Intermetallic hydrides: A review with ab initio aspects

Progress in Solid State Chemistry 38 (2010) 1e37

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Intermetallic hydrides: A review with ab initio aspects Samir F. Matar CNRS, University of Bordeaux, ICMCB. 87 avenue du Dr Albert Schweitzer, 33600 Pessac, France

a b s t r a c t Keywords: Intermetallic Hydrides MgH2 DFT VASP ASW

The review aims to provide a coverage of different classes of intermetallic systems, which have the ability of absorbing hydrogen in different amounts, like binary and ternary Laves phases and Haucke-type intermetallics. Such intermetallic hydrides are attractive for applied research as potential candidates for on-board vehicular use (engines, batteries, etc.). Focus is made here on the fundamental features regarding the physical and chemical properties obtained from the first-principles e ab initio, for a better understanding of the role played by inserted hydrogen. Beside establishing the equation of state, the binding energetics, the electronic band structure, the magneto-volume effects, the hyperfine field etc., we endeavor answering the relevant question raised by solid state chemistry: “where are the electrons?”. This is approached through different schemes calling for a description of the chemical bonding, of the electron localization as well as the charge density mappings and the numerical Bader charge analysis scaling the iono-covalence of hydrogen within the lattice. For the sake of a complete scope we extend the studies to characteristics regarding the valence state changes in cerium based hydrided phases and the magnetism (spin-only, spin-orbit coupling, magnetic order of the ground state) in hydrogen modified ternary uranium intermetallics. Ó 2010 Elsevier Ltd. All rights reserved.

Contents 1. 2. 3. 4.

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 The problem of MgH2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Different classes of intermetallics and their hydrides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Theoretical framework and computational methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 4.1. The density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.2. Accounting for the spin dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.3. The computational methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.3.1. Pseudo-potential methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.3.2. All-electrons methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.4. Examining the chemical bonding properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Binary equiatomic intermetallics and their hydrides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 5.1. ZrNi and its hydrides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.1.1. Geometry optimization and energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.1.2. Description of the band structure and the chemical bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 AB2 Laves phases and hydrides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 6.1. The interest in Laves phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 6.2. YFe2 Laves phase and its hydride: lattice distortion effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 6.2.1. YFe2 versus volume and distortion effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 6.2.2. Magnetic properties of YFe2 and “covalent magnetism” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6.2.3. YFe2 and hydrogen insertion effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6.3. ScFe2 and its di-hydride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.3.1. Geometry optimization from pseudo-potentials calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

E-mail address: [email protected]. 0079-6786/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.progsolidstchem.2010.08.003

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6.3.2. Analysis of electron localization with ELF function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.3.3. All-electrons calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.4. Hydrides of pseudo Laves phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.4.1. Calculating electronic structure of RE based intermetallics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.4.2. Insertion sites of hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.4.3. Geometry optimization and equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.4.4. Mixed occupation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Hydrides of AB5 intermetallics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24 7.1. Questions around the crystal structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7.2. Geometry optimization results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7.3. Equation of state and derived quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 7.4. Analysis of the electron localization function (ELF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 7.5. All-electrons calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7.5.1. Analysis of the density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7.5.2. Analysis of the instability toward magnetic polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7.5.3. Bonding characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7.5.4. Specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Hydrides of ternary cerium based intermetallics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 8.1. Search of cerium valence change threshold: CeRhSn and its hydride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 8.2. Crystal structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 8.3. Geometry optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 8.4. All-electrons calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 8.5. Spin degenerate (non-magnetic) calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 8.5.1. Site projected density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 8.5.2. Analysis of the DOS within Stoner theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8.6. Spin polarized (magnetic) calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Case study of a uranium ternary intermetallics and its hydride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 9.1. Structural considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 9.2. Spin degenerate calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 9.3. Spin polarized calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 9.3.1. Ferromagnetic hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 9.3.2. Anti-ferromagnetic configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 9.3.3. Spin orbit coupling effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1. Introduction It is well known that power from fossil fuels, such as oil, gas and coal, relies on resources that are being used up. Besides, a significant production of various forms of pollution results when power is made. This problem has incited the development of sustainable energy forms (solar, geothermal, wind, wave, tidal, nuclear and thermonuclear), among which the solar source of energy offers facilities (photovoltaic and thermal) allowing its use at public scale, despite the costly initial investment. People need to learn practicing reducing waste and improving energy efficiency. Such training will decrease the impact of our energy use on the environment in order to have a future with a clean planet for our offspring. Free and responsible power-on-demand should become eventually, the citizens’ pledge. Along with these applications mainly meant for static usage, metal hydrides are major candidates for on-board vehicular applications. Besides the wearing out of fossil fuels, the relative ease of hydrogen production, with respect to the other energy sources cited above, is the main reason, such as with the electrolysis reaction: H2O þ Energy / H2 þ ½O2 Where ‘Energy’ is supplied by an electrical power source such as a battery. In this case, since water is a dielectric medium, an

electrolyte needs to be added such as a strong acid (H2SO4) or a strong base (NaOH, KOH). The resulting change in the Gibbs free energy is: DG ¼ DH  TDS w286 kJ  49 kJ ¼ 237 kJ/mol. The electrolysis process leads to entropy increase explained by the splitting an organized system, water, into its constituting parts, oxygen and hydrogen. Then the environment helps the process by contributing the amount TDS. Now, since we know the necessary amount of energy, DG, for the reaction, other forms of energy can be envisaged to induce the process such as by solar energy. Australian scientists obtain this through using the power of the sun to produce hydrogen fuel from splitting water, based on special titanium oxide ceramics. Another consequence of the splitting of water molecule at high temperature, e.g. in nuclear reactors, is in the secondary reactions of the metallic elements of the structures with nascent, highly reactive O2 and H2. This leads to the formation of complex mixtures of oxides and hydrides and other chemical compounds within the metal lattice leading eventually to its wearing out. This problem of solid state chemistry and materials science justifies the investigation of the fundamental physical and chemical properties at all levels of experiment and simulation/modeling. In the reaction equation above, replacing the / arrow by “)” pertains to the reverse process and describes the mechanism of the hydrogen fuel cell. In fact, hydrogen can be used to generate electric energy in fuel cells with a much larger efficiency than oil operating vehicular engines. The fuel energy is converted to electrical energy with an efficiency of 83%. This is far greater than the ideal efficiency

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

of a generating facility with burning the hydrogen and using the heat to power a rotor or a turbine. Despite the fact that actual fuel cells do not approach that ideal operating rate, they are still much more efficient than any electric power plant, which burns a fuel [1]. However “there is no free lunch” and the cost of efficient high energy fuel cells is the catalytic e expensive-platinum needed for its operation e which will be hardly available at a large worldwide scale. Nonetheless, opposing to the advantages of hydrogen as an energy source, many drawbacks can be traced out, mainly those related to the difficulties of its storage. Compressed hydrogen gas requires very large volumes compared to petrol, for producing the same amount of energy. On the other side, liquid hydrogen is ten times denser than the gas, but its production and conditioning are expensive. Then the reversible chemical conversion of molecular hydrogen within a metallic lattice presents an attractive and secure alternative method of its storage in the solid state. Among the 92 stable elements of the periodic table, nearly fifty combine with hydrogen in large quantities. This makes them to play a role of “hydrogen sponge”. Therefore there is, in principle a large choice of candidates for hydrogen storage. Hydrogen “goes into the solid” through five steps: 1. Molecular hydrogen (di-hydrogen, H2) is adsorbed at the surface; this is called “physisorption”. In this mechanism the forces involved are intermolecular forces (van der Waals forces); they are weak with respect to intramolecular forces and do not involve a significant change in the electronic orbital patterns of the species involved. For this reason, physisorption can also be called van der Waals adsorption. 2. Reduction by hydrogen of possibly existing passivated oxide (nitrides, sulphides) layers. This is the activation process. 3. Decomposition of H2 into H atomic constituents. 4. Chemisorption with the formation of mono-layer hydride; i.e., contrary to physic-sorption, actual chemical bonding involving sharing electrons is at stakes. 5. Diffusion of the hydrided layer toward the bulk. The latter step is reversible, allowing hydride H species to move either to the surface or to the bulk, according to the temperature and the pressure [2]. So what is a “hydride” ion? It is the simplest anion, formally H, consisting of two electrons and a proton. It may be found in the free state, like free protons (Hþ) only under extreme conditions. The reason is the high charge/radius ratio. With a relatively low electron affinity, w72.8 kJ/mol, hydride anion reacts exothermically with protons: H þ Hþ / H2; DH ¼ 1676 kJ/mol. The low electron affinity of hydrogen and the strength of the HeH bond (DHbind. ¼ 436 kJ/mol) imply that hydrides in general, can be used in chemical reactions as strong reducing agents: H2 þ 2e # 2H, with a standard electrode potential of 2.25 V. This is the case of compounds displayed in Table 1.1 for main group elements combinations with hydrogen. The systems are classified as Lewis acid/base chemicals. For a reminder, a Lewis base is an electron donor through a lone pair, e.g. NH3; while a Lewis acid is electron deficient, i.e. acceptor of electrons, e.g. BH3. A Lewis acid interacts with a Lewis base to give a complex with a polar covalent, two-electron-chemical bond. This is the case of borazane chemical compound: H3BeNH3. From the molecular chemistry viewpoint, this involves frontier molecular orbitals (highest occupied molecular orbital, HOMO and lowest unoccupied molecular orbital, LUMO). Then a Lewis acid interacts with a Lewis base via its LUMO leading to the bonding. Consequently, many of the compounds shown in Table 1.1 are used in chemical synthesis and less for hydrogen storage. However, at the nano-scale (109 m) large specific surface materials have been tested for uptaking large

3

quantities of hydrogen such as carbon nanotubes; this is formulated as CHx in Table 1.1. However tests at industry scale signaled little efficiency for such a storage medium. The first metal hydride was discovered in 1866 by Thomas Graham who found that heating up an airtight palladium H2 container results in a decrease of the pressure. This is explained by the formation of palladium di-hydride, PdH2. Note that many transition metals form hydrides, mainly metastable e.g. CoeH [3]. But the archetype hydride of a metal is MgH2, meant for purposes of hydrogen storage. However the difficulties of its use as is (cf. next section) led to find solutions for hydrogen storage in intermetallic systems. Also, for other instances of metal hydrides, there are stable nf hydrides: in Table 1.1 we mention rare earth (RE) La, Ce as well as uranium hydrides. For instance La reacts readily with hydrogen gas at ambient pressure with little heating, producing a black solid which is pyrophoric in air and highly reactive in water. Cerium forms cubic di-hydride and tri-hydride with simple structures derived from fluorite (CaF2); the study of the cerium valence change is of fundamental issue in these hydrides, its investigation is underway. UH3 is the most important hydride of the actinide metals partly and mainly for strategic reasons: anecdotally, uranium hydride bomb was a variant design of the atomic bomb which was suggested by Robert Oppenheimer in first tests, back in 1939; but it did not work. UH3 has a primitive cubic structure changing with pressure [4]. It presents as highly pyrophoric black powders. Alloys and intermetallics. For a definition alloys exhibit a continuous composition range of a solid solution between two metals A and B: A1xBx, 0 < x < 1 such as between two alkali metals. On the contrary, an intermetallic has a finite composition such as AB2 (e.g. YFe2), AB5 (e.g. LaNi5) etc. Adjoining two metals in a solid solution or forming an intermetallic brings new physical properties; e.g. tin plating of steel surface through FeSn2, enhances the chemical and electrochemical resistance against corrosion. Concerning hydrogen uptake ability, YFe2 intermetallic shows a large

Table 1.1 Main group elements compounds with hydrogen. Yellow: acid/base Lewis complexes, red: Lewis acids, blue: Lewis bases. Also mentioned are hydrides of lanthanum, cerium and uranium. 3

Li

4

Be

5

B

6

C

7

N

8

O

9

F

LiH

BeH2

BH3 B2H6

CHx CH4

NH3 N2H4

H 2O

HF

11

12

13

14

15

16

17

Na

Mg

Al

Si

P

S

Cl

NaH MgH2

AlH3

SiH4

PH3

H2S

HCl

19

31

32

33

34

35

K

20

Ca

Ga

Ge

As

Se

Br

KH

CaH2

GaH3

GeH4

AsH3 SeH2

HBr

37

38

49

50

51

Te

53

HI

Rb

Sr

In

Sn

Sb

52

RbH SrH2

InH3

SnH4

SbH3

TeH2

55

°

57

58

92

Cs

56

Ba

CsH BaH2

La

LaH3

Ce

U

CeH2 UH3 CeH3

I

4

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

hydrogen capacity close to 5 H atoms per formula unit (fu), while there are no stable Fe hydrides. Beside hydrogen storage for on-board fuel applications, at the industrial scale, hydrogen storage in an intermetallic has one major application in electrochemical energy storage. The development of Ni/Metal Hydride (MH) batteries based on MH negative electrodes is one of the most important areas of electrochemical study today. Batteries based on such hydride materials have some major advantages over the more conventional leadeacid and nickele cadmium [5] ones. Furthermore, Pb and Cd are known to be unsafe for health and environment. In Ni/MH, intermetallic systems such as AB2 (Laves) and AB5 (Haucke) (cf. next sections) are used as multi components e composite-mixtures [6]. On the fundamental side, the interaction of hydrogen with the atoms of the host crystal structure leads to different phenomena as the modification of the long-range structural order, the chemical bonding of hydrogen and its binding energy within the metallic lattice, the volume expansion effects such as those influencing the magnetic order and the valence state of the rare earth (mainly cerium), etc. The most indicated framework for investigating these physicochemical properties is that of the quantum density functional theory (DFT) [7] due to its reliability in accounting for energies and derived quantities, as it is illustrated in this review article. Since its formulation in the 1960s, several methods were constructed within the DFT with different degrees of required information on the system [8]. The objective of this review is to provide a general coverage of different classes of intermetallic systems and their hydrides for which, we provide calculated electronic structure ebased properties and confront them with experimental data when available. The paper organizes as follows: after this introduction, the different classes of intermetallic systems and their hydrides are detailed with a particular attention to MgH2. Then the theoretical framework is briefly presented focusing on the two main computational methods used. After afore mentioned chemical hydrides, a brief review of metal hydrides is presented; then we develop on hydrides of binary equiatomic intermetallics (ABHx), Laves (AT2Hx) and Haucke (AT5Hx) phases. In the last two sections, ternary cerium CeTXHx and uranium U2T2XHx compounds are discussed. In these formulae, A is an alkaline-earth, a lanthanide, an actinide or a transition metal of groups 3B and 4B (e.g. Sc, Y and Ti, Zr) and X are respectively a transition metal and a p-element of groups 3A and 4A (e.g. Ga, In and Si, Ge, Sn). 2. The problem of MgH2 Metallic magnesium is a good candidate for the storage of hydrogen. Both Mg and H are cheap and abundant whence the attractiveness to MgH2 which can be assigned the label of archetype metal hydride. It has a large mass capacity of 7.6 wt% H owing to Mg light weight: 12Mg24. The Mg þ H2 / MgH2 reaction is strongly exothermic (DHhydr ¼ 75 kJ/mol H2), whence the stability of the hydride. The desorption temperature is consequently high: w300  C at 1 bar. On the other hand the absorption kinetics is very slow [9,10]. Several solutions were envisaged to circumvent these drawbacks. MgH2 can be destabilized by LiAlH4 in MgH2eLiAlH4 composite mixtures. Then the onset dehydrogenation temperature is lowered by w50  C. Another solution is to add Ni e which plays a catalytic role e leading to materials with reversible hydrogen adsorption ability such as in Mg2NiH4 [11]. Somehow this is related with the iono-covalent character of hydrogen whereby its net charge varies from one system to another. From a Bader charge analysis (cf. next section), it is fully anionic in MgH2, i.e. “H1”,

while in Mg2NiH4 we find an iono-covalent H0.8 meaning that the electron is shared between H and the metal sublattice, here Ni. Furthermore the systems become metallic. It will be shown throughout this manuscript that the covalent character is reinforced in intermetallic hydrides. Lastly, the association of RENi2 (RE ¼ Y, Ce, Gd) with Mg can be attractive because of the electropositive character of Mg and its close radius to RE e.g. Gd, then a partial substitution of Gd by Mg can occur, leading to (RE, Mg)Ni2, i.e. REMgNi4, called pseudo-Laves phases. Magnesium metal is hexagonal with P63/mmc space group (SG) but the absorption of hydrogen induces a structural change into the tetragonal rutile-type structure (P42/mnm SG) and a large volume increase of DV/V z 25%. The electronic structure, calculated for the purpose of this review (Fig. 2.1), clearly characterizes this hydride as a large band gap insulator with an ionic character. In addition, the presented results in Fig. 2.1 agree with former theoretical work [11]. Further, the high ductility of magnesium is reduced after hydrogen absorption, due to the change of its chemical bonding properties; however, unlike titanium (or rare earth hydrides) there is no ductile / fragile transition upon hydrogenation. This is actually observed in transition metal intermetallics. TiCo3 shows a good ductility over a wide temperature range. However, this property is hindered by hydrogen uptake leading to the embrittlement of the hydrogen containing system [12]. This can be explained by a mechanism of generation of lattice strains whereby the distortion from cubic to tetragonal symmetry occurs as we have shown from first-principles computations [13]. The stability resulting in a strong MgeH chemical bond makes difficult the direct use of the hydride in devices. Furthermore, another factor, the activation treatment, hinders its use. MgH2 usually requires several activation treatments, which are quite time and energy demanding; these include both hydriding and de-hydriding cycles under the control of temperature and hydrogen pressure. Further, magnesium is not hydrided completely under these conditions because of the low reaction rate and the non-recoverable poisoned part through the formation of MgO. For these reasons Mg or MgH2 is admixed as composite or within an intermetallic system in the search for new candidates for hydrogen storage. 3. Different classes of intermetallics and their hydrides At the microscopic scale, several metals and intermetallics possess interstitial sites in their crystal structure within which small size atoms, mainly of the 1st period (X ¼ H, B, C, N) can be inserted. The resulting synthetic interstitial metals and intermetallics called respectively, hydrides, borides, carbides and nitrides with chemical terminologies, in the sense of fully or partially negatively charged non-metal Xd, have been proven of both fundamental and applied interests, throughout academic and industrial research works especially on intermetallic hydrides. In these systems hydrogen is inserted in the interstices of the metallic lattice and shares its electron with the host conduction band (CB). CB should only be partially occupied to enable the migration of electrons coming from reacting H2 towards the valence band (VB) leading to the formation of the metalehydrogen bond. For this reason, candidates for hydrogen uptake (absorption) are elements or systems with a metallic behavior (metals, alloys and intermetallics). However the particularity of hydrogen is that it does not insert in all the available crystallographic sites of the metallic system. There are rules relating to steric and electrostatic criteria: According to Westlake [14], the coordination sphere of hydrogen atom has to be larger than 0.4 Å on one hand and following Switendick [15], the HeH distance should be larger than 1 Å, on the other hand. The other consequences for the hydrogen insertion at the interstitial sites, are the large modifications of the material at both the micro and macroscopic levels.

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

5

Fig. 2.1. Electronic structure of rutile MgH2. The band structure along major directions of tetragonal Brillouin zone shows a band gap ofw4eV with a large dispersion of the bands signaling the s-like character. This is equally exhibited in the site projected density of states PDOS on the right hand side of the band structure. The intense PDOS of H is not only due to the twice larger number of H in the structure but also to the transfer of electrons Mg / H leading to an ionic hydride as it can be seen from the electron localization ELF map projected over a plane at 0 0 ½ (xOy) for 4 adjacent cells with red H spheres characterized by ELF ¼ 1 (strong localization, see the ruler) and ELFw0 around Mg (blue spheres). This is also obtained from a Bader charge analysis giving Mgþþ and H. (For ELF and Bader cf. Section 4). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).

