Magnon-mediated Dzyaloshinskii-Moriya torque in homogeneous ferromagnets

In thin magnetic layers with structural inversion asymmetry and spin-orbit coupling, the Dzyaloshinskii-Moriya interaction arises at the interface. When a spin-wave current ${\mathbf{j}}_m$ flows in a system with a homogeneous magnetization m, this interaction produces an effective fieldlike torque of the form ${\mathbf{T}}_{\mathrm{FL}}\propto \mathbf{m}\times (\mathbf{z}\times {\mathbf{j}}_m)$ as well as a dampinglike torque, ${\mathbf{T}}_{\mathrm{DL}}\propto \mathbf{m}\times [(\mathbf{z}\times {\mathbf{j}}_m)\times \mathbf{m}]$, the latter only in the presence of spin-wave relaxation $(\mathbf{z}$ is normal to the interface). These torques mediated by the magnon flow can reorient the time-averaged magnetization direction and display a number of similarities with the torques arising from the electron flow in a magnetic two-dimensional electron gas with Rashba spin-orbit coupling. This magnon-mediated spin-orbit torque can be efficient in the case of magnons driven by a thermal gradient.

Figure 1

Schematics of the magnetized stripe studied in this work. The magnetization is initially oriented along $\mathbf{x}$ and spin waves are generated by an ac field applied in the center of the stripe at $x=0$. Due to DMI, the spin-wave flow induces effective fields, ${\mathbf{B}}_{\mathrm{FL}}\propto \mathbf{y}$ and ${\mathbf{B}}_{\mathrm{DL}}\propto \mathbf{m}\times \mathbf{y}$, resulting in deviations of the background magnetization $\Delta m_{y,z}$.

Figure 2

Different systems of coordinates, describing the general direction of a unitary vector $\mathbf{m}$. (a) In Cartesian coordinates, $\mathbf{m}=m_x{\mathbf{e}}_x+m_y{\mathbf{e}}_y+m_z{\mathbf{e}}_z$ and $m_z=\sqrt{1-m_x^2-m_y^2}$. (b) In spherical coordinates, $\mathbf{m}={\mathbf{e}}_r$, where ${\mathbf{e}}_r=cos\beta sin\alpha {\mathbf{e}}_x+sin\beta sin\alpha {\mathbf{e}}_y+cos\alpha {\mathbf{e}}_z$. (c) In the system of coordinates we adopt in this work, the magnetization is defined as $\mathbf{m}=s_r{\mathbf{e}}_r+s_{\theta }{\mathbf{e}}_{\theta }+s_{\varphi }{\mathbf{e}}_{\varphi }$, where $s_r=\sqrt{1-s_{\theta }^2-s_{\varphi }^2}$. Here, ${\mathbf{e}}_r$ is the direction of the axis about which the magnetization precesses. (d) Definition of the polar and azimuthal parts of the magnetization in (c), in a plane normal to the rotation axis ${\mathbf{e}}_r$.

Figure 3

Numerical results for magnon-mediated Dzyaloshinskii-Moriya fieldlike and dampinglike torques. (a) Spatial distribution of the normalized $y$ component of magnetization (=$m_y)$ for $D=0$ (blue) and $D=1.5$ ${\mathrm{erg}/\mathrm{cm}}^2$ (red). (b) Normalized magnetization tilting $\Delta m_y$ for various damping constants $\alpha$ when the magnetization initially lies along $+\mathbf{x}$ for $D=1.5$ ${\mathrm{erg}/\mathrm{cm}}^2$ calculated numerically (open symbols) and using Eq. (8) (solid lines). The results with $\alpha ={10}^{-5}$ are multiplied by 1/10. (c) Normalized magnetization tilting $\Delta m_z$ for various damping constants $\alpha$ when the magnetization initially lies along $+\mathbf{x}$, for $D=1.5$ ${\mathrm{erg}/\mathrm{cm}}^2$ calculated numerically (open symbols) and using Eq. (9) (solid lines). Here we assume $H_d=0$. When the magnetization is switched to the $-\mathbf{x}$ direction, panel (b) remains unchanged while the curves in panel (c) are mirrored with respect to the x axis.

Figure 4

(a) Numerical results of spin-wave-induced magnetization tilting $\Delta m_y$ and $\Delta m_z$ with $\alpha =0$. Symbols and lines are obtained from numerical simulation and Eqs. (30) and (31), respectively. (b) $\Delta m_y$ and $\Delta m_z$ obtained with $\alpha =0.01,H_k=5$ Oe, $H_{ac}=100$ Oe, $H_d=10023$ Oe, $D=1.5$ ${\mathrm{erg}/\mathrm{cm}}^2$, and $f=0.8$ GHz. The magnitude of the deviation is about 1%.

Figure 5

Magnetization tilting induced by uniform temperature gradient. Spatial distribution of time-averaged (a) $\langle m_y\rangle$ and (b) $\langle m_z\rangle$. Corresponding histograms of (c) $\langle m_y\rangle$ and (d) $\langle m_z\rangle$.

Figure 6

(Top panel) Three magnetic configurations investigated in this work and their corresponding anisotropy energy constants. At rest (i.e., in the absence ofmagnon flow), the magnetization is assumed to be aligned either along the easy axis, which is $x$ (in-plane longitudinal anisotropy), $y$ (in-plane transverse anisotropy), or $z$ (perpendicular anisotropy). (Bottom panel) The table provides the values of anisotropy constants for the three cases depicted above.