  We study the singlet-triplet relaxation due to the spin-orbit coupling
assisted by the electron-phonon scattering in two-electron SiGe/Si/SiGe double
quantum dots in the presence of an external magnetic field in either Faraday or
Voigt configuration. By explicitly including the electron-electron Coulomb
interaction and the valley splitting induced by the interface scattering, we
employ the exact-diagonalization method to obtain the energy spectra and the
eigenstates. Then we calculate the relaxation rates with the Fermi golden rule.
We find that the transition rates can be effectively tuned by varying the
external magnetic field and the interdot distance. Especially in the vicinity
of the anticrossing point, the transition rates show intriguing features. We
also investigate the electric-field dependence of the transition rates, and
find that the transition rates are almost independent of the electric field.
This is of great importance in the spin manipulation since the lifetime remains
almost the same during the change of the qubit configuration from $(1,1)$ to
$(2,0)$ by the electric field.
