Accurate electronic free energies of the $3\mathit{d},4\mathit{d}$, and $5\mathit{d}$ transition metals at high temperatures

Free energies of bulk materials are nowadays routinely computed by density functional theory. In particular for metals, electronic excitations can significantly contribute to the free energy. For an ideal static lattice, this contribution can be obtained at low computational cost, e.g., from the electronic density of states derived at $T=0$ K or by utilizing the Sommerfeld approximation. The error introduced by these approximations at elevated temperatures is rarely known. The error arising from the ideal lattice approximation is likewise unexplored but computationally much more challenging to overcome. In order to shed light on these issues we have computed the electronic free energies for all $3d,4d$, and $5d$ transition elements on the ideal lattices of the bcc, fcc, and hcp structures using finite-temperature density-functional theory. For a subset of elements we have explored the impact of explicit thermal vibrations on the electronic free energies by using ab initio molecular dynamics simulations. We provide an analysis of the observed chemical trends in terms of the electronic density of states and the canonical $d$ band model and quantify the errors in the approximate methods. The electronic contribution to the heat capacities and the corresponding errors due to the different approximations are studied as well.

Figure 1

Ideal lattice SCF electronic free energies in meV/atom at (a)–(c) 1500 K and (d)–(f) the respective melting points $(T^{\mathrm{melt}}$, cf. Table 2) referenced with respect to the internal energy at $T=0$ K. (g)–(i) show the electronic free energy change upon a volume increase of $8\%$ with respect to the equilibrium volume $V_0$. The insets in (a)–(c) show the electronic DOS (arbitrary units) for some of the elements (emphasized by slightly larger symbols in the respective plots) in the three different structures (bcc, fcc, hcp). The Fermi levels ${\varepsilon }_{\mathrm{F}}$ are indicated by the red dashed lines. For bcc Nb and fcc Rh, the numbers 1–4 indicate peaks that are relevant for the discussion in the text. For bcc Mo, the entropy distribution $s(\varepsilon ,T=1500\mathrm{K})$ [Eq. (9)] is plotted (green line). Throughout the figure, results correspond to nonmagnetic calculations, except for the ferromagnetic (FM) results for bcc Fe, fcc Co, and fcc Ni (blue open circles). Exact numbers for all free energy values shown in (a)–(f) are given in Table 2.

Figure 2

Influence of volume and temperature on the electronic DOS. (a) Change of the DOS for bcc Nb upon increasing the volume from $V_0$ (equilibrium volume at $T=0$ K, black line and gray shading) to $1.08V_0$ (red line). The resulting compression is emphasized by the arrows, and the Fermi level (the same for both DOS') is marked by the black dashed line. (b) Change of the DOS directly at the Fermi level, ${\varepsilon }_{\mathrm{F}}$, with volume for two examples. (c) Change of the DOS for bcc Ta upon increasing the (electronic) temperature from $T=0$ K (black line and gray shading) to $T^{\mathrm{melt}}=3290\mathrm{K}$ (red dashed line) within the self-consistent finite temperature DFT calculation. The black and red dashed lines mark the respective Fermi levels, the red line for the $T^{\mathrm{melt}}$ calculation being shifted by about 0.1 eV with respect to the black one. (d) Fermi level shift with temperature for three examples showing different dependencies (positive, small, negative).

Figure 3

(a)–(c) Error in the fixed DOS and (d)–(f) Sommerfeld approximation at an electronic temperature corresponding to the respective melting point $T^{\mathrm{melt}}$ for all $3d$ (blue dots), $4d$ (red squares), and $5d$ (green triangles) transition metals in the (a), (d) bcc, (b), (e) fcc, and (c), (f) hcp structures. The error is defined as the deviation from the SCF electronic free energy for an ideal lattice. In (d)–(f) the thick lines with symbols correspond to a smearing parameter $\sigma =0.2$ eV and the thin lines without symbols to $\sigma =0.1$ eV, used to obtain the corresponding DOS at the Fermi level. Throughout the figure, results correspond to nonmagnetic calculations, except for the data shown by the blue open circles which represent ferromagnetic (FM) calculations for bcc Fe, fcc Co, and fcc Ni.

