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, we 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 spin-orbit coupling (SOC) influences the relative stability of the two structures in both the Pb/Si(111) and 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, which gives rise to a relatively larger Rashba spin splitting of surface states as well as a relatively larger pseudo-gap opening compared to the T$_4$ structure. By the nudged elastic band calculation, we obtain a sizable energy barrier from the H$_3$ to the T$_4$ structure as ${\sim}$0.59 and ${\sim}$0.27 eV for Pb/Si(111) and Pb/Ge(111), respectively. It is thus likely that the two energetically competing structures can coexist at low temperatures.