Structural and vibrational properties of lithium under ambient conditions within density functional theory

We apply a general first-principles approach to derive the phase diagram of metallic lithium at ambient pressure between 0 and 350 K, including identification of candidate phases. We use ab initio random structure searching to identify competing phases and supplement the results with calculations of vibrational properties and relevant derived neutron diffraction patterns. Strong quantum nuclear effects are present, prompting a careful treatment of vibrations. We directly map the Born-Oppenheimer surface of Li, allowing the extension of the normal quasi-harmonic treatment of vibrations to a “quasi-anharmonic” approach, where the effects of anharmonicity are included. The Gibbs free energies of the fcc, bcc, hcp, and 9R phases are derived using a variety of equations of state. We find that the anharmonic contribution to the Gibbs free energies of Li phases is of the same order as the differences between phases. Anharmonicity also makes a noticeable difference to the 0-K phonon dispersion of the bcc phase, with the largest difference at the $N$-point phonon. The ordering of phase transitions that we find agrees with the calculations of Ackland et al. [Science 356, 1254 (2017)], even when anharmonic effects are included, suggesting that a quasi-harmonic treatment is sufficient for correct phase behavior. We show explicitly that the martensitic phase transition from close-packed to bcc lithium upon heating is driven by entropic contributions to the phonon free energy.

Figure 1

The possible hexagonal-layer stacking sequences from 2 to 5 layers, showing the stacking sequence, the space group, and the electronic energy per atom relative to the fcc (ABC stacking) electronic ground state. Note that the energy differences are on the meV scale. The inset is an illustration of the A-, B-, and C-type layers. Structures which are not close packed as a result of having the same first and last layer in the sequence are labeled as such.

Figure 2

Simulated and experimental neutron diffraction patterns for Li. The experimental pattern ($<20$ K at ambient pressure) from [27] is shown, as well as patterns derived by thermodynamically weighting close-packed Li structures and AIRSS structures using DFT energies [see Eq. (6)]. For the close-packed patterns, all unique close-packed sequences consisting of fewer than 16 hexagonal layers are included. A Gaussian broadening was applied to the derived patterns, with a width derived from a best fit to the experimental data.

Figure 3

The Helmholtz free energy per atom vs volume per atom at 300 K for fcc and bcc lithium. The solid lines show the Birch-Murnaghan fit used to derive the 300-K data in Figs. 4 and 5.

Figure 4

The Gibbs free energy of the fcc and bcc Li phases at a range of temperatures. This data was obtained by fitting the Birch-Murnaghan equation of state to data for $F(V,T)$ from DFT calculations (see main text). The standard fitting error is shown as a shaded region. The result of including the anharmonic correction from Fig. 6 is also shown.

Figure 5

From the same calculation as Fig. 4, but showing the derived equilibrium volume and thermal expansion.

Figure 6

The correction to the Gibbs free energy of bcc and fcc Li that results from including the effects of anharmonic vibrations via Eq. (4).

Figure 7

The Gibbs free energy of Li phases at a range of temperatures. The data was obtained by fitting the Birch-Murnaghan equation of state to data for $F(V,T)$, obtained using DFT calculations (see main text). The standard fitting error is shown as a shaded region.

Figure 8

As Fig. 7, but neglecting the entropic contribution to the phonon free energy. With this modification we see that the phase transitions in the 0–300 K range disappear.

Figure 9

Phonon contributions to the Gibbs free energy of bcc Li.

Figure 10

The 0-K vibrational band structure of bcc Li. The solid line shows the normal harmonic phonon dispersion relation from Eq. (2). The dotted line shows the lowest eigenvalue of the anharmonic Hamiltonian given in Eq. (4). The Brillouin zone positions are given in terms of primitive reciprocal lattice vectors. The shapes of the mode potentials at points $N$ and $H$ in the Brillouin zone are shown in the inset with quadratic fits (red dashed lines) to illustrate the anharmonicity at the $N$ point.

Figure 11

As Fig. 10, but for fcc Li. In this case the anharmonicity is negligible.