Magnetic bimerons as skyrmion analogues in in-plane magnets
A magnetic bimeron is a pair of two merons and can be understood as the in-plane magnetized version of a skyrmion. Here we theoretically predict the existence of single magnetic bimerons as well as bimeron crystals, and compare the emergent electrodynamics of bimerons with their skyrmion analogues. We show that bimeron crystals can be stabilized in frustrated magnets and analyze what crystal structure can stabilize bimerons or bimeron crystals via the Dzyaloshinskii-Moriya interaction. We point out that bimeron crystals, in contrast to skyrmion crystals, allow for the detection of a pure topological Hall effect. By means of micromagnetic simulations, we show that bimerons can be used as bits of information in in-plane magnetized racetrack devices, where they allow for current-driven motion for torque orientations that leave skyrmions in out-of-plane magnets stationary.
Over the last years magnetic skyrmions [Fig. 1(a) top] Skyrme (1962); Bogdanov and Yablonskii (1989); Bogdanov and Hubert (1994); Rößler et al. (2006); Mühlbauer et al. (2009); Nagaosa and Tokura (2013) have attracted immense research interest, as these small spin textures possess strong stability, characterized by a topological charge . Skyrmions offer a topological contribution to the Hall effect Bruno et al. (2004); Neubauer et al. (2009); Schulz et al. (2012); Kanazawa et al. (2011); Lee et al. (2009); Li et al. (2013); Hamamoto et al. (2015); Lado and Fernández-Rossier (2015); Göbel et al. (2017, 2017); Ndiaye et al. (2017); Yin et al. (2015), commonly measured in skyrmion crystals, and can be stabilized as individual quasiparticles in collinear ferromagnets. They can be driven by currents in thin films Jonietz et al. (2010); Sampaio et al. (2013); Nagaosa and Tokura (2013); Zang et al. (2011); Iwasaki et al. (2013); Jiang et al. (2017); Litzius et al. (2017); Tomasello et al. (2014); Göbel et al. (2018) allowing for spintronics applicability. The stabilizing interaction in most systems is the Dzyaloshinskii-Moriya interaction (DMI) Dzyaloshinsky (1958); Moriya (1960), while theoretical simulations also point to other stabilizing mechanisms, e. g. frustrated exchange interactions Okubo et al. (2012); Leonov and Mostovoy (2015).
Magnetic bimerons 111Note that recently the term ’bimeron’ has also been used for elongated skyrmions Ezawa (2011); Du et al. (2013); Silva et al. (2014), instead of the original object, found in dual layer two-dimensional electron gases and quantum Hall systems Moon et al. (1995); Yang and MacDonald (1995); Brey et al. (1996); Bourassa et al. (2006); Côté et al. (2010). Throughout this Letter we always refer to the latter. [Fig.1(a) bottom] are the combination of two merons [red and blue] and can be understood as in-plane magnetized versions of magnetic skyrmions. Instead of the out-of-plane component of the magnetization it is an in-plane component which is radial symmetric about the quasiparticle’s center; being aligned with the saturation magnetization of the ferromagnet at the outer region of the bimeron and pointing into the opposite direction in the center. Recently, Kharkov et al. showed that isolated bimerons can be stabilized in an easy-plane magnet by frustrated exchange interactions Kharkov et al. (2017). In DMI dominated systems (as is the case for all experimentally known skyrmion-host materials) bimerons have only been shown to exist as unstable transition states Heo et al. (2016); Zhang et al. (2015).
In this Letter, we show that bimerons in frustrated magnets can also be stabilized in an array, the bimeron crystal. Furthermore, we propose a structural configuration that allows for DMI stabilizing isolated bimerons and bimeron crystals. We compare fundamental properties of skyrmions and bimerons and find that both show the same topological Hall effect, whereas the bimeron allows for a pure detection, that is without superposition of the anomalous and ordinary Hall effects. Elaborating on the spintronics applicability of bimerons in in-plane racetrack memory devices, we find that bimerons can be driven by spin currents, similar to skyrmions. However, they extend the class of materials and spin-torque configurations for building spintronics devices.
