2.6.3.2 Current-driven switching experiment. Analytical and numerical results

In the sequel, we apply the perturbation technique outlined above to a special case relevant to the spintronics applications: self-oscillations and reversal of magnetization driven by the current in absence of applied field ( $\textbf{h}_a=0$). In the discussion we will assume that the injected spin polarization $\textbf{p}$ is aligned with the easy axis of the magnetic free layer ( $\textbf{p}=\mathbf{e}_x$). More general cases, with nonzero applied field and arbitrary orientation of $\textbf{p}$, can be treated by following a very similar line of reasoning [49]. We suppose that the system is initially in the potential well around the equilibrium state $\textbf{{m}}=\mathbf{e}_x$. This region is characterized by magnetization states with energies values

$$\frac{D_x}{2} \leq g_0 \leq \frac{D_y}{2} \quad.$$ (2.117)

In the latter equation $g_0=D_x/2$ is the energy of $\textbf{{m}}=\mathbf{e}_x$, while $g_0=D_y/2$ corresponds to the saddles points $\textbf{{m}}=\pm \mathbf{e}_y$. The analytical solution of the unperturbed dynamics in this region is given by [50]

$$m_x(t) = k_{g} \textrm{dn}( \Omega_{L} t, k_{L} ) \quad,$$ (2.118)

$$m_y(t) = ({k_{g}^{\prime}}/{k^{\prime}}) \textrm{sn}( \Omega_{L} t, k_{L})\quad,$$ (2.119)

$$m_z(t) = - k_{g}^{\prime} \textrm{cn}( \Omega_{L} t, k_{L}) \quad,$$ (2.120)

where $\textrm{sn}(u,k_{L})$, $\textrm{dn}(u,k_{L})$, $\textrm{cn}(u,k_{L})$ are the Jacobi elliptic functions^2.5 of modulus $k_{L}$. The following relationship hold for the quantities appearing in Eqs. (2.118)-(2.120):

$$k_g^2= (D_z - 2 g_0)/(D_z -D_x)\quad,$$ (2.121)

$$k^2=(D_z - D_y)/(D_z - D_x)\quad,$$ (2.122)

$$k_g^{\prime2}=1 - k_g^2\quad,$$ (2.123)

$$k^{\prime2}=1-k^2 \quad,$$ (2.124)

$$k_L=(k k_g^\prime)/(k_g k^\prime) \quad,$$ (2.125)

$$\Omega_L=k_g k^\prime (D_z -D_x) \quad.$$ (2.126)

The period of the solution is given by the following formula:

$$T_{g_0}= 4 K(k_{L})/\Omega_{L}\quad,$$ (2.127)

where $K(k_{L})$ is the complete elliptic integral of the first kind. By using the above expressions in Eq. (2.116), the following analytical formulas can be derived

$$\begin{split} M^{\alpha}(g_0)= 4 \Omega_L \left[ \textrm{K}(... ...ac{(2 g_0 -D_x)}{(D_z - D_x)} \qquad \qquad \qquad \end{split}$$ (2.128)

where $\textrm{E}(k_L)$ is the complete elliptic integral of the second kind. Notice that $M^{\alpha}(g_0)$ and $M^{\beta}(g_0)$ are positive functions, zero only for $g_0=D_x/2$ (equilibrium at $\textbf{{m}}=\mathbf{e}_x$). Plots of the $M(g_0)$ versus $g_0$ for different values of $\beta$ are reported in Fig. 2.22. The interpretation of Melnikov function in terms of energy balance, expressed by Eq. (2.107) and Eq. (2.116), can help one to qualitatively understand the dynamics and the stability of limit cycles. In fact, zeros of the Melnikov function determine trajectories on which there is average balance between power dissipation and absorption. Conversely, a positive (negative) value of the Melnikov function indicates, according to Eq. (2.107), that the system has to move towards periodic trajectories with lower (upper) energy^2.6.

