Crossover from global to local rule for the Coulomb blockade in small tunnel junctions

Abstract

We have investigated several single and double aluminum tunnel junctions in the nonsuperconducting state. Current-voltage (I-V) characteristics were measured up to voltages corresponding to several hundred times e/C, where C is the capacitance of each junction. The individual junctions had capacitances in the order of 0.5 fF and resistances of 50--150 k\ensuremath{\Omega}. The offset voltage, defined as ${\mathit{V}}_{\mathrm{off}}$=V-I/(dI/dV), is resistance independent for given junction capacitances, but depends on the bias voltage. For a single junction the offset voltage is very small for low voltages, but it approaches the local-rule value of e/2C at high voltages. For a double junction the global rule applies at low voltages and the offset voltage equals ${\mathit{V}}_{\mathrm{off}}$=e/${\mathit{C}}_{\mathrm{\ensuremath{\Sigma}}}$, where ${\mathit{C}}_{\mathrm{\ensuremath{\Sigma}}}$=2C+${\mathit{C}}_{0}$. At large voltages the local rule applies and each junction contributes with a voltage of e/2C to the offset voltage, so that ${\mathit{V}}_{\mathrm{off}}$=e/C. This paper shows that a careful analysis of the offset voltage has to be made to determine the values of the capacitances involved.