2.3 The dynamic equation

The solution of Brown's equations gives us the magnetization distribution in equilibrium. If we are interested in the dynamic properties and time evolution of the magnetization, we have to consider the precession of the magnetization in a magnetic field [16].

The torque $\mathbf{l}$ is given by the rate of change of angular momentum $\mathbf{g}$ with time

$$\frac{d \mathbf{g}}{dt}= \mathbf{l} \quad, \quad \vert\mathbf{g}\vert=\hbar \quad.$$

The torque acting on a magnetic moment $\mathbf{m}$ in a magnetic field $\mathbf{H}$ is given by

$$\mathbf{l} = \mathbf{m} \times \mathbf{H} \quad.$$

The magnetic moment is linked to the angular momentum by the gyromagnetic ratio $\gamma$

$$\mathbf{g}=-\frac{\mu_0 \mathbf{m}}{\gamma} \quad, \quad \... ... = 2.210173\times 10^{5} \frac{\mathrm{m}}{\mathrm{As}}\quad.$$

$g \approx 2$ is the Landé factor, $\vert e\vert$ the elementary charge, and $m_\mathrm{e}$ the electron's mass. The magnetic field constant $\mu_0$ has been pulled into $\gamma$. By the above definition $\gamma$ is positive, but the electron's charge is negative. As a result, we obtain

$$\frac{d \mathbf{m}}{dt}=-\gamma \mathbf{m} \times \mathbf{H}$$ (2.21)

as the equation of motion for the magnetic moment of the electron.

We can replace the magnetic moment of the electrons by the magnetization. The magnetic field, which drives the precession, can be identified with the effective field (), but we simply write $\mathbf{H}$. Thus, we obtain

$$\frac{d \mathbf{M}}{dt}=-\gamma\mathbf{M} \times \mathbf{H} \quad.$$

This equation describes the undamped precession of the magnetization vector $\mathbf{M}$ about the field direction. It is the well known Larmor precession with the Larmor frequency $\omega=\gamma H$. From experiments it is known, that changes in the magnetization decay in finite time. As this damping cannot be derived rigorously from basic principles, it is just added by a phenomenological term. In reality it is caused by a complex interaction of the electron's magnetic moment with the crystal lattice.

Gilbert [17] proposed a damping term of the form

$$\frac{\alpha}{M_\mathrm{s}} \mathbf{M} \times \frac{d \mathbf{M}}{dt}$$

with the dimensionless damping parameter $\alpha$.

It is equivalent to a an older form of Landau and Lifshitz [18], which is usually written as

$$-\frac{\lambda \gamma'}{M_\mathrm{s}} \mathbf{M} \times (\mathbf{M} \times \mathbf{H})$$

with the dimensionless damping parameter $\lambda$.

The relationship between $\alpha$ and $\lambda$ can be derived as follows. First, we apply $\mathbf{M} \cdot$ to both sides of the Gilbert equation

$$\frac{d \mathbf{M}}{dt} = -\gamma\mathbf{M} \times \mathbf... ...M_\mathrm{s}} \mathbf{M} \times \frac{d \mathbf{M}}{dt} \quad.$$ (2.22)

Since the right hand side vanishes, we obtain

$$\mathbf{M} \cdot \frac{d \mathbf{M}}{dt} = 0$$

or

$$\frac{d M^2}{dt} = 0 \quad.$$

Thus, it is ensured, that the saturation magnetization $\vert\mathbf{M}\vert = M_\mathrm{s}$ remains constant during the motion, as assumed in ().

When we apply $\mathbf{M} \times$ to both sides of Gilbert's equation (), we get

$$\mathbf{M} \times \frac{d \mathbf{M}}{dt} = -\gamma \mathbf{M} \times (\mathbf{M} \times \mathbf{H}) +\frac{\... ...rm{s}} \mathbf{M} \times \left(\mathbf{M} \times \frac{d \mathbf{M}}{dt}\right)$$

$$= -\gamma \mathbf{M} \times (\mathbf{M} \times \mathbf{H}) +\frac{\... ..._\mathrm{s}} \left( \mathbf{M} \cdot \frac{d \mathbf{M}}{dt} \right) \mathbf{M}$$

$$-\alpha \frac{d\mathbf{M}}{dt}$$

$$= -\gamma \mathbf{M} \times (\mathbf{M} \times \mathbf{H}) -\alpha \frac{d \mathbf{M}}{dt}$$ (2.23)

If we substitute this result in (), we arrive at

$$\left(1+\alpha^2\right)\frac{d \mathbf{M}}{dt} = -\gamma\m... ...rm{s}} \mathbf{M} \times (\mathbf{M} \times \mathbf{H}) \quad.$$

With

$$\lambda=\alpha \quad \mathrm{and} \quad \gamma'=\frac{\gamma}{1+\lambda^2}$$ (2.24)

we get the Landau-Lifshitz equation in Gilbert form

$$\frac{d \mathbf{M}}{dt} = -\gamma'\mathbf{M} \times \mathb... ...m{s}} \mathbf{M} \times (\mathbf{M} \times \mathbf{H}) \quad.$$ (2.25)

Figure: Larmor precession with damping