Micromagnetic modelling of ferro-, ferri-, and antiferromagnetic materials.

Abstract

The manipulation of magnetic textures in ferromagnets (FM) has been the focus of intense research in recent years. More recently, this focus has been displaced towards the use of antiferromagnets (AFM), and ferrimagnets (FiM), which are more robust against external perturbation and can sustain faster magnetization dynamics.In this context, simplified models have been proved very useful to describe and clarify the phenomenology on FM [1], AFM [2], and FiM [3] materials giving in some cases also good quantitative predictions. Nevertheless, AFM and FiM are described by two sublattices strongly couple through an exchange interaction while FM are described by only one magnetic lattice. In order to recover the FM case from the FiM description, one needs not only to set the second sublattice magnetization to zero but also its corresponding energy densities. It is possible to describe all these cases in a unified framework by a simple renormalization of the energy densities to the squared of the sublattices magnetization modulus. To give an example, considering Ku1 and Ku2 the common uniaxial anisotropy parameters, we substitute them by ku1*(MS1)2 and ku2*(MS2)2 respectively. This procedure is followed for all the energy density terms and included in Petaspin micromagnetic solver [4].Here, we focus on the statics and dynamics of domain walls (DWs) when excited by spin-orbit torque (SOT). Regarding the static states of AFM, we show that as in FM, AFM also may exhibit periodic patterns due to the interfacial Dzyaloshinskii-Moriya interaction (iDMI) [5]. In this case, the ‘effective’ inhomogeneous exchange accounting for the two intralattice inhomogeneous exchange contributions and the one interlattice inhomogeneous exchange plays the role of the exchange interaction in FM. Correspondingly, the ‘effective’ iDMI to consider is the sum of the iDMI parameter of the two sublattices.Additionally, we show that similarly to FM, iDMI promotes the nucleation of new domains at the edges of and AFM strip even for moderate current densities [6]. Differently from FM, in AFM this effect is more efficient due to the robustness of the Néel DW against SOT excitation dynamics. This effect is one of our main results since it could limit the maximum DW velocity in and AFM racetrack memory because the nucleation of new domains would change the stored information, thus limiting the maximum current density that can be applied.We have also compared full micromagnetic simulations with a collective coordinates (CC) model which also includes the DW width changes due to the SOT and the spatial tilting of the domain wall [7] for different saturation magnetization ratios. The CC model properly predicts the domain wall width at equilibrium. On the other hand, when the magnetic texture is excited by a current through the SOT we can find three scenarios:(i) when the exchange and anisotropy parameters are larger enough, the DW dynamics can be described by the CC model even if we neglect the DW tilting and width variation. When we neglect these effects we talk about the ‘rigid CC model’.(ii) The cases where the exchange is lower but the anisotropy is large enough cannot be properly described by the rigid CC model. In these cases, the effect of the tilting and the DW width reduced the stationary DW velocity and, thus, they should be taken into account to describe the dynamics. This is shown in Fig. 1 where the solid linear line corresponding to the rigid CC model cannot predict the stationary DW velocities from the full micromagnetic simulations. Nonetheless, the dashed line corresponding to the CC model follows the saturation of the velocity due to the DW tilting.(iii) For low anisotropy parameters the excitation of self-oscillation threshold can be found for moderate current densities (red line in Fig. 2). We observe that before this threshold, where the Gilbert damping is almost compensated by the antidamping term induced by the SOT, the DW velocity increases faster than the usual linear trend being able to reach much larger velocities. Neither the rigid CC model nor the CC model can describe this effect and these cases required a full micromagnetic approach.From a fundamental point of view, this super-linear trend is another of our main results. The threshold current for self-oscillations depends on the anisotropy [8] which can be tuned, opening a path to exploit the super-linear DW mobility under moderate current densities. **