Probing details of spin--orbit coupling through Pauli spin blockade

Spin--orbit interaction (SOI) plays a fundamental role in many low-dimensional semiconductor and hybrid quantum devices. In the rapidly evolving field of semiconductor spin qubits, SOI is an essential ingredient that can allow for ultrafast qubit control. The exact manifestation of SOI in a given device is, however, often both hard to predict theoretically and probe experimentally. Here, we develop a detailed theoretical connection between the leakage current through a double quantum dot in Pauli spin blockade and the underlying SOI in the system. We present a general analytic expression for the leakage current, which allows to connect experimentally observable features to both the magnitude and orientation of the effective spin--orbit field acting on the moving carriers. Motivated by the large recent interest in hole-based quantum devices, we further zoom in on the case of Pauli blockade of hole spins, assuming a strong transverse confinement potential. In this limit we also find an analytic expression for the current at low external magnetic field, that includes the effect of hyperfine coupling of the hole spins to randomly fluctuating nuclear spin baths. This result can be used to extract detailed information about both hyperfine and spin--orbit coupling parameters for hole spins in devices with a significant fraction of non-zero nuclear spins.