Strains are created leading to the so-called “decrepitation phenomenon”, meaning that the pristine ingot is reduced to powder on one hand and to significant changes of atomic positions leading in some cases to structural reorganization [16,17], on the other hand. But the first intermetallic hydrides were only evidenced almost 100 years later in the course of developing such compounds for specific uses as neutron moderator in e early e small nuclear reactors [18]. In the 1970s, the accidental discovery of the LaNi5 hydride at Philips Company while attempting the modification of the magnetic properties of Haucke phases (AB5) by inserting hydrogen is noteworthy. This started an era of intensive research on intermetallic hydrides for their large technological potentialities as hydrogen storage materials, thermal machines, electrochemical applications, etc. [19]. The objectives being the increase of the storage capacities, the improvement of the absorption 4 desorption reversibility and the cycle life curve [20] as well as the reaction kinetics. The main families of hydrided intermetallics have the ATn formulation, i.e. combining generally a RE with a transition nd

metal (n ¼ 3, 4, 5). Their crystallographic characteristics were given by Yvon and Fischer [21]. For addressing the potential materials for hydrogen insertion/storage, they can be classified into families as follows:  The equiatomic AT systems. They generally lead to stable hydrides at room temperature. For instance hydriding ANi (A ¼ Ce, Y, Zr) [22] leads to tri-hydrides with 3/2 H atoms for each metal. Among these hydrides, recent reports claim even larger hydrogen content. YNiH4 composition [23] with 2 H per metal (like MgH2) has the largest capacity of w3 wt% hydrogen.  AT2 Laves phases crystallize into two major structural types: face centered cubic (C15) and hexagonal (C14) beside a minority type: C36 (cf. Fig. 3.1). In this category, the C15b cubic pseudo-Laves phases REMgNi4 with MgCu4Sn type structure are considered as good candidates for hydrogen storage [24]. These compounds can absorb reversibly

6

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

about 1 wt% H2 (i.e. 3e4 H/fu) within a few minutes at room temperature. From a volumetric point of view, they can uptake more than 120 g/L, which is more than the liquid hydrogen storage (71 g/L) but also higher than the volumetric capacities of other hydrides (Mg: 109 g/L; LaNi5: 107 g/L, .).  AT5 intermetallics, also called Haucke phases are the most studied candidates for hydrogen storage. The archetype system is LaNi5 which can accommodate up to 6 H atoms per formula unit (fu), i.e. one H per metal constituent. Although weight capacities are low w1.8 wt%, due to the presence of heavy elements (here La), the interest in these systems with respect to Laves phases, AT2, dwells in their resistance to corrosion in alkaline medium. This is further enhanced by the presence of Co replacing Ni.  RE based hydrogenated intermetallic systems. Those with RE ¼ cerium are particularly interesting from the fundamental point of view, despite a low wt% capacity. The ternary equiatomic CeTX systems with a transition metal T of 3d or 4d period and X a p-metal of groups 3A and 4A have been extensively studied in our Institute by B. Chevalier et al. because of their interesting structural and magnetic properties.

Fig. 3.1. The MgNi2 Laves C36 structure is defined in the same space group as C14: P63/ mmc. It contains eight formula units per cell, i.e. 24 atoms. For the archetype shown here, A(Mg) atoms are distributed over two crystal sites: Mg1 and Mg2. The Mg1 is similar to A sublattice in C15-AB2 and Mg2 is identical to A sublattice in C14-AB2. B (Ni) atoms are distributed over 3 crystal positions; they form two types of tetrahedra making chains along the hexagonal vertical c-axis and connected to each other by Ni (Ni1 in the figure). Within one chain, tetrahedra of different kinds are interconnected through a corner Ni (Ni3) and tetrahedra of same kind interconnect through a triangular face along the x, y-plane, forming trigonal pyramids. With this description, the C36 Laves phase can be considered as composed of the two major ones: C14 and C15.

 Ternary uranium U2T2Sn (T ¼ Co, Ni) intermetallic systems uptake 2 hydrogen atoms per formula unit. A case study is presented in the last section on the fundamental aspects of H insertion such as the enhancement of the uranium magnetic polarization and the chemical bonding features.

4. Theoretical framework and computational methodology 4.1. The density functional theory In quantum theory, the adiabatic approximation which consists of treating the motions of the electrons separately from those of the nuclei is commonly known as the Born-Oppenheimer approximation. The first solution to the single electron problem is known as the Hartree method in which the intractable solution of the Nelectrons system is treated as a product of single electron wave functions; but the resulting wave function is not ensured to be orthogonal. Fock then introduced a determinant to treat exchange which plays the major role of accounting for the fact that two electrons with the same spin cannot approach each other indefinitely. This actually introduces the well known Pauli Exclusion Principle. Such a treatment would seem exact since a further sophistication is brought to the inter-electron interaction; but in fact there is a missing contribution due to correlation: according to the Hartree-Fock (HF) approximation no correlation exists for the trajectories of electrons with opposite spins which are then allowed to be in the same region of space; and that cannot be. This leads to neglect a correlation hole around the electron which is best visualized as an impenetrable space surrounding it. Thus the energy associated with correlation appears as the difference between the total energy of the N-electron system and HF energy. Improvements on HF including such a correlation were done, such as in “configuration interaction” methods [25]. Nevertheless a major alternative was brought in 1964 by Hohenberg and Kohn and in 1965 by Kohn and Sham by the density functional theory (DFT) [7] which describes the ground state electronic structure and the related quantities derived from it. The DFT has been very successful since its formulation with many computational methods constructed around it and a huge number of published research works. Also concerning the theory itself, several articles have been devoted to the mathematical presentation [26] of the DFT. Hence it is not our purpose here to give an exhaustive account of this theory but only to present those details to make a clear presentation. Hohenberg and Kohn demonstrated two theorems which can be formulated as follows: (i) The total energy of the system is a unique functional [27] of the electron density and (ii) its minimum corresponds to the energy of the ground state (we implicitly calculate 0 K properties of the system). Notice that the use of the electron density as a variable was not “invented” by these authors since it was introduced in the early years of quantum mechanics by Thomas [28] and Fermi [29] who, nevertheless, could not formulate the problem in a usable way as the authors of the DFT did. The underlying approach is now shown: Kohn and Sham expressed the total energy functional for an interacting electron gas in a static potential v(r) (bold face r designates a vector) as follows: E [r(r)] ¼ F [r(r)] þ !r(r) v (r) d3r; where, F [r(r)] ¼ T [r(r)] þ ½ ! !r(r)r(r’)/jr e r0 jd3r d3r0 þ Exc [r(r)] is a universal functional (unknown because of the right hand side third quantity) whose terms are relative to the kinetic energy (T [r(r)]), Hartree interaction energy (½ ! ! r(r)r(r’)/jrer0 jd3rd3r0 ) and exchange-correlation (XC) energy (Exc[r(r)]). Hence the XC term appears as the difference between the true kinetic energy and that of a non-interacting electron gas system plus the difference between the true interaction energy and that of Hartree. The lacking information on the actual chemical system is embedded in it and its

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

accurate calculation is the cornerstone of modern computational methods. Thus one can see that by construction the DFT is an “exact theory” if the actual Exc[r(r)] were known. But this is not so and the solution is a hard task which is only possible through approximations as we show hereafter. One major approximation involves the way the electron density, r, is accounted for. Assuming that the density varies slowly with r, Kohn and Sham (KS) proposed a development for Exc[r(r)]: Exc [r(r)] ¼ ! r 3xc(r) d3r þ ! jVrj2 3xc(r)d3r þ . The local density approximation (LDA) consists of considering Exc [r(r)] as a local functional of the electron density r(r), i.e., it depends on the density at r. The gradient term, Vr(r), as well as higher order terms, are then omitted. Thus Exc [r(r)] ¼ ! r 3XC (r) d3r where 3XC is the contribution of exchange and correlation to the total energy per electron in a homogeneous, but interacting electron gas. Consequently we limit the missing information of the “true” electron density to this term. Then the KS equations can be solved self-consistently to obtain the KS orbitals: ðð1=2ÞDi þ Veff ðrÞ  3i Þji ðrÞ ¼ 0 where Di is the Laplacian (V2i ) operator and Veff is the effective potential detailed below. The ground state electron density is then conPN 2 structed according to: rðrÞ ¼ i ¼ 1 jji ðrÞj . Note that the singleparticle HF and KS equations do not have the same physical meaning: Although exchange is included non-locally in HF formalism, within DFT we are dealing with a “quasi-particle”, i.e., the electron with its accompanying correlation hole as added to exchange giving the exchange-correlation hole (see web site of Matthew Foulkes [30] for pictures of exchange-correlation hole in silicon). How is that done? Whereas the exchange part of 3XC, 3X, is obtained exactly in as far as it stems from the Pauli Exclusion Principle, the correlation part 3C is numerically parameterized using quantum Monte Carlo calculations [31]. Throughout the years different schemes have been formulated for the parameterization of 3XC such as the early homogeneous gas based ones [32,33]. They lead to a high numerical accuracy of 3XC which makes Exc [r(r)] the major approximation of the calculations undertaken in the framework of this theory. The corresponding XC potential Vxc(r) ¼ dExc/dr(r) is included into the overall potential as follows: V(r) ¼ Vions(r) þ Veff.(r) with the effective potential: Veff.(r) ¼ Vext. (r) þ !r(r’)/jrer0 jdr0 þ Vxc(r). Kohn and Sham did not expect the LDA to be efficient in describing real chemical systems where the density does not vary slowly between the sites. However in many cases results in good agreement with experiment are obtained. Nevertheless there are shortcomings to this approximation such as overestimated binding energies and too small energy gaps for insulating materials which led to try improvements on the LDA. The GGA (generalized gradient approximation) [34], where gradient terms such as the one in the above development of Exc[r (r)] are included, is one of them. Another attempt to improve on the LDA is the LDA þ U correction [35] which allows for a better account of properties such as the insulating character of transition metals monoxides. In this work both LDA and GGA are used according to the case study. Borrowing Jacob’s ladder image of ascending angels, we follow John Perdew, one of the prominent actors of DFT world (cf. for instance Ref. [34]), by placing Hartree-Fock methodology as Earth level (Hartree approach would be in a cavern), then the LDA on the first rung and the GGA on 2nd rung. Meta and hybrid functionals occupy the 3rd up to 5th rungs. As an example of hybrid functional the so-called B3LYP is among best known and most used in inorganic chemistry (see for instance Ref. [36] and therein cited works). It mixes non-local (HF-type) with local (LDA-type) exchange. Expectedly, one would place at the top of Jacob’s ladder the sought “grail” functional, i.e. that which is totally non-local. The only requirements to carry out calculations within the DFT for crystalline solids are the lattice symmetry and the atomic

7

coordinates. One starts from the neutral atomic configurations (valence states explicitly) which only serve as a first guess of the crystal electron density. This underlines the predictive nature of the calculations. The procedure is then self-consistently carried out because the potential and the wave functions are interdependent. The self-consistent cycle is sketched in Fig. 4.1. The solution uses the variational principle which is a powerful mathematical tool commonly used in theoretical chemistry. It enunciates that if a given system can be described by an ensemble of parameters representing its ground state, then it is this set which minimizes the total energy. The variational solution of the problem is made easier in linearized methods (cf. Section 4.3). The energy dependence of the wave function jkL(r) (L: regrouping l and m quantum numbers) is lifted and the variational procedure conducted on the energy dependent coefficients ckL in jkL(r) ¼ ckL cL(r). 4.2. Accounting for the spin dependence A generalization of the LDA to spin polarized cases, leading to the LSDA, consists of assuming r[ and rY in the density matrix and writing the spin (a) dependent XC potential, Vxca(r), with a ¼ [, Y-spin dependence is also introduced for 3XC (r[, rY). The wave equation shows then an explicit dependence on the spin of the wave functions and of the potential. Also, the KS equation shows the spin (a) dependence: ðð1=2ÞDi þ V aeff ðrÞ  3ai Þjai ðrÞ ¼ 0. The explicit spin-dependence of the wave equation leads to a solution for each spin direction independently with spin polarized

Fig. 4.1. The self-consistent field cycle used in calculations.

8

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

(SP) calculations. One usually starts with a charge imbalance for each atomic species, i.e. with a non-zero magnetic moment defined as m ¼ n[  nY and Zv ¼ n[ þ nY (Zv is the number of valence elecP 2 3 trons). na ¼ !V kjjak(r)j d r is the integrated charge density (a ¼ [ or Y). The sum extends over lower states filled up to a common Fermi energy EF for both spin populations. Self-consistency can either lead to m ¼ 0 or to finite magnetic moments which are generally found in good agreement with experiment. When a stable magnetic state is found the band structures are different for the two spin directions. From this one can define two quantities hDEi, the energy difference between [ and Y spin populations, called exchange splitting e most significant for d and f bands e and I, exchange-correlation integral such that hDEi ¼ I.m. Despite this usual procedure to self-consistently calculate magnetic systems, it is enlightening to start from a non-magnetic (NM) configuration (total spins, Zv) and ask about the conditions to be met for the occurrence of a magnetic moment. If one starts by creating a slight imbalance of the spin density, a small moment m appears. This involves a spin flipping whereby the spin [ population increases and the spin Y one decreases. This causes kinetic energy to rise by ½ m/n(EF) (n(EF) is the non-magnetic density of states at Fermi energy) and there is a gain in exchange energy of ½ Im2. The total energy counted from the non-magnetic state then amounts to: E ¼ ½ {m2/n(EF)}{1  1/In(EF)}. If In(EF)  1 the system is unstable in a non-magnetic configuration and a magnetic moment should develop via intra-band polarization. The Stoner integral was tabulated for the elements by Janak [37]. This is the Stoner criterion for band ferromagnetism [38]. Ferromagnetism is implicit here because we are dealing with one site at a time. This is different from longrange magnetic orders (ferro-, ferri-, antiferro- or even spin spirals) which hierarchies through total energy minima. 4.3. The computational methods Several computational methods were built within the DFT since its birth more than forty years ago. They may be placed within two main categories according to the way the Z atomic number electron population is accounted for. Here we focus on the methods aimed at electronic structure investigation of organized extended crystalline solids i.e. contrary to molecular methods [39]. In most case studies presented here we firstly call for a geometry optimization and a search of equilibrium properties through establishing the equation of state; also an ab initio search of atomic positions is done especially for hydrogen. Note that deuterium often replaces hydrogen for such determinations by X-rays (H barely interacts with them) and neutron diffraction/scattering. Ordinary hydrogen has a large incoherent neutron cross section, which is nil for deuterium and substituting H by D reduces scattering noise. Then a full study of the electronic band structure is done for the detailed analysis of the density of states and the chemical bonding properties. 4.3.1. Pseudo-potential methodology Firstly calculations of the optimized geometries and relative stabilities are carried out in the framework of a pseudo-potential (PP) approach within the Vienna ab initio simulation package (VASP) code [40]. The interactions between the ions and the electrons are described by using ultra-soft Vanderbilt pseudo potential (US-PP) [41] and the electroneelectron interaction is treated either within the LDA [42] or the GGA [34]. When rare earth or actinide elements are present, we use Projector Augmented Wave (PAW) potentials [43], which account for nf states (n ¼ 4, 5). In the plane wave pseudo potential approach, the rapid variation of the potential near the nuclei is avoided by substituting the Hamiltonian near the atoms with a smoother pseudo-Hamiltonian which reproduces

the valence energy spectrum. Using PP’s allows for a considerable reduction of the necessary number of plane waves per atom; thus force and full stress tensor can be easily calculated and used to relax atoms into their ground state. In order to establish energy trends, the equation of state (EOS) is needed. This is because the calculated total energy pertains to the cohesive energy within the crystal, since the solution of the KS equations yields the energy with respect to infinitely separated electrons and nuclei. In as far as the zero of energy depends on the choice of the pseudo-potentials, somehow it becomes arbitrary; i.e. it is shifted, not scaled. However the energy derivatives as well as the equations of state remain unaltered. For this reason one needs to establish the EOS and extract the fit parameters for an assessment of the equilibrium values. The energy versus volume, E(V), curves are plotted around experimental volume. In general they present a quadratic variation and they can be fitted with Birch EOS to the 3rd order [44]:

h i2 EðVÞ ¼ Eo ðVo Þ þ ½9=8Vo Bo ð½ðVo Þ=VÞ½2=3 1 i3  h  þ ½9=16Bo B0  4 Vo ð½ðVo Þ=VÞ½2=3 1 where Eo, Vo, Bo and B0 are the equilibrium energy, the volume, the bulk modulus and its pressure derivative, respectively. As for technical details, we mention equally the use of the conjugate-gradient algorithm scheme [45] to relax the ions of the systems into their ground state. Optimization of the structural parameters is performed until the forces on the atoms are less than 0.02 eV/Å and all stress components are less than 0.003 eV/Å3. The calculations are performed with an energy cut-off for the plane wave basis set whose magnitude depends on the case study. The tetrahedron method with Blöchl [46] corrections is applied for both geometry relaxation and total energy calculations. Brillouin-zone integrals were approximated using the special k-point sampling of Monkhorst and Pack [47]. 4.3.2. All-electrons methodology All-electrons methods such as full potential LAPW method [48], allow carrying out spectroscopy studies of 1s core level shifts and core ionization energies [49,50] as well as hyperfine field such as its Fermi contact term HFC [51]. However the cost of such high precision computations is their large time consuming. In the scope of this review we do not indulge into core electrons spectroscopic properties. On the other side, HFC results from the core ns polarization by 3d magnetic moments. The effective magnetic field Heff acting on the nucleus is usually written as the sum of four contributions: Heff ¼ Hi þ HFC þ Horb þ Hdip. Hi, the internal field is the magnetic field at the nucleus generated from externally applied field. Horb and Hdip are the fields arising from the orbital magnetic moments (small for 3d elements) and from the dipole interactions with surrounding atoms. HFC, is the Fermi contact term of hyperfine interaction arising from unbalanced spin density of ns-electrons at the nucleus (in nonrelativistic description): HFC ¼ 8p/3 gN [{V[(0)}2  {VY(0)2}], where gN is the nuclear gyromagnetic ratio and the quantities between brackets are the densities of ns electrons at the nucleus (i.e. r ¼ 0) for [ and Y spin. This quantity is a major contribution to Heff and it can be obtained from non-frozen core type of calculations such as the augmented spherical wave ASW method [52] which has been largely developed by Volker Eyert [53] especially for the properties of chemical bonding through different schemes (cf. next paragraphs). In this review we present all-electrons results based on this method. Likewise the computations are based on DFT and its regular functionals: LDA and GGA, as well as LDA þ U and GGA þ U when needed, such as for computing the electronic

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

structures of oxide insulators e.g. antiferromagnetic NiO. In the ASW method, the wave function is expanded in atom-centered augmented spherical waves, which are Hankel functions and numerical solutions of Schrödinger’s equation, respectively, outside and inside the so-called augmentation spheres. In the minimal ASW basis set the outermost shells represent the valence states and the matrix elements are constructed using partial l-waves (l is the secondary quantum number), up to lmax. þ 1 ¼ 4 for RE and uranium, lmax. þ 1 ¼ 3 for transition metals and lmax. þ 1 ¼ 2 for p-elements and H. The ASW method uses the atomic sphere approximation ASA within which the volume of the cell is equal to the sum of the atomic spheres AS, so that AS overlap. This approximation is unproblematic for alloys and intermetallic systems, which have compact structures and the overlap is at minimum. For open structures, the overlap would become large. To optimize the basis set, additional augmented spherical waves are placed at carefully selected interstitial sites (empty spheres ES). ES are pseudo-atoms with zero atomic numbers. The choice of these sites as well as the augmentation radii are automatically determined using the spheregeometry optimization algorithm [54]. The BZ integrations are performed using the linear tetrahedron method with up to 576 kpoints within the irreducible wedge [46]. The computational procedure follows a protocol through which a non-magnetic (NM) configuration is firstly assumed, meaning that spin degeneracy is imposed for all valence states with equal spin occupations. This configuration does not represent a paramagnet whose configuration requires calling for the construction of a large cell, i.e. a supercell with 8a  8b  8c; a, b and c being the initial cell constants and large scale computations. However such NM computations are relevant at two levels: (i) for carrying out an analysis of the partial atom projected densities of states (PDOS) at the Fermi level within a mean field theory, searching for a possible magnetic instability in such a configuration and (ii) for analyzing the chemical bonding between the different atomic constituents. Subsequent spin polarized (SP) calculations with different initial spin populations, i.e. majority [-spin and minority Y-spin populations, can lead at self-consistency to finite or zero local moments within an implicit long-range ferromagnetic order. When needed, antiferromagnetic (SP-AF) calculations can be carried out by assuming half of the atoms as UP-spin and the other half as DOWNspin while each magnetic subcell contains spin polarized atomic species. More elaborate non-collinear calculations can be carried out for antiferromagnetic spin spiral structures [55]. 4.4. Examining the chemical bonding properties The results obtained from the calculations are relevant to total energies and energy derived quantities as well as to the nature and energy position of the states with respect to the Fermi level, etc. However the chemist needs a more elaborate tool of information about the nature of the interaction between atomic constituents as well as the respective quantum states involved. In order to answer the basic question, “where are the electrons?”, Roald Hoffmann introduced the COOP (crystal orbital overlap populations), in early extended-Hückel type calculations [56]. The COOP are calculated based on the overlap matrix elements c*ni(k) Sijcnj(k) ¼ c*ni(k) < cki(r)jckj(r) > cnj(k) where Sij is an element of the overlap matrix formed of the valence basis functions c and cnj(k) are the expansion coefficients entering the expression of the wave function of the nth band. The partial COOP coefficients, Cij(E),

