- Email: [email protected]
Ab initio calculations study of the electronic, optical and thermodynamic properties of NaMgH3, for hydrogen storage
Journal of Physics and Chemistry of Solids 71 (2010) 1264–1268
Contents lists available at ScienceDirect
Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
Ab initio calculations study of the electronic, optical and thermodynamic properties of NaMgH3, for hydrogen storage Youcef Bouhadda a,n, Youcef Boudouma b, Nour-eddine Fennineche c, Abdelouahab Bentabet d ´ nergies Renouvelables, BP 88 Ghardaı¨a, Algeria ´e en E Unite´ de Recherche Applique Faculte´ de Physique, USTHB, Algiers, Algeria c LERMPS, UTBM, Belfort, France d Institute of Sciences and Technology, Bordj-Bou-Arreridj University Center, Algeria a
b
a r t i c l e in f o
a b s t r a c t
Article history: Received 14 October 2009 Received in revised form 21 February 2010 Accepted 13 May 2010
The structural stability, electronic structure, optical and thermodynamic properties of NaMgH3 have been investigated using the density functional theory. Good agreement is obtained for the bulk crystal structure using both the local density approximation (LDA) and the generalized gradient approximation (GGA) for the exchange-correlation energy. It is found from the electronic density of states (DOS) that the valence band is dominated by the hydrogen atoms while the conduction band is dominated by Na and Mg empty states. Also, the DOS reveals that NaMgH3 is a large gap insulator with direct band gap 3.4 eV. We have investigated the optical response of NaMgH3 in partial band to band contributions and the theoretical optical spectrum is presented and discussed in this study. Optical response calculation suggests that the imaginary part of dielectric function spectra is assigned to be the interband transition. The formation energy for NaMgH3 is investigated along different reaction pathways. We compare and discuss our result with the measured and calculated enthalpies of formation found in the literature. & 2010 Elsevier Ltd. All rights reserved.
Keywords: A. Alloy C. Ab initio calculations D. Electronic structure D. Optical properties D. Thermodynamic properties
1. Introduction Hydrogen is expected to be a clean and recyclable energy carrier after the fossil fuel: it does not produce any polluting or greenhouse gases or toxic chemicals but water. However, the hydrogen storage is one of the most powerful technology barriers to the widespread acceptance of hydrogen as an energy vector [1]. Conventional ways to store hydrogen such as gas compression and liquefaction are massive, expensive and raise important safety issues, and therefore several different ways to store hydrogen have been proposed [1,2]. An alternative safer and more economical technique is to use a material that can absorb large amounts of hydrogen. The identification and the characterization of such materials are currently of great interest in the global scientific area [3]. The technical challenge is to find materials that exhibit the best combination of thermodynamics and kinetics for hydrogen desorption and absorption, and have the ability to store a sufficient percentage of hydrogen by weight and volume [4,5]. In recent years, much attention has been paid to the investigation on specific materials properties of Mg-based hydrides for the development of new functional materials [6].
Some Mg-based hydrides exhibit the perovskite structures expressed as AMgH3 where A is an alkali element [7–10]. The material functions and also fundamental properties of the perovskite-type ‘‘hydrides’’ have not been widely investigated and clarified yet. Li et al. [11] studied the electronic structure of LiMgH3, NaMgH3 and LiCaH3, and reported that LiMgH3 is an insulator whereas the later compounds are metals. While, Fornari et al. [12] and Vajeeston et al. [13] found theoretically that NaMgH3 must be an insulator. They tried to reproduce theoretically the experimental values of the standard enthalpy of formation measured by Bouamrane et al. [14] and Ikeda et al. [15]. The hydride NaMgH3 has high gravimetric and volumetric H densities (rG ffi6% and rV ffi88 Kg/m3) and reversible hydriding and dehydriding reactions [15]. In this work, we will contribute to the investigation of the perovskite hydride NaMgH3 by studying the electronic, crystal, optical and thermodynamic properties. At the end of this paper, we compare and discuss our result with the measured and calculated enthalpies of formation found in the literature.
2. Calculation details n
Corresponding author. Tel.: + 00 213 29870126; fax: + 00 213 29870152. E-mail address: [email protected] (Y. Bouhadda).
