Phase diagrams of ferromagnet-superconductor multilayers with misaligned exchange fields

We study the influence of misalignment of the ferromagnetic exchange field on the equilibrium properties of hybrid structures, composed of superconducting (S) and ferromagnetic (F) parts. In particular, we study numerically the superconducting critical temperature $T_c$ in F-S-F trilayers and in F-S-F-S-F Josephson junctions as a function of the misalignment angle $\theta$ of the ferromagnetic magnetization. We discuss the corresponding phase diagrams for these hybrid structures. For the Josephson junctions, a transition between the zero-phase and the $\pi$-phase ground states as a function of $\theta$ takes place under certain conditions. Within the quasiclassical Green’s function technique in the diffusive limit, we introduce a fast and effective method for calculating $T_c$ in such multilayer structures.

Figure 1

(Color online) Geometry of the asymmetric ${\mathrm{F}}_1-\mathrm{S}-{\mathrm{F}}_2$ structure. The moments ${\mathbf{J}}_1$ (in ${\mathrm{F}}_1$) and ${\mathbf{J}}_2$ (in ${\mathrm{F}}_2$) may have different amplitudes and point in different directions (the relative orientation angle is denoted $\theta$).

Figure 2

(Color online) The critical temperature $T_c$ of a trilayer for a variety of thicknesses $d_{f1}$ and exchange fields $J$. The model parameters are given in Table . Note that $T_c∕T_{c0}$ for $d_{f1}=0=J$ is suppressed to $\approx 0.86$ because $d_{f2}=0.5{\xi }_s$.

Figure 3

(Color online) Critical temperature $T_c$ of a trilayer versus the thicknesses of the ferromagnetic layers. The model parameters are given in Table .

Figure 4

(Color online) Critical temperature $T_c$ of a trilayer versus the interface resistance parameter ${\gamma }_b={\gamma }_{b1}={\gamma }_{b2}$ at a few thicknesses $d_{f1}$ corresponding to points on the $J=20T_{c0}$ curve in Fig. 2 to the left, inside, and to the right of the $T_c=0$ region. The model parameters are given in Table .

Figure 5

(Color online) Critical thickness $d_s$ for which $T_c\to 0$ versus the conductivity mismatch $\gamma$ for a number of values of the interface resistance parameter ${\gamma }_b$. The lines separate the normal phase (below) from the superconducting phase (above). The model parameters are given in Table .

Figure 6

(Color online) Critical temperature $T_c$ of a trilayer versus the misorientation angle $\theta$ between the exchange fields in ferromagnets ${\mathrm{F}}_1$ and ${\mathrm{F}}_2$. The model parameters are given in Table .

Figure 7

(Color online) (a) The spatial dependence of the order parameter for several interface resistance parameters ${\gamma }_b={\gamma }_{b1}={\gamma }_{b2}$ on the solid curve $(d_{f1}=0.1{\xi }_s)$ in Fig. 4. (b) The Fourier components in Eq. (10). Note that $\Delta$ is normalized to the first component ${\Delta }_0$, which remains unknown in a linearized theory.

Figure 8

(Color online) The ${\mathrm{F}}_2∕\mathrm{S}∕{\mathrm{F}}_1∕\mathrm{S}∕{\mathrm{F}}_2$ pentalayer structure. We consider two types of misalignment of the outer exchange fields relative to the exchange field in the center layer. First, as shown here, $J_2$ is rotated by the same angle $\theta$. The second possibility is when $J_2$ is rotated in opposite directions, $-\theta$ in the left ${\mathrm{F}}_2$ and $+\theta$ in the right ${\mathrm{F}}_2$.

Figure 9

(Color online) Critical temperature $T_c$ of a pentalayer versus the center ferromagnet layer thickness $2d_{f1}$ for several barrier transparencies. The curves come in pairs, solid line for the 0 junction and dashed line for the $\pi$ junction, from top to bottom for ${\gamma }_{b1}={\gamma }_{b2}={\gamma }_b=\{2,1,0.8,0.7\}$, respectively. The model parameters are given in Table .

Figure 10

(Color online) (a) The order parameter profile for the parameters in Fig. 9 for ${\gamma }_b=0.8$ at $2d_{f1}=0.4{\xi }_s$, for which the critical temperature for the 0 and $\pi$ junctions are, respectively, $T_c\approx 0.3T_{c0}$ and $T_c=0.16T_{c0}$. (b) The corresponding Fourier components.

