Asymmetric ferromagnetic resonance, universal Walker breakdown, and counterflow domain wall motion in the presence of multiple spin-orbit torques

We study the motion of several types of domain wall profiles in spin-orbit coupled magnetic nanowires and also the influence of spin-orbit interaction on the ferromagnetic resonance of uniform magnetic films. Whereas domain wall motion in systems without correlations between spin space and real space is not sensitive to the precise magnetization texture of the domain wall, spin-orbit interactions break the equivalence between such textures due to the coupling between the momentum and spin of the electrons. In particular, we extend previous studies by fully considering not only the fieldlike contribution from the spin-orbit torque, but also the recently derived Slonczewski-like spin-orbit torque. We show that the latter interaction affects both the domain wall velocity and the Walker breakdown threshold nontrivially, which suggests that it should be accounted for in experimental data analysis. We find that the presence of multiple spin-orbit torques may render the Walker breakdown universal in the sense that the threshold is completely independent on the material-dependent Gilbert damping $\alpha$, nonadiabaticity $\beta$, and the chirality $\sigma$ of the domain wall. We also find that domain wall motion against the current injection is sustained in the presence of multiple spin-orbit torques and that the wall profile will determine the qualitative influence of these different types of torques (e.g., fieldlike and Slonczewski-like). In addition, we consider a uniform ferromagnetic layer under a current bias, and find that the resonance frequency becomes asymmetric against the current direction in the presence of Slonczewski-like spin-orbit coupling. This is in contrast with those cases where such an interaction is absent, where the frequency is found to be symmetric with respect to the current direction. This finding shows that spin-orbit interactions may offer additional control over pumped and absorbed energy in a ferromagnetic resonance setup by manipulating the injected current direction.

Figure 1

Schematic setup: a spin-polarized current is passed through a domain wall magnetic nanowire with spin-orbit coupling. The spin-orbit interaction may be either intrinsic or induced via a heavy metal proximate host. We consider several types of domain wall configurations, since the presence of spin-orbit coupling qualitatively distinguishes the domain wall motion with one type of magnetization texture from another. More specifically, we consider two types of Bloch-domain walls relevant for perpendicular magnetic anisotropy systems in addition to a head-to-head domain wall with an in-plane magnetization easy anisotropy.

Figure 2

Threshold value for the magnitude of the spin-orbit coupling above which there is no Walker breakdown. In the more general scenario where both types of spin-orbit torques are accounted for, the threshold value for ${\tilde{\alpha }}_R$ is constant. When only the fieldlike torque is considered, the threshold is strongly increased in the regime $\alpha /\beta <0.5$. In the limit $\alpha /\beta \to \infty$, the asymptote is $2/\pi$.

Figure 3

Domain wall velocity for a Bloch ($z$) wall plotted against the injected current. We have chosen $\sigma =1$ without loss of generality (see text) and set $\beta =0.01$ and ${\mathcal{H}}_{\perp }$ = 0.5. In (a) $\alpha >\beta$ ($\alpha =0.02$) whereas in (b) $\alpha <\beta$ ($\alpha =0.005$). Note the inverted sign of the $y$ axis, which simply corresponds to the direction of the wall motion.

Figure 4

Plot of left-hand side (lhs) (dashed line) and right-hand side (rhs) (full lines) of Eq. (9) in order to illustrate the intersection points. When there is no intersection between the lines, Walker breakdown has occurred. We have set $\beta =0.01$, $\alpha =0.005$, and consider (a) ${\tilde{\alpha }}_R=0.01$ and $u$ ranging from 0.24 to 0.30 along the direction of the arrow, in addition to (b) ${\tilde{\alpha }}_R=1$ and $u$ ranging from 0.10 to 0.16 along the direction of the arrow. The black arrow between the circles in (b) highlights how the intersection point changes abruptly upon increasing $u$. Insets: Time evolution of the tilt angle for the same choices of $u$.

Figure 5

The domain wall velocity $\langle v_{\text{DW}}\rangle$ as a function of the current density $u$ for various chiralities and spin-orbit coupling strengths. (a) Positive chirality $\sigma =+1$ and $\alpha >\beta$ ($\alpha =0.02$). (b) Negative chirality $\sigma =-1$ and $\alpha >\beta$ ($\alpha =0.02$). (c) Positive chirality $\sigma =+1$ and $\alpha <\beta$ ($\alpha =0.005$). (d) Negative chirality $\sigma =-1$ and $\alpha <\beta$ ($\alpha =0.005$). For all plots, we have used $\beta =0.01$ and ${\mathcal{H}}_{\perp }=0.5$.

Figure 6

Critical velocity $u_c/{\mathcal{H}}_{\perp }$ that triggers Walker breakdown. We have chosen $\beta =0.02$ as a representative value which demonstrates the fundamental behavior of $u_c$. For sufficiently low damping $\alpha <\beta$ shown in (a), the threshold velocity is lowered monotonically as the spin-orbit interaction ${\tilde{\alpha }}_R$ is increased. When the damping becomes stronger such that $\alpha >\beta$, $u_c$ is strongly enhanced in a limited interval of ${\tilde{\alpha }}_R$.

Figure 7

Schematic setup of the free ferromagnetic (FM) layer with a general saturation magnetization direction ${\overrightarrow{\stackrel{̈}{\mathcal{M}}}}_S$, described by polar and azimuthal angles ${\theta }_{\ddot{M}}$ and ${\phi }_{\ddot{M}}$, respectively. The thickness of free ferromagnetic layer is denoted by $d$. The externally applied static magnetic field ${\overrightarrow{\stackrel{̈}{H}}}_0$, polarization vector of injected charge current $\overrightarrow{\stackrel{̈}{S}}$, spin-orbit coupling torque vector ${\overrightarrow{\stackrel{̈}{H}}}^{so}$, and finally normal unity vector $\overrightarrow{\stackrel{̈}{n}}$ are shown. The ferromagnetic film is located in the $\ddot{x}\ddot{y}$ plane so that $\ddot{z}$ axis is normal to the ferromagnetic film. The spin-orbit coupling is assumed to be induced via a substrate layer into the free ferromagnetic layer. The double dot represents the vector quantities in the nonrotated coordinate system (laboratory framework).