A spatially indirect exciton in vertically coupled quantum dots:1/Q-expansion

A spatially indirect exciton in vertically coupled quantum dots is considered with the use of 1/Q-expansion, where Q is the dimensionless quantum parameter determined by the ratio of characteristic Coulomb energy of electron-hole interaction to the energy of one-particle transition in a confining potential. Analytical expressions for the energies and the wave functions are derived. They are asymptotically exact when Q much greater than I and the parameter of separation of quantum dots d is much larger than the size of a direct exciton a*(B). It is shown, however, that even the first four terms in the 1/Q-expansion provide one percent accuracy for the energies of relative motion in the ground and the first excited states of the exciton for Q greater than or similar to 2 and d greater than or similar to a*(B). It is also shown that the Pade summation of the first four terms in the 1/Q-expansion provides accuracy no worse than 5% even for Q similar to 0.2. We use the perturbation theory with respect to Coulomb interaction to calculate the energies for smaller values of Q and thus obtain analytical expressions for them in the whole range of variation of the parameter Q.