Stabilization of bimerons and bimeron crystals.
A bimeron [see Fig. 1(a) bottom] consists of two subtextures: a meron and an antimeron (with mutually reversed components of the magnetic moments ). Still, the bimeron itself is the quasiparticle in in-plane magnets, since merons and antimerons can not exist individually in a ferromagnet. The topological charge density
is distributed radially symmetrically around the center of the bimeron and integrates to ; meron and antimeron carry a topological charge of each Tretiakov and Tchernyshyov (2007).
The recurring idea of this Letter is a geometric comparison of skyrmions, bimerons, and antiskyrmions: all three magnetic textures are related by a rotation of each spin around an in-plane axis (in this Letter always ). A bimeron is constructed by rotating each spin of a skyrmion by [cf. Fig. 1(a)]; for an antiskyrmion the spins have to be rotated by another .
To find stable bimerons or bimeron crystals one can therefore start from any system that stabilizes skyrmions and rotate every vectorial term in the Hamiltonian. The most effortless approach is to consider skyrmions stabilized by frustrated exchange . If the scalar constants for nearest and next-nearest neighbor interactions have opposite signs the ground state of the system is a spin-spiral phase Okubo et al. (2012). When an external magnetic field and easy-axis anisotropy are present pointing out-of-plane, skyrmions and skyrmion crystals may be stabilized.
Following this idea, bimerons and bimeron crystals are stabilized in a system where both and are rotated in-plane [cf. Fig. 1(a)]. Then, the Hamiltonian
gives the same energy as for the skyrmion phase before the rotation. The results of Monte Carlo simulations confirming this finding are presented in the Supplemental Material Sup (). The analogy of the two systems does also hold for other typical phases: For low fields we find a spin-spiral state, for medium fields the bimeron crystal and for high fields the system is fully magnetized. At the transition we find isolated bimerons in an in-plane magnetized background.
To illustrate the geometric equivalence of bimeron and skyrmion we used an easy-axis anisotropy along an in-plane direction, even though such quantity is commonly small. Our results also hold for systems without anisotropy or with an easy-plane anisotropy (as in Ref. Kharkov et al., 2017, where isolated meta-stable bimerons have been considered), since the applied magnetic field makes the two in-plane directions inequivalent (see Sup ()).
It is a relativistic energy contribution due to spin-orbit coupling and broken inversion symmetry. The DMI vectors obey Moriya’s symmetry rules Moriya (1960) and can be estimated from the Levy-Fert rule Fert and Levy (1980); points into the direction , i. e., it is perpendicular to the plane of the two lattice sites and the nearest heavy-metal atom (HM). Similar to the frustrated exchange interactions the DMI leads to spin canting, but since it is vectorial it strictly dictates the type of magnetic texture: Skyrmions can not be turned into bimerons by rotating and only, have to be adjusted as well.
At interfaces of layered systems [Fig. 1(b)] heavy-metal atoms (green and blue) induce DMI vectors between neighboring magnetic atoms (black). Typically, the DMI vectors form a toroidal arrangement and produce Néel skyrmions (e) or Néel skyrmion crystals (h). Rotating the HM atoms around the bond in direction, the are rotated in the same way according to the Levy-Fert rule. If now external field and anisotropy are oriented along the direction, as in the frustrated exchange case, bimerons or bimeron crystals are stabilized for the same parameters (in magnitude) as for the skyrmion phase, see Fig. 1(f) and (i). This approach is confirmed by Monte Carlo simulations and atomistic simulations of the Landau-Lifshitz-Gilbert equation (LLG) Landau and Lifshitz (1935); Gilbert (1955) (see Supplemental Material Sup ()). To complete this picture, we point out that for the stabilization of antiskyrmions (g, j) the indicated HM atoms (green) have to be rotated another around the bond in direction (g) — a configuration recently found experimentally Nayak et al. (2017) in the Heusler alloy MnPtPdSn. The corresponding DMI is called ‘anisotropic’ Güngördü et al. (2016); Huang et al. (2017); Hoffmann et al. (2017).
