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

Abstract

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 H-3 and T-4 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 T-4 structure is energetically favored over the H-3 structure by Delta E = 25 meV for Pb/Si(111) and 22 meV for Pb/Ge(111), but the inclusion of SOC reverses their relative stability as Delta E = -12 and -7 meV, respectively. Our analysis shows that the SOC-induced switching of the ground state is attributed to a more asymmetric surface charge distribution in the H-3 structure compared to the T-4 structure, which is associated with the hybridization of the Pb p(x), p(y), and p(z) 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 H-3 structure. By the nudged elastic-band calculation, we obtain a sizable energy barrier from the H-3 to the T-4 structure as similar to 0.59 and similar to 0.27 eV 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.