Correlation of phase fluctuations in granular superconductors

An expansion in 1/z is used to study the correlation of phase fluctuations in granular superconductors. To zeroth order in 1/z, the expectation value of any operator is given by mean-field (MF) theory, which neglects the coupling of phase fluctuations. The first-order correction involves the coupling of phase fluctuations over an infinite number of grains. The 1/z correction to the transition temperature $T_c^{\mathrm{*}}$=$T_c$/zJ is negative and diverges at the MF value of the critical grain diameter ${\alpha }_c$, signaling a shift in ${\alpha }_c$ away from the MF value. For α=zJ/U less than ${\alpha }_c$, the phase coherence is destroyed by quantum fluctuations. The 1/z correction to ${\alpha }_c$ is positive, so that the coupling of phase fluctuations increases the critical grain diameter. This expansion technique is used to calculate the short-range-order parameter 〈cos(${\phi }_1$-${\phi }_2$)〉, where grains 1 and 2 are nearest neighbors. In the normal state the short-range order is enhanced at the temperature $T^{\mathrm{*}}$≊1/α, when thermal fluctuations allow a Cooper pair to overcome the Coulomb energy barrier to tunneling between grains. This enhancement is directly related to the observed minimum in the resistivity of two-dimensional granular films. The resistivity minimum first appears when U≊$T_c$, which can be simply explained in this model. Using the results for the short-range-order parameter, the 1/z corrections to the free energy and the specific heat are calculated. The fluctuation specific heat also shows signs of an enhanced phase coherence at a temperature inversely related to the grain size.