First-principles study of hydrogen adsorption behavior in C15 Laves phase compound ZrV₂

With the rapid development of electronic technology, obtaining better vacuum conditions has become one of the essential requirements in the research and development (R&D) of electric vacuum devices. For sealed-off vacuum devices, acquisition and long-term maintenance of a high vacuum level are requisites for good performance.^1,2 Consequently, the removal of residual gases in vacuum devices is critical for their service life and performance. In particular, the residual hydrogen in an ultra-high vacuum (UHV) system cannot be easily pumped dynamically due to its small atomic mass.³ Because the use of getter materials that can absorb H₂ and other active gases would be an effective solution to achieve the desired vacuum degree and ensure the long-term stability and reliability of such devices,⁴ various getter materials are of major interest for hydrogen adsorption and have been extensively investigated.^5–7 

One type of promising hydrogen absorption material is AB₂-type intermetallic compounds with a Laves phase-type structure, which can establish stable hydrogen reactions relative to the hydrides of the component metals. Typically, this class of the intermetallic Laves phase includes the cubic C15 (MgCu₂-type), hexagonal C14 (MgZn₂-type), and hexagonal C36 (MgNi₂-type) structures.^9,10 It has been verified that only C14 and C15 type materials can be used to absorb hydrogen.^10–12 Zr-based alloys, consisting either the C14 or C15 type Laves phase, usually exhibit outstanding structural stability and high hydrogen-absorbing capacity. Inside Zr-based AB₂-type compounds, Zr is typically located on the A site and acts as the “hydride formation” element, whereas the B site is usually occupied by V, Mn, Cr, Fe, and Ni elements, playing a catalytic role in the formation of hydrides.^13,14

From studies on getter materials, it is known that ZrV₂ alloy materials with C15 type structures present a high hydrogen absorption capability by forming interstitial hydrogen solid solutions, namely, ZrV₂H_x.¹⁵ Both theoretical and experimental studies have been carried out to investigate ZrV₂ compounds. For instance, Zhang et al.¹⁶ studied the annealing effects on AB₂-type hydrogen storage compounds, it was found that annealing treatment decreases the lattice parameters of the C14 phase and increases those of the C15 phase. Furthermore, Lototsky et al.¹⁷ carried out experimental investigation on Zr–V hydrogen getters, including the hydrogen adsorption characteristics and vacuum thermodesorption of hydrogen. Obviously, it is difficult to further study the microscopic mechanism of ZrV₂ hydrogen adsorption from the atomic scale in the experimental study. However, on the theoretical side, the first-principles calculations can be used to illustrate the intrinsic reaction mechanism from the basic quantum mechanical electronic structure. Recently, Zhang et al.¹⁸ studied the equilibrium structure, the electronic and elastic constants, and some thermodynamic properties of the ZrV₂ compounds at various pressures using first-principles calculations. Hong and Fu¹⁹ investigated the binding energy and corresponding magnetic structures in ZrX₂H_0.5 (X = V, Cr, Mn, Fe, Co, and Ni) compounds by the first-principles approach. Therefore, a lot of efforts have been made on trying to investigate the routine properties of the ZrV₂ Laves phase compound and its hydride. However, the hydrogen absorption behavior of ZrV₂ compounds has not yet been investigated theoretically.

In this paper, the favorable interstitial sites and diffusion behavior of H atoms within the ZrV₂ lattice, as well as the stability and electronic structure of ZrV₂H_x (x = 0.5, 1, 2, 3, 4, 6, 7, and 12), were systematically investigated using the first-principles calculations. The aim is to conduct a comprehensive study on the hydrogen absorption behavior of ZrV₂ compounds and provide beneficial theoretical support for further experimental studies.

