Excess conductivity in high-$T_c$ superconducting thin films: Role of smooth doping disorder

While the time-dependent Ginzburg-Landau theory successfully describes a number of experimental results for homogeneous high-$T_c$ superconductors, the proper value of the characteristic depairing electric field was recently found unreasonably small for Bi-Sr-Ca-Cu-O thin films. We present numerical simulations which show that these features can be the consequence of smooth spatial inhomogeneity of the local critical temperature. Because of the interplay of percolation and superconducting transition, even a small disorder affects strongly the nonlinear excess conductivity in the transition region, causing considerable rescaling of the characteristic electric field.

Figure 1

Part of the square lattice modeling a smoothly inhomogeneous thin film. (a) At a given temperature $T$, cells are superconducting ($T_c^{\mathcal{L}}>T$, in gray), or normal ($T_c^{\mathcal{L}}<T$, in white). (b) Same part of the system discretized in nodes and bonds. The four bonds starting from a cell-center node (e.g., $C_0$) have the same conductivity (2). All the sites connected by a superconducting path (e.g., $A^{\prime },B^{\prime },C^{\prime }$), and all the sites inside the superconducting domain (edged by $A^{\prime },B^{\prime },C^{\prime }$) are at the same electric potential and form an iso-potential group.

Figure 2

Reduced excess conductivity vs temperature, for two $60\times 60$ systems. Left-hand column: numerical data (dots) for $s^{\mathcal{L}}=0.1$, and three values of the electric field, $E=0.001,0.004,0.016$. Continuous curves are fits with Eq. (4) and the values $({\overline{E}}_0,{\overline{T}}_c)=(0.272,0.983),(0.429,0.985),(0.674,0.992)$ respectively. The Schmid formula for the homogeneous case (${\overline{E}}_0=1$, ${\overline{T}}_c=1$, $E=0.001$) is the dotted line on the top left figure. Right-hand column: similar plots for $s^{\mathcal{L}}=0.3$. The values for $({\overline{E}}_0,{\overline{T}}_c)$ are respectively (0.089, 0.927), (0.197, 0.935), (0.398, 0.952). The top right figure exhibits nonmonotonic behavior due to successive fragmentations of superconducting domains when increasing $T$.

Figure 3

Power-law dependence of the characteristic depairing field ${\overline{E}}_0$ vs the in-plane electric field $E$ for three values of the standard deviation $s^{\mathcal{L}}$ of the local critical temperature $s^{\mathcal{L}}=0.1,0.2,0.3$ from top to bottom. The lines are the slopes 0.44, 0.45, and 0.46 respectively. $⟨T_c^{\mathcal{L}}⟩=1$, $E_0=T_c^{\mathcal{L}}$, and 27 independent systems of size $60\times 60$ were used for each set of data. Note the strong decrease of the magnitude of ${\overline{E}}_0$ when increasing $s^{\mathcal{L}}$. This is also seen when comparing the two examples of Fig. 2 for a same value of $E$.