H ordering in hcp M–H systems (M=Sc, Y; Ti, Zr)

international journal of hydrogen energy 35 (2010) 6025–6030

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H ordering in hcp M–H systems (M[Sc, Y; Ti, Zr) Jorge Garce´s a,*, Peter Vajda b a b

Centro Ato´mico Bariloche, Comisio´n Nacional de Energı´a Ato´mica, Avenida Bustillo 9500, 8400 Bariloche, Rio Negro, Argentina Laboratoire des Solides Irradie´s, CNRS- CEA, Ecole Polytechnique, þF- 91128 Palaiseau, France

article info

abstract

Article history:

The H-sublattice ordering and the effect of local atomic relaxations due to H interstitial

Received 24 November 2009

atoms on the structural stability of hexagonal hydrogen-M solid solutions (M ¼ Sc, Ti, Y, Zr)

Accepted 16 December 2009

is studied by means of the density functional theory. A first-principle based model is

Available online 13 January 2010

presented which, by considering the relaxation of the host metal cell, yields for the first time a suitable explanation for the observed chain-like short range ordering through

Keywords:

a coherent stress field along the chain as well as its limitation to only six hcp metals,

Hydrogen

namely Sc, Y, Ho, Er, Tm, and Lu, and the absence of a mono-hydride phase in the same

Ordering

elements.

Solid solution

ª 2010 Professor T. Nejat Veziroglu. Published by Elsevier Ltd. All rights reserved.

Transition metal First principle

1.

Introduction

The rare earth metals (R), including Sc and Y, absorb hydrogen readily and form – depending on concentration and temperature – solid solutions (a-phase) and/or hydrides (b- and/or g-phases), with often wide existence ranges around the stoichiometric compositions [1]. These deviations from stoichiometry are responsible for various interesting physical phenomena such as H-sublattice ordering and metal-insulator transitions, which are visible in many structural and electronic properties. A particularly intriguing characteristic is the presence of a meta-stable low-T solid solution a*  RHx phase observed in Sc, Y, Ho, Er, Tm, and Lu up to rather high x-concentrations and manifest for example as anomalies in resistivity around 150 K or peaks in internal friction. Neutronscattering experiments [2–4] have permitted to establish the a*-phase as a short range ordered structure formed of hydrogen pairs occupying second-neighbour tetrahedral (T) sites around a metal atom along the c-axis, H–R–H, and arranging into zig-zagging (probably helicoidal) pairs in the c-direction, the pairs being shifted along the b-axis of the hcp

cell forming a chain-like structure as it is shown in Fig. 1a. The chain lengths varied from one system to another and seemed to correlate with the elastic anisotropy of the metal. (A review of the experimental situation has been given in [1]). This peculiar chain-like structure has intrigued the theoreticians soon after its discovery more than 20 years ago [5–7]. However, a complete understanding of the phenomenon was only recently available [8] and important questions remained open without a definite answer: e.g. the solubility limit in the a*-phase varying from xmax(a*) ¼ 0.03 H/R in Ho to 0.35 H/R in Sc or the fact that the chain-like structure is observed in hcp systems where the mono-hydride phase is not present in the phase diagram. Indeed, the chain structure is observed in Sc and Y and, the mono-hydride is observed in Ti and Zr (see Fig. 1b). In this study, we shall present evidence regarding the source of the chain stability and the relation between its exclusive occurrence in six metals and the absence of a monohydride in the same elements, through a first-principle calculation at 0 K. We will focus our attention on the role of local atomic relaxations inside the cell and assume that they

* Corresponding author. Tel./fax: þ54 2944 445191. E-mail address: [email protected] (J. Garce´s). 0360-3199/$ – see front matter ª 2010 Professor T. Nejat Veziroglu. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijhydene.2009.12.079

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international journal of hydrogen energy 35 (2010) 6025–6030

a

cnnn c-axis H

a-axis

basal Plane cnn

b

Fig. 1 – (a) HCP unit cell of MHx in the a*-phase showing two adjacent H–M–H pairs (dashed lines) shifted along the b-axis. Also shown are the basal plane, the nearest (cnn) and the next nearest neighbours (cnnn) to the hydrogen atom labelled H. (b) Unit cell of the mono-hydride phase showing two H–M–H pairs (dashed lines).

play an essential part in the hydrogen behaviour and the ensuing structural stability of the M-H systems in question (M ¼ Sc, Ti, Y, Zr).

