The effect of relaxation on the structural and electronic ...

Chinese Journal of Physics 55 (2017) 2157–2164

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The effect of relaxation on the structural and electronic properties of a terbium superstoichiometric dihydride Zahia Ayat a,∗, Aomar Boukraa a, Bahmed Daoudi a, Abdelouahab Ouahab b a

Univ Ouargla, Fac. Des Mathématiques et des Sciences de la Matière, Lab. Développement des énergies nouvelles et renouvelables dans les zones arides et sahariennes, Ouargla 30 000, Algeria b Université Mohamed Khider Biskra, Département des Sciences de la Matière, Faculté des Sciences Exactes et des Sciences de la Nature et de la Vie, 07000 Biskra, Algeria

a r t i c l e

i n f o

Article history: Received 16 March 2017 Revised 3 June 2017 Accepted 29 June 2017 Available online 30 August 2017 Keywords: Rare-earth dihydrides TbH2.25 Density functional theory Ab initio calculations WIEN2k Relaxation

a b s t r a c t The electronic structure and equilibrium properties for the cubic rare earth superstoichiometric dihydride TbH2.25 are calculated using the full-potential linearized augmented plane wave method (FP-LAPW) within the density functional theory (DFT) in the generalized gradient approximation (GGA) and local density approximation (LDA) as implemented in the WIEN2k simulation code at 0 K. The structure of TbH2.25 is stabilized by local atomic relaxations in both approximations. The atomic positions under relaxation calculated with the GGA are closer to the ideal atomic positions than those calculated with the LDA. The GGA calculated equilibrium lattice constant is in excellent agreement with the available experimental data. The calculation of the density of states, electronic charge density and energy band structures show that this superstoichiometric dihydride has a metallic character with a mixed covalent and ionic bonding. © 2017 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.

1. Introduction The depletion of fossil energies has encouraged the search for new energy sources and energy vectors. Hydrogen as a handy clean energy carrier can be used in a variety of applications [1], which may be a viable solution to the problem of possible future energy shortages [2]. However, one of the major problems encountered in this case is related to the storage of hydrogen. This inflammable gas can be stored in gaseous, liquid or metal hydride forms. The latter way to store hydrogen is the safest method and presents several advantages [3], mainly the volume requirements for storing hydrogen is much smaller and it has a potential to work near ambient pressures and temperatures. One example of metal hydrides, the rareearth hydrides, have received much attention due to their potential application for hydrogen storage technology [4,5]; they have an ability to absorb and store hydrogen under moderate conditions of temperature and pressure [6]. Since 1950, the rare earth hydrides have been the object of many investigations due to their interesting properties [7-11]. Comprehensive reviews have been given by Vajda [12,13] and Schöllhammer [14]. The rare earth metal reacts directly with hydrogen to form RHy hydrides, which are normally nonstoichiometric [7]. In this respect the character of the bulk heavy rare-earth could be changed from a metallic hexagonal close-packed (HCP) (in most cases) solid solution, or α -phase (y ≈ 0.1), to an even more metallic CaF2 -type dihydride in a β -phase (y ≈ 2), to an insulating or semiconducting HoD3 -type trihydride in a γ -phase (y ≈ 3) [15,16]. ∗

Corresponding author. E-mail address: [email protected] (Z. Ayat).

http://dx.doi.org/10.1016/j.cjph.2017.06.017 0577-9073/© 2017 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.

