New insight into the intrinsic instability of fcc ZrH₂ by energy-resolved local bonding analysis

Abstract

The electronic-driven instability of fcc ZrH₂ due to the reduction of density of states (DOS) at E_F under the tetragonal distortion were used to be explained by Jahn–Teller effect or the shift of von Hove singularity. Here we explain this intrinsic instability with energy-resolved local bonding analysis by means of first-principles calculations. Our local bonding analysis reveals that this intrinsic instability stems from the peak of T_2g and E_g orbitals at E_F with the former contributing much more. Tetragonal distortion lifts the T_2g and E_g degenerate orbitals, causing the change to local Zr–H and Zr–Zr bonding. These two fct structures share similar Zr–H bonding but different Zr–Zr bonding due to the different Zr–Zr distances. For all these three structures, Zr-4s and 4p electrons do not contribute to any bonding but partially Zr-5s electrons participate in the Zr–H bonding. We discuss the hybridization of Zr and H orbitals for these three ZrH₂ structures. Presented calculations support the Jahn–Teller type effect and provide a comprehensive understanding of the intrinsic instability of Zr dihydrides.

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1 Introduction

Hydrogen in Zr and its alloy has received much interest in the past few decades due to hydride precipitation in the nuclear industry^1–3 and the interaction of Zr and H in Zr hydrides is of great interest in solid state physics.^4–6 CaF₂ type Zr dihydride is one of the typical Zr hydrides and exhibits instability under the tetragonal distortion, resulting in a double minimum structure, i.e., fct with c/a < 1 and fct with c/a > 1. There has been many attempts to elucidate this mechanism,^5,6 however, controversy still exists over the basic understanding of the intrinsic instability from fcc to fct structures.

The most popular explanation is Jahn–Teller effect, i.e., the splitting of the bands at the Fermi level (E_F).^4,5 Kularkova et al. have shown that the weakly dispersive segments of the energy bands along Γ–L does not play an exclusive role in the strong reduction in N(E_F) during the cubic–tetragonal distortion and they have concluded that the reduction in the DOS at E_F must be supplemented by a shift in energy of the band along the Γ–K direction.⁷ However, a recent calculation within a full-potential linearized augmented plane-wave (FP-LAPW) method has suggested two common features along the two major symmetry directions of the Brillouin zone, Γ–L and Γ–K directions, a nearly flat doubly degenerate band, and a van Hove singularity, respectively.⁶ They have concluded that the tetragonal distortion produces a strong reduction in the density of states (DOS) at E_F resulting mainly from the splitting of the doubly-degenerate bands in the Γ–L direction and the shift of the van Hove singularity to above the E_F. Particularly, band structure for fcc-HfH₂ including the spin orbit interaction shows that the degeneracy of the band at E_F in the Γ–L direction is lifted by this interaction. Thus, within this context, the Jahn–Teller model is no longer valid to explain the tetragonal distortion in HfH₂.

We notice that both the Jahn–Teller model and the von Hove singularity model accept the fact that the DOS at E_F is very high for fcc ZrH₂, and the DOS at E_F locates near the pseudogap for fct ZrH₂. Here we calculate the electronic structures of the fcc ZrH₂ and two fct ZrH₂ structures through first-principles calculations, and explain the reduction of DOS from fcc to fct transformation from the chemical bonding viewpoint, i.e. Crystal Orbital Hamilton Population (COHP) analysis.⁸ Recently, this method has been further developed to projected COHP and successfully applied to covalent, ionic and metallic bonding analysis.^9,10 Using this method we can obtain the energy-resolved COHP(E) plots which can indicate bonding, nonbonding, and anti-bonding contributions in a solid. Our calculation reveals that electron spin plays a less important role in these ZrH₂ compounds and there exists strong Zr–H and Zr–Zr bonding. Zr-4s and 4p electrons do not participate in the bonding process. The intrinsic instability of fcc ZrH₂ stems from high occupancy of T_2g and E_g with the former contributing much more. The contribution of different 4d orbitals to Zr–Zr bonding results in the two fct structures. The paper is organized as follows. Details of the computational methods have been described in Section 2. Results and discussions are provided in Section 3. Lastly, concluding remarks are presented in Section 4.

