Interplay of Magnetic and Superconducting Proximity Effects in Ferromagnet-Superconductor-Ferromagnet Trilayers

We present theoretical results on the interplay of magnetic and superconducting orders in diffusive ferromagnet-superconductor-ferromagnet trilayers. The induced triplet superconducting correlations throughout the trilayer lead to an induced spin magnetization. We include self-consistency of the order parameter in the superconducting layer at arbitrary temperatures, arbitrary interface transparency, and any relative orientation of the exchange fields in the two ferromagnets. We propose to use the torque on the trilayer in an external magnetic field as a probe of the presence of triplet correlations in the superconducting phase.

Figure 1

(a) The trilayer consists of a superconductor of thickness $d_S$ and two ferromagnets of thicknesses $d_{F1}$ and $d_{F2}$. (b) The exchange fields of the ferromagnets, ${\overrightarrow{J}}_1$ and ${\overrightarrow{J}}_2$, are confined to the $y\mathrm{\text{−}}z$ plane, but are misaligned by an angle ${\theta }_J$. For torque measurements, a small magnetic field is applied in the $y\mathrm{\text{−}}z$ plane at an angle $\phi$.

Figure 2

Spatial dependence of the induced magnetization at various $T\in [0.05,0.55]T_{c0}$ in steps of $0.1T_{c0}$ for (a) parallel (${\theta }_J=0$, ${\overrightarrow{J}}_1\uparrow \uparrow {\overrightarrow{J}}_2$) and (b) antiparallel (${\theta }_J=\pi$, ${\overrightarrow{J}}_1\uparrow \downarrow {\overrightarrow{J}}_2$) configurations. The $S$ region is shaded. (c) Dependence of $\delta M_z$ on $T$ at the left interface (in $S$) for both configurations. (d) Spatial dependence of the normalized LDOS in $F_2$ at $ϵ=0$ and at $ϵ=\Delta (x=0)=1.16T_{c0}$ for the parallel configuration. Here, $d_{F1}=d_{F2}={\xi }_S$, $d_S=4{\xi }_S$, $J_1=J_2=20T_{c0}$, ${\sigma }_F={\sigma }_S$, $D_F=D_S$, and both interfaces have $r_b=0.1$ and $\mathcal{T}=1$.

Figure 3

Spatial dependence (a) of the singlet (${\Phi }_s$) and triplet (${\overrightarrow{\Phi }}_t$) pairing amplitudes, and (c) of the induced magnetization $\delta \overrightarrow{M}$ for perpendicular configuration (${\overrightarrow{J}}_1\perp {\overrightarrow{J}}_2$, ${\theta }_J=\pi /2$). Here, $d_{F1}=8{\xi }_S$, $d_{F2}=d_S=4{\xi }_S$, and $T=0.1T_{c0}$. The other parameters are as in Fig. 2. The $S$ region is shaded. (a) The singlet ${\Phi }_s(x)$ (dotted line) leaks into the $F$ regions and oscillates and decays on a short length scale. The triplets ${\Phi }_{\mathrm{ty}}(x)$ (full line) and ${\Phi }_{\mathrm{tz}}(x)$ (dashed line) are short range or long range depending on their respective projections on ${\overrightarrow{J}}_1$ or ${\overrightarrow{J}}_2$. Inset: spatial dependence of $\Delta (x)$ in the $S$ region. (c) There is an asymmetry in the oscillation period between $\delta M_y$ (full line) and $\delta M_z$ (dashed line) in both $F$ regions. (b) and (d) the angles ${\theta }_{\Phi }$ and ${\theta }_{\delta M}$ relative to the $z$ axis that quantify the directions of ${\overrightarrow{\Phi }}_t$ and $\delta \overrightarrow{M}$ in the $y\mathrm{\text{−}}z$ plane.

Figure 4

(a) Change in magnetic moment $\delta \overrightarrow{m}$ of a trilayer as function of the layer thickness $d_{F1}$, for $d_S=4{\xi }_S$, $d_{F2}=0.1{\xi }_S$, $T=0.1T_c$, and $J_1=J_2=20T_c$. The other parameters are as in Fig. 2. (b) The temperature dependence of $\delta \overrightarrow{m}$ for parameters as in (a), but with $d_{F1}=0.2{\xi }_S$. The vertical lines in (a) are referred to in Fig. 5.

Figure 5

(a)–(c) The change in the torque relative to that in the normal state for parallel (${\theta }_J=0$), antiparallel (${\theta }_J=\pi$), and perpendicular (${\theta }_J=\pi /2$) orientations. The parameters are the same as in Fig. 4 with the three numbered curves corresponding to the three layer thicknesses $d_{F1}=\{0.01,0.035,0.21\}{\xi }_S$ (indicated by dashed vertical lines in Fig. 4). In (d) we compare the total torque in the normal state (dashed line) with the one in the superconducting state [solid line from curve 2 in (c)] within the itinerant ferromagnet model.