Figure 4

Electronic DOS for (a) fcc Ru and (b) bcc Ir at $T=0$ K, obtained with two different smearing parameters, $\sigma$, of 0.1 eV (red dashed lines) and 0.2 eV (black lines and gray shading) in Eq. (16). The blue solid lines indicate the Fermi-Dirac occupation function at the respective melting points (Ru: 2606 K and Ir: 2719 K).

Figure 5

Temperature dependence of the SCF electronic internal energy $U^{\mathrm{el}}$ (black lines) and the SCF electronic free energy $F^{\mathrm{el}}$ (blue lines) for (a) fcc Ru and (b) bcc Ir up to the respective melting points, $T^{\mathrm{melt}}$. The red dashed lines and orange dash-dotted lines show results of the corresponding Sommerfeld approximation [Eqs. (10) and (12)] for two different smearing parameters $(\sigma =0.1$ eV and $\sigma =0.2$ eV).

Figure 6

Comparison of the electronic free energies in meV/atom extracted from AIMD simulations (red bars) with those computed via the SCF finite temperature DFT approach for an ideal static lattice (black bars). The blue bars correspond to the Sommerfeld approximation but using the effective DOS from the AIMD simulations as input.

Figure 7

Influence of lattice vibrations on the electronic DOS for a few selected elements at their respective melting temperatures, extracted from AIMD simulations. The black lines indicate the mean value from a statistically converged set of uncorrelated AIMD snapshots and the orange gradient shows the corresponding standard deviation. The white solid lines show the DOS obtained from static calculations for the ideal lattices at an electronic temperature corresponding to the melting point. The white dotted lines indicate the Fermi level.

Figure 8

Temperature dependence of the change in the electronic free energy due to thermal lattice vibrations for bcc W (black circles), fcc Pt (blue triangles), and hcp Ru (red squares). The temperature axis has been normalized by the respective melting temperature. The inset shows the temperature dependence of the effective electronic DOS (in units of states/$\mathrm{eV}·\mathrm{atom})$ for bcc W from the AIMD simulations.

Figure 9

(a)–(c) Ideal lattice SCF electronic contribution to the constant volume heat capacity and (d)–(f) error of the Sommerfeld approximation at the melting point $\left(T^{\mathrm{melt}}\right)$ and $1.08V_0$ for all $3d$ (blue dots), $4d$ (red squares), and $5d$ (green triangles) transition metals in the (a), (d) bcc, (b), (e) fcc, and (c), (f) hcp structures. The ideal lattice SCF heat capacity was obtained from a finite difference of the temperature dependence of the electronic internal energy (using 1 K steps). The Sommerfeld heat capacity was calculated with Eq. (13), using $\sigma =0.1$ eV for smoothing the required DOS at the Fermi level. Throughout the figure, results correspond to nonmagnetic calculations, except for the data shown by the blue open circles which represent ferromagnetic (FM) calculations for bcc Fe, fcc Co, and fcc Ni.

Figure 10

Temperature dependence of the constant volume electronic heat capacity for bcc W (black), fcc Pt (blue), hcp Ru (red) with (solid lines) and without (dashed lines) the impact of vibrations. To obtain an accurate heat capacity from the coarse set of free energies (see Fig. 8), we used a physically motivated fit as introduced in Ref. [25] with a second-order polynomial for the energy independent electronic density of states. For consistency, the same fit was used to obtain the temperature dependence of the ideal lattice SCF heat capacities (dashed lines). The temperature axis has been normalized by the respective melting temperature.

Figure 11

Electronic densities of states for all $3d$ (blue), $4d$ (red), and $5d$ (green) elements on the ideal lattices of bcc, fcc, and hcp in the nonmagnetic state, and at an electronic temperature of $T=0$ K and a volume of $1.08V_0$, referenced with respect to the Fermi level (marked by the vertical dashed lines). The shift due to an increase in the electronic temperature up to the melting point is negligible on the shown scale. The black solid lines show the effective mean DOS' from the AIMD simulations at the melting temperature for the investigated cases. All DOS' were obtained with a smoothing parameter of $\sigma =0.1\mathrm{eV}$.