This is the accepted manuscript made available via CHORUS. The article has been
published as:

Current-Driven Dynamics of Frustrated Skyrmions in a
Synthetic Antiferromagnetic Bilayer
Jing Xia, Xichao Zhang, Motohiko Ezawa, Zhipeng Hou, Wenhong Wang, Xiaoxi Liu, and
Yan ZhouPhys. Rev. Applied 11, 044046 — Published 16 April 2019
DOI: 10.1103/PhysRevApplied.11.044046

Current-Driven Dynamics of Frustrated Skyrmions in a Synthetic Antiferromagnetic
Bilayer
Jing Xia,1, ∗ Xichao Zhang,1, ∗ Motohiko Ezawa,2, † Zhipeng Hou,3, 4 Wenhong Wang,4 Xiaoxi Liu,5 and Yan Zhou1, ‡1School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China
2
Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-8656, Japan
3
South China Academy of Advanced Optoelectronics,
South China Normal University, Guangzhou 510006, China
4
Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
5
Department of Electrical and Computer Engineering,
Shinshu University, 4-17-1 Wakasato, Nagano 380-8553, Japan
We report the current-driven dynamics of frustrated skyrmions in an antiferromagnetically exchange coupled bilayer system, where the bilayer skyrmion consists of two monolayer skyrmions
with opposite skyrmion numbers Q. We show that the in-plane current-driven bilayer skyrmion
moves in a straight path, while the out-of-plane current-driven bilayer skyrmion moves in a circularpath. It is found that the in-plane current-driven mobility of a bilayer skyrmion is much better
than the monolayer one at a large ratio of β/α, where α and β denote the damping parameter andnon-adiabatic spin transfer torque strength, respectively. Besides, the out-of-plane current-driven
mobility of a bilayer skyrmion is much better than the monolayer one when α is small. We also
reveal that one bilayer skyrmion (consisting of monolayer skyrmions with Q = ±2) can be separated
to two bilayer skyrmions (consisting of monolayer skyrmions with Q = ±1) driven by an out-of-planecurrent. Our results may be useful for designing skyrmionic devices based on frustrated multilayer
magnets.
PACS numbers: 75.10.Hk, 75.70.Kw, 75.78.-n, 12.39.Dc
Keywords: skyrmions, frustrated magnet, synthetic antiferromagnet, spintronics

I.

INTRODUCTION

The magnetic skyrmion is a topologically non-trivial
object [1–3], which can be regarded as quasi-particles [4]and promises advanced electronic and spintronic applications [5–8]. For examples, recent studies have suggested that skyrmions can be used as building blocks
for racetrack-type memories [9–12], logic computing devices [13], and bio-inspired applications [14–17]. The
magnetic skyrmion was first experimentally observedin ferromagnetic (FM) materials with DzyaloshinskiiMoriya (DM) interactions in 2009 [18]. The DM interaction is an essential energy term stabilizing skyrmion
textures [18–20]. It can also be induced at the interface
between the heavy metal and ferromagnet [21], which
promotes recent studies of skyrmions in magnetic bilayer
and multilayer structures [3, 22–28].
However, some recent studies have demonstrated thatskyrmions can be stabilized in frustrated magnets even in
the absence of DM interaction [29–39], where skyrmions
are stabilized by competing exchange interactions [29,30, 33, 34]. Frustrated skyrmions have many unique
physical properties compared with skyrmions in DM ferromagnets. For examples, both skyrmions and antiskyrmions can exist in a frustrated magnet as metastable∗ These authors contributed equally to this work.
† E-mail: ezawa@ap.t.u-tokyo.ac.jp‡ E-mail: zhouyan@cuhk.edu.cn

