Magnetic field driven domain-wall propagation in magnetic nanowires

Abstract

A global picture of magnetic domain wall (DW) propagation in a nanowire driven by a magnetic field is obtained: A static DW cannot exist in a homogeneous magnetic nanowire when an external magnetic field is applied. Thus, a DW must vary with time under a static magnetic field. A moving DW must dissipate energy due to the Gilbert damping. As a result, the wire has to release its Zeeman energy through the DW propagation along the field direction. The DW propagation speed is proportional to the energy dissipation rate that is determined by the DW structure. An oscillatory DW motion, either the precession around the wire axis or the breath of DW width, should lead to the speed oscillation.Magnetic domain-wall (DW) propagation in a nanowire due to a magnetic field [1,2,3,4,5] reveals many interesting behaviors of magnetization dynamics. For a tail-to-tail (TT) DW or a head-to-head (HH) DW (shown in Fig. 1) in a nanowire with its easy-axis along the wire axis, the DW will propagate in the wire under an external magnetic field parallel to the wire axis. The propagation speed v of the DW depends on the field strength [3,4]. There exists a so-called Walker's breakdown field H W [6]. v is proportional to the external field H for H < H W and H ≫ H W . The linear regimes are characterized by the DW mobility µ ≡ v/H. Experiments showed that v is sensitive to both DW structures and wire width [1,2,3]. DW velocity v decreases as the field increases between the two linear H-dependent regimes, leading to the so-called negative differential mobility phenomenon. For H ≫ H W , the DW velocity, whose time-average is linear in H, oscillates in fact with time [3,6].Schematic diagram of a HH DW of width ∆ in a magnetic nanowire of cross-section A. The wire consists of three phases, two domains and one DW. The magnetization in domains I and II is along +z-direction (θ = 0) and -zdirection (θ = π), respectively. III is the DW region whose magnetization structure could be very complicate. H is an external field along +z-direction.It has been known for more than fifty years that the magnetization dynamics is govern by the LandauLifshitz-Gilbert (LLG) [7] equation that is nonlinear and can only be solved analytically for some special problems [6,8]. The field induced domain-wall (DW) propagation in a strictly one-dimensional wire has also been known for more than thirty years[6], but its experimental realization in nanowires was only achieved [1,2,3,4,5] in recent years when we are capable of fabricating various nano structures. Although much progress [9,10] has been made in understanding field-induced DW motion, it is still a formidable task to evaluate the DW propagation speed in a realistic magnetic nanowire even when the DW structure is obtained from various means like OOMMF simulator and/or other numerical software packages. A global picture about why and how a DW propagates in a magnetic nanowire is still lacking.In this report, we present a theory that reveals the origin of DW propagation. Firstly, we shall show that no static HH (TT...

Domain wall pinning in notched nanowires

“…Results in two extremes can be justified. One has h c = 0 when a notch does not exist, i.e., w 1 = 0, and there is no DW pinning, a well-known result [29]. The other is w 1 ≈ w, where a wire consists of detached parts.…”

“…(2) does not have a DW solution. In early publications [29], it was shown that Eq. (2) does not have a DW solution for a homogeneous nanowire in the presence of an external field along the wire.…”

Spin wave emission in field-driven domain wall motion

“…The DW plane precesses with frequency H while the DW center Z oscillates periodically with frequency 2H and moves slowly and simultaneously along the field direction. The oscillatory DW motion can easily be explained by energy dissipation theory [15] though it is widely understood by the well known collective coordinate model [18][19][20][21], in which the DW is assumed to have a constant azimuthal angle φ(t) (no twisting). The polar angle θ n (t), with DW width (t), follows the Walker form:…”

“…The nice DW breathing motion and SW emission exist clearly in the presence of damping. Because of the energy conservation [13,15], the average DW speed connects directly to the SW emission power density P by P = 2H v. Thus, Figs. 2(b) and 2(c) give also H and D x dependencies of P with a proper factor of 2H .…”

“…The interplay between SWs and DWs has also received much attention [6][7][8][9][10][11][12][13][14], including DW propagation driven by externally generated SWs and SW generation by a moving DW. Field-driven DW propagation under a longitudinal field is originated from the energy dissipation that is compensated by the Zeeman energy released by the DW motion [15]. In the well-known Walker solution [4], the Zeeman energy is dissipated by phenomenological damping.…”