Enhanced spin transfer torque effect for transverse domain walls in cylindrical nanowires

Recent studies have predicted extraordinary properties for transverse domain walls in cylindrical nanowires: zero depinning current, the absence of the Walker breakdown, and applications as domain wall oscillators. In order to reliably control the domain wall motion, it is important to understand how they interact with pinning centers, which may be engineered, for example, through modulations in the nanowire geometry (such as notches or extrusions) or in the magnetic properties of the material. In this paper we study the motion and depinning of transverse domain walls through pinning centers in ferromagnetic cylindrical nanowires. We use (i) magnetic fields and (ii) spin-polarized currents to drive the domain walls along the wire. The pinning centers are modelled as a section of the nanowire which exhibits a uniaxial crystal anisotropy where the anisotropy easy axis and the wire axis enclose a variable angle ${\theta }_{\mathrm{P}}$. Using (i) magnetic fields, we find that the minimum and the maximum fields required to push the domain wall through the pinning center differ by $30\%$. On the contrary, using (ii) spin-polarized currents, we find variations of a factor 130 between the minimum value of the depinning current density (observed for ${\theta }_{\mathrm{P}}=0^{\circ }$, i.e., anisotropy axis pointing parallel to the wire axis) and the maximum value (for ${\theta }_{\mathrm{P}}={90}^{\circ }$, i.e., anisotropy axis perpendicular to the wire axis). We study the depinning current density as a function of the height of the energy barrier of the pinning center using numerical and analytical methods. We find that for an industry standard energy barrier of $40k_{\mathrm{B}}T$, a depinning current of about $5\mu \mathrm{A}$ (corresponding to a current density of $6\times {10}^{10}\mathrm{A}/{\mathrm{m}}^2$ in a nanowire of $10\mathrm{nm}$ diameter) is sufficient to depin the domain wall. We reveal and explain the mechanism that leads to these unusually low depinning currents. One requirement for this depinning mechanism is for the domain wall to be able to rotate around its own axis. With the right barrier design, the spin torque transfer term is acting exactly against the damping in the micromagnetic system, and thus the low current density is sufficient to accumulate enough energy quickly. These key insights may be crucial in furthering the development of novel memory technologies, such as the racetrack memory, that can be controlled through low current densities.

Figure 1

Sketch of the system. The arrows on the cylinder axis represent the magnetization. The transverse DW on the left is pushed through the barrier on the right. The barrier is modeled as a region where an additional uniaxial anisotropy pins the magnetization along the direction determined by ${\theta }_{\mathrm{P}}$.

Figure 2

Examples of asymmetric (a) and symmetric (b) barriers for the domain wall motion obtained through localized variations of the nanowire geometry. Note that barriers may also be engineered as regions of the wire with different magnetic properties.

Figure 3

The components of the normalized magnetization, $\vec{m}=\vec{M}/M_{\mathrm{s}}$, along the axis of the nanowire before (dashed lines) and after (solid lines) the preliminary relaxation computation. The blue curves (light in grayscale) show the $x$ (axial) component of $\vec{m}$, while the dark red curves (dark in grayscale) show the component of $\vec{m}$ orthogonal to the axis.

Figure 4

The relaxed magnetization configurations for ${\theta }_{\mathrm{P}}=0^{\circ }$ (a) and ${\theta }_{\mathrm{P}}={90}^{\circ }$ (b). The sketches on the corners of the two figures show that the DW moves into the barrier to allow the magnetization to align along the pinning direction in the barrier.

Figure 5

$H_{\mathrm{crit}}$, the field required in order to push the DW throughout the barrier, is plotted as a function of the pinning angle, ${\theta }_{\mathrm{P}}$.

Figure 6

$j_{\mathrm{crit}}$, the current required in order to push the DW through the barrier, is plotted as a function of the pinning angle ${\theta }_{\mathrm{P}}$.

Figure 7

Comparison of the current-driven dynamics of the spatially averaged magnetization $\langle \vec{m}\rangle$ for ${\theta }_{\mathrm{P}}=0^{\circ }$ (top) and ${\theta }_{\mathrm{P}}={20}^{\circ }$ (bottom). The inset in the bottom plot shows the oscillatory motion of the DW induced by the pinning in the barrier. The discontinuities in $\langle m_x\rangle$ correspond to changes in the applied current density (see Fig. 8) following the procedure described in the text. Two different scales are used for the time axis of the bottom plot in order to give clear information on the dynamics during the two different phases of the switching process. Note that the dynamics of the preliminary relaxations is omitted in these figures.

Figure 8

The applied current density for the two simulations of Figs. 7 and 9. The vertical lines in the plot mark a change in the scale of the time axis.

Figure 9

Evolution of the energy for the two cases ${\theta }_{\mathrm{P}}=0^{\circ }$ (dashed line) and ${\theta }_{\mathrm{P}}={20}^{\circ }$ (solid line) considered in Figs. 7 and 8. Units of $k_{\mathrm{B}}T=4.14\times {10}^{-21}\mathrm{J}$ ($T=300K$) are used for the energy. The time axis uses different scales as in Fig. 8.

Figure 10

Current-driven trajectories of the average magnetization $\langle \vec{m}\rangle$ projected on the $yz$ plane (orthogonal to the nanowire axis). The dynamics is in response to the application of a current density $j=2\times {10}^{10}\mathrm{A}/{\mathrm{m}}^2$ in the case of asymmetric barrier, ${\theta }_{\mathrm{P}}={20}^{\circ }$ (a) and symmetric barrier, ${\theta }_{\mathrm{P}}=0^{\circ }$ (b). The line thickness is proportional to the magnitude of the energy which is being pumped in (red in color, dark in grayscale) or out (green in color, light in grayscale) at the considered point. The two black dots indicate where the trajectories begin. Note that side (a) shows a few precessions of $\langle m_x\rangle$, while side (b) shows only one precession. This is just to avoid the overlapping of the line.

Figure 11

Study of the critical current $j_{\mathrm{crit}}$ required to push the DW through a rotationally symmetric barrier as a function of the barrier height ($\Delta E$). The top $x$ axis shows the values of the anisotropy constant $K_1$ corresponding to the values of $\Delta E$ shown on the bottom axis.