First-principles calculations based on the density functional theory (DFT) were carried out using the plane-wave pseudopotential technique implemented in a CASTEP package.²⁰ The generalized gradient approximation (GGA) introduced by Perdew–Burke–Ernzerhof (PBE)²¹ and the ultrasoft pseudopotentials²² were employed to describe the exchange correlation energy and the interactions between electrons and ions, respectively. Based on the convergence tests, the cutoff energy for the plane-wave basis was set to 350 eV for all calculations. The irreducible Brillouin zone (IBZ) was sampled by using the Monkhorst–Pack mesh²³ with a k-point grid of 5 × 5 × 5. Geometry optimization was performed using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm.²⁴ All the structures were fully relaxed during optimization, and the convergence criteria were set as follows: the energy changes on each atom are less than 5.0 × 10⁻⁶ eV/atom, the maximum force is less than 0.01 eV/Å, the displacement of atoms is less than 5.0 × 10⁻⁴ Å, and the maximum stress is less than 0.02 GPa.

For this work, the absorption energy of the hydrogen atom in the ZrV₂ compound can be defined as

where $EZrV2Hx$⁠, $EZrV2$⁠, and $E(H2)$ are the total energies of equilibrium ZrV₂H_x, ZrV₂, and H₂, respectively. According to this definition, a positive value of E_abs indicates an endothermic (unstable) reaction, and negative E_abs indicates an exothermic (stable) reaction. In addition, the complete linear synchronous transit-quadratic synchronous transit (LST-QST) method was performed in the searching transition state (TS) to investigate some possible diffusion paths and diffusion barriers of hydrogen atoms in the ZrV₂ crystalline lattice.

In order to validate that the selected computational method and parameters are reliable for studying the hydrogen absorption behavior of ZrV₂ compounds, the equilibrium lattice parameter of ZrV₂ has been computed as a test. Using the present computational method, the optimized equilibrium lattice parameter has been determined as 7.356 Å, which is in excellent agreement with the computed results from other authors (7.349 Å)¹¹ and the experimental value (7.442 Å).²⁵ Consequently, these results demonstrate that the selected computational method and parameters are appropriate for describing the present systems.

The crystal structure of the C15 Laves phase compound ZrV₂ with the Fd$3̄$m space group is shown in Fig. 1. In the AB₂ Laves phases, there are three types of tetrahedral interstitial sites: the 2A2B site formed by two A atoms and two B atoms, the 1A3B site formed by one A atom and three B atoms, and the 4B site formed by four B atoms, which correspond to the 96g, 32e, and 8b Wyckoff sites, respectively. In the process of hydrogen adsorption, hydrogen atoms are likely to occupy these three kinds of interstitial sites.

To study the occupancy of the hydrogen atom in these sites, we have computed the absorption energy for a ZrV₂ unit cell with a single H atom at 2A2B, 1A3B, and 4B sites, and the calculated results are summarized in Table I. From the calculations, the absorption energies of the H atom at 2A2B and 1A3B sites are −0.750 and −0.688 eV, respectively, which have a larger absolute value of the hydrogen absorption energy thcan the present computed value of −0.352 eV at 4B sites. Therefore, it can be assumed that 2A2B and 1A3B will be the two favorable interstitial sites in ZrV₂ when the hydrogen atoms are introduced into ZrV₂. For comparison, previous reported calculations and experimental values of the hydrogen absorption energy are presented in the same table. Our calculated hydrogen absorption energies at 2A2B and 1A3B sites for ZrV₂ are in good agreement with the computed results from other authors^11,19 and experiment values of −0.798 to −0.829 eV, respectively.²⁶ In addition, we can see that among 2A2B and 1A3B sites, the 2A2B interstitial site is more favorable for hydrogen adsorption. The results can be attributed to the space-filling effect, that is, the larger interstitial space can lead to more stable hydrogen atoms inside interstitial sites. However, the largest and smallest interstitial hole size in these three kinds of interstitial sites are 2A2B and 4B sites, respectively.

