Double quantum dot systems in the spin blockade regime exhibit leakage currents that have been attributed to the hyperfine interaction. We model weakly coupled double-dot transport using a rate equation approach which accounts for hyperfine flip-flop transitions. The rate equations allow us to self-consistently obtain the time evolution for electronic charge occupations and for the nuclei polarizations in each dot. We analyze the current in the spin blockade region as a function of magnetic field and observe hysteretic behavior for fields corresponding to the crossing between triplet and singlet states.
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