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

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... ructure by ΔE = 25 meV for Pb/Si(111) and 22 meV for Pb/Ge(111), but the inclusion of SOC reverses their relative stability as ΔE = -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 ∼0.59 and ∼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.