9

are then obtained by integrating the above expression over the Brillouin zone (BZ): Cij(E) ¼ Cji(E) ¼ 1/UBZ Sn !BZ d3k Re{c*ni(k) Sij cnj(k)} d(E  3nk); where UBZ is the Brillouin zone volume, d is the Dirac delta notation, serving as counter of states and Re refers to the real part of following expression. In this context Cij(E) can be grossly designated as a density of states (DOS) function, modulated by the overlap population. The total COOP, C(E), are evaluated as a summation over the non-diagonal elements: C(E) ¼ Sij(isj) Cji(E). Another scheme uses the Hamiltonian based populations with the COHP (crystal orbital Hamiltonian populations) [57], partly because the COOP tend to exaggerate the intensity of anti-bonding states. It is described similarly to the COOP while replacing the overlap matrix by Hamiltonian one, Hij. Lastly more recent developments led to use both approaches to propose the “covalent bond energy” criterion, ECOV [58]. A partial ECOVij is then proportional to COHPij(E)  COOPij(E). This brings together both advantages of the two methods and allows for mutual corrections. Just like the COOP, the ECOV were implemented within the ASW method [59]. In the plots, negative, positive and nil ECOV magnitudes (unit less) indicate bonding, anti-bonding and non-bonding interactions respectively. We use the ECOV as well as integrated iECOV schemes to provide a qualitative description of the bonding in this presentation. Still elaborating on the question “where are the electrons?” one may map the localization of the electrons on a slice within the crystal, such as along lattice planes containing selected atoms. While the COOP and COHP are calculated in the reciprocal space, electron localization plots of the ELF (electron localization function) introduced by Becke and Edgecomb [60] are obtained from real space calculations: ELF ¼ (1 þ cs2)1 with 0  ELF  1. In this expression the ratio cs ¼ Ds/Ds where Ds ¼ ss  Vs  ¼(Vrs)2/rs and Ds ¼ 3/5 (6p2)2/3 rs5/3 correspond respectively to a measure of Pauli repulsion (Ds) of the actual system and to the free electron gas repulsion (Ds ) and ss is the kinetic energy density. The ELF function is normalized between 0 (zero localization) and 1 (strong localization) with the value of ½ corresponding to a free electron gas behavior (cf. Fig. 1.1). Finally it is important to mention the theoretical concept of “atoms in molecules and crystals” introduced by Richard Bader [61], developing an intuitive way of dividing molecules into atoms as based purely on the electronic charge density. Bader uses what are called ‘zero flux surfaces’ to divide atoms. A zero flux surface is a 2D surface on which the charge density is a minimum perpendicular to the surface. Typically in molecular systems, the charge density reaches a minimum between atoms and this is a natural place to separate atoms from each other. Besides being an intuitive scheme for visualizing atoms in molecules, Bader’s definition is often useful for charge analysis. For example, the charge enclosed within the Bader volume is a good approximation to the total electronic charge of an atom. In MgH2 one actually finds the ionic picture, Mg2þ and H1, while for Mg2NiH4, identified as a better candidate for hydrogen source, the charge on H is calculated less ionic with H0.8 from the Bader analysis of the charge density output file. In this case Ni plays a major role in reducing the ionic behavior of MgH2. 5. Binary equiatomic intermetallics and their hydrides Equiatomic binary intermetallics based on nickel are an interesting class of materials for a diversity of physical properties such as the shape memory intermetallic systems-like TiNi [62] on one hand and INVAR-related magnetic properties for FeNi [63], on the other hand. On the side of RE based Ni intermetallics, they are known to improve the hydriding kinetics of Mg even at relatively low temperatures [64]. Also they have the capacity of absorbing large amounts of hydrogen [65,66]. For instance CeNi and ZrNi absorb up to 3 H/fu [22]. However, recent reports on YNi and other RENi

10

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

intermetallics indicate a higher absorption amount of w4 H per fu [23]. Despite a monoclinic structure proposed for YNi, these intermetallics systems are found in two main crystal types with orthorhombic symmetry: the CrB-type structure with Cmcm SG and the FeB-type structure with Pnma SG. The latter structure is less common and several RENi (RE ¼ Ce, Sm, Gd) intermetallics crystallize in the former Cmcm SG, like the corresponding tri- and tetrahydrides. For the purpose of illustrating the insertion characteristics of hydrogen within equiatomic intermetallics, we discuss in this section the ZrNi case [67]. 5.1. ZrNi and its hydrides 5.1.1. Geometry optimization and energies Like ZrNi, ZrNiH3 crystallizes in the Cmcm space group (SG). There are 4 formula units (fu) per cell, but due to the base centered SG, only two fu are explicitly accounted for in the calculations. Table 5.1 regroups the experimental data as well as the calculated ones. The uptake of 3 hydrogen atoms per fu giving ZrNiH3, results in an isotropic expansion of the cell. Hydrogen atoms are accommodated within two different sets of lattice sites: H1 (4c) and H2 (8f). H1 and H2 are then found in Zr3Ni2 trigonal bipyramids and Zr3Ni tetrahedra respectively. The coordination polyhedra of the two hydrogen sites are shown in Fig. 5.1. Starting from the experimental crystal data in Table 5.1, a full geometry relaxation was carried out for both ZrNi and ZrNiH3 systems using pseudo potentials approach with PAW-GGA method (cf. Section 4). Further, in order to enable comparisons and trends instability, the mono- and di-hydrides were also examined; the first one by considering the occupation of (4c) H1 positions, the second with only (8f) H2 ones. For ZrNiH2 model preliminary calculations were also attempted assuming full (4c) and partial (8f) occupations in order to obtain the 1:1:2 stoichiometry; but the partial occupation of (8f) sites led to the loss of orthorhombic symmetry leading to a triclinic one with an energy destabilization. After geometry optimization the orthorhombic symmetry is kept for all systems within the base centered Cmcm SG. The calculated values (Table 5.1) show small deviations with respect to the experimental lattice volume and internal coordinates as given in Table 5.1. The lattice constants and volumes of the intermediate hydrides are calculated in between those of ZrNi and ZrNiH3. The trend is toward a progressive increase of cell volume upon H insertion. In a second step a set of calculations was done around optimized minima to obtain the equilibrium ground state energy/volume values from the quadratic fit of the curves with a 3rd order Birch equation of state (EOS). Fig. 5.2 shows the E ¼ f (V) curves for ZrNi Table 5.1 Geometry optimization results for ZrNi and its hydrides. Space group Cmcm, No63. Experimental values are in italics. System

ZrNi [65]

ZrNiH

ZrNiH2

ZrNiH3 [65]

a-latt. const. (Å) b/a c/a Zr, Ni and H1 atoms at 4c (0,y,1/4) yNi yZr yH1 H2 at 8f (0,y,z) y z Vol./4 fu (Å3) Equil. vol./4 fu (Å3) Equil. energy/4 fu (eV) Bulk modulus (GPa)

3.287/3.268 3.07/3.04 1.23/1.26

3.43 2.98 1.20

3.41 3.06 1.27

3.498/3.53 2.97/2.97 1.23/1.218

0.362/0.361 0.083/0.082 e

0.42 0.14 0.916

0.435 0.141 e

0.426/0.43 0.139/0.14 0.931/0.956

e e 134.1/133.14 135.94/133.14 59.08 138

e e 144.3 145.68 73.74 149

0.313 0.687 154.1 155.98 89.54 147

0.312/0.298 0.687/0.507 156.8/159.08 158.57/159.08 105.12 155

Fig. 5.1. Coordination of hydrogen within ZrNiH3 Cmcm orthorhombic structure. Large (red) and medium (blue) spheres are for Zr, Ni. Small black and white spheres are H1 in Zr3Ni2 and H2 in Zr3Ni polyhedra, respectively.

and the tri-hydride. The equilibrium values show close volumes to experiment. An energy decrease from ZrNi down to ZrNiH3 is observed. The energies for mono- and di-hydrides are in between. This can be expected from the extra electron brought by additional H. The bulk moduli B0 are also obtained from the fit results. The trend is toward a slight increase of B0 from the intermetallic system toward the tri-hydride. It can be suggested that this relative ’hardening’ is due to the formation of increasing number of hydrogen-metal bonds within the intermetallic lattice. This feature will be developed hereafter when discussing the chemical bonding. The E0, V0 equilibrium values are then plotted in Fig. 5.3 for all 4 systems. With respect to a line joining ZrNi and ZrNiH3, the intermediate hydrides are found off the line; this would agree with their hypothetic existence. However one can notice a larger deviation, i.e. a destabilization of ZrNiH2 with respect to the mono-hydride which is found closer to the line. This could point to a stronger bonding of hydrogen when it is in (4c) positions. This is detailed in next section describing the chemical bonding. The energy results then allow examining the stability of hydrogen using the expression:

EH ¼ EðZrNiHxÞ  EðZrNiÞ  xEðH2Þ given for 2 fu: While the two first terms of the right-hand-side equality are the equilibrium values obtained from the calculations, E(H2) is derived from PP-PAW-GGA calculations of H2. This is done by placing dihydrogen in a cube box with a ¼ 4.5 Å. The resulting energy, E (H2) ¼ 6.651 eV is the total electronic energy. It includes twice the energy of monohydrogen and needs to be corrected by the zero point energy (ZPE). In order to establish comparison with experiment calculations were carried out for 1 H in a similar box, resulting in energy of 0.95 eV. For H2, ZPE amounts to w0.28 eV as calculated by the same method. The binding energy of H2 is then 4.47 eV which comes close in magnitude to the dissociation energy (inverse sign) of the molecule as obtained from fluorescence excitation spectroscopy: w36118 cm1, i.e. w4.48 eV. Then with E (H2) and the equilibrium energy values we extract the stabilization ¼ energies of hydrogen in the different hydride lattices: EZrNiH H 2 ¼ 1:928 eV for 4H, i.e. 0.964 per H2 0:679 eV, per H2; EZrNiH H 3 and EZrNiH ¼ 3:067 eV for 6H, i.e. 1.02 per H2. Consequently H H is found most stable within ZrNiH3, in agreement with its experimental existence. However, if this compound were to be used in hydrogen storage devices, we suggest that H release should be

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

a

ZrNi E0 = -29.59 eV B0 = 138 GPa

-20

Energy (eV)

Lastly from the charge density results of the hydrides, treated with the Bader charge analysis [61], we find the net charge on hydrogen as: 0.63 (ZrNiH); 0.53 (ZrNiH2) and 0.54/0.45 for the two hydrogen sites in ZrNiH3. This shows the iono-covalent character of hydrogen in as far as the charge is close to H0.50 (fully negatively ionized anionic hydrogen being charged 1: H1). However it is interesting to note the trend to larger covalence for the tri-hydride especially for hydrogen at H2 position. If one limits the reasoning to these criteria, i.e. beside other considerations based on experiment, H at H2 position would be the first to decompose out of the lattice.

-16 -18

V0 = 68.62 Å3 B' = 4.03

-22 -24 -26 -28 -30 40

50

60

70

80

90

100

110

100

110

3

Cell volume (Å )

b

-20

ZrNiH3 E0 = -52.51 eV B0 = 155 GPa

Energy (eV)

-25 -30

3

V0 = 80.76 Å B' = 4.15

-35 -40 -45 -50 -55 40

50

60

70

80

11

90 3

Cell volume (Å ) Fig. 5.2. Energy versus volume curves with respective equilibrium values obtained from Birch quadratic fit; a) ZrNi; b) ZrNiH3.

stepwise because the energy cost to expel all 3 hydrogen atoms (ZrNiH3 / ZrNi) would be too high compared to an energy difference from one system to the other going up the line in Fig. 5.3. In this context the di- and mono-hydrides can be considered as metastable. While further experimental evidence is needed in this context, the energies involved in these mechanisms compare fairly well with those computed for H within TiCo3 with DE ¼ 0.305 eV (cf. Ref. [13]) as well as Section 6 on LaNi5:H.

Fig. 5.3. Graphic display of the different equilibrium E0, V0 with connecting lines added as guide for the eye, cf. text.

5.1.2. Description of the band structure and the chemical bonding For ZrNi and ZrNiH3, using the calculated data in Table 5.1 we examine the electronic structure and properties of chemical bonding with all-electrons calculations using the GGA functional as before. At energy self-consistent convergence charge exchange with small amounts of w0.3 electrons is found from Zr toward Ni in ZrNi and from Zr to Ni, H and ES in ZrNiH3. This is in agreement with the larger electronegativity of Ni, c ¼ 1.91 with respect to Zr c ¼ 1.33 at Pauling scale. The slight charge transfer toward the interstitial spheres (ES) over s, p valence basis sets is a sign of covalence ensured by their insertion within ZrNi and points to the completeness of the electronic representation of the system. The site projected density of states, PDOS, is shown in Fig. 5.4 in which the Fermi level, EF is taken as zero energy. For ZrNi the valence band (VB) is mainly dominated by Ni d states centered well below EF because Ni is a late element in the 3d series; whereas Zr which is an early element in the 4d series has its states broader and mainly centered above EF within CB. From the similar peak shapes there is a quantum mixing between the two constituents within the VB due to the itinerant (delocalized) states. When H is introduced (Fig. 5.4bed), extra states are created in the lower part of the VB which becomes increasingly extended. The peak as just below EF will be shown here below to correspond to anti-bonding states. Simultaneously the intensity of Zr PDOS within the VB becomes smaller. This pertains to the chemical role of H which should bind differently to the Zr and Ni metal sublattices through its s and p valence basis set. Besides volume expansion, one may expect changes in the inter-metal bonding upon going from the intermetallic to the saturated hydride. The chemical bonding based on the covalent bond energy (ECOV) criterion is then used to illustrate the relevant features. Fig. 5.5a and b show the three possible kinds of interactions within the metal sublattices in ZrNi and its tri-hydride. ZreNi interaction is mainly of bonding character for pristine intermetallic so that it contributes to its stability but it starts to have larger antibonding behavior in the hydride because some of the metal states are involved in bonding with H. Negligible ZreZr bonding is observed due to the large separation (dZrZr w 3.25 Å). The NieNi interaction is half bonding/half anti-bonding within the VB, due to the large filling of Ni d states. It is enhanced within the hydride system, which could be due to the smaller NieNi separation. Turning to the metal-H interactions, these are detailed in Fig. 5.5c for the two hydrogen sites H1 and H2 with Zr and Ni. The major part of the VB is of bonding character (negative ECOV intensities), i.e. up to 2 eV where anti-bonding states start to occur. The main contribution to the bonding arises from NieH interactions due to systematically larger ZreH separations: hdZrHi w1.95 Å while hdNiHi w1.75 Å. However the larger intensity for NieH1 versus NieH2 bonds cannot be due to distance criteria because of their close magnitudes: d(NieH1)/d(NieH2) ¼ 1.783/1.772 Å. It can be rather suggested that the coordination polyhedra are at the origin of this feature (cf. Fig. 5.1): Zr3Ni2 trigonal bipyramids for H1 and Zr3Ni tetrahedra for H2, i.e. with twice more Ni nearest neighbors

12

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

Fig. 5.4. Site projected DOS for a) ZrNi and b) ZrNiH3 and respective inter-metal bonding in c) and d).

for H1. Within a conceptual molecular orbital (MO) scheme one may expect s NieH bonding involving low energy lying itinerant states (s, p-like) up to 4 eV, then p-type in the range [4, 2 eV] involving d states; this is followed by the anti-bonding counterparts: p* from 2 to 1 eV above EF and lastly s* from 2 to

6 eV. Needless to say that this schematic view shows little separation between the levels due to their merging into bands on one hand and to the covalent character of the system; i.e. this MO like scheme would stand better for an ionic hydride such as MgH2.

Fig. 5.5. Inter-metal bonding in a) ZrNi, b) ZrNiH3 and c) metal-hydrogen bonding in ZrNiH3.

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

6. AB2 Laves phases and hydrides

Table 6.2 Atomic positions of A and B metals and H insertion sites in cubic C15 Laves phases.

No doubt, the Laves phases are the largest family of binary intermetallics. It was discovered in 1927 by James Friauf then intensively studied by Fritz Laves in the 1930s [68]. With AB2 stoichiometry they crystallize in three types defined as follows in Strukturbericht designation: the hexagonal C14, the face centered cubic C15 and the less common di-hexagonal form C36 (cf. Fig. 3.1). Only systems with the first two types will be considered here. The differentiation between the C14 and C15 types originates from the stacking sequence of the atomic spheres. Therefore the transformation of one structure into the other is possible for certain phases, mainly with thermal activation. Table 6.1 presents different Laves phases with available information on the experimental structure type [69]. In both structural types the parameters of particular positions (mainly H) are generally determined by neutron diffraction by replacing H by deuterium; but it will be shown that geometry optimization calculations can lead to accurate results as well.

6.1. The interest in Laves phases Laves phases are largely used in industry partly because of their good mechanical properties as well as unique magnetic and superconducting properties. But the most extended field of investigation in recent decades is their applications in hydrogen storage. This is due to relatively favorable kinetics of absorption/desorption as well as a high capacity of insertion thanks to the numerous available interstices within the cell as shown in Tables 6.2 and 6.3. As a matter of fact there are 17 such sites in C14 and C15 structures, dispatched into three types of tetrahedral interstices according to the atomic environment. They are formed by two A atoms and 2 B atoms B (A2B2 sites), one A atom and 3B atoms (AB3 sites) or by 4B atoms (sites B4). Consequently in theory one could form hydrides as rich as AB2H17 but the electrostatic forces between H neighbors as well as the large volume expansion are limitation factors, reducing the maximum number of six hydrogen atoms per AB2 motif (AB2H6) such as for YMn2 [70]. Further the hydrogenemetal bond is less strong (iono-covalent) than in ionic hydrides such as MgH2 thus enabling the observation of reversible absorption/ desorption of hydrogen around room temperature and ambient pressure. Beside the experimental investigations, there are several efforts by us and others to understand the modifications of the electronic properties brought by the insertion of hydrogen. Mainly, it is desired to obtain accurate information on the MeH bond and on which lattice sites are preferred for the insertion. This would enable elucidating the static configuration of H network and the ordering of the sublattice. Information on the dynamics of the system is also needed for a better understanding of the diffusion phenomena of H within the metal lattice. Table 6.1 Structural sorting of a collection of Laves phases. AB2

A

3d A-element

Sc Ti

4d A-element

V Y Zr Nb

13

B Cr

Mn

Fe

Co

Ni

e C15: low T C14: high T e e C14, C15 C15: low T C14: high T

C14 C14

C14 C14

C15 C15

e e

e C15 C14 C14

e C15 C15 C14

e C15 C15 C15

e C15 e e

Atom A (Y) B (Fe) H

Local environment

Site symmetry

Atomic positions

A2B2 AB3 B4

8a 16d 96g 32e 8b

0, 0, 0 5/8, 5/8, 5/8 x, x, z x, x, x ½, ½, ½

For an illustration of the specific properties of Laves phases and their hydrides we discuss in this section three representative members of the C15, C14 and C15b; namely YFe2, ScFe2 and GdMgNi4. 6.2. YFe2 Laves phase and its hydride: lattice distortion effects YFe2 crystallizes in the cubic C15 type structure described in the Fd3m space group with 8 AB2 formula units. But the F centering of the lattice imposes Z ¼ 4 by symmetry, then there are 8  4 ¼ 32 total number of atoms per cell. The arrangement of the A atoms in C15 structure, shown in Fig. 6.1 is close to that of carbon in diamond structure. The B atoms form tetrahedra around them. When H is inserted, three lattices sites are available as shown on the right hand side of Fig. 6.1 and described in Table 6.2. Experimentally, YFe2 is a ferromagnet with an atomic moment, 1.8 mB (Bohr magneton) carried by iron and a Curie temperature TC ¼ 560 K. Hydrogen absorption leads to the formation of hydrides YFe2Hx with a large compositional range 1.3  x  5. The main effect is a regular increase of the cell volume versus x. Also this is accompanied by various structural distortions of the initially cubic lattice. The lowering of the crystal symmetry to orthorhombic then down to monoclinic for highest H content of 5 per fu can be related to hydrogen order in preferential tetrahedral interstitial sites A2B2 and AB3 (cf. Table 6.2). The hydrides recover the cubic C15 structure through a first order transition above a characteristic temperature TS, which decreases with the increase of the hydrogen content. Fig. 6.2 shows the deuterium composition dependence of TC and TS; deuterium is often used instead of hydrogen for physical characterizations such as neutron diffraction investigations [17]. The Curie temperature TC first increases from 560 to 720 K (x ¼ 1.3), then decreases linearly down to 363 K for x ¼ 3.5 (Fig. 6.2). For x w 4, the hydride is ferromagnetic from 4.2 to 140 K, then there is a sharp decrease of the magnetization. In order to understand the changes of the Fe moment band structure calculations were done for the hydrogen free intermetallic and on the hydrides. This is examined within two complementary approaches relevant to the influence of volume expansion on the magnetic properties within the C15 structure and the distorted lattice of YFe2 and the chemical role of hydrogen and its influence on the changes of the magnetization of the hydride lattice for increasing amounts of hydrogen: for YFe2H3 in the cubic structure, YFe2H4 in the orthorhombic Pmn21 SG and YFe2H5 in monoclinic lattice symmetry. 6.2.1. YFe2 versus volume and distortion effects Firstly we address the volume effects induced by hydrogen insertion. All-electrons calculations within LDA were carried out at the experimental and expanded cell volume of the C15 Fd3m type structure. The derived lattice constant from the hydride is 8.14 Å; the experimental lattice constant being 7.35 Å. Further, assuming the orthorhombic Pmn21 SG of YFe2H5, another calculation was done with the actual hydride orthorhombic lattice constants given in Table 6.3. These calculations are meant to simulate both the volume and structure distortion effects brought by the insertion of hydrogen.

