Ab initio study of the thermodynamic properties of nonmagnetic elementary fcc metals: Exchange-correlation-related error bars and chemical trends

Thermal properties of an extensive set of fcc metals (Al, Pb, Cu, Ag, Au, Pd, Pt, Rh, and Ir) have been studied using density-functional theory in combination with the quasiharmonic approximation. Systematic convergence checks have been performed to ensure an accuracy greater than $1\mathrm{meV}$/atom in the free energies. Phonon dispersion relations, Grüneisen parameters, free energies, thermal expansions, and heat capacities have been calculated for the two popular exchange-correlation functionals: Local density approximation and Perdew-Burke-Ernzerhof generalized gradient approximation [Phys. Rev. Lett. 77, 3865 (1996)]. The results are found to be in excellent agreement with both experimental and CALPHAD data.

Figure 1

(Color online) Convergence of the averaged Grüneisen parameter $\gamma$ with respect to the augGrid (see text) in terms of grid points per atom (gp/atom) for Cu and Ag. The lines $L_1$ (similar for Cu and Ag on this scale) and $L_{2,\mathrm{Cu}∕\mathrm{Ag}}$ correspond to grid sizes being 8 and 64 times larger than the basic grid at a converged cutoff (see Table ). The values were obtained using a $3^3$ supercell and a kp mesh of $7\times {10}^3\mathrm{kp}\bullet \text{atom}$.

Figure 2

(Color online) (a) AugGrid convergence of the averaged Grüneisen parameter $\gamma$ for Ag for the $1^3$ and $3^3$ supercells (sc). The black squares show the convergence of the $1^3$ sc after rescaling it using Eq. (14). (b) Similar to (a) but with the plane wave energy cutoff $E^{\mathrm{cut}}$ as the convergence parameter.

Figure 3

(Color online) AugGrid convergence of the averaged Grüneisen parameter $\gamma$ and the vibrational free energy $F^{\mathrm{vib}}$ for all investigated elements. The lines $L_1$ (falls together with the $y$ axis on this scale) and $L_2$ (dashed line) are defined as in Fig. 1. Line $L_3$ (dot-dashed line) corresponds to the augGrid size used in the present study (Sec. 5). The numbers inside each graph show the error range of the Grüneisen parameter at line $L_1,L_2,L_3$. The values have been obtained using a kp mesh of $7\times {10}^3\mathrm{kp}\bullet \text{atom}$ and a $1^3$ supercell. The scaling procedure [Eq. (14)] has been employed to rescale the values to correspond to large supercells. Al and Pb are separated to distinguish them from the transition metals.

Figure 4

(Color online) kp convergence of the averaged Grüneisen parameter $\gamma$ and the vibrational free energy $F_{\mathrm{vib}}$ for all investigated elements. The line $L_1$ (falls together with the $y$ axis on this scale) corresponds to the smallest investigated kp mesh of $2\times {10}^3\mathrm{kp}\bullet \text{atom}$ and the line $L_2$ (dot-dashed line) to the kp meshes used for the calculations presented in Sec. 5. The numbers inside each graph show the error range of $\gamma$ at lines $L_1$,$L_2$. The values have been obtained using a $2^3$ supercell.

Figure 5

(Color online) kp convergence of the phonon dispersion $\omega \left(\vec{q}\right)$ ($E_{\omega }=\hslash \omega$ with $\hslash$ the Planck constant) along some high symmetry directions for Ir, Al, and Pb for the $4^3$ supercell. The red circle marks unphysical phonon instabilities (imaginary frequencies; see text).

Figure 6

(Color online) Supercell (sc) size convergence of the phonon dispersion for Pd using a $k$-point mesh of $7\times {10}^3\mathrm{kp}\bullet \text{atom}$. The red ticks mark the exact $q$ points (see text) for the specific supercell.

Figure 7

(Color online) Supercell (sc) size convergence of the phonon dispersion for Pb. The $k$-point mesh size was $32\times {10}^3\mathrm{kp}\bullet \text{atom}$ for the $5^3$ supercell and $7\times {10}^3\mathrm{kp}\bullet \text{atom}$ for the other supercells. Exact $q$ points are marked by red ticks.

Figure 8

(Color online) Supercell (sc) size convergence of the averaged Grüneisen parameter $\gamma$ and the vibrational free energy $F_{\mathrm{vib}}$ for all investigated elements. The $k$-point mesh is $7\times {10}^3\mathrm{kp}\bullet \text{atom}$ for all elements. The numbers in the graphs show values for the $1^3$ supercell when the corresponding data point lies outside the graph.

