APPLIED PHYSICS LETTERS 97, 022511 共2010兲
Asymmetric ground state spin configuration of transverse domain wall
on symmetrically notched ferromagnetic nanowires
Dede Djuhana,1,2 Hong-Guang Piao,1 Sang-Hyuk Lee,1 Dong-Hyun Kim,1,a兲
Sung-Min Ahn,3 and Sug-Bong Choe3
1
Department of Physics, Chungbuk National University, Cheongju 361-763, Republic of Korea
Department of Physics, University of Indonesia, Depok 16424, Indonesia
3
Department of Physics, Seoul National University, Seoul 151-742, Republic of Korea
2
共Received 17 March 2010; accepted 13 June 2010; published online 16 July 2010兲
We report that a ground state spin configuration around a notch of ferromagnetic nanowires can have
either symmetric or asymmetric transverse domain wall structure depending on the notch geometry
by means of micromagnetic simulation with a systematic variation in the notch aspect ratio. An
asymmetric off-centered domain wall configuration becomes stable for a certain range of the notch
aspect ratio. © 2010 American Institute of Physics. 关doi:10.1063/1.3459965兴
Understanding and controlling domain wall 共DW兲 behavior in ferromagnetic nanowires have been extensively explored in recent few years for realization of spin devices
based on DWs such as magnetic logic devices.1–4 It has been
found that an inner structure of DW plays an essential role in
governing the DW dynamics in ferromagnetic nanowires.5
The DW inner structure is determined mostly by the wire
geometry. Introducing a structural constraint such as a notch
onto ferromagnetic nanowires, thereby controlling DW
propagation behavior in nanowires has been intensively examined. Numerous studies have been reported on the DW
dynamics driven by an external magnetic field6–9 or by a spin
torque10–13 in notched ferromagnetic nanowires. The spin
configuration around the notched wire or the nanoconstriction is reported to show another kind of magnetic wall, besides the Bloch and Néel walls,14 since the inner spin structure of DW is determined mostly by the geometry of the
constriction rather than the material parameters due to the
much smaller constriction cross section than the cross section of the wide wire region far from the constriction.
Several experimental observations have been reported to
investigate a spin configuration of DW around notches or
antinotches by means of an electron holography,15 a
magneto-optical Kerr effect,9,16 a Lorentz microscopy,17,18
and a magnetic force microscopy.19–21 An idea has been proposed to utilize the DW pinning behavior around the notch as
a multibit memory cell.16 The detailed dynamic behavior of
the DW propagating under an applied field or a spin torque
through the notch has been investigated,9,16,19,20,22,23 whereas
relatively few studies have been addressed to identifying the
DW ground configuration around the notch. In most cases,
the DW configuration has been considered to be symmetric
around a symmetric notch or antinotch. Interestingly, it has
been reported recently that the DW configuration could be
asymmetric even around a symmetric notch. Asymmetric
DW has been experimentally evidenced by an electron
holography15 and a Lorentz microscopy,17 where the asymmetric DW configuration is considered to be originated from
irregularities such as the edge roughness15 or from the effect
of nonuniform magnetization to the spin curling structure at
the antinotch.17 However, no systematic investigation has
a兲
Electronic mail: [email protected].
0003-6951/2010/97共2兲/022511/3/$30.00
been devoted to the symmetric or antisymmetric ground spin
configuration of the DW around notches, which could be
very essential in designing spin devices utilizing the DW
behavior.
