Quantum mechanical theory of current-induced domain wall torques in ferromagnetic materials

Calculations are made of the spin transfer torque acting on the magnetization in a ferromagnetic domain wall when an electric current traverses the wall. The magnetization is assumed to lie in a plane, as in current experiments on flat nanowires. A nonadiabatic out-of-plane torque is found, which can be regarded as an addition to the Zhang-Li out-of-plane torque that does not appear in our calculation since we neglect spin-orbit coupling. There is also an in-plane nonadiabatic component of torque together with a correction to the standard adiabatic torque. It is shown that the calculated nonadiabatic torques are fitted rather well by two additional terms in the phenomenological Landau-Lifshitz-Gilbert equation. The electronic structure of the ferromagnetic metal is treated within the tight-binding approximation both for a one-band model and for realistic multiband models of Co and Permalloy, but with spin-orbit coupling neglected. The torque is deduced from calculations of spin current in the ballistic limit by using the Keldysh formalism, as in previous work on magnetic trilayers. The wall dynamics due to the current-induced torque is investigated and the out-of-plane torque contributes an additional term to the current-driven wall velocity. The sign of this term differs between Co and Permalloy, and it can be significant in the case of a narrow wall and/or weak damping.

Figure 1

Schematic picture of a domain wall in a nanowire

Figure 2

Magnetization direction of atomic planes in a domain wall.

Figure 3

Dimensionless spin current as a function of position $n$ in the domain wall for a single-band system. The symbols are the calculated spin currents, the solid curves in (a) are the $x$ and $z$ components of the magnetization $\widehat{\mathbf{m}}$, and the solid curves in (b) are the phenomenological spin currents discussed in Sec. 2.

Figure 4

(Color online) (A) shows the out-of-plane spin current as a function of ${\mathit{k}}_{\parallel }$ for the single-band model. (B) shows the in-plane spin current as a function of ${\mathit{k}}_{\parallel }$ for the same system.

Figure 5

(Color online) Dependence of the out-of-plane spin current on the number of $k$ points in the direction perpendicular to the wire plane.

Figure 6

The calculated dimensionless ballistic out-of-plane torque for an electronlike single-band model and the profile of the phenomenological out-of-plane torque terms $F$ and $F_1$. The magnitude of $F_1$ is obtained by fitting the phenomenological curve to the calculated one.

Figure 7

The calculated dimensionless ballistic out-of-plane torque for a holelike single-band model and the profile of the phenomenological out-of-plane torque terms $F$ and $F_1$. The magnitude of $F_1$ is obtained by fitting the phenomenological curve to the calculated one.

Figure 8

Dependence of the calculated dimensionless out-of-plane torque on the wall width $n_w$ (solid curve) and the dependence $Q\left(n_w\right)=\mid \mathit{T}\mid n_w^3$ (broken curve).

Figure 9

(Color online) Out-of-plane spin current as a function of ${\mathit{k}}_{\parallel }$ for Co. The black lines in the lower plot are the edges of the two-dimensional projection of the Fermi surface sheets.

Figure 10

The calculated dimensionless ballistic out-of-plane torque for Co and the profile of the phenomenological out-of-plane torque term $F_1$. The magnitude of $F_1$ is obtained by fitting the phenomenological curve to the calculated one.

Figure 11

(Color online) Out-of-plane spin current as a function of ${\mathbf{k}}_{\parallel }$ for Py. Solid curves in the lower plot are the edges of the two-dimensional projection of the Py Fermi surface sheets.

Figure 12

The calculated dimensionless ballistic out-of-plane torque for Py and the profile of the phenomenological out-of-plane torque term $F_1$. The magnitude of $F_1$ is obtained by fitting the phenomenological curve to the calculated one.