2 Computational method

We employed the Vienna Ab initio Simulation Package (VASP)^11–13 by utilizing the projector augmented wave (PAW) method^14,15 within the framework of the density functional theory (DFT).^16,17 The description of the exchange-correlation adopted the Perdew, Burke and Ernzerhof (PBE)¹⁸ generalized gradient approximation (GGA). For optimizations, k⃑-space integrations were performed with the 1 order Methfessel–Paxton method with SIGMA = 0.1, and for the static calculation, k⃑-space integrations with incompletely filled orbitals were performed with the tetrahedron method¹⁹ with Blöchl correction.²⁰ Optimizations were achieved by minimizing forces and total energies. The convergence criteria of the total energy and the force were set to be 0.01 meV and 0.001 eV Å⁻¹, respectively. We used a plane wave cut-off energy of 500 eV, which was sufficient for precise energetics for all the elements considered here. For bulk properties, 15 × 15 × 15 k⃑-mesh samplings have been applied in the Brillouin zone according to the Monkhorst–Pack scheme.²¹ All the calculations were performed with spin polarization treatment.

Since Zr has partially filled d electrons, we also performed the calculations above within the DFT+U and hybrid functional (HSE)^22,23 framework. The DFT+U method²⁴ was introduced with J = 0.51 and U_eff varying from 1.0 to 4.0. In these calculations, a sufficient k-mesh sampling was used for fcc and fct phases of ZrH₂ until the total energy satisfied the convergence criterions above. The HSE employs an admixture of Hartree–Fock-like nonlocal exchange interaction and Perdew–Burke–Ernzerhof (PBE) exchange in the construction of the many-body exchange (x) and correlation (c) functional as follows,

where (sr) and (lr) refer to the short- and long-range parts of the respective exchange interactions, whereas μ controls the range separation of the Coulomb kernel, varying between 0.2 and 0.3 Å⁻¹. We used μ = 0.2 Å⁻¹. The HSE functional is largely self-interaction free thus improving over the standard DFT description and enables us to achieve a correct understanding of correlated electronic systems, e.g., d and f electronic systems.^25,26 In the case of HSE06 calculations, a 9 × 9 × 9 k-mesh sampling was used for fcc and fct phases of ZrH₂.

Since there was no significant difference in the electronic structure within GGA, GGA+U and HSE framework, as shown later, thus, we only performed the energy-resolved local bonding analysis within GGA framework, i.e., project crystal orbital Hamilton population (pCOHP) and integrated crystal orbital Hamilton population (ICOHP), implemented in LOBSTER,^8–10 and Bader charge integration of Zr and H atoms were calculated within Y–T method²⁷ implemented in Critic2 software.^28,29

3 Results and discussions

3.1 Energetics of the tetragonal distortion for fcc ZrH₂

We start with the energetics calculation of the tetragonal distortion for fcc ZrH₂. Fig. 1 shows the energetics of fcc ZrH₂ as a function of c/a within the GGA and GGA+U framework. We find that Coulomb interaction has little effect on the energetics of fcc ZrH₂, suggesting the weak Coulomb interaction in Zr–H systems. Both of them can reproduce the double minimum structures with the ground-state fct structure with c/a < 1.0. For convenience, the fct structures with c/a < 1 are denoted as FCT1 and the fct structure with c/a > 1 are FCT2. Importantly, Coulomb interaction has no significant effect on lattice parameters of fcc and fct structures, which is consistent with the available experimental findings (cf. Table 1). Further detailed analysis reveals that these two fct structures are both tetragonal structures with space group 139 but with different c/a ratio. In these tetragonal structures, Zr atoms occupy the Wyckoff 2a and H atoms occupy the Wyckoff 4d sites. For fcc ZrH₂, every Zr atom loses ca. 1.5 electrons H atoms, while for fct ZrH₂ structures, every Zr atom loses the same electrons to H atoms within both GGA and GGA+U frameworks (cf. Table 1), suggesting these three ZrH₂ structures may be similar in Zr–H bonding, which is evidenced by subsequent local bonding analysis.

Fig. 1 Tetragonal deformation of ZrH₂ from fcc to fct within the GGA and GGA+U framework with an effective Coulomb interaction of 2.0.