0022-3697/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2010.05.007
In this study, all the computations have been done by the use of the ABINIT code [16] that is based on pseudopotentials and
Y. Bouhadda et al. / Journal of Physics and Chemistry of Solids 71 (2010) 1264–1268
planewaves within density functional theory (DFT) [17]. It relies on an efficient fast Fourier transform algorithm [18] for the conversion of wavefunctions between real and reciprocal space, on the adaptation to a fixed potential of the band-byband conjugate-gradient method [19] and on a potential-based conjugate-gradient algorithm for the determination of the selfconsistent potential [20]. We performed generalized gradient approximation (GGA-PBE) to DFT [21]. Fritz-Haber Institute GGA pseudopotentials [22] are used to represent atomic cores. To study the effect of the exchange-correlation energy in the bulk structure, the local density approximation (LDA) exchange correlation functional was used too. We employed Hartwigsen– Goedecker–Hutter (HGH) pseudopotentials, within the LDA adopting Teter–Pade parameterization [23]. We carefully tested the convergence of our calculations with respect to the plane wave cut-off and k-point mesh. An energy cutoff of 50 Hartree and a 8 8 8 grid for k-point was used.
3. Results and discussions 3.1. Structural properties The crystal structure of NaMgH3 is known and this has a distorted perovskite structure analogous to the GdFeO3 type: it has an orthorhombic structure with the space group of Pnma [9,24]. Moreover, it has been reported that NaMgH3 has 20 atoms (Fig. 1) and contains two occupation sites of hydrogen (4c, 8d) and that both of them are surrounded by elemental Na (4c) and Mg (4b) [9,24]. Consistent with the ABX3 perovskite structure, each Na (A-site) cation is surrounded by 12 H anions, while each Mg (B-site) cation is coordinated with 6H anions. Therefore, each H anion is bonded with 2 Mg and 4 Na cations. The crystal structure and bonding in NaMgH3 can be discussed in terms of the tolerance factor, octahedral tilting, lattice distortion and the MgH6 octahedral distortion. More details can be found in Ref. [24]. We relaxed the atomic positions; this relaxation lowered the total energy. The final structure obtained within the LDA and PBE GGA approximations is given in Table 1.
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Our relaxed structure is in good agreement with the reported structure from the experiment. The deviations between the experimental and the calculated unit-cell parameters a, b and c, are estimated by less than 1% using GGA and 4.2% using LDA. These deviations are acceptable and are typical for the state-of-art approximation of the density functional theory. We also note that in our case both LDA and GGA underestimate the lattice parameters, and GGA is more accurate than LDA. These results show our method is reliable and we will use the optimized lattice constants to calculate other properties. 3.2. Electronic density of states In Fig. 2, total density of states (DOS) of NaMgH3 is shown. These results agree qualitatively with DOS of NaMgH3 in the ideal cubic structure reported by Khowash et al. [25] using the linear muffin tin orbital method, and by Fornari et al. [12] using linear augmented plane wave, and by Vajeeston et al. [13] using projector augmented wave method. There is an energy gap of 3.4 eV between valence and conduction band which is similar to 3 and 3.5 eV by LDA using LAPW [12] and GGA by PAW [13], respectively. It is well known that both LDA and GGA underestimate the true gap, and the calculated gap with LDA [12] is less than the gap calculated with GGA in this work and in other work [13]. So we think that our calculated gap with GGA is closer to the true gap for this compound. This gap (3.4 eV) indicates that NaMgH3 is an insulator. The contribution by Na in valence band is very small (Fig. 3), which shows that Na is ionized to Na + (valence electrons are transferred from the Na site to the H site). The Mg-s states predominantly occur in the lower energy range whereas, the Mg-p states occur in the upper energy range of the valence band. The Na-s and Na-p state PDOS is spread over the entire valence band range (Fig. 3). As more H-s states are present in the valence band (VB) than in the conduction band (CB), the interaction between Mg and H has an ionic character. On the other hand, the valence band consists mainly of the H-s state hybridizing with the Mg-s and Mg-p states (Fig. 3), which is a favourable situation for the formation of the covalent bonds within the anionic [MgH6] octahedra. Thus, the covalent bond nature between Mg and H ions still remains. According to this remark, we can conclude that the bonding nature of the hydrides NaMgH3 does not exhibit a simple ionic or covalent character. In fact, the bonding interaction in this compound is quite complicated. The interaction between Na and MgH3 is ionic and that between Mg and H comprises both ionic and covalent characters. 3.3. Optical properties The optical properties of matter can be described by means of the dielectric function e(o). The dielectric function has the following form:
eðoÞ ¼ e1 ðoÞ þ ie2 ðoÞ
ð1Þ
In this study, the imaginary part of the dielectric function is given as follows [26]: Z 4p2 e2 X 2 e2 ðoÞ ¼ 2 2 ð2Þ 9/i9P9jS9 ðfi ð1fj ÞÞdðEf Ei _oÞ d3 k m o i,j
Fig. 1. NaMgH3 structure (pink: Mg, green: Na, small and blue: H). Locations of H, Na and Mg are labeled in the illustration. Mg is located at the center of the octahedral. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
where P is the momentum operator, e and m are the electron charge and mass, respectively. o is the frequency of the photon. i and j are the eigenfunctions with eigenvalues Ei and Ej, respectively. fi and fj are the Fermi distribution for i and j states, respectively.
