by Yingdong Guan

Introduction

Surface energy is an important property for crystals. In most cases, the lowest surface energy determines the most stable surface and will thus influence the crystal growth process. [1] Also, there are many interesting phenomena happening on the surface of thin films and on the interfaces of multiple different thin films. Some of those phenomena are related to surface energy. In this post, we used the Density Functional Theory (DFT) calculation to calculate surface energy for different surfaces of Au crystal, namely (100) and (111). Slab models [2] are used to simulate different surfaces of Au crystal. The surface energy of different surfaces are calculated based on convergence tests and are compared to determine which surface is energetically favored.

Bulk Au

Au lattice with the cubic close-packed structure was constructed. The crystal structure of Au is shown in Figure 1.

Parameters used to build Au lattice is listed below: [3]

space group: Fm-3m (No.225)

Cell parameters:

a: 4.0782 Å

b: 4.0782 Å

c: 44.0782 Å

α: 90.000°

β: 90.000°

γ: 90.000°

Our calculation used plane-wave bases with on the fly generated ultrasoft (OTFG-ultrasoft) pseudopotentials in CASTEP. PBE-GGA was used as the functional. Convergence tests for both k-point mesh and energy cutoff are done. The results of convergence tests is shown in Figures 2.1 and 2.2. Blue and red dots are data points acquired from DFT calculation and the curves are B-spline fitting of corresponding data points. Total energy for the conventional cell is calculated using a k-point mesh of 10*10*10 and an energy cutoff of 500 eV. The core radius for ultrasoft pseudopotential for Au is 2.4 Bohr (~1.27 Å). The ultrasoft pseudopotential was generated with 25 valence electrons (4f14 5s2 5p6 5d10 6s1). The calculated total energy is -56515.8988 eV and energy for each gold atom is -14128.9747 eV. We used equation Ecoh = ( Ebulk – N*Eatom ) / N to calculate cohesive energy. Ebulk is the final energy for bulk Au, Eatom is the final energy for Au atom and N is the number of atoms in the unit cell. The cohesive energy here is -5.6961 eV.

(100) and (111) surfaces

FCC unit cell was built and was cleaved along the Miller indices of (100) and (111). The thickness of atom layers was chosen to be 2.5 for (100) surface and 5 for (111) surface so that five layers of Au atoms were in the slab. This difference in thickness is due to the different stacking style of Au atoms in different directions. The vacuum distance was set to 10 Å to prevent interactions between the top layer and the bottom layer of the slab. The bottom two layers of Au atoms were chosen to be frozen layers to simulate bulk Au crystal. The slabs are shown in Figure 3.

Convergences tests for k-point mesh and energy cutoff were done for both (100) slab and (111) slab. Results were shown in Figures 4.1, 4.2, 5.1, 5.2. Three to four calculations were done for each convergence tests here due to the limited computational power of my laptop. However, it is clear the both (111) slab and (100) slab converges with a k-point mesh of 9*9*1 and an energy cutoff of 500 eV.

The energy was calculated for (100) surface slabs with different vacuum thickness (10 Å, 15 Å, 20 Å). The result of surface energy dependence on vacuum thickness is shown in Figure 6. As can be seen, the convergence problem does exist here. However, the difference in surface energy decrease to below 0.0005 eV after vacuum thickness reached 15 Å. Thus, we chose to use a vacuum thickness of 15 Å to make sure both accuracy and calculation time are in an acceptable range.

Attempts on calculating layer dependence of energy were also made. However, after we increase atom layers in the slab, the calculation became extremely slow. We waited for a long period of time but the second optimization step in geometry optimization still didn’t show up. Thus, we have to stop the calculation.

For (111) surface, we simply change vacuum thickness to 15 Å to determine its surface energy due to the extremely long calculation time.

The equation we use to calculate surface energy is from “Density Functional Theory, A Practical Introduction” book:

σ(surface)= 1/2A*[E(slab)-n*E(bulk)]

Here, σ(surface) is the surface energy per area. A is the area of used slab and n is the number of atoms in the slab. E(slab) is the total energy of the slab and E(bulk) is the energy per atom in bulk Au.

The final result is listed below:

The surface energy for (100) surface is 0.0544 eV/Å^2.

The surface energy for (111) surface is 0.0316 eV/Å^2.

Here we compare to the surface energy calculated by Tran, R., Xu, Z., Radhakrishnan, B. et al. Their surface energy for (100) surface is 0.054 eV/Å^2 and their surface energy for (111) surface is 0.044 eV/Å^2. The difference in surface energy for (111) surface may due to difference in vaccum thickness and atom layer.

The formation of a new surface by cleaving form a bulk crystal requires the breaking of atom bonds on two side of the cleavage plane. Surface with lower energy is usually considered more stable since lower surface energy means smaller amount of work to be done when cleaving along this surface. Thus, the Au (111) surface is energetically more favored than the (100) surface. This is reasonable since the more closely stacked surface is usually more favored.

References:

[1] Tran, R., Xu, Z., Radhakrishnan, B., Winston, D., Sun, W., Persson, K. A., & Ong, S. P. (2016). Surface energies of elemental crystals. Scientific data, 3(1), 1-13.

[2] Sholl, David, and Janice A. Steckel. Density functional theory: a practical introduction. John Wiley & Sons, 2011.

[3] Maeland, A., & Flanagan, T. B. (1964). Lattice spacings of gold–palladium alloys. Canadian journal of physics, 42(11), 2364-2366.

[4] Tran, R., Xu, Z., Radhakrishnan, B. et al. Surface energies of elemental crystals. Sci Data 3, 160080 (2016).