Magnetization dynamics has been routinely studied using the Landau-Lifshitz-Gilbert (LLG) equation. Although the LLG equation has a quantum mechanical origin and reflects the fact that magnetization and angular momentum of electrons are really the same thing, it has been used most of the time as a classical equation. The reason for it is that magnetism can be viewed as a condensate of the underlying quantum mechanical degrees of freedom, through which their wavefunctions become an identical macroscopic quantity and acquire the physical meaning of the order parameter, similar to the reason why light is formed by photons but one can still describe it classically in daily life.
However, in certain cases the quantum mechanical nature of the underlying electrons contributing to the magnetism can still manifest in the magnetization dynamics. In a recent paper by Bangguo Xiong, Xiao Li, and Qian Niu (UT Austin) and Hua Chen (CSU), we have given examples of such nontrivial modifications to the classical LLG equation due to the coupling between the magnetization and the microscopic electronic degrees of freedom [1]. The theory is based on the semiclassical wavepacket approach, which captures the slow, long-wavelength dynamics of electrons in a solid by constructing wavepackets which respond to electromagnetic fields as classical point-like objects. There are, however, remnant quantum effects appearing in their equations of motion through the so-called Berry curvatures. We find that in the absence of external fields the gyromagnetic ratio appearing in the LLG equation acquires an additional correction inversely proportional to one kind of Berry curvatures of the electrons. In the presence of an electric field, we find that the electric field modifies the “effective magnetic field” coupled to the magnetization, the gyromagnetic ratio, as well as the damping in the LLG equations. Electric field (or current) effects on magnetization dynamics are usually described as spin-orbit torques in the spintronics community, which are conventionally calculated as the linear response of spin density to electric fields. However, our theory points out that such an approach misses some important contributions to the magnetization dynamics that are due to the dynamical coupling between the magnetization and the electrons. One interesting example of such new contributions, due to the modification to the damping term, is that a current may turn a hard axis of the magnetization into an easy axis and offer a new scenario for electrically-induced magnetic switching. Such an effect is demonstrated using a model of ferromagnet/topological insulator bilayer.
[1] Bangguo Xiong, Hua Chen, Xiao Li, and Qian Niu, “Geometric Dynamics of Magnetization: Electronic Contribution”, Phys. Rev. B 98, 035123 (2018)
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