states [29, 30, 33, 34]. The frustrated skyrmions withskyrmion number of Q = ±1 can merge and form
skyrmions with higher skyrmion number of Q = ±2 [34].
Besides, the center-of-mass dynamics of a frustrated
skyrmion is coupled to its helicity dynamics [29, 30, 33],
which results in the circular motion of a skyrmion [30, 34].
Therefore, frustrated skyrmions can be used as information carriers, which have multiple degrees of freedom thatcan be utilized to store information [29, 30, 33, 34].
So far, most studies on frustrated skyrmions are focused on the monolayer system, while bilayer and multilayer systems play an important role in developing
nanoscale devices [3, 22–28, 40–47]. On the other hand,recent theoretical works have suggested that the monolayer antiferromagnetic (AFM) skyrmion has different
dynamics and improved mobility in comparison with the
FM one [48–50]. For examples, the skyrmion Hall effect can be eliminated in the AFM monolayer, leading
to ultra-fast straight motion of a skyrmion that is usefulfor practical applications [48, 49]. However, the dynamic
properties of frustrated skyrmions in bilayer and multilayer AFM systems remain elusive.
Here we report the dynamics of frustrated skyrmions
with Q = ±1 and ±2 in the synthetic AFM bilayer
system driven by an in-plane or out-of-plane spin current. We also numerically investigate the current-induced
separation of a frustrated skyrmion. We show thatthe current-driven frustrated skyrmions in the synthetic
AFM bilayer system have better mobility than that in
the monolayer system.

2
II.

METHODS

We consider two FM layers with competing Heisenberg exchange interactions based on the J1 -J2 -J3 model
on a simple square lattice [30, 34], of which the Hamiltonian is given in Ref. 51. As shown in Fig. 1(a), thetwo frustrated FM layers are coupled by an AFM interfacial exchange coupling, which can be realized by utilizing
the Ruderman-Kittel-Kasuya-Yosida interaction [26, 40–
43, 47]. The background magnetization direction of thebottom FM layer is assumed to be pointing along the+z direction. The simulation is performed by using the
Object Oriented MicroMagnetic Framework [52] with our
extension modules for the J1 -J2 -J3 classical Heisenberg
model [30, 34], where the time-dependent spin dynamics
is described by the Landau-Lifshitz-Gilbert (LLG) equation (see Ref. 51 for modeling details, parameters, andparameter dependency diagrams).
We consider two geometries for the injection of spinpolarized current. For the current-in-plane (CIP) geometry, we assume that an in-plane spin current flows along
the +x direction in both top and bottom FM layers. The
adiabatic [τ1 = u (m × ∂x m × m)] and non-adiabatic[τ2 = −βu (m × ∂x m)] spin transfer torque (STT) terms
are considered, where m represents the normalized spin,jPu = | µγ00~e | 2M
is the STT coefficient and β is the strengthSof the non-adiabatic STT torque. ~ is the reduced Planck
constant, e is the electron charge, γ0 is the absolute gyromagnetic ratio, µ0 is the vacuum permeability constant,
j is the applied driving current density, P = 0.4 is thespin polarization rate [9], and MS is the saturation magnetization.
For the current-perpendicular-to-plane (CPP) geometry, we assume a heavy-metal substrate layer underneaththe bottom FM layer, in which a charge current flows
along the +x direction and leads to an out-of-plane spincurrent propagating into the bottom FM layer due tothe spin Hall effect [27]. In such a case, the damping-like
STT term [τd = ua (m × p × m)] is considered, wherea = 0.4 nm is the thickness of a single FM layer andp = +ŷ stands for the unit spin polarization direction. Note that we ignore the field-like STT term for
simplicity as its contribution to the skyrmion dynamics is minor. For the CPP geometry, we set P = 0.1,
which is a typical value of spin Hall angle [10, 27]. The
skyrmion
R number in a single FM layer is defined by
Q = − d2 r · m · (∂x m × ∂y m) /4π. The internal structure of a skyrmion is described by its helicity numberη = [0, 2π) (see Refs. 2 and 53).
III.
A.