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Table 1 Optimized crystal structure of NaMgH3, compared to the experimental structure from Ref. [9]. Perovskite
NaMgH3 (62 Pnma)
Atomic position
˚ Cell parameters (A) Exp. [9]
Cal:LDA
Cal:GGA
Exp. [9]
Theory:LDA
Theory:GGA
a¼ 5.463 b ¼7.703 c ¼5.411
a¼ 5.233 b ¼7.378 c ¼5.182
a ¼5.410 b ¼ 7.628 c ¼ 5.358
Mg(4b) (0,0,0.5) Na(4c) (0.021,0.25,0.006) H1(4c) (0.503,0.25,0.093) H2(8d)(0.304,0.065,0.761)
(0,0,0.5) (0.032,0.25, 0.005) (0.467,0.25,0.089) (0.298,0.047,0.701)
(0,0,0.5) (0.027,0.25,0.005) (0.474,0.25,0.081) (0.293,0.043,0.706)
Fig. 4. The calculated imaginary part e2(o) of the dielectric function of NaMgH3.
Fig. 2. Electronic density of state for NaMgH3. The Fermi level is set to zero energy.
Fig. 5. The calculated band structure along the high symmetry points of NaMgH3. Fig. 3. Projected density of state for NaMgH3. The Fermi level is set to zero energy.
The real part of the dielectric function can be extracted from e2(o) using the Kramers–Kroning relation: Z 2 1 e2 ðo0 Þo0 do0 e1 ðoÞ ¼ 1 þ ð3Þ p 0 o02 o2 In Fig. 4, the imaginary (absorptive) part of the dielectric function e2(o) is plotted as a function of photon energy for NaMgH3. e2(o) has major peaks and minor peaks. The peaks in the
optical response are caused by the electric-dipole transitions between the valence and conduction bands. It is noted that a peak in e2(o) does not correspond to a single interband transition since many direct transitions may be found in the band structure with an energy corresponding to the same peak. It would be useful to identify the transitions that are responsible for the peaks in e2(o) using our calculated band structure (Fig. 5), along the high symmetry point: X(1/2,0,0), Gamma (0,0,0), Y(0,1/2,0), T(0,1/2,1/2), S(1/2,1/2,0).
Y. Bouhadda et al. / Journal of Physics and Chemistry of Solids 71 (2010) 1264–1268
In order to identify these peaks, we decompose the optical spectrum to its contributions from each pair of valence vi and conduction cj bands (vi–cj), and plotting the transition from valence to conduction band: the transition energy E ¼ Ecj Evi . These techniques allow the knowledge of the bands that contribute more to the peaks and their locations in the BZ. The positions of the peaks and the corresponding interband transitions and their locations in the BZ are reported in Table 2 for NaMgH3. Following is an analysis of the optical spectrum (Fig. 4), in partial band to band contributions. The first critical point in the curve, which is attributed to the threshold for the direct optical transition ! -! between the
Table 2 Optical interband transitions in NaMgH3. Energy of optical structure peak position (eV)
Major contribution transition Transition
Energy (eV)
4.5
v1–c1; S–Y–T
4.49
5.5
v2–c3; X v3–c2; S–Y–T v4–c1; T–G
5.65 5.74 5.78
6
v5–c3; S–Y–T v1–c6; G–X
6.01 6.16
6.5
v5–c3; Y v1–c6; T–G
6.50 6.82
Fig. 6. The transition energy band structure. vi–cj is the transition from valence band (vi) to Conduction band cj.
