Two-electron spin relaxation in a double quantum dot*

Motivated by recent experiments done on GaAs gate-defined double quantum dots (DQD)\footnote{J. R. Petta, \textit{et al}. Science 309,2180(2005)}, we study the two-electron spin relaxation in DQD at various energy detuning $\varepsilon$ and magnetic fields. In particular, we focus on two limits of interdot energy detuning. For $\varepsilon>0$, we investigate the spin blockade by evaluating the electron spin relaxation rate from low energy (1,1) triplet states to the ground (0,2) singlet state. For $\varepsilon<0$, we calculate the electron spin relaxation time at the S(1,1)-$\mathtt{T}{}_{0}$ degeneracy to identify the leakage of an ST$_{0}$ qubit. Particularly, we explore the relaxation channel due to the hyperfine interaction and the electron-phonon interaction.