Temperature-driven phase transitions from first principles including all relevant excitations: The fcc-to-bcc transition in Ca

The temperature-driven fcc-to-bcc phase transition in calcium is examined by a fully ab initio-based integrated technique including all relevant finite-temperature excitation mechanisms. The approach is based on density-functional-theory calculations with a controlled numerical stability of below 0.5 meV/atom for the electronic, quasiharmonic, and structural excitations and better than 1 meV/atom for the explicitly anharmonic contribution. The latter is achieved by successfully utilizing the recently developed hierarchical upsampled thermodynamic integration using Langevin dynamics method. This approach gives direct access to a numerically highly precise volume- and temperature-dependent free-energy surface and derived properties. It enables us to assign the remaining deviations from experiment to inherent errors of the presently available exchange-correlation functionals. Performing the full analysis with both of the conventional functionals, local density approximation and generalized gradient approximation, we demonstrate that—when considered on an absolute scale—thermodynamic properties are dictated by a strikingly similar free energy vs volume curve. Further, we show that, despite an error in the $T=0$ K energy difference between the two phases ($\approx$6 meV in the present case), an excellent agreement of the temperature dependence of the Gibbs energy difference with experimentally derived data is feasible. This makes it possible, for instance, to unveil unreliable and possibly erroneous experimental input used in popular thermodynamic databases as we explicitly demonstrate for the isobaric heat capacity of calcium.

Figure 1

(a) Supercell size convergence and (b) electronic temperature (Fermi broadening) dependence of the quasiharmonic free energy of the bcc phase for the GGA-PBE functional at a lattice temperature of 716 K (experimental transition point[31]). In (a) the supercell size convergence (in terms of the conventional cubic bcc unit cell) is shown for three different electronic temperatures and corresponds to frequencies on a dense Fourier mesh. In (b) both curves refer to a $3\times 3\times 3$ bcc supercell; one is, however, calculated using only exact frequencies ${\omega }_q$ while the other is based on a dense ${\omega }_q$ mesh.

Figure 2

Volume dependence of the anharmonic Helmholtz free energy for both investigated xc functionals (LDA and GGA-PBE) and phases (fcc and bcc). The temperatures are (see main text for details) $T_{\text{md}}=800$ K, $T_{\mathrm{dyn}}^{\mathrm{el}}=T_{\mathrm{stat}}^{\mathrm{el}}=1160$ K, and $T_{\mathrm{ref}}^{\mathrm{el}}=1160$ K (for blue and orange curves) and $T_{\mathrm{ref}}^{\mathrm{el}}=0$ K (gray curves). Open circles correspond to a $3\times 3\times 3$ bcc supercell (54 atoms), closed circles to a $4\times 4\times 4$ bcc supercell (128 atoms), open triangles to a $2\times 2\times 2$ fcc supercell (32 atoms), and closed triangles to a $3\times 3\times 3$ fcc supercell (108 atoms). Statistical errors are of a similar size as the symbols indicating calculated values (not shown). Lines in between symbols are a guide to the eye. The arrows at the top indicate equilibrium volumes at $T=0$ K and the melting temperature, respectively, for both xc functionals and the experimental $T=0$ K equilibrium volume.

Figure 3

Illustration of the performance of the fit based on Eq. (21). In (a) the $\lambda$ dependence of the thermodynamic average ${\langle \cdots \rangle }_{NVT,\lambda }$ is shown for a fcc and bcc structure both at an MD temperature of 1000 K. Gray lines indicate a fit based on Eq. (21) and the red line is a third-order (cubic) polynomial fit. In (b) the root-mean-square error (RMSE) of the explicitly calculated points from the respective fit is shown for the bcc structure at various MD temperatures. At a single temperature different points correspond to different lattice constants.

Figure 4

Phonon dispersion of fcc Ca at 295 K along high-symmetry directions. The corresponding lattice constant $a$ for the theoretical calculations (LDA and GGA-PBE) is shown in the legend. The T${}_1$ and T${}_2$ branches correspond to the $\langle 1\overline{1}0\rangle$ and $\langle 001\rangle$ polarizations, respectively. The experimental data including error bars are from Ref. 39.

Figure 5

Analysis of the dynamical instability in the calcium bcc phase. (a) Phonon dispersion for LDA and GGA-PBE at the consistent $T=0$ K equilibrium volume $V_{0\mathrm{K}}^{\mathrm{eq}}$ (Table ). Solid lines show results for $T^{\mathrm{el}}=0$ K and dashed and dotted lines those for two increased electronic temperatures (GGA-PBE only). The inset enlarges the [110] direction with the upper red ticks indicating exact wave vectors for the bcc supercells used in the quasi/anharmonic calculations (numbers correspond to the three bcc supercells given in Table ) and with the lower red ticks indicating the exact wave vectors employed in the special treatment of the T${}_1$[110] description (Sec. 3f). The gray line shows the qualitative result of phonon-phonon interactions on the instability (Sec. 3f) for GGA-PBE at an MD temperature of 250 K. The T${}_1$[110] branch corresponds to the $\langle 1\overline{1}0\rangle$ polarization, whereas T${}_2$[110] corresponds to the $\langle 001\rangle$ one. (b) Effect of electronic temperature $T^{\mathrm{el}}$ and applied hydrostatic pressure $P$ (the other parameter being kept fixed at zero) on the GGA-PBE energy dependence for a shear of the unit cell $\delta$ corresponding to the long-wavelength limit of the T${}_1$[110] branch in (a) (see Sec. 3f for definition of $\delta$). The applied pressures of 0.9, 1.9, and 2.9 GPa correspond to lattice constants of 4.3, 4.23, and 4.175 Å, respectively. (c) Effect of electronic temperature $T^{\mathrm{el}}$, applied hydrostatic pressure $P$, and the shear $\delta$ on the electronic density of states close to the Fermi level $E_{\mathrm{F}}$. The curves for $T^{\mathrm{el}}$ and $P$ are shifted up for clarity by 1 and 2 states/eV$·$atom, respectively.