2.

Computational method

The full-potential LAPW method based on density functional theory as it is implemented in the WIEN2k code [9] was used for the calculations. This code uses the fullpotential APW þ lo method that makes no shape approximation to the potential or density. The generalized gradient approximation of Perdew, Burke and Ernzerhof [10] was

used for the correlation and exchange as no experimental or theoretical evidence of strong correlations was found on the systems studied here. The atomic sphere radii, RMT, selected for the M and H elements were 2.1–2.4 bohr and 1.0–1.2 bohr, respectively. Local orbital extensions were included for the semicore states of M elements. The basis set size RMTK max (RMT is the smallest atomic sphere radius inside the cell and K max is a cutoff for the basis function wave vector) were chosen, respectively, as 9 and 4–4.5 for pure M and the M–H systems studied. The charge density cutoff Gmax was selected as 75–82 eV1/2. The maximum l values for partial waves inside the spheres and for the non-muffin-tin matrix elements were selected to lmax ¼ 12 and lmax ¼ 6, respectively. A mesh of 250–300 special k-points was taken in the whole Brillouin zone. The iteration process is repeated until the calculated total energy, charge and force components ˚3 converge to less than 7  104 eV/cell, 3  104 electrons/A ˚ , respectively. The minimization proceand 2  102 eV/A dure includes the optimization as a function of the lattice parameters a and the c/a ratio, followed by the relaxation of the internal atomic coordinates in the optimized values of the lattice parameters. A necessary additional step includes checking for changes in the lattice parameters against atomic displacements, which could result in further minimization of the internal parameters. In our work, good agreement was found between the first and the second step in the minimization procedure. For example, the energy difference between these two steps is only 0.9 meV/cell in ScH1/3 in the Pmma space group. The respective lattice parameter changes are less than 0.1%.

3.

Results and discussion

3.1.

Local atomic relaxation and one H atom

In order to study the effect of one H atom on its local tetrahedral environment, a hexagonal supercell (2  2  3 hcp unit cell) in the P1 space group with MH1/25 composition and the lattice parameters of the pure elements have been chosen. The octahedral site was not studied in this work as it had been found in earlier experimental and theoretical work that the tetrahedral site is the most favourable [1,11]. The theoretical results clearly show that there are two different relaxation patterns concerning the force components around the H atoms: i) pattern I for Sc and Y, as Fig. 2a shows, where the nearest neighbor (nn) metal atoms feel small radial forces while the next nearest neighbor atom along the c-axis (cnnn) feels the biggest one, similar in magnitude for the two ˚ ) and, elements (w0.5 eV/A ii) pattern II for Ti and Zr, as Fig. 2b shows, where all atoms ˚ ) being felt feel big radial forces, the biggest one (w1.4 eV/A by the nearest neighbor atom along the c-axis (cnn). The energy gain after internal coordinate relaxation is 38 meV/cell and 64 meV/cell for ScH1/25 and TiH1/25, respectively. In all cases, the H atom moves toward the basal plane of the tetrahedron, in agreement with experimental results [12].

international journal of hydrogen energy 35 (2010) 6025–6030

1.4 0.26

0.13

0.4

0.5

respectively. In addition, a mixing with the cnnn atom states is observed, which is smaller in Ti and Zr. The H–s state has no contribution near the Fermi level in all cases.

3.2.

0.13

0.75

0.2 0.44 ˚ ) on Fig. 2 – Schematic view of the atomic forces (in eV/A the neighbour of one H atom for: a) ScH1/25 and b) TiH1/25. YH1/25 and ZrH1/25 present the same force component pattern as ScH1/25 and TiH1/25, respectively. Only ˚ are displacements due to forces bigger than 0.1 eV/A shown.

