Breakdown of the Arrhenius Law in Describing Vacancy Formation Energies: The Importance of Local Anharmonicity Revealed by Ab initio Thermodynamics

We study the temperature dependence of the Gibbs energy of vacancy formation in Al and Cu from $T=0\mathrm{K}$ up to the melting temperature, fully taking into account anharmonic contributions. Our results show that the formation entropy of vacancies is not constant as often assumed but increases almost linearly with temperature. The resulting highly nonlinear temperature dependence in the Gibbs formation energy naturally explains the differences between positron annihilation spectroscopy and differential dilatometry data and shows that nonlinear thermal corrections are crucial to extrapolate high-temperature experimental data to $T=0\mathrm{K}$. Employing these corrections—rather than the linear Arrhenius extrapolation that is commonly assumed in analyzing experimental data—revised formation enthalpies are obtained that differ up to 20% from the previously accepted ones. Using the revised experimental formation enthalpies, we show that a large part of the discrepancies between DFT-GGA and unrevised experimental vacancy formation energies disappears. The substantial shift between previously accepted and the newly revised $T=0\mathrm{K}$ formation enthalpies also has severe consequences in benchmarking ab initio methods against experiments, e.g., in deriving corrections that go beyond commonly used LDA and GGA exchange-correlation functionals such as the AM05 functional.

Popular Summary

The properties and performance of modern materials critically depend on the presence of point defects. Even minute amounts of such defects are known to dramatically impact the doping efficiency in optoelectronic or microelectronic semiconductor devices or the mechanical strength of structural materials. A key quantity to describe such defects is their formation energy. Getting accurate measures for this quantity has been an outstanding challenge for both experiment and theory. Experimentally detectable and equilibrated concentrations are obtained only close to the melting point. Theoretical state-of-the-art ab initio approaches, which would allow an accurate description of such defects, have previously been restricted to low temperatures. In this study we are, for the first time, able to bridge the large temperature gap between theoretical and experimental results.

The keys to this success were methodological advances that allowed us to include anharmonic lattice vibrations in the first-principles determination of free energies. Using this novel approach for two prototype elemental solids, Al and Cu, we have been able to reconcile all theoretical and experimental data into a single unified picture. Having the highly accurate formation energies available over the entire temperature window reveals that the concept of a temperature-independent formation entropy—an essential assumption behind the traditionally and universally employed empirical Arrhenius law—must be replaced by an entropy linear in temperature.

This insight has fundamentally significant consequences for the “official” point-defect data as, e.g., compiled in the Landoldt-Börnstein series: A general revision of formation energies will now be critical and possible to make, and should lead to significantly lower (by as much as 20%) formation energies.

Figure 1

Experimental (black symbols) and DFT [blue/orange (LDA/GGA-PBE) lines] Gibbs energy of formation of vacancies in (a) Al and (b) Cu. Experiments ($\mathrm{PAS}=\text{positron}$ annihilation spectroscopy [1], $\mathrm{DD}=\text{differential}$ dilatometry [1, 2]) are limited to a region (gray shaded) close to the melting point, $T_{\mathrm{Al}/\mathrm{Cu}}^{\text{melt}}$. Extrapolations of available PAS [1, 3, 4, 5, 6, 7] and DD data [2, 1, 3, 8, 9, 10, 11, 12] to $T=0\mathrm{K}$ using the common Arrhenius ansatz, $G^f(T)=H^f-T{S}^f$, introduce scatter in the reported values (filled/empty black bars mark corresponding intervals). Formation energies computed by common ab initio approximations such as the $T=0\mathrm{K}$ (dotted line) and the electronic-plus-quasiharmonic ($\mathrm{el}+\mathrm{qh}$; dashed line) approach are shown. The full curve ($\mathrm{el}+\mathrm{qh}+\mathrm{ah}$) includes all free-energy contributions in particular anharmonicity. The error resulting when assuming the Arrhenius extrapolation, ${\mathrm{\Delta }}^{\mathrm{Arr}}$, is marked by the orange arrow at $T=0\mathrm{K}$.

Figure 2

Entropy of formation, $S^f=-d{G}^f/dT$, for the Cu monovacancy (solid line) and divacancy (dashed line). The thinner lines indicate the entropy of formation considering only the electronic and quasiharmonic free-energy contribution ($\mathrm{el}+\mathrm{qh}$). The thick curves include all contributions in particular anharmonicity ($\mathrm{el}+\mathrm{qh}+\mathrm{ah}$). Green lines represent numbers suggested in Ref. [8] for explaining non-Arrhenius behavior within a $\text{monovacancy}+\text{divacancy}$ model. Experimental PAS [37] and DD entropies (filled/empty black bars) are derived from the experimental data shown in Fig. 1.

Figure 3

(a) Harmonic (black) and anharmonic (orange) distribution ${\rho }_{V,T}(x,y)$ according to Eq. (5) for Cu at $T^{\text{melt}}=1360\mathrm{K}$. The vacancy center is placed at (0, 0), and the equilibrium position of the first nearest neighbor is marked by a green cross. The points (black/orange) show the molecular-dynamics trajectory of the atom at discrete time steps of 10 fs. The region close to the equilibrium position is densely populated, and thus the individual points are not resolvable on this scale. The harmonic data are obtained from thermodynamic integration runs at zero coupling constant. (b) Distribution function ${\rho }_{V,T}(d)$ (dashed lines) according to Eq. (6), i.e., projection of ${\rho }_{V,T}(x,y)$ onto the [110] direction indicated in (a), and corresponding effective potential according to Eq. (7) (solid lines). The zero line of the distribution function is shifted upwards by the energy $k_{\mathrm{B}}{T}^{\text{melt}}$ according to the temperature at which ${\rho }_{V,T}(d)$ was calculated. The orange diamonds mark the shift of the center of mass of the anharmonic ${\rho }_{V,T}(d)$ shown and additional anharmonic distributions (not explicitly shown) towards the vacancy at the following temperatures: 250, 450, 800, 1100, 1250 K (related to the energy axis by $k_{\mathrm{B}}T$). For clarity, the latter curve is scaled by a factor of 10 on the $d$ axis.