Orientation-dependent surface and step energies of Pb from first principles

The orientation-dependent surface energies of 35 low-index and vicinal Pb surface orientations, located in the [001], $\left[\overline{1}10\right]$, and $\left[01\overline{1}\right]$ zones, have been calculated by density-functional theory within the local-density approximation. The highest surface energy anisotropies in these zones are at the (210), (110), and (311) directions. Surface relaxation decreases the surface energy anisotropy significantly. For misorientations smaller than 12° the (projected) surface energy in a given zone increases linearly with step density, while curvature is found at higher misorientations, indicative of repulsive step-step interactions. These results are fully consistent with the orientation-dependent surface energy predicted by the statistical mechanics of the terrace-step-kink model of vicinal surfaces. The step formation energies and surface and step relaxation energies are derived and analyzed. There is good agreement with available experimental data. The calculated surface energies in eV/atom correlate linearly with the number of broken surface bonds. Deviations from perfect linearity are found to be essential for a proper description of the equilibrium crystal shape of Pb.

Figure 1

Plot of $\gamma \left(\theta \right)∕{\gamma }_{\left(111\right)}$ versus $\theta$ in the ⟨001⟩ and ⟨110⟩ zones for relaxed and unrelaxed Pb surfaces, respectively. Maxima in each zone are located at (210), (110), and (311). Real cusps exist at (111), (100), and (320) while a semicusp is noticed at the (110) orientation for relaxed surfaces. There are also local minima at (411) and (551) which are dual-step structures (see text). Experimental data taken at $323\mathrm{K}$ (Ref. 3) and $473\mathrm{K}$ (Ref. 13) are shown for comparison.

Figure 2

Plot of $f\left(\theta \right)$ versus $\tan \left(\theta \right)$ for relaxed surfaces in the (a) $\left[01\overline{1}\right]$ and (b) $\left[\overline{1}10\right]$ zones, representing (111) vicinal $A$ and $B$ steps, and (c) the $\left[01\overline{1}\right]$ and [001] zones, representing the (100) vicinal $B$ and $C$ steps, respectively. The lines are linear fits to points up to $\tan \left(\theta \right)=0.22$. Points deviating from the lines at higher misorientation angles indicate positive curvature due to repulsive step-step interaction.

Figure 3

Surface relaxation energies of Pb(111) vicinal orientations versus terrace width $d$ in the $\left[01\overline{1}\right]$ and $\left[\overline{1}10\right]$ zones. Curves are calculated according to a $1∕d$ dependence of the surface relaxation energy. The surface relaxation energy of the low-index Pb(111) is $1.56\mathrm{meV}∕{\mathrm{\AA }}^2$.

Figure 4

Step relaxation energies of $A$ and $B$ steps of vicinal Pb(111) surfaces versus step density. Note the oscillation of the step relaxation energy at large tilt angle. Also note that the dual step structures (551) and (411) have much lower step relaxation energy.

Figure 5

Plot of theoretical surface energy $F\left(hkl\right)$ in eV per surface atom versus number of surface bonds which are broken in forming an $\left(hkl\right)$ surface. DFT results are shown for unrelaxed and relaxed surfaces of Pb. The data are best fit by straight lines.

Figure 6

Comparison of calculated $\gamma \left(\theta \right)$ versus $\theta$ functions in the ⟨001⟩ and ⟨110⟩ zones for the ideal broken surface bond model and the full DFT results for relaxed Pb surfaces. The two data sets are matched at the (111) orientation. Note the local changes in slope for the DFT data at the indicated vicinal orientations.