ⌿兴tanh共␨/2兲兴,
兩␭2兩2
,
k*2兲2
␭1␭*2
共k1 + k*2兲2
␭
关sin共⌽ + ⌿兲 − sin ⌿ + 关sin共⌿ + ⌽兲
⌳
my =
2
共k1 + k*1兲2
共k2 +
␭
关cos共⌽ + ⌿兲 − cos ⌿ + 关cos共⌿ + ⌽兲
⌳
mx =
mz =
,
兩␭1兩2兩␭2兩2兩k*1 − k*2兩4
共k1 + k*1兲2共k2 + k*2兲2兩k1 + k*2兩4
,
共10兲
and ␩i = ki关x + 共bJt0 / l0 − iki兲t兴, k1, k2, ␭1, and ␭2 are four complex parameters and determined by the initial conditions. In
the limits t → ± ⬁, the two-soliton solution is reduced to two
single solitons similar to Eq. 共8兲. That is to say, the twosoliton solution is combined with two one-solitons asymptotically. Asymptotical analysis indicates that the two solitons collide without amplitude switching, but the associate
phase shift is ␦␾ = 共R3 − R2 − R1兲 / 2, namely, the center of the
solitons displace with ␦␾. We show this two-soliton solution
in Fig. 2, where we use the materials parameters of CoPt3:
HK = 33778 Oe, A = 1.0⫻ 10−6 erg/ cm, M s = 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 mx
and my vary periodically with time because of the oscillating
1
关4共1 − ␭2兲 + P2 − P2 tanh2共␨/2兲兴,
4⌳
共13兲
where ⌳ = 关4共1 + ␭2兲 − P2 + P2 tanh2共␨ / 2兲兴 / 4. In this solutions,
in addition to the change of velocity, spin-polarized current
induces the oscillation of the mx and my components.
Also, it is easy to find the two-soliton solutions of Eq.
共12兲 ␺ = −␭ei⌿共1 + g1 + g2兲 / 共1 + f 1 + f 2兲, and the nonlinear evolution of magnetization like that of Eq. 共9兲. The functions in
the solution are as follows:
g1 = ei⌽1 exp ␨1 + ei⌽2 exp ␨2 ,
g2 = Cei共⌽1+⌽2兲exp共␨1 + ␨2兲,
f 1 = exp ␨1 + exp ␨2 ,
f 2 = A exp共␨1 + ␨2兲,
where
064410-3
␨ j = P j关x − 共冑4␭2 − P2j − 2k + kbJt0/l0兲t兴 + ␨ j0 ,
⌽ j = arctan关P j冑4␭2 − P2j /共P2j − 2␭2兲兴,
j = 1,2,
PHYSICAL REVIEW B 72, 064410 共2005兲
P.-B. HE AND W. M. LIU
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.
C=
共P1 − P2兲2 + 共冑4␭2 − P21 − 冑4␭2 − P22兲2
共P1 + P2兲2 + 共冑4␭2 − P21 − 冑4␭2 − P22兲2
,
with P1, P2, ␨10, and ␨20 are real constants. Using asymptotical analysis, we find that the two solitons collide with phase
shift ␦␾ = ln C. After the collision, the centers of solitons
displace with ln A. In Fig. 3, we plot this two-soliton solution
with material parameters of permalloy: HK = 10 Oe, A = 1.3
⫻ 10−6erg/ cm, M s = 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 However, some special solutions can be found. For example, we
consider the case that Hext = Hxex + Hzez and Q ⬍ 1. In the
presence of spin-polarized current, there exists a dynamic
soliton solution
mx =
1 − D2 cosh2 ␰
cos共␥Hzt0t兲,
1 + D2 cosh2 ␰
my =
1 − D2 cosh2 ␰
sin共␥Hzt0t兲,
1 + D2 cosh2 ␰
mz =
2D cosh ␰
,
1 + D2 cosh2 ␰
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parameters as above. The depth and width of the trough decrease as Hx increases. The amplitudes of mx and my oscillate
with frequency ␥Hz. According to this solution, when the
external field perpendicular to wire’s axis vanishes, the magnetization rotates purely in a plane parallel to the wire’s axis.
IV. CONCLUSION
Using a stereographic projection, we transform the modified LLG equation with a spin torque into a nonlinear equation of complex function. By the Hirota bilinear method, we
get one- and two-soliton solutions describing nonlinear magnetization in ferromagnetic nanowire with spin-polarized
current, and these solitons propagate along the wire’s axis.
These results indicate that the soliton velocity varies due to
the 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 frequency of alternating currents. In the case of multisoliton
solutions, magnetic solitons collide elastically with phase
shift.
ACKNOWLEDGMENTS
where ␰ = 冑1 − ␥t0Hx共x + bJt0 / l0t兲, D = 冑共␥Hxt0兲 / 共1 − ␥Hxt0兲.
This solution is plotted in Fig. 4 with the same material
1 J.
FIG. 4. Two neighbor bright solitons of mz component with
Hy = 1000 Oe.
Peng-Bin He thanks Lu Li and Zai-Dong Li for helpful
discussions. This work was supported by the NSF of China
under Grants Nos. 60490280, 90403034, and 90406017.
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