The relaxation patterns were confirmed by using a supercell of 3  3  3 hcp unit cells. The effect of the H atom, for example ˚. in the ScH1/55 supercell, extends as far as 5.85 A The Density of Sates (DOS) was studied in order to analyze the atomic interaction between the hydrogen atom and its neighbor atoms. The total DOS has been decomposed into its partial waves (s,p,d) components around the H, the basal plane, cnn and cnnn atomic sites (see Fig. 1a). The analysis of the DOS around the H peak in MH1/25 shows that the main H bonding is realized with the cnn and basal plane atoms through the s,d and p states of Sc and Y while, in Ti and Zr, the order is d, s and p states, as shown in Fig. 3a and b,

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Local atomic relaxation and two H atoms

Previous theoretical works found the pair H–M–H as the most stable configuration in the Y–H system [5–7], but the effect of local relaxations or other elements like Sc, Ti or Zr were not treated. Therefore, the addition of a second H atom will allow us to evaluate the host lattice reaction in the presence of two H atoms. A hexagonal supercell (2  2  3 hcp unit cell) in the P3m1 or P1 space group with composition MH2/26 and the theoretical equilibrium lattice parameters are used in this study. The theoretical results show that the pair is the most stable configuration and local atomic relaxations stabilize this configuration even more in the four systems studied in this work. For example, the pair in the unrelaxed ScH2/26 system is more stable than two random H atoms by 77 meV/cell while, after atomic relaxation, this difference is 86 meV/cell. The same differences in TiH2/26 are 88 meV/cell and 39 meV/cell for the unrelaxed and relaxed situations, respectively. Recently, Tao et al. [13] have published first-principle VASP calculations in the Ti–H system. The authors found that the random hydrogen atom is the most stable configuration, in apparent contradiction with the results obtained in our work. The source of the discrepancy is the small 2  1  1 supercell used by these authors which prevents an analysis of dilute solid solutions due to interactions between H atoms mainly in the c-axis direction. Thus, even if the force pattern shows initially the same basic features observed for all MH1/25 cases, the behavior of the host lattice in the presence of H atoms is quite different if relaxations are included in the description. In Ti and Zr, the pair extends mainly to the nearest neighbor along the c-axis,

Fig. 3 – Site and partial waves analysis of the DOS for: a) ScH1/25 and b) TiH1/25 . The atoms are labelled according to Fig. 1a. s-state: solid line, p-state: dashed line and d-state: dot-dashed line. The bonding strength in ScH1/25 decreases in the order s, d and p for the cnn and basal plane atomic states respectively, while in TiH1/25 it is d, s and p for the same atoms. A small mixing with the cnnn atom orbitals is also observed.

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Fig. 4 – Evolution of the total DOS with the composition for ScHx (x [ 0, 1/25, 2/26, 4/28). The figure shows the changes in the shape of the H peaks as it is the only contribution below L5 eV. The most relevant information is the change from one peak for one H atom to two peaks once the H–Sc–H pair is formed. The dashed line is the Fermi energy set to zero.

while the influence of the H atoms upon the next nearest cnnn atoms in Sc and Y implies that they also must be taken into account for a proper description of the pairs and the overall relaxation behavior. Again, the analysis of the DOS allows us to understand the atomic bonding between the H atoms and their neighbors. The most relevant feature obtained for this composition is the split

of the H peak, centered around 5.5 eV for one H atom, into two peaks for each H atom for higher concentrations, as Fig. 4 shows. The calculations also show an unexpected drastic change in the bonding characteristics between the H s-state and the first neighbor electronic state in the structure, as shown in Fig. 5a and b for ScH2/26 and TiH2/26, respectively. The higher energy peak (to the left) in MH2/26 mixes mainly with the s and d states belonging to the cnn atom while the lower energy peak (to the right) mixes exclusively with the cnn p states. The basal plane states mix equally with both peaks. The small mixing with the cnnn states remains unchanged. Under relaxation, the former peak does not move while the latter moves slightly, by about 55 meV, to higher energies. This peculiar bonding structure is probably the reason for the high stability of the H–M–H pair observed in different systems of H in transition metals. Remarkably, the chain formation is observed only in the group of elements with the first relaxation pattern, namely Sc and Y, while Ti and Zr, with a relaxation pattern shown in Fig. 2b, exhibits experimentally no isolated H-pairs or chain formation but only mono- or/and dihydride phases [11].

