Corrections to the universal behavior of the Coulomb-blockade peak splitting for quantum dots separated by a finite barrier

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 small-$g$ correction that results from an interdot tunneling barrier that is not a $\delta$ function but, rather, has a finite height $V_0$ and a nonzero width $\xi$ and can be approximated as parabolic near its peak. The finiteness of the barrier leads to a small upward shift of the $f$-versus-$g$ curve for $g\ll 1$. The shift is a consequence of the fact that the tunneling matrix elements vary exponentially with the energies of the states connected. For a parabolic barrier, the energy scale for the variation is $ħ\omega$, where $\omega$, which is proportional to $\sqrt{V_0}/\xi$, is the harmonic oscillator frequency of the inverted parabolic well. In the limit $g\to 0$, the finite-width $f$-versus-$g$ curve behaves like $(U/ħ\omega )/|\ln g|$, where $U$ is the energy cost associated with moving electrons between the dots.