Summarizing up to this point we predict the existence of isolated bimerons and bimeron crystals by frustrated exchange and DMI. Next, we discuss implications of the in-plane magnetized bulk systems and thin films with bimerons for electronic properties and spintronic applications.
Pure topological Hall effect of electrons.
When an electric field is applied to a metal, a current flows according to Ohm’s law . For a skyrmion crystal the transverse element of the resistivity tensor
is decomposed into an ordinary Hall contribution Hall (1879) due to an external magnetic field , an anomalous Hall contribution Nagaosa et al. (2010) due to spin-orbit coupling and a net magnetization , and a topological Hall contribution due to the local topological charge density [Eq. (1)] that acts like an emergent field . For skyrmions , , and point along the direction.
For a bimeron the spin rotation renders the component of magnetic quantities zero, [cf. Fig. 1(a)], but since is invariant under global spin rotation remains. For this reason only the topological Hall effect emerges in a sample with bimerons (see Fig. 2). This hallmark for real-space topology can be detected in an isolated manner making bimeron crystals a playground for investigating fundamental physics. In the Supplemental Material Sup () we numerically validate the equivalence of the topological Hall effect for skyrmion and bimeron crystals following Refs. Hamamoto et al., 2015; Göbel et al., 2017, 2017; Ndiaye et al., 2017; Yin et al., 2015; Göbel et al., 2018, in which the energy-dependent conductivity is discussed.
Current-driven motion in thin film.
In the following, we show that bimerons can be utilized as topologically protected information carriers in in-plane magnetized thin films and discuss similarities and differences to skyrmion racetrack devices Sampaio et al. (2013); Fert et al. (2013); Parkin (2004); Parkin et al. (2008); Parkin and Yang (2015).
In the spin-transfer toque (STT) scenario Sampaio et al. (2013) a current of spin-polarized electrons is applied along the ferromagnet. Since the electron spin at site is given by the magnetic texture itself, the torque is rotated in the same way as the magnetization, which leads to identical motion of bimerons and skyrmions under STT.
A more efficient way to drive skyrmions is the spin-orbit torque (SOT) scenario Sampaio et al. (2013): spins are injected perpendicularly to the ferromagnetic film, via (i) a spin-polarized current traversing a second ferromagnetic layer with a distinct magnetization or via (ii) a charge current in an adjacent heavy-metal layer, which is transformed into a spin current by the spin Hall effect ( in cubic systems). The perpendicularly injected spins are independent of the magnetization in the actual racetrack layer, and large torques can be generated.
The motion of magnetic textures in nanostructured samples is simulated within the micromagnetic approach, that models magnetic textures on a larger length scale compared to the atomistic simulations presented in Fig. 1. We solve the LLG equation (see Sup ()) for each micromagnetic moment with the in-plane spin torque Slonczewski (1996) proportional to
where is the layer thickness, is the saturation magnetization, and is the spin polarization of a perpendicular current for (i) or spin Hall angle for (ii). For comparability the parameters of Co/Pt are taken from Ref. Sampaio et al., 2013 (they are specified in the Supplemental Material Sup ()). The DMI that stabilizes bimerons is derived from the vectors of Fig. 1(c)
For the SOT scenario (i) skyrmions in a magnetized ferromagnet can be driven by injected spins . Due to the global rotation of spins a bimeron in an magnetized ferromagnet can be driven by spins and remain stationary for , see Fig. 3(a).
