  The interplay of dynamical nuclear polarization (DNP) and leakage current
through a double quantum dot in the spin-blockade regime is analyzed. A finite
DNP is built up due to a competition between hyperfine (HF) spin-flip
transitions and another inelastic escape mechanism from the triplets, which
block transport. We focus on the temperature dependence of the DNP for zero
energy-detuning (i.e. equal electrostatic energy of one electron in each dot
and a singlet in the right dot). Our main result is the existence of a
transition temperature, below which the DNP is bistable, so a hysteretic
leakage current versus external magnetic field B appears. This is studied in
two cases: (i) Close to the crossing of the three triplet energy levels near
B=0, where spin-blockade is lifted due to the inhomogeneity of the effective
magnetic field from the nuclei. (ii) At higher B-fields, where the two
spin-polarized triplets simultaneously cross two different singlet energy
levels. We develop simplified models leading to different transition
temperatures T_TT and T_ST for the crossing of the triplet levels and the
singlet-triplet level crossings, respectively. We find T_TT analytically to be
given solely by the HF couplings, whereas T_ST depends on various parameters
and T_ST>T_TT. The key idea behind the existence of the transition temperatures
at zero energy-detuning is the suppression of energy absorption compared to
emission in the inelastic HF transitions. Finally, by comparing the rate
equation results with Monte Carlo simulations, we discuss the importance of
having both HF interaction and another escape mechanism from the triplets to
induce a finite DNP.
