# Current-driven dynamics of frustrated skyrmions in a synthetic antiferromagnetic bilayer

###### Abstract

Magnetic skyrmions can be stabilized and manipulated in frustrated magnets, and are useful for building nonvolatile magnetic memories and logic devices. Here we study the current-driven dynamics of frustrated skyrmions in an antiferromagnetically exchange coupled bilayer system. The bilayer skyrmions consisting of two monolayer skyrmions with opposite skyrmion numbers are studied. We show that the in-plane current-driven bilayer skyrmion moves in a straight path (showing no skyrmion Hall effect), while the out-of-plane current-driven bilayer skyrmion moves in a circle path. 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 , where and denote the damping parameter and non-adiabatic spin transfer torque strength, respectively. Besides, we reveal that one bilayer skyrmion (consisting of monolayer skyrmions with ) can be separated to two bilayer skyrmions (consisting of monolayer skyrmions with ) driven by an out-of-plane current. Our results may be useful for designing skyrmion devices based on frustrated multilayer magnets.

###### pacs:

75.10.Hk, 75.70.Kw, 75.78.-n, 12.39.Dc^{†}

^{†}preprint:

^{†}

^{†}thanks: These authors contributed equally to this work.

^{†}

^{†}thanks: These authors contributed equally to this work.

Introduction. The magnetic skyrmion is a topologically non-trivial object Roszler_NATURE2006 (); Nagaosa_NNANO2013 (); Wanjun_PHYSREP2017 (), which can be regarded as quasi-particles Lin_PRB2013 () and promises advanced electronic and spintronic applications Finocchio_JPD2016 (); Kang_PIEEE2016 (); Fert_NATREVMAT2017 (); Zhou_NSR2018 (). For examples, recent theoretical and experimental studies have suggested that skyrmions can be used as building blocks for racetrack-type memories Sampaio_NNANO2013 (); Tomasello_SREP2014 (); Guoqiang_NL2017 (); Muller_NJP2017 (), logic computing devices Xichao_SREP2015B (), and bio-inspired applications Yangqi_NANO2017 (); Lisai_NANO2017 (); Prychynenko_PRAPPL2018 (); Bourianoff_AIP2016 (). The magnetic skyrmion was first experimentally observed in ferromagnetic (FM) materials with Dzyaloshinskii-Moriya (DM) interactions almost ten years ago in 2009 Muhlbauer_SCIENCE2009 (), where the DM interaction is an essential energy term stabilizing skyrmion textures Muhlbauer_SCIENCE2009 (); Yu_NATURE2010 (); Du_NCOMMS2015 (). The DM interaction can also be induced at the interface between the heavy metal and ferromagnet Yang_PRL2015 (), which promotes recent studies of skyrmions in magnetic bilayer and multilayer structures Wanjun_PHYSREP2017 (); Wanjun_NPHYS2017 (); Litzius_NPHYS2017 (); Xichao_NCOMMS2016 (); Woo_NMATER2016 (); MoreauLuchaire_NNANO2016 (); Pollard_NCOMMS2017 (); Woo_NatElect2018 ().

However, by inducing the DM interaction is not the only way to stabilize skyrmions. Most recently, theoretical and experimental studies have demonstrated that skyrmions can be stabilized in frustrated magnets even in the absence of DM interaction Okubo_PRL2012 (); Leonov_NCOMMS2015 (); Leonov_NCOMMS2017 (); Lin_PRB2016A (); Hayami_PRB2016A (); Yuan_PRB2017 (); Xichao_NCOMMS2017 (); Rozsa_PRL2016 (); Rozsa_PRB2017 (); Kharkov_PRL2017 (); Sutcliffe_PRL2017 (); Hou_AM2017 (); Liang_NJP2018 (), where skyrmion textures are stabilized by competing exchange interactions Leonov_NCOMMS2015 (); Leonov_NCOMMS2017 (); Lin_PRB2016A (); Xichao_NCOMMS2017 (). Frustrated skyrmions have many unique physical properties compared with their counterparts in DM ferromagnets. For examples, both skyrmions and antiskyrmions can exist in a frustrated magnet as metastable states Leonov_NCOMMS2015 (); Leonov_NCOMMS2017 (); Xichao_NCOMMS2017 (); Lin_PRB2016A (). The frustrated skyrmions with skyrmion number of can merge and form skyrmions with higher skyrmion number of Xichao_NCOMMS2017 (). Besides, the center-of-mass dynamics of a frustrated skyrmion is coupled to its helicity dynamics Leonov_NCOMMS2015 (); Leonov_NCOMMS2017 (); Lin_PRB2016A (), which results in the rotational motion of a skyrmion Lin_PRB2016A (); Xichao_NCOMMS2017 (). Therefore, frustrated skyrmions can be used as information carriers, which have multiple degrees of freedom that can be utilized to store information Leonov_NCOMMS2015 (); Leonov_NCOMMS2017 (); Lin_PRB2016A (); Xichao_NCOMMS2017 (). Indeed, a recent experimental work demonstrates that the helicity of a frustrated skyrmion can be switched by an electric current pulse Hou_Helicity2018 (), which offers the possibility to design helicity-based skyrmion memories. So far, most studies on frustrated skyrmions are focused on their dynamics in the monolayer system, while bilayer and multilayer systems play an important role in developing nanoscale devices Wanjun_PHYSREP2017 (); Wanjun_NPHYS2017 (); Litzius_NPHYS2017 (); Xichao_NCOMMS2016 (); Woo_NMATER2016 (); MoreauLuchaire_NNANO2016 (); Pollard_NCOMMS2017 (); Woo_NatElect2018 (); Parkin_NNANO2015 (); Xichao_NCOMMS2016 (); Prudnikov_IEEEML2018 ().

