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 Tc*=Tc/zJ is negative and diverges at the MF value of the critical grain diameter αc, signaling a shift in αc away from the MF value. For {α}=zJ/U less than αc, the phase coherence is destroyed by quantum fluctuations. The 1/z correction to αc 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(φ₁-φ₂)〉, where grains 1 and 2 are nearest neighbors. In the normal state the short-range order is enhanced at the temperature T*{}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{}Tc, 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.