Figure 6

(a) Gibbs energy difference between bcc and fcc $\Delta G^{\mathrm{bcc}–\mathrm{fcc}}$ at ambient pressure (0.1 GPa). The orange dashed line corresponds to the GGA-PBE result shifted by $-$6 meV/atom. The vertical lines indicate the experimental fcc-to-bcc transition temperature $T_{\exp }^{\mathrm{fcc}\to \mathrm{bcc}}=716$ K (Ref. 31) and melting temperature $T_{\exp }^{\mathrm{melt}}=1115$ K (Ref. 31). calphad values are obtained from Saunders et al. (Ref. 42) and from the SGTE unary database (Ref. 43). The gray dotted line indicates a linear extrapolation of the SGTE data (Ref. 42) to $T=0$ K, since the original parametrization diverges at low temperatures. (b) Influence of the various Gibbs energy contributions on the GGA-PBE phase transition: $0\mathrm{K}=(T=0\mathrm{K})$ enthalpy difference, $\mathrm{h}=\mathrm{harmonic}$, $\mathrm{q}=\mathrm{quasi}$, that is, expansion influence, $\mathrm{el}=\mathrm{electronic}$, $\mathrm{ah}=\mathrm{anharmonic}$, and $\mathrm{vac}=\mathrm{vacancies}$.

Figure 7

Analysis of the temperature dependence of $\Delta G^{\mathrm{bcc}–\mathrm{fcc}}$ for LDA. In (a), the solid blue curve reproduces the LDA Gibbs energy difference at $P=0.1$ GPa from Fig. 6a ($\mathrm{full}=\mathrm{all}$ excitations included). The dashed curve has all but the electronic excitations included. The dotted curves show the electronic Gibbs energies for each phase separately. The shaded regions indicate whether fcc (gray) or bcc (green) is stabilized by the electronic excitations. In (b) and (c), the LDA and GGA-PBE electronic Helmholtz free energy is shown as a function of the volume $V$ for fcc and bcc, respectively. For each combination (e.g., LDA fcc), the free energy is plotted for the five given electronic temperatures. The vertical lines indicate the equilibrium volumes at $T=0$ K (dashed; black dash-$\mathrm{dotted}=\mathrm{experiment}$) and at the melting temperature (solid). The curves with open circles show the continuation of the free energy beyond the normal volume range into the regime of the other functional. The blue dotted curves show the temperature dependence of the Gibbs energy ($P=0.1$ GPa) and correspond therefore to the dotted curves from (a). The green pluses indicate FLAPW results [wien2k (Ref. 22)]. (d) Similar to (c) but this time showing the comparison between the full electronic free energy (darker curves) and the ideal electronic entropy term $-T{S}^{\mathrm{el}}/2$ (light shaded curves). Part (d) is discussed in Sec. 3b.

Figure 8

Thermodynamic properties at ambient pressure (0.1 GPa) derived from the free-energy surface including all studied excitation mechanisms. The vertical lines indicate the experimental fcc-to-bcc transition temperature $T_{\exp }^{\mathrm{fcc}\to \mathrm{bcc}}=716$ K (Ref. 31) and melting temperature $T_{\exp }^{\mathrm{melt}}=1115$ K (Ref. 31). The experimental data are from $•$ Anderson (Ref. 44); $\circ$ Touloukian (Ref. 50); $\blacksquare$ Bernstein (Ref. 52) (including error bars); and $\square$ Schulze (Ref. 49). Shown are (a) the thermal linear expansion $\epsilon \left(T\right)$ [Eq. (4)], (b) the linear expansion coefficient $\alpha \left(T\right)$ [Eq. (4)], and (c) the isothermal bulk modulus $B_T\left(T\right)$ [Eq. (2)] relative to its $T=0$ K value in the fcc phase $B_{0\mathrm{K}}^{\mathrm{fcc}}$.

Figure 9

Volume dependence of the quasiharmonic contribution to the thermal pressure $P^{\mathrm{qh}}$ at different temperatures for the (a) fcc and (b) bcc phase. The vertical lines indicate the equilibrium volumes at $T=0$ K (dashed; black dash-$\mathrm{dotted}=\mathrm{experiment}$ at $T=0$ K) and at the melting temperature (solid). The curves with open circles in (b) show the continuation of the pressure beyond the normal volume range into the regime of the other functional.

Figure 10

(a) Isobaric heat capacity $C_P$ [Eq. (3)] of fcc and bcc Ca at ambient pressure (0.1 GPa) containing all studied excitation mechanisms and (b) the influence of the latter for the GGA-PBE functional. The vertical lines indicate the experimental fcc-to-bcc transition temperature $T_{\exp }^{\mathrm{fcc}\to \mathrm{bcc}}=716$ K (Ref. 31) and melting temperature $T_{\exp }^{\mathrm{melt}}=1115$ K (Ref. 31). calphad data are taken from the SGTE database (Ref. 43). Experimental values are from $\square$ Ditmars (Ref. 56); $▸$ Robie (Ref. 57); $\blacksquare$ Ulyanov (Ref. 58); $•$ Kubaschewski (Ref. 59); $▵$ Jauch (Ref. 60); $\circ$ Clusius (Ref. 61); $▴$ Zalesinski (Ref. 62); $⧫$ Eastman (Ref. 63). See Fig. 6 for further notation.