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valence-band maximum and the conduction-band minimum, occurs around 3.4 eV. We note that there is a peak at 4.5 eV contributed by many transitions but the most distinctive transition is from the first valence band (v1) to the first conduction band (c1) (4.49 eV) along S–Y–T directions. Also, there is a peak at 5.5 eV mainly due to the following: a. The transition v2–c3 at 5.65 along X direction. b. The transition v3–c2 at 5.74 along S–Y–T direction. c. The transition v4–c1 at 5.78 along T–G direction. Other peaks and band to band transitions are listed in Table 2, and transition energy band structure is plotted in Fig. 6. To the best of our knowledge, no experimental optical spectrum is available for this compound. It seems for us, that the theoretical optical spectrum is presented in this study for the first time. 3.4. Standard enthalpy of formation The formation energy DH has been calculated according to the following reaction equations: NaH þ MgH2 -NaMgH3
ð4Þ
Naþ MgH2 þ 12H2 -NaMgH3
ð5Þ
NaH þ Mgþ H2 -NaMgH3
ð6Þ
Naþ Mg þ 32H2 -NaMgH3
ð7Þ
The enthalpy of the formation is calculated by taking the difference in total electronic energy of the products and the reactants. The total energies of NaH, Na and Mg have been computed for the ground-state structures, viz. in space group Fm3m for NaH, Im3m for Na, P63/mmc for Mg and P42/mnm for MgH2, with full geometry optimization. Calculated enthalpies for the reactions (4–7) are shown in Table 3. The calculated enthalpies are in good agreement with the theoretical findings [12,13,27]. Also, our calculated enthalpy using the reactions (6) and (7) is in good agreement with the experimental value [15,28,29]. For the NaMgH3 perovskite, the experimental standard enthalpy of formation Hf0 has been measured by Ikeda et al. [15,28] and by Bouamrane et al. [14]. However, the values reported by Ikeda et al. [15,28] differ markedly from the value reported by Bouamrane et al. [14]. This disagreement can be interpreted by the fact that calorimetric measurements [14] are more direct than the extraction of enthalpies from pressure– composition–temperature (PCT) isotherms [15,28].
Table 3 Calculated hydride formation energy (DH) according to Eqs. (4–7). Enthalpy of reactions (kJ/mol)
D H4 D H5 D H6 D H7 a
Ref. [13]. Ref. [27]. c Ref [12]. d Ref. [15]. e Ref. [28]. f Ref. [29]. b
This work
Calculated
10.88 54.53 71 151.83
a
Experiment b
c
11 , 10 , 10 56.01a 74.80a 179.70a, 147b
887 0.9d, 93.9e, 947 15f 145.05d, 147e
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Y. Bouhadda et al. / Journal of Physics and Chemistry of Solids 71 (2010) 1264–1268
Table 4 The calculated and experimental standard enthalpy of formation of NaMgH3.
Standard enthalpy of formation DHf of NaMgH3 in (kJ/mol) a b c
Calculated
Experimental
151.83
145.05a, 147b, 232c
Ref. [15]. Ref. [28]. Ref. [14].
In the following, we want to restrict ourselves to the concept of the standard enthalpy of formation. Following Ref. [30] it can be stated: the standard enthalpy of formation DHf is the energy exchange that takes place when 1 mole of a substance is formed from the basic elements in their natural state. For elements in their natural state the standard enthalpy of formation is considered to be zero. Corresponding to the above definition, reaction (7) determines directly this standard enthalpy for the formation of NaMgH3 since it is formed solely by the basic constituents Na, Mg and H. According to the data for the standard enthalpy mentioned separately in Table 4, the result from the measurements of Bouamrane et al. [14] disagrees with ours, and with the one obtained by Ikeda et al. [15,28]. It should be noted that our calculation using DFT is at T¼0 K and P¼0, and further studies might depend on temperature corrections such as zero-point energy, polaritons and vibration excitation. However, the influence of these corrections on the calculated results was previously found to be minor [31,32]. On the other hand, our results reproduce the experimental results of Ikeda et al. [15,28], which make us think that the experimental results of Bouamrane et al. [14] may contain errors. In fact, Bouamrane et al. [14] determined the standard enthalpy of formation of NaMgH3 via the reaction NaMgH3(cr) +3HCl(aq)-NaCl(aq)+ MgCl2(aq)+ 3H2(g)
(8)
Using the values of the other compounds’ formation enthalpies known in the literature, Bouamrane et al. [14] obtained the standard enthalpy of formation of NaMgH3, without adding corrections of solution energies of compounds in water. Additional experimental measurements would be useful in determining the correct value.
4. Conclusion In this paper, the crystal structure, electronic structure, optical and thermodynamic properties of NaMgH3 have been studied by first-principles calculations within DFT. The optimized structure is obtained from both LDA and GGA total-energy minimizations. From the single-particle wave functions, we calculate the dielectric functions.
