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 $H_3$ 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 $\mathrm{\Delta }E=25\mathrm{meV}$ for Pb/Si(111) and 22 meV for Pb/Ge(111), but the inclusion of SOC reverses their relative stability as $\mathrm{\Delta }E=-12$ and $-7\mathrm{meV}$, 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 $\sim 0.59$ and $\sim 0.27\mathrm{eV}$ 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.

Figure 1

Top and side views of the optimized (a) $H_3$ and (b) $T_4$ structures of Pb/Si(111). The green and gray circles represent Pb and Si atoms, respectively. For distinction, the Si atoms in the subsurface layers are drawn with small circles. The dashed line indicates the $\sqrt{3}\times \sqrt{3}R{30}^{\circ }$ unit cell where the four Pb atoms are numbered. The ${\mathrm{Pb}}_4$ atom in (a) and (b) is located at the $H_3$ and $T_4$ sites, respectively. The $\mathbf{x},\mathbf{y}$, and $\mathbf{z}$ axes point along the $\left[11\overline{2}\right],\left[1\overline{1}0\right]$, and [111] directions, respectively. The height difference between the $H_3\left(T_4\right)$ Pb atom and the first Si-substrate layer is labeled as $\mathrm{\Delta }h$.

Figure 2

Band structures of the (a) $H_3$ and (b) $T_4$ structural models, obtained using DFT. The corresponding ones obtained using DFT + SOC are given in (c) and (d), respectively. The bands projected onto the Pb $p_x,p_y$, and $p_z$ orbitals are displayed with circles whose radii are proportional to the weights of such orbitals. The energy zero represents $E_F$. The inset in (a) shows the surface Brillouin zones of the $\sqrt{3}\times \sqrt{3}R{30}^{\circ }$ unit cell. In (c) and (d), the contour plots of the charge density of the lowest unoccupied surface state at the $K$ point are drawn on the vertical plane containing the $H_3$ and $T_4$ Pb atoms, respectively, where the first line is at $0.3\times {10}^{-3}{\mathrm{electrons}/\AA }^3$ with spacings of $0.5\times {10}^{-3}{\mathrm{electrons}/\AA }^3$. A close-up of the spin splitting of the surface state at the $K$ point (just above $E_F$) and the $M$ point (just below $E_F$) is displayed for the $H_3$ and $T_4$ structures in (c) and (d), respectively, together with including their time-reversal counterparts.

Figure 3

Helical spin textures of the (a) $H_3$ and (b) $T_4$ structures of Pb/Si(111) along the Fermi surface. The SAM vectors with the negative, zero, and positive $S_z$ components are represented with increasing brightness of the arrows. The total spin-polarization $\left|\mathit{S}\right|=\sqrt{S_x^2+S_y^2+S_z^2}$ and the $S_z$ component along the outer and inner Fermi surfaces are given in (c) and (d), respectively.

Figure 4

Calculated energy profile along the transition pathway from the $H_3$ to the $T_4$ structure in Pb/Si(111). The atomic geometry of the TS is given. The numbers denote the total energies of TS and the $T_4$ structure relative to the $H_3$ structure. The corresponding energy values for Pb/Ge(111) are given in parentheses.