Zero-field magnetic skyrmions in exchange-biased ferromagnetic–antiferromagnetic bilayers

We report on the stabilization of ferromagnetic skyrmions in zero external magnetic fields, in exchange-biased systems composed of ferromagnetic–antiferromagnetic (FM-AFM) bilayers. By performing atomistic spin dynamics simulations, we study cases of compensated, uncompensated, and partly uncompensated FM-AFM interfaces, and investigate the impact of important parameters such as temperature, inter-plane exchange interaction, Dzyaloshinskii–Moriya interaction, and magnetic anisotropy on the skyrmions appearance and stability. The model with an uncompensated FM-AFM interface leads to the stabilization of individual skyrmions and skyrmion lattices in the FM layer, caused by the effective field from the AFM instead of an external magnetic field. Similarly, in the case of a fully compensated FM-AFM interface, we show that FM skyrmions can be stabilized. We also demonstrate that accounting for interface roughness leads to stabilization of skyrmions both in compensated and uncompensated interface. Moreover, in bilayers with a rough interface, skyrmions in the FM layer are observed for a wide range of exchange interaction values through the FM-AFM interface, and the chirality of the skyrmions depends critically on the exchange interaction.


INTRODUCTION
Modern society currently produces an ever-increasing amount of data, leading to pressing challenges for datastorage technologies, one of which is storage density, i.e. storing bigger amount of data in smaller space.Moreover, data centers already use around one percent of global electricity demand, calling for less energyhungry data storage technologies.Magnetic skyrmions are nanoscale topological spin structures [1][2][3], that are very promising candidates for data-carrying or for logic elements in developing spintronic devices [3].Skyrmions are stable due to the combination of Heisenberg exchange interactions, asymmetric exchange interactions, so-called Dzyaloshinskii-Moriya (DMI) interaction, and an external magnetic field.DMI, in contrast with the Heisenberg exchange interaction, tends to favour perpendicular magnetic moment orientations and introduces chirality to the system, which is central for the creation of skyrmions.
Skyrmions are often stabilized using external magnetic fields [3], however, for full exploration of spintronic devices involving skyrmions it is desirable to stabilize skyrmions in the absence of a magnetic field.Many works have been devoted to skyrmion stabilization in zero applied field, e.g. in bilayers and multilayers [4][5][6][7][8][9].It was shown that the DMI interaction can be transferred from an antiferromagnet (AFM) to a ferromagnet (FM) in FM-AFM bilayers to observe skyrmions.Recently, AFM skyrmions in zero magnetic were predicted in FM-AFM bilayers on triangular lattice by Monte Carlo simulations [10].
Moreover, it was experimentally shown that skyrmions can be stabilized in zero external magnetic field by us-ing the exchange bias phenomena (EBP) in ferromagnets [11][12][13] and antiferromagnets [7].EBP [14] manifest itself in the shift of magnetic hysteresis along the field axis, and appears in systems with contacting FM-AFM, and other systems.The simplest model of exchange bias involves an uncompensated antiferromagnetic interface [15,16] (non-zero magnetic moment of the AFM interface layer), therefore, an effective field acts from the AFM to the FM layer.The resulting effective field can be used to stabilize magnetic skyrmions in a FM, instead of an external applied magnetic field.However, this uncompensated model is known to lead to an overestimation of the shift of the hysteresis loop, when compared with experimental studies.Therefore, other models assumed a compensated FM-AFM interface (both AFM sublattices are presented at the interface and couple with equal strength to the FM) [17].Koon [18,19] suggested that due to the competition between couplings in AFM with a FM, the interface spins could deviate from the easy axis, and form canted states, leading to the small effective field observed in experiments, that is manifested by EBP with rather small shifts of the hysteresis curve.
Another important factor to consider, while studying EBP, is the interface roughness [19].In most experiments some roughness or geometrical defects are present at the interface.Also, it was shown that exchange interaction through the interface is strongly impacted by the interface roughness [20].It was demonstrated experimentally in Ref. [21] that only about 7% of the spins at the interface are uncompensated and contribute to the EBP.It was confirmed theoretically in Ref. [22,23] that only several percent of uncompensated moments are enough to induce EBP.Also, it was shown that roughness and defects impacts interfacial DMI [24] and therefore skyrmion stability.In addition, roughness has a strong impact on skyrmion dynamics, and results for rough and perfect interface cases can significantly differ [25], which makes accounting of interface defects and roughness vital.
In this work, we study the possibility of stabilizing skyrmions in zero external magnetic fields using the exchange bias phenomena.We investigate various types of FM-AFM interfaces and the range of parameters, such as exchange interaction, DMI, magnetic anisotropy, and temperature allowing skyrmion stabilization.The paper is organized as follows: we start with describing details of atomistic spin dynamics simulations, then we investigate the case of an perfect interface.Finally, we study the impact of disorder or (geometrical) roughness of the interface on the magnetic structure of FM and AFM films.

