Landau–Lifshitz–Gilbert equation

In a ferromagnet, the magnitude of the magnetization at each spacetime point is approximated by the saturation magnetization (although it can be smaller when averaged over a chunk of volume). The LLG equation describes the rotation of the magnetization in response to the effective field and accounts for not only a real magnetic field but also internal magnetic interactions such as exchange and anisotropy. An earlier, but equivalent, equation (the Landau–Lifshitz equation) was introduced by : {{NumBlk|:|\frac{\mathrm{d}\mathbf{M}}{\mathrm{d} t}= -\gamma \mathbf{M} \times \mathbf{H_\mathrm{eff}} - \lambda \mathbf{M} \times \left(\mathbf{M} \times \mathbf{H_{\mathrm{eff}}}\right)|}} where is the electron gyromagnetic ratio and is a phenomenological damping parameter, often replaced by :\lambda = \alpha \frac{\gamma}{M_\mathrm{s}}, where is a dimensionless constant called the damping factor. The effective field is a combination of the external magnetic field, the demagnetizing field, and various internal magnetic interactions involving quantum mechanical effects, which is typically defined as the functional derivative of the magnetic free energy with respect to the local magnetization . To solve this equation, additional conditions for the demagnetizing field must be included to accommodate the geometry of the material. ==Landau–Lifshitz–Gilbert equation==

In 1955 Gilbert replaced the damping term in the Landau–Lifshitz (LL) equation by one that depends on the time derivative of the magnetization: {{NumBlk|:|\frac{\mathrm{d} \mathbf{M}}{\mathrm{d} t}=-\gamma \left(\mathbf{M} \times \mathbf{H}_{\mathrm{eff}} - \eta \mathbf{M}\times\frac{\mathrm{d} \mathbf{M}}{\mathrm{d} t}\right)|}} This is the Landau–Lifshitz–Gilbert (LLG) equation, where is the damping parameter, which is characteristic of the material. It can be transformed into the Landau–Lifshitz equation: ==Landau–Lifshitz–Gilbert–Slonczewski equation==

In 1996 John Slonczewski expanded the model to account for the spin-transfer torque, i.e. the torque induced upon the magnetization by spin-polarized current flowing through the ferromagnet. This is commonly written in terms of the unit moment defined by : :\dot{\mathbf{m}}=-\gamma \mathbf{m}\times \mathbf{H}_{\mathrm{eff}}+\alpha \mathbf{m}\times \dot{\mathbf{m}}+\tau _{\parallel}\frac{\mathbf{m}\times (\mathbf{x}\times \mathbf{m})}{\left|\mathbf{x}\times \mathbf{m}\right|}+\tau _{\perp}\frac{\mathbf{x}\times \mathbf{m}}{\left|\mathbf{x}\times \mathbf{m}\right|} where \alpha is the dimensionless damping parameter, \tau_\perp and \tau_\parallel are driving torques, and is the unit vector along the polarization of the current. {{Cite journal ==Application to the response to an oscillating magnetic field==

The response of a magnetic material to an oscillating magnetic field (e.g. the magnetic component of an electromagnetic wave) can be found using the LLG equation. Without loss of generality, the magnetization equilibrium magnetization vector \mathbf{M}_s, which is aligned with the static magnetic field \mathbf{H}_s can be taken to be in the z-direction. One can then consider a small oscillating magnetic field \mathbf{h}\propto e^{i\omega t}, which causes a small change in the magnetization \textbf{m}. Thus, the total magnetization reads \mathbf{M}=\mathbf{M}_s+\mathbf{m}, and the total magnetic field is \mathbf{H}=\mathbf{H}_s+\mathbf{h}. Filling this in to the LLG equation (neglecting the damping terms) gives :\frac{d}{dt}\left( \mathbf{M}_s+\mathbf{m} \right)=\gamma (\mathbf{M}_s + \mathbf{m})\times (\mathbf{H}_s+\mathbf{h}). The equilibrium magnetization does not vary over time. Additionally, the \mathbf{M}_s\times \mathbf{H}_s term is zero because they are parallel. Finally, the equation is considered in leading order, so the \mathbf{m}\times \mathbf{h} term is neglected because it is the product of two small terms. Thus, the equation becomes \frac{d\mathbf{m}}{dt}=\gamma(\mathbf{m}\times \mathbf{H}_s + \mathbf{M}_s\times \mathbf{h}). The z-component of this equation is zero, meaning that the magnetization does not respond to an magnetic field oscillating in the z-direction. Assuming that \mathbf{m} oscillates together with \mathbf{h}, the time derivative turns into i\omega. The x- and y-components can then be written in matrix form as :\begin{pmatrix}\gamma H_s & i\omega \\ -i \omega & \gamma H_s\end{pmatrix}\begin{pmatrix}m_x\\m_y\end{pmatrix} = -\gamma M_s\begin{pmatrix}h_x\\h_y\end{pmatrix}. Inverting this matrix gives the permeability tensor :\begin{pmatrix}m_x\\m_y\end{pmatrix}= \frac{\gamma M_s}{\gamma^2 H_s^2 - \omega^2} \begin{pmatrix}\gamma H_s & -i\omega \\ i\omega & \gamma H_s\end{pmatrix}\begin{pmatrix}h_x \\ h_y\end{pmatrix}. Thus, there is a resonance at the ferromagnetic resonance frequency. ==References and footnotes==