We have investigated detailed spin configurations of the
DW in the notched ferromagnetic nanowires by means of
public micromagnetic simulaton software, OOMMF based on
the Landau–Lifshitz–Gilbert equation.24 The length of ferromagnetic nanowires in the present study is 2000 nm, the
width is 200 nm, and the thickness is 5 nm. The simulation
has been carried out with a systematic variation in triangular
notch aspect ratio between the bottom and the height of the
triangular notch. The height of notch d is fixed to be 50 nm
for comparison and the length of bottom s has been varied
from 30 nm to 400 nm to have a corresponding change in the
aspect ratio d / s from 1.67 to 0.125. Transverse DW with a
head-to-head spin structure is initially prepared to be positioned at the center of the notch and the system is relaxed
spontaneously to reach a minimum energy without an applied external magnetic field. Considering the width and the
thickness of the wires, the transverse DW structure is energetically preferred in all configuration of s and d / s of the
present study.25 The cell size of the simulation is 2.5⫻ 2.5
⫻ 5 nm3 and the damping constant ␣ is set to be 0.01. The
material parameters of permalloy are used with the saturation
magnetization Ms of 8.6⫻ 105 A / m, the exchange stiffness
coefficient A of 13⫻ 1012 J / m, and zero magnetocrystalline
anisotropy.
Without a notch, the straight wire of 200 and 100 nm
width with the same material parameters as the notched
wires in the present study exhibits a symmetric transverse
DW, as expected. With an introduction of the notch, the
notch plays as a strong pinning site and the DW is stabilized
by being positioned around the notch. The representative
ground spin configurations of the DW around the notch with
variation in s from 50 to 400 nm are listed in Fig. 1. Since
we consider fully symmetric notched wires keeping the leftright and up-down symmetries, it is expected to have the
symmetric DW configuration as demonstrated in the figure
for the case of s = 50 nm, which holds as well for the cases
of s less than 50 共s = 40 and 30 nm兲. Interestingly, it is observed in the figure that the DW spin configuration becomes
97, 022511-1
© 2010 American Institute of Physics
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
022511-2
Djuhana et al.
Appl. Phys. Lett. 97, 022511 共2010兲
FIG. 1. 共Color online兲 Examples of the ground spin configuration around the
notch with a variation of the notch bottom length from 50 to 400 nm. Color
code represents a spin direction at each pixel. Arrows guide spin direction.
off-centered from the notch center, for example, in case of
s = 75 nm, where the tilted DW spin configuration appears
obviously with the broken left-right and up-down symmetries. The trend of spontaneous broken symmetry is observed
for the cases of s = 75, 100, and 160 nm in Fig. 1, while the
DW spin configuration then becomes symmetric again for
the larger s 共400 nm兲.
To understand the transitional behavior of the DW configuration around the notch, we have analyzed the magnetic
energies of the system with variation of s from 30 to 400 nm,
as depicted in Fig. 2共a兲. Since we adopted the material parameters of Permalloy, the magnetocrystalline anisotropy is
set to be zero. The Zeeman energy is also zero because there
is no external field. Thus, the total magnetic energy is the
sum of the magnetostatic energy and the exchange energy. It
is immediately noted in the figure that the magnetic energy
densities are mostly dominated by the magnetostatic energy
rather than the exchange energy.
On the other hand, the magnetostatic energy density exhibits a significant variation with respect to the s value. For a
comparison, we plotted total, magnetostatic, and exchange
energy densities for a case of 200 nm width straight wire, as
in Fig. 2共a兲. The energy density for notched wires are all less
than the energy density of the straight wire, which is originated from the energy gain of the DW positioned around the
notch. By being positioned around the notch, the gradually
varying transverse components of the DW can be more stabilized by being more parallel to the notch edge. This reduces the generation of magnetic free poles on the wire edge,
which reduces the magnetostatic energy significantly. The
trend of the magnetostatic energy density is directly reflected
in the trend of the total energy density, which implies that the
magnetostatic energy gain is the main reason of the existence
of the spontaneous off-centered asymmetric DW.