To study the hydrogenation-induced lattice expansion at 2A2B, 1A3B, and 4B sites, the lattice parameters for the ZrV₂ unit cell with a single H atom at these interstitial sites have been calculated, and the calculations are listed in Table I. During the calculations of lattice expansion after introducing hydrogen atoms, it is obligatory to allow relaxation of the ZrV₂ crystal lattice with respect to both atomic positions of all atoms and lattice parameters. From the calculated results, it can be seen that the extent of lattice expansion exhibits a linear increase during hydrogenation corresponding to the occupied interstitial by the hydrogen atom: 2A2B, 1A3B, and 4B sites. This is due to the fact that 2A2B sites with large space are easier to accommodate hydrogen atoms, which means that the lattice is not prone to distortion, i.e., the extent of hydrogenation-induced lattice expansion is smaller. In contrast, insertion of hydrogen atoms inside the 4B interstitial sites is difficult because it leads to a large volume expansion.

The diffusion of the hydrogen atoms inside the ZrV₂ crystal lattice is important for the study of hydrogen absorption behavior for alloys. It has been demonstrated that hydrogen diffusion in the C15 Laves-phase hydrides is realized by serial discrete jumps between nearest-neighbor interstices.²⁷ Since 2A2B and 1A3B interstitial sites are likely to be occupied by hydrogen atoms, it is essential to provide the fictitious structure formed by these two sites for investigating the different diffusion behaviors of the hydrogen atoms in the ZrV₂ crystal.

The spatial arrangement of the 2A2B and 1A3B sites is presented in Fig. 2. It shows that 2A2B and 1A3B sites together form a network that is somewhat analogous to the clathrate-type structure.²⁸ In the ZrV₂ unit cell, four 16-face polyhedra have symmetrical characteristics. It can also be seen that the polyhedral cages are a combination of the six-membered rings formed by 2A2B sites and five-membered rings formed by four 2A2B sites and one 1A3B site, in which the 1A3B site connects the six-membered rings. This is consistent with the previously reported experimental results for ZrV₂D_x, and it was considered that the hydrogen atoms themselves may form some ordered structures in the ZrV₂ Laves phase compounds.^29–31 

There are four possible diffusion paths for hydrogen atoms between two adjacent interstitial sites, as indicated by the arrows in Fig. 2. Path 1 with a distance of 1.127 Å is from one 2A2B site to another 2A2B site within the same six-membered rings (intra-ring). Paths 2 and 3, with different distances of 1.302 and 1.952 Å, respectively, are from the 2A2B site to another 2A2B site in the adjacent six-membered ring (inter-ring). Path 4 having the same initial and final site as path 3 is from the 2A2B site to the 1A3B site and 2A2B site along one side of the five-membered ring, with a total distance of 2.258 Å. The computed barrier energies of hydrogen atoms along these diffusion paths are illustrated in Fig. 3, and the diffusion paths of hydrogen atoms are indicated by numbers in Fig. 2.

It can be seen from Fig. 3 that paths 1, 2, and 4, corresponding to the barrier energy value of 0.150, 0.266, and 0.349 eV, respectively, are the most likely diffusion paths for hydrogen atoms. In the previous experimental study of the ZrV₂H_x hydride, the reported diffusion barriers for hydrogen atoms vary from 0.179 to 0.217 eV.³² The diffusion behavior of the hydrogen atoms inside the C15 Laves phase compound TiCr₂ has been studied, and the barrier energies are 0.17 and 0.44 eV for the path of 2A2B–2A2B (inter-ring) and 2A2B–2A2B (inter-ring), respectively.⁸ Therefore, our calculated barrier energies are in good agreement with the previously reported ones. In addition, the diffusion barrier energy of path 1 with minimum distance is the smallest, compared with the diffusion barrier of other paths. In other words, hydrogen atoms located in the 2A2B site prefer hopping within the same six-membered rings formed by the 2A2B site. It is worth noting that the diffusion barrier of hydrogen atoms exhibits a linear increase with the increase in path distance for paths 1, 2, and 4. Compared with path 4, path 3 has the same initial and final site, but its path distance is shorter. However, the diffusion barrier of path 3 is much larger than that of path 4; in other words, diffusion of hydrogen atoms along path 3 is more difficult. Consequently, it can be concluded that the existence of the 1A3B site can facilitate the diffusion of hydrogen atoms in the adjacent six-membered rings. We argue that the 1A3B site plays a bridge role in hydrogen diffusion although it is not the most favorable interstitial site for hydrogen absorption. Since the inter-ring jump has a virtual contribution to the overall diffusion of hydrogen atoms within the metal lattice, paths 2 and 4 are very important diffusion paths for the H diffusion in the ZrV₂ crystal.

