  We examine electronic transport in a spin-blockaded double quantum dot. We
show that by tuning the strength of the spin-orbit interaction the current
flowing through the double dot exhibits a dip at zero magnetic field or a peak
at a magnetic field for which the two-electron energy levels anticross. This
behaviour is due to the dependence of the singlet-triplet mixing on the field
and spin-orbit amplitude. We derive approximate expressions for the current as
a function of the amplitudes of the states involved in the transport. We also
consider an alternative model that takes into account a finite number of
nuclear spins and study the resulting coupled dynamics between electron and
nuclear spins. We show that if the spin ensemble is in a thermal state there
are regular oscillations in the transient current followed by quasi-chaotic
revivals akin to those seen in a thermal Jaynes-Cummings model.