The NaMgH3 is a large gap insulator with direct band gap (! -! ) 3.4 eV. The valence band is dominated by hydrogen atoms. The bonding within the NaMgH3 involves mixing between Mg and H orbitals but shows a strong ionic character. The theoretical optical spectrum is presented in this study, and some peaks in the linear optical spectrum have been identified from the band structure. To the best of our knowledge, no experimental optical spectrum is available for this compound. The theoretical optical spectrum has been presented in this study for the first time. Formation energy for NaMgH3 is calculated for different possible reaction pathways. Our result of the standard enthalpy of formation is in good agreement with the experiment. Reference [1] Y. Bouhadda, A. Rabehi, Y. Boudouma, N. Fenineche, S. Drablia, H. Meradji, Int. J. Hydrogen Energy 34 (2009) 4997. [2] J. Zhang, D.W. Zhou, L.P. He, P. Peng, J.S. Liu, J. Phys. Chem. Solids 70 (2009) 32. [3] P. Dantzer, Topics in Applied Physics, Hydrogen in Metals III, vol. 73, Springer, Berlin, 1997, p. 279. [4] M. Gupta, L. Schlapbach, Electronic properties of metal hydrides, in: L. Shlapbach (Ed.), Topics in Applied Physics, Hydrogen in Intermetallic Compounds, vol. 63, Springer, Berlin, 1988, pp. 139–217. [5] L. Schlapbach, A. Zuttel, Nature 414 (2001) 358. [6] B. Sakintuna, F. Lamari-Darkrim, M. Hirscher, J. Int., Hydrogen Energy 32 (2007) 1121. ¨ [7] A. Zaluska, L. Zaluski, J.O. Strom-Olsen, J. Alloys Compd. 307 (2000) 157. [8] F. Gingl, T. Vogt, E. Akiba, K. Yvon, J. Alloys Compd. 282 (1999) 125. [9] A. Bouamrane, J.P. Laval, J.P. Soulie, J.P. Bastide, Mater. Res. Bull. 35 (2000) 545. ¨ [10] B. Bertheville, T. Herrmannsdorfer, K. Yvon, J. Alloys Compd. 325 (2001) L13. [11] Y. Li, B.K. Rao, T. McMullen, P. Jena, P.K. Khowash, Phys. Rev. B 44 (1991) 6030. [12] M. Fornari, A. Subedi, D.J. Singh, Phys. Rev. B 76 (2007) 214118. [13] P. Vajeeston, P. Ravindran, A. Kjekshus, H. Fjellvag, J. Alloys Compd. 450 (2008) 327. [14] A. Bouamrane, C. De Brauer, J.P. Soulie, J.M. Letoffe, J.P. Bastide, Thermochim. Acta 326 (1999) 37. [15] K. Ikeda, Y. Kogure, Y. Nakamori, S. Orimo, Scr. Mater. 53 (2005) 319. [16] X. Gonze, J.M. Beuken, R. Caracas, F. Detraux, M. Fuchs, G.M. Rignanese, L. Sindic, M. Verstraete, G. Zerah, F. Jollet, M. Torrent, A. Roy, M. Mikami, Ph. Ghosez, J.Y. Raty, D.C. Allan, Comput. Mater. Sci. 25 (2002) 478. [17] W. Kohn, L.J. Sham, Phys. Rev. A 140 (1965) 1133. [18] S. Goedecker, SIAM J. Sci. Comput. 18 (1997) 1605. [19] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, J.D. Joannopoulos, Rev. Mod. Phys. 64 (1992) 1045. [20] X. Gonze, Phys. Rev. B 54 (1996) 4383. [21] J.P. Perdew, k Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [22] M. Fuchs, M. Scheffle, Comput. Phys. Commun. 119 (1999) 67. [23] S. Goedecker, M. Teter, J. Hutter, Phys. Rev B 54 (1996) 1703. ¨ [24] E. Ronnebro, D. Nore´us, K. Kadir, A. Reiser, B. Bogdanovic, J. Alloys Compd. 299 (2000) 101. [25] P.K. Khowash, B.K. Rao, T. McMullen, P. Jena, Phys. Rev. B 55 (1997) 1454. [26] C. Ambrosch-Draxl, J.O. Sofo, Comput. Phys. Commun. 175 (2006) 1. [27] A. Klaveness, O. Swang, H. Fjellvag, Europhys. Lett. 76 (2006) 285. [28] K. Ikeda, S. Kato, Y. Shinzato, N. Okuda, Y. Nakamori, A. Kitano, H. Yukawa, M. Morinaga, S. Orimo, J. Alloys Compd. 446–447 (2007) 162. [29] K. Komiya, N. Morisaku, R. Rong, Y. Takahashi, Y. Shinzato, H. Yukawa, M. Morinaga, J. Alloys Compd. 453 (2008) 157. [30] R.E. Sonntag, G.J. Van Wylen, C. Borgnakke, Fundamentals of Thermodynamics, fifth ed., John Wiley & Son, Inc., New York, 1998. [31] X. Ke, I. Tanaka, Phys. Rev. B 71 (2005) 024117. [32] A. Klaveness, H. Fjellvag, A. Kjekshus, P. Ravindran, O. Swang, J. Alloys Compd. 469 (1–2) (2009) 617–622.