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In rare earth superstoichiometric dihydride systems RH2+x , it may be observed that the repulsion between the x octahedral hydrogen (Hoct ) atoms may lead to their eventual ordering. For low enough temperatures and high enough concentrations of excess hydrogen (x ≥ 0.10), the order appeared to be long range, whereas the arrangement appeared to be short range ordered at small x values (x < 0.10) [17]. Khachaturyan [18] has predicted theoretically two possible superstructures for the RH2.25 composition: the first one is tetragonal and doubled along the c axis and the second is cubic. The latter structure was used to study CeH2.25 [19], PuH2.25 [20] and GdH2.25 [21]. However, none of these first-principles theoretical works had included the effect of local atomic relaxation inside the cell, an effect which was included in the ab initio works of Jorge Garcés et al. to study YHx in 2005 [22], more α -RH systems (R = Sc, Y, Ho, Er, Tm, and Lu) in 2009 [23] and HCP M-H systems (M = Sc, Y, Ti, Zr) in 2010 [24]. In the case of the superstoichiometric TbH2.25 dihydride, which is the subject of the present work, no first principles study has been carried out to investigate its electronic structure for the hydrogen sublattice symmetry (in the second superstructure of Khachaturyan), even though the experimental lattice parameters were determined by X-ray diffractometry (XRD) on TbH2+x (x < 1) [25] together with GdH2+x , DyH2+x and YH2+x materials, but without suggesting any type of structure. It is known that the density-functional theory (DFT) [26,27] is not only used to understand the observed behavior of condensed matter, but increasingly more often to predict properties of compounds that have not yet been determined experimentally. Consequently, it is expected that these predictions will be used to plan future experiments in a rational way or to analyze recent measurements. Therefore, in this paper, we shall present the complete results of a detailed electronic structure calculation of the superstoichiometric compound TbH2.25 , based on ab initio calculations at 0 K within density functional theory (DFT). In addition, we shall look for the role of local atomic relaxation inside the unit cell using both the GGA and LDA approximations. 2. Computational method Density functional theory calculations were performed using the WIEN2k package [28] with the full potential linearised augmented plane wave (FP-LAPW) method. The generalized-gradient approximation of Perdew, Burke, and Ernzerhof (GGA 96) [29] and the local density approximation (LDA) [30] were adopted to describe the exchange and correlation effects. The cut-off energy, which defines the separation energy between the core and valence states, was taken to be −8 Ry, while the Tb (5s2 5p6 5d1 6s2 ) and H (1s1 ) orbitals were treated as valence states. We did not treat the f orbitals of Tb as valence electrons but as core electrons because, in rare earths, the 4f electrons, being very close to the core, are expected to be chemically inert, i.e. they cannot hybridize with the other s, p, and d valence electrons anymore and are perfectly localized [31]. Self consistency is obtained using 10 0 0 k-points in the irreducible Brillouin zone (IBZ), without considering the ˚ The self-consistent calculations are spin polarization. All atoms were fully relaxed until the forces were less than 0.01 eV/A. considered to be converged only when the total energy of the system changes by less than 10−3 Ry. 3. Results and discussion 3.1. Equilibrium properties The conventional unit cell for the superstoichiometric dihydride TbH2.25 with the P m3¯ m space group (No. 221) is shown in Fig. 1 (by using Xcrysden [32]). Table 1 summarizes the results for TbH2.25 obtained before (unrelaxed state) and after (relaxed state) the geometrical optimization of the internal variables with both the GGA and LDA approximations. We can see that the atomic positions after relaxation calculated with the GGA are closer to the ideal atomic positions than those calculated with the LDA. The total energy versus volume is fitted by the non-linear Murnaghan equation of state [33]. Under relaxation, the energy vs. volume curves of TbH2.25 for both the LDA and GGA approximations are shown in Fig. 2(a) and 2(b), respectively, whereas those for the unrelaxed states are plotted in Fig. 2(c) and 2(d). From this fit, we can obtain the equilibrium lattice parameter (a0 ), the bulk modulus (B0 ), its first order pressure derivative (B0 ’ ), and the total energy (E0 ). The calculated structural parameters of TbH2.25 before and after the relaxation are reported in Table 2 together with other available data. It can be seen that this relaxation lowers the total energy as expected. To the best of our knowledge, the experimental or theoretical bulk moduli of this material have not been reported, and there is no direct ab initio theoretical information available for TbH2.25 related to the effects of interstitial H atoms on their local atomic environment. Our calculated values can thereby be considered as a prediction for future investigations. We find that the structure of TbH2.25 is stabilized by local atomic relaxations in both approximations, in agreement with the previous first-principles computational work in other hydrides [22-24]. Thus, the energy gain (relaxed minus unrelaxed minimum energy) after internal coordinate relaxation is 0.964 meV/cell within the GGA and 2.673 meV/cell within the LDA approximation. It is easily seen from Table 2 that the equilibrium lattice parameter after relaxation is in better agreement with the experimental value in both approximations, but the GGA equilibrium lattice parameter a0 value is closest to the experimental value and is smaller than that for the calculated lattice parameters of TbH2 and GdH2.25 . This accords with the experimental observations reported by Ref. [12], as we know that a0 decreases with increasing octahedral hydrogen content x [38-41] and the rare earth atomic number, respectively. As a general rule, Hoct atom addition leads to lattice contraction whereas Htet