RESULTS AND DISCUSSION

Skyrmions driven by an in-plane current

We first study in-plane current-driven skyrmions in asynthetic AFM bilayer [see Fig. 1(a)]. A bilayer skyrmion
is first relaxed in the sample, which consists of a top

(a )

(b )
z
y
x
T o p la y e r
B o tto mla y e r

(c )

Q

(d )

= + 1

Q

= -1

Q

= + 2

Q

= -2

FIG. 1.
(a) Schematic of the simulated synthetic AFM
bilayer model. The top and bottom FM layers are coupled
via an AFM interfacial exchange coupling. Illustrations of(b) the synthetic AFM bilayer skyrmion, (c) the monolayer
skyrmion with Q = ±1, and (d) the monolayer skyrmion with
Q = ±2. The arrows represent the magnetization directions.
The out-of-plane component of magnetization (mz ) is color
coded: blue is into the plane, red is out of the plane, white isin-plane.

skyrmion with Q = −1 and η = 3π/2 and a bottom
skyrmion with Q = +1 and η = π/2 [see Fig. 1(b)].
The top and bottom skyrmions are strongly coupled in
an AFM manner. In this work, we refer to this bilayer
skyrmion as a bilayer skyrmion with Q = ±1. The diameter of the relaxed bilayer skyrmion with Q = ±1 equals
2 nm. An in-plane spin current is then injected into thebilayer with the CIP geometry, which drives the bilayer
skyrmion into motion. We also study the skyrmion motion in a single FM monolayer for the purpose of comparison. The monolayer skyrmion structures with Q = ±1
are illustrated in Fig. 1(c). Note that the diameters of therelaxed monolayer and bilayer skyrmions are identical.
Figure 2(a) shows the trajectories of the monolayer andbilayer skyrmions at the damping parameter α = β/2 =0.3. The bilayer skyrmion with Q = ±1 straightly moves
along the +x direction without showing a transverse
shift. However, the monolayer skyrmions with Q = +1
and Q = −1 show transverse shifts toward the +y and −ydirections, respectively. The reason is that the monolayer
skyrmion experiences a topological Magnus force [26–28],
which is perpendicular to the skyrmion velocity vector.
Since the directions of Magnus forces acted on monolayer skyrmions with Q = +1 and Q = −1 are opposite,
the bilayer skyrmion consisting of a bottom monolayer
skyrmion with Q = +1 and a top monolayer skyrmion
with Q = −1 experiences zero net Magnus force and thus
can move in a straight path. Note that the transverse
shift of a skyrmion is referred as the skyrmion Hall effect,
which has been observed in experiments [27, 28]. The
skyrmion Hall effect of the monolayer skyrmion driven
by an in-plane current can be eliminated when α = β(see Ref. 51 and 54), which is difficult to be realized inreal materials. However, the bilayer skyrmion driven by

3
2 5

2 5

(a )

1 0

B ila y e r Q
5

M o n
o
0


-5

la y e



r Q

-2

3 0
x (n m )

= −π
/2

4 0

= π/ 2 ,  T o p = - π/ 2

la y e= 0 .3 0 ,  = 0 .6 0 , j = 5 0 M A c m



-5

(d )

= 0 .1 5= 0 .3 0
 = 0 . 6 0
 = 0 . 3 0

1 0

2 0

= (+ 2 ) + (-2 )


r Q

= −π
/2

= -2

-2

3 0
x (n m )

4 0

5 0

3 0
B ila y e r Q = ( + 1 ) + ( - 1 )M o n o la y e r Q = + 1
B ila y e r Q = ( + 2 ) + ( - 2 )M o n o la y e r Q = + 2

2 5



2 0

3 0

1 5

v (m

v (m

s -1 )

0
5 0



4 0

M o n
o

= + 2
Q

a y e rB o tto m

B ila y e r Q

= -1

6 0
5 0

1 0

s -1 )(c )2 0= (+ 1 ) + (-1 )

5= 0 .3 0 ,  = 0 .6 0 , j = 5 0 M A c m

1 0

o l
M o n

1 5
= π/ 2 ,  T o p = - π/ 2

B o tto m

2
= π/



y (n m )y (n m )