Since the 2A2B interstitial site is the most favorable one for accommodating hydrogen atoms, hydrogen atoms were set in 2A2B sites to simulate ZrV₂ hydrides with different hydrogen concentrations. In order to make the calculations have a physical meaning, especially for the larger hydrogen occupancies, the Switendick criterion,³³ i.e., the distance of H–H atoms should be larger than 2.1 Å, was fulfilled to minimize the short-range repulsive interaction between hydrogen atoms. In addition, it is significant to evenly distribute the hydrogen atoms and maintain the symmetry of the crystal lattice as much as possible when multiple hydrogen atoms are placed in the interstitial sites because hydrogen atoms form a chemical bond not only with each other but also with metal atoms within the host lattice.³⁴ 

The absorption energies of ZrV₂H_x (x = 0.5, 1, 2, 3, 4, 6, 7, and 12) hydrides were computed and are shown in Fig. 4. In these adsorption configurations with different H concentrations, the absorption energies are negative when x is in the range of 0.5–6, which indicates that it is an exothermic (stable) reaction. However, with the accommodation of more H atoms, the adsorption energy becomes positive and increases nearly linearly. In other words, an endothermic (unstable) reaction is caused in the range of 7 ≤ x ≤ 12. Therefore, it can be argued that the hydrogen content reaches the maximum for the range of 6 < x < 7. The present calculations for the maximum hydrogen content of the ZrV₂ crystal show excellent agreement with the previous experimental values,^35,36 which have illustrated that ZrV₂ metallic hydrides absorb hydrogen up to x = 6.

Lattice expansion induced by the hydrogenation for ZrV₂H_x was observed and is presented in Fig. 5. It can be seen that the extent of volume expansion for ZrV₂H_x increases almost linearly with the increase in hydrogen content. It has been confirmed that the lattice expansion effect due to hydrogenation can lower the diffusion barrier of hydrogen atoms remarkably.⁸ That is to say, lattice expansion of the metal host can assist H diffusion. Thus, the hydrogen absorption capacity of ZrV₂H_x hydrides would be improved by increasing the extent of lattice expansion.

As we all know, it is of great importance to study the electronic structure before and after hydrogenation for hydrogen absorption materials. In order to further understand the microscopic mechanism of hydrogen adsorption, the orbital-resolved partial densities of states (PDOS) for the ZrV₂H_x hydrides were calculated, as shown in Fig. 6. Electron s and p states of V and Zr participate in bonding between the transition metal and hydrogen but to a much less extent.³⁴ Therefore, H-1s, V-3d, and Zr-4d orbitals are presented in PDOS plots. In general, the Fermi level (E_F) is set to zero as a reference and labeled with a black line. It can be seen from Fig. 6 that the hydrogen concentration significantly affects the PDOS shape and the peak positions. Compared to the electronic structure of ZrV₂ before hydrogenation, there are some s-bands for ZrV₂H_x hydrides below the energy of −4.0 eV, and the electronic states around E_F, consisting of metal d orbitals, are remarkably modified. For ZrV₂H_0.5 intermetallic hydrides, the s-bands associated with H atoms have main peaks at about −6.7 and −5.9 eV. Besides, there are two little new peaks for metal states, aligning with the energy of the H-1s orbital. This indicates a weak hybridization contribution to the hydrogen–metal interaction. It can be further seen that with the increase in hydrogen concentration, the discrete peaks for the H-s states gradually overlap with each other and form a continuous s-band shifted toward the low-energy region. Meanwhile, both the V-3d and Zr-4d bands more involved in bond formation are significantly broadened and stretch out into the H-1s band, and the hybridization of the H-1s orbital with the d band also becomes stronger. These features reveal that the charge is redistributed and the bonding contribution from the interaction between H-1s and metal d bands is strengthened as x increases. In addition, further increasing x generates stronger V-3d and weaker Zr-4d bands in the low-energy region. This reveals the fact that the interaction between V and H atoms is stronger than that between Zr and H atoms for ZrV₂H_x hydrides.

