  We theoretically study the nuclear spin dynamics driven by electron transport
and hyperfine interaction in an electrically-defined double quantum dot (DQD)
in the Pauli-blockade regime. We derive a master-equation-based framework and
show that the coupled electron-nuclear system displays an instability towards
the buildup of large nuclear spin polarization gradients in the two quantum
dots. In the presence of such inhomogeneous magnetic fields, a quantum
interference effect in the collective hyperfine coupling results in sizable
nuclear spin entanglement between the two quantum dots in the steady state of
the evolution. We investigate this effect using analytical and numerical
techniques, and demonstrate its robustness under various types of
imperfections.