2 0

= + 1
e r Q

la y

o
M o n

1 5

(b )

2
= π/



2 0

2 0

1 0

1 0



= 0 .3 0 , j = 5 0 M A c m

-2

5
0
0

2 0

4 0

6 0
j (M A c m

-2

8 0
)

1 0 0

0

0 .1 5

0 .3 00 .6 0

FIG. 2. (a) Trajectories of monolayer and bilayer skyrmions
with Q = ±1 driven by an in-plane current at α = β/2 = 0.30.
The arrow denotes the motion direction, and the dot denotes
the skyrmion center. Insets are the top-view snapshots of simulated skyrmions. (b) Trajectories of monolayer and bilayer
skyrmions with Q = ±2 driven by an in-plane current. (c)Bilayer skyrmion (Q = ±1 and ±2) velocity as a function of
in-plane driving current density. (d) Comparison of the monolayer skyrmion velocity and bilayer skyrmion velocity induced
by an in-plane current.

an in-plane current shows no skyrmion Hall effect evenwhen α 6= β, which is useful for building racetrack-typedevices [10, 26].
As shown in Fig. 2(b), we also simulate the motionof a bilayer skyrmion with Q = ±2, which consists of abottom monolayer skyrmion with Q = +2 and η = π/2
and a top monolayer skyrmion with Q = −2 and η =3π/2 [see Fig. 1(d)]. Similarly, the bilayer skyrmion with
Q = ±2 driven by an in-plane current also straightly
moves along the +x direction. The monolayer skyrmions
with Q = +2 and Q = −2 show transverse shifts towardthe +y and −y directions, respectively. Note that thediameter of the relaxed bilayer skyrmion with Q = ±2
equals 2.8 nm, which is slightly larger than that of theskyrmion with Q = ±1.
Figure 2(c) shows the bilayer skyrmion velocity v as afunction of in-plane driving current density j for α = 0.3
and β = 0.15 ∼ 0.60. Note that the v-j relation of thebilayer skyrmion with Q = ±1 is identical to that of thebilayer skyrmion with Q = ±2 at a given α and β. For
both bilayer skyrmions with Q = ±1 and Q = ±2, thevelocity increases linearly with j. At certain values of α
and j, larger β will result in larger velocity. By comparing the velocities of bilayer and monolayer skyrmions atcertain values of α and j [see Fig. 2(d)], it is found thatbilayer skyrmions move faster than monolayer skyrmions
when α < β, while bilayer skyrmions move slower thanmonolayer skyrmions when α > β. When α = β, the velocities for bilayer and monolayer skyrmions are almost
identical.

The center-of-mass dynamics of skyrmions can also
be described by the Thiele equation [51, 55]. For
the monolayer skyrmion, the longitudinal (along the±x directions) and transverse (along the ±y directions) velocities induced
current
 by an in-planeare vxm = u Q2 + αβD2 / Q2 + α2 D2 and vym =RRu [(β − α) QD] / Q2 + α2 D2 , where D =(∂x m) ·(∂x m) dxdy/4π is the dissipative force tensor [51]. Notethat we numerically find that D equals ∼ 1.23 and ∼ 2.02
for skyrmions with Q = ±1 and Q = ±2 in a single FMlayer, respectively. For the bilayer skyrmion, the longitudinal and transverse velocities induced by an in-planecurrent are vxb = u (β/α) and vyb = 0. Hence, basedon the Thiele equation solutions [51], vxm and vym of theskyrmion with Q = 1 is 1.03 and 1.17 times larger thanthat of the skyrmion with Q = 2 at 2α = β = 0.6, for example. At the same conditions, vxb is 1.79 and 1.85 times
larger than vxm of skyrmions with Q = 1 and Q = 2.
These theoretical results agree well with the simulation
results [51].
Several experiments suggest that β is considerablylarger than α (i.e., α  β) [56, 57], which means
the mobility of bilayer skyrmion is better than that ofmonolayer skyrmion in real materials. The monolayer
skyrmion speed is proportional to β/α in the absence ofthe skyrmion Hall effect [9], which is as same as the bilayer skyrmion [51]. However, it should be noted that amerit of the bilayer skyrmion is the enhancement of velocity in large-β materials with the CIP geometry, which ispromising for future information processing applications
with ultrahigh operation speed.

