PHYSICAL REVIEW B VOLUME 43, NUMBER 7 1 MARCH 1991 Charge and vortex dynamics in arrays of tunnel junctions Rosario Fazio and Gerd Schon Department of Applied Physics, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands (Received 7 August 1990) Arrays of tunnel junctions can be fabricated with small nearest-neighbor capacitances between the islands and with even smaller self-capacitances of the islands. It has been suggested by Mooij et a1. that electric charges on the islands of such an array interact logarithmically over large dis- tances, and as a consequence, a Kosterlitz-Thouless-Berezinskii transition can occur where charge- anticharge pairs dissociate. Furthermore, in superconducting junction arrays the more familiar vortex-unbinding transition can occur. %'e investigate the effect of single-electron and Cooper-pair tunneling on these transitions and discuss the competition between the charge and the vortex un- binding. I. INTRODUCTION In recent years much attention has been devoted to the study of tunnel-junction arrays. ' Even if at low tempera- tures the individual islands of the array are supercon- ducting, it may require still lower temperatures for global phase coherence to establish across the whole array. In classical, two-dimensional (2D) arrays this transition is of the Kosterlitz-Thouless-Berezinskii (KTB) type. It separates a superconducting low-temperature phase, where vortices and antivortices are bound in pairs, from a resistive high-temperature phase with free vortices. The transition temperature T, is of the order of the Josephson coupling energy EJ of the junctions. When the dimensions of the islands and the capaci- tances involved are small, the charging energy with characteristic scale Ec is large, and quantum fiuctuations of the phase gain importance. They suppress the vortex- unbinding transition temperature; and for small values of EJ/E~ ~ 1 even at T=O only a disordered phase ex- ists. The tunneling of quasiparticles or Ohmic shunts in- troduces dissipation. The most striking consequence, observed in experiments involving granular films and in regular 2D networks, is the existence of a critical value of the normal state conductance ct, : — h /(4e R„) = l above which the superconductivity is recovered even for small values of EJ /Ec. Recently Mooij et al. pointed out a different collective efFect involving the charge on the islands of a 2D array. If the junctions are of high quality the charges of the is- lands can change only due to tunneling, and the total charge on each island is an integer multiple of the ele- mentary charge e. Furthermore, it is possible to fabricate junction arrays where the capacitance C between the is- lands, i.e. , the one associated with the junctions, is significantly larger than the capacitance Co of the islands to the ground. In this case the electrostatic interaction of two charges +e separated by r is characterized by the po- tential energy U(r )=(2E&/m)lnr, up to distances of the order A=QC/Co. The unit of the length is the lattice spacing, the charging-energy scale is defined as Ec — — e /2C. In the ideal situation where Co =0 the loga- rithmic interaction extends to infinity. The system is thus a physical realization of a two-dimensional Coulomb gas. This implies that a KTB transition can occur at a tem- perature T, of the order Ec where charge-anticharge pairs dissociate. This charge-unbinding KTB transition differs from the mentioned vortex-unbinding KTB transi- tion: If the charges are bound in pairs, the array is insu- lating; if the charges are free, the array has a finite con- ductance. In contrast, the vortex-unbinding transition separates a superconducting from a resistive phase. In superconducting arrays the charge-unbinding transi- tion can involve either single electrons (charge e) or Cooper pairs (charge 2e). Since the charging energy differs by a factor of 4 in these two cases, we expect a cor- responding difference in the transition temperature. Furthermore, in superconducting arrays both the charge- and the vortex-unbinding transition can occur, depending on the ratio between the energies EJ and Ec. The experi- ments reported in Ref. 7 show a transition of normal junction arrays from insulating to conducting, consistent with the picture of the charge-KTB transition. Further- more, in superconducting arrays a similar transition, but at a higher temperature, is observed. Also, arrays with larger Ez/Ec show the vortex-unbinding transition be- tween a superconducting and a resistive state. It is clear that in a real experiment the charge-unbinding transition is washed out due to the finite range of the logarithmic interaction (for Co&0), but if Co/C is small enough a fairly sharp crossover remains observable. The finite size of the array has a similar effect. The junction array is equivalent to a classical 2D Coulomb gas as long as the single-electron and Cooper- pair tunneling is weak. The tunneling allows the relaxa- tion to thermal equilibrium and determines the response of the system. We will discuss the weak-tunneling limit in Sec. II. If the tunneling is strong, the quantum Auc- tuations associated with it further affect the charge dy- namics and the charge-unbinding transition. In Secs. II and IV we will discuss this inhuence for normal and su- perconducting arrays, extending our previous work. ' 43 5307 Qc1991 The American Physical Society