Table 1 The lattice parameter (in Å) and Bader charge for Zr and H atoms in fcc and fct ZrH₂ within GGA and GGA+U, as compared with available experimental findings. For convenience, effective Coulomb interactions of 1, 2, 3 and 4 are denoted for U1, U2, U3 and U4

	a	c	c/a	Zr	H	Note
fcc ZrH2	4.822	—	1.0	+1.48	−0.74	GGA
fcc ZrH2	4.8215	—	1.0	+1.49	−0.75	GGA+U1
fcc ZrH2	4.8214	—	1.0	+1.49	−0.75	GGA+U2
fcc ZrH2	4.8214	—	1.0	+1.49	−0.75	GGA+U3
fcc ZrH2	4.8212	—	1.0	+1.49	−0.75	GGA+U4
fcc ZrH2	4.82	—	1.0	—	—	Expt^30
FCT1	5.0053	4.4476	0.889	+1.46	−0.73	GGA
FCT2	4.6473	5.1699	1.112	+1.46	−0.73	GGA
FCT1	5.0044	4.4489	0.889	+1.46	−0.73	GGA+U2
FCT2	4.6467	5.1709	1.113	+1.48	−0.74	GGA+U2
FCT1	4.985	4.430	0.889	—	—	Expt^6

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3.2 Electronic structure of fcc and fct ZrH₂

Now we focus on the electronic structure of these three ZrH₂ structures. First of all, we explore the electronic structure of fcc ZrH₂ within the GGA and HSE frameworks (cf. Fig. 2). It is clear that the Fermi level (E_F) locates at the peak of density of states (DOS), consistent with previous calculations.³¹ Calculations within HSE only shift H-s, Zr-p and Zr-d orbitals to lower energy with the hybridization between the orbitals above unchanged, suggesting the cause of the intrinsic instability of fcc ZrH₂ is unique. We also plot the band structure of fcc ZrH₂ with H and without H to investigate the electronic interaction of H and Zr atoms (cf. Fig. 3). It shows that the presence of H lowers the energy of d orbitals in fcc Zr. Particularly, there exists a strong hybridization between the H and Zr atoms in the energy range −10.0 to −5.0 eV. We find that Zr-s and p orbitals are shifted to lower energy and interact with the H-s orbital, suggesting the formation of Zr–H bonding. It also shows that the interaction among d orbitals in Zr is stronger, suggesting the presence of Zr–Zr bonding. In addition, we do not find the von Hove singularity along the Γ–K direction within the standard DFT framework, as a previous publication suggested within FP-LAPW.⁶

Fig. 2 Electronic structure of fcc ZrH₂ within GGA (a) and HSE (b) framework.

Fig. 3 Band structure of fcc ZrH₂ (left panel) and fcc ZrH₂ without H (right panel). The contribution of H-s, Zr-s, p and d orbitals are indicated by text with corresponding colors.

Then, we calculate the electronic structure of fct ZrH₂ within GGA and GGA+U framework (cf. Fig. 4). We find that the Coulomb interaction has a weak effect on the electronic structure of FCT structures and the E_F locates near the pseudogap of the total DOS for these two fct structures. Similar to fcc ZrH₂, there are two hybridization energy ranges, i.e., H and Zr from −10.0 to −3.5 and d orbitals of Zr between −3.5 and E_F, suggesting that strong H–Zr and Zr–Zr interactions still exist. The difference in the electronic structures of FCT1 and FCT2 is the shape of the H-s orbital in the energy range from −10.0 to −3.5 eV and the d orbitals in the energy range from −3.5 eV to E_F. These changes reveal that the tetragonal distortion lifts the degeneracy of the d orbitals in Zr and then changes the local bonding in ZrH₂. The details will be discussed in the following section.

Fig. 4 Electronic structure of fct ZrH₂ within GGA and GGA+U framework with an effective Coulomb interaction of 2.0.

3.3 Discussion

In the present work, we have calculated the electronic structure of these three ZrH₂ structures and it reveals that there exists hybridization between H and Zr atoms, and d orbitals of Zr, suggesting the presence of H–Zr and Zr–Zr bonding. Thus, it is necessary to analyse the local bonding and its change under the tetragonal distortion.