B.

Skyrmions driven by an out-of-plane current

We also study out-of-plane current-driven skyrmionsin a synthetic AFM bilayer. A spin current is perpendicularly injected into the bottom FM layer with the
CPP geometry, which drives the skyrmion in the bottom FM layer into motion. The skyrmion in the top FM
layer is simultaneously dragged into motion as the bottom and top skyrmions are strongly coupled. We firststudy the bilayer skyrmion with Q = ±1 as well as monolayer skyrmions with Q = +1 and Q = −1.
The dynamics of a bilayer skyrmion with Q = ±1 [seeFig. 3(a)] and its monolayer counterparts (see Ref. 51)
driven by an out-of-plane current are in stark contrast
to the case driven by an in-plane current. It is foundthat the bilayer skyrmion with Q = ±1 and monolayer
skyrmions with Q = +1 and Q = −1 driven by an outof-plane current move in a circular path. The bilayer
skyrmion with Q = ±1 moves in the clockwise direction, while the monolayer skyrmions with Q = +1 and
Q = −1 move in the counterclockwise and clockwise directions, respectively. As pointed out in Ref. 30, themotion direction depends on the skyrmion helicity. For
both the bilayer and monolayer skyrmions, the diameters of their circular trajectories increase with increasing

4
6 0

(a )

1 0 0

(c )

5 0
5 0



= 0 .1 5= 0 .3 0= 0 .6 0

s -1 )





3 0

v (m

y (n m )

0= 0 .1 5= 0 .3 0
 = 0 . 6 0


4 0

2 0

-5 0

1 0
-1 0 0
0

2 .0

(b )

5 0



1 0 0
1 5 0
x (n m )= 0 .1 5= 0 .3 0

0

2 0 0



0

2 0

6 0
j (M A c m

-2

8 0

1 0 0

)

B ila y e r Q = ( + 1 ) + ( - 1 )M o n o la y e r Q = + 1
j = 8 0 M A c m -2

2 0

s -1 )

1 .5

4 0

(d )

2 5

= 0 .6 0

( π)

