Coulomb blockade of strongly coupled quantum dots studied via bosonization of a channel with a finite barrier

A pair of quantum dots, coupled to each other through a point contact, can exhibit Coulomb blockade effects that reflect the presence of an oscillatory term in the dots’ total energy whose value depends on whether the total number of electrons on the dots is even or odd. The effective energy associated with this even-odd alternation is reduced, relative to the bare Coulomb blockade energy U_ρ for uncoupled dots, by a factor (1-f) that decreases as the interdot coupling is increased. When the transmission coefficient for interdot electronic motion is independent of energy and is the same for all channels within the point contact (which are assumed uncoupled), the factor (1-f) takes on a universal value determined solely by the number of channels N_ch and the dimensionless conductance g of each individual channel. When an individual channel is fully opened (the limit g⃗1), the factor (1-f) goes to zero. When the interdot transmission coefficient varies over energy scales of the size of the bare Coulomb blockade energy U_ρ, there are corrections to this universal behavior. Here we consider a model in which the point contact is described by a single orbital channel containing a parabolic barrier potential, with ω_P being the harmonic oscillator frequency associated with the inverted parabolic well. We calculate the leading correction to the factor (1-f) for N_ch=1 (spin-split) and N_ch=2 (spin-degenerate) point contacts, in the limit where g is very close to 1 and the ratio 2πU_ρ/ħω_P is not much greater than 1. Calculating via a generalization of the bosonization technique previously applied in the case of a zero-thickness barrier, we find that for a given value of g, the value of (1-f) is increased relative to its value for a zero-thickness barrier, but the absolute value of the increase is small in the region where our calculations apply.