Since the electronic structure calculations within GGA, GGA+U and HSE framework are similar, we only plot the COHP curves and corresponding DOS of fcc ZrH₂ within GGA in Fig. 5. The minus value in COHP curves means bonding states and the positive value suggests anti-bonding states. We find that bonding states between Zr–H atoms locate from −10.0 to −3.5 eV and most of the Zr–Zr bonding states is in the energy range between −3.5 and E_F. It also shows that Zr-4s and Zr-4p orbitals do not participate in any bonding, which can be explained by their relatively lower energy and fully occupancy. Most of Zr-5s electrons participate in Zr–H bonding. Most of Zr–H bonding stems from the hybridization of H-1s and Zr-T_2g. Thus, we conclude that Zr atoms give partially 5s electrons (ca. 1.48 electrons) to H atoms, forming the Zr–H bonding in fcc ZrH₂ (cf. Table 1). Most importantly, we notice that both T_2g and E_g contribute to the peak at E_F with the former contributing much more. In the fcc ZrH₂, Zr atoms are octahedrally coordinated by H atoms. Thus, the crystal field of the H ligand splits the Zr-d orbitals into T_2g and E_g, like octahedral type crystal field splitting. Obviously, the present calculations reveal that the intrinsic instability of fcc ZrH₂ stems from the hybridization of Zr–Zr orbitals, especially the T_2g at E_F.

Fig. 5 COHP curves of Zr–H (red) and Zr–Zr (black) bonding and corresponding DOS of fcc ZrH₂ within the GGA framework.

When compared with fcc ZrH₂, we find that tetragonal distortion further splits T_2g and E_g orbitals, among which the d_yz and d_xz are degenerate, like square planar type crystal field splitting. For the FCT1 structure, the hybridization among H-s, Zr-5s, Zr-4d_x², Zr-4d_yz, Zr-4d_xz forms the Zr–H bonding. We also find that Zr-4d_z² and Zr-4d_xy contributes to much of the Zr–Zr bonding states with the Zr-4d_yz, Zr-4d_xz contributing to the peak near the E_F (cf. Fig. 6). For the FCT2 structure, the Zr–H bonding still presents but 4d_x² shifts to a higher energy level and doubly-degenerate orbitals shift to a lower energy level (cf. Fig. 7). The strength of Zr–H bonding (ICOHP value) for FCT1 (−0.764 eV) and FCT2 (−0.763 eV) are nearly the same due to the similar Zr–H distance (2.0899 Å for FCT1 and 2.0903 Å for FCT2). We also note that the strength of Zr–H bonding in fcc ZrH₂ is −0.766 eV (2.0877 Å), similar to that in these two fct structures, evidencing similar Zr–H bonding, as previously suggested. However, in terms of Zr–Zr bonding we find that the strength of Zr–Zr bonding increases from −0.445 to −0.274 eV, suggesting the Zr–Zr interaction is weakened due to the increasing Zr–Zr distance (3.4750 Å), compared with the FCT1 structure (3.3455 Å). Fig. 6 and 7 show that this weakness depends on the occupancy of the doubly-degenerate orbital i.e.,4d_yz, 4d_xz (FCT1) or 4d_z² and 4d_x² (FCT2) near the E_F. The calculations above reveal that these three ZrH₂ structures share similar Zr–H bonding, but different Zr–Zr bonding. The intrinsic instability originates from the peaks of T_2g and E_g with the former contributing more. The increasing strength of Zr–Zr bonding under the tetragonal distortion results in the fct structures with c/a < 1. Conversely, the decrease will result in the fct structures with c/a > 1.

Fig. 6 COHP curves of Zr–H (red) and Zr–Zr (black) bonding and corresponding DOS of FCT1 within the GGA framework.

Fig. 7 COHP curves of Zr–H (red) and Zr–Zr (black) bonding and corresponding DOS of FCT2 within the GGA framework.

4 Conclusions

In summary, we calculated the electronic structures of fcc and fct ZrH₂ by first-principles calculation and analysed the corresponding chemical bonding using the energy-resolved local bonding analysis. Our calculations revealed that the intrinsic instability stems from Zr–Zr bonding, particularly the T_2g orbitals. The tetragonal distortion splits the T_2g and E_g orbitals in fcc ZrH₂. Zr-5s electrons participate in the Zr–H bonding. The stronger Zr–Zr bonding results in the stability of FCT1 while the weaker one results in FCT2. These calculations reveal the intrinsic instability of fcc ZrH₂ and provide a basic understanding of the Zr–H interactions.

Acknowledgements

This work is supported by NSFC under the grant No. 91226203, 11404299 and Science and Technology on Surface Physics and Chemistry Laboratory (STSPCL) under the grant No. ZDXKFZ201410. We also acknowledge X.-Q. Chen at IMR for his helpful discussion.

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