II. ATOMISTIC SPIN DYNAMICS SIMULATIONS
Atomistic spin dynamics (ASD) is governed by Landau-Lifshits-Gilbert equation: where m i represents an atomic magnetic moment, m i and γ are the saturation magnetization and the gyromagnetic ratio correspondingly.We obtain an effective exchange field B i = −∂H SD /∂m i from the spin Hamiltonian, H SD .The Hamiltonian used in this work includes FM, AFM parts and interaction between FM and AFM subsystems: Concretely: where J ij , J kl , and J ik are the exchange interaction in FM, AFM, and through FM-AFM interface correspondingly.H ext denotes external magnetic field.We take into account DMI interaction D FM ij and magnetocrystalline anisotropy K AFM for AFM.The anisotropy term is included mainly to ensure that the AFM layers stay pinned during the simulations, and is thus larger than what is normally expected in real systems.
In these Langevin-type simulations we employ stochastic fields, B fl i as white noise with properties In our simulations, we use D M = αk B T /γm, D L = νM k B T , where T and k B are temperature and Boltzmann constant respectively (for details see e.g.Ref. [26]).
The atomistic spin dynamics simulations are complemented by Monte Carlo (MC) calculations on the spin Hamiltonian.Since both LLG and MC simulations are based on the same Hamiltonian, they exhibit the same energetics and give the same ground state.However, the ground state search for the simulated systems is often complicated by the large energy barriers present in the systems due to the inclusion of strong DMI and anisotropies.To improve the efficiency of the simulations we therefore consider several different starting states for our simulations, including random, ferromagnetic, and spin-spiral states.
In our simulations we use the spin simulation package UppASD [26,27] and study the FM-AFM bilayer with square lattice consisting of three atomic layers for AFM and one atomic layer for FM.The simulation cell size vary between models, due to the reasons given above, and it is between 120 × 120 × 4 and 160 × 160 × 4. The value of magnetic moment in all simulations is the same for FM and AFM, and equals 1 µ b /atom.The value of exchange interactions in FM and AFM is fixed in all simulations and the ratio is |J AFM /J FM | = 1.Most simulations are performed at zero temperatures, unless stated otherwise.
The topological charge Q ab i is calculated using the method proposed in [28], since the model is atomistic and the expression for the continuum limit cannot be used.In particular, each unit cube of the lattice is divided into 12 triangles, two for each face.Then the chirality enclosed by the unit cube is given by the expression: where is the chirality of the triangle formed by three neighboring spins.Spins are taken in in counter-clockwise direction according to [28].

