Scaling law for 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 (${\alpha }_{\mathrm{R}}$) that the magnitude of ${\alpha }_{\mathrm{R}}$ 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.

Figure 1

(a) Experimental geometry of SARPES. (b) ARPES intensity map for 12 ML Ag/Au(111). (c) Spin polarization and intensity map with the two-dimensional color code [19], obtained for the red rectangle shown in (b). (d) SARPES spectra taken at different angles. The red (blue) lines show spin-up (down) spectra with peak positions indicated by red (blue) bars. (e) Momentum-dependent energy splitting ($\mathrm{\Delta }E$ vs $k$) and a linear fit to obtain ${\alpha }_{\mathrm{R}}$.

Figure 2

[(a)–(c)] ARPES intensity maps obtained for 9, 21, 30 ML Ag/Au(111). (d) Evolution of spin-splitting of $\nu =1$ QWSs as the film thickness increases. $E_1$ is the center of the peak positions of spin-up and spin-down spectra. [(e)–(g)] Spin-resolved photoemission spectra for QWSs in 9, 21, and 30 ML Ag/Au(111). Red (Blue) bars denote the peak position of spin-up (down) spectra. [(h)–(j)] $k$-dependent energy splitting obtained from the spin-resolved spectra in (e)–(g). Black circles indicate the size of spin splitting and the red lines are linear fits to obtain ${\alpha }_R$.

Figure 3

Rashba parameters for $\nu =1,2,3,4$ QWSs obtained by SARPES (triangles) and DFT calculations (circles) for each film thickness.

Figure 4

(a) Contribution of each atomic layer to the spin splitting (top) and the envelope functions of charge density distributions at $k_x=0.043{\AA }^{-1}$ (bottom) for $\nu =1$ QWSs (left) and $\nu =2$ QWSs (right) obtained by DFT calculations. Kinklike features in $|{\psi }_{\mathrm{env}}{|}^2$ are due to the surface or interface effects. The blue or green dashed lines indicate the representative positions where $|{\psi }_{\mathrm{env}}{|}^2$ has its maximum values. (b) Schematic of the relation between the charge density distribution of QWS and the layer-resolved Rashba effect. Blue curves indicate the charge density distributions and its envelope is shown by the black curve. The red curves in the lower figures denote the potential gradient $\partial V/\partial z$ which becomes larger near the nuclei. (c) Schematic of the decomposition of ${\alpha }_{\mathrm{R}}$ to the contributions from the Au substrate and the Ag film. (d) Local charge density distribution of QWSs at the interface Au layer ($|\psi \left(z_0\right){|}^2$) and the interface Ag layer ($|\psi \left(z_1\right){|}^2$) together with the magnitude of ${\alpha }_{\mathrm{R}}$ shown by the black dashed lines. (e) Rashba parameters of QWSs in Ag/Au(111) as a function of local charge density at the interface layer.