First-principles study of hydrogen ordering in lanthanum hydride and its effect on the metal-insulator transition

We discuss the structural details and the ordering of hydrogen in LaH${}_x$ for $2⩽x⩽3$. To this end, we combine first-principles calculations with the cluster-expansion method. This approach allows us to follow the H occupation of the interstitial sites within the face-centered cubic matrix of La atoms. We find that LaH${}_x$ clearly favors the fluorite structure at $x=2$ and adds excess H atoms at the octahedral interstitial sites. The ground-state behavior of the system is discussed and is found in agreement with experimentally observed structures at compositions LaH${}_{2.25}$ and LaH${}_{2.50}$; an additional ground state at composition LaH${}_{2.71}$ is predicted. The cluster expansion also permits an extensive scan of LaH${}_x$ structures with two octahedral vacancies per unit cell. For energetically favorable configurations, this scan yields a vacancy percolation threshold at LaH${}_{2.75}$ that possibly drives the concentration-dependent metal-insulator transition: The band gap calculated for isolated vacancy pairs disappears for percolating vacancy chains. This transition from metallic to insulating state is also experimentally observed near to the composition LaH${}_{2.8}$ and gives rise to the “switchable mirror” phenomenon.

Figure 1

This ground-state diagram combines approximately 250 DFT energies with $969770$ energies from a CE scan with up to eight La atoms for $2⩽x⩽3$. The ground-state line is determined by the DFT energies. Its five vertices, the ground states, are marked by blue arrows. (For our definition of “formation energy” see Sec. 2a.)

Figure 2

The CE energetics of Fig. 1 is overlaid by a color code that displays the number of unoccupied tetrahedral interstitial sites.

Figure 3

The ground-state structures at LaH${}_{2.25}$ (left) and LaH${}_{2.50}$ (right), found by density functional theory (DFT) as a result of the cluster expansion (CE) and confirmed by experiments (Refs. 10 and 11). The space groups are $I4/mmm$ and $I{4}_1/amd$, respectively. Both structures can be described as $1\times 1\times 2$ body-centered supercells of the conventional cubic unit cell. All tetrahedral sites are occupied and are not shown. For LaH${}_{2.25}$, the octahedral occupation can be viewed as a D$0_{22}$ structure, and for LaH${}_{2.50}$, the octahedral occupation can be associated with a 40 structure.

Figure 4

The figure shows the energetics of a CE scan over all configurations with exactly two H vacancies, La${}_{N_{\mathrm{La}}}$H${}_{3N_{\mathrm{La}}-2}$, for selected $N_{\mathrm{La}}⩽68$. The color code reveals the formation of vacancy chains: It indicates the number of nearest-neighbor octahedral vacancy-vacancy “bonds” in each structure; see the illustration on the left. At a hydrogen concentration $x⩾2.78$, the two H vacancies in each structure form exactly one nearest-neighbor pair, and those pairs are isolated from each other. When the H concentration decreases, $x⩽2.75$, more nearest-neighbor vacancy pairs form (gray boxes). Since each structure's unit cell comprises precisely two vacancies, more nearest-neighbor pairs are tantamount to the formation of vacancy chains, created by the translational symmetry of the cells.

Figure 5

The octahedral vacancy bonds discussed in Fig. 4 are shown for five selected structures, which are the energetically most favorable structures at the given concentrations, determined by the CE scan of Fig. 4. All structures are either ground states ($x=2.50$, $x=2.71$) or deviate only slightly ($<1\mathrm{meV}/$La) from the DFT ground-state line of Fig. 1. The chosen concentrations are above (left) or below (right) the percolation threshold near $x=x_{\mathrm{c}}=2.75$. For clarity, no La or H atoms are shown, and all the structures are depicted within a $3\times 3\times 3$ supercell of the unit cell with the least volume. In this way, the bonds become more clearly visible. Owing to the periodicity, the bonds within each structure are all alike. Note that bonds crossing the boundaries of the supercell are not shown.

Figure 6

The figure shows the electronic DOS for four selected La${}_{32}$ configurations inside a $2\times 2\times 2$ superlattice of the conventional cubic unit cell. The structures differ in the way the octahedral vacancy pairs are aligned. For clarity, no H atoms are shown in the given $\left(100\right)$ projections, and the octahedral vacancies are depicted as large gray spheres with orange nearest-neighbor bonds (cf. Figs. 4 and 5). In the plot of the DOS, the Fermi level is indicated by a dashed line. (a) In the two topmost structures, the vacancy pairs are isolated. The farther away from each other the pairs are, the more pronounced is the band gap at the Fermi level. The band gap in the La${}_{32}$H${}_{92}$ structure, where the vacancy pairs are relatively close together, hardly develops at all, and a small fraction (roughly $1\%$) of one electronic state is present up to the Fermi level. (b) As soon as the vacancy pairs percolate through the system—e.g., by formation of one-dimensional structures like a zigzag line or a linear arrangement—the band gap vanishes.