14

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

Table 6.3 Atomic positions in cubic, orthorhombic and monoclinic YFe2/Dx. Deuterium (D) replaces H for experimental investigations purposes. Space group

F-43m

Pmn21

P1n1

Lattice parameters

Fe

16d

½, ½,½

c0 z c (Å) c ¼ 8.083 x,y,z 0, 0.262, 0 0, 0.762, 0.77 0.255, 0.257, 0.637

b ¼ 90

x, y, z 1/8,1/8,1/8

orthorhombic a0 z b0 z a*O2 (Å) a ¼ 5.437 b ¼ 5.850 Site Y1:2a Y2:2a Fe1:4b

monoclinic a0 ,b0 ,c0 (Å) a ¼ 5.437 b ¼ 5.850

Atom Y

Cubic a (Å) 8.011 Site 8a

Fe2:2a Fe3:2a D1:4b

0, 0.027, 0.395 0,0.487, 0.382 0.04, 0.235, 0.273

D2:4b

0.428, 0.28, 0

D3:2a D4:2a D5:4b

0,0.388, 0.719 0,0.883, 0.033 0.214, 0.580, 0.953

D6:4b

0.270, 0.930, 0.304

D7:2a D8:2a

0,0.063, 0.588 0,0.57, 0.178

Y1:2a Y2:2a Fe1:2a Fe1b:2a Fe2:2a Fe3:2a D1a:2a D1b:2a D2a:2a D2b:2a D3:2a D4:2a D5a:2a D5b:2a D6a:2a D6b:2a D7:2a D8:2a

0,0.262, 0 0,0.762, 0.77 0.255, 0.257, 0.637 0.745, 0.257, 0.637 0,0.027, 0.395 0, 0.487, 0.382 0.04, 0.235, 0.273 0.96, 0.235, 0.273 0.428, 0.28, 0.0 0.527, 0.28, 0.0 0, 0.388, 0.719 0,0.883, 0.033 0.214, 0.580, 0.953 0.786, 0.580, 0.953 0.27, 0.93, 0.304 0.73, 0.930, 0.304 0, 0.063, 0.588 0, 0.57, 0.178

D (A2B2)

D (AB3)

96g

32e

x, x, z x ¼ 0.3368 z ¼ 0.1556

x, x, x x ¼ 0.7118

At self-consistent convergence using a high precision integration of the face centered cubic first Brillouin zone, charge transfer shows a departure of 0.35 electron from Y spheres to Fe spheres. This slight transfer, not significant of ionic effects e rarely observed in the framework of such ab initio calculations e, signals a redistribution of the 2 s electrons of yttrium over its three valence basis sets (s, p, d) thus providing it with a d-character arising from its mixing with Fe(3d). Therefore one can conclude that the major effect is that of the hybridization of the different valence states, not the charge transfer. The site projected DOS (PDOS) for an assumed non-magnetic, spin degenerate configuration (non-spin polarized NSP) accounting for site multiplicity (twice more Fe than Y) are shown in Fig. 6.3 at the experimental as well as at the hydride volume. The origin of energies along the x-axis is taken with respect to the Fermi energy; this is equally followed in all other plots. Looking firstly at the general shape of the DOS one can observe the predominance of the Fe states in the close neighborhood of the Fermi level (EF), i.e., with respect to very low intensity yttrium states. The major effect of a larger lattice spacing can be seen from the narrower valence band and the increase of the density of states at the Fermi level n(EF) from 15 to 20 eV1. This signals a magnetic instability of this configuration in the Stoner mean field analysis; such an instability increasing upon expanding the lattice. The similar shapes between the PDOS signaling the mixing between Fe and Y states can be seen

c ¼ 8.083

at the lower part of the valence band (VB), with mainly s, p-like states between 6 and 2 eV, as well as toward the top of VB (d states). Such mixing will be further analyzed in the next paragraph regarding the chemical bonding. Within the conduction band (CB), Y(4d) states are found dominant as it can be expected from their emptiness. Similar features are obtained for the expanded YFe2 lattice in the orthorhombic structure of the hydride so that the discussion follows from above. The analysis of the chemical bonding is done using the ECOV approach presented above. The corresponding plots are shown in Fig. 6.4; we do not show the corresponding plots for the orthorhombic YFe2 expanded lattice which look like those of the expanded lattice in C15 symmetry. Partial ECOV are given for the interactions of one species of each kind for the YeFe, FeeFe and YeY bonds. The YeY interaction is found to be the lowest in magnitude. On the contrary the magnitude of the FeeFe interaction is dominating with respect to YeFe and YeY in the VB as well as in the CB. However its magnitude is qualitatively lower when the lattice is expanded meaning there is smaller overlap between FeeFe for instance. This should have consequences on the magnitude of the atomic moment carried by Fe within the expanded lattice as it will be shown in next section. The anti-bonding character of the FeeFe interaction toward the top of the VB and at EF points to the instability of the system in the non-magnetic NSP configuration. On the contrary, due to the lower occupation of

Fig. 6.1. Cubic C15 Laves phase with the three sites available for hydrogen insertion (cf. Table 6.2).

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

Fig. 6.2. YFe2: Deuterium composition dependence of TC and TS (cf. text).

yttrium states, its interaction with Fe as well as with Y atom gives bonding states up to EF and beyond within the CB; this YeFe bonding ensures for the stability of the system. The electrons in the d band crossed by the Fermi level are not all anti-bonding meaning that there is a part of them which becomes non-bonding in the neighborhood of EF and participate to the onset of the magnetic moment. 6.2.2. Magnetic properties of YFe2 and “covalent magnetism” As it can be expected for the ferromagnetic ground state of YFe2 at the experimental lattice constant, there is an energy stabilization of DE w0.95 eV per unit cell with respect to the NSP calculation when SP calculations are done. The magnetic moments are M (Y) ¼ 0.50 mB and M(Fe) ¼ 1.85 mB with a total magnetization Mcalc cell ¼ 6.37 mB, i.e. 3.18 mB per formula unit. This value is close to the one found experimentally: Mexp YFe2 ¼ 2.9 mB and to the calculated value by Mohn and Schwarz [71]. Consequently YFe2 is a ferrimagnet in its ground state, contrary to experiment which announces it as a ferromagnet with a zero moment on yttrium [72]. Another significant result that we extract from these calculations is the Fermi contact term of the hyperfine field (HFC) based on the spin density at the nucleus for the ns caused by the polarization of the s electrons by the d moments: Hcalc FC (Fe) ¼ 250 kGauss. The experimental value Hexp FC (Fe) ¼ 220 kGauss [72] points to a good agreement and comforts the computed electronic structure. The expanded lattice at the volume of YFe2H5 shows an enhanced magnitude of the moments for both constituents: M(Y) ¼ 0.81 mB and M(Fe) ¼ 2.42 mB with a total magnetization Mexpand ¼ 8.08 mB, cell i.e. 4.04 mB per formula unit. This stresses the magneto-volume

15

Fig. 6.4. Chemical bonding for pair interactions within non-magnetic YFe2.

effects. However it will be shown that the chemical interaction between iron and hydrogen decreases the magnetic polarization. The magnetization results are illustrated by the site projected SP-DOS (twice more Fe than Y) Fig. 6.5a which can help to further assess them. For both PDOS calculated at the experimental (a) and the expanded (b) volumes, the general trend is that the weights of the DOS at [ (majority) and Y (minority) spin populations are not the same; meaning that it is not a rigid-band shift which rules the magnetism of this system like in aeFe but it is rather that of a covalent magnetism [71]. Following the scheme used by Peter Mohn [38] to describe ZrFe2 magnetic behavior, there is a systematic antiparallel spin orientation for intermetallics made of a magnetic transition metal such as Fe here, with an A element from the beginning of the series, i.e. with small number of valence electrons such as Y and Sc (cf. case study C14-ScFe2 Laves phase in next section). The consequence is that the moment of A (yttrium) is provided by the covalent YeFe bond rather than a rigid energy shift of non-magnetic DOS, whence its negative sign -notice the YeFe overlap around 1 eV for Y DOS. For the expanded cell, beside the energy shift between the two spin populations causing the onset of the magnetic moment, there can be seen from Fig. 6.5b an enhanced localization of the Fe states with sharper PDOS peaks and less FeeFe overlap which explains the larger moment magnitudes. Lastly the effect of lattice distortion of YFe2 in the orthorhombic Pmn21 SG group leads to closely the same magnitude of the ¼ 16.45 mB, i.e. 4.11 mB per formula magnetization: Mexpanddistorted cell unit with the following distribution of the atomic moments over the 2-fold Y and 3-fold Fe lattice sites: M(Y) ¼ 0.721/0.724 and

Fig. 6.3. Non-magnetic PDOS for YFe2 at experimental volume a) and at expanded volume b).

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S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

Fig. 6.5. Site and spin projected density of states of YFe2: a) at experimental volume; b) at expanded volume of the hydride.

M(Fe) ¼ 2.44/2.39/2.39 mB. The agreement with the C15 expanded YFe2 results above points to a driving effect of the volume as with respect to the distortion one. 6.2.3. YFe2 and hydrogen insertion effects For the model structures accounting for the insertion of hydrogen within the YFe2 lattice with an ordered manner we have followed experimental findings regarding lattice expansions and distortions. Three different structural setups were assumed at increasing concentrations of hydrogen. The highest H content in A2B2 (96g) sites in C15 structure avoiding too small HeH distances (dHH  2.1 Å) is 3, i.e. YFe2H3. Then in the Pmn21 orthorhombic distorted lattice we considered YFe2H4 and lastly the monoclinic P1n1 structure was used for the hydrogen rich hydride YFe2H5 (Table 6.3). Calculations were carried out for the magnetic spin polarized (SP) configuration using the experimental lattice constants; for the YFe2H4 intermediate H concentration composition. The calculations were carried out at a volume computed in between those of YFe2H3 and YFe2H5. In all systems we observe charge transfer to occur from the atomic species (Y, Fe and H) to the interstitial empty spheres ES introduced due to the less compact structure of the hydrides. Its small amount of w 0.2 electron per ES/atomic species points to the covalent character of the chemical bonding. YFe2H3. The spin polarized calculations for the C15 YFe2H3 hydride give a magnetization per YFe2 formula unit: M ¼ 3.8 mB in agreement with experiment (3.9 mB). This magnetization enhancement is mainly driven by the increase of the moment carried by iron (2 mB versus 1.85 mB in the intermetallic system) which equally leads to an increase of the absolute value of the Fermi contact term of the hyperfine field HFC ¼ 232 kGauss. Despite this result which favors the volume effect of hydrogen (larger localization of the DOS similarly to the expanded intermetallic system, see Fig. 6.3b), the onset of an FeeH chemical bonding, significantly larger than the YeFe one partly due to the shorter average FeeH distance, was pointed out [73]. This is further developed for hydrogen rich systems YFe2H4 and YFe2H5 hereafter. YFe2H4 and YFe2H5. At self-consistency the magnetization of YFe2H4: M ¼ 3.35 mB per formula unit is found to be slightly smaller than that of YFe2H3. This is due to the decrease of the 4-fold Fe(4b) moment (1.48 mB) as with respect to the two 2-fold Fe(2a) moments (2.22 and 2.46 mB respectively) which arise from the changes of the FeeH distances computed as largest for the Fe(2a) sites, i.e. 1.72 Å (Table 6.3). This leads to a lower average Fe moment of w1.9 mB.

On the contrary the calculation of the magnetic structure of monoclinic YFe2H5 gives a vanishing magnetization with no polarization of the atomic species. Although this agrees with experiment, an assessment is orderly with the help of the PDOS. Fig. 6.6 gives the corresponding plots of the two hydrides YFe2H4 and YFe2H5 respectively. The PDOS are gathered for each atomic species to make the presentation clear (i.e. we do not show the splitting effects into the different crystallographic sites in the orthorhombic and monoclinic distortions). Although they show a large contribution in the lower part of the valence band, hydrogen states are found in a broad energy range of the DOS; furthermore they show a similar skyline to the Fe (as well as Y) PDOS. This feature should be further assessed by the analysis of the chemical bonding in next section. The largest contribution to the DOS arises from Fe states. While spin polarization causes energy shift between majority ([) and minority (Y) spin states within YFe2H4, there is no energy shift for the YFe2H5 in agreement with the absence of magnetization. The other feature we mention is the splitting of the DOS into massifs whose isolation with respect to each other can be seen most pronounced for YFe2H5: s-like from 10 to w7.5 eV, s, p-like from 7 to 3 eV, Fe(d)  t2g-like from 3 to 0.5 eV and Fe (d)  eg-like from 0.5 to w3 eV together with Y(4f) empty states within the CB. This could point to the increasing ionic-like behavior of the hydrogen-rich system. Fig. 6.7 shows the spin resolved pair interactions for metale metal (YeFe, FeeFe) and metal-hydrogen (YeH, FeeH) bonds, regrouping crystallographic sites contributions with their multiplicities; this explains the different y-axis scale as with respect to YFe2 which presents less atomic species. In as far as both spin channels are identical within YFe2H5, Fig. 6.7c shows the chemical bonding for one of the two spin channels. We remind that along the y-axis, negative, positive and zero ECOV values correspond respectively to bonding, anti-bonding and non-bonding contributions. Looking firstly at the metalemetal bonding, FeeFe and YeFe interactions have similar features to those discussed above for the intermetallic system. The new result is in the bonding with hydrogen which appears in the lower part of the VB [6, 4 eV] and it can be seen to prevail for FeeH with little contribution from YeH for both hydrides. Further FeeH interaction becomes as large as FeeFe one in YFe2H5 and the anti-bonding contribution from FeeFe becomes smaller toward the top of the VB but much larger above EF. From this the bonding of FeeH can be seen to prevail when hydrogen is inserted within YFe2 system at increasing amounts. The underlying mechanism is that of spin pairing of Fe with H leading to a progressive loss of the Fe moment. We underlined this feature in former calculations on iron nitrides such as FexN (8  x  2) [74].

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

17

magneto-volume effect due to the volume expansion which should lead to an enhancement of the localization of the iron 3d states due to the reduced ded overlap, on one hand and to the interactions between the valence states of scandium and iron (at both sites) with hydrogen on the other hand. Further the different roles played by the two Fe sublattices through their specific magnetic moments and the Fermi contact HFC term of the effective hyperfine field are examined. In addition, a detailed atom-resolved study of the magnetism is provided and the nature of the non-rigid-band behavior within ScFe2 and its di-hydride are assessed.

Fig. 6.6. Spin polarized PDOS for a) YFe2H4 and b) YFe2H5.

6.3. ScFe2 and its di-hydride C14 ScFe2 adopts the P63/mmc SG. There are 4 AB2 motifs per cell, i.e. 12 atoms. Since the P (primitive) centering of the lattice imposes Z ¼ 1, then there is 12  1 ¼ 12 total number of atoms. The A atoms are located at carbon sites in the hexagonal diamond structure also called Lonsdaleite. Contrary to the C15 structure there are two non-equivalent positions for the B atoms. This is exhibited in Fig. 6.8 with sites (2a) and (6h) B (Fe1) and B0 (Fe2) atoms respectively found at corners and edge middles on one hand and two planes separated by A (Sc) and B atoms on the other hand. Some hydrogen insertion sites are also shown. Contrary to YFe2, the hydrogen uptake in ScFe2, which adopts the C14 structure in its ground state, is only 2 per fu, i.e. ScFe2H2. Smit and Buschow [75] have studied the synthesis of this Laves phase and its hydride. As detailed in Table 6.4, the interesting structural peculiarity is that Fe is found in two distinct sublattices which are likely to interact differently with interstitial H. 57Fe Mössbauer spectroscopic measurements for both the average magnetic moment and the effective hyperfine field Heff of iron showed an increase of the magnitudes of these two quantities upon hydrogen insertion, without specifically assigning a role for each one of the two iron sites. Further an anisotropic structural change accompanies the formation of the hydride: c/a (ScFe2) ¼ 1.636 and c/a (ScFe2H2) ¼ 1.611. In this section hydrogen insertion effects within ScFe2 are investigated with complementary approaches relevant to the

6.3.1. Geometry optimization from pseudo-potentials calculations Like ScFe2, ScFe2H2 crystallizes in the hexagonal C14-type structure (P63/mmc SG). In this structure (Fig. 6.8; Table 6.4), iron atoms are in general 2a and particular 6h Wyckoff positions, Fe1(0, 0, 0) and Fe2(uFe2, 2uFe2, 1/4) while Sc are in 4-fold particular positions, 4f at (1/3, 2/3, uSc). Fe2, Sc and Fe1 atoms have an occupancy ratio of 3:2:1. The choices for hydrogen insertion sites were done based on the neutron diffraction studies on the deuterated C14 ZrMn2 [76]. In this system, H atoms occupy interstitial tetrahedral A2B2 (A ¼ Sc and B ¼ Fe2) voids with three possible positions: 6h, 12k and 24l which are partially occupied according to experimental studies. Therefore preliminary calculations of interatomic distances testing different structural setups with these positions were firstly done. Similar trends for spacing of H with the different sublattices were identified, i.e., dFe2H < dFe1H (see Table 6.5). Consequently in the present work, band structure calculations were performed with the assumption that H is located within 6h partially filled sites: H1 at (uH1, 2uH1, 1/4) and H2 at (uH2, 2uH2, 1/4) so that the resulting stoichiometry corresponds to the dihydride, Sc4Fe8H8. Starting values for a, c as well as the u internal coordinates were taken from experimental data. Keeping in mind the relatively large difference in atomic volume between Sc and Fe, the choice of the H insertion sites is in agreement with the Westlake criterion that imposes a minimum interstitial hole size of 0.40 Å. Too small HeH distances (dHH  2.1 Å) were avoided in order to respect the Switendick criterion (cf. Table 6.5). From the results, the hexagonal symmetry was preserved for both ScFe2 and its hydride. The initial and final u parameters given in Table 6.5 show slight differences; thus confirming the assumed setup. In order to obtain the equilibrium volumes and confront them with experiment, (energy, volume) values were computed around the experimental values. The resulting equilibrium volumes, respectively 40.6 and 47.1 Å3 for ScFe2 and for the hydride are w6 and w8% smaller than experimental volumes which amount to 43.4 and 51.3 Å3. Such a result is expected within the LDA approximation used, as it is known to under estimate lattice distances. The bulk modulus B0, was obtained from (E,V) curve fitting with a Birch 3rd order EOS. Magnitudes of 166 and 176 GPa of bulk modulus for ScFe2 and ScFe2H2 respectively, are within range of other intermetallics such as ZrNi and they indicate a larger compressibility of the hydride phase with respect to ScFe2. Using the total energy values extracted from the geometry optimization results a binding energy magnitude of 0.02 eV, in favor of bonded hydrogen within the lattice. 6.3.2. Analysis of electron localization with ELF function Fig. 6.9a and b shows the electron localization function mapping for the planes at z ¼ 0.25 and x w 0.77 respectively. In Fig. 6.9a the structure and the tetrahedral A2B2 void within which hydrogen is inserted are reproduced as in Fig. 6.8 to enable comparison. The strongest localization is around hydrogen sites while around the Sc and Fe2 locations there is weak localization. This agrees with the hydride’ chemical picture of the system, i.e., with negatively charged hydrogen which amounts to w0.4

18

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

Fig. 6.7. Chemical bonding in hydrogen rich YFe2. Spin resolved in YFe2H4. a) majority spins ([), b) minority spins (Y) and c) spin degenerate in non-magnetic YFe2H5.

electrons from a Bader charge analysis. This indicates a covalently hydrogen within the intermetallic. Between the atomic species the green areas point to a smeared electron distribution within the system similarly to a free electron-like character expected within a metallic network. From Fig. 6.9b showing the nearest neighbors of hydrogen, the ELF of Sc and Fe2 are seen distorted towards H. This is due to the interactions between Sc and Fe2 on one hand

and H on the other hand. This is concomitant with the interatomic distances listed in Table 6.5 in as far as Sc and Fe2 have the shortest separations with H. Such a feature will be further developed within the chemical bonding section. 6.3.3. All-electrons calculations The crystal parameters provided by the geometry optimization initial processing (Table 6.5) were used for the input of all-electrons calculations. For ScFe2H2, the positions of the lacking hydrogen atoms within the 6h interstices were considered as interstitial sites where augmented spherical waves were placed. The resulting breaking of initial crystal symmetry was accounted for from the differentiation of the crystal constituents in the calculations. Since both ScFe2 and ScFe2H2 systems are experimentally known to have a magnetic behavior, spin polarized calculations were carried out. Further, other sets of computations were performed for hydrogenfree models, at the same volume of the experimental hydride. This simulates the manner in which the volume expansion (negative pressure) affects the magnetic behavior of the different atomic Table 6.4 Atomic positions of A and B (dispatched into two sublattices) metals and H insertion sites in hexagonal C14 Laves phase (here ScFe2). Atom

Environment

B (Fe) A (Sc) H Fig. 6.8. The hexagonal crystal structure of ScFe2 AB2 C14 Laves phase with the two B sites occupied by Fe1 (red spheres) and Fe2 (blue spheres) e cf. Table 6.4). In the dihydride, H can be inserted at positions signaled by small black spheres. Connected dashed lines represent the tetrahedral A2B2 environments of hydrogen formed by two Sc (purple spheres) and two Fe2.

