First order $0\text{–}\pi$ phase transitions in superconductor/ferromagnet/superconductor trilayers

We study the thermodynamics of the diffusive SFS trilayer composed of thin superconductor (S) and ferromagnet (F) layers. On the basis the self-consistent solutions of nonlinear Usadel equations in the F and S layers we obtain the Ginzburg–Landau expansion and compute the condensation free energy and entropy of the 0 (even) and $\pi$ (odd) order parameter configurations. The first order $0\text{–}\pi$ transition as a function of temperature $T$ occurs, which is responsible for a jump of the averaged magnetic field penetration depth $\lambda \left(T\right)$ recently observed in experiments [N. Pompeo et al., Phys. Rev. B 90, 064510 (2014)]. The generalized Ginzburg-Landau functional was proposed to describe the SFS trilayer for arbitrary phase difference between the superconducting order parameters in the S layers. The temperature dependence of the SFS Josephson junction critical current demonstrates the strong anharmonicity of the corresponding current-phase relation in the vicinity of the $0\text{–}\pi$ transition. In an rf superconducting quantum interference device (SQUID), coexistence of stable and metastable 0 and $\pi$ states provides integer and half-integer fluxoid configurations.

Figure 1

The schematic behavior of the pair wave function $F\left(x\right)=F_{s,f}$ inside the SFS trilayers. The red solid line represents approximately the behavior of the pair wave function in an even mode (0 state). Due to symmetry the derivative ${\partial }_x{F}_f$ is zero at the center of the F layer. The pair wave function in the odd mode (blue dashed line) vanishes at the center of the F layer, and $F\left(x\right)$ has a $\pi$ shift in diametrically opposite points ($\pi$ state).

Figure 2

The typical dependence of the critical temperature $T_c^{0,\pi }$ on the thickness of the F layer $d_f$ for even mode (0 phase) (solid red line) and for odd mode ($\pi$ phase) (dashed blue line). Here we choose $d_s=2{\xi }_s$; ${\sigma }_f/{\sigma }_s=0.12$; ${\xi }_s/{\xi }_f=3 \left(\varepsilon =0.18\right)$; $h{\tau }_s=7$.

Figure 3

The typical dependence of the coefficients of the Ginzburg-Landau expansion $a^{0,\pi },b^{0,\pi }$ on the thickness of the F layer $d_f$: $a^0 \left(b^0\right)$: closed (open) red circles; $a^{\pi } \left(b^{\pi }\right)$: closed (open) blue triangles. Here we choose the parameters for Fig. 2.

Figure 4

The $\left(T,d_f\right)$ phase diagram for the SFS trilayers in the vicinity of crossing $\left(d_f\approx d_f^{*}\right)$ of the curves $T_c^0\left(d_f\right)$ and $T_c^{\pi }\left(d_f\right)$: (a) $d_f^{*}\simeq 1.95{\xi }_f$; (b) $d_f^{*}\simeq 5.31{\xi }_f$. At temperature $T_0$ the first-order transition between the 0 and $\pi$ states takes place. Here we choose the parameters for Fig. 2. The inset gives the dependence of the value of the superconducting gap jump ${\mathrm{\Delta }}_0^2\left(T_0\right)/{\mathrm{\Delta }}_{\pi }^2\left(T_0\right)$ (37) and the latent heat $Q$ (36) on the transition temperature $T_0$.

Figure 5

Schematic dependence of the gap ${\mathrm{\Delta }}_{0,\pi }^2$ (31) and the Ginzburg–Landau energy $F_{GL}\left({\mathrm{\Delta }}_{0,\pi }\right)$ (33) on the temperature $T$: $T_c^{\pi }<T_c^0$ and ${(a^2/b)}^{\pi }>{(a^2/b)}^0$. The inset gives the schematic temperature dependence of the penetration depth $\lambda \sim 1/{\mathrm{\Delta }}_{0,\pi }$.

