Energy, metastability, and optical properties of anion-disordered $R{\mathrm{O}}_x{\mathrm{H}}_{3-2x}$ $(R=\mathrm{Y},\mathrm{La})$ oxyhydrides: A computational study

In this paper, we investigate by ab initio DFT how the O:H ratio influences the formation and lattice energy, metastability, and optical properties of Y and La anion-disordered $R{\mathrm{O}}_{\mathrm{x}}{\mathrm{H}}_{3-2\mathrm{x}}$ oxyhydrides. To achieve this, a set of special quasirandom structures (SQS) is introduced to model anion-disorder along the whole ${R\mathrm{H}}_3-R_2{\mathrm{O}}_3$ composition line. A comparison with an extensive set of anion-ordered polymorphs of the same composition shows the comparable energy of the anion-disordered phase, which, in particular, in the H-rich composition interval showed the lowest relative energy. In turn, the metastability of the anion-disordered phase depends on the cation size (Y versus La), which determines the maximum H content above which the ${\mathrm{CaF}}_2$-type structure itself becomes unstable. To overcome the accuracy limitations of classical DFT, the modified Becke-Johnson (mBJ) scheme is employed in the study of the electronic properties. We show that major differences occur between H-rich and O-rich R oxyhydrides, as the octahedral ${\mathrm{H}}^{-}$ present for $x<1$ form electronic states at the top of the valence band, which reduce the energy band gap and dominate the electronic transitions at lower energies, thus increasing the refractive index of the material in the VIS-nIR spectral range. Comparing the DFT results to experimental data on photochromic Y oxyhydride films reinforces the hypothesis of anion-disorder in the H-rich films ($x<1$), while it hints towards some degree of anion ordering in the O-rich ones ($x>1$). Our paper exemplifies a strategy to calculate ab initio the electronic/optical properties of a wide range of materials with occupational disorder.

Figure 1

Trend upon O:H ratio of the occupation of tetrahedral (yellow polygon) and octahedral (green polygon) anion sites in anion-disordered $R{\mathrm{O}}_{\mathrm{x}}{\mathrm{H}}_{3-2\mathrm{x}}$ oxyhydrides with ${\mathrm{CaF}}_2$-type structure [13]. Colour scheme: R, black; O, red; H, blue.

Figure 2

Selected examples of the structures considered in this paper. Colour scheme: R, black; O, red; H, blue. (a) Previously reported anion-ordered compounds. This includes the binary ${R\mathrm{H}}_3$ trihydride ($P{6}_3/mmc$, $Fm\overline{3}m$) and $R_2{\mathrm{O}}_3$ sesquioxide ($P\overline{3}m1$, $I\overline{3}a$), and the 1:1:1 $R\mathrm{OH}$ oxyhydride ($P4/nmm$, $Pnma$, and $P{2}_1/m$). The structural difference between $Pnma$ and $P{2}_{1}m$ is minimal, and in the figure it appears as a minor translation of the anions relative to each other. (b) Examples of anion-ordered oxyhydrides based on the ${\mathrm{CaF}}_2$ motif. Here, the anions preferentially occupy the tetrahedral sites (yellow polygons), while the octahedral ones (green polygons) only host the excess hydrogen for $x<1$. In this paper we considered all schemes of anion ordering that are possible within one fcc unit cell. The three structures here reported are the only ones that resulted in an energy stabilization compared to the anion-disordered lattice, in view of their particular symmetry that mitigates Coulomb repulsion between neighboring ${\mathrm{O}}^{2-}$ anions. In $R3m\text{−}R{\mathrm{O}}_{0.25}{\mathrm{H}}_{2.5}$ the only octahedral empty site is adjacent to the only oxygen atom; in $F\overline{4}3m\text{−}R\mathrm{OH}$ the anions alternate throughout the tetrahedral sites; and in $P\overline{4}2m\text{−}R{\mathrm{O}}_{1.25}{\mathrm{H}}_{0.5}$ all the next-nearest neighbours of the empty tetrahedral site are oxygen anions. (c) Examples of special-quasirandom-structures to model anion-disordered $R{\mathrm{O}}_{\mathrm{x}}{\mathrm{H}}_{3-2\mathrm{x}}$ oxyhydrides.

Figure 3

Energy difference along the $R{\mathrm{O}}_{\mathrm{x}}{\mathrm{H}}_{3-2\mathrm{x}}$ (R = Y, La) composition line between the anion-disordered and the most stable anion-ordered structure among those considered in this paper. For each composition, the space group of the most stable anion-ordered structure is indicated next to the point.