A2B2

AB3 B4

Site symmetry

Atomic positions

2a 6h 4f 24l 12k 6h1 6h2 12k 4f 4e

0, 0, 0 x, 2x, 1/4 1/3, 2/3, z x, y, z x, 2x, z x, 2x, 1/4 x, 2x, 1/4 x, 2x, z 1/3, 2/3, z 0, 0, z

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

19

Table 6.5 Structural results for ScFe2 and its di-hydride. Volumes per fu are given as equilibrium/experimental. u internal coordinates are given as starting/final values from the geometry optimization calculations. ScFe2

H-free ScFe2H2

H-free ScFe2H2

ScFe2H2

c/a ¼ 1.636

c/a ¼ 1.636

c/a ¼ 1.611

c/a ¼ 1.611

a (Å) c (Å) Volume (Å3) uSc uFe2 uH1 uH2 dScFe1

4.965 8.125 40.6/43.4 0.066/0.065 0.836/0.828 e e 2.915

5.279 8.507 e e e e e 3.100

5.279 8.507 47.1/51.3 0.066/0.046 0.836/0.830 0.463/0.462 0.202/0.212 3.100

dScFe2

2.899 2.952 2.470 e e e e e

5.250 8.592 e e e e e 3.084 3.110 3.052 3.084 2.592 e e e e e

dFe1Fe2 dFe1H (Å) dFe2H (Å) dScH (Å) dHH (Å)

3.068 3.100 2.603 e e e e e

3.068 3.100 2.603 2.800 1.730 1.730 2.385 2.052

species. These effects can be important in such intermetallic systems in as far as the onset of the magnetic moment is due to interband spin polarization, i.e., it is mediated by the electron gas in a collective electrons approach. This is opposite to other systems, such as insulating oxides where the magnetization is of intra-band character, and hence, is less affected by volume changes such as those induced by pressure. For the sake of addressing anisotropy effects, an additional expanded hydrogen-free ScFe2 model was calculated with the experimental value of the c/a ratio of the intermetallic system. 6.3.3.1. Spin degenerate calculations. A slight charge transfer of w0.104 electron is seen from Fe2 towards Sc and Fe1. However its amount is not significant of an ionic behavior e rarely observed in the framework of ab initio calculations for such systems. Therefore it can be argued that the bonding is not due to charge transfer but rather imposed by the hybridization between the different valence states. The PDOS of ScFe2 is given in Fig. 6.10a. Fe1 and Fe2 PDOS cross the Fermi level with a larger intensity than Sc states. The similar shapes of the partial PDOS indicate a mixing between Fe2, Fe1 and Sc states, mainly at the lower part of the valence band (VB), with mainly s, p-like states between 6 and 2.5 eV, as well as towards the top of VB (d states). Such mixing will be analyzed later with the chemical

Fig. 6.10. Non-magnetic PDOS for a) ScFe2 and b) ScFe2H2.

bonding. Within the conduction band (CB), Sc(3d) states are found dominant as it may be expected from the location of scandium at the very beginning of the 3d period, with mainly empty d states. For the hydride PDOS shown in Fig. 6.10b, narrower and more localized peaks are found within the energy range from 5 eV to EF. This is due to reduced ped overlap between the Sc, Fe and Fe1. Further, two additional regions appear within VB, from 5 to 8 eV and from 10 to 12 eV, respectively. One can observe within these regions a quantum mixing between Sc, Fe2 and H. This is significant of interactions between Sc and Fe2 on one hand and H on the other hand. This is concomitant with the character of the ELF described earlier. It is then expected that Fe1 have a different magnetic

Fig. 6.9. Electron localization function (ELF) maps along a) horizontal plane at z ¼ 0.25 and b) vertical plane at x w 0.77. Connected full lines represent the tetrahedral A2B2 H environments formed by two Sc and two Fe2.

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behavior within the hydride with respect to pristine ScFe2 as it is the less interacting species with H. 6.3.3.2. Analysis within Stoner theory of band ferromagnetism. In as far as Fe 3d states were treated as band states by our calculations; the Stoner theory of band ferromagnetism, described above, can be applied to address the spin polarization. From Janak [37], I (Fe) ¼ 0.46 eV and the calculated In(EF) values of Fe1 and Fe2 are given in Table 6.6 for all the computed models. This amounts to a magnitude of w1.409 and 1.519 for Fe1 and Fe2 respectively within ScFe2 on one hand and w1.645 and 1.799 for Fe1 and Fe2 respectively within ScFe2H2. This indicates a magnetic instability of for both systems. This prediction is concomitant with the experiment where both ScFe2 and ScFe2H2 are found magnetic with an average magnetic moment measured for iron. From the spin polarized calculations, finite magnetic moments are expected to be carried by both Fe1 and Fe2. Also, Table 6.6 gives the values of the Stoner product for hydrogen-free models. Again, a tendency towards magnetic order is found with magnitude of their Stoner products larger with respect to the intermetallic and the di-hydride models, which emphasizes interplay of magneto volume versus H chemical bonding effects. This will be quantitatively discussed within the spin polarized calculations sections where magnitudes of the magnetic moments and the spin occupation of the Fe1(3d) and Fe2(3d) will be presented. 6.3.3.3. Analysis of the chemical bonding. As before, we analyze the chemical bonding using the ECOV approach. The corresponding plots for pristine ScFe2 and its di-hydride are shown in Fig. 6.11a and b respectively. Partial ECOV are given for the atomic pair interactions of SceFe1, SceFe2 and Fe1eFe2 bonds; this is followed for all ECOV. In both cases Fe1eFe2 interaction is found to have the largest bonding peak between 4 eV and 1 eV. This feature is mirrored by an anti-bonding peak at the top of VB up to EF which points to the instability of the system in the non-magnetic configuration. Then it can be suggested that the strongest bonding interaction within VB arises from SceFe1, followed by weakly bonding SceFe2. These contribute to the stability of both systems respectively. The change in bonding strength is proportional to the distance magnitudes given in Table 6.5, i.e., the shortest interatomic distances characterize the strongest interactions. This applies for the SceFe1 and SceFe2 interactions in as far as dScFe1 < dScFe2 for both intermetallic and its di-hydride. Further, one can notice that the electrons in the d band crossed by the Fermi level for SceFe1 and SceFe2 interactions are not all anti-bonding. A part of those electrons becomes non-bonding in the neighborhood of EF, thus participating to the onset of the magnetic moment. This is expected for iron in as far as the experiment determines magnetic ScFe2 and ScFe2H2. We also suggest the possibility of an ordered magnetic moment to be carried by Sc, since it takes part of both non-bonding

Fig. 6.11. Inter-metal bonding within a) ScFe2 and b) ScFe2H2.

interactions at EF. This proposition will be examined within the spin polarized calculations as a finite moment is expected for Sc. 6.3.3.4. Spin polarized SP calculations. As it can be expected for the spin polarized configuration of all models (see Table 6.6), there is an energy stabilization with respect to the non-spin polarized configuration. This agrees with the experimental data whereby ScFe2 and its di-hydride are identified both magnetically ordered as ferromagnets. On the other hand, the relative energies (Erel) magnitudes, which represent the difference between NSP or SP energies and the non-magnetic energy of the intermetallic (E0 ¼ 89859.40896 eV), indicate that the magnetic configuration of the hydride system is the most stable. Further, both expanded H-free models are unstable. This can be expected in as far as such hypothetical systems were not evidenced by the experiment.

Table 6.6 Results of magnetic calculations for ScFe2 and ScFe2H2. In(EF) magnitudes are given for Fe1/Fe2 sublattices respectively. Erel in eV units and per fu, represents the NSP/SP energies with respect to the non-magnetic energy of ScFe2 (E0 ¼ 89859.40896 eV). Fermi contact term HFC of hyperfine field and its core part HFCcore are in kGauss.

In(EF) Erel mSc (mB) mFe1 (mB) mFe2 (mB) hmFei (mB) M (mB) Htotal FC (Fe1) Htotal FC (Fe2) Hcore FC (Fe1) Hcore FC (Fe2)

ScFe2

H-free ScFe2H2

H-free ScFe2H2

ScFe2H2

c/a ¼ 1.636

c/a ¼ 1.636

c/a ¼ 1.611

c/a ¼ 1.611

1.409/1.519 0.000/0.215 0.482 1.468 1.560 1.514 2.600 157 162 159 164

1.794/1.914 2.560/1.906 0.676 2.229 2.101 2.165 3.591 240 229 252 232

1.931/1.923 2.598/1.917 0.682 2.279 2.118 2.198 3.635 246 226 255 231

1.645/1.799 24.318/24.754 0.347 2.486 1.967 2.226 3.791 203 232 272 214

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The PDOS curves for the spin polarized configuration of ScFe2 are shown in Fig. 6.12a. Within the VB two energy regions can be identified, from 7.5 to 5 eV, low intensity itinerant s,p states of all constituents are found; this is followed by larger intensity peaks mainly due to 3d(Fe) up to and above EF. Exchange splitting can be seen to mainly affect the latter as it is expected from the above analysis of the magnetizations. Majority [-spin states, for both Fe1 and Fe2 at EF, are concentrated in sharp and narrow PDOS peaks, contrary to minority Y-spin states that are found in PDOS minima. Fe1 and Fe2 peaks within the energy range (1.3, 0.3) eV for [-spin states are similar to those corresponding to (0,1) eV for minority spins (Y-spin) states. This shift in spectral weight for [-spin states below the Fermi level and for Y-spin states above EF corresponds to the onset of magnetic moments carried by Fe1 and Fe2. One can attribute this to Stoner rigid-band magnetism at first sight. But PDOS weights for [and Y-spin populations are not the same. This mismatch between both spin populations is mainly due to the Fe2 (3d) states peaks at w1 eV for the Y-spin states. One also notices that the peaks at w0.7 eV for Y-spin Sc (3d) states are more intense than those for [-spin states. This indicates the possibility of a magnetic moment carried by Sc with and opposite direction to those carried by Fe1 and Fe2. The moment of scandium is provided by the covalent SceFe1 bond, rather than by a rigid energy shift of non-magnetic PDOS, whence its negative sign -notice the SceFe1 overlap around 0.7 eV for Y-spin PDOS. 6.3.3.5. Covalent magnetism approach of the DOS. Generally, the magnetism of metals and alloys is correctly provided within the Stoner model whereby the magnetic moment is the result of a rigid-band shift of an initially non-magnetic (i.e., with total spins) system to low energy majority spin ([) DOS below the Fermi level

21

and to minority spin (Y) DOS at higher energy (above EF). This is the case of body centered cubic ferromagnetic a-Fe. However early magnetic band calculations, allowed to identify a new feature exhibited by many intermetallic systems such as ZrFe2, where the weights of the two spin populations DOS are not equal as proposed by the Stoner rectangular rigid-band model. The model of ‘covalent magnetism’ was therefore proposed to provide an explanation going beyond the basic Stoner model. Schematically, the model borrows the concept of chemistry, by using a molecular orbital type sketching of the way spins of two magnetic species are arranged in the lattice [71]. The result is a larger majority DOS weight with respect to the minority DOS on one hand and different partial DOS (PDOS) intensities in the spin down channel, on the other hand. Interestingly the systems under investigation are described in this framework too. Fig. 6.12b shows Sc-Fe1 interactions for [ and Y spin states curves for ScFe2. These are responsible for the magnetic moment carried by scandium which was announced earlier on within the non-magnetic ECOV section. One can notice that Y-spin interaction is the strongest. This explains the negative sign of the moment carried by Sc. In particular, the peak at w0.7 eV is concomitant with the overlap between Sc and Fe1 3d states in Fig. 6.12a. 6.3.3.6. ScFe2 and magneto-volume effects. Magnetic moments are obtained from the charge difference between [-spin and Y-spin of all valence states; their calculated values are listed in Table 6.6. The computed magnitudes for both the average magnetic moment of iron and the magnetization per fu for ScFe2 are 1.514 and 2.600 mB respectively. These are within the range of experiment, where magnitudes of 2.9 and 1.45 mB are measured for the average magnetic moment of iron and the magnetization per fu respectively. Also Sc carries a negative magnetic moment of 0.482 mB. Thus ScFe2 is closer to a ferrimagnet than to a ferromagnet suggested by experimental results. The same feature of antiparallel magnetic alignment between the two constituents was equally observed for YFe2 in preceding section Moreover, the Sc (3d) states have a calculated value of 0.347 mB which stands out as the largest contribution within the magnetic moment. The second largest contribution is that of the Sc (4p) states with a value of 0.103 mB. The magnitude of the average magnetic moment for iron within ScFe2H2 is of 2.226 mB in agreement with experimental value of 2.23 mB. This computed value corresponds to an increase of 67% with respect to ScFe2. The volume expansion induces an increase of the average magnetic moment of iron as seen for the expanded H-free model in (Table 6.6). 6.3.3.7. The role of hydrogen within ScFe2H2. The PDOS accounting for site multiplicities within ScFe2H2 are shown in Fig. 6.13a. In comparison with the pristine intermetallic system ScFe2, the VB of the hydride is 4 eV wider. Consequently the energy intervals given above are found here too with the difference of widened itinerant part which now includes H s-like states, i.e., 11, 6 eV. The similar skylines between the PDOS of the different atomic species in the energy regions describe the hybridization of the metallic species with hydrogen. The sharper and narrower nature of these PDOS peaks, compared to those of ScFe2 (Fig. 6.12a) point to a larger localization of the states in the hydride system. In particular, one can notice a sharper localization of Fe1 PDOS, within the energy range between 5 eV and EF, where the PDOS magnitude falls to w0, while the PDOS for Fe2 has a finite magnitude at EF. This feature is significant of peculiar magnetic behaviors of the two Fe sublattices.

Fig. 6.12. SPeScFe2. a) Site and spin projected DOS; b) Spin resolved ECOV (UPspin ¼ majority spin channel [; DOWN-spin ¼ minority spin channel Y).

6.3.3.8. Strong and weak ferromagnetism within ScFe2H2. Within the hydride ScFe2H2, hydrogen insertion is seen to further increase

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S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

are sketched. Fe1 interactions with hydrogen are weak with a slightly anti-bonding character in Fig. 6.13b (majority spins) and negligible bonding magnitude in Fig. 6.13c (minority spins) within the VB. The second less weak bonding is the SceH one; it is bonding throughout the VB in both panels. But the major contribution to the bonding comes from strong Fe2eH bonds in both panels. These results illustrate the discussion in earlier sections for the inversion and the hyperfine field. This is in agreement with the interatomic distances given in Table 6.5 which shows shorter SceH and Fe2eH separations with respect to Fe1eH. Thus it can be considered that the Fe1 sublattice is shielded from the other atomic species by hydrogen atoms. Turning to the di-hydride, this is no more the case and these populations become unbalanced, i.e., 3d4.6 for Fe1 and 3d4.3 for Fe2. An electron population close to 5 for Fe1ed([) corresponds to a nearly half filled d band while this is not so for Fe2. One can further confirm this trend by simulating a hypothetical tri-hydride ScFe2H3 by filling up all interstitial 6h positions. The resulting majority spin populations are found increasingly unbalanced, with 3d4.8(Fe1) and 3d3.7(Fe2); for minority spins these numbers are such as d1.8(Fe1) and 3d3.3(Fe2). The resulting magnetic moments for Fe1 and Fe2 are w3 mB and 0.4 mB respectively. A lowering of the overall magnetization, i.e., 1.95 versus 3.79 mB is then obtained. This can be explained by an increasingly e magnetically e isolated Fe1 with respect to Fe2 which loses its spins by pairing with neighboring 12 H. This particular situation is found in magnetic systems which present simultaneously strong and weak ferromagnetic constituents such as the anti-perovskite iron nitride g-Fe4N [74]. In this system Fe atoms which are close to N, at face centers behave as weak ferromagnets with M(Fefc) ¼ 2 mB, while those away from N at cube corners behave as strong ferromagnets with an enhanced magnetic moment, M(Fec) ¼ 3 mB. A strong ferromagnet in this definition has a completely filled subband, here d([) and the Fermi level falls in a DOS minimum; this is also observed from the Fe1 PDOS in Fig. 6.13. Consequently, the hydride is characterized by the simultaneous presence of strongly and weakly magnetic behaving iron species, Fe1 and Fe2 respectively. We emphasize that this is a known feature characterizing INVAR alloy, Ni0.35Fe0.65 as well (Ni,Pd)Fe3N nitrides [51].

Fig. 6.13. SPeScFe2H2. Site and spin projected DOS a) and chemical bonding b) ECOV for spin[ and c) ECOV for spin Y.

the average moment magnitude. But looking at the relative magnitudes of the iron magnetic moment there is an interesting feature relevant to the inversion of magnitudes for Fe1 and Fe2 magnetic moments, i.e., 2.486 and 1.967 mB respectively. This can be due to the closer Fe2eH (1.73 Å) separation than Fe1-H (2.8 Å) one as it will be further analyzed in the spin resolved chemical bonding section. Nevertheless, this pertains to relevant physics of magnetism which is obtained from a detailed analysis of the electron populations. Considering the majority spins ([), there is close amount of charge distribution in both Fe d states within the pristine intermetallic system, i.e., 3d4.05 for Fe1 and 3d4.07 for Fe2. The different FeeH interactions are resolved for the two spin orientations as shown in Fig. 6.13b and c. These are plotted for all atoms contrary to the other ECOV where atom to atom interactions

6.3.3.9. Fermi contact term of the hyperfine field. The calculations of the Fermi contact term of the hyperfine field HFC for ScFe2 provide HFC magnitudes of 157 and 162 kGauss for Fe1 and Fe2 respectively. 57Fe Mössbauer spectroscopic investigations, reported experimental magnitudes for Heff such as 167 and 174 kGauss without assigning them to the corresponding iron site. The computational results permit to establish the following trend by assigning the former magnitude to Fe1 and the latter to Fe2. As for the difference with respect to experiment, it can be related to different origins relevant to (i) drawbacks of the LDA in treating with sufficient accuracy the polarization of core wave functions, and (ii) the non-stoichiometry of the experimentally prepared intermetallics and the subsequent disorder within the solid solutions. The calculated HFC magnitudes for the hydride system are 203 and 232 kGauss for Fe1 and Fe2 respectively. Experimental findings point to magnitudes of 239 and 300 kGauss for Heff. These experimental values were not assigned to the different iron sites. Before doing so, one must treat the striking feature of the large departure from experimental values. For this purpose we suggest to split HFC into its two major contributions namely the one arising from the core 1s, 2s and 3s electrons (Hcore FC ) and the one due to the core valence 4s electrons (Hval FC ). While HFC is usually strictly proportional to the magnetic moment, Hval FC contains large contributions from the neighboring atoms. The calculated Htotal FC values are the sum of these two parts. Since Fe1 interacts weakly with other constituents, its

