Field-induced superconducting phase in superconductor–normal metal and superconductor-superconductor bilayers

We study the proximity effect in a superconductor (S)–normal metal (N) bilayer system under in-plane magnetic field and demonstrate that a compensation between the Zeeman effect and the energy splitting between bonding and antibonding levels may lead to a magnetic-field-induced superconducting phase well above the standard paramagnetic limit. It occurs that the nonuniform Fulde-Ferrell-Larkin-Ovchinikov superconducting state also exists in the field-induced phase. The presence of the impurity scattering shrinks the region of field-induced superconductivity existence in S-N and S-S bilayers.

Figure 1

Mechanism of compensation of the Zeeman effect by the degeneracy lifting between the bonding and antibonding states.

Figure 2

A superconducting (S)–normal metal (N) bilayer with in-plane magnetic field $H$.

Figure 3

Graph of $T_c/T_{c0}$ as a function of $t/T_{c0}$ (solid line). For $t\gg T_{c0}$, the critical temperature of the superconductor decreases to zero.

Figure 4

$(h/T_{c0},t/T_{c0})$ diagram for the S-N bilayer in the clean limit $(\tau \to \infty )$ at $T=0$ K (solid line). The uniform superconducting state is presented in gray. The line $h={\Delta }_0/2$ presents the critical magnetic field for a second-order superconducting phase transition for a single superconducting layer. The line $h={\Delta }_0$ corresponds the FFLO paramagnetic limit for a single superconducting layer. The line $h={\Delta }_0/\sqrt{2}$ represents the first-order paramagnetic limit for a single superconducting layer.

Figure 5

$(h/T_{c0},t/T_{c0})$ diagram for the S-N bilayer in the clean limit $(\tau \to \infty )$ (solid line). The uniform superconducting state is presented in the gray region. The nonuniform superconducting (FFLO) phase in the S-N bilayer is presented in the dotted region. The line $h={\Delta }_0/2$ presents the critical magnetic field for a second order superconducting phase transition for a single superconducting layer. The line $h={\Delta }_0$ represents the FFLO paramagnetic limit for a single superconducting layer. The line $h={\Delta }_0/\sqrt{2}$ represents the first-order paramagnetic limit for a single superconducting layer.

Figure 6

$(h/T_{c0},T_c/T_{c0})$ phase transition diagram calculated for $t=1.35T_{c0}$ with the second-order transition line (solid line) and FFLO state to normal state transition line (dotted line). We see below $T_c\simeq 0.2T_{c0}$ that the transition line is deformed. The compensation between the Zeeman effect and the bonding and antibonding states becomes relevant at low temperature.

Figure 7

$(h/T_{c0},T_c/T_{c0})$ phase transition diagram calculated for $t=2T_{c0}$ with the second-order transition line (solid line) and FFLO state to normal state transition line (dotted line). The inset presents a zoom of the superconducting reentrance phase around $h\simeq t\simeq 2T_{c0}$.

Figure 8

$(h/T_{c0},T_c/T_{c0})$ phase transition diagram calculated for $t=3T_{c0}$ with the second-order transition line (solid line) and FFLO state to normal state transition line (dotted line). The inset presents a zoom of the superconducting reentrance phase around $h\simeq t\simeq 3T_{c0}$.

Figure 9

Graph of $T_c/T_{c0}$ as a function of $t/T_{c0}$. The clean case $(1/2\tau =0)$ is presented by the solid line. The impurities are plotted with $(1/2\tau )/T_{c0}=0.01,0.05,0.097,1$ in dotted, dashed, dashed-dotted, and dashed-dotted-dotted lines, respectively. We see that the impurities enhance the superconducting transition temperature for weak interlayer coupling. On the other hand superconducting critical temperature decreases quickly in the presence of impurities at strong interlayer coupling.

Figure 10

$(h/T_{c0},t/T_{c0})$ diagram for the S-N bilayer in the clean case $(\tau \to \infty )$ (solid line) and with $(1/2\tau )/T_{c0}=0.05,0.097,1$ in dashed, dotted, dashed-dotted lines, respectively.

Figure 11

$(h/T_{c0},T_c/T_{c0})$phase transition diagram for the S-N bilayer calculated for $t=2T_{c0}$ with the second-order transition line in the clean case $[(1/2\tau )/T_{c0}=0]$ (solid line) and with $(1/2\tau )/T_{c0}=0.01,0.05,0.097,1$ in dotted, dashed, dashed-dotted, dashed-dotted-dotted lines, respectively. The inset presents a zoom of the superconducting reentrance phase around $h\simeq t\simeq 2T_{c0}$.

Figure 12

$(h/T_{c0},t/T_{c0})$ diagram for the S-S bilayer in the clean case $[(1/2\tau )/T_{c0}=0]$ (solid line) and with $(1/2\tau )/T_{c0}=0.15,0.194,0.5,1$ in dashed, dotted, dashed-dotted, and dashed-dotted-dotted lines, respectively. The lines $h={\Delta }_0/2$, $h={\Delta }_0/\sqrt{2}$, and $h={\Delta }_0$ present respectively the critical magnetic field for a second-order superconducting phase transition, the critical magnetic field for a first-order phase transition, and the FFLO paramagnetic limit for a single S layer. The close-dashed line is the FFLO paramagnetic limit in the S-S bilayer in the clean limit.

Figure 13

$(h/T_{c0},T_c/T_{c0})$ phase transition diagram for the S-S bilayer calculated for $t=2T_{c0}$ with the second-order transition line in the clean case $[(1/2\tau )/T_{c0}=0]$ (solid line) and with $(1/2\tau )/T_{c0}=0.01,0.05,0.1,0.125,0.15,0.175$ in dotted, dashed, dashed-dotted, dashed-dotted-dotted, close-dashed, close-dotted lines, respectively. The inset presents a zoom of the superconducting reentrance phase around $h\simeq t\simeq 2T_{c0}$.