Standard, inverse, and triplet spin-valve effects in $F_1$${\mathrm{/S/F}}_2$ systems

We demonstrate that, contrary to common belief, the critical temperature $T_c$ of clean ${\mathrm{F}}_1/\mathrm{S}/{\mathrm{F}}_2$ spin valves can depend nonmonotonically on the angle between the magnetic moments of the ferromagnetic ${\mathrm{F}}_1$ and ${\mathrm{F}}_2$ layers. Depending on the system parameters the minimum of $T_c$ may correspond to a parallel, antiparallel, or noncollinear mutual orientation of magnetic moments. Such anomalous behavior can reveal itself only if the ferromagnetic layers differ from each other and it completely disappears in the dirty limit.

Figure 1

Sketch of a multilayered ${\mathrm{F}}_1/\mathrm{S}/{\mathrm{F}}_2$ spin valve. The magnetic moments (shown with arrows) in the ferromagnetic layers form the angle $\theta$ with each other.

Figure 2

The dependencies of the critical temperature of the clean ${\mathrm{F}}_1/\mathrm{S}/{\mathrm{F}}_2$ spin valve for parallel (blue dashed curves marked $\uparrow \uparrow$) and antiparallel (red solid curves marked $\uparrow \downarrow$) orientations of magnetic moments on the width $d_2$ of the ${\mathrm{F}}_2$ ferromagnetic layer. The width of the ${\mathrm{F}}_1$ layer takes the values (a) $d_1={\xi }_f$ and (b) $d_1=2{\xi }_f$ (here ${\xi }_f=v_f/4\pi T_{c0}$). Other parameters are $h_1=h_2=2\pi T_{c0}$, $4\pi T_{c0}{d}_s/v_s=0.1$, and $p_f=0.2p_s$.

Figure 3

The dependence of the value $100\times (T_c^{AP}-T_c^P)/T_{c0}$ on the thicknesses of the ferromagnetic layers. The red (blue) areas correspond to the standard (inverse) spin-valve effect. The notations and parameters are the same as in Fig. 2.

Figure 4

The dependencies of the critical temperature of the clean ${\mathrm{F}}_1/\mathrm{S}/{\mathrm{F}}_2$ spin valve on the angle $\theta$ between the magnetic moments in the ferromagnetic layers. Dashed red and blue curves are monotonic, while solid green, brown, and black curves demonstrate the triplet spin-valve effect. The parameters are $d_1={\xi }_f$, $h_1=h_2=2\pi T_{c0}$, $4\pi T_{c0}{d}_s/v_s=0.1$, and $p_f=0.2p_s$, and the values of the thickness $d_2$ of the $F_2$ layer are shown near the corresponding curves.