Tuning vortex fluctuations and the resistive transition in superconducting films with a thin overlayer

It is shown that the temperature of the resistive transition $T_r$ of a superconducting film can be increased by a thin superconducting or normal overlayer. For instance, deposition of a highly conductive thin overlayer onto a dirty superconducting film can give rise to an “antiproximity effect,” which manifests itself in an initial increase of $T_r\left(d_2\right)$ with the overlayer thickness $d_2$ followed by a decrease of $T_r\left(d_2\right)$ at larger $d_2$. Such a nonmonotonic thickness dependence of $T_r\left(d_2\right)$ results from the interplay of the increase of a net superfluid density mitigating phase fluctuations and the suppression of the critical temperature $T_c$ due to the conventional proximity effect. This behavior of $T_r\left(d_2\right)$ is obtained by solving the Usadel equations to calculate the temperature of the Berezinskii-Kosterlitz-Thouless transition, and the temperature of the resistive transition due to thermally activated hopping of single vortices in dirty bilayers. The theory incorporates relevant material parameters such as thicknesses and conductivities of the layers, the interface contact resistance between them, and the subgap quasiparticle states, which affect both phase fluctuations and the proximity effect suppression of $T_c$. The transition temperature $T_r$ can be optimized by tuning the overlayer parameters, which can significantly weaken vortex fluctuations and nearly restore the mean-field critical temperature. The calculated behavior of $T_r\left(d_2\right)$ may explain the nonmonotonic dependence of $T_r\left(d_2\right)$ observed on (Ag,Au,Mg,Zn)-coated Bi films, Ag-coated Ga and Pb films, or NbN and NbTiN films on AlN buffer layers. These results suggest that bilayers can be used as model systems for systematic investigations of optimization of fluctuations in superconductors.

Figure 1

The BKT transition temperature as a function of the resistance ratio $r=8R/\pi \zeta R_0$ at different values of the DOS broadening parameter $\mathrm{\Gamma }/2\pi T_{c1}$ calculated from Eq. (11).

Figure 2

A perpendicular vortex in a superconducting bilayer. The horizontal black line represents either a weakly coupled planar Josephson junction or an interface with a sheet contact resistance $R_B$. The bottom panel shows a complete core of a single-quantized vortex in a phase-locked bilayer (left) and a partial core of a fractional vortex (right).

Figure 3

Critical temperature $T_{c0}\left(d_2\right)$ of the N-S bilayer calculated from Eq. (31) for ${\lambda }_1=0.5$, ${\lambda }_2=0$, and different values of the contact resistance parameter $2\pi \beta T_{c1}=0.1,3,10$.

Figure 4

Critical temperature $T_c\left(d_2\right)$ of the N-S bilayer calculated from Eqs. (8), (9), (28), and (29) for ${\lambda }_1=0.5$, ${\lambda }_2=0$, and different values of ${\gamma }_1=\mathrm{\Gamma }/2\pi T_{c1}$.

Figure 5

BKT transition temperature $T_b\left(d_2\right)$ calculated from Eqs. (48) and (49) for different film resistances, $r=8R/\pi \zeta R_0$, ${\lambda }_1=0.7$, ${\lambda }_2=0.2$, ${\mathrm{\Omega }}_2=2{\mathrm{\Omega }}_1$, and (a) $D_2=0.5D_1$ and (b) $D_2=50D_1$. The dashed line shows the proximity-effect-limited $T_{c0}\left(d_2\right)$ in the absence of the BKT fluctuations.

Figure 6

BKT transition temperature $T_b\left(d_2\right)$ calculated from Eqs. (7), (27) and (47) at $D_2=50D_1$, ${\lambda }_1=0.7$, ${\lambda }_2=0.2$, and ${\mathrm{\Omega }}_2=2{\mathrm{\Omega }}_1$ for different values of the DOS broadening parameter $\mathrm{\Gamma }/2\pi T_{c0}$ and the resistance ratios $r=0,8$ (a) and $r=2$ (b). The dashed line shows the proximity-effect-limited $T_{c0}$ of the bilayer in the absence of the BKT fluctuations.

Figure 7

Temperature of the resistive transition in the NS bilayer as a function of the overlayer thickness calculated from Eqs. (55) at the resistance criterion ${\mathcal{R}}_v=0.1\mathcal{R}$, ${\lambda }_1=0.5$, $D_2=100D_1$, $w=10{\xi }_1$, and different values of $r=8R/\pi \zeta R_0$: 0.5, 1, and 2. The dashed line shows the mean-field $T_c\left(d_2\right)$.

Figure 8

Top: Thermally activated hopping of the vortex along a chain of pinning centers shown as blue regions. Bottom: Sketch of the local energy of the vortex $U\left(x\right)$. The dashed line shows $U\left(x\right)$ in a uniform bridge calculated from Eq. (50) at $w=10\tilde{\xi }$. The solid line shows $U\left(x\right)$ given by Eq. (50) plus the pinning potential modeled by three Lorentzian wells, $U_p\left(x\right)=-{\sum }_i{U}_i{\xi }^2/[{(x-x_i)}^2+{\xi }^2]$ with $U_i=(0.3,0.6,0.4){\epsilon }_0$ at $x_i=(0.2,0.5,0,7)w$. The London core singularities at $x=0$ and $x=w$ were regularized to provide zero vortex energy at the edges, $U\left(0\right)=U\left(w\right)=0$.

Figure 9

The surface plot of $\chi (x,y)$ calculated from Eq. (C6) with the branch cut at $x=0$ and $-\infty <y<0$ for a vortex at $u=0.3w$.