From giant magnetoresistance to current-induced switching by spin transfer

We develop a model of the current-induced torque due to spin transfer in a layered system consisting of two ferromagnetic layers separated by a nonmagnetic layer. The description is based on (i) the classical spin diffusion equations for the distribution functions used in the theory of current-perpendicular-to-plane giant magnetoresistance (CPP-GMR), (ii) the relevant boundary conditions for the longitudinal and transverse components of the spin current in the situation of quasi-interfacial absorption of the transverse components in a magnetic layer. The torque is expressed as a function of the usual parameters derived from CPP-GMR experiments and two additional parameters involved in the transverse boundary conditions. Our model is used to describe qualitatively normal and inverse switching phenomena studied in recent experiments. We also present a structure for which we predict only states of steady precession above a certain critical current. We finally discuss the limits of a small angle between magnetic moments of the ferromagnetic layers and of vanishing imaginary part of the mixing conductance.

Figure 1

Schematic structure of the system studied in this paper. The system consists of thick $\left(F1\right)$ and thin $\left(F2\right)$ ferromagnetic films, separated by a nonmagnetic $\left(N\right)$ layer. The thick magnetic film (similarly as the right nonmagnetic lead) is assumed to be semi-infinite, while the thin nonmagnetic and ferromagnetic films have thicknesses $d_0$ and $d_2$, respectively. The arrows indicate orientation of the net spin of the magnetic films, with $\phi$ being the angle between the spins.

Figure 2

Normalized in-plane torque ${\tau }_{\phi }=\mid {\tau }_{\Vert }\mid$ acting on the thin ferromagnetic film due to spin transfer, calculated as a function of the angle $\phi$ for parameters typical for $\mathrm{Co}∕\mathrm{Cu}$ system, as described in the text. The insets indicate the similarity between the angular dependence of the parameter $a$ of the expression ${\tau }_{\Vert }=a\widehat{\mathbf{s}}\times (\widehat{\mathbf{s}}\times \widehat{\mathbf{S}})$ and that of the spin accumulation amplitude $g$ (note that $g$ is normalized to $\mid I_0\mid$ whereas $a$ to $I_0$). The other parameters are $E_F=7\mathrm{eV}$, $d_0=10\mathrm{nm}$, $d_2=10\mathrm{nm}$.

Figure 3

Normalized out-of-plane torque, calculated as a function of the angle $\phi$ for the parameters typical for $\mathrm{Co}∕\mathrm{Cu}$ system. The inset shows the corresponding parameter $b$. The other parameters are the same as in Fig. 2.

Figure 4

Torque as a function of $\phi$ for the four situations described in the text, and for indicated values of the spin asymmetry parameters. The other parameters are the same as in Fig. 2.

Figure 5

Torque for the case when the spin asymmetry factors of the thick ferromagnetic film, ${\beta }_1=0.1$ and ${\gamma }_1=-0.1$, are significantly different from those of the thin one, ${\beta }_2=0.51$ and ${\gamma }_2=0.76$. The torque is now normalized to $\mid I_0\mid$ and the two curves correspond to $I_0>0$ and $I_0<0$, as indicated. The angle ${\phi }_c$ corresponds to the point where the torque due to spin transfer vanishes. Inset shows variation of the angle ${\phi }_c$ with the spin asymmetry factor ${\gamma }_1$ for several values of ${\beta }_1$ indicated on the curves. The other parameters are as in Fig. 2.

Figure 6

Amplitude of spin accumulation at the interface (a), the angle ${\phi }_g$ (b), the factor $\sin (\phi -{\phi }_g)∕\sin \phi$ (c), and the torque ${\tau }_{\phi }$ and parameter $a$ (both normalized to $\hslash I_0∕\mid e\mid$) (d). All curves are calculated for $\mathrm{Im}{G}_{\uparrow \downarrow }=0$ and for negative current, $I_0<0$. The other parameters are as in Fig. 2.