Modeling magnetic domain-wall evolution in trilayers with structural inversion asymmetry

The one-dimensional motion of magnetic domain walls in a thin ferromagnetic nanostrip sandwiched between a heavy metal and a metal oxide is investigated analytically in the framework of the extended Landau-Lifshitz-Gilbert equation. The trilayer system under investigation exhibits structural inversion asymmetry and exploits the combined effects of spin-transfer-torque and spin-orbit-torque to optimize the domain-wall propagation along the nanostrip. Through the traveling-wave formalism, an explicit expression for the key features involved in both steady and precessional regimes is provided, with a particular emphasis on the role played by the two spin-orbit-torque contributions, Rashba and Spin-Hall. In particular, it is shown how the domain-wall velocity and mobility, the direction of propagation, the depinning threshold and the Walker breakdown can be controlled via a suitable combination of Rashba and Spin-Hall coefficients. A comparison between analytical results and numerical data extracted from literature is also addressed revealing a qualitative agreement between them. Additional information on spin-orbit-torque-driven DW dynamics is extracted from such an analysis and, in particular, a linear dependence between the spin-Hall angle and the azimuthal angle is outlined as a possible mechanism responsible for the reversal of propagation direction.