Charge accumulation on a Luttinger liquid

The average charge Q on a quantum wire, modeled as a single-channel Luttinger liquid (LL), connected to metallic leads and coupled to a gate is studied theoretically. We find that the behavior of the charge as the gate voltage $V_G$ varies depends strongly on experimentally adjustable parameters (length, contact transmission, temperature, $\dots ).\mathrm{}$ When the intrinsic backscattering at the contacts is weak (i.e., the conductance is close to ${2e}^2/h$ at high temperature), we predict that this behavior should be described by a universal function. For short such wires, the charge increases roughly linearly with $V_G,$ with small oscillations due to quantum interference between electrons scattered at the contacts. For longer wires at low temperature, Coulomb blockade behavior sets in, and the charge increases in steps. In both limits $\partial Q/\partial V_G,$ which should characterize the linear-response conductance, exhibits periodic peaks in $V_G.$ We show that due to Coulomb interactions the period in the former limit is twice that of the latter, and describe the evolution of the peaks through this crossover. The study can be generalized to multichannel LL’s, and may explain qualitatively the recent observation by Liang et al. [Phys. Rev. Lett. 88, 126801 (2002)] of a four-electron periodicity for electron addition in single-walled carbon nanotubes.