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Fig. 1. The compound crystallizes in the CaF2 fluorite type structure: the large spheres represent rare earth atoms and the small spheres hydrogen atoms occupying tetrahedral sites and the central octahedral site. Table 1 Unrelaxed and relaxed positions of equivalent atoms for the TbH2.25 structure (space group) in units of lattice parameters (a, b, c) for the (x, y, z) coordinates, respectively. Unrelaxed x

Tb1 Tb2

H3

H4

Relaxed y

z

x

y

Z

LDA

GGA

LDA

GGA

LDA

GGA

LDA

GGA

LDA

GGA

LDA

GGA

0.00 0.50 0.50 0.00 0.25 0.75 0.75 0.25 0.25 0.75 0.25 0.75 0.50

0.00 0.50 0.50 0.00 0.25 0.75 0.75 0.25 0.25 0.75 0.25 0.75 0.50

0.00 0.50 0.00 0.50 0.25 0.25 0.75 0.75 0.75 0.75 0.25 0.25 0.50

0.00 0.50 0.00 0.50 0.25 0.25 0.75 0.75 0.75 0.75 0.25 0.25 0.50

0.00 0.00 0.50 0.50 0.25 0.25 0.25 0.25 0.75 0.75 0.75 0.75 0.50

0.00 0.00 0.50 0.50 0.25 0.25 0.25 0.25 0.75 0.75 0.75 0.75 0.50

0.00 0.50 0.50 0.00 0.24445753 0.75554247 0.75554247 0.24445769 0.24445769 0.75554247 0.24445769 0.75554247 0.50

0.00 0.50 0.50 0.00 0.24925265 0.7507435 0.7507435 0.24925265 0.24925265 0.7507435 0.24925265 0.7507435 0.50

0.00 0.50 0.00 0.50 0.24445769 0.24445769 0.75554231 0.75554247 0.75554247 0.75523367 0.24445769 0.24445769 0.50

0.00 0.50 0.00 0.50 0.24925265 0.24925265 0.7507435 0.7507435 0.7507435 0.7507435 0.24925265 0.24925265 0.50

0.00 0.00 0.50 0.50 0.24445769 0.24445769 0.24445769 0.24445769 0.75554231 0.75554231 0.75554247 0.75554231 0.50

0.00 0.00 0.50 0.50 0.24925265 0.24925265 0.24925265 0.24925265 0.7507435 0.7507435 0.7507435 0.7507435 0.50

atoms lead to lattice expansion. The LDA value is clearly smaller than the experimental one, indicating the occurrence of over-binding in this latter method. This shows that the GGA is the most reliable for the optimized lattice constants. Conversely, for the bulk modulus in both the relaxed and unrelaxed states, it is the GGA value which is lower than that of the LDA as a result of the over-binding characteristic of the LDA. Furthermore, in the GGA, the bulk modulus value of TbH2.25 is larger than that of TbH2 ; this behaviour is similar to that found experimentally in the ErHx system (as shown in Table 2) [37] and theoretically by Refs. [14,20,21], which indicates a cell volume contraction with increasing hydrogen content. Hence in both the relaxed and unrelaxed states, the GGA overestimates the lattice parameter whereas it underestimates the bulk modulus (B0 ) in comparison with the LDA, a feature also observed in several similar systems in other simulation works [42]. It can be concluded that the LDA, with or without relaxation, may incorrectly capture the electronic structure of cubic TbH2.25 . 3.2. Electronic properties From the last section we find that with both approximations the relaxed structure of TbH2.25 is more stable and the relaxed equilibrium lattice parameter agrees better with the experimental value. For this reason, all figures presented in this section are for the relaxed state.

