Effect of an inhomogeneous exchange field on the proximity effect in disordered superconductor-ferromagnet hybrid structures

We investigate the effect of an inhomogeneous exchange field on the proximity effect in superconductor-ferromagnet hybrid structures within the quasiclassical theory of superconductivity. As an example, we study a superconductor-ferromagnet bilayer with an in-plane spiral magnetic order in the ferromagnet. The superconducting proximity effect induces in this case triplet correlations in the bilayer that are sensitive to the local quantization axis of the exchange field in the ferromagnet. The coexistence of singlet and triplet pair correlations in the bilayer results into a sensitivity of the superconducting transition temperature on the spatial variation of the exchange field in the ferromagnetic layer. We show that the inhomogeneity also tends to suppress the oscillating behavior of the pair amplitudes in the ferromagnet.

Figure 1

(Color online) The exchange field rotates in the ferromagnetic layer in the plane $\left(xy\right)$ with a constant wave vector $Q$. The period of rotation in the $y$ direction is $2\pi ∕Q$.

Figure 2

(Color online) $T_c$ versus the spiral wave vector $Q$ for different harmonic solutions. The highest critical temperature at $Q\ne 0$ is obtained for $p=0$. Here ${\gamma }_b=0.3$, $d_f=2.5\mathrm{nm}$, and $d_s=7\mathrm{nm}$.

Figure 3

(Color online) $T_c$ versus $d_f$ for $Q{\xi }_s=0$ and $Q{\xi }_s=1$. The characteristic nonmonotonic dependence of $T_c\left(d_f\right)$ found at $Q=0$ is suppressed by the moment inhomogeneity in F. Here $J=20T_{c0}$, ${\xi }_s={\xi }_f$, $d_s=2{\xi }_s$, $\gamma =0.15$, and ${\gamma }_b=0$.

Figure 4

(Color online) $T_c$ versus the spiral wave vector $Q$ for different values of the interface parameter ${\gamma }_b$. Here $d_f=5\mathrm{nm}$ and $d_s=7\mathrm{nm}$. The remaining parameters are the same as in Figs. 2 and 3 of Ref. 23.

Figure 5

(Color online) Singlet and triplet pair amplitudes at the S/F interface versus the spiral wave vector $Q$. An increase of $Q$ is accompanied by an enhancement of the singlet amplitude together with a reduction of the (negative) triplet amplitude. Here, we took the same parameters as in Fig. 3 with $d_f=2{\xi }_s$.

Figure 6

(Color online) Spatial dependence of singlet and triplet amplitudes for $Q{\xi }_s=0$ and $Q{\xi }_s=1$. Here we used the same parameters as in Fig. 3 with $d_f=2{\xi }_s$.