Towards utilization in a racetrack device the current-driven motion () is the most relevant aspect of SOTs. Using a cubic heavy metal material for scenario (ii) (e. g. Pt), i. e., , skyrmions and bimerons are propelled equally in a system with their favoring easy-axis anisotropy ( for the skyrmion and for the bimeron) and DMI [Fig. 1(b) for the skyrmion and Fig. 1(c) for the bimeron], as long as the demagnetization field is neglected, cf. Fig. 3(b).
The demagnetization field acts effectively as an inhomogeneous in-plane magnetic field for both textures, leading to an increase of the skyrmion size and a decrease of the bimeron size. Consequently the skyrmion velocity is larger than that of the bimeron [cf. Fig. 3(c) and see Supplement Sup () for a complementary Thiele equation analysis]. Still, the bimeron can reach similar velocities as the skyrmion since the bimeron allows for larger applied currents densities. While a bimeron is still stable at , the skyrmion is already annihilated at the edge of the racetrack for . Both quasiparticles can reach velocities of around . In the Supplemental Material Sup () we show that current-driven motion is also possible for a material with an easy-plane anisotropy (), when a magnetic field is applied in-plane to generate a preferred direction.
Conclusion and perspective.
In this Letter we have demonstrated how to generate isolated bimerons and bimeron crystals via DMI and frustrated exchange interactions. Since the magnetic moments of a bimeron are merely rotated moments of a skyrmion, the topological properties of the two objects are unchanged and the topological Hall effects due to both of them are identical. Nevertheless, the fact that all magnetic quantities (net magnetization and stabilizing field) are rotated, while the emergent field is not, allows for the pure and therefore unambiguous detection of the topological Hall effect in bimeron systems, and for the development of future spintronic devices based on this effect.
We have shown that magnetic materials with in-plane magnetization can be used to build racetrack storage devices with magnetic bimerons as carriers of information. In these materials, the current-induced dynamics of bimerons can be accomplished similarly to that of skyrmions in conventional racetracks. Furthermore, in-plane ferromagnets allow us to use different orientations of injected spins for the propulsion of bimerons as well as for their generation (for in analogy to Refs. Sampaio et al., 2013; Romming et al., 2013). A technological advantage of these materials is the stackability of the quasi-one-dimensional racetracks, since the dipolar energy of two in-plane magnets is smaller than that of two out-of-plane magnets. The smaller stray fields in a bimeron-based racetrack allow for a denser array of tracks in three dimensions and thus a higher storage density.
The established analogy between skyrmions and bimerons can be carried over to all types of skyrmion-related spin textures such as higher-order skyrmions Ozawa et al. (2017); Leonov and Mostovoy (2015), biskyrmions Yu et al. (2014); Peng et al. (2017), multi- skyrmions Bogdanov and Hubert (1999); Zhang et al. (2016a); Zheng et al. (2017); Zhang et al. (2018), bobbers Rybakov et al. (2015); Zheng et al. (2018), and topologically trivial bubbles. Regarding applicability in spintronics the antiferromagnetic skyrmions Barker and Tretiakov (2016); Zhang et al. (2016b, c); Bessarab et al. (2017); Göbel et al. (2017); Shen et al. (2018); Akosa et al. (2018) that become antiferromagnetic bimerons (two mutually reversed bimerons on different sublattices), stand above all, since they allow for SOT-driven dynamics precisely in the middle of the racetrack at speeds of up to several . Very recently the existence of such bimerons has been confirmed experimentally in synthetic antiferromagnets Kolesnikov et al. (2018).
Acknowledgements.We are grateful to Steffen Trimper and Stuart S. P. Parkin for fruitful discussions. This work is supported by Priority Program SPP 1666 and SFB 762 of Deutsche Forschungsgemeinschaft (DFG). O.A.T. acknowledges support by the Grants-in-Aid for Scientific Research (Grant Nos. 17K05511 and 17H05173) from MEXT (Japan), by the grant of the Center for Science and Innovation in Spintronics (Core Research Cluster), Tohoku University, and by JSPS and RFBR under the Japan-Russian Research Cooperative Program.
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