  We study spin relaxation in a two-electron quantum dot in the vicinity of the
singlet-triplet crossing. The spin relaxation occurs due to a combined effect
of the spin-orbit, Zeeman, and electron-phonon interactions. The
singlet-triplet relaxation rates exhibit strong variations as a function of the
singlet-triplet splitting. We show that the Coulomb interaction between the
electrons has two competing effects on the singlet-triplet spin relaxation. One
effect is to enhance the relative strength of spin-orbit coupling in the
quantum dot, resulting in larger spin-orbit splittings and thus in a stronger
coupling of spin to charge. The other effect is to make the charge density
profiles of the singlet and triplet look similar to each other, thus
diminishing the ability of charge environments to discriminate between singlet
and triplet states. We thus find essentially different channels of
singlet-triplet relaxation for the case of strong and weak Coulomb interaction.
Finally, for the linear in momentum Dresselhaus and Rashba spin-orbit
interactions, we calculate the singlet-triplet relaxation rates to leading
order in the spin-orbit interaction, and find that they are proportional to the
second power of the Zeeman energy, in agreement with recent experiments on
triplet-to-singlet relaxation in quantum dots.
