Spin-Injection-Generated Shock Waves and Solitons in a Ferromagnetic Nanowire

Abstract

A promising means for long-distance transport of angular momentum is spin superfluidity [1-4]. In this work, the unsteady nonlinear dynamics of spin transport in an easy-plane, thin-film, one-dimensional ferromagnetic channel is studied analytically. We formulate an initial-boundary value problem (IVBP) for the Landau-Lifshitz equation subject to spin injection into a static, uniformly magnetized nanowire. The long-time steady-state configuration in such channels with zero applied field has been identified as either a dissipative exchange flow (DEF) or a contact soliton DEF (CS-DEF) [5], depending on the strength of the injection. Here, we study the injection-induced highly nonlinear, transient magnetization dynamics that occur prior to the damping-dominated steady-state is reached and include a constant applied field. In particular, we focus on the 1 ns timescale where damping is weak, and the spin dynamics are dominated by spin injection, applied field, exchange interaction, and demagnetizing fields. In our analysis, we recast the Landau-Lifshitz equation into its dispersive hydrodynamic (DH) form [6], in terms of the longitudinal spin density and a magnetic analog of the fluid velocity that is proportional to the spin current. The spin injection is characterized by a nonzero fluid velocity at one end of the channel, hence can be viewed as a spin hydrodynamic “piston”. The DH form captures two important effects in the ferromagnetic system: nonlinearity and dispersion, which are inherited from anisotropy and exchange, respectively. These effects lead to the generation of unsteady nonlinear waves when large gradients occur. A rapid rise in the fluid velocity due to spin injection leads to large gradients that result in a new kind of spin shock wave that is an example of a dispersive shock wave (DSW). DSWs are expanding, highly oscillatory, nonlinear excitations that realize a coherent transition between two states in a variety of dispersive media [7]. Spin wave envelope DSWs have been observed in a Yttrium Iron Garnet feedback ring [8]. We provide a full analytical classification of nonlinear wave solutions to this IVBP in terms of the applied field and injection strength, using the Whitham modulation theory [9]. The solution components are rarefaction waves (RWs), DSWs, and solitons (see Fig. 1). Both RWs and DSWs have been identified as solutions to a pure initial value problem (the Riemann problem) in a two-component Bose-Einstein condensate [10], whose governing equations are equivalent to the DH form of the Landau-Lifshitz equation we study here. In contrast, we show that the injection boundary plays a prominent role in the dynamics. Depending on the interplay between the spin injection intensity and the applied magnetic field magnitude, different types and combinations of RWs, DSWs, and solitons arise. The generation of a soliton is a distinct feature due to the boundary and is identified with an important concept [10]: the magnetic sonic condition. Solitons only occur at the spin injection end and only in the supersonic injection regime. Our analytical solutions are confirmed by micromagnetic simulations. **