Roles of chiral renormalization on magnetization dynamics in chiral magnets

In metallic ferromagnets, the interaction between local magnetic moments and conduction electrons renormalizes parameters of the Landau-Lifshitz-Gilbert equation, such as the gyromagnetic ratio and the Gilbert damping, and makes them dependent on the magnetic configurations. Although the effects of the renormalization for nonchiral ferromagnets are usually minor and hardly detectable, we show that the renormalization does play a crucial role for chiral magnets. Here the renormalization is chiral, and as such we predict experimentally identifiable effects on the phenomenology of magnetization dynamics. In particular, our theory for the self-consistent magnetization dynamics of chiral magnets allows for a concise interpretation of domain-wall creep motion. We also argue that the conventional creep theory of the domain-wall motion, which assumes Markovian dynamics, needs critical reexamination since the gyromagnetic ratio makes the motion non-Markovian. The non-Markovian nature of the domain-wall dynamics is experimentally checkable by the chirality of the renormalization.

Figure 1

(a) Magnetization dynamics described by the unrenormalized LLG equation. The dynamics of magnetization and that of electrons are coupled to each other by the exchange interaction. (b) After tracing out the electron degree of freedom, the gyromagnetic ratio ($\zeta \gamma$) and the magnetic damping ($\mathcal{G}$) are chirally renormalized [Eq. (1)].

Figure 2

Chiral dynamics of a DW between domains with $\mathrm{m}=\mp \widehat{\mathrm{z}}$ (red and blue, respectively). The DW chirality is characterized by the DW tilting angle $\varphi$ [the positivity (negativity) of $\varphi$ corresponds to the left-handed (right-handed) chirality] and can be controlled by an in-plane field ($H_x$). The DW motion is driven by an applied field ($H_z$). Measuring the DW velocity as a function of $\varphi$ (or $H_x$), the difference between $v\left(\varphi \right)$ and $v(-\varphi )$ gives the information of the chiral renormalization.

Figure 3

The effective dynamical parameters ${\alpha }_{\mathrm{eff}}^X$ (the red, solid curve), ${\alpha }_{\mathrm{eff}}^{\varphi }$ (the red, dashed curve), and ${\zeta }_{\mathrm{eff}}^{-1}$ (the blue curve) as a function of the DW tilting angle $\varphi$. We take the phenomenological expression of spin torque in magnetic bilayers [21, 22, 23, 30], which is a typical example with large SOT: $\mathrm{T}=(\gamma \hslash /2e{M}_s)\{({\mathrm{j}}_s·\mathbf{\nabla })\mathrm{m}-{\beta }_1\mathrm{m}\times ({\mathrm{j}}_s·\mathbf{\nabla })\mathrm{m}+k_{\mathrm{SO}}(\widehat{\mathrm{z}}\times {\mathrm{j}}_s)\times \mathrm{m}-{\beta }_2{k}_{\mathrm{SO}}\mathrm{m}\times [(\widehat{\mathrm{z}}\times {\mathrm{j}}_s)\times \mathrm{m}]\}$ where each term refers to the adiabatic STT [44], nonadiabatic STT [45, 46], fieldlike SOT [47, 48], and dampinglike SOT [30, 49, 50, 51], induced by the spin current ${\mathrm{j}}_s$. Here, $M_s=1000\mathrm{emu}/{\mathrm{cm}}^3$ is the saturation magnetization, $e>0$ is the (negative) electron charge, $\widehat{\mathrm{z}}$ is the interface normal direction, $k_{\mathrm{SO}}=1.3{\left(\mathrm{nm}\right)}^{-1}$ characterizes the strength of the SOT. We take ${\beta }_1=0.05,{\beta }_2=5,\lambda =8\mathrm{nm}$, and the electrical conductivity ${\sigma }_0^{-1}=6\mu \mathrm{\Omega }\mathrm{cm}$. The parameters are on the order of the typical values for Pt/Co systems [28, 45, 52].