Figure 6

(a) Current-phase relation of the SFS junction $I\left(\phi \right)$ (45) for several values of the temperature $T/T_{c0}=0.30,0.31,0.32$ in the vicinity of the $0\text{–}\pi$ transition for $d_f=1.94{\xi }_f$ ($T_c^0/T_{c0}\simeq 0.332,T_c^{\pi }/T_{c0}\simeq 0.327,T_c/T_{c0}\simeq 0.329$). The case $T=T_0\simeq 0.311T_{c0}$ is shown by the dash-dotted curve $[I_{\mathrm{\Delta }}=I_0{\mathrm{\Delta }}_0^2\left(T\right)$, where ${\mathrm{\Delta }}_0\left(T\right)=1.764T_{c}tanh(1.74\sqrt{T_c/T-1}$ is BCS superconducting gap for the temperature $T]$. (b) Dependence of the critical current $I_c\left(T\right)=\max \left|I(T,\phi )\right|$ (red solid line) and the amplitudes of harmonics $I_1,I_2$ (blue dashed lines) on temperature $T$. The inset gives the temperature dependence of a relative amplitude of the first harmonic $\kappa =I_1/I_2$. Here we choose the parameters of Fig. 3: $d_s=2{\xi }_s$; ${\sigma }_f/{\sigma }_s=0.12$; ${\xi }_s/{\xi }_f=3 (d_f^{*}\simeq 1.946{\xi }_f,\varepsilon =0.18)$; and $h=10T_{c0},h{\tau }_s=7$.

Figure 7

(a) Current-phase relation of the SFS junction $I\left(\phi \right)$ (45) for several values of the temperature $T/T_{c0}=0.28,0.29,0.30$ in the vicinity of the second $0\text{–}\pi$ transition for $d_f=5.3{\xi }_f$ ($T_c^0/T_{c0}\simeq 0.317,T_c^{\pi }/T_{c0}\simeq 0.318,T_c/T_{c0}\simeq 0.3175$). The case $T=T_0\simeq 0.29T_{c0}$ is shown by the dash-dotted curve $[I_{\mathrm{\Delta }}=I_0{\mathrm{\Delta }}_0^2\left(T\right)$, where ${\mathrm{\Delta }}_0\left(T\right)=1.764T_{c}tanh(1.74\sqrt{T_c/T-1}$ is BCS superconducting gap for the temperature $T]$. (b) Dependence of the critical current $I_c\left(T\right)=\max \left|I(T,\phi )\right|$ (red solid line) and the amplitudes of harmonics $I_1,I_2$ (blue dashed lines) on temperature $T$. Here we choose the parameters of Fig. 3: $d_s=2{\xi }_s$; ${\sigma }_f/{\sigma }_s=0.12$; ${\xi }_s/{\xi }_f=3 (d_f^{*}\simeq 5.31{\xi }_f,\varepsilon =0.18)$; and $h=10T_{c0},h{\tau }_s=7$.

Figure 8

The shape of the Josephson vortex ${\phi }_{0,\pi }\left(x\right)$ [Eqs. (48) and (49)] for two values of the relative amplitude of the first harmonic $\kappa =I_1/I_2$ in the vicinity of the $0\text{–}\pi$ transition: blue dashed line, $\left|\kappa \right|=0.05$; red solid line, $\left|\kappa \right|={10}^{-4}$.

Figure 9

Magnetic flux through the single-junction loop $\mathrm{\Phi }$ as a function of the external flux ${\mathrm{\Phi }}_e$ for the normalized inductance $L_I=500$ and several values of the temperature $T/T_{c0}=0.28,0.29,0.3,0.31,0.32$ in the vicinity of the $0\text{–}\pi$ transition for $d_f=1.94{\xi }_f$. The case for $T=T_0\simeq 0.31T_{c0}$ is shown by the dash-dotted curve. Here we choose the parameters of Fig. 6.