3.3.

Chain structure or mono-hydride formation

The different behaviour of the host lattice in the presence of interstitial H poses the question about the behaviour of the pairs when two of them approach each other at low temperature. For this purpose, we studied the local atomic relaxations in the system MH1/7 with two H pairs, in a hexagonal (2  2  3) supercell in the Cmcm and P1 space group, arranged in two different configurations: i) two pairs in the chain-like configuration (see Fig. 6a), and ii) two pairs as close as possible in the hcp structure (compact pairs structure) (see Fig. 6b). The investigation of the stability of the local chain structure and its limitation to six metals can be reduced to that of the competition between these two kinds of pair arrangements

Fig. 5 – Site and partial waves analysis of the DOS for: a) ScH2/26 and b) TiH2/26 . The atoms are labelled according to Fig. 1a. s-state: solid line, p-state: dashed line and d-state: dot-dashed line. The higher energy peak (to the left) mixes mainly with the s and d states belonging to the cnn atom while the lower energy peak (to the right) mixes exclusively with the cnn pstates. The basal plane and the cnnn states mix equally with both peaks.

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Fig. 6 – Two H–M–H pairs arranged in: (a) chain-like structure and (b) compact pair structure. The 2 3 1 3 1 cell of Tao et al. is drawn by thick lines.

which, ultimately, is the competition between the tendency to form chain-like structures or mono- and/or dihydride phases, as will be shown in this section. It is easily seen in Fig. 6b that the cell used by Tao et al. [13] is too small to study the occupation order of the indicated interstitial sites. However, the occupancy order e–f–g–h found by these authors confirms the compact pair formation. Table 1 shows the effect of local atomic relaxations on the structural stability of the chain-like and the compact pairs structures in MH1/7 systems (M ¼ Sc, Ti, Y and Zr). Fig. 7 shows the effect of local relaxations on the relative stability between these two structures versus lattice parameter. Although the relaxed chain-like structure in ScH1/7 (YH1/7) is stabilized with respect to the relaxed compact pairs at the theoretical equilibrium lattice parameters, a volume effect was found since the latter become more stable under compression as shown in Fig. 7. That means the chain-like structure stabilization occurs due to atomic relaxations together with electronic effects via the equilibrium volume. On the other hand, the compact pairs configuration is always the most stable structure for the TiH1/7 and ZrH1/7 systems and is strongly stabilized under relaxation, in agreement with experimental results where no evidence of single pairs or chain structures were found at any temperature [11] and the

Table 1 – Effect of atomic relaxations on the structural stability of MH1/7 systems (M [ Sc, Ti, Y and Zr). Energy gain differences (in meV/cell) between the relaxed and unrelaxed chain-like and compact pair structures. The total energy calculations were made in a hexagonal supercell (2 3 2 3 3 unit hcp cell) in the Cmcm space group. The calculation for ScH1/7 was also made in the P1 space group. DE (meV/cell)

ScH1/7

TiH1/7

YH1/7

ZrH1/7

Chain

137 148 (P1) 120

276

146

215

351

109

269

Compact pairs

Fig. 7 – Energy difference (in eV/cell) between the chain-like and compact pairs structures versus lattice parameter for: squares: ScH1/7, circles: TiH1/7; up triangle: ZrH1/7 and down triangle: YH1/7. Open signs: unrelaxed structure differences. Full signs: relaxed structure differences. The arrows are drawn at the theoretical equilibrium lattice parameter a, and their directions show the effect of relaxation upon the energy difference.

mono-hydride is observed in the phase diagram. The monohydride phase can be interpreted as a regular assembly of compact chains in the (110) plane alternating pure metal (110) planes in such a way that each Ti atom shares 4 H atoms in a planar arrangement and each H atom shares 4 Ti atoms in a diamond like arrangement, as shown in Fig. 1b. On the other hand, the dihydride could be described as a regular assembly only of compact chains in the (110) planes. Thus, the monoand the dihydride phases are structures built up with a basic block such as the compact pair configuration of Fig. 6b. Therefore, the latter could be interpreted as the seed for the hydride formation. Our calculations show that the distances and the angles in this hydride seed are very close in the hcp and in the cubic structures. Thus, for example, the distance ˚ and Ti–Ti is 2.96 A ˚ , while H–Ti in the mono-hydride is 1.87 A the distances in the compact pairs structure in the hcp phase, ˚ , Ti–Ti: 2.90 A ˚ and 2.975 A ˚ . On after relaxation, are H–Ti : 1.80 A the other hand, the angle formed by the two metal atoms and the H located in the vertex of a pair is 143.7 deg in the compact pair structure and 142.4 deg in the mono-hydride phase. The

