Definitions

Surface energy

For a given slab model of a facet with Miller index $(hkl)$ , the surface energy $\gamma^\sigma_{hkl}$ is given by:

$\gamma^\sigma_{hkl} = \frac{E^{hkl,\sigma}_{slab} - E^{hkl}_{bulk} \times n_{slab}}{2 \times A_{slab}}$

where, $E^{hkl,\sigma}_{slab}$ is the total energy of the slab model with termination $\sigma$ , $E^{hkl}_{bulk}$ is the energy per atom of the bulk OUC, $n_{slab}$ is the total number of atoms in the slab structure, $A_{slab}$ is the surface area of the slab structure, and the factor of 2 in the denominator accounts for the two surfaces in the slab model.

Work function

Similarly, each slab model of a facet has a work function $\Phi^\sigma_{hkl}$ given by:

$\Phi^\sigma_{hkl} = V_{vac} - E_{F}$

where $V_{vac}$ is the electrostatic potential in the middle of the vacuum region of the slab model and $E_{F}$ is the Fermi energy of the material.

Wulff shape and area fraction

The Wulff construction gives the crystal shape under equilibrium conditions. In this construction, the distance of a facet from the crystal center is directly proportional to the surface energy of that facet, and the inner convex hull of all facets form the Wulff shape.
We define the weighted surface energy $\bar{\gamma}$ using this fraction as given by the following equation:

$\bar{\gamma} = \frac{\sum_{\{hkl\}}\gamma_{hkl}A_{hkl}}{\Sigma A_{hkl}}=\sum_{\{hkl\}}\gamma_{hkl}f^A_{hkl}$

where $\gamma_{hkl}$ is the surface energy for a unique facet existing in the Wulff shape, $A_{hkl}$ is the total area of all facets in the $\{hkl\}$ family in the Wulff shape, and $f^A_{hkl}$ is the area fraction of the $\{hkl\}$ family in the Wulff shape.

Similarly the weighted work function $\bar{\Phi}$ can be defined as:

$\bar{\Phi} = \frac{\sum_{\{hkl\}}\Phi_{hkl}A_{hkl}}{\Sigma A_{hkl}}=\sum_{\{hkl\}}\Phi_{hkl}f^A_{hkl}$

where $\Phi_{hkl}$ is the work function for a unique facet existing in the Wulff shape.

Surface anisotropy

The most commonly used general measure of anisotropy is the shape factor $\eta$ , which is given by the following equation:

$\eta = \frac{A}{V^{2/3}}$

where $A$ and $V$ are the surface area and volume of the Wulff shape, respectively. The shape factor is a useful quantity in determining the critical nucleus size. Typically, the shape factor is compared against that of an ideal sphere ( $\eta = (36\pi)^{\frac{1}{3}}$ ), and a larger $\eta$ indicates greater anisotropy.

An alternative definition of surface energy anisotropy $\alpha_{\gamma}$ used in this database given by the following equation:

$\alpha_{\gamma} = \frac{\sqrt{\sum_{\{hkl\}}(\gamma_{hkl} - \bar{\gamma})^2 f^A_{hkl}}}{\bar{\gamma}}$

$\alpha_{\gamma}$ can effectively be viewed as a coefficient of variation of surface energies that is normalized for comparison across crystals with different average surface energies. A perfectly isotropic crystal would have $\alpha_{\gamma} = 0$ . $\alpha_{\gamma}$ is comparable across different crystal systems and accounts for all surfaces based on their relative importance (in terms of contribution to the Wulff shape).