1 5

v (m

1 .0



α. Under the same j and α, the trajectory diameterof the bilayer skyrmion is significantly larger than thatof the monolayer skyrmion, which means the frustrated
skyrmion can be delivered farther in the synthetic AFM
bilayer.
On the other hand, the helicity numbers η of bilayer
skyrmion with Q = ±1 [see Fig. 3(b)] and monolayer
skyrmions with Q = +1 and Q = −1 (see Ref. 51)
driven by an out-of-plane current are coupled to theircenter-of-mass dynamics. Namely, η changes linearlywith time during the skyrmion motion. Such a phenomenon has been reported for monolayer skyrmions [30],and we show that it also happens for bilayer skyrmions
with Q = ±1. As pointed out in Ref. 30, the frustrated
monolayer skyrmion has a translational mode and a rotational mode, which can be excited and hybridized by thedamping-like STT. Namely, when the skyrmion in thebottom FM layer is driven by the damping-like STT, it
moves along a circle with rotating helicity. At the sametime, it drives the skyrmion in the top FM layer into
circular motion with rotating helicity as the bottom andtop skyrmions are coupled.
Figure 3(c) shows the velocity of the bilayer skyrmion
with Q = ±1 as a function of out-of-plane driving current
density. At a given α, the velocity of bilayer skyrmion
with Q = ±1 increases linearly with j. At a given j,
the velocity of bilayer skyrmion with Q = ±1 is inversely
proportional to α. By comparing Fig. 2(c) and Fig. 3(c)at same j and α, it is found that the bilayer skyrmion
with Q = ±1 driven by an in-plane current can move
much faster than the one driven by an out-of-plane current if β is large. For example, when j = 100 MA cm−2
and α = 0.3, the bilayer skyrmion with Q = ±1 driven
by an in-plane current can reach a velocity of v = 53 m
s−1 at β = 0.6, while the one driven by an out-of-planecurrent only reaches a velocity of v = 15 m s−1 . However, it should be noted that the driving force provide by
the in-plane current is larger than that provided by theout-of-plane current at the same j in this work, as the
CIP spin-polarization rate is four times larger than the
CPP spin Hall angle. The bilayer skyrmion driven by an
out-of-plane current can move faster when the value ofthe spin Hall angle is increased.
Besides, by comparing the velocities of bilayer
skyrmion with Q = ±1 and monolayer skyrmions with
Q = +1 and Q = −1 at a given j [see Fig. 3(d)], it isfound that the bilayer skyrmion velocity remarkably decreases with increasing α, while the monolayer skyrmion
velocity slightly decreases with increasing α. The bilayer skyrmion velocity is much larger than monolayer
skyrmion velocity at small α (e.g., the bilayer skyrmion
is 2.2 times faster than the monolayer one at α = 0.15),
while it is slightly smaller than monolayer skyrmion velocity at large α (e.g., α = 0.60).
The circular motion and helicity dynamics of themonolayer and bilayer skyrmions with Q = ±1 induced by an out-of-plane current can also be analyzed
by the Thiele equation [51, 55]. First, for the mono-

1 0

0 .5

5
0 .0
0

3

6

9
1 2
t (n s )

1 5

1 8

0

0 .1 5

0 .3 00 .6 0

FIG. 3. (a) Trajectories of bilayer skyrmions with Q = ±1
driven by an out-of-plane current. The arrow denotes the
motion direction, and the dot denotes the skyrmion center.
(b) Helicity as a function of time during the motion of bilayer skyrmions with Q = ±1. Note that the helicity of theskyrmion in bottom FM layer is shown. (c) Bilayer skyrmion(Q = ±1) velocity as a function of out-of-plane driving current
density. (d) Comparison of the monolayer skyrmion velocity
and bilayer skyrmion velocity induced by an out-of-plane current.

layer skyrmion with Q = ±1 [51], the velocity can be expressed as v m = R (−η) v m
η=0 , where R (−η) is a counterclockwise rotation matrix determined by η [51], andvm
η=0 is the velocity at η = 0. The transverse and lonm
gitudinal components of v m
η=0 are found to be vx =
2 2
2 2
2
m
(uIαD)/(α D + Q ) and vy = (uIQ)/(α D + Q2 ),where I = πrsk /4a with rsk being the skyrmion radius. Namely,
the magnitude of the skyrmion velocitypv m = uI/ α2 D2 + Q2 , which is independent of η.
Second, for the bilayer skyrmion with Q = ±1 [51],the velocity can also be expressed as v b = R (−η) v bη=0 .
The transverse and longitudinal components of v bη=0 are
found to be vxb = uI/2αD and vyb = 0. Similarly, themagnitude of the skyrmion velocity v b = uI/2αD. It canbe seen that the monolayer skyrmion speed v m decreases
from uI to uI/1.56 as α increases from 0 to 1. The bilayer
skyrmion speed v b is also inversely proportional to α. It
is found that the bilayer skyrmion moves faster than themonolayer one when α < 0.47. These theoretical results
are consistent with the simulation results [see Fig. 3(d)].
We continue to numerically study the bilayer skyrmion
with Q = ±2 and monolayer skyrmions with Q = +2
and Q = −2 driven by an out-of-plane current. The
bilayer skyrmion with Q = ±2 does not move toward acertain direction when an out-of-plane current is applied(see Fig. 4). Instead, it is elongated and then separated
to two bilayer skyrmions with Q = ±1. The reason is5that the bilayer skyrmion with Q = ±2 is topologicallyequal to a combination of two bilayer skyrmions with
Q = ±1 and opposite η. Such two bilayer skyrmions with
Q = ±1 tend to move along opposite directions upon theapplication of an out-of-plane current, as indicated byyellow arrows in Fig. 4.
Similarly, when an out-of-plane current is applied, themonolayer skyrmion with Q = +2 is separated to twomonolayer skyrmions with Q = +1 and the monolayer
skyrmion with Q = −2 is separated to two monolayer
skyrmions with Q = −1 (see Appendix A). The most obvious difference between the out-of-plane current-induced
separation of a bilayer skyrmion with Q = ±2 and theseparation of a monolayer skyrmion with Q = +2 or
Q = −2 is that the separation of a monolayer skyrmion
is accompanied with an obvious clockwise or counterclockwise rotation of the skyrmion structure. The reason
is that the out-of-plane current-induced circular motion
diameter of the monolayer skyrmion with Q = +1 or
Q = −1 is much smaller than that of the bilayer skyrmion
with Q = ±1. Hence, the separation of the monolayer
skyrmion with Q = ±2 shows very obvious rotation
caused by the circular motion of monolayer skyrmions
with Q = +1 and Q = −1. Note that if an out-ofplane current is applied after the separation of the bilayer skyrmion with Q = ±2, it will also drive the bilayer
skyrmions with Q = ±1 into motion along a large circle.

