Effects of fluctuations and Coulomb interaction on the transition temperature of granular superconductors

We investigate the suppression of superconducting transition temperature in granular metallic systems due to (i) fluctuations of the order parameter (bosonic mechanism) and (ii) Coulomb repulsion (fermionic mechanism) assuming large tunneling conductance between the grains $g_T⪢1$. We find the correction to the superconducting transition temperature for $3d$ granular samples and films. We demonstrate that if the critical temperature $T_c>g_T\delta$, where $\delta$ is the mean level spacing in a single grain, the bosonic mechanism is the dominant mechanism of the superconductivity suppression, while for critical temperatures $T_c<g_T\delta$ the suppression of superconductivity is due to the fermionic mechanism.

Figure 1

Diagrams (a)–(c) describe the correction to the superconducting transition temperature due to Coulomb repulsion, and (d) describes correction to the transition temperature due to superconducting fluctuations. All diagrams are shown before averaging over the impurities. The solid lines denote the electron propagators, the dashed lines denote screened Coulomb interaction, and the wavy lines denote the propagator of superconducting fluctuations.

Figure 2

Diagram (a) defines the Cooperon propagator (shaded rectangle) [Eq. (8)] in terms of the single-grain Cooperon $C_0$ and (b) describes the renormalization of the BCS interaction vertex because of impurities. The superconducting propagator [Eq. (9)] is represented by the thick wavy line and is defined by (c), where the thin wavy line denotes the bare superconducting propagator. The solid lines denote the propagator of electrons, and the tunneling vertices are denoted as circles.

Figure 3

Diagrams describing correction to the transition temperature due to superconducting fluctuations (bosonic mechanism). The diagrams were obtained after averaging Fig. 1d over the disorder. The solid lines denote the propagator of electrons, the dotted lines denote the elastic interaction of electrons with impurities, and the wavy lines denote the propagator of superconducting fluctuations. The shaded rectangle and triangle denote the Cooperon [see Eq. (8)] and impurity vertex of granular metals, respectively. The tunneling vertices are denoted as circles.

Figure 4

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 dotted line denotes the elastic interaction of electrons with impurities. The tunneling vertices are denoted as circles.

Figure 5

Diagrams (a) define the diffusion propagator $D({\Omega }_n,\mathbf{q})$ [Eq. (17)] in terms of the single-grain diffusion propagator $D_0\left({\Omega }_n\right)$, (b) defines the renormalization of the Coulomb vertex due to impurity scattering, and (c) defines the dynamically screened Coulomb interaction [Eq. (18)]. The thin dashed lines represent the bare Coulomb interaction, whereas the thick lines denote the screened Coulomb interaction.

Figure 6

Diagrams obtained from Figs. 1a, 1b after disorder averaging. The solid lines denote the propagator of electrons, the dashed lines denote screened Coulomb interaction, and the dashed-dotted lines denote the elastic interaction of electrons with impurities. The shaded rectangle and triangle denote the Cooperon [see Eq. (8)] and impurity vertex of granular metals, respectively. The indices $i$ and $j$ stand for the grain numbers. The tunneling vertices are denoted as circles.

Figure 7

Diagrams describing vertex renormalization obtained from Fig. 1c after averaging over the disorder. All notations are the same as in Figs. 2, 4.

Figure 8

Diagrams (a) describe “bosonic” Hikami box [Eq. (A1)]. (b) Using Hikami box the sum of Figs. 3a, 3b, 3c can be conveniently represented as a single diagram. All notations are the same as in Fig. 3.

Figure 9

Diagrams (a) describe “fermionic” Hikami box. (b) Using Hikami box the sum of Figs. 6g, 6h, 6i can be conveniently represented as a single diagram. All notations are the same as in Fig. 6.