Fluctuation spectroscopy in granular superconductors with application to boron-doped nanocrystalline diamond

We perform a detailed calculation of the various contributions to the fluctuation conductivity of a granular metal close to its superconducting transition. We find three distinct regions of power law behavior in reduced temperature, $\eta =(T-T_c)/T_c$, with crossovers at $\mathrm{\Gamma }/T_c$ and $E_{\text{Th}}/T_c$, where $\mathrm{\Gamma }$ is the electron tunneling rate, and $E_{\text{Th}}$ is the Thouless energy of a grain. The calculation includes both intergrain and intragrain degrees of freedom. This complete theory of the fluctuation region in granular superconductors is then compared to experimental results from boron-doped nanocrystalline diamond, using the assumption of a constant phase breaking rate ${\tau }_{\varphi }^{-1}$. We find a semiquantitative agreement between the theoretical and experimental results only in the case of large phase breaking. We argue that there may be a phase breaking mechanism in granular metals worthy of further experimental and theoretical investigation.

Figure 1

Dyson equation for the granular electron Green's function.

Figure 2

Dyson equation for the granular diffuson.

Figure 3

Leading order contributions to the electrical conductivity due to superconducting fluctuations in granular systems. (a) and (c) are density of states (DOS) diagrams; (b) and (d) are Maki-Thompson (MT) diagrams; (e) is the Aslamazov-Larkin (AL) diagram. Diagrams (c) and (d) cancel exactly in the DC conductivity limit and so are not considered.

Figure 4

Fluctuation conductivity contributions with no phase breaking. The black vertical lines show where the power law crossovers were seen in Klemencic et al.'s experiment [13]. The dashed lines act as guides for power law behavior. (a) Total fluctuation conductivity; (b) DOS plus regular MT contribution; (c) AL contribution; (d) anomalous MT contribution.

Figure 5

Experimental measurements of the fluctuation conductance from the work of Klemencic et al. [13] with power law fittings in each region. These measurements were taken on a 329 nm thick BNCD film. The shaded areas show the approximate regions for the two crossovers.

Figure 6

Fluctuation corrections with various constant phase breaking rates, ${\tau }_{\varphi }^{-1}$. The dashed black line acts as a guide for ${\eta }^{-3}$ behavior. (a) shows the total fluctuation conductivity using parameters based upon Klemencic et al.'s experiment [13]. (b) shows the total fluctuation conductivity using a custom set of parameters.