Contents lists available at ScienceDirect
Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
Ab initio calculations study of the electronic, optical and thermodynamic properties of NaMgH3, for hydrogen storage Youcef Bouhadda a,n, Youcef Boudouma b, Nour-eddine Fennineche c, Abdelouahab Bentabet d ´ nergies Renouvelables, BP 88 Ghardaı¨a, Algeria ´e en E Unite´ de Recherche Applique Faculte´ de Physique, USTHB, Algiers, Algeria c LERMPS, UTBM, Belfort, France d Institute of Sciences and Technology, Bordj-Bou-Arreridj University Center, Algeria a
b
a r t i c l e in f o
a b s t r a c t
Article history: Received 14 October 2009 Received in revised form 21 February 2010 Accepted 13 May 2010
The structural stability, electronic structure, optical and thermodynamic properties of NaMgH3 have been investigated using the density functional theory. Good agreement is obtained for the bulk crystal structure using both the local density approximation (LDA) and the generalized gradient approximation (GGA) for the exchange-correlation energy. It is found from the electronic density of states (DOS) that the valence band is dominated by the hydrogen atoms while the conduction band is dominated by Na and Mg empty states. Also, the DOS reveals that NaMgH3 is a large gap insulator with direct band gap 3.4 eV. We have investigated the optical response of NaMgH3 in partial band to band contributions and the theoretical optical spectrum is presented and discussed in this study. Optical response calculation suggests that the imaginary part of dielectric function spectra is assigned to be the interband transition. The formation energy for NaMgH3 is investigated along different reaction pathways. We compare and discuss our result with the measured and calculated enthalpies of formation found in the literature. & 2010 Elsevier Ltd. All rights reserved.
Keywords: A. Alloy C. Ab initio calculations D. Electronic structure D. Optical properties D. Thermodynamic properties
1. Introduction Hydrogen is expected to be a clean and recyclable energy carrier after the fossil fuel: it does not produce any polluting or greenhouse gases or toxic chemicals but water. However, the hydrogen storage is one of the most powerful technology barriers to the widespread acceptance of hydrogen as an energy vector [1]. Conventional ways to store hydrogen such as gas compression and liquefaction are massive, expensive and raise important safety issues, and therefore several different ways to store hydrogen have been proposed [1,2]. An alternative safer and more economical technique is to use a material that can absorb large amounts of hydrogen. The identification and the characterization of such materials are currently of great interest in the global scientific area [3]. The technical challenge is to find materials that exhibit the best combination of thermodynamics and kinetics for hydrogen desorption and absorption, and have the ability to store a sufficient percentage of hydrogen by weight and volume [4,5]. In recent years, much attention has been paid to the investigation on specific materials properties of Mg-based hydrides for the development of new functional materials [6].
Some Mg-based hydrides exhibit the perovskite structures expressed as AMgH3 where A is an alkali element [7–10]. The material functions and also fundamental properties of the perovskite-type ‘‘hydrides’’ have not been widely investigated and clarified yet. Li et al. [11] studied the electronic structure of LiMgH3, NaMgH3 and LiCaH3, and reported that LiMgH3 is an insulator whereas the later compounds are metals. While, Fornari et al. [12] and Vajeeston et al. [13] found theoretically that NaMgH3 must be an insulator. They tried to reproduce theoretically the experimental values of the standard enthalpy of formation measured by Bouamrane et al. [14] and Ikeda et al. [15]. The hydride NaMgH3 has high gravimetric and volumetric H densities (rG ffi6% and rV ffi88 Kg/m3) and reversible hydriding and dehydriding reactions [15]. In this work, we will contribute to the investigation of the perovskite hydride NaMgH3 by studying the electronic, crystal, optical and thermodynamic properties. At the end of this paper, we compare and discuss our result with the measured and calculated enthalpies of formation found in the literature.
2. Calculation details n
Corresponding author. Tel.: + 00 213 29870126; fax: + 00 213 29870152. E-mail address: [email protected] (Y. Bouhadda).
