Scaling law for the Rashba-type spin splitting in quantum well films

We use laser-based spin- and angle-resolved photoemission spectroscopy (laser-SARPES) with high-resolution, and experimentally determine, for the first time, the Rashba-parameters of quantum well states (QWSs) systematically changing with the film thickness and the quantum numbers, through the observation of the Ag films grown on an Au(111) substrate. The data are very well reproduced by the theoretical calculations based on the density functional theory. Most importantly, we find a scaling law for the Rashba parameter ($\alpha_{\rm R}$) that the magnitude of $\alpha_{\rm R}$ is scaled by the charge density at the interface and the spin-orbit coupling ratio between the film and the substrate, and it is expressed by a single straight line regardless of the film thickness and the quantum numbers. The new finding not only is crucial to understand the Rashba effect in QWSs but also gives a foundation of film growth engineering to fine-tune the spin splitting in 2D heterostructure systems.