Vacancy formation enthalpy of filled $d$-band noble metals by hybrid functionals

First-principles determination of the vacancy formation enthalpies has been long-term believed to be highly successful for metals. However, a widely known fact is that the various conventional density functional theory (DFT) calculations with the typical semilocal approximations show apparent failures to yield accurate enthalpies of Ag and Au. Recently, the previously commonly assumed linear Arrhenius extrapolation to determine the vacancy formation enthalpies at $T=0$ K from the high-temperature measured concentration of thermally created vacancies has been demonstrated to have to be replaced by the non-Arrhenius local Grüneisen theory (LGT) [A. Glensk, B. Grabowski, T. Hickel, and J. Neugebauer, Phys. Rev. X 4, 011018 (2014)]. The large discrepancies between the conventional DFT-PBE data and the unrevised experimental vacancy formation enthalpies disappear for Cu and Al. Even by following the same LGT revisions for Ag, the large discrepancies still remain substantial at $T=0$ K. Here, we show that the hybrid functional (HSE), by including nonlocal exchange interactions to extend the conventional DFT method, can further correct these substantial failures. Upon a comparison of the experimental valence-band spectra for Cu, Ag, and Au, we have determined the HSE exchange-correlated mixing parameters $\alpha$ of 0.1, 0.25, and 0.4, and further derived the HSE enthalpies of vacancy formation of 1.09, 0.94, and 0.72 eV, respectively; in nice agreement with available LGT-revised experimental data. Our HSE results shed light on how to improve the theoretical predictions to accurately determine the defect formation energies and related thermodynamical properties.

Figure 1

Experimental (solid symbols), theoretical linear Arrhenius interpolated (dashed lines) and non-Arrhenius LGT fitting (solid curves) Gibbs energies of vacancy formation as a function of temperatures in fcc-type Cu and Ag. The experimental PAS (positron annihilation spectroscopy, open circles) and DD (differential dilatometry, solid circles) data [12] are all located in the high-temperature region close to their melting point.

Figure 2

(Left) Density of states of Au by increasing the xc mixing parameter $\alpha$ within the HSE hybrid functional framework, along with the experimental valence-band spectrum [26]. (Right) Optimized lattice constants of fcc-type Au as a function of the HSE mixing parameter of $\alpha$.

Figure 3

The density of states (DOSs) of Cu and Ag by increasing the xc mixing parameter $\alpha$ within the HSE hybrid functional framework, as compared with available experimental valence-band spectra [27, 28, 29].

Figure 4

The Cu-Ag-Au trend of PBE, LDA, and HSE calculated enthalpies of vacancy formation as compared with available experimentally measured data (experimentally recommended data and other reported Expt-1 and Expt-2 data [4, 5, 33]) and the non-Arrhenius LGT revised value at $T=0$ K according to the previous experimental measurements. The lines are guides to the eye.