We divided the Fig. 2共a兲 into three regions depending on
the symmetric/antisymmetric ground DW configuration. Region I represents a region where the symmetric DW configuration is the ground state for smaller s 共30–50 nm兲. Region II
represents a region where the asymmetric DW becomes the
ground state for s from 60 to 150 nm. Region III represents a
FIG. 2. 共Color online兲 共a兲 The total magnetic energy density together with
the magnetostatic and the exchange energy densities. Dashed lines are dividing into three regions as labeled by region I, II, and III. Dotted lines show
data of the 200 nm width straight wire for comparisons. The normalized
transverse 共My兲 magnetization component profiles of the DW for cases of
共b兲 s = 50 and 共c兲 s = 150 nm. Zero DW position is set to be the horizontal
center of the notch. The DW width ␦ is shown as the full-width halfmaximum of the profile. 共d兲 The DW width and position are plotted with
respect to s. Dashed lines represents the three regions as in Fig. 2. Dotted
lines show the DW width of 100 and 200 nm straight wires.
region where the symmetric DW becomes the ground state
again for larger s 共⬎160 nm兲. At each region boundary, an
abrupt change of the energy density is observed. For example
in the boundary of the region I and II, the magnetostatic
energy density and thus, the total magnetic energy density
abruptly decreases as s changes from 50 to 60 nm.
To understand further, we have analyzed the DW inner
structure by characterizing the DW width. The horizontal
line profiles of My component normalized by the saturation
magnetization Ms are plotted for s = 50 nm 关Fig. 2共b兲兴 and
s = 150 nm 关Fig. 2共c兲兴, where the horizontal line is selected
to be parallel to the wire axis and the My is the magnetization
component transverse to the wire axis. In case of s = 50 nm
共region I兲, the line profile of the DW transverse component
has a distribution with a peak almost centered at the notch as
in the figure, where the zero DW position is select to be the
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022511-3
Appl. Phys. Lett. 97, 022511 共2010兲
Djuhana et al.
FIG. 3. 共Color online兲 共a兲 Magnetic energy excluding
Zeeman energy with respect to the DW position for s
= 50 nm and 共b兲 s = 150 nm. Inset figures show the
zoomed in energy landscapes around the wire center.
horizontal center of the wire. In case of s = 150 nm 共region
II兲, the line profile has a distribution with an obviously offcentered peak. The position of the DW is determined by
averaging the peak positions while the width 共␦兲 of the DW
is determined by taking the full-width half-maximum of the
averaged line profiles of the transverse component. The position and the width of the DW with respect to s are plotted
in Fig. 2共d兲.
It should be noted that there exist distinctive three regions both in the DW width and the DW position. The DW
position is around zero in region I, shifted about few tens of
nanometers in region II, and then converges to zero again in
region III, which is exactly corresponding to the DW inner
structure changes from the symmetric to the asymmetric and
finally to the symmetric spin configuration around the notch.
We like to stress that the DW width, ␦, exhibits also three
distinctive phases. In region I, ␦ is about 150 nm, which is
comparable to the value of 200 nm straight wire without
notch. In region III, ␦ is about 100 nm and seems to converge to the value of 100 nm straight wire without notch,
which is expected because the larger s means the wire with a
narrower width. In region II, ␦ is in the middle of the values
of region I and III. ␦ suddenly decreases to about 130 nm and
stays almost constant throughout the region, which means
the inner structure of the DW is stabilized in this region with
the off-centered asymmetric DW spin configuration.
We have further carried out the micromagnetic simulation for determination of potential wells and depinning fields
in the notched ferromagnetic nanowires using the method
described in Ref. 11. By slowly varying the external field
then subtracting the Zeeman energy, it has been found that
the energy landscape around the notch exhibits a symmetric
shape of potential well overall as demonstrated in Figs. 3共a兲
and 3共b兲 for two representative cases of s = 50 nm 共symmetric DW兲 and s = 150 nm 共asymmetric DW兲. Interestingly,
there exists a small energy barrier in the center of the potential well in case of the asymmetric DW ground state. This
tiny center hill potential barrier provides a double well shape
of which two minimum ground states are displaced off from
the wire center, as shown in the inset of Fig. 2共b兲. The double
well structure clearly explains the origin of asymmetric DW
configuration.
This work was supported by the Korea Research Foundation Grant funded by the Korean government 共MEST兲
共Grant No. 2009-0072927兲.
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