Spin splitting and strain in epitaxial monolayer WSe$_2$ on graphene

We present the electronic and structural properties of monolayer WSe$_{2}$ grown by pulsed-laser deposition on monolayer graphene (MLG) on SiC. The spin splitting in the WSe$_{2}$ valence band at $\overline{\mathrm{K}}$ was $\Delta_\mathrm{SO}=0.469\pm0.008$ eV by angle-resolved photoemission spectroscopy (ARPES). Synchrotron-based grazing-incidence in-plane X-ray diffraction (XRD) revealed the in-plane lattice constant of monolayer WSe$_{2}$ to be $a_\mathrm{WSe_2}=3.2757\pm0.0008 \mathrm{\r{A}}$. This indicates a lattice compression of -0.19 % from bulk WSe$_{2}$. By using experimentally determined graphene lattice constant ($a_\mathrm{MLG}=2.4575\pm0.0007 \mathrm{\r{A}}$), we found that a 3$\times$3 unit cell of the slightly compressed WSe$_{2}$ is perfectly commensurate with a 4$\times$4 graphene lattice with a mismatch below 0.03 %, which could explain why the monolayer WSe$_{2}$ is compressed on MLG. From XRD and first-principles calculations, however, we conclude that the observed size of strain is negligibly small to account for a discrepancy in $\Delta_\mathrm{SO}$ found between exfoliated and epitaxial monolayers in earlier ARPES. In addition, angle-resolved, ultraviolet and X-ray photoelectron spectroscopy shed light on the band alignment between WSe$_{2}$ and MLG/SiC and indicate electron transfer from graphene to the WSe$_{2}$ monolayer. As further revealed by atomic force microscopy, the WSe$_{2}$ island size depends on the number of carbon layers on top of the SiC substrate. This suggests that the epitaxy of WSe$_{2}$ favors the weak van der Waals interactions with graphene while it is perturbed by the influence of the SiC substrate and its carbon buffer layer.