Current driven dynamics of domain walls constrained in ferromagnetic nanopillars

We investigate the effects of an electric current on the domain wall formed inside a cylindrical ferromagnetic nanopillar as a consequence of the pinning of the magnetization at its ends. We first present the results of three-dimensional and one-dimensional micromagnetic simulations and show that the system approaches a stationary equilibrium, where the domain wall is compressed in the direction of the electron flow and rotates around the nanopillar axis with constant frequency in the microwave frequency range. We obtain the dependence of the rotation frequency on the length of the nanopillar and on the magnitude of the applied current density. We then introduce a one-dimensional analytical model and find a formula for the rotation frequency in two current regimes: a low current regime, where the frequency is linearly proportional to the current density and a high current regime, where the frequency is quadratically proportional to the current density. Good agreement is found with the results of the simulations. The system may have possible applications as a nanosized microwave generator, which could operate without external magnetic fields and whose emission frequency could be controlled by a dc current.

Figure 1

(Color online) A sketch of the system. The arrows on the cylinder axis represent the magnetization, pinned in opposite directions at the nanopillar ends.

Figure 2

(Color online) The evolution of the components of the average normalized magnetization $⟨\mathbf{m}⟩=⟨\mathbf{M}⟩/M_{\text{s}}$ as a function of time. The nanopillar length is $L=40\text{nm}$.

Figure 3

(Color online) The magnetization configuration for the simulation of Fig. 2 is shown at $t=0\text{ns}$ (a) and $t=6.6\text{ns}$ (b).

Figure 4

The time dependence of the frequency for the rotation of the domain wall around the $x$ axis for a three-dimensional micromagnetic simulation of a nanopillar with $L=40\text{nm}$.

Figure 5

(Color online) The frequency as a function of $j_P$ for different nanopillar lengths $L$, as obtained from three-dimensional micromagnetic simulations.

Figure 6

(Color online) The frequency as a function of $j_P$ for different nanopillar lengths $L$, as obtained from one-dimensional micromagnetic simulations. The dotted and dashed curves show the results obtained for the three-dimensional system (Fig. 5) in the case $L=20$ and $L=45$, respectively.

Figure 7

(Color online) Comparison between the numerical values for $\left|\Omega \left(V\right)\right|$ obtained from the one-dimensional micromagnetic simulations (crosses) and the low current (dotted line) and high current (solid line) analytical solutions.