Figure: (left) Melnikov function for different values of $\beta$ ( $\alpha=5\cdot 10^{-3}$, $D_x=-0.1$, $D_y=0$, $D_z=1$). $\beta_{c1}=-3\cdot10^{-3}$, $\beta_{c2}=-3.338\cdot10^{-3}$. (right) Frequency ($1/T_{g_0}$) and amplitude of self-oscillations of the limit cycles vs the value of $\vert\beta\vert$ in the interval $[\beta_{c1}, \beta_{c2}]$ ( $\alpha=5\cdot 10^{-3}$, $D_x=-0.1$, $D_y=0$, $D_z=1$).

It is clear from Fig. 2.22 that $M(g_0)$ has another zero for $\beta$ negative and such that $\vert\beta \vert>\vert\beta _{c1}\vert$ where $\beta_{c1}$ is the value such that the derivative of $M(g_0)$ in $D_x/2$ is equal to zero. This corresponds to the condition of tangency of the Melnikov function diagram at $g_0=D_x/2$:

$$\frac{dM}{d g_0}(D_x/2)=0 \quad.$$ (2.129)

By using Eq. (2.128) and simple algebra one can derive that

$$\beta_{c1}=-\frac{\alpha}{2}[(D_z - D_x ) + (D_y -D_x)] .$$ (2.130)

This value of $\beta_{c1}$ correspond to an Hopf bifurcation [43] of the equilibrium $\textbf{{m}}=\mathbf{e}_x$: the equilibrium becomes unstable and a stable limit cycle around the equilibrium is created (see Fig. 2.23). If we further increase the value $\vert\beta\vert$ the zero of $M(g_0)$ will move toward larger value of energy. Since the limit cycle is $\epsilon-$close to the conservative trajectory corresponding the zero of $M(g_0)$, we can derive its properties from the property of the corresponding conservative trajectory. In Fig. 2.22 the amplitude and the frequency of the limit cycle as function of $\vert\beta\vert$ are reported. Notice that due to the thin film geometry the amplitude of the oscillation of $m_z(t)$ remains rather small. When $\vert\beta\vert$ is increased further we arrive to the value $\beta_{c2}$ such that $M(D_y/2)=0$, which is given by

$$\beta_{c2}=- \alpha \frac{M^\alpha (D_y/2)}{M^\beta (D_y/2)} \quad .$$ (2.131)

At this value of $\beta$ the system undergoes an homoclinic connection bifurcation [43] in which the limit cycle first degenerates in a homoclinic connection and then is destroyed (see Fig. 2.24). The system then relaxes toward the equilibrium $\textbf{{m}}=-\mathbf{e}_x$. In order to the test the accuracy of the perturbation technique we have carried out numerical simulations of Eq. (2.105) for different values of $\beta$. The results are presented in Fig. 2.25. Notice that the two bifurcations, Hopf and Homoclinic connection, occur at critical values that are very close to the ones predicted by the theory.

Figure: Mechanism of the Hopf bifurcation. (up left) The focus $F$ is a stable equilibrium. (up right) For $\beta=\beta_{c1}$ the limit cycle $L$ coincides with the equilibrium $F$. (bottom) For $\vert\beta \vert>\vert\beta _{c1}\vert$ the focus $F$ becomes unstable and a stable limit cycle $L$ is created. This mechanism justifies the onset of self-oscillations driven by the spin-transfer torque term.

Figure: Mechanism of the homoclinic bifurcation. (up left) For $\vert\beta_{c1}\vert<\vert\beta\vert<\vert\beta_{c2}\vert$ there is a stable limit cycle $L$ around the unstable equilibrium $F$. (up right) For $\beta =\beta _{c2}$ the limit cycle $L$ disappears and a homoclinic connection $\Gamma^+\equiv\Gamma^-\equiv L$ is created. (bottom) For $\vert\beta\vert>\vert\beta_{c2}\vert$ the homoclinic connection disappears but the mutual position of the separatrices $\Gamma ^+$ and $\Gamma ^-$ is exchanged. The region around the unstable equilibrium $F$ is now in the basin of attraction of the reversed state. This mechanism permits the switching of the free layer.

Figure: Numerical simulation of Eq. (2.105) with $\alpha=5\cdot 10^{-3}$, $D_x=-0.1$, $D_y=0$, $D_z=1$. Top: $\beta=-2.9\cdot 10^{-3}$. Middle: $\beta=-3.336\cdot 10^{-3}$. Bottom: $\beta=-3.339\cdot 10^{-3}$.