  We study the triplet-singlet relaxation in two-electron semiconductor quantum
dots. Both single dots and vertically coupled double dots are discussed. In our
work, the electron-electron Coulomb interaction, which plays an important role
in the electronic structure, is included. The spin mixing is caused by
spin-orbit coupling which is the key to the triplet-singlet relaxation. We show
that the selection rule widely used in the literature is incorrect unless near
the crossing/anticrossing point in single quantum dots. The triplet/singlet
relaxation in double quantum dots can be markedly changed by varying barrier
height, inter-dot distance, external magnetic field and dot size.