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Fig. 2. Calculated total energy curves for TbH2.25 as a function of cell volume in the LDA approximation: (a) relaxed state, (c) unrelaxed state. In the GGA approximation: (b) relaxed state, (d) unrelaxed state.

The calculated electronic band structures at the equilibrium lattice constant for different high-symmetry points in the Brillouin zone and the total density of states DOS (measured in states per electron-Volt) of TbH2.25 in GGA and LDA at 0 K are shown in Figs. 3 and 4, respectively, where the line at zero eV indicates the Fermi energy. In Figs. 3 and 4, it is clear that TbH2.25 possesses a metallic ground-state, because several bands cross the Fermi level (EF ), in agreement with the electrical resistivity measurement interpretations [12]. In both approximations the energy band structures are qualitatively similar. Indeed, the crossings of bands with the Fermi level are nearly the same, where in the relaxed state the values of the Fermi energy are 0.56402 Ry in the GGA and a higher 0.59989 Ry in the LDA (as seen in Table 3). Another significant feature of the band structures in the two approximations is the different positions of the valence bands (at  ), where, in the LDA, these shift towards higher energies at the top of the valence band, and towards lower energies at the bottom of the valence band. All of these indicate a slight increase in the bandwidth, as a consequence of the reduced lattice parameter. We now turn our attention to the calculated total density of states of TbH2.25 which has similar features in both the GGA and LDA (see Figs. 3 and 4 respectively), especially at the Fermi level. However, these figures show small but non-negligible differences as the peaks in the GGA are fairly sharper and narrower than those of the LDA, and the total DOS in the LDA moves a little towards lower energies compared to the GGA. Therefore, it is concluded that the width of the valence states in the LDA is increased due to the lattice contraction. The present study shows that the density of states on the Fermi level N(EF ) in both the relaxed and unrelaxed states is not negligible, as seen in Table 3, and is smaller in the LDA than in the GGA, because the latter causes an under-binding effect of the crystal, yet the crystal structure is more compact in the LDA. Also from Table 3, it can be remarked that when the equilibrium lattice parameter decreases, the Fermi energy increases in both approximations. In order to explain the chemical bonding in TbH2.25 , we have calculated the total and partial density of states (PDOS), as shown in Fig. 5. From Fig. 5, the conduction band is mainly dominated by unoccupied s, p and d states from terbium and s from the hydrogen in octahedral sites (Hoct ). The Fermi level is completely dominated by the D-eg states of terbium. The upper valence states situated between −1.95949 eV and EF in the GGA and −1.93051 eV and EF in the LDA are dominated by the

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Table 2 ˚ bulk modulus B0 (in GPa), its first order presCalculated equilibrium lattice constant a0 (in A), sure derivative B0 ’, and total energy (Ry), for the GGA and LDA compared to other available data. Method TbH2.25

GGA LDA

TbH2.24 TbH2 GdH2.25 GdH2.26 (6) GdH2 ErH1.95 ErH2.091 a b c d e f

Exp. GGA GGA LDA Exp. GGA Exp. Exp.

Unrelaxed Relaxed Unrelaxed Relaxed

Unrelaxed Unrelaxed

a0

B0

B0 ’

E0

5.2558 5.2548 5.1032 5.1056 5.2308a 5.2993b 5.299c 5.143c 5.284d 5.326e

61.631 61.7312 72.5047 72.3416

3.526 4.8942 2.6947 3.5847

−93,762.408612 −93,762.409576 −93,700.517516 −93,700.520189

59.9989b 62.4385c 80.3544c

1.9613b 3.0152c 3.1906c

53.1873e 67 ± 3f 73 ± 4f

4.0861e 9 fixedf 8 fixedf

Ref. [12]-Expt.; Ref. [34]; Ref. [21]; Ref. [35]-Expt.; Ref. [36]; Ref. [37]-Expt.