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

contact with them is reduced. Consequently only the core contribution of HFc must be considered for Fe1. In contrast, Fe2 atoms interactions with neighboring species are larger, hence both core and valence contributions of HFC must be accounted for. The computed value of Hcore FC for Fe1 results into a magnitude of 272 kGauss. Now a comparison can be established between the computed core and total HFC magnitudes of 272 and 232 kGauss for Fe1 and Fe2 respectively on one side and the experimentally measured magnitudes of 300 and 239 kGauss on the other side. It follows that the larger experimental value of 300 kGauss for Heff can be assigned to Fe1 and the smaller one of 239 kGauss to Fe2. Also, the orders of magnitudes for HFC (see Table 6.6) are proportional to those of the magnetic moments in both ScFe2 and ScFe2H2. Thus the inversion of the order of magnitudes is also observed for the hyperfine field. 6.4. Hydrides of pseudo Laves phases It is now admitted that cubic pseudo-Laves phases C15b are serious candidates for hydrogen storage [24]. With general formulation REMgNi4 they crystallize in F-43m SG with MgSnCu4type structure. They can reversibly absorb w1 wt% H2 (i.e. 3e4 H/ fu) under moderate conditions. Depending on the rare earth used, these intermetallics have very different behaviors upon hydrogenation. For RE ¼ Y, La, Nd, Gd, an orthorhombic distortion occurs [16]. For RE ¼ Ce, no absorption is noticed and the compound is decomposed under hydrogen at high temperature (i.e. 250e300  C) [77]. As these rare earths have close metallic radii, a conclusion based on a steric hindrance effect cannot be taken into account. RE based intermetallics are known to exhibit special properties regarding RE electronic configuration. For instance, cerium based compounds have interesting structural, magnetic and magnetovolume properties and the valence state of Ce is a key point (cf. Section 8). In as far as the study of the role played by RE within REMgNi4H4 is not easy to assess from an experimental point of view, the modeling approach within DFT can bring further information. In this section we examine GdMgNi4 and its corresponding hydride. Despite the experimental evidence of an orthorhombic distortion of the intermetallic ternary system upon hydrogenation, the cubic symmetry is also considered for sake of comparison and validation of the theoretical approach [78]. 6.4.1. Calculating electronic structure of RE based intermetallics Discussing the electronic band structure of a rare earth based intermetallic with RE from the middle of the 4f series such as Gd, is problematic because of the positioning of the 4f states in the calculations. In other words, as far as the 4f subshell is half-filled (7 electrons occupying half of the available 14 positions) and being part of the valence basis set, it is placed by the DFT-LDA/GGA calculations on the Fermi level. This is not correct because the degree of localization of the 4f states becomes important and the treatment of itinerant as well as localized electrons within the same theoretical framework imposes the use of self-interaction corrections (SIC). SIC pertains to the interaction of an electron with itself as obtained by the LDA. In all-electrons calculations this problem is circumvented using open-core f states. PAW method accounts for the f states in the generated pseudo-potentials. Here we use PAW potentials as built within GGA which was preferred over the LDA approximation because it gives better agreement for lattice constants versus LDA which is over binding, i.e. giving too small parameters. However trends of the LDA versus GGA are casesensitive and care should be taken when using them. Then a search of the equilibrium quantities is done from a fit of energy versus volume curves around experimental lattice constant (w7 Å) with Birch EOS. The resulting equilibrium values are E0 ¼ 30.52 eV,

23

V0 ¼ 347.3 Å3 and B0 ¼ 121.0 GPa. The calculated equilibrium a lattice parameter is in close agreement with experiment. The bulk modulus B0 is within the range of intermetallic systems. 6.4.2. Insertion sites of hydrogen Considering the semi-empirical criteria of Westlake (rHsite > 0.4 Å) and Switendick (dH-H  2.1 Å), not all the crystal interstices within the C15b cubic lattice are available for hosting hydrogen. The sketch of the structure with the available tetrahedral sites for hydrogen: RE2Ni2, REMgNi2, RENi3 and Ni4 are shown in Fig. 6.14. 6.4.3. Geometry optimization and equation of state Although the hydride GdMgNi4H4 is orthorhombic; we make the hypothesis that it remains cubic upon hydrogenation. Four hydrogen atoms are placed alternately in these sites, and the corresponding energies show that the more suitable sites to host hydrogen atoms are the RENi3 sites. Such a conclusion is consistent with the works of Hong and Fu [79] who have shown that the transition metal has an important role in the location of hydrogen in ZrT2 Laves phases. According to them, for T ¼ Ti to Fe, the hydrogen atoms are located in Zr2T2 sites whereas for T ¼ Co to Cu, the hydrogen atoms are placed in ZrT3 sites. Considering that the hydrogen atoms are located in RENi3 tetrahedra, optimizing the geometry is done for obtaining the actual position of the hydrogen atoms within the cell. Indeed, if hydrogen is exactly at the center of the tetrahedron, then xH ¼ 0.8611 in the 16e site (x,x,x). After a full geometry optimization, the hydrogen position is xH ¼ 0.839. This involves a displacement of H toward Ni atoms rather than to Gd (Fig. 6.15). Such a conclusion is consistent with the works of Guénée et al. [16] who have shown that hydrogen was closer to nickel rather than to lanthanum in Ni4H4 clusters derived from the LaMgNi4H3.6 hydride. For an illustration we can see from the ELF slice in Fig. 6.16 that the electron cloud of the H- entity is no longer centered on the ion, but it is distorted towards the nickel atoms. The red color for the contour plots around hydrogen atom is consistent with the fact that it carries a negative charge contrary to the nickel, whose contour is drawn in blue. Mg ELF is smeared out over the whole cell in a free electron-like behavior (ELF ¼ ½ green areas). For GdMgNi4H4 in face centered cubic structure, the starting lattice parameter is a ¼ 7.35 Å. The fully relaxed structure gives a ¼ 7.46 Å. Note that GGA overestimates distances, contrary to the

Fig. 6.14. C15b cubic structure of pseudo-Laves phases REMgNi4. a), b), c) and d) indicate respectively RE2Ni2, REMgNi2, RENi3 and Ni4 sites for H insertion.

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S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

Fig. 6.15. Location of hydrogen (black spheres) within a GdNi3 tetrahedron a) before geometry optimization and b) after geometry optimization (cubic hypothesis). Distances are in units of Angströms (Å)

over-binding LDA. The position of hydrogen is kept as xH ¼ 0.839. Then, following the procedure above for the intermetallic, the energies of the system were plotted for different cell volumes at Fig. 6.17a with the equilibrium parameters deduced from Birch EOS in the insert. From the energy values a hydriding enthalpy of: DHhydr. ¼ 0.015 eV ¼ þ1.45 kJ/mol H2 is computed. According to this positive value, the cubic hydride GdMgNi4H4 is unstable. Considering the orthorhombic structure (Pmn21 S.G.), the starting parameters were the crystallographic data provided for LaMgNi4H4 for which an orthorhombic distortion was reported [16]. Fig. 6.17b gives the E(V) curve and the EOS fit values established for the hydride. The enthalpy of this hydride is then DHhydr. ¼ 0.2775 eV ¼ 26.8 kJ/mol H2. From an energetic point of view, the orthorhombic structure is more stable. These trends are also observed from the equilibrium energy values in the inserts of (43.64) > ½ Eortho (89.62) enote the twice larger Fig. 6.17: Ecubic 0 0 fu per cell in the orthorhombic structure-. Further there is a hardening of the hydride when it is in orthorhombic symmetry meaning that the structure is more compact than the cubic one. A more quantitative assessment can be obtained from the analysis of the charge density file obtained at self-consistent convergence. For a reminder, in MgH2 we find the ionic picture of Mg2þ and H1, while for Mg2NiH4, identified as a better candidate for hydrogen source, the charge on H is calculated less ionic with H0.8. Clearly Ni plays a major role in reducing the ionic behavior of MgH2. Now, in presently studied systems Mg is equally ionized and there is an equal number of Ni in both forms of GdMgNi4H4. But the structure distortion from cubic to orthorhombic leads to a change on the charge carried by H: w0.39 in the cubic to w0.55 in the orthorhombic form. Then the stabilization of the orthorhombic

Fig. 6.17. Energy versus volume curves for GdMgNi4H4 in a) cubic structure hypothesis and b) orthorhombic structure hypothesis.

form can be due, at least partly, to the change from covalently bonded H in the cubic form to a more iono-covalent H in the orthorhombic form. 6.4.4. Mixed occupation An exchange ratio has been experimentally observed in the material elaborated by mechanical alloying, then subsequently heat treated to 600  C for 1 h. In order to confirm such a phenomenon, a supercell was created, with 8 GdMgNi4 cells. The energy of the system has been calculated with an energy of 30.52 eV. Then an atom of gadolinium was exchanged with an atom of magnesium in order to obtain the (Gd7Mg)(Mg7Gd)Ni32 configuration, and therefore simulate an exchange ratio of 12.5% (i.e. 1/8) which is close to the ratio noticed experimentally (i.e. s ¼ 15%). The results of total energy e30.30 eV are less favorable than without exchange. This was also observed for RE ¼ Y, Ce. Then the compounds with exchange ratio, prepared by mechanical alloying are metastable. Finally, it is worth pointing out that the calculated energy for the “disordered” compounds are always slightly higher (DE ¼ 0.1e0.2 eV). This means that the “disordered” compounds should be considered as metastable compounds. Nevertheless, as the energy difference between both configurations is very small, it can be assumed that a long heat treatment can output a null exchange ratio. Such experimental developments are currently underway. 7. Hydrides of AB5 intermetallics

Fig. 6.16. Electron localization function ELF slice, with all atomic species in GdMgNi4H4 (cf. text). The ruler on the right indicates colors of full localization (1) and zero localization (0).

Haucke phases AB5 (A ¼ rare earth and B ¼ d- and p-block elements), are known to absorb hydrogen up to the approximate composition of 6 H atoms per formula unit at mild conditions of

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temperature and pressure (w298 K and w2  105 Pa) [79]. This enables their use in rechargeable metal hydride batteries [5]. The hydrogen storage capacity for LaNi5 amounts to w1.38 wt% which exceeds that of liquid hydrogen by w40%. Nevertheless its gravimetric capacity is still small with respect to that of archetype hydride for applications, MgH2, which amounts to w7.6 wt%. It is also important to mention the remarkable feature of the rapid kinetics of the LaNi5eH2 system [80]. The experiment reports the formation of two hydrogen solutions, namely primary a-LaNi5H0.5 and more filled b-La2Ni10H12 (Z ¼ 1) [81]. The a phase crystallizes in the same SG as the pure intermetallic system whereas the b phase is characterized in two different SGs, namely P63mc and P31c. For the latter hydride phase, neither the experiment nor previous ab initio calculations were able to discard either SGs. On the other hand, LaNi5 is reported to be paramagnetic by the experiment [82]. With respect to this specific point, previous ab initio calculations have been performed but the resulting ground state configuration was different to the one reported by the experiment. For instance, early calculations performed within the augmented plane wave method (APW) showed that LaNi5 is a “very week ferromagnet” with a total moment of 0.69 mB. Recent calculations gave total moment values such as 1.25 and 1.15 mB with the self-consistent linear muffin-tin orbital (LMTO) within the atomic sphere approximation (ASA) and the full potential (FP) LMTO methods respectively [83]. The authors commented the results by considering that the discrepancy of the total moment with respect to the experiment may be due to a delicate problem of calculation of the total energy when the material is in a transitional state between a ferromagnetic and a paramagnetic state. In this section the electronic structure, energetics and magnetic ground state configuration of LaNi5 and its saturated hydrogen b solution are analyzed from the first-principles as an illustration of this class of intermetallic hydrides. For the calculation procedure, we first revisit the symmetry of the b models to check for the most stable SG from both a structural and energetic points of view. We also study the binding of hydrogen within the host lattice. The latter property is further evaluated by computing an additional hypothetic intermediate hydride model LaNi5H3 at the SGs of both the pure intermetallic and the b hydride. Secondly we examine the ground state of LaNi5 by calculating the electronic band occupations in both non-spin polarized (NSP) and spin polarized (SP) configurations provided by all-electron method. Finally, we study the energetics of hydrogen uptake in the b hydride as far as it is generally controlled by a size factor and an electronic factor. The volume expansion brought by hydrogen insertion in the host lattice usually induces a magnetization in such systems anticipated namely at the d- and p-block elements constituents. In the case of LaNi5 one can presume the onset of a local magnetic moment carried by nickel atoms. It is important to study the magnetization of the hydride system relevant to the interplay of magneto-volume effects versus the chemical bonding of hydrogen with the different atomic constituents, especially with nickel atoms.

25

and/or 12n sites have been frequently considered as the most favorable occupation sites. This may be explained in the context of the atomic environment of hydrogen within the octahedral site on one hand and the reported hole size of the latter tetrahedral site of w0.408 Å on the other hand which is concomitant to the Westlake criterion imposing a minimum interstitial hole size of 0.40 Å. Consequently, the 6m site cannot be ruled out with its hole radius of w0.551 Å. However a saturated hydride (the b phase) exists with two possible ordered structures as reported by neutron diffraction data [81]. These have the P63mc and the P31c symmetries respectively. They both show a doubled unit cell along the c-axis. Within P31c SG, La atoms are located in special 2a position at (0, 0, z). Ni atoms are found within three non-equivalent positions: Ni1 in 6-fold 6c position at (x, y, z) and Ni2 and Ni3 in 2-fold 2b position at (1/3, 2/3, z). There are fourteen hydrogen atoms at three different crystallographic sites, namely 6c (H1) with an octahedral surrounding and 2b (H2) and 6c (H3) with a tetrahedral surrounding. The difference between the two SG’s is relevant to the presence of a mirror plane within P63mc for the site 6c which leads to a particular position (x, x, z) with respect to (x, y, z) in P31c. Thus Ni and H atoms are found within the hydride with the P63mc symmetry in the same locations and with the same folding except for Ni1 which occupies a position that only differs from its P31c homologue by a mirror symmetry shift along the yordinate direction. In the following, the hydride phases crystallizing with SGs P63mc and P31c will be referred to as b1 and b2 respectively. A sketch of the b1 structure is given in Fig. 7.3c. 7.2. Geometry optimization results LaNi5 as well as its a and b hydride phases are characterized by the experiment. However two possible SGs for the b phase are proposed: P31c and P63mc. It is then necessary to optimize the geometry of both systems prior to further calculations. From energy differences this should help pointing to the favorable symmetry for the saturated hydride as the one with the most stable energy between the two b models. It is worth mentioning that a similar attempt has been made before from first-principles computations without being able to discard either of the two SGs with respect to the total energy values of the optimized structures. Nevertheless, the internal parameters for both models were very close to those generated with SG P63mc whereby the authors concluded that the latter SG is more favorable.

7.1. Questions around the crystal structures LaNi5 system crystallizes in the hexagonal CaCu5 -type, P6/ mmm SG, Haucke structure. A sketch is shown in Fig. 7.1 with La atoms occupying Wyckoff general position 1a at (0, 0, 0), while Ni atoms are found in two non-equivalent crystallographic sites: 2c at (1/3, 2/3, 0) and 3g at (1/2, 0, ½); they will be referred to as Ni2 and Ni3. When hydrogen is absorbed into the lattice, a primary solid solution (the a phase) is formed without a change in the general crystal symmetry. For P6/mmm SG, there are five possible hydrogen sites: Octahedral 3f: La2Ni22Ni32 and tetrahedral 4h: Ni22Ni33, 6m: La2Ni32, 12 : LaNi2Ni32 and 12n: LaNi32Ni3. The 3f

Fig. 7.1. LaNi5 hexagonal structure (CaCu5-type, P6/mmm SG). Blue spheres of Ni are of two kinds: Ni2c and Ni3g (cf. text).(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

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Also calculations performed by the same authors without geometry optimization were energetically favorable with respect to P63mc SG. The pure intermetallic was also optimized to establish an equation of state allowing to confront quantities of equilibrium volume and bulk modulus with the hydride models. Primary a phase has the composition LaNi5H0.5 with several possible hydrogen insertion sites as discussed in next paragraphs. A hydride model was optimized with a fully occupied 3f site resulting into a composition such as LaNi5H3 in order to avoid any breaking of the symmetry. This model can be considered as an intermediate hydride phase and compared to the pure intermetallic with respect to the pressure effects brought by the insertion of hydrogen in the host lattice. From the latter intermediate hydride a model was built with a doubled cell along the c-axis resulting into a composition: La2Ni10H6 (Z ¼ 2). An additional model with the same SG of the b1 hydride model was also considered. Within this model H occupies the octahedral 6c sites resulting into an intermediate hydride La2Ni10H6 (Z ¼ 1). This choice of H occupation site was done with respect to the other intermediate model which holds H in an octahedral site. As far as the difference between b1 and b2 models does not affect the surroundings of hydrogen atoms in their insertion sites there was no need to consider a model with the same SG of the b2 hydride model. The latter two models are then important if one wishes establishing a comparison with the saturated hydride system with respect to the binding of hydrogen. For the geometry optimization, we use PAW-GGA potentials. The starting internal parameters and lattice constants (a ¼ 5.017 Å; c ¼ 3.986 Å) for LaNi5 are taken from the experiment. The resulting structure is found similar to the experimental one with respect to the symmetry and the atomic species positions. The optimized lattice parameters: a ¼ 5.001 Å and c ¼ 3.963 Å, are less than 1% smaller lower than experimental values. This gives confidence to the optimization scheme and the used potentials. Now more complicated structures such as the hydride phases can be optimized. The crystal structures of both b phase hydrides are described in next paragraphs. The initial parameters for the special positions of the different atomic constituents are those given by the experiment [81]. A detailed description of the initial internal parameters for the starting optimization calculations is given in Table 7.1. An analysis of the optimized parameters can be then performed based on the listed information. The value of the ratio c/a will be considered for the discussion of the optimized lattice constants with respect to the experiment. This ratio is 1.8 and 0.5% larger than its experimental values for the b1 and b2 models respectively. The internal parameters in both models are closer to the symmetry of SG P63mc with respect to P31c. In previous ab initio calculations [84] showing the same trends, the authors pointed towards P63mc as the favorable SG for the hydride system. Nevertheless this must be confirmed by a comparison of the total energies. 7.3. Equation of state and derived quantities In order to obtain the equilibrium volumes and confront them with the experiment, (energy, volume) values were computed around the experimental data. The resulting E(V) curves are plotted in Fig. 7.2a for the b1 and b2 hydrides models and in Fig. 7.2b for LaNi5 and the tri-hydride. The resulting equilibrium volumes for the saturated hydride models are 218.6 and 219.0 Å3 for b1 and b2 respectively. These values are close to the experimental volume which amounts to 217.9 Å3. On the side of LaNi5H3 model the volume is larger than that of the pure intermetallic by w12 Å3. To compare the volume expansion brought by H insertion within the intermediate hydride with that of the saturated hydride models, one needs to consider

Table 7.1 Internal parameters of the b1 and b2 models for the saturated hydride La2Ni10H14 (Z ¼ 1) from geometry optimization calculations with PAW-GGA calculations. Cell parameters, a and c, are given in Å. P63mc experiment

La Ni1 Ni2 Ni3 H1 H2 H3

P63mc optimized

a

c

a

c

5.409

8.600

5.332

8.626

x

y

z

x

y

z

0 0.498 1/3 1/3 0.504 1/3 0.160

0 x 2/3 2/3 x 2/3 x

0.022 0.250 0.002 0.489 0.056 0.814 0.280

0 0.499 1/3 1/3 0.506 1/3 0.149

0 x 2/3 2/3 x 2/3 x

0.022 0.252 0.006 0.482 0.059 0.819 0.275

P31c experiment

La Ni1 Ni2 Ni3 H1 H2 H3

P31c optimized

a

c

a

c

5.409

8.600

5.315

8.492

x

y

z

x

y

z

0 0.517 1/3 1/3 0.510 1/3 0.154

0 0.486 2/3 2/3 0.490 2/3 0.168

0.004 0.250 0.006 0.497 0.057 0.830 0.299

0 0.500 1/3 1/3 0.506 1/3 0.146

0 0.502 2/3 2/3 0.494 2/3 0.852

0.025 0.258 0.011 0.488 0.066 0.825 0.278

both La2Ni10H6 (Z ¼ 1) and (Z ¼ 2) models. Fig. 7.2a shows that these two intermediate models have very close E(V) curves in shape and in energy. This results into very close equilibrium volumes of w196 Å3. Then the volume differences with the saturated hydride yields a b model volume larger by w23 Å3 with respect to La2Ni10H6 models. This amount should be divided by two in order to compare it with the difference between the pure intermetallic and LaNi5H3. This results into a value of w11.5 Å3 which is close to the volume difference between LaNi5 and LaNi5H3. The stability of hydrogen within the hydride is another significant quantity that can be investigated using the equilibrium energy values. EH describes the binding of hydrogen within the lattice and is expressed as follows:

EH ¼ 1=n½EðLaNi5 Hn Þ  EðLaNi5 Þ  1=2EðH2 Þ; where n is the amount of hydrogen present in the host lattice. E (LaNi5Hn), E(LaNi5) represent the total energies of the hydride models and the intermetallic system respectively. These are given in the inserts of Fig. 7.2. E(H2) is the energy of the di-hydrogen molecule (6.595 eV) (cf. Section 3 on ZrNi). The obtained EH magnitudes are such as 0.099, 0.301 and 0.299 eV for LaNi5H3, b1 and b2 models respectively. The latter two values compare well with the heat of formation given by the literature, which ranges from 0.332 to 0.363 eV/mol H2 [85]. As for the intermediate phase, it is clear that hydrogen is less bonded to the host lattice. This is relevant to the fact that it is (i) a hypothetic model and (ii) an intermediate phase, which explains the formation of a saturated hydride where H is more bonded. Nevertheless EH values for the La2Ni10H6 models are 0.218 and 0.195 eV for the hydrides in the P63mc and P6/mmm SGs respectively. This provides an explanation for the cell doubling upon the uptake of more than one H per LaNi5. This cell doubling ensures a more bonded H in the host lattice and the more negative energy for SG P63mc indicates the contribution of the latter to the stability of the hydride phase. In the following the intermediate model with SG P63mc, i.e. La2Ni10H6 (Z ¼ 1), will be considered for comparison with both LaNi5 and b1(2) models. On the other hand, calculations

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27

Fig. 7.2. Energy versus volume curves and parameters from Birch EOS fit; a) b model hydrides and intermediate phases La2Ni10H6, b) LaNi5 and its tri-hydride.

of the bulk modulus B0 are possible through E(V) curve fitting with a Birch 2nd order EOS. The obtained magnitudes are w134.9, 129.8, 122.9 and 122.7 GPa for LaNi5, La2Ni10H6 (Z ¼ 1), b1 and b2 models respectively. This is indicative of the larger compressibility of the hydride phase, with respect to LaNi5, under hydrostatic pressure. B0 decreases by w5 and 7 GPa while passing from the intermetallics to the saturated hydride through the intermediate hydride phase. This is related to volume expansion discussed earlier. Recent calculations performed for B0 [86] resulted into a value of 126.4 GPa for LaNi5 which is closer to the experiment (126.8 GPa) [87] than the value reported here. In order to evaluate the b phase SG we considered the relative energy DE which is the difference between the equilibrium energies resulting from E(V) curves of both b1 and b2 models. DE favors the b1 model by 0.015 eV. This value is 1.5 times larger than the one given by previous calculations with a similar approach [84] which amounts to 102 eV, favoring the b2 model. Thus neither of the two SGs can be discarded by a PP approach which slightly favors the b1 model in the present calculations.