In this Letter, we report the dynamics of frustrated skyrmions with and in an antiferromagnetically exchange coupled bilayer system driven by either an in-plane or an out-of-plane spin current. The antiferromagnetically exchange coupled bilayer system can be regarded as a synthetic antiferromagnet, which is promising for building racetrack-type memories Parkin_NNANO2015 (); Xichao_NCOMMS2016 (). We also numerically investigate the current-induced separation of a frustrated skyrmion in the synthetic antiferromagnetic (AFM) bilayer system. We show that the current-driven frustrated skyrmions in the synthetic AFM bilayer system have better mobility than the monolayer skyrmions.

Methods. We consider two frustrated FM layers with competing Heisenberg exchange interactions based on the -- model on a simple square lattice Lin_PRB2016A (); Xichao_NCOMMS2017 (). As shown in Fig. 1(a), the two frustrated FM layers are coupled by an AFM interfacial exchange coupling, which can be realized by utilizing the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction Parkin_NNANO2015 (); Xichao_NCOMMS2016 (); Prudnikov_IEEEML2018 (). For example, the interlayer AFM coupling between two FM layers can be mediated by an ultrathin metallic spacer layer Parkin_NNANO2015 (); Prudnikov_IEEEML2018 (). We assume the background magnetization direction in the bottom FM layer to be pointing along the direction. The total Hamiltonian for the synthetic AFM bilayer is given in Ref. SM, . The simulation is performed by using the standard micromagnetic simulator Object Oriented MicroMagnetic Framework (OOMMF) OOMMF () with our home-made extension modules for the -- classical Heisenberg model Lin_PRB2016A (); Xichao_NCOMMS2017 (), where the time-dependent spin dynamics is described by the Landau-Lifshitz-Gilbert (LLG) equation (see Ref. SM, for modeling details and parameters).

We consider two geometries for the injection of spin-polarized current. For the current-in-plane (CIP) geometry, an in-plane spin current is injected to both top and bottom FM layers, and the adiabatic () and non-adiabatic () spin transfer torque (STT) terms are considered, given as

(1) |

where represents the normalized spin, is the STT coefficient and is the strength of the non-adiabatic STT torque. is the reduced Planck constant, is the electron charge, is the applied driving current density, is the spin polarization rate Sampaio_NNANO2013 (), and is the saturation magnetization. For the current-perpendicular-to-plane (CPP) geometry, an out-of-plane spin current is only injected to the bottom FM layer, which can be realized by harnessing the spin Hall effect in a heavy-metal substrate underneath the bottom FM layer Wanjun_NPHYS2017 (). In such a case, the damping-like STT term is considered, given as

(2) |

where nm is the thickness of a single FM layer and 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 , which is a typical value of spin Hall angle Tomasello_SREP2014 (); Wanjun_NPHYS2017 (). The skyrmion number in a single FM layer is defined by . The internal structure of a skyrmion is also described by its helicity number , of which the definition is given in Refs. Nagaosa_NNANO2013, ; Koshibae_NCOMMS2016, .

Skyrmions driven by an in-plane current. We first study in-plane current-driven skyrmions in a synthetic AFM bilayer [see Fig. 1(a)]. At first, a bilayer skyrmion is relaxed in a synthetic AFM bilayer, which consists of a top skyrmion with and and a bottom skyrmion with and [see Fig. 1(b)]. The monolayer skyrmion structures with are illustrated in Fig. 1(c). In this work, we refer to this bilayer skyrmion as a bilayer skyrmion with . The top and bottom skyrmions are strongly coupled as a result of the AFM interfacial exchange coupling. The relaxed diameter of the bilayer skyrmion with equals nm. Note that the diameters of the relaxed skyrmions in monolayer and bilayer systems are identical. An in-plane spin current is then injected into both top and bottom FM layers (i.e., 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.