IV.

CONCLUSION

In conclusion, we have studied the current-driven dynamics of frustrated bilayer skyrmions in a synthetic
AFM bilayer. It is found that the bilayer skyrmion with
Q = ±1 or Q = ±2 moves in a straight path driven
by an in-plane current, while the bilayer skyrmion with
Q = ±1 moves in a circular path driven by an out-ofplane current. Therefore, the bilayer skyrmion driven by
an in-plane current is promising for building racetracktype applications, as it shows no skyrmion Hall effectand can be delivered in a straight line even when α 6= β.
Especially, as the synthetic AFM bilayer skyrmions with
Q = ±1 and Q = ±2 show the same velocity-current relation (i.e., vxb = uβ/α) and no skyrmion Hall effect (i.e.,
vyb = 0), therefore, it is possible to build a racetrack-type
memory based on the in-line motion of skyrmions with
Q = ±1 and Q = ±2, where the elementary binary information digits “1” and “0” are encoded and carried byskyrmions with Q = ±1 and Q = ±2, respectively.
The elementary binary information digits in traditionalmonolayer skyrmion-based racetrack-type memory designs are encoded by the skyrmion and ferromagnetic
background. However, because the spacing between moving skyrmions is easy to be changed due to the pinningeffect [4, 58], the information stored in the racetrack may
be distorted or destroyed. Obviously, such a problem canbe reliably avoided by using the synthetic AFM bilayer
skyrmions with different Q, as the spacing between bi-

P u ls e o n

P u ls e o n
Q

P u ls e o n

P u ls e o n
Q

= + 2
Q

= + 1

t = 0 p s

t = 2 0 p s

t = 4 0 p s

t = 6 0 p s

P u ls e o n

P u ls e o ff

P u ls e o ff

P u ls e o ff
Q

= + 1
Q

t = 8 0 p s

t = 1 0 0 p s

t = 1 5 0 p s

= + 1

= + 1

t = 2 0 0 p s

FIG. 4. Out-of-plane current-induced separation of one bilayer skyrmion with Q = ±2 to two bilayer skyrmions with
Q = ±1. Snapshots are top views of the bottom FM layer.
Here, a current pulse of j = 400 MA cm−2 is applied for 100
ps to force the separation.

