Coulomb blockade threshold in inhomogeneous one-dimensional arrays of tunnel junctions

A general expression is given for the change in free energy when a charge tunnels through a junction in a one-dimensional array of N metallic islands with arbitrary capacitances and arbitrary background charges. This is used to obtain expressions for the (average) threshold voltage of the Coulomb blockade for a few characteristic geometries. We find that including random background charges has a large effect on the N dependence of the threshold voltage: In an array with identical junction capacitances C and gate capacitances ${\mathrm{C}}_{\mathrm{g}}$, the threshold voltage, averaged over the background charge, is proportional to ${\mathrm{N}}^{\mathrm{a}}$, where a crosses over from to 1 when N becomes larger than 2.5$\sqrt{C/C_g}$.