Figure 11

(Color online) (a) The same curves as in Fig. 9 for ${\gamma }_b=0.8$. For thicknesses $d_{f1}$ between the two vertical lines the $0\to \pi$ transition can be tuned by the relative orientation of the exchange fields in ${\mathrm{F}}_1$ and ${\mathrm{F}}_2$. (b) The switch $0\to \pi$ appears at a critical angle ${\theta }_c\approx 84°$ for the thickness $2d_{f1}=0.65{\xi }_s$. For the larger thickness $2d_{f1}={\xi }_s$, outside the window indicated in (a), the largest $T_c$ is obtained for the $\pi$-junction.

Figure 12

(Color online) (a) Phase diagram of the $0\to \pi$ transition in the window indicated in Fig. 11a. In (b) we show the phase diagram for a smaller exchange field, $J=10T_{c0}$. The range of thicknesses $d_{f1}$ for which there is a switching by changing $\theta$ is larger in this case.

Figure 13

(Color online) (a) Next to the region where $T_c$ is suppressed to zero, the curve $T_c\left(d_{f1}\right)$ can contain a back-bend. This latter could signal the occurrence of a first-order transition, which is, however, beyond the scope of the present theory. For the pentalayer, the $\pi$-phase can interfere and the first order transition might be avoided. In (b) we study the dependence on the exchange field misorientation angle $\theta$ for two particular thicknesses $d_{f1}$ in (a). Clearly, the back-bend behavior can occur also as function of $\theta$. The model parameters are given in Table .

Figure 14

(Color online) For thick center films (large $d_{f1}$) the communication between the two superconductors is taken over by long-range nonoscillatory equal spin triplet correlations and the junction is stabilized at phase difference 0 or $\pi$ depending on the exchange field orientation: in (a) a $\pi$ junction for $\theta =90°$ is obtained for exchange field rotation type 1 ($+\theta$ in both the left and right outer ferromagnets ${\mathrm{F}}_2$, as illustrated in Fig. 8), while in (b) a 0 junction for $\theta =90°$ is obtained for rotation type 2 ($-\theta$ in the left ${\mathrm{F}}_2$ and $+\theta$ in the right ${\mathrm{F}}_2$). The difference in $T_c$ between the 0 and $\pi$ cases is quite small for large $d_{f1}$. For $\theta =0$ we use solid lines and for $\theta =90°$ we use dashed lines. Inset in (a): the differences in $T_c$ for various exchange field orientations are due to the interaction between the ferromagnetic layers $F_1$ and $F_2$. Inset in (b): spatial dependence inside the central $6{\xi }_s$ thick ${\mathrm{F}}_1$ layer of the long-range triplet $f_{ty}$ induced for $\theta =90°$. The parameters are given in Table .

Figure 15

(Color online) (a) Critical temperature versus the technical cutoff ${\epsilon }_c$. To achieve good accuracy for the critical temperature we need ${\epsilon }_c$ of order $1000T_{c0}$ (dashed line). When the high-energy tail is summed to infinity, as described for the Fourier method in Eq. (F10), the convergence is more acceptable (solid line). (b) The eigenvalue of the gap equation (12) versus temperature for two different thicknesses. The zero-crossing determines $T_c$. When $T_c$ is suppressed, $\lambda \left(T\right)$ can become a flat function of $T$ which makes it important to compute $\lambda$ with high accuracy to avoid numerical errors in $T_c$. The parameters in (a) were chosen as in Fig. 13, 0 junction, at $2d_{f1}=0.7$ and $\theta =0$. In (b) the two thicknesses are indicated in the legend. See also Table .

Figure 16

(Color online) Critical temperature versus the number $p_c$ of included Fourier coefficients in Eq. (10). The variation in absolute numbers is shown by circles (vertical scale to the left), while the variation in relation to the corresponding value for $T_c$ at a high cutoff $p_c=100$ is shown by squares (vertical scale to the right). The even-odd variation is due to the choice of parameters: the junction is almost symmetric and the even-number Fourier components corresponding to symmetric cos functions contribute more to $T_c$. The model parameters were chosen as in Fig. 15, upper panel (see also Table ).