A. Uncompensated FM-AFM interface
We start with an uncompensated model of the AFM interface.This means that the magnetic moments in each AFM layer are parallel to each other with J AFM inlayer > 0 with an antiparallel coupling between layers, J AFM interlayer < 0 (the so-called A-type AFM).Therefore, the total magnetic moment of the AFM interface layer in contact to the FM layer is non-zero and there is an effective exchange field that acts on FM film (see Fig. 1a).Moreover, we assume a sizeable magnetic anisotropy K AFM in each AFM layer, so AFM magnetic moments are almost "frozen" at their easy axis positions during the reversal of an external magnetic field [15,16].Due to the effective field acting on the FM layer, we observe an exchange bias phenomenon with the shift of the magnetic hysteresis depending on the strength of exchange interaction J FM/AFM through the FM-AFM interface and on the magnetic anisotropy in the AFM layer.The appearance and stabilization of skyrmions will also depend on the strengths of this effective field, and therefore, the exchange coupling J FM/AFM .In Fig. 2 we show the magnetic structure of the FM layer for various values of DMI, and exchange interaction through the FM-AFM interface J FM/AFM from simulations without an external magnetic field.One can see that in the absence of exchange interaction through the FM-AFM interface we observe stripe domains in FM (Fig. 2, first column).However, when the exchange interaction, J FM/AFM , is increased, single skyrmions, then a skyrmion lattice, appears, as can be seen in Fig. 2 (for example, the second figure in the second row).A further increase of the exchange interaction leads to a collapse of parts of the skyrmions (Fig. 2).A continued increase of the exchange interaction, J FM/AFM , causes the effective field to be sufficiently strong to overcome the DMI and the magnetization of FM layer will become uniform.
Another important aspect to consider, while studying skyrmions is the impact of the DMI on the magnetic structure of the FM layer.We should note that the skyrmion lattice presented in Fig. 2 is very sensitive to the DMI value.As can be seen from Fig. 2, the increase of DMI value in FM leads to skyrmions observation for larger values of exchange interactions through the interface, since a stronger effective field is required to overcome DMI.
In Fig. 3 the magnetic hysteresis loops are shown for two values of interplane exchange interactions, J FM/AFM /J F M = -0.4 and -0.8.It can be seen that in both cases the hysteresis loops are shifted and the shift is increasing with the increase of the exchange interaction J FM/AFM .We notice that hysteresis loops consist of several loops, something that was also observed experimentally in Ref. [6].At zero fields the inclined part of the hysteresis loop corresponds to non-collinear magnetization structure in the film.
The results in Fig. 2 are obtained for very low tem-peratures.However, it is known that the stability and inner structure of skyrmions depend on the temperature, which is especially pronounced near the Curie temperature (please see Ref. [29] and reference therein).Therefore, we have studied the impact of temperature on the skyrmion lattice.For low temperatures the magnetic structure remains similar to the one presented in Fig. 2, however, with the rise of the temperature the magnetic structure changes (see, for example, Fig. 4a) and skyrmions disappear for sufficiently high temperatures.
The explicit temperature dependence of the skyrmion number of the simulation cell is presented in Fig. 4c, showing that the topological magnetic order disappears at a temperature just above 2.5 k B /J F M .

B. Compensated FM-AFM interface
Next we study the case of afully compensated AFM interface.The structure of the model can be seen in Fig. 1(b).In this case, the total magnetic moment of the AFM interface atoms in contact with the FM layer is zero and therefore there is no effective exchange field acting on the magnetic moments in the FM layer that stems from atoms in the AMF layer, at least if the AFM moments are frozen in perfect AFM order with vanishing fluctuations.This is expected to not result in the formation of skyrmions.However, fluctuations of magnetic moments will generate an effective exchange field across the AFM-FIG.3. Exchange-biased magnetic hysteresis of a FM film for J FM/AFM /J FM = −0.4(blue), and J FM/AFM /J FM = −0.8(red).

FM interface, with the possibility to induce skyrmions.
To allow for this we have for the simulations presented in this section used a magnetic anisotropy in the AFM layers that is half of that used in the uncompensated model.We present the magnetic structure of FM and AFM layers in zero magnetic field, for and lower magnetic anisotropy in the AFM layer, see Fig. 5.In the absence of exchange through the FM-AFM interface, the magnetic structure of FM film is naturally similar to uncompensated model (see Fig. 2 for J FM/AFM /J FM = 0).It can be seen from Fig. 5(a), that there are skyrmions in the FM layer.We would like to note that no skyrmions would be observed for high values of the magnetic anisotropy, for the same values of other parameters.Moreover, if we study closely the magnetic structure of the AFM layers, as displayed in Fig. 5(b), an imprint of FM magnetic structure, in this case, of skyrmions in the AFM layer can be observed.The transfer of the magnetic structure of FM to AFM in exchange-biased bilayers was shown experimentally in Ref. [7].Such random topological magnetic textures could be useful for e.g.reservoir computing [30].Finally, it was shown experimentally in [7], that induction of skyrmions in AFM by transferring FM magnetic order to AFM by exchange bias, is a very promising technique.Our findings support these experimental observations.Antiferromagnetic skyrmions are very promising candidates for memory devices because they are expected to move without a skyrmion Hall effect.

