We develop a systematic asymptotic description for domain wall motion in one-dimensional magnetic nanowires under the influence of small applied magnetic fields and currents and small material anisotropy. The magnetization dynamics, as governed by the Landau-Lifshitz-Gilbert equation, is investigated via a perturbation expansion. We compute leading-order behaviour, propagation velocities and first-order corrections of both travelling waves and oscillatory solutions, and find bifurcations between these two types of solutions. This treatment provides a sound mathematical foundation for numerous results in the literature obtained through more ad hoc arguments.

Under a magnetic field along its axis, domain wall motion in a uniaxial nanowire is much slower than in the fully anisotropic case, typically by several orders of magnitude (the square of the dimensionless Gilbert damping parameter). However, with the addition of a magnetic field transverse to the wire, this behaviour is dramatically reversed; up to a critical field strength, analogous to the Walker breakdown field, domain walls in a uniaxial wire propagate faster than in a fully anisotropic wire (without transverse field). Beyond this critical field strength, precessional motion sets in, and the mean velocity decreases. Our results are based on leading-order analytic calculations of the velocity and critical field as well as numerical solutions of the Landau-Lifshitz-Gilbert equation.

We study existence and properties of one-dimensional edge domain walls in ultrathin ferromagnetic films with uniaxial in-plane magnetic anisotropy. In these materials, the magnetization vector is constrained to lie entirely in the film plane, with the preferred directions dictated by the magnetocrystalline easy axis. We consider magnetization profiles in the vicinity of a straight film edge oriented at an arbitrary angle with respect to the easy axis. To minimize the micromagnetic energy, these profiles form transition layers in which the magnetization vector rotates away from the direction of the easy axis to align with the film edge. We prove existence of edge domain walls as minimizers of the appropriate one-dimensional micromagnetic energy functional and show that they are classical solutions of the associated Euler-Lagrange equation with Dirichlet boundary condition at the edge. We also perform a numerical study of these one-dimensional domain walls and uncover further properties of these domain wall profiles.

We present an analysis of edge domain walls in exchange-biased ferromagnetic films appearing as a result of a competition between the stray field at the film edges and the exchange bias field in the bulk. We introduce an effective two-dimensional micromagnetic energy that governs the magnetization behavior in exchange-biased materials and investigate its energy minimizers in the strip geometry. In a periodic setting, we provide a complete characterization of global energy minimizers corresponding to edge domain walls. In particular, we show that energy minimizers are one-dimensional and do not exhibit winding. We then consider a particular thin film regime for large samples and relatively strong exchange bias and derive a simple and comprehensive algebraic model describing the limiting magnetization behavior in the interior and at the boundary of the sample. Finally, we demonstrate that the asymptotic results obtained in the periodic setting remain true in the case of finite rectangular samples.

We study the magnetization dynamics of thin-film magnetic elements with in-plane magnetization subject to a spin-current flowing perpendicular to the film plane. We derive a reduced partial differential equation for the in-plane magnetization angle in a weakly damped regime. We then apply this model to study the experimentally relevant problem of switching of an elliptical element when the spin-polarization has a component perpendicular to the film plane, restricting the reduced model to a macrospin approximation. The macrospin ordinary differential equation is treated analytically as a weakly damped Hamiltonian system, and an orbit-averaging method is used to understand transitions in solution behaviors in terms of a discrete dynamical system. The predictions of our reduced model are compared to those of the full Landau-Lifshitz-Gilbert-Slonczewski equation for a macrospin.