  We analyze the low-energy spectrum of a two-electron double quantum dot under
a potential bias in the presence of an external magnetic field. We focus on the
regime of spin blockade, taking into account the spin orbit interaction and
hyperfine coupling of electron and nuclear spins. Starting from a model for two
interacting electrons in a double dot, we derive a perturbative, effective
two-level Hamiltonian in the vicinity of an avoided crossing between singlet
and triplet levels, which are coupled by the spin-orbit and hyperfine
interactions. We evaluate the level splitting at the anticrossing, and show
that it depends on a variety of parameters including the spin orbit coupling
strength, the orientation of the external magnetic field relative to an
internal spin-orbit axis, the potential detuning of the dots, and the
difference between hyperfine fields in the two dots. We provide a formula for
the splitting in terms of the spin orbit length, the hyperfine fields in the
two dots, and the double dot parameters such as tunnel coupling and Coulomb
energy. This formula should prove useful for extracting spin orbit parameters
from transport or charge sensing experiments in such systems. We identify a
parameter regime where the spin orbit and hyperfine terms can become of
comparable strength, and discuss how this regime might be reached.
