Ballistic current induced effective force on magnetic domain wall

C. Wang and K. Xia* The collective dynamics of magnetic domain wall under electric current is studied in the form of spin transfer torque (STT). The out-of-plane STT induced effective force is obtained based on the Landau-Lifshitz-Gilbert (LLG) equation including microscopic STT terms. The relation between microscopic calculations and collective description of the domain wall motion is established. With our numerical calculations based on tight binding free electron model, we find that the non adiabatic out-of-plane torque components have considerable non-local properties. It turns out that the calculated effective forces decay significantly with increasing domain wall widths.

effective force is obtained.

EFFECTIVE FORCE AND OUT-OF-PLANE STT
The current-driven domain wall dynamics is studied based on the Landau-Lifshitz-Gilbert equation, the connection between effective force and out-of-plane STT on each site is obtained. The LLG equation reads, where the magnetization M can be written as the function of polar angles as indicated in Fig. 1 We assume that the magnitude of the magnetization will not change when the DW moves. We can rewrite the on sites LLG equations Eq. (1) with sphere polar angle coordinates (θ i , φ i ) and then apply the Walker ansatz so that Euler angles are treated as functions of position and time.
The DW is parallel to the z direction, θ and ϕ are the function of position and time. θ =θ(z −X(t)), ϕ =ϕ(z, t), X is defined as the wall center's position within the "rigid wall" approximation, and ϕ(z, t) is simply reduced to ϕ(t) in the model used in following derivation. The variable X and ϕ can serve as collective coordinates to describe the wall motion.
Namely, we can get the dynamic information of the domain wall if we know the time evolution of (X(t), ϕ(t)). And spin transfer torques can be explicitly expressed as,  The above two sets of equations can be solved numerically for arbitrary STT form and effective field. However, when current and applied magnetic fields are not very large, rigid wall is a good approximation to study the DW motion. Applying Walker's ansatz [16], the localized spins in Neel domain wall structure (Shown in Fig. 2) expressed in spin polar coordination satisfies ) ) the localized macroscopic spin S r satisfy the relationship: , we can express this relationship by pole angles： And we can see the time dependent behavior of θ is completely determined by the wall(center) position's dynamics property: . We can immediately come to some interesting results after comparing Eq. (8) (9) with effective forces and effective torques [9]. It turns out that the out-plane spin torque , which appears in Eq. (9) corresponds to the force on domain wall, while the in-plane spin torque contributes to the z component of torque on the wall as a whole: Although it is widely accepted that different components of STTs can be written in the form of Eq. (4), the origin of out-of-plane STT could be stem from different physics. In our calculation, we focused on the nonadiabatic term pointed out by J. Xiao and M. Stiles [12].

NUMERICAL CALCULATION AND DISCUSSION
In the following, numerical calculation is used to DOI: 10.5101/nml.v1i1.p34-39 http://www.nmletters.org investigate the torque as the function of domain wall width λ.
After introducing the calculation methods we used, we subsequently investigate the in-plane and out-plane STTs. The obtained calculations are in accordance with our model and can be helpful to the discussion about effective force.

A. Calculation of STT in Domain Wall
The structure used in the calculation is current in plane (CIP) domain walls in ferromagnetic materials. We apply free electron model in our calculations where the current is along (110) direction in fcc structure using the lattice constant of Cobalt a Co = 3.549 Å. And we set the energy split between majority and minority spins in free electron energy band as 1.69eV, which equals the value of exchange energy split of bulk Cobalt. Our numerical approach is based on the Tight-binding linear muffin-tin orbital formulism [13,14]. Scattering wave function is obtained by the wave function matching method [15]. The rigid potential approximation is employed to simulate the DW. Here STTs in our study can be defined as the difference between the incoming and outgoing spin current at R site.

B. Out-plane STT and Effective Force
Let us focus on the out-plane component of STTs. From our numerical calculations, we find a decaying oscillation for out-of-plane STT as shown in Fig. 4. From the inset in Combining the calculated results, we will discuss the DW velocities expressed in Eq. (11). When the rigid wall still holds, it is suitable to express the terminal velocities v(t→∞)=−c j /α.
Considering c j is not constant but oscillating, we may use the average to express the terminal velocity. When the wall is thicker, the increase of inertia or wall mass makes it harder to move, thus the velocities get smaller and consequently larger driving currents are required. In our model, the velocities will indeed drop significantly when the out plane torques (c j terms) vanish in thicker walls, as depicted in Fig. 6. However, we need to keep in mind that the Walker's model we use to describe rigid walls will break down under high external field [16,17] or high current density [9]. In that case, the domain wall would be oscillating so the definition of DW velocities is no longer available.
The forces at different wall structures are calculated from integration within the narrowed region of c j (z) using ) ( ) ( Wang for useful discussions and his codes that we used for calculating spin transfer torques.