Triplet-singlet spin relaxation in a vertical electrostatic quantum dot containing two interacting electrons is studied with a realistic model that includes the effects of finite thickness and screening. The spin relaxation mechanism is taken to be spin mixing via the part of the Dresselhaus Hamiltonian that is linear in the lateral momentum. The electron-electron interaction enhances relaxation and finite thickness suppresses it. The relaxation rate varies extremely rapidly with well width, w, approximately like 1/w^18 in zero magnetic field and 1/w^10 in a nonzero magnetic field. In addition, the rate oscillates with w and this leads to further suppression of relaxation. If w is sufficiently large the linear spin mixing mechanism is suppressed to the extent that it is no longer dominant
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