Electron-Electron Interaction in Linear Arrays of Small Tunnel Junctions

We have calculated the spatial distribution of the electrostatic potential created by an unbalanced charge q in one of the conducting electrodes of a long, uniform, linear array of small tunnel junctions. The distribution describes, in particular, the shape of a topological single-electron soliton in such an array. An analytical solution obtained for a circular cross section model is compared with results of geometrical modeling of a more realistic structure with square cross section. These solutions are very close to one another, and can be reasonably approximated by a simple phenomenological expression. In contrast to the previously accepted exponential approximation, the new result describes the crossover between the linear change of the potential near the center of the soliton to the unscreened Coulomb potential far from the center, with an unexpected “hump” near the crossover point. Typeset using REVTEX 1 Recent theoretical and experimental studies have resulted in considerable progress in understanding correlated single-electron transfer in ultra-small tunnel junctions (for reviews, see Refs. 1–3). These phenomena may be used as a background for a new generation of analog and digital devices. The most common component of single-electronic devices is a one-dimensional array of small tunnel junctions (Fig. 1(a)). Thus it is very important to gain a quantitative understanding of the Coulomb interaction potential, U(r), between single electrons in such an array. To our knowledge, all previous works on this topic (see, e.g., Ref. 4 and references therein) have used a simple model in which the complete matrix [C] of mutual capacitances between conducting “islands” of the array is truncated to tridiagonal form. In this form of the matrix, the only non-zero elements are (a) the diagonal elements Ci,i = Co, representing the stray capacitances of the islands, and (b) the nearest-neighbor elements Ci,i±1 = C, dominated by tunnel junction capacitances. Electron-electron interaction in the tridiagonal model is described by a simple exponential law: Ut(r) = Ut(0) exp(−m/mo), (1) where m = r/a is the distance between the two electrons, in units of the array period a (i.e., in number of islands). The parameters Ut(0) and mo depend on the C/Co ratio, and in the most important limit of Co ≪ C: Ut(0) = e 2 √ CCo , (2)