C. Partly uncompensated interface. Impact of interface roughness
In this section, we study the case of a partly uncompensated interface, where uncompensated spins appear due to, for example, interface roughness or disorder.It was shown in Ref. [30] that skyrmions pinned by inhomogeneities can be useful for the implementation of reservoir computing.The scheme of the model is presented in Fig. 1(c).We start with a perfectly compensated AFM interface (checkerboard AFM pattern) and then assume that one out of four AFM magnetic moments on the interface is uncompensated (the orange one in Fig. 1(c)).We study cases when coupling ferromagnetic J FM/AFM uncomp > 0, or antiferromagnetic J FM/AFM uncomp < 0 to the FM layer and to its neighbors in the layer.The other AFM layers are ordered as perfect AFM.Therefore, in this system, the EBP, and, therefore, the effective field will depend on the ratio between coupling in an AFM interface layer, and through the FM-AFM interface.
In particular, in Fig. 6 we present the case when the coupling of uncompensated AFM magnetic moments (orange ones in Fig. 1c) through the interface is much stronger than the coupling of compensated AFM spins (green ones in Fig. 1c) J FM/AFM uncomp > J FM/AFM comp .At zero exchange through the interface, similarly, to compensated or uncompensated model, we observe stripe domains in FM layer (see Fig. 6(b)).Then, with an increase of J FM/AFM uncomp exchange coupling we can observe the formation of a skyrmion lattice in the FM layer (please see Fig. 6(c)).Skyrmions become smaller with increasing exchange interaction Fig. 6(d) and then, skyrmions lattice collapse.
FIG. 6. Ferromagnetic skyrmions in FM-AFM bilayer with partly uncompensated interface in zero external magnetic field for various values of exchange interaction J FM/AFM through the FM-AFM interface.
In particular, (a) J Above, we have studied the case of positively coupled through the FM-AFM interface uncompensated AFM spins J FM/AFM uncomp > 0, while were J FM/AFM comp < 0. However, one can observe skyrmions in this model even for negative values of exchange interaction through the interface of uncompensated AFM spins J FM/AFM uncomp < 0. For small negative values values of J FM/AFM uncomp the magnetization of FM layer is similar to Fig. 6, however, with the rise of exchange interaction value, we can observe skyrmion lattice, as presented in Fig. 6(a).Importantly, for negative values of coupling J FM/AFM uncomp skyrmions change chirality, and one can observe a lattice of skyrmions (Fig. 6(a)), with negative chirality.With further increase of coupling number of skyrmions decrease (Fig. 6b) and then they disappear.The dependence of skyrmions number as a function of exchange interaction J FM/AFM uncomp is presented in Fig. 7.One can observe, that there is the critical value of exchange interaction allowing skyrmions formation, however, skyrmions are observed in a wide range of exchange interactions through the interface, both positive and negative ones.We can conclude, that with only 25% of spins uncompensated at the AFM interface, it takes enormous couplings to saturate magnetization in an FM film, and therefore, skyrmions exist in a much wider range of exchange interaction values than in the case of fully uncompensated interface considered above.In the previous section, we studied a model when a perfectly compensated interface became partly uncompensated due to, for example, disorder.In that case, the uncompensated spins were distributed periodically at the interface.However, in real systems with rough/disordered interfaces, these defects would be distributed randomly.
In this section, we demonstrate the impact of interface roughness/disorder by introducing randomly distributed magnetic moments with positive and negative exchange interaction through the interface.The case when all of the spins have J FM/AFM > 0 (or J FM/AFM < 0) would correspond to the uncompensated interface, studied above.In turn, the situation when the number of spins with J FM/AFM > 0 and J FM/AFM < 0 is the same corresponds to a compensated interface.However, unlike for the perfectly compensated interface above, here, spins from two AFM sublattices are distributed randomly, and therefore, effective fields can arise and lead to skyrmions stabilization.The other ratio between two AFM sublat-tices will lead to cases close to the partly uncompensated interface.
The magnetic structure of a FM layer is presented in Fig. 8.We study various percentage of spins from two AFM sublattices presented at the AFM interface layer, which corresponds to various percentage of positively of negatively coupled spins at the interface.In Fig. 8a,c,d we illustrate the cases when most of the spins either negative or positively coupled.It can be seen from the figure that the behavior of the system is not symmetric, concerning the amount of positive (negative) coupling through the AFM-AFM interface.In particular, one can observe the coexistence of labyrinth domains and skyrmions for case Fig. 8(a), however, in case Fig. 8(d) we obtain stable skyrmions lattice.It is due to the sign of DMI, and also to the exchange bias effect itself.In Fig. 8(b) we present the stabilization of skyrmions in FM-AFM bilayer, when the amount of positively and negatively coupled spins at the interface is the same.In all presented cases, one can observe the imprint of FM magnetic structure in the AFM interface layer, for example, skyrmions lattice from Fig. 8(d) will be imprinted in the AFM interface layer, leading to the similar periodic pattern, with a bigger deviation of AFM magnetic moments in places corresponding to skyrmions positions.

