We study the relation between the barrier conductance and the Coulomb blockade peak splitting for two electrostatically equivalent dots connected by tunneling channels with bandwidths much larger than the dot charging energies. We note that this problem is equivalent to a well-known single-dot problem and present solutions for the relation between peak splitting and barrier conductance in both the weak and strong coupling limits. Results are in good qualitative agreement with the experimental findings of F. R. Waugh et al.

We extend earlier results on the relation between the dimensionless tunneling channel conductance g and the fractional Coulomb blockade peak splitting f for two electrostatically equivalent dots connected by an arbitrary number N ch of tunneling channels with bandwidths W much larger than the two-dot differential charging energy U2. By calculating f through second order in g in the limit of weak coupling (g → 0), we illuminate the difference in behavior of the large-N ch and small-N ch regimes and make more plausible extrapolation to the strong-coupling (g → 1) limit. For the special case of N ch = 2 and strong coupling, we eliminate an apparent ultraviolet divergence and obtain the next leading term of an expansion in (1 − g). We show that the results we calculate are independent of such band structure details as the fraction of occupied fermionic single-particle states in the weak-coupling theory and the nature of the cut-off in the bosonized strong-coupling theory. The results agree with calculations for metallic junctions in the N ch → ∞ limit and improve the previous good agreement with recent two-channel experiments.

Building upon earlier work on the relation between the dimensionless interdot channel conductance g and the fractional Coulomb-blockade peak splitting f for two electrostatically equivalent dots, we calculate the leading correction that results from an interdot tunneling barrier that is not a delta-function but, rather, has a finite height V0 and a nonzero width ξ and can be approximated as parabolic near its peak. We develop a new treatment of the problem for g ≪ 1 that starts from the single-particle eigenstates for the full coupled-dot system. The finiteness of the barrier leads to a small upward shift of the f -versus-g curve for g ≪ 1. The shift is a consequence of the fact that the tunneling matrix elements vary exponentially with the energies of the states connected. Therefore, when g is small, it can pay to tunnel to intermediate states with single-particle energies above the barrier height V0. For a parabolic barrier, the energy scale for the variation in the tunneling matrix elements is ω, where ω, which is proportional to √ V0/ξ, is the harmonic oscillator frequency of the inverted parabolic well. The size of the correction to previous zero-width (ξ = 0) calculations depends strongly on the ratio between ω and the energy cost U associated with moving electrons between the dots. In the limit g → 0, the finite-width f -versus-g curve behaves like (U/ ω)/| ln g|. The correction to the zero-width behavior does not affect agreement with recent experiments in which 2πU/ ω ≃ 1 but may be important in future experiments.

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 Uρ 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 g → 1), 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 Uρ, 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 ωP 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 2πUρ/ ωP 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.