Spin-orbit interaction (SOI) in low-dimensional systems results in the fascinating property of spin-momentum locking. In a Rashba system the inversion symmetry normal to the plane of a two-dimensional (2D) electron gas is broken, generating a Fermi surface spin texture reminiscent of spin vortices of different radii which can be exploited in spin-based devices. Crucial for any application is the possibility to tune the momentum splitting through an external parameter. Here we show that in Pb quantum well states (QWS) the Rashba splitting depends on the Si substrate doping. Our results imply a doping dependence of the Schottky barrier which shifts the Si valence band relative to the QWS. A similar shift can be achieved by an external gate voltage or ultra-short laser pulses, opening up the possibility of terahertz spintronics.