PHYSICAL REVIEW B VOLUME 10, NUMBER 11 1 DEC EMBER 1974 Supercontiucting phase transitions in granular systems G. Deutscher» Department of Physics and Astronomy, Tel-Aviv University, Ramat Aviv, Israel Y. Imry» Department of Physics and Astronomy, Tel-Aviv University, Ramat Aviv, Israel and Quantum Institute, University of California, Santa Barbara, California 93106 L. Gunther Department of Physics, Tufts University, Medford, Massachusetts 02155 (Received 20 June 1974) We consider a model for granular superconductors consisting of an array of small superconducting particles interacting by Josephson coupling through insulating barriers. %'e obtain systematically the various critical regions, critical temperature shifts, and crossover regions between zero- and three-dimensional behavior as functions of measurable sample parameters. The qualitative behavior of the system in the various regimes is analyzed and results for the specific heat and fluctuation conductivity in the Gaussian region above T, are obtained. The possibility of obtaining large critical regions is emphasized. The conditions for observing the phase-locking transition distinct from quasiordering within the grains are found. Theoretical predictions are compared with existing experimental results. I. INTRODUCTION The physics of small superconducting particles has recently attracted considerable attention. The behavior of isolated, so-called zero-dimensional (O-D), particles is quite well understood theoret- ically, at least insofar as their size is not in the microscopic limit. ' The predictions for the magnetic behavior have recently received a beauti- ful experimental confirmation. However, our un- derstanding of systems that are composed of small and weakly coupled particles is of a rather pre- liminary nature. The reason for the interest in such systems is that by varying both the particle size and the interparticle coupling one may span a very wide range of physically interesting situations. In particular, the crossover between 0-D and 3-D (or 2-D for thin films) criticai behaviors may be systematically studied. Recent theories' have emphasized the dependence of the critical behavior on the dimensionality of the system, and the im- portance of understanding the crossover between different dimensionalities. Furthermore, there is evidence that the systems under consideration can be realized experimentally with grain sizes and inter-grain couplings kept under control. One can therefore hope that theoretical predictions may be directly checked by experiment. The calculations that we present here are a. de- velopment of two preliminary notes' '" where the basic physical ideas and the relevant model were presented. Here we emphasize the similarity of our model, based on Landau-Ginzburg theory for the grains and Josephson coupling 4 among the grains, to modern general models for critical behavior. In fact, the granular sys- tems appear to be a physical discrete realiza- tion of the field-theoretical. models used by Wilson' and others, with widely varying ranges of the parameters. In this paper we do not em- phasize the critical behavior (e. g. the values of critical indices), we focus our attention on ques- tions like what are the magnitudes of the various regions and shifts of the critical temperature due to fluctuations, "' and what determines whether the system is 0-D or 3-D. These questions will be dealt with in Sec. II and III. It will turn out that, from the theoretical point of view, the dis- cussion presents a number of relevant cutoff prob- lems, our answers for which can be checked ex- perimentally. The most relevant result is per- haps that we compute the critical region as a func- tion of measurable quantities and show that one can easily make it large enough to be experimentally accessible for a superconducting system. We al- so discuss the possibility of two distinct transi- tions' and give well-defined criteria for its occur- rence. In Sec. IV we calculate the specific heat and the paraconductivity in the Gaussian regime and discuss the 3-D to 0-D crossover. Section V is devoted to a summary and to some concluding remarks. II. THE MODEL, CORRELATION LENGTHS, AND THE VARIOUS TEMPERATURE REGIONS Our model (see Fig. i) for the granular system has the following'3 free energy a. s a function of the order parameters of the grains, g, : 10 4598