Figure 1 shows the trajectories of the monolayer and bilayer skyrmions at the damping parameter . It is found that the bilayer skyrmion with straightly moves along the direction without showing a transverse shift. However, the monolayer skyrmions with and show transverse shifts toward the and directions, respectively. The reason is that the monolayer skyrmion experiences a topological Magnus force Wanjun_NPHYS2017 (); Litzius_NPHYS2017 (); Xichao_NCOMMS2016 (), which leads to its transverse shift. Since the directions of Magnus forces acted on monolayer skyrmions with and are opposite, the bilayer skyrmion consisting of a bottom monolayer skyrmion with and a top monolayer skyrmion with experiences zero net Magnus force and thus can move in a straight path. It is worth mentioning that the transverse shift of skyrmion is referred as the skyrmion Hall effect, which has been directly observed in experiments Wanjun_NPHYS2017 (); Litzius_NPHYS2017 (). A previous study has reported the skyrmion Hall effect of the monolayer skyrmion driven by an in-plane current can be eliminated only when Xichao_IEEE2017 (), which is difficult to be realized in real materials. However, the bilayer skyrmion driven by an in-plane current shows no skyrmion Hall effect even when , which is useful for building racetrack-type devices Xichao_NCOMMS2016 (); Tomasello_SREP2014 ().

As shown in Fig. 2(b), we also simulate the motion of a bilayer skyrmion with , which consists of a bottom monolayer skyrmion with and and a top monolayer skyrmion with and [see Fig. 1(d)]. Similarly, the bilayer skyrmion with driven by an in-plane current also straightly moves along the direction. The monolayer skyrmions with and show transverse shifts toward the and directions, respectively. Note that the relaxed size of skyrmions with equals nm, which is slightly larger than that of skyrmions with .

Figure 2(c) shows the bilayer skyrmion velocity as a function of in-plane driving current density for and . It should be noted that the relation of the bilayer skyrmion with is identical to that of the bilayer skyrmion with at a given and . For both bilayer skyrmions with and , the velocity increases linearly with the in-plane driving current density. At certain values of and , larger will result in larger velocity of the bilayer skyrmion. By comparing the velocities of bilayer and monolayer skyrmions at certain values of and , it is found that bilayer skyrmions move faster than monolayer skyrmions when , while bilayer skyrmions move slower than monolayer skyrmions when . When , the velocities for bilayer and monolayer skyrmions are almost identical. It is worth mentioning that several experiments suggest that is considerably larger than (i.e., ) Heyne_PRL2010 (); Pollard_NCOMMS2012 (), which means the mobility of bilayer skyrmion is better than that of monolayer skyrmion. The monolayer skyrmion speed is proportional to in the absence of the skyrmion Hall effect Sampaio_NNANO2013 (), which is the same case for the bilayer skyrmion studied here.

Skyrmions driven by an out-of-plane current. We also study out-of-plane current-driven skyrmions in a synthetic AFM bilayer. Namely, a spin current is perpendicularly injected into the bottom FM layer (i.e., 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 of a synthetic AFM bilayer skyrmion are strongly coupled. We first study the bilayer skyrmion with as well as monolayer skyrmions with and driven by an out-of-plane current.

As shown in Fig. 3, the dynamics of a bilayer skyrmion with and its monolayer counterparts (see Ref. SM, ) driven by an out-of-plane current are in stark contrast to the case driven by an in-plane current. It can be seen that the bilayer skyrmion with [see Fig. 3(a)] and monolayer skyrmions with and (see Ref. SM, ) driven by an out-of-plane current move in a circle path. The bilayer skyrmion with moves in the clockwise direction, while the monolayer skyrmions with and move in the counterclockwise and clockwise directions, respectively. As pointed out in Ref. Lin_PRB2016A, , the motion direction depends on the skyrmion helicity. For both the bilayer and monolayer skyrmions, the diameter of circle trajectory increases with increasing . Under the same out-of-plane driving current density and , the diameter of the bilayer skyrmion trajectory is significantly larger than that of monolayer skyrmion trajectory, which means the frustrated skyrmion can be delivered farther in the bilayer system driven by an out-of-plane current.

On the other hand, the helicity numbers of bilayer skyrmion with [see Fig. 3(b)] and monolayer skyrmions with and (see Ref. SM, ) driven by an out-of-plane current are coupled to their center-of-mass dynamics. Namely, the helicity number changes linearly with time during the motion of skyrmions. Such a phenomenon has been reported for monolayer skyrmions in Ref. Lin_PRB2016A, , and we show that it also happens for bilayer skyrmions with . As pointed out in Ref. Lin_PRB2016A, , the frustrated monolayer skyrmion has a translational mode and a rotational mode, which can be excited and hybridized by the damping-like STT due to the spin Hall effect. Namely, when the skyrmion in the bottom FM layer is driven by the damping-like STT, it moves along a circle with rotating helicity. At the same time, it drives the skyrmion in the top FM layer into circular motion with rotating helicity due to the AFM interlayer coupling.

