Inhomogeneous domain walls in spintronic nanowires

Abstract

In case of a spin-polarized current, the magnetization dynamics in nanowires are governed by the classical Landau-Lifschitz equation with Gilbert damping term, augmented by a typically non-variational Slonczewski term. Taking axial symmetry into account, we study the existence of domain wall type coherent structure solutions, with focus on one space dimension and spin-polarization, but our results also apply to vanishing spin-torque term. Using methods from bifurcation theory for arbitrary constant applied fields, we prove the existence of domain walls with nontrivial azimuthal profile, referred to as inhomogeneous. We present an apparently new type of domain wall, referred to as non-flat, whose approach of the axial magnetization has a certain oscillatory character. Additionally, we present the leading order mechanism for the parameter selection of flat and non-flat inhomogeneous domain walls for an applied field below a threshold, which depends on anisotropy, damping, and spin-transfer. Moreover, numerical continuation results of all these domain wall solutions are presented. * Universität Bremen, lars.siemer@uni-bremen.de; Corresponding author † Universität Hamburg ‡ Universität Bremen 2 Model equations and coherent structure formThe classical model for magnetization dynamics was proposed by Landau and Lifschitz based on gyromagnetic precession, and later modified by Gilbert [15, 16]. See [17] for an overview. The Landau-Lifschitz-Gilbert equation for unit vector fields m(x, t) ∈ S 2

Domain wall motion in axially symmetric spintronic nanowires

“…e.g. [41,49]. More precisely, allowing only for perturbations tangential to the sphere at the poles, i.e., n = (n 1 , n 2 , 0) ∈ T m S 2 , we obtain from ( 16) with s = Ω = 0 the eigenvalue problems L + n = λn associated to +e 3 as well as L − n = λn associated to −e 3 , where…”

“…In case both asymptotic states are stable, the existence of a DW and the selection of the speed s and frequency Ω is equivalent to the construction of a heteroclinic orbit in the underlying coherent structure ODE (12). It was shown in [49] that such a heteroclinic connection is generically 'codimension-2' in the sense that both s and Ω need to be selected in order to close the connection. It was also proven there that these DWs can be continued to inhomogeneous DWs in the parameter c cp and it was further shown numerically that this holds for the entire interval c cp ∈ (−1, 1), see also Figure 1 (b)-(c).…”

“…To start, we recall that (LLGS) reduces to (LLG) in case β = 0 which is equivalent to c cp = 0 with the additional shift in the constant applied field h → h − β/α [41,49]. Let us now denote the shifted applied field by h and set c cp = 0.…”

“…Figure 1 (a), and will be discussed in § 2.1. A first classification of more general DWs is based on the local wavenumber q := φ (ξ), where a DW with q ≡ 0 is called homogeneous and inhomogeneous otherwise [49]; in particular m 0 is homogeneous. It is known that DW solutions to (LLGS) for c cp = 0 cannot be homogeneous [41].…”

“…An additional classification of DWs is based on the convergence behavior of the local wavenumber q as ξ → ±∞: we refer to DWs as flat, if |q(ξ)| has a limit in R for |ξ| → ∞, and non-flat otherwise; homogeneous DWs are flat by definition. Utilizing perturbations methods, it has been proven in [49] that non-flat DWs also exists in case c cp = 0, and more importantly that flat as well as non-flat DWs exists for |c cp | 1 sufficiently small, where the applied field strength can be arbitrary. Examples of inhomogeneous DWs for different applied fields and c cp away from zero are presented in Figures 1 (b) and (c), which were obtained by direct long-time simulations of (LLGS) starting from a Heaviside step function initial condition in (b) and a compact perturbation of the unstable state −e 3 in (c).…”