7.4. Analysis of the electron localization function (ELF) Fig. 7.3 shows the electron localization function mapping for LaNi5 along <110> a) and <100> b) planes. A diagonal plane along the c-axis cuts through H and La in c) and La and Ni in b). For LaNi5, the yellow distorted pentagon shapes between lanthanum and nickel, that there is a charge transfer between the La and Ni. Further, one can notice the green areas between the atomic species (Fig. 7.3a and b). This indicates a smeared electron distribution within the system similarly to a free-electron-like character expected within a metallic network. The presence of the red spots around La is the result of using PAW potentials which include 4f states in the basis set. But the strong localization is found around hydrogen sites within the hydride. This is shown in Fig. 7.3c. From a Bader charge analysis the anionic character of hydrogen is H0.3. This indicates a rather covalent-like behavior contrary to ionic MgH2, and larger than the average charged H0.5 found in ZrNi hydrides discussed earlier.

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Fig. 7.3. Electron localization slice maps for LaNi5, a) and b) and the b1 model for the saturated hydride c) (cf. text).

7.5. All-electrons calculations From the calculations, the energy difference which follows the expression DE ¼ Eb1  Eb2 shows that the b1 model is w1.65 eV more stable. This difference points to P63mc as the most probable space group for the b phase hydride. It is important to mention that first-principles calculations derived from a tight-binding linear muffin-tin orbital method (TB-LMTO) without optimizing the structure comforted this last hypothesis [88]. Further, other sets of computations were performed for hydrogen-free models, at the same volume of the b1 model. This simulates the manner in which the volume expansion (negative pressure) affects the magnetic behavior of the different atomic species. These effects can be important in such intermetallic systems as far as the onset of the magnetic moment is due to interband spin polarization, i.e. it is mediated by the electron gas in a collective electrons approach contrary to oxide systems where the magnetization is of intra-band character, and hence, is less affected by volume changes such as those induced by pressure [89]. 7.5.1. Analysis of the density of states The PDOS of all the atoms from the constituents present in the lattice, for LaNi5 and the b1 model are shown in Fig. 7.4a and b. Looking firstly at the general shape of the PDOS for LaNi5, the Fermi level crosses the peaks of both Ni sites with predominance in terms of intensity for the states of Ni3g. This can be relevant to the fact that Ni at 2c and 3g sites are found with the occupation ratio 2:3 respectively. The similar skylines between the partial PDOS pointing to the overlap between Ni and La states can be seen at the lower part of the valence band (VB), with mainly s, p-like states between 6 and 2.5 eV, as well as towards the top of VB between both Ni d states. Lastly, within the conduction band (CB), lanthanum 4f states are found dominant which is due to their emptiness. On the lower DOS panel, the characteristic features of the saturated hydride system are shown with hydrogen PDOS artificially magnified by a factor of 10 for a clear view. A significant enlargement to the extent of 2 eV is noticed for the VB with respect to

LaNi5. This is due to the presence of hydrogen s states. The energy range from 10 to 4.7 eV comprises hydrogen and lanthanum s states as well as nickel s and p states. An important feature of higher the intensity of H1 and H3 PDOS with respect to H2 is relevant to the larger site multiplicities of the former two which amount to 6 for both with respect to a multiplicity of 2 for the latter. Further the large Ni2 PDOS intensity with respect to all the other atomic constituents is related to the presence of Ni2 in all three hydrogen insertion sites as shown in Fig. 7.3c. This is contrary to Ni2 and Ni3 which are found in two and one hydrogen absorption interstices respectively. From this it can expected to have dominant Ni1-H interaction which will be checked in the chemical bonding section. Further, the similar shapes of PDOS peaks of the different species within this lower part of the VB indicate their quantum mixing. Within the itinerant part of the VB, i.e. from 4 eV up to EF , Ni1 3d states PDOS are of dominant intensity. Interestingly Ni2 and Ni3 3d states show very low intensity at EF which is indicative of a stability of the system in a degenerate spin configuration in as far as nickel atoms are responsible for the onset of magnetic moments in case of a magnetic ground state. A quantum mixing is also observed for the PDOS peak shapes of the latter states throughout the itinerant part of the VB. This peculiar aspect could be related to their similar site multiplicities as well as to their positioning within the octahedral hydrogen absorption hole (see Fig. 7.3c). Lanthanum 4f states are found, as expected, above EF. These states are more localized and intense from their sharper and higher intensity shape as with respect to pure LaNi5. 7.5.2. Analysis of the instability toward magnetic polarization The Stoner exchange correlation integral for Ni is I ¼ 0.5 eV [37]. The computed n(EF) values of Ni 3d states at the 2c and 3g sites for pure LaNi5 are to the extent of 1.55 and 1.86 eV1. These result into calculated values for the Stoner criterion In(EF) of 0.78 and 0.94, for Ni2c and Ni3g sites respectively; i.e. In(EF) < 1. Then a magnetic instability cannot be expected within LaNi5 despite the high value for Ni3g PDOS. This agrees with the paramagnetic ground state of this system as found by the experiment. On the other hand, the

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29

Fig. 7.4. Spin degenerate (non-magnetic) density of states for a) LaNi5 and b) the b1 model for the saturated hydride.

computations performed for the b1 model result into n(EF) values such as 1.52, 0.59 and 0.71 eV1 for Ni1, Ni2 and Ni3 respectively. The calculated Stoner products for these species amount to 0.76, 0.29 and 0.36 respectively which indicate that the hydride system ground state is NM, from this analysis. 7.5.3. Bonding characteristics Fig. 7.5 a and b shows the ECOV plots for inter-metal interactions within LaNi5 and the b1 model. In LaNi5, the dominant interaction within the VB occurs between Ni(2c) and Ni(3g). Although this interaction is of a bonding character within the energy interval ranging from 5 to 1.5 eV it shows a larger anti-bonding behavior from 1.5 eV and up to EF. These two opposite binding strengths compensate each other. It remains to discuss the interactions of lanthanum states with nickel at 2c and 3g sites respectively which are both bonding within the VB. Such a feature can be commented with respect to the interatomic distances whereas the smaller the distance is the stronger is the interaction. The calculated LaeNi(2c) and LaeNi(3g) separations are to the extent of 2.89 and 3.2 Å respectively. This is significant of a stronger LaeNi(2c) bond with respect to LaeNi(3g) but the larger number of Ni(3g) atoms within the lattice compensates the distance factor and explains the found similarity. Also one can notice the almost nil values for both interactions at EF which point to their contribution to the onset of a magnetic moment if the predictions by the Stoner criterion from the former section are correct. Turning to the b1 model (Fig. 7.5b) a similar conclusion can be drawn around the characters of all NieNi interactions. As for LaeNi1/Ni2/Ni3 interactions respectively, they show bonding characters from 7 to 1.5 eV which change into anti-bonding from 1.5 eV and up to EF. LaeNi1 is the strongest in the bonding interval but its higher anti-bonding intensity rules it out from being responsible for the stability of the system. The latter is relevant to LaeNi2/Ni3 interactions in which LaeNi2 is slightly dominant. This is concomitant with the interatomic distances amounting to 3.07 and 3.09 Å for the La-Ni2 and La-Ni3 separations respectively (Fig. 7.6).

electron-phonon enhancement effects in the experiment that are not accounted for here. 8. Hydrides of ternary cerium based intermetallics In 1964 J. Kondo showed that below a critical low temperature, TK a magnetic impurity (e.g. Ce) introduced in a non-magnetic metallic lattice is screened by a spin polarized electronic cloud of conduction electrons (c). The correlation between the electrons of the open f subshell of the impurity and c stabilizes the whole system in a ground state which is non-magnetic by definition. The

7.5.4. Specific heat Another significant parameter can be extracted from the calculations and compared with the experiment namely the electronic specific-heat coefficient g which is given by the following expression:

h

i

g ¼ 2=3p2 k2B  nðEF Þ: The computed value of g for LaNi5 is evaluated to be 23.12 mJ K2 mol1. Compared with the experimental values of 34.3 mJ K2 mol1 [90] a discrepancy is found. This is due to the

Fig. 7.5. Chemical bonding for pair metal interactions in a) LaNi5 and b) the b1 model for the saturated hydride.

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S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

simultaneously trivalent and intermediate valence Ce such as CeRuSn [91b]. Furthermore, several ternary equiatomic intermetallics possess the ability of absorbing hydrogen. Hydrogenation modifies the physical properties of the intermetallic with two main consequences:

Fig. 7.6. Chemical bonding for La/Ni bonding with hydrogen in the saturated hydride.

fec overlap is represented by a coupling constant, Jcf obtained inter alia by the impurity model of Anderson. Jcf reflects the degree of delocalization of the f electrons as a function of two types of interactions:  The RKKY inter site magnetic interactions (named after the authors: Rudermann, Kittel, Kasuya, Yosida) which introduces a long range order. These magnetic interactions are defined by a characteristic temperature, TRKKY, proportional to J2cf;  The demagnetizing local Kondo effect, equally defined by a characteristic temperature TK, proportional to expð1=nðEF ÞJcf Þ. With n(EF) representing the density of states at the Fermi level for f states. The competition between these two opposite effects gives a rich variety of physical properties such as intermediate valence cerium, magnetic Kondo compounds, heavy Fermions’ systems, spin glass, BCS superconductors etc. Jcf tightly depends on the electronic configuration and the interatomic distances. Then the physical properties can be modified by i) Compressing the lattice with applying pressure. This leads to a decrease of interatomic distances inducing an increase of Jcf, ii) Expanding the volume either by substituting a component by a larger size one or by hydrogenation. In the latter manner, the insertion of hydrogen modifies the volume, thus acting as a negative pressure and the density of states at the Fermi level (chemical bonding effects). The magnitude of Jcf coupling was assessed for 1D lattice by Doniach [91a] in a magnetic phase diagram involving a competition between the RKKY mechanism, which favors the paramagnetic state in the weak-coupling region (small Jcf), and the Kondo screening, which leads to the formation of tightly bound singlets in the strong-coupling region (large Jcf). Between these two regimes, magnetic (Kondo) systems are found with reduced magnetic moments due to the Kondo effect. A sketch of a Doniach phase diagram is given for a few cerium based intermetallics and hydrides in Fig. 8.1. Ternary CeTX equiatomic intermetallics where T is an nd transition metal (n ¼ 3, 4 or 5) and X a p-element, form a rich family with a large panel of physical properties. These are mainly governed by the valence character of cerium which can be of intermediate valence, i.e. in between trivalent Ce3þ (4f1) and tetravalent Ce4þ(4f0). Also some intermetallics contain

i) the cell expansion leads to an enhanced localization of the 4f (Ce) states as a consequence of the reduced fef overlap such as in CeRuSi which undergoes a transition from heavyfermion to antiferromagnetic behavior in CeRuSiH. Hydrogenation changes the moderate heavy-fermion compound behavior to an antiferromagnet one. In other words, the Hinsertion diminishes the influence of the Kondo effect. This can be understood in terms of the classical Doniach diagram (cf. Fig. 8.1) where the hydrogenation plays a role of negative pressure. The expansion of the lattice induced by H insertion into CeRuSi is much more important than the role of CeeH chemical bonding observed in other hydrogenated compounds as CeCoSiH (cf. Fig. 8.2 and Ref. [92] with therein cited experimental works). ii) the quantum mixing between the valence states of the chemical species in presence Ce, T, X and H, leads to changes of the chemical bonding strength between the metal sublattices on one hand and induces in some cases a decrease (or loss) of the magnetic polarization. This is the case of the transition from the antiferromagnetic character of CeCoSi (TN ¼ 8.8 K) to intermediate valence behavior in CeCoSiH [93]. The hydrogenation involves an expansion of the unit cell volume of 7.8%. The Investigation of the hydride CeCoSiH by magnetization, specific heat and electrical resistivity reveals an intermediate valence character exhibited by a broad maximum in the magnetic susceptibility curve present at w70 K and characteristic of valence fluctuating systems. This means that the by absorbing hydrogen, the valence state of cerium changes from trivalent in pristine CeCoSi, to intermediate valence state in CeCoSiH. This is also evidenced from the temperature dependence of the thermoelectric power S(T) of CeCoSi/CeCoSiH. The demagnetization of Ce at low temperatures in CeCoSiH can be associated with the strong CeeH interaction, which is bonding throughout the conduction band as we have shown from the band theoretical investigations [94]. Being the first element in RE series, cerium is considered as a border case where the degree of delocalization of 4f states depends on the applied pressure as well as on the crystal

Fig. 8.1. Doniach phase diagram with a selection of cerium based intermetallics and corresponding hydrides.

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31

a trivalent ground state of Ce was evidenced from the measurements of the magnetic susceptibility, but no detailed H positions were reported; neither the H composition at which the change of valence of Ce from intermediate valence to trivalence occurs. Consequently a study of a range of compositions with discrete hydrogen amounts is done with full geometry optimizations in order to define the threshold of the valence change of Ce. 8.2. Crystal structures

Fig. 8.2. Magnetic measurements for CeCoSi and CeCoSiH: The presence of a sharp peak at TN ¼ 8.8 K in the temperature dependence of the magnetic susceptibility of CeCoSi (see the arrow) suggests an antiferromagnetic ordering. CeCoSiH shows a different behavior whereby above 150 K, the inverse susceptibility c1, follows a CurieeWeiss law with an effective moment meff. ¼ 2.72 mB. The presence of a broad minimum around 70 K is a characteristic feature of valence fluctuating systems [93].

environment. In electronic structure calculations this delicate situation is addressed through various approaches treating the 4f states either as atomic-like core states or as itinerant in the framework of DFT-LDA/GGA. This duality was experimentally evidenced in a combined analysis of mSR (m on spin relaxation) and neutron experiments on cerium intermetallic systems which reveals the existence of magnetic excitations due to both conduction electrons at the Fermi level and well localized f-electrons (cf. Ref. [92] and therein cited works). Regarding the lattice environment, the quantum mixing (hybridization) of the 4f states with those of the ligand states can have large effects as well. This involves chemical bonding properties, which are dependent of the crystal lattice properties (structure and interatomic distances). As discussed in the Section 6.4.3 regarding Gd-based intermetallics, there are difficulties carrying on all-electron calculations with valence basis set comprising 4f states due to their enhanced localization.

8.1. Search of cerium valence change threshold: CeRhSn and its hydride As stated above CeRhSn can be considered as a homogeneous intermediate-valent system [91b]. Recent experimental studies [95] report the formation of a hydride CeRhSnH0.8, within which

Like intermetallic CeRhSn, the corresponding hydride systems CeRhSnHx crystallize in the hexagonal ZrNiAl -type structure (P62m SG). In this structure, Rh atoms are in 1a and 2d Wyckoff general positions, Rh1 (0, 0, 0) and Rh2 (1/3, 2/3, ½), while Ce and Sn are in 3-fold particular positions, 3f at (uCe, 0, 0) and 3g at (uSn, 0, ½) respectively. Since precise values for uCe and uSn are unknown for the hydride system, starting positions for the subsequent optimization were assumed as those of the equiatomic stannides, i.e., uCe ¼ 0.414 and uSn ¼ 0.750 [96]. We note here that such positions are likely to change as a function of hydrogen contents as it will be detailed in the geometry optimization section. Among 3g, 4h, 6i and 12l interstitial particular positions within the hexagonal structure, hydrogen is identified in the most favorable 4-fold 4 h one, at (1/3, 2/3, uH) in a tetrahedral coordination with 3 Ce and one Rh2 atoms. Based on the works of Yartys et al. on CeNiIn deuterides, [97], uH varies between 0.076 and 0.174 according to the amount of deuterium, i.e., 0.48 up to 1.23, respectively. In the preliminary computations for CeRhSnH, uH was taken as 0.176. However, the complete filling of such arranged sites with hydrogen has been considered unlikely on the basis of the experimental results [95]. In the present work, a discrete filling of 4h sites up to a full occupancy was performed. 8.3. Geometry optimization To optimize the starting structures for different hydride compositions we used LDA-built projector augmented wave (PAW) potentials, necessary for an account of Ce 4f states. In as far as no structural determination for the particular Ce, Sn and H positions were available, trends in cell volumes and ground state crystal structures for the different amounts of H were necessary in the first place. The equilibrium structures were obtained starting from CeRhSn structural setup for uCe and uSn internal parameters for Ce and Sn positions. As for H, our starting guesses were based on the experimental study performed by Yartys et al. [97] on CeNiInDx structures. An initial input for the lattice a constant for the different compositions was done on the basis of the experimental increase of a between the 1:1:1 intermetallic and the experimental hydride composition, i.e., (Da)/a ¼ 1.5% [95]. The hexagonal c/a ratio of 0.547 of CeRhSnH0.8 taken from the experimental data [95] was preserved throughout the calculations, thus allowing for an isotropic volume increase of the different structures

Fig. 8.3. Charge density maps for CeRhSnH1.33: the iso-surface, and volume slice are sketched on the left and right hand sides respectively. Both are drawn with c hexagonal axis along the paper sheet.

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S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

upon the intake of discrete amounts of hydrogen. After that, computations were carried out for the hydride systems with different amounts of H. From the results, the hexagonal symmetry was preserved for the optimized geometries of all systems. The new values for uCe, uSn and uH internal parameters are given in Table 8.1. As it can be expected, uCe, uSn are close to starting values. As for uH all results are in agreement with starting ones except for CeRhSnH0.33 where the small value of 0.028 is obtained. This value is close to zero, which allows assuming that H within CeRhSnH0.33 is relaxed into the general 2-fold position 2c at (1/3, 2/3, 0). According to the total energies values extracted from the PP calculations, which are shown in Table 8.1, stabilization is proportional to the amount of the absorbed hydrogen. Further, the values of DE, which is the relative energy of the hydride system with respect to CeRhSn, confirm this tendency. However, to investigate the origin of this stability, the energetic amount n/2 EH2 was subtracted from E. Here n is an integer ranging from 0 to 4, and E(H2) the energy of the di-hydrogen molecule (6.59 eV) computed by considering a cubic supercell of lattice parameter 4.5 Å. This computed value of the energy of H2 compares well with the experimental value of 6.65 eV. From Table 8.1 these values point to the chemical bonding of H with other atomic constituents as responsible of this stability. Furthermore, we notice that E  ½ n E(H2) value for CeRhSnH0.66 is smaller than that of CeRhSn, thus indicating a difficulty to obtain such a hydride experimentally. One can also analyze the electron charge density around the chemical species. Fig. 8.3 shows the charge iso-surface as well as the corresponding slice of the charge density both divided by the unit cell volume within the saturated hydride system. This illustration points to charge localization between Rh2 and H along the c-axis, from which the strong Rh2-H bonding can be inferred. This is detailed further in the next section regarding the chemical bonding.