layer skyrmions with Q = ±1 and Q = ±2 is irrelevantto the information encoding. Besides, at same j, themobility of the bilayer skyrmion driven by an in-planecurrent is better than that of the monolayer skyrmion atlarge β/α.
On the other hand, the monolayer and bilayer
skyrmions with Q = ±1 can be driven into circular motion by an out-of-plane current. First, it is noteworthy
that the maximum speed of the monolayer skyrmion with
Q = ±1 induced by a given j approaches uI as α → 0.
However, there is no maximum speed limit for the bilayer
skyrmion with Q = ±1 induced by a given j, because its
speed is determined by uI/2αD. Namely, the bilayer
skyrmion can move in an extremely fast manner when α
is very small.
Form an application point of view, the circular motionof skyrmions can be harnessed for building skyrmionbased oscillators [59, 60], which generate high-frequency
microwave signals. As the helicity of the skyrmion with
Q = ±1 driven by the out-of-plane current is coupled to
its translational motion, which means the helicity can becontrolled by manipulating the motion of the skyrmion.
With this feature, it is also possible to build skyrmion
helicity-based multi-state memory devices [34], where different skyrmion helicity numbers stand for different elementary information digits. For example, the Bloch-typeskyrmions with η = π/2 and η = 3π/2 can be used tocarry different information.
In addition, the out-of-plane current can lead to theseparation of a bilayer skyrmion with Q = ±2 to twobilayer skyrmions with Q = ±1. The separation of abilayer skyrmion with Q = ±2 does not show the rotation of skyrmion structure, while the monolayer skyrmion
with Q = +2 or Q = −2 is rotating until it is separated
to two monolayer skyrmions with Q = +1 or Q = −1.
The forced separation of skyrmions with higher Q (i.e.,
excited states) to skyrmions with lower Q (i.e., groundstates) could be an important operation in future multistate memory devices, which are based on the manipulation of Q. Our results are useful for understanding bi-

6layer skyrmion physics in frustrated magnets and could
provide guidelines for building skyrmion-based devices.

ACKNOWLEDGMENTS

X.Z. acknowledges the support by the Presidential
Postdoctoral Fellowship of the Chinese University ofHong Kong, Shenzhen (CUHKSZ). M.E. acknowledges
the support by the Grants-in-Aid for Scientific Research from JSPS KAKENHI (Grant Nos. JP18H03676,
JP17K05490 and JP15H05854), and the support by
CREST, JST (Grant Nos.
JPMJCR1874 and JPMJCR16F1). Z.H. and W.W. acknowledge the support
by the National Key R&D Program of China (Grant
Nos. 2017YFA0303202 and 2017YFA206303), the National Natural Science Foundation of China (Grant Nos.
11604148 and 11874410), and the Key Research Programof the Chinese Academy of Sciences (Grant No. KJZDSW-M01). X.L. acknowledges the support by the Grantsin-Aid for Scientific Research from JSPS KAKENHI(Grant Nos. JP17K19074, 26600041 and 22360122). Y.Z.
acknowledges the support by the President’s Fund of
CUHKSZ, the National Natural Science Foundation ofChina (Grant No. 11574137), and Shenzhen Fundamental Research Fund (Grant Nos. JCYJ20160331164412545
and JCYJ20170410171958839).

APPENDIX A: VIDEOS OF SKYRMIONDYNAMICS

Three videos are attached. Video 1 shows the out-ofplane current-induced separation of one bilayer skyrmion
to two bilayer skyrmions. Video 2 shows the out-of-planecurrent-induced separation of one monolayer skyrmion
with Q = +2 to two monolayer skyrmions with Q = +1.
Video 3 shows the out-of-plane current-induced separation of one monolayer skyrmion with Q = −2 to twomonolayer skyrmions with Q = −1.

7Video 2. Out-of-plane current-induced separation of one
monolayer skyrmion with Q = +2 to two monolayer
skyrmions with Q = +1.

Video 1. Out-of-plane current-induced separation of one
bilayer skyrmion (consisting of two monolayer skyrmions
with Q = ±2) to two bilayer skyrmions (consisting ofmonolayer skyrmions with Q = ±1).

Video 3. Out-of-plane current-induced separation of one
monolayer skyrmion with Q = −2 to two monolayer
skyrmions with Q = −1.

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