IV. CONCLUSIONS
We have performed atomistic spin dynamic simulations to study the stabilization of magnetic skyrmions in an exchange-biased FM-AFM bilayer in a zero external magnetic field.We have investigated the cases of compensated, uncompensated FM-AFM interfaces, and the impact of geometrical roughness at the interface.We have shown that, in the case of an uncompensated interface, the effective field acting from AFM to FM film leads to the appearance of a skyrmions and even a skyrmion lattice.However, either too weak or too strong an exchange through the FM-AFM interface destroys skyrmions.The fully compensated AFM interface results in the appearance of skyrmions, however, only for low values of magnetic anisotropy in AFM.Moreover, due to the transfer by exchange bias of the magnetic structure of a FM to an AFM, one can observe imprint of FM skyrmions in AFM, which was also observed experimentally.In the case of a partly uncompensated interface, where only 25% of the spins at the AFM interface are uncompensated, due to competition between compensated and uncompensated spins, skyrmions can be observed both for positive and negative exchange interaction through the FM-AFM interface in the wide range of exchange values and skyrmions chirality depend of the exchange interactions through the FM-AFM interface.We show that the interface roughness can lead to skyrmions lattice stabilization in the FM layer for the cases of both compensated and uncompensated interfaces.The results of this work could be useful in reducing the energy consumption of perspective memory devices based on skyrmions and for skyrmion-based reservoir computing applications.

V. DATA AVAILABILITY
The data used to produce results presented in paper, are obtained using UppASD software, available to download, from the UppASD web-page [27].

FIG. 1 .
FIG. 1.The scheme of uncompensated (a), compensated (b), and partly-uncompensated (c) AFM interface models.In all our calculations we use one FM layer and three AFM layers.

FIG. 2 .
FIG. 2. Ferromagnetic skyrmions in FM/AFM bilayer in zero external magnetic field for various values of exchange interaction through the FM/AFM interface and the strength of the DMI for interaction.Note that in the figure these two interactions are divided by the strength of the exchange interaction of the ferromagnetic layer.Magnetic anisotropy use din calculations is K AFM /J FM = −0.5.

FIG. 7 .
FIG. 7. Skyrmion number depends on the exchange interaction ratio between compensated and uncompensated spins in the AFM interface.The minus/plus sign in the skyrmion number denotes skyrmion chirality.