$^{\mathrm{4}}$He adsorption on a $^{\mathrm{3}}$He-plated graphite surface

Path-integral Monte Carlo (PIMC) calculations have been performed for $^{\mathrm{4}}$He atoms on top of the $^{\mathrm{3}}$He first layer on graphite. For this we ignore Fermi statistics of solidified $^{\mathrm{3}}$He adatoms while Bose statistics of $^{\mathrm{4}}$He atoms are fully incorporated. We first find that the first $^{\mathrm{3}}$He layer exhibits a 7/12 commensurate solid structure at the areal density of 0.111 {\AA}$^{\mathrm{-2}}$, which turns out to be identical to the experimental value for its completion density. Additional adsorption of $^{\mathrm{4}}$He atoms above the complete first $^{\mathrm{3}}$He layer is found to sustain the underlying $^{\mathrm{3}}$He commensurate structure and the second $^{\mathrm{4}}$He layer is observed to display the 4/7 commensurate structure with respect to the first-layer commensurate $^{\mathrm{3}}$He solid at the areal density of 0.0636 {\AA}$^{\mathrm{-2}}$. Furthermore, it is found that the 4/7 commensurate structure of the second-layer $^{\mathrm{4}}$He atoms can be formed above a mixture of the first-layer $^{\mathrm{3}}$He and $^{\mathrm{4}}$He atoms on graphite. These PIMC results suggest that the 4/7 commensurate structure of the second-layer $^{\mathrm{4}}$He atoms on graphite, whose existence on top of the first $^{\mathrm{4}}$He layer has long been in dispute, may be realized on a $^{\mathrm{3}}$He-plated graphite surface. This could lead to a new approach to observe two-dimensional supersolidity in $^{\mathrm{4}}$He on graphite.