Figure 3(c) shows the velocity of the bilayer skyrmion with as a function of out-of-plane driving current density. At a given , the velocity of bilayer skyrmion with increases linearly with driving current density . At a given , the velocity of bilayer skyrmion with is inversely proportional to the value of . By comparing Fig. 2(d) and Fig. 3(c) at same and , it is found that the bilayer skyrmion with driven by an in-plane current can move much faster than the one driven by an out-of-plane current if the value of is large. For example, when MA cm and , the bilayer skyrmion with driven by an in-plane current can reach a velocity of m s at , while the one driven by an out-of-plane current only reaches a velocity of m s. However, it should be noted that the driving force provide by the in-plane current is larger than that provided by the out-of-plane current at the same in this work, as the CIP spin-polarization rate is four times larger than the CPP spin Hall angle. The bilayer skyrmion driven by the out-of-plane current can move faster when the value of the spin Hall angle is increased.

Besides, by comparing the velocities of bilayer skyrmion with and monolayer skyrmions with and at a given [see Fig. 3(d)], it is found that the bilayer skyrmion velocity remarkably decreases with increasing , while the monolayer skyrmion velocity slightly increases with increasing . The significant decrease of bilayer skyrmion velocity with increasing may be caused by the enhancement of the dissipative force provided by the skyrmion in the top FM layer due to the increasing damping parameter. The bilayer skyrmion velocity is much larger than monolayer skyrmion velocity at small (e.g., ), while it is smaller than monolayer skyrmion velocity at large (e.g., ).

We continue to study the bilayer skyrmion with and monolayer skyrmions with and driven by an out-of-plane current. As shown in Fig. 4, it is found that the bilayer skyrmion with does not move toward a certain direction when an out-of-plane current is applied. Instead, it is elongated and then separated to two bilayer skyrmions with . The reason is that the bilayer skyrmion with is topologically equal to a combination of two bilayer skyrmions with and opposite helicity numbers. Such two bilayer skyrmions with tend to move along opposite directions upon the application of the out-of-plane current, as indicated by yellow arrows in Fig. 4. Similarly, when an-out-of-plane current is applied, the monolayer skyrmion with is separated to two monolayer skyrmions with and the monolayer skyrmion with is separated to two monolayer skyrmions with (see Ref. SM, ). The most obvious difference between the out-of-plane current-induced separation of a bilayer skyrmion with and the separation of a monolayer skyrmion with or 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 or is much smaller than that of the bilayer skyrmion with . Hence, the separation of the monolayer skyrmion with shows very obvious rotation caused by the circular motion of monolayer skyrmions with and . Note that if the an out-of-plane current is applied after the separation of the bilayer skyrmion with , it will also drive the bilayer skyrmions with into motion along a large circle [see Fig. 3(a)].

Conclusion. In conclusion, we have studied the dynamics of frustrated bilayer skyrmions driven by an in-plane or out-of-plane spin-polarized current in a synthetic AFM bilayer. It is found that the bilayer skyrmion with or moves in a straight path driven by an in-plane current, while the bilayer skyrmion with moves in a circle path driven by an out-of-plane current. Therefore, the bilayer skyrmion is promising for building racetrack-type applications, as it shows no skyrmion Hall effect and can be delivered in a straight line even when . At same and , the mobility of the bilayer skyrmion with driven by an in-plane current is better than that driven by an out-of-plane current at large . The out-of-plane current can lead to the separation of a bilayer skyrmion with to two bilayer skyrmions with . The separation of a bilayer skyrmion with does not show the rotation of skyrmion structure, while the monolayer skyrmion with or is rotating until it is separated to two monolayer skyrmions with or . Our results are useful for understanding bilayer skyrmion physics in frustrated magnets and could provide guidelines for building skyrmion devices.

Acknowledgments. This work was supported by the Presidential Postdoctoral Fellowship and the President’s Fund of CUHKSZ, the Shenzhen Fundamental Research Fund (Grant Nos. JCYJ20160331164412545 and JCYJ20170410171958839), the National Natural Science Foundation of China (Grant Nos. 11574137, 11604148 and 11874410), the National Key R&D Program of China (Grant Nos. 2017YFA0303202 and 2017YFA206303), the Key Research Program of the Chinese Academy of Sciences (Grant No. KJZD-SW-M01), the Grants-in-Aid for Scientific Research from JSPS KAKENHI (Grant Nos. JP18H03676, JP17K05490, JP15H05854 and JP17K19074), and CREST, JST (Grant Nos. JPMJCR1874 and JPMJCR16F1).

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