0022-3697/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2010.05.007
In this study, all the computations have been done by the use of the ABINIT code [16] that is based on pseudopotentials and
Y. Bouhadda et al. / Journal of Physics and Chemistry of Solids 71 (2010) 1264–1268
planewaves within density functional theory (DFT) [17]. It relies on an efficient fast Fourier transform algorithm [18] for the conversion of wavefunctions between real and reciprocal space, on the adaptation to a fixed potential of the band-byband conjugate-gradient method [19] and on a potential-based conjugate-gradient algorithm for the determination of the selfconsistent potential [20]. We performed generalized gradient approximation (GGA-PBE) to DFT [21]. Fritz-Haber Institute GGA pseudopotentials [22] are used to represent atomic cores. To study the effect of the exchange-correlation energy in the bulk structure, the local density approximation (LDA) exchange correlation functional was used too. We employed Hartwigsen– Goedecker–Hutter (HGH) pseudopotentials, within the LDA adopting Teter–Pade parameterization [23]. We carefully tested the convergence of our calculations with respect to the plane wave cut-off and k-point mesh. An energy cutoff of 50 Hartree and a 8 8 8 grid for k-point was used.
3. Results and discussions 3.1. Structural properties The crystal structure of NaMgH3 is known and this has a distorted perovskite structure analogous to the GdFeO3 type: it has an orthorhombic structure with the space group of Pnma [9,24]. Moreover, it has been reported that NaMgH3 has 20 atoms (Fig. 1) and contains two occupation sites of hydrogen (4c, 8d) and that both of them are surrounded by elemental Na (4c) and Mg (4b) [9,24]. Consistent with the ABX3 perovskite structure, each Na (A-site) cation is surrounded by 12 H anions, while each Mg (B-site) cation is coordinated with 6H anions. Therefore, each H anion is bonded with 2 Mg and 4 Na cations. The crystal structure and bonding in NaMgH3 can be discussed in terms of the tolerance factor, octahedral tilting, lattice distortion and the MgH6 octahedral distortion. More details can be found in Ref. [24]. We relaxed the atomic positions; this relaxation lowered the total energy. The final structure obtained within the LDA and PBE GGA approximations is given in Table 1.
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Our relaxed structure is in good agreement with the reported structure from the experiment. The deviations between the experimental and the calculated unit-cell parameters a, b and c, are estimated by less than 1% using GGA and 4.2% using LDA. These deviations are acceptable and are typical for the state-of-art approximation of the density functional theory. We also note that in our case both LDA and GGA underestimate the lattice parameters, and GGA is more accurate than LDA. These results show our method is reliable and we will use the optimized lattice constants to calculate other properties. 3.2. Electronic density of states In Fig. 2, total density of states (DOS) of NaMgH3 is shown. These results agree qualitatively with DOS of NaMgH3 in the ideal cubic structure reported by Khowash et al. [25] using the linear muffin tin orbital method, and by Fornari et al. [12] using linear augmented plane wave, and by Vajeeston et al. [13] using projector augmented wave method. There is an energy gap of 3.4 eV between valence and conduction band which is similar to 3 and 3.5 eV by LDA using LAPW [12] and GGA by PAW [13], respectively. It is well known that both LDA and GGA underestimate the true gap, and the calculated gap with LDA [12] is less than the gap calculated with GGA in this work and in other work [13]. So we think that our calculated gap with GGA is closer to the true gap for this compound. This gap (3.4 eV) indicates that NaMgH3 is an insulator. The contribution by Na in valence band is very small (Fig. 3), which shows that Na is ionized to Na + (valence electrons are transferred from the Na site to the H site). The Mg-s states predominantly occur in the lower energy range whereas, the Mg-p states occur in the upper energy range of the valence band. The Na-s and Na-p state PDOS is spread over the entire valence band range (Fig. 3). As more H-s states are present in the valence band (VB) than in the conduction band (CB), the interaction between Mg and H has an ionic character. On the other hand, the valence band consists mainly of the H-s state hybridizing with the Mg-s and Mg-p states (Fig. 3), which is a favourable situation for the formation of the covalent bonds within the anionic [MgH6] octahedra. Thus, the covalent bond nature between Mg and H ions still remains. According to this remark, we can conclude that the bonding nature of the hydrides NaMgH3 does not exhibit a simple ionic or covalent character. In fact, the bonding interaction in this compound is quite complicated. The interaction between Na and MgH3 is ionic and that between Mg and H comprises both ionic and covalent characters. 3.3. Optical properties The optical properties of matter can be described by means of the dielectric function e(o). The dielectric function has the following form:
eðoÞ ¼ e1 ðoÞ þ ie2 ðoÞ
ð1Þ
In this study, the imaginary part of the dielectric function is given as follows [26]: Z 4p2 e2 X 2 e2 ðoÞ ¼ 2 2 ð2Þ 9/i9P9jS9 ðfi ð1fj ÞÞdðEf Ei _oÞ d3 k m o i,j
Fig. 1. NaMgH3 structure (pink: Mg, green: Na, small and blue: H). Locations of H, Na and Mg are labeled in the illustration. Mg is located at the center of the octahedral. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
where P is the momentum operator, e and m are the electron charge and mass, respectively. o is the frequency of the photon. i and j are the eigenfunctions with eigenvalues Ei and Ej, respectively. fi and fj are the Fermi distribution for i and j states, respectively.
