Spin-orbit coupling effects on the stability of two competing structures in Pb/Si(111) and Pb/Ge(111)

Using first-principles density-functional theory (DFT) calculations with/without including the spin-orbit coupling (SOC), we systematically investigate the (4/3)-monolayer structure of Pb on the Si(111) or Ge(111) surface within the two competing structural models termed the H3 and T4 structures. We find that the SOC influences the relative stability of the two structures in both the Pb/Si(111) and the Pb/Ge(111) systems, i.e., our DFT calculation without including the SOC predicts that the T4 structure is energetically favored over the H3 structure by ΔE=25meV for Pb/Si(111) and 22 meV for Pb/Ge(111), but the inclusion of SOC reverses their relative stability as ΔE=-12 and -7meV, respectively. Our analysis shows that the SOC-induced switching of the ground state is attributed to a more asymmetric surface charge distribution in the H3 structure compared to the T4 structure, which is associated with the hybridization of the Pb px,py, and pz orbitals. This asymmetry of surface charge distribution gives rise to a relatively larger Rashba spin splitting of surface states as well as a relatively larger pseudogap opening in the H3 structure. By the nudged elastic-band calculation, we obtain a sizable energy barrier from the H3 to the T4 structure as ∼0.59 and ∼0.27eV for Pb/Si(111) and Pb/Ge(111), respectively. Based on the predicted thermodynamics and kinetics of Pb/Si(111) and Pb/Ge(111), we suggest not only the coexistence of the two energetically competing structures at low temperatures, but also the order-disorder transition at high temperatures.