In a granular Nm, modeled by a Josephson-coupled lattice, virtual tunneling of quasiparticles results in a critical value Rg for the normal-state sheet resistance. R6 is a universal upper bound for zero-temperature superconductivity and is computed in descending powers of s, the number of grain neighbors. Mean-Seld theory for a square lattice gives the leading term RS 5.7 kO. For the fractional correction from the next term in the expansion, the Bethe approximation gives -3/5zz, or -4% for z 4.Ever since the discovery of quantum mechanics, its applicability to macroscopic systems has been a subject of great interest. The recently reported universal criterion' for the onset of superconductivity in granular Slms, we assert, serves to confirm the validity of quantum mechanics in describing the collective behavior of the grains, each of which contains on the order of 10 -10s electrons. In this Rapid Communication we put forward a simple quantum-mechanical treatment of the collective eH'ect of the Josephson coupling between neighboring grains in a granular Slm. Our computation predicts a threshold resistance per square of approximately 5.7 ko, above which the Slm will not be superconducting.In view of the various approximations on which this prediction is based, we think that it is in satisfactory accord with the reported' empirical critical resistance of Ro 6.5 kQ.Our study is limited to the ground state of the granular film at a temperature T 0. The "macroscopic"' collective variables are pt, where p; is the phase of the Bardeen-Cooper-Schrieffer (BCS) ground-state wave function within the ith grain. The "velocities" are given by the usual quantum-mechanical expression where -e, 2trh h, and Vt are the electron charge, Planck's constant, and the electrostatic potential of the ith grain, respectively. Substituting Eq. (1) into the total electrostatic energy of the array of grains yields the "kinetic energy" 2 Qct Vt + 2 ghc;J(v; -V/) where C; is the capacitance of the ith grain to ground and d, c~/ is the additional mutual capacitance between neighboring grains. The double sum is taken over all neighboring pairs. The total energy of losephson coupling between neighboring pairs is U -QE Jt'cos(p;pj) . Denoting the charge on the ith grain by Q;, we 6nd that -c;v, +g~c;, (v;v, ) 'C; j;+gactj (j;j, ) '. e 2ed 8W 5 Ch Q',.the second line corresponding to the definition of capacitance as a linear response function. The sum is over the neighbors of the ith grain. The third and fourth lines follow from the substitution of Eqs. (1) and (2), respectively. Comparison of the Srst line of Eq. (4) and the fourth line reveals that it is nothing other than the Euler-Lagrange equation based on the Lagrangian L 8'-U.Passing from L to the corresponding Hamiltonian leads to a complicated nonlinear quantum-mechanical manybody problem. Some aspects of this problem, particularly its long-wavelength behavior, have been discussed by Chakravarty, Kivelson, Zimanyi, and Halperin. 2 The occurrence of Goldstone modes in the limiting case that we...