Table 8.1 CeRhSn and CeRhSnHx model systems: uCe, uSn and uH are particular positions for Ce, Sn and H refined from geometry optimization calculations using PAW-GGA potentials. CeRhSn a (Å) c (Å) Volume (Å3) E (eV) E[n/2]EH2 (eV) DE (eV) uCe uSn uH

CeRhSnH0.33 CeRhSnH0.66 CeRhSnH CeRhSnH1.33

7.448 7.465 4.080 4.136 196 197 4.35 4.62 4.35 4.37 0 0.489 e 0.412 e 0.743 e 0.028

7.532 4.136 202 4.83 4.34 0.489 0.412 0.744 0.130

7.598 4.136 208 5.12 4.39 0.771 0.404 0.744 0.131

7.664 4.136 213 5.36 4.38 1.016 0.395 0.743 0.139

8.5. Spin degenerate (non-magnetic) calculations 8.5.1. Site projected density of states The characteristic features of the site-projected DOS (PDOS) for the saturated hydride system CeRhSnH1.33 plotted in Fig. 8.4 together with pristine 1:1:1 intermetallic, enable discussing the bonding in the CeRhSnHx models. In the PDOS three energy regions can be distinguished. The first one, from 12 to 5 eV, comprises 5s (Sn) and 1s (H) states. Then from 6 eV up to EF shows 5p (Sn) states which hybridize with 4d (Rh) states and itinerant Ce states. Finally, Ce 4f states are found at and above EF. These states are more localized within the hydride from their sharper and higher intensity shape as with respect to CeRhSn. This is concomitant with

8.4. All-electrons calculations The crystal parameters obtained from geometry optimizations (Table 8.1) were used for the input of all-electron calculations. For CeRhSnHx compositions lower than that of the saturated hydride system CeRhSnH1.33, the positions of the lacking hydrogen atoms within the 4h interstices were considered as interstitial sites where augmented spherical waves are placed. The resulting breaking of initial crystal symmetry was accounted for from the differentiation of the crystal constituents in the calculations. In a first step the calculations were carried out assuming nonmagnetic configurations (non-spin polarized NSP), meaning that spin degeneracy was enforced for all species. However, such a configuration does not describe a paramagnet, which could be simulated for instance by a supercell entering random spin orientations over the different magnetic sites. Subsequent spin polarized calculations (spin-only) lead to an implicit long-range ferromagnetic ordering. In order to provide a thorough description of the magnetic system an antiferromagnetic (AF) configuration was also considered. Since neutron diffraction data were not available, we have constructed ad hoc a double unit cell along the c-axis. This provides one possible model of a long-range AF spin structure which could be validated with a possible ground state configuration from the relative energies of the band theoretical calculations. Lastly, other sets of computations were performed for hydrogen-free models within ASW, starting from the previous four hydride systems with an additional model at the same volume of the experimental hydride. This procedure evaluates the manner in which the volume expansion affects the magnetic behavior of cerium.

Fig. 8.4. Non-magnetic site projected DOS for a) CeRhSn and b) CeRhSnH1.33. For the sake of clear presentation hydrogen PDOS are multiplied by 5.

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

8.5.2. Analysis of the DOS within Stoner theory In as far as 4f (Ce) states were treated as band states by our calculations, the Stoner theory of band ferromagnetism presented at the beginning of this review, can be applied to address the spin polarization. From our former work on Ce and U intermetallics [98], I(Ce-4f) w 0.27 eV and the computed n(EF) values of Ce (4f) for all the model systems are given in Table 8.2. The calculated values for In(EF) of 0.53, 0.41, 1.05, and 2.30 for CeRhSnH0.33 , CeRhSnH0.66 , CeRhSnH, and CeRhSnH1.33 , respectively, point to a magnetic instability of for the latter two hydride systems. This prediction for the behavior of the valence of Ce will be checked within the spinpolarized calculations for further confirmation, whereby finite magnetic moments are expected to be carried by 4f (Ce) states of CeRhSnH and CeRhSnH1.33. Also, Table 8.2 gives the values of the Stoner product for hydrogen-free models. Among the computed hydrogen-free models, CeRhSnH and CeRhSnH1.33 are the only ones which are likely to present a tendency towards magnetic order, but the magnitude of their Stoner products of 0.93 and 0.96, lower than 1 (In(EF)  1), with respect to their hydride model analogues, emphasizes the contribution of hydrogen chemical bonding to the arising of the trivalent character of cerium over the volume expansion. Lastly, an additional hydrogen-free model at the same volume of the experimental hydride [95] was computed. The calculated value of its Stoner product is equal to 0.76, which points to a non-favorable trend for magnetic instability. This is a further confirmation of the dominant role of hydrogen as a chemical species interfering with the bonding.

0.173 and 0.377 mB carried by 4f (Ce) states for both CeRhSnH and CeRhSnH1.33 respectively. These results are illustrated at Fig. 8.5a showing the site and spin projected density of states of CeRhSnH1.33. The exchange splitting is observed for cerium. Its magnitude extracted from the energy difference between the Hankel spherical functions for l ¼ 3 (Ce 4f), amounts to 0.016 Ryd. In order to check for the nature of the magnetic ground state, AF calculations were carried out using a supercell built from two simple cells along the c-axis. These two structures were used to distinguish between the up- and down-spin atoms. The site and spin projected DOS of antiferromagnetic CeRhSnH1.33 are sketched in Fig. 8.5b. An overall similarity with the PDOS for the ferromagnetic hydride is observed, but there is full compensation between [ and Y spin populations, i.e. there is no energy shift between them, with a spin moment of Ce of 0.336 mB, lower than in the SPeF configuration. This is concomitant with a value of 2.04 eV for the exchange splitting at Ce, smaller than in the ferromagnetic configuration. Then, at self-consistency, the total energy difference per unit cell, DE ¼ EFerroEAF ¼ 0.03 eV, favors the ferromagnetic ordering, thus pointing to a ferromagnetic ground state for the hydride. Further experimental investigations are underway.

a

20 Ce Rh1 Rh2 Sn 5*H

15 10 DOS (1/eV)

a higher density of states at EF, indicative of the instability of the system in a degenerate spin configuration as it is discussed next. In comparison with CeRhSn new states are formed upon hydrogenation around 12 to 8 eV. Hence the valence band (VB) is shifted to lower energies and EF is pushed upwards due to the additional electrons brought by H, resulting in a larger width of the VB with respect to CeRhSn.

8.6. Spin polarized (magnetic) calculations

5 0 -5 -10

From the NSP calculations and their analysis within the Stoner mean field theory of band ferromagnetism, it has been established that the hydride system is unstable in such a configuration for H content close to the experiment, i.e. with xH ¼ 1 up to 1.333. Consequently, spin polarized calculations were carried out, assuming implicitly a hypothetic ferromagnetic order. The experimental finding suggested a trivalent character of Ce within CeRhSnH0.8. From spin-polarized calculations this trend is confirmed by identifying finite spin-only magnetic moments of

-15 -20 -12

b

-10

dCeRh1 dCeRh2 dCeSn dRh1Rh2 dRh1Sn dRh2Sn dCeH dRh1H dRh2H dSnH

CeRhSnH0.33

CeRhSnH0.66

CeRhSnH

CeRhSnH1.33

0.69 0.69 3.084 3.031 3.227 3.375 4.756 2.761 2.846 e e e e e

0.53 0.81 3.095 3.036 3.232 3.386 4.772 2.767 2.851 2.306 4.338 1.534 4.576 3.237

0.41 0.69 3.121 3.063 3.264 3.412 4.814 2.793 2.878 2.328 4.380 1.545 4.613 3.264

1.05 0.93 3.148 3.089 3.290 3.444 4.856 2.814 2.904 2.343 4.417 1.561 4.655 3.296

2.30 0.96 3.174 3.116 3.317 3.475 4.898 2.841 2.925 2.365 4.454 1.571 4.697 3.322

-6

-4 -2 (E - EF) (eV)

0

2

4

6

-6

-4 -2 (E - EF) (eV)

0

2

4

6

Ce Rh1 Rh2 Sn 5*H

15

DOS (1/eV)

CeRhSn

-8

20

10 Table 8.2 CeRhSn and CeRhSnHx model systems. In(EF) values in italics are relevant to the hydrogen-free hydride models. Interatomic distances are given in Å.

In(EF)

33

5 0 -5 -10 -15 -20 -12

-10

-8

Fig. 8.5. Site and spin projected DOS of CeRhSnH1.33 in the ferromagnetic state a) and in the antiferromagnetic state b). For the sake of clear presentation H PDOS are multiplied by 5.

34

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

Fig. 9.1. Spin degenerate density of states of a) U2Ni2Sn and b) U2Ni2SnH2.

9. Case study of a uranium ternary intermetallics and its hydride The families of ternary U2T2X intermetallics (T ¼ 3d, 4d transition metal; X ¼ Sn, In p-metal) are well known for exhibiting a wide variety of electronic and magnetic properties [99,100]. In these compounds the formation of magnetic moments is governed by the degree of hybridization of the electronic valence states of uranium and those of the respective T and X ligands. The physical reasons for the varying hybridization strength can be seen in the bond lengths and in the crystal structure characteristics of these compounds. In uranium based intermetallic systems, the mechanism of intra-band spin polarization of the 5f states depends on the so-called Hill critical distance, i.e., dUU ¼ 3.5 Å [101]. Generally there is no intraband spin polarization below this value as the 5f (U) band broadens due to the direct overlap of the 5f wave functions. Recently it was shown that the 2:2:1 Sn-based intermetallic systems can absorb hydrogen with different amounts depending on the nature of T [102]. The highest hydrogen uptake occurs for U2Ni2Sn with w2 H per formula unit. In this case we note that the hydrogen storage capacity for the material does not exceed 0.3 wt%, which is too low to be envisaged for energy storage for mobile applications, but might be relevant in stationary ones. However issues in materials science fundamentals can justify their study. With respect to U2Ni2Sn intermetallic, the expansion of the lattice due to hydrogen absorption should lead to a larger separation of 5f (U) states due to the reduction of the 5f  5f overlap. For instance, spin-fluctuation U2Co2X shows different changes pertaining to an onset of long distance magnetic order upon hydrogenation [103]. However the chemical interaction between the valence states of U, T and Sn on one hand and H on the other hand may induce a decrease of the magnetic polarization and, eventually, a loss of magnetization. The interplay of such effects is addressed here within the newly found U2Ni2SnH2 system.

9.2. Spin degenerate calculations At self-consistent convergence little charge transfer was observed between the atomic species. A departure of w0.2 electron occurs from U spheres to other constituents spheres. This indicates a redistribution of the two s electrons of uranium over its three valence basis sets thus providing it with p and d characters arising

9.1. Structural considerations U2Ni2Sn has a primitive tetragonal structure with P4/mbm SG and 4 fu/cell, with a ¼ 7.296 Å; c/a ¼ 0.747. The hydride has the same structure and an expanded cell: a0 ¼ 7.445 Å and c0 /a0 ¼ 0.506. Within the structure U, Ni and Sn are located respectively at 4h, 4g and 2a Wyckoff positions while H atoms occupy half of the 8k positions in U3Ni-like tetrahedra. This leads to expect that hydrogen will mainly interact with Ni and U rather than with Sn. For U2Ni2Sn and U2Ni2SnH2 lattice parameters given above, ab initio self-consistent calculations were carried out considering spin degenerate configuration firstly.

Fig. 9.2. Chemical bonding in U2Ni2SnH2. a) inter-metal bonding, b) integrated ECOV between metal and hydrogen.

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37 Table 9.1 Calculated results for U2Ni2Sn and U2Ni2SnH2: DE values are for total energy differences with respect to spin degenerate non spin polarized (NSP) configuration. Magnetic (spin-only) moments of U, Ni and Sn are also presented for the different configuration. System

U2Ni2Sn

U2Ni2SnH2

U2Ni2Sn,2

DENSP (eV) DESP-F (eV) DESP-AF (eV) MUSP-F (mB) MUSP-AF (mB) MNiSP-F (mB) MNiSP-AF (mB) MSnSP-F (mB) MSnSP-AF (mB)

0 0.30 0.34 1.50 1.67 0.055 0 0.010 0

0 0.35 0.38 1.41 1.68 0.072 0.019 0.015 0.001

0 0.41 0.47 1.78 1.83 0.081 0 0.006 0

from its mixing with 3d (Ni) and 5p (Sn). A further redistribution is identified when hydrogen is inserted, giving a larger s character within the valence band (VB) as it is discussed within the DOS section. Therefore one can be argue that the major effect is that of the hybridization of the different valence states, not the charge transfer. The site-projected DOS (PDOS) for the intermetallics and the hydride are given in Fig. 9.1. From Fig. 9.1b showing the U2Ni2SnH2 PDOS, the overall feature is that of a larger localization of U and Ni metal states with respect to U2Ni2Sn (Fig. 9.1a). Extra states are created in the lower part of the VB in the hydride where the quantum mixing between the different atomic constituents is observed. The uranium PDOS are seen to prevail through the large peak around EF mainly due to the 5f states. This is a signature of a magnetic instability as it is confirmed from the Stoner products: I.n(EF) amount to 4 and 5 for the intermetallic and for the hydride respectively. However this needs to be checked with spin polarized calculations in next section. The 3d (Ni) states are closer to the Fermi level in the hydrogenated intermetallic. This is caused by the additional electrons brought by H, resulting in a larger width of the VB with respect to U2Ni2Sn. 5p (Sn) states are found in the energy range from 5 eV up to EF characterized by their weak PDOS intensities with respect to U and Ni. They mix with 3d (Ni) states within the lower part of this region. Low energy lying 5s (Sn) states are observed around 8 eV within an energy range comprising itinerant U and Ni states as well as hydrogen broad states. The presence of hydrogen increases the itinerant part within the VB, especially for 7s (U) occupation which increases upon hydrogenation. However the major part of the bonding will be seen to occur in the energy range 6 eV, EF as it is detailed below. Chemical bonding properties can be addressed on the basis of the spin-degenerate calculations because to a large extent, the spin-polarized electronic bands result from the spin-degenerate

35

bands by a rigid energy splitting. Fig. 9.2 a and b shows the bonding characteristics in the hydride. From Fig. 9.2a, the dominant interaction within the VB results from the UeNi interaction which is bonding throughout the VB, then the less strong UeSn interaction followed by NieSn which is weakest. Thus the dominant feature within VB is the bonding nature of the inter-metal interaction; whence the stability of the compound. Fig. 9.2b shows the integrated ECOV (iECOV) plots for metalhydrogen interactions within U2Ni2SnH2. Like the ECOV plots, negative iECOV areas indicate bonding characteristics and allow for comparisons. The larger the area underneath the curve, the stronger the bond is. The NieH bonding is the largest, followed by UeH; SneH is very weakly bonding. All interactions are of bonding character and this indicates the stabilizing role brought by hydrogen inserted within U2Ni2Sn. A Bader charge analysis of the charge density results provides a hydrogen charge of H0.3. This indicates a covalently bonded hydrogen within the intermetallic. It is less than the charge transfer on H in ZrNi hydrides studied above at the beginning of this review, where the Bader charge analysis gives w H0.5. 9.3. Spin polarized calculations 9.3.1. Ferromagnetic hypothesis Assuming firstly a hypothetic ferromagnetic order the charges and the magnetic moments are self-consistently converged. The relative energy difference (DESP-F) of the spin-polarized calculations for U2Ni2SnH2, with respect to its non-magnetic -spin degenerate- non spin polarized (NSP) energy, given in Table 9.1, favors the ferromagnetic state. This further confirms the analysis of the NSP PDOS for the trend to magnetic instability. Finite spinonly magnetic moments develop on uranium with a slightly smaller magnitude for the hydrogenate intermetallic probably due to the spin pairing with hydrogen. This is supported by the calculation of U2Ni2Sn at the hydride volume, leading to an enhancement of uranium f moment up to 1.78 mB carried by 5f (U) states. These results are illustrated in Fig. 9.3 showing the site and spin projected density of states of U2Ni2SnH2. The exchange splitting is observed for uranium. Its magnitude extracted from the energy difference between the Hankel spherical functions which designate the middle of the band in ASW formalism, for l ¼ 3 (U 5f), amounts to 0.7 eV. 9.3.2. Anti-ferromagnetic configuration The experimental finding suggested an antiferromagnetic (SPAF) ground state configuration for U2Ni2SnH2. In order to check for the nature of the magnetic ground state, AF calculations were

Fig. 9.3. Site and spin projected DOS for U2Ni2Sn and U2Ni2SnH2: a) Ferromagnetic hypothesis; b) Antiferromagnetic ground state.

36

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

carried out by using a supercell built from two simple cells along the c-axis. These two structures were used to distinguish between the up- and down-spin atoms. The site and spin projected DOS of antiferromagnetic U2Ni2SnH2 are given in Fig. 9.3b. An overall similarity with the PDOS of the ferromagnetic H-based system (Fig. 9.3a) is observed, when it comes to Ni, Sn and H PDOS shapes. But for U PDOS a less intense and narrower peak is observed at EF. This is significant of a more important localization which can be explained from a detailed analysis of the electron populations. Also, there is full compensation between [ and Y spin populations, i.e., no energy shift between them, with a spin moment of U of 1.68 mB, higher than in the SP configuration. This is concomitant with a value of w0.7 eV for the exchange splitting at U, higher than in the ferromagnetic configuration. Then, at self-consistency, the total energy difference, DE ¼ ESPeF  ESPeAF, has a value of 0.03 eV/fu, favoring the antiferromagnetic order. Furthermore, the energy differences values for H-free U2Ni2SnH2 (Table 9.1) show that the volume expansion is favorable with respect to an AF ground state. 9.3.3. Spin orbit coupling effects It is known that the relativistic effects-like spin-orbit coupling (SO) have considerable influence on the formation and the magnitude of magnetic moments in narrow-band systems such as those based on 5f elements. In fact, the size of the SO splitting is within the order of magnitude of the 5f bandwidth as it is shown in Fig. 9.4. Moreover, the atomic magnetic moment treated in such a framework consists of contributions from spin as well as orbital moments. While the former is obtained from our ferro- and antiferromagnetic calculations, i.e., 1.41 and 1.68 mB respectively, the latter is obtained from a former work on 2:2:1 uranium based systems [98] using fully relativistic calculations, mU L ¼ 2.7 mB. It should be mentioned that the orbital moment of U stems from an f occupation of about 2.74 (2.75) electrons, for ferromagnetic calculations, whose orbital moment (2.7 mB) comes close to that of an atomic orbital, namely 3 mB as expected from Hund’s second rule. This reflects the atomic-like character of the 5f (U) shell. The total moment can then be calculated from the sum of both spin and orbital moments which align oppositely due to Hund’s 3rd rule for less than half filled f subshell. The resulting values are such as 1.3 and 1.02 mB for ferro- and antiferromagnetic configurations respectively. The latter compares well with the experimental value of 0.83 mB [102]. Furthermore, applying these LS corrections to U2Ni2Sn intermetallic system, results into a magnetic moment carried by uranium amounting up to 1.03 mB which is close to the experiment (1.05 mB) [102]. This moment has a slightly larger magnitude with respect to the H-based system, which comforts the lower Néel temperature for the AF ordering of the intermetallic system (TN ¼ 26 K) with respect to U2Ni2SnH2 (TN ¼ 87 K) [103,104].

10. Conclusion In this review, different classes of intermetallic hydrides e binary (equiatomic, Laves, Haucke) and ternary (pseudo-Laves, cerium and uranium based) e have been examined. While a large experimental literature coverage exists in the context of their potential applications for hydrogen storage applications, focus was made here on the fundamental issues of the changes brought by hydrogen in the pristine intermetallic system. For the purpose of interpreting properties and predicting ones, we used up-to-date complementary computational tools within the framework of the quantum density functional theory DFT. Such physical and chemical properties are: i) the magneto-volume effect due to the cell expansion (negative pressure due to hydrogen uptake), leading to an enhancement of the localization of the nd or nf states due to the reduced ded or fef overlap. The latter, in cerium based systems, leading possibly to a change of its valence state; ii) the interactions between the metal outer electrons with hydrogen through a description of the chemical bonding properties, iii) a direct space mapping of electron localization within the crystal lattice, iv) the lattice distortion and crystal anisotropy occurring upon the hydrogenation, v) the different roles played by the magnetic sublattices when they are present such as in ScFe2 and its hydride, vi) the covalent band magnetism explaining the antiparallel spin alignment between A and B in AB2 Laves phases, here C15YFe2 and C14-ScFe2, vii) the Fermi contact HFC term of the hyperfine field, in connection with Mössbauer characterizations, viii) the change of iono-covalent character of hydrogen from one system to the other, from 1 to w0.3 studied by charge density analysis, . They were examined in illustrative case studies selected from each family, to provide the reader with an as wide as possible scope of intermetallic hydrides. Throughout this work we have endeavored showing the usefulness of analysis tools of chemical bonding, electron localization and charge density analyses, issued from of self-consistent electronic structure computations to better assess the behavior and the role of hydrogen within intermetallics. For the solid state chemist and the materials scientist both the fundamental and the application levels are involved in the ab initio investigation, such as the magneto-volume versus chemical effects and the established trends from fully ionic hydrogen in MgH2 to increasingly covalent behavior in connection with the search for candidates for hydrogen absorption/desorption. Lastly we propose that calling for computational science in interpreting and predicting properties of materials increasingly stands as complementary tool in research.

Acknowledgments

Fig. 9.4. Sketch of comparative spin-orbit splitting (DSO) magnitude versus band width (W) for the transition metal 3d, 4d, 5d and 5f and 4f series.

Exchange and discussions on the topic of hydrides with experimentalists of the ICMCB institute, Dr Bernard Chevalier and Pr. Jean-Louis Bobet; in France, Dr Valérie Paul-Boncour (CNRSParis/Thiais); in Europe, Pr. Rainer Pöttgen (Univ. Münster, Germany) as well as in Lebanon, Dr Michel Nakhl (Lebanese Univ.) are gratefully acknowledged. Special thanks are addressed to Priv. Doz. Dr Volker Eyert (Univ. Augsburg, Germany) for fruitful collaboration on modeling and theory issues and for his development of the now worldwide

S.F. Matar / Progress in Solid State Chemistry 38 (2010) 1e37

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