Multilayer films with perpendicular magnetic anisotropy (PMA) are highly active media in modern magnetic technologies and research. Domain walls in such materials are usually treated either in 2D, similarly to homogeneous magnetic layers (via the so-called effective medium model, in which all the magnetic constants are effectively scaled), or as in multilayers, but with a trivial assumption of a fixed domain wall width Δ and angle ψ across the layers. However, recently it has been argued that the actual equilibrium configuration of the domain walls in multilayers is rather different. Both Δ and ψ, as revealed from the explicit multilayer simulations in these studies, vary from layer-to-layer, with the Néel-like caps of opposite chirality developing at the bottom and the top layers.
In this work, we rigorously prove that such a twist is indeed a ground state in PMA multilayers and find that it persists even for films with high Dzyaloshinsky-Moriya Interaction (DMI). The key aspect of our work is that it — for the first time — provides an accurate and complete calculation of the magnetostatic energy in multilayers, including the contributions that were inherently ignored in the well-known effective medium model.
By solving the exact magnetostatic integrals, we evaluate the total micromagnetic energy density of the domain walls in multilayers analytically and derive the equilibrium Δ and ψ in every magnetic layer. We find that the value DMI at which all the layers become Néel (threshold DMI) is underestimated by the earlier 2D model and provide the exact numerical relations for the new 3D model that contains the wall twist. We analyze the exact influence of this twist on the size of the domains and skyrmions and detect notable differences from the expressions provided by the 2D model. We also find that the extraction of DMI from the domain width measurements is highly inaccurate in the region of small and intermediate values of DMI, where the wall twist usually persists.
Finally, we identify the impact of the domain wall twist on the dynamics of skyrmions in multilayers under the influence of injected currents carrying the spin-orbit torque. We show that the skyrmion velocity and skyrmion hall angle derived from the exact multilayer theory can vary significantly compared with the predictions of the 2D model. We provide the numerical multilayer relations that are valid at low and intermediate values of current density (j). We also explore the high-j regime with the help of multilayer micromagnetic simulations and reveal new physical phenomena, such as the domain wall precession that result in the impeded skyrmion motion.
Our findings are confirmed with explicit multilayer micromagnetic simulations. The corresponding paper is under the preparation to be submitted to Physical Review Letters journal.