2.6.3 Analytical investigation of self-oscillating behavior
and current-induced switching

In the following we will present an analytical approach to study magnetization self-oscillations and reversal in the free layer of a trilayers structure traversed by a spin-polarized electric current perpendicular to the layers plane (see Fig. 2.20). According to the derivation performed in section 2.6.1, the model equation which describe the dynamics of the free layer is:

$$\frac{d\textbf{{m}}}{dt}-\alpha \textbf{{m}}\times \frac{d\textbf... ...s \left( \textbf{h}_{\text{eff}}+\beta \textbf{{m}}\times \textbf{p}\right) .$$ (2.105)

which is written in dimensionless form with usual normalizations introduced in section 1.3.4; $\textbf{p}$ is the direction of spin polarization and $\beta=\beta(\textbf{{m}})$ is the dimensionless function describing the intensity of the spin-transfer torque. We model the free layer as a flat ellipsoidal particle in order that the effective field is given by the usual expression

$$\textbf{h}_{\text{eff}}= \textbf{h}_a - D_x m_x\mathbf{e}_x - D_y m_x\mathbf{e}_y - D_zm_z\mathbf{e}_z \quad ,$$ (2.106)

where $\textbf{h}_a$ is the applied field, $\mathbf{e}_x$, $\mathbf{e}_y$, $\mathbf{e}_z$, are cartesian unit vectors and $D_x \leq D_y \leq D_z$ take into account both shape and crystalline anisotropy. As far as the anisotropy field is concerned, most publications on Co-Cu-Co trilayers report value of H$_$an in the range of $10 \div 100$ mT which correspond to value of the normalized anisotropy constant $\kappa_$an around $10^{-2} \div 10^{-1}$. In the analysis below we will assume that $\beta$ is constant, which is a condition reasonably verified for $P\leq 0.1$. A more general analysis including the dependence of $\beta$ on $\textbf{{m}}$ has been performed in Ref. [49]. To start our discussion, let us consider the energy balance equation associated to Eq. (2.105):

$$\frac{dg(\textbf{{m}}) }{d t}= - {\mathcal{P}}(\textbf{{m}})= -\a... ... \left( \textbf{{m}}\times \textbf{p}\right) \cdot\frac{d\textbf{{m}}}{dt} ,$$ (2.107)

where

$$g(\textbf{{m}})=\frac{D_x}{2} m_x^2 + \frac{D_y}{2} m_y^2 +\frac{D_z}{2} m_z^2 - \textbf{h}_ \cdot \textbf{{m}}$$ (2.108)

is the free energy of the magnetic body and ${\mathcal{P}}(\textbf{{m}})$ is the ``absorbed power'' function. Equation (2.107) has very interesting implications: in appropriate conditions the spin-transfer torque term may provide energy to the system and counterbalance dissipation associated to the Gilbert term. If this is the case, the dynamical system (2.105) may exhibit limit cycles i.e. periodic self-oscillation.