Tight-binding modeling of interstitial ordering processes in metals: Application to zirconium hydrides

We present here a theoretical study of ordering processes in metal-hydrogen compounds based on a generalized perturbation method and on tight-binding coherent potential approximation. This approach is illustrated for zirconium hydrides, in which case we demonstrate that a cluster expansion of the ordering energy can be limited to effective pair interactions, the leading one being between hydrogen atoms in third-neighbor positions. These results are quantitatively confirmed by comparison to density functional theory calculations and qualitatively interpreted through orbital symmetry analysis. The method is then applied first to draw a preliminary Zr-H phase diagram and then to characterize the effect of lattice deformation on the ordering processes in zirconium hydrides.

Figure 1

Multiplet interactions under consideration (only H atoms on the interstitial SC sublattice are shown).

Figure 2

TB-GPM multiplet interactions (in eV) for structures described in Fig. 1. Note the different scales used for the different multiplets. The calculations have been performed for $c=1.0$, with 3280 $\vec{k}$ points and 200 points in energy.

Figure 3

Hydrogen-hydrogen pairwise interactions $V_i\left(c\right)$ (eV) as a function of hydrogen concentration $c$.

Figure 4

Elementary cell ${\text{ZrH}}_c{\square }_{2-c}$ ordered structures (drawn with ovito [45] with the color code being blue for Zr, green for H, and red for vacancy), used for TBIM validation.

Figure 5

Comparison between total energy sequences of structures represented in Fig. 4, in TBIM and DFT (vasp).

Figure 6

Walsh diagram of ZrH tetrahedrons for the ${\mathrm{A}}_{1.0}$, ${\mathrm{F}}_{1.0}$, and ${\mathrm{D}}_{1.0}$ structures. The relevant $C_{2v}$ group of symmetries is illustrated at the top. Only the lower occupied levels are illustrated.

Figure 7

Hydrogen-hydrogen pairwise interactions $V_i\left(c\right)$ (eV) issued by structure reverse method from DFT calculations compared to those calculated directly through TB-GPM.

Figure 8

Formation energies of the various structures relative to the stablest one, $\mathrm{\Delta }{E}_{\mathrm{form}}$, reconstructed with effective pair interactions either calculated directly within TB-GPM (red dashed line) or derived from DFT total energies (blue dashed line) as a function of the actual DFT calculations. The small solid red and blue symbols refer to the different ordered structures used in the fitting procedure. The large solid red (open green) dots recall the TB-GPM calculations (exact DFT values) for the previous limited set of structures A, B, C, $\cdots$ represented in Fig. 4.

Figure 9

Phase diagram of the ${\text{ZrH}}_c{\square }_{2-c}$ system obtained by Monte Carlo simulation, compared with experimental data [53]. The Monte Carlo results are displayed by blue solid lines and black symbols; the experimental ones are displayed by red and yellow lines. The red line shows the limit of fcc-fct hydride existence domains.

Figure 10

Stress effect on ordering for $c=1.0$ : variation of $V_i$ as a function of the lattice parameter from TB-GPM calculations.

Figure 11

Variation of the formation energies of the ${\mathrm{A}}_{1.0}$, ${\mathrm{B}}_{1.0}$, ${\mathrm{C}}_{1.0}$, ${\mathrm{E}}_{1.0}$, and ${\mathrm{F}}_{1.0}$ structures referred to the ${\mathrm{D}}_{1.0}$ one as a function of the lattice parameter, either reconstructed from the TB-GPM pair interactions of Fig. 10 (left) or directly calculated through DFT (right).