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=250.28em0exmeV for Pb/Si(111) and 22 meV for Pb/Ge(111), but the inclusion of SOC reverses their relative stability as ΔE=-12 and -70.28em0exmeV, 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,0.28em0expy, 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 ~0.270.28em0exeV 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.