Figure 4

(a) Born-Haber cycle of R oxyhydride. The arrows are at scale for the 1:1:1 composition and indicate: Atomization energy ($\mathrm{\Delta }{H}_{\mathrm{at}.}$), R ionization energies, O and H electron affinities, oxyhydride lattice energy ($\mathrm{\Delta }{H}_L$), binding energy ($\mathrm{\Delta }{H}_b$), and formation energy ($\mathrm{\Delta }{H}_f$). [(b), (c)] Influence of O:H ratio on the formation energy of Y and La oxyhydrides, respectively. The insets focus on the stoichiometric 1:1:1 composition. [(d), (e)] Influence of O:H ratio on the lattice energy of Y and La anion-disordered oxyhydrides, respectively. For $x\le 1$, the (negative) lattice energy increases mainly due to a shift of charge from the less-stable octahedral to the more-stable tetrahedral anion sites. For $x>1$, further stabilization results from a larger freedom for structural distortion related to the presence of empty tetrahedral sites. For comparison, the lattice energy evaluated with the Born-Landé equation is also reported for three different structural models of increasing resemblance to the real material: (i) idealized anion-disordered model based on $Fm\overline{3}m$ symmetry (dotted lines), (ii) nonrelaxed SQS structures (solid lines), and (iii) relaxed SQS structures (dashed lines).

Figure 5

$R-\mathrm{H}$ radial pair distribution functions (top panels) and their cumulative values (bottom panels) for the relaxed SQS of $\mathrm{Y}$ and $\mathrm{La}$ anion-disordered oxyhydrides. The dashed-vertical lines refer to the thermodynamically stable $P{6}_3/mmc\text{−}{\mathrm{YH}}_3$ and $Fm\overline{3}m\text{−}{\mathrm{LaH}}_3$ hydrides. Yellow and green vertical lines mark the position of tetrahedral and octahedral interstitial sites for an ideal fcc structure with lattice constant tuned to match the $R{\mathrm{O}}_{0.25}{\mathrm{H}}_{2.5}$ composition. The pink region highlights peaks that do not match the ${\mathrm{CaF}}_2$-type structure. To help visualization, lines in the top panels are shifted downward by a constant value of $-0.2$.

Figure 6

Example of the effect of structural relaxation on the anion-disordered structure of ${\mathrm{YO}}_{0.25}{\mathrm{H}}_{2.5}$ and ${\mathrm{LaO}}_{0.25}{\mathrm{H}}_{2.5}$ oxyhydrides. The yellow polygons indicate the tetrahedral anion sites. Irrespective of the composition, the displacement of $R^{3+}$ and ${\mathrm{O}}^{2-}$ is minimal. However, a shift of the tetrahedral ${\mathrm{H}}^{-}$ to the empty octahedral sites is observable in the case of Y (see text). The pink arrows indicate the interatomic distances that correspond to the two additional peaks in the PDF (Fig. 5). In the case of La, no major reorganization happens and the ions only relax within the space of their initial site.

Figure 7

(a) Projected mBJ density of states around the Fermi level for Y and La anion-disordered $R{\mathrm{O}}_{\mathrm{x}}{\mathrm{H}}_{3-2\mathrm{x}}$ oxyhydrides. The valence band has hybrid character, with the extend of the hybridization depending on the lattice dimension and on the rearrangements of the H sublattice involving tetrahedral and octahedral sites (see text). A Gaussian smearing of 0.2 eV has been used in the figure to aid visualization. (b) Dependency of the direct and indirect energy band gaps upon the O:H ratio in Y and La anion-disordered $R{\mathrm{O}}_{\mathrm{x}}{\mathrm{H}}_{3-2\mathrm{x}}$ oxyhydrides. The energy band gap widens upon increasing O:H content, with a bigger jump at $x\ge 1$, where the octahedral anion sites are all empty. Lines are a guide for the eyes. (c) Direct/Indirect band gaps (filled/open points) of the Y and La anion-ordered oxyhydrides of lower energy compared to the anion-disordered phase. Reference values for the optical gap of $R_2{\mathrm{O}}_3$ oxides (corresponding to $x=1.5$) are also reported, showing the excellent predictive ability of the mBJ scheme ($\delta E_g<10\%$).

Figure 8

Imaginary and real mBJ dielectric functions of Y and La anion-ordered $R{\mathrm{O}}_{\mathrm{x}}{\mathrm{H}}_{3-2\mathrm{x}}$ oxyhydrides upon increasing O:H content. The electronic transitions responsible for the main features are indicated in the insets, which report the pDOS of ${\mathrm{YO}}_{0.25}{\mathrm{H}}_{2.5}$ and ${\mathrm{LaO}}_{0.25}{\mathrm{H}}_{2.5}$, respectively.

Figure 9

Comparison between the experimental optical gap of photochromic Y oxyhydride thin films [3] and the direct/indirect energy band gaps computed for anion-disordered ${\mathrm{YO}}_{\mathrm{x}}{\mathrm{H}}_{3-2\mathrm{x}}$. Lines are a guide for the eyes.

Figure 10

Comparison between the experimental refractive index of photochromic Y oxyhydride thin films [15, 16] and the ones computed for anion-disordered ${\mathrm{YO}}_{\mathrm{x}}{\mathrm{H}}_{3-2\mathrm{x}}$.