Suppression of Superconductivity in Granular Metals

We investigate the suppression of the superconducting transition temperature due to Coulomb repulsion in granular metallic systems at large tunneling conductance between the grains, $g_T\gg 1$. We find the correction to the superconducting transition temperature for 3D granular samples and films. We demonstrate that, depending on the parameters of superconducting grains, the corresponding granular samples can be divided into two groups: (i) the granular samples that belong to the first group may have only insulating or superconducting states at zero temperature depending on the bare intergranular tunneling conductance $g_T$, while (ii) the granular samples that belong to the second group in addition have an intermediate metallic phase where superconductivity is suppressed while the effects of the Coulomb blockade are not yet strong.

Figure 1

Diagrams describing the first order self-energy corrections, $\delta {\lambda }_1$, to the Cooperon channel coupling constant $\lambda$ of granular metals due to the electron-electron repulsion. The solid lines denote the propagator of electrons, the wavy lines describe screened Coulomb interaction, and the dashed lines describe the elastic interaction of electrons with impurities. The shaded rectangles and triangles denote the renormalized Cooperon [see Eq. (5)] and the impurity vertex of granular metals, respectively. The indices $i$ and $j$ stand for the number of the grains. The tunneling vertices are described by the circles. All diagrams should be taken with a factor of 2. The final result for the sum of the diagrams (a)—(f) is given by Eq. (7).

Figure 2

Diagrams describing the first order vertex corrections, $\delta {\lambda }_2$, to the Cooperon channel coupling constant $\lambda$ of granular metals due to the electron-electron repulsion. All notations are the same as in Fig. 1. The diagrams (b)–(d) should be taken with an additional factor of 2. After that the final result for the sum of the diagrams (a)–(d) is given by Eq. (10).

Figure 3

Diagrams describing self-energy corrections to the single electron propagator due to (a) elastic interaction of electrons with impurities and (b) electron hopping. The solid lines denote the bare propagator of electrons and the dashed line describes the elastic interaction of electrons with impurities. The tunneling vertices are described by the circles.