Figure 9

(Color online) Supercell (sc) size convergence of the averaged Grüneisen parameter $\gamma$ and the vibrational free energy $F_{\mathrm{vib}}$ (at $T=400\mathrm{K}$) in absolute values for all investigated elements. The used $k$-point mesh was $7\times {10}^3\mathrm{kp}\bullet \text{atom}$ for all elements. Transition metals in the same row in the Periodic Table are given the same color. Al and Pb are separated for reading convenience. The experimental data are taken from Ref. 43.

Figure 10

(Color online) Correlation between the deviation from experiment for the lattice constants $\Delta a_{\mathrm{i}}$ and the bulk moduli $\Delta B_{\mathrm{i}}$ (Table ). All quantities contain the influence of zero-point vibrations. The dashed lines are a guide for the eye and emphasize the linear trend for each functional.

Figure 11

(Color online) Phonon dispersion $\omega \left(\vec{q}\right)$ [$E=h\nu =h\omega ∕\left(2\pi \right)$ with $h$ the Planck constant] along some high symmetry directions calculated at the ab initio obtained equilibrium lattice constant for the given temperature. The corresponding lattice constant for each functional is given in the legend. The insets in the Pd and Pt phonon dispersions magnify the anomalies (the straight dashed line is a guide for the eye). A $5^3$ supercell has been used for Pb and a $4^3$ supercell for all other elements (see Sec. 3B for further technical details; these parameters have also been used to obtain the results presented in all following figures). The experimental values are taken from Ref. 56 for Rh, from Ref. 59 for Ir, and from Ref. 41 for all other elements. Al and Pb are slightly separated to emphasize the Periodic Table correspondence of the other elements.

Figure 12

(Color online) Linear thermal expansion $\epsilon$ (right axis) and its coefficient $\alpha$ (left axis) as a function of temperature $T$ [Eq. (16)]. For Al and Pb, the experimental melting temperature $T_{\mathrm{MP}}$ is below the upper limit of the temperature interval considered here and included as a dashed line. The experimental data are taken from Ref. 47.

Figure 13

(Color online) Heat capacity $C_P$ at $P=0\mathrm{Pa}$ as a function of temperature $T$ in units of the Boltzmann constant $k_B$. For Al and Pb, the melting temperature $T_{\mathrm{MP}}$ is included (black dashed line). For GGA, the dashed-dotted lines show $C_V$ and the dashed lines the electronic contribution $C_P^{\mathrm{el}}$. The inset in the Cu heat capacity enlarges the low temperature region. The experimental data are taken from Ref. 62.

Figure 14

(Color online) Free energy $F$ at $P=0\mathrm{Pa}$ as a function of temperature $T$ for both functionals in comparison with values obtained with the CALPHAD method (using THERMOCALC version Q and the “PURE4 -SGTE Pure Elements Database”). For Al and Pb, the melting temperature $T_{\mathrm{MP}}$ is included (dashed line). The numbers inside each graph give $F^{\mathrm{CA}}-F^{\mathrm{DFT}}$ at $T=900\mathrm{K}$ ($T=600\mathrm{K}$ for Pb) for LDA and GGA $(1\mathrm{meV}∕\text{atom}\approx 0.023\mathrm{kcal}∕\mathrm{mol})$.

Figure 15

(Color online) Phonon dispersion of Rh at room temperature ($4^3$ supercell) calculated (a) fully ab initio at lattice constants as given for Rh in Fig. 11 and (b) at the experimental room temperature lattice constant of $3.803\mathrm{\AA }$.

Figure 16

(Color online) Linear thermal expansion $\epsilon$ (right axis) and its coefficient $\alpha$ (left axis) as a function of temperature $T$ using the mixed approach [see Eq. (19) and text]. For a comparison with the fully ab initio data, see Fig. 12.

Figure 17

(Color online) Heat capacity $C_P$ at constant pressure as a function of temperature $T$ using the mixed approach. For a comparison with the fully ab initio data, see Fig. 13.

Figure 18

(Color online) Free energy $F$ at $P=0\mathrm{Pa}$ as a function of temperature $T$. The numbers inside each graph give $F^{\mathrm{CA}}-F^{\mathrm{DFT}}$ at $T=900\mathrm{K}$ ($T=600\mathrm{K}$ for Pb) for LDA and GGA $(1\mathrm{meV}∕\text{atom}\approx 0.023\mathrm{kcal}∕\mathrm{mol})$. For a comparison with the fully ab initio data, see Fig. 14.