  We analyzed the electronic transport through a double quantum dot in the spin
blockade regime. Experiments of current rectification by Pauli exclusion
principle in double quantum dots were discussed. The electron and nuclei spin
dynamics and their interplay due to the Hyperfine interaction were
self-consistently analyzed within the framework of rate equations. Our results
show that the current leakage experimentally observed in the spin-blockade
region, is due to spin-flip processes induced by Hyperfine interaction through
Overhauser effect. We show as well how a magnetic field applied parallel to the
current allows excited states to participate in the electronic current and
removes spin blockade. Our model includes also a self-consistent description of
inelastic transitions where the energy is exchanged through interactions with
acoustic phonons in the environment. It accounts for spontaneous emission of
phonons which results in additional features in the current characteristics. We
develop a microscopical model to treat the Hyperfine interaction in each dot.
Using this model we study the dynamical nuclear polarization as a function of
the applied voltage.
