From orbital to Pauli-limited critical fields in granular aluminum films

The temperature dependence of the perpendicular upper critical field of superconducting granular aluminum films has been measured for samples approaching the metal-to-insulator transition. Analysis of the results shows a shift from an orbital to a Pauli-limited critical field, which we propose is made possible by electron mass renormalization. In that regime, the critical field transition becomes of the first order, as predicted by Fulde, Ferrel, Larkin, and Ovchinikov. The phase-coherence length ${\xi }_{\mathrm{phase}}$ and the superconducting gap $\mathrm{\Delta }$ obtained from the analysis are consistent with a Bardeen-Cooper-Schrieffer–Bose-Einstein condensate crossover region, which we propose is triggered by the Mott transition of the granular films.

Figure 1

Temperature dependence of the resistivity, normalized to its value at 4.2 K, showing the SC transition. The legend corresponds to the sample number.

Figure 2

Temperature dependence of the upper critical field for several of the studied samples. The legend corresponds to the sample number. The data were fitted to Eq. (2). The inset shows part of the data from the main figure, but with bars which mark the value of the field where the resistance has decreased by 90 and 10 percent relatively to the normal-state value.

Figure 3

The orbital critical field $H_{c2}^{\mathrm{orb}}$ vs resistivity (circles) as obtained from fitting the data to WHH theory along a linear fit to the low-resistivity data.

Figure 4

Resistivity magnetic field and temperature dependence of the samples 10 and 11, measured down to 50 mK. (a),(b) High-resistivity sample with ${\rho }_{4.2\text{K}}=3400\mu \mathrm{\Omega }\text{cm}$. (c),(d) Low-resistivity sample with ${\rho }_{4.2\text{K}}=452\mu \mathrm{\Omega }\text{cm}$. Note that in sharp contrast, the high-resistivity sample transition narrows dramatically as the temperature decreases. Note in (b) that in a narrow field range around 4.25 T, the resistivity goes through a minimum, at a temperature compatible with the occurrence of a maximum in $H_{c2}\left(T\right)$. This is where the transition changes from second to first order.

Figure 5

Sample 11 paraconductivity analysis. (a) $H_{c2}\left(T\right)$ as extracted from fitting the $\rho \left(H\right)$ data to Fulde and Maki theory [29], as described in the text. (b) Conductivity $\sigma$ vs temperature at different values of magnetic field. Circles are the data as extracted from the $\rho \left(H\right)$ curves and the thick lines are linear fits to the Fulde and Maki theory, $\sigma ={\sigma }_n+aT/\sqrt{H-H_{c2}\left(T\right)}$. ${\sigma }_n$ was chosen as the intercept at the highest field measured (5 T) and was fixed for all fits.

Figure 6

Magnetic field dependence of the resistivity, showing $\rho \left(H\right)$ isotherms of four samples. Note that for the high-resistivity samples 6 and 8, the transition becomes narrow as the temperature decreases.

Figure 7

$\rho \left(H\right)$ isotherms of sample 11, zoomed into the regime $H>H_{c2}$. The arrow marks the direction of decreasing temperature.