Singlet-triplet splitting in double quantum dots due to spin-orbit and hyperfine interactions

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 an 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.