Fig. 3. Density of states (right panel) and electronic band structure along the high-symmetry directions (left panel) of TbH2.25 in the GGA, the Fermi energy being at 0 eV. Table 3 Fermi energy (Ry) and density of states at the Fermi level (in states/Ry) for TbH2.25 in the two approaches GGA and LDA.

GGA LDA

Unrelaxed Relaxed Unrelaxed Relaxed

Fermi energy

N(EF )

Ns (EF )

Np (EF )

Nd (EF )

Nd-eg (EF )

Nd-t2g (EF )

NHtet-s (EF )

NHoct-s (EF )

0.56297 0.56402 0.60011 0.59989

39.69 39.80 37.21 37.87

0.00 0.00 0.00 0.00

0.17 0.17 0.20 0.19

2.14 2.13 2.18 2.13

1.93 1.92 1.95 1.91

0.21 0.21 0.23 0.22

0.08 0.08 0.07 0.07

0.16 0.16 0.14 0.13

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Fig. 4. Density of states (right panel) and electronic band structure along the high-symmetry directions (left panel) of TbH2.25 in the LDA, the Fermi energy being at 0 eV.

Tb D-eg states. Whilst it can be seen that in low energy levels ranging between −8,70,792 eV and −1.95949 eV in the GGA and between −8,89,663 and −1.93051 eV in the LDA, TbH2.25 dihydride exhibits hybridization mainly between the Tb d-t2g states and both H 1s orbitals, implying directional (covalent) bonding in Tb D-t2g -Htet (bonding distance is 2.2672 A˚ in the GGA and 2.1653 A˚ in the LDA) and a degree of ionic character in Tb D-t2g - Hoct (bonding distance is 2.6258 A˚ in the GGA and 2.5537 A˚ in the LDA). This can be clearly understood from the electronic charge density contours along the (110) plane, as shown in Fig. 6. The charge-density was presented only for the GGA method, because it is similar to that of the LDA method with a small difference. In Fig. 6, it is clear that appreciable charge density exists in the outer regions of Tb and Htet atoms with a slight deformation in the direction of these nearest-neighboring atoms. This feature confirms that the bonding between Tb and Htet atoms is certainly covalent, a fact confirmed by the hybridization analysis. At the same time, it is clear that very little electronic charge is shared between Tb and Hoct , where most of the valence electrons of Hoct are tightly bound around their atoms, and this implies that the bond has some ionic character (the electron density is much weaker in the middle of the bond). Therefore, the Tb-H bonds in TbH2.25 have a mixed (covalent-ionic) character, as is found in several metal hydrides [19-21,43-45]. Another point of interest is the existence of a little charge in the interstitial regions away from the bonds, which gives a metallic character to this compound. 4. Conclusion In the present study, ab-initio calculations were performed to investigate the structural and electronic properties for the cubic superstoichiometric TbH2.25 dihydride using the FP-LAPW method in the local density approximation (LDA); and the generalized gradient approximation (GGA) for the exchange correlation as implemented in the WIEN2k code at 0 K. The atomic positions under relaxation calculated with the GGA are closer to the center of the ideal atomic positions than those calculated with the LDA. The structure of TbH2.25 is stabilized by local atomic relaxations. The equilibrium lattice parameter under relaxation is in better agreement with the experimental value in both approximations, the GGA value being the best. The bulk modulus is determined by fitting the energy vs. cell volume curve to the Murnaghan equation of state. Under relaxation, the GGA value is smaller by 14.67% than that in the LDA. The electronic band structure, the density of states and electronic charge density confirm the metallic character of TbH2.25 . The total density of states in both approximations shows important differences in the vicinity of the valence band, where, in both the relaxed and unrelaxed states, the GGA presents a tendency for under-binding in this material. Atomic relaxation shows that the Fermi energy changes in all cases

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Fig. 5. The calculated total and partial density of states for TbH2.25 in the GGA (right panel) and in the LDA (left panel), the Fermi energy being at 0 eV.

Fig. 6. Calculated valence-electron-charge density contour (in electrons per A˚ 3 ) of TbH2.25 in the (110) plane.

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