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We theoretically study the $(1,1)$ triplet to $(0,2)$ singlet relaxation ratein a lateral gate-defined double quantum dot tuned to the regime of Pauli spinblockade. We present a detailed derivation of the effective phonon density ofstates for this specific charge transition, keeping track of the contributionfrom piezoelectric as well as deformation potential electron-phonon coupling.We further investigate two different spin-mixing mechanisms which can couplethe triplet and singlet states: a magnetic field gradient over the double dot(relevant at low external magnetic field) and spin-orbit interaction (relevantat high field), and we also indicate how the two processes could interfere atintermediate magnetic field. Finally, we show how to combine all results andevaluate the relaxation rate for realistic system parameters.

Spin-flip phonon-mediated charge relaxation in double quantum dots