To further investigate the intensity of interaction between hydrogen and metal atoms, Mulliken population analysis was carried out in ZrV₂H_x hydrides. As listed in Table II, the bond orders (BOs), bond lengths (BLs) and BO^s (scaled bond orders) between H and Zr (or V) atoms were considered. Among them, the BO parameter denotes the overlapping charge distribution between two given atoms, and BO^s is usually used to evaluate the covalent bonding strength, which can be calculated by BO^s = BO/BL. Generally, a positive BO^s value indicates covalent interaction. The higher the BO^s value, the stronger the interatomic interaction. As shown in Table II, the BO^s values of H–V are always larger than that of H–Zr for each ZrV₂H_x hydride. From the general trend, all the BO^s values between H and Zr (or V) become higher as the hydrogen concentration increases. This means that the H–V bonds are stronger than the H–Zr bonds. Both types of bonds are covalent interactions and strengthened as x is increased. It is consistent with our previous PDOS analysis for ZrV₂H_x hydrides and the results from other authors^37,38 that the weaker (or non-)hydride forming elements B (e.g., V) form stronger bonds with H atoms than the hydride formation element A (e.g., Zr) for AB₂-type Laves phase compounds. Such an effect can be partly attributed to the fact that the H–V bonds are obviously shorter than the H–Zr bonds (shown in Table II), in which the atomic radius of the B-type V atom is smaller than that of the A-type Zr atom.

In this study, the favorable interstitial sites for hydrogen adsorption, the barrier energies for atomic H diffusion inside the ZrV₂ lattice, and the stability and electronic structure of ZrV₂H_x (x = 0.5, 1, 2, 3, 4, 6, 7, and 12) hydrides were systematically investigated through the first-principles calculations. The following conclusions can be drawn:

1. Absorption energies for a ZrV₂ unit cell with a single H atom at different sites, namely, 2A2B, 1A3B, and 4B, were computed. Results show that 2A2B and 1A3B are two favorable interstitial sites in ZrV₂ for hydrogen atoms, with the 2A2B interstitial site being the preferred one. In addition, the extent of hydrogenation-induced lattice expansion of the 2A2B site is also the smallest among these three kinds of interstitial sites.

2. The 2A2B and 1A3B sites can form some ordered clathrate-type structures, which will provide diffusion paths for hydrogen atoms. According to the calculated barrier energies for H atoms along different diffusion paths, it is found that hydrogen atoms prefer intra-ring diffusion to inter-ring diffusion, despite the inter-ring diffusion being non-negligible to the overall diffusion of hydrogen atoms within the ZrV₂ lattice. Moreover, the existence of 1A3B sites can facilitate the diffusion of hydrogen atoms in the adjacent six-membered rings.

3. Absorption energies of ZrV₂H_x hydrides were computed, and the results show that the hydrogen adsorption reaction is stabilized, with x ranging from 0.5 to 6, and further increasing x (e.g., to 12) results in instability. In other words, the maximum hydrogen content in a stable state is in the range of 6 < x < 7, which can provide beneficial theoretical support for experimental studies on hydrogen adsorption.

4. The electronic structures were studied by the PDOS and the Mulliken population. The bonding interaction between H-1s and V (or Zr) d bands is strengthened as x increased, and hydrogen atoms make a stronger covalent bond with V atoms rather than with Zr atoms.

All authors of this manuscript declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work.

The data that support the findings of this study are available from the corresponding author upon reasonable request.