The Influence of Magnetic Anisotropy on Current-Induced Spindynamics

Abstract

The chapter provides a short and intuitive introduction to the basic concept of spin-transfer torque and the field of spin-torque driven magnetization dynamics in nanopillar systems. The influence of spin-polarized currents on magnetic nano-objects may lead to current-induced magnetization reversal as well as current-driven magnetization dynamics. The quantities that determine the critical currents for magnetization switching and the influence of the relative orientation of magnetization and current polarization are discussed. We focus on the nanopillar geometry and address the influence of magnetic anisotropy on the spin-torque driven spindynamics. Selected experimental examples are given to illustrate the interplay between magnetic anisotropy and spin-transfer torque.

Appendix

In the following we show that Eq. (13) and Eq. (16) are indeed eqvivalent. We start from Eq. (13),

$$\begin{aligned} \frac{{\text{d}} {\mathbf M}}{{\text{d}t}} = - \left| \gamma \right| {\mathbf M} \times {\mathbf B}_{{\text{eff}}} + \frac{\alpha }{M} {\mathbf M} \times \frac{{\text{d}} {\mathbf M}}{{\text{d} t}} - \left| \gamma \right| \frac{a_J}{M} {\mathbf M} \times \left( {\mathbf M} \times {\mathbf m}_p \right). \end{aligned}$$

(28)

Inserting $\text{d} {\mathbf M} / \text{d} t$ on the right hand side yields:

$$\begin{aligned} \frac{{\text{d}} {\mathbf M}}{{\text{d}t}}&= - \left| \gamma \right| {\mathbf M} \times {\mathbf B}_{{\text{eff}}} - \left| \gamma \right| \frac{a_J}{M} {\mathbf M} \times \left( {\mathbf M} \times {\mathbf m}_p \right)+ \frac{\alpha }{M} {\mathbf M} \nonumber \\&{}\times \left[ - \left| \gamma \right| {\mathbf M} \times {\mathbf B}_{{\text{eff}}} + \frac{\alpha }{M} {\mathbf M} \times \frac{{\text{d} {\mathbf M}}}{{\text{d} t}} - \left| \gamma \right| \frac{a_J}{M} {\mathbf M} \times \left( {\mathbf M} \times {\mathbf m}_p \right) \right]. \end{aligned}$$

(29)

Using the relation ${\mathbf a} \times ({\mathbf b} \times {\mathbf c}) = {\mathbf b} ({\mathbf a} \cdot {\mathbf c}) - {\mathbf c} ({\mathbf a} \cdot {\mathbf b})$ and considering that ${\mathbf M} \perp \dot{{\mathbf M}}$ one easily derives

$$\begin{aligned} {\mathbf M} \times \left( {\mathbf M} \times \frac{{\text{d}} {\mathbf M}}{{\text{d} t}} \right)&= {\mathbf M} \left( {\mathbf M} \cdot \frac{{\text{d}} {\mathbf M}}{{\text{d} t}} \right) - \frac{{\text{d}} {\mathbf M}}{{\text{d} t}} \cdot {\mathbf M}^2 = - \frac{{\text{d}} {\mathbf M}}{{\text{d} t}} \cdot M^2 \nonumber \\ {\mathbf M} \times \left( {\mathbf M} \times {\mathbf m}_p \right)&= {\mathbf M} \left( {\mathbf M} \cdot {\mathbf m}_p \right) - {\mathbf m}_p \cdot M^2 \nonumber \\ {\mathbf M} \times \left[ {\mathbf M} \times \left( {\mathbf M} \times {\mathbf m}_p \right) \right]&= - {\mathbf M} \times {\mathbf m}_p \cdot M^2 . \end{aligned}$$

(30)

This in turn yields:

$$\begin{aligned} \frac{{\text{d}} {\mathbf M}}{{\text{d}t}} =&-\left| \gamma \right| {\mathbf M} \times {\mathbf B}_{{\text{eff}}} - \left| \gamma \right| \frac{\alpha }{M} {\mathbf M} \times \left( {\mathbf M} \times {\mathbf B}_{{\text{eff}}} \right) - \alpha ^2 \frac{{\text{d}} {\mathbf M}}{{\text{d}t}} \nonumber \\&+ \left| \gamma \right| \alpha a_J {\mathbf M} \times {\mathbf m}_p - \left| \gamma \right| \frac{a_J}{M} {\mathbf M} \times \left( {\mathbf M} \times {\mathbf m}_p \right). \end{aligned}$$

(31)

Reorganizing the terms finally leads to

$$\begin{aligned} \frac{1}{\left| \gamma \right|} \frac{{\text{d}} {\mathbf M}}{{\text{d}t}} = - A \cdot {\mathbf M} \times \left( {\mathbf B}_{{\text{eff}}} - \alpha a_J {\mathbf m}_p \right) - A \cdot \frac{{\mathbf M}}{M} \times \left[ {\mathbf M} \times \left( \alpha {\mathbf B}_{{\text{eff}}} + a_J {\mathbf m}_p \right) \right],\nonumber \\ \end{aligned}$$

(32)

which is Eq. (16).