Error estimates for density-functional theory predictions of surface energy and work function

Density-functional theory (DFT) predictions of materials properties are becoming ever more widespread. With increased use comes the demand for estimates of the accuracy of DFT results. In view of the importance of reliable surface properties, this work calculates surface energies and work functions for a large and diverse test set of crystalline solids. They are compared to experimental values by performing a linear regression, which results in a measure of the predictable and material-specific error of the theoretical result. Two of the most prevalent functionals, the local density approximation (LDA) and the Perdew-Burke-Ernzerhof parametrization of the generalized gradient approximation (PBE-GGA), are evaluated and compared. Both LDA and GGA-PBE are found to yield accurate work functions with error bars below 0.3 eV, rivaling the experimental precision. LDA also provides satisfactory estimates for the surface energy with error bars smaller than 10%, but GGA-PBE significantly underestimates the surface energy for materials with a large correlation energy.

Figure 1

The surface entropy $S$ as a function of temperature in the simplified model of Tyson and Miller [36]. The first linear increase is due to the activation of surface-related vibrational contributions. The second linear increase is associated with surface roughening.

Figure 2

To model a surface on the atomic scale in a 3D periodic code, slabs of atomic layers are constructed. As an example the $\left\{100\right\}$ surface of a face-centered cubic material is shown. The slab unit cell is indicated by the red box.

Figure 3

An illustration of the meaning of the SER. It is a variable, residual error (green dotted line) on top of a predictable deviation (solid blue line).

Figure 4

Results of the Bayesian model selection (BMS) analysis. Left: LDA work functions (a) and surface energies (c). Right: PBE work functions (b) and surface energies (d). A linear fit is clearly the most likely model for both LDA and PBE work functions. For surface energies, the different error character of group 1 and group 2 materials as opposed to the $d$ block and $p$ block materials result in the quadratic model competing with the linear model.

Figure 5

Analysis of the residual errors for LDA work functions: histograms with fitted normal distributions before (a) and after (c) the removal of the outliers Be and Ti, accompanied by QQ plots before (b) and after (d) outlier removal.

Figure 6

Linear fits of experimental values to DFT predictions for all four evaluated data sets. Left: LDA work functions (a) and surface energies (c). Right: PBE work functions (b) and surface energies (d). For clarity, an inset is provided of the low surface energy range for both LDA (c) and PBE (d). The data points are color coded per material class [legend in (a)]. The linear fit is indicated by a red dashed line, with the first bisector in black, corresponding to the ideal theory, serving as reference. Materials of particular interest are indicated with a label.

Figure 7

Overview of the residual errors of the materials for which a comparison with experiment can be made. Top: LDA (a) and PBE (b) work functions. Bottom: LDA (c) and PBE (d) surface energies. A darker shade represents a larger absolute value of the weighted residual error ${\epsilon }_i$. There are four shades in total, corresponding with the intervals 0–1, 1–2, 2–3, and 3–$\infty$ (see legend). Be and Ti were removed from the work function data set, Cr and Ge were removed from the surface energy data set (see Sec. 3b). Materials for which no comparison is made have a gray symbol. Mn, Be, Se, and Te were not calculated because of their crystallographic complexity. There are experimental work function data for Zr and As, but the accuracy of these values is unconfirmed [40].