  We formulate the effective Hamiltonian of Rashba spin-orbit coupling (RSOC)
in $\mathrm{LaAlO_3/SrTiO_3}$ (LAO/STO) heterostructures. We derive analytical
expressions of properties, e.g., Rashba parameter, effective mass, band edge
energy and orbital occupancy, as functions of material and tunable
heterostructure parameters. While linear RSOC is dominant around the
$\Gamma$-point, cubic RSOC becomes significant at the higher-energy
anti-crossing region. We find that linear RSOC stems from the structural
inversion asymmetry (SIA), while the cubic term is induced by both SIA and bulk
asymmetry. Furthermore, the SOC strength shows a striking dependence on the
tunable heterostructure parameters such as STO thickness and the interfacial
electric field which is ascribed to the quantum confinement effect near the
LAO/STO interface. The calculated values of the linear and cubic RSOC are in
agreement with previous experimental results.