Table 2 – Energy differences (in meV/cell) between the relaxed and unrelaxed chain-like structures for MH1/7 and MH1/4 (M [ Sc,Y). The energy gain is decomposed in two contributions: one due to the cnnn and H atomic movement along the c-axis, the other due to the movement of the remaining atoms belonging mainly to the tetrahedron basal plane. DE (meV/cell) relaxed–unrelaxed (cnnn, H)–unrelaxed (cnnn, H)–relaxed

ScH1/7

YH1/7

ScH1/4

YH1/4

148 144 4

146 120 26

113 99 14

126 110 16

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Fig. 8 – A model for the chain formation. The elastic short range interaction between the atoms cnnn belonging to two nearest neighbor pairs creates the basis for the chain stability.

analysis of the hydride seed distances and angles in the hcp and in the cubic structures reveals that only a small change in angles and distances is needed to transform the basic block in the hcp structure into a seed for the mono-hydride phase. Thus, if local relaxations are included in the description of M–H systems, the exclusive existence of the chain-like structure in only six hcp metals can be easily understood. It would only be observed in systems with the local atomic relaxation pattern I: namely, Sc and Y, where the atom cnnn shows strongest displacement along the c-axis. The competition between the chain-like structure and the compact pairs structure, and the resulting a*- or mono-hydride phase formation, reveals ultimately the different reaction of the host lattice to the presence of H atoms and the particular way in which two H–M–H pairs interact.

3.4.

A model for the chain structure stability

Although it is difficult to compute with enough confidence the energy contributions due to small atomic displacements, it is possible to identify qualitatively the main source of the chain stability. The main energy gain from relaxation is due to the displacements of the cnnn and the H atoms along the c-axis as was obtained previously. But, there is a small secondary energy gain due to the atomic movements (principally due to distortion of the tetrahedron basal plane) produced by the interactions between pairs that can be considered as a source of the chain stability. This energy gain due to interaction between nearest neighbour pairs depends on the element and on composition, as it is shown in Table 2. Thus, the atom cnnn, associated with each H atom and also with one of the tetrahedron basal plane of the nearest neighbour pair (see Fig. 1a), could be considered as the nexus between pairs and responsible for the chain structure formation. It is possible that an extra energy gain is provided by lateral interactions between chains. The short range interaction provided through the cnnn atoms supports the idea of a coherent stress field along the chain suggested in [14]. Its main features will ultimately depend on the different response of the electron density inside each tetrahedron to the presence of an interstitial H atom. Fig. 8 shows schematically the arrangement between the elastic deformation and the hydrogen H–M–H pairs. The result of the interaction between them is the chain-like structure.

4.

Conclusions

The results presented in this paper show that local atomic relaxations are the key to understand the behaviour of H atom ordering in the hcp phase of M–H systems (M ¼ Sc, Ti, Y, Zr)

and the chain-like structure formation in only six hcp elements. It will only occur in systems where the next nearest neighbour to the H atom along the c-axis (cnnn) exhibits ˚ . The short range intera displacement of the order of 0.1 A action between the cnnn atoms belonging to two nearest neighbour pairs provides a coherent stress field along the chain and is the principal mechanism for its formation. This elastic interaction between two M–H–M pairs is absent in other hcp systems, such as Ti and Zr, and as a consequence the two pairs can be located as close as possible to form a seed for the mono-hydride phase.

Acknowledgements The authors wish to thank Prof. K. Schwarz, P. Blaha and A. Kokalj for sharing their codes. This work was supported by the National Agency of Scientific and Technological Promotion (Argentina) under project BID/OC 38268 PICT 2005.

references

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