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Table 1 Optimized crystal structure of NaMgH3, compared to the experimental structure from Ref. [9]. Perovskite
NaMgH3 (62 Pnma)
Atomic position
˚ Cell parameters (A) Exp. [9]
Cal:LDA
Cal:GGA
Exp. [9]
Theory:LDA
Theory:GGA
a¼ 5.463 b ¼7.703 c ¼5.411
a¼ 5.233 b ¼7.378 c ¼5.182
a ¼5.410 b ¼ 7.628 c ¼ 5.358
Mg(4b) (0,0,0.5) Na(4c) (0.021,0.25,0.006) H1(4c) (0.503,0.25,0.093) H2(8d)(0.304,0.065,0.761)
(0,0,0.5) (0.032,0.25, 0.005) (0.467,0.25,0.089) (0.298,0.047,0.701)
(0,0,0.5) (0.027,0.25,0.005) (0.474,0.25,0.081) (0.293,0.043,0.706)
Fig. 4. The calculated imaginary part e2(o) of the dielectric function of NaMgH3.
Fig. 2. Electronic density of state for NaMgH3. The Fermi level is set to zero energy.
Fig. 5. The calculated band structure along the high symmetry points of NaMgH3. Fig. 3. Projected density of state for NaMgH3. The Fermi level is set to zero energy.
The real part of the dielectric function can be extracted from e2(o) using the Kramers–Kroning relation: Z 2 1 e2 ðo0 Þo0 do0 e1 ðoÞ ¼ 1 þ ð3Þ p 0 o02 o2 In Fig. 4, the imaginary (absorptive) part of the dielectric function e2(o) is plotted as a function of photon energy for NaMgH3. e2(o) has major peaks and minor peaks. The peaks in the
optical response are caused by the electric-dipole transitions between the valence and conduction bands. It is noted that a peak in e2(o) does not correspond to a single interband transition since many direct transitions may be found in the band structure with an energy corresponding to the same peak. It would be useful to identify the transitions that are responsible for the peaks in e2(o) using our calculated band structure (Fig. 5), along the high symmetry point: X(1/2,0,0), Gamma (0,0,0), Y(0,1/2,0), T(0,1/2,1/2), S(1/2,1/2,0).
Y. Bouhadda et al. / Journal of Physics and Chemistry of Solids 71 (2010) 1264–1268
In order to identify these peaks, we decompose the optical spectrum to its contributions from each pair of valence vi and conduction cj bands (vi–cj), and plotting the transition from valence to conduction band: the transition energy E ¼ Ecj Evi . These techniques allow the knowledge of the bands that contribute more to the peaks and their locations in the BZ. The positions of the peaks and the corresponding interband transitions and their locations in the BZ are reported in Table 2 for NaMgH3. Following is an analysis of the optical spectrum (Fig. 4), in partial band to band contributions. The first critical point in the curve, which is attributed to the threshold for the direct optical transition ! -! between the
Table 2 Optical interband transitions in NaMgH3. Energy of optical structure peak position (eV)
Major contribution transition Transition
Energy (eV)
4.5
v1–c1; S–Y–T
4.49
5.5
v2–c3; X v3–c2; S–Y–T v4–c1; T–G
5.65 5.74 5.78
6
v5–c3; S–Y–T v1–c6; G–X
6.01 6.16
6.5
v5–c3; Y v1–c6; T–G
6.50 6.82
Fig. 6. The transition energy band structure. vi–cj is the transition from valence band (vi) to Conduction band cj.
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valence-band maximum and the conduction-band minimum, occurs around 3.4 eV. We note that there is a peak at 4.5 eV contributed by many transitions but the most distinctive transition is from the first valence band (v1) to the first conduction band (c1) (4.49 eV) along S–Y–T directions. Also, there is a peak at 5.5 eV mainly due to the following: a. The transition v2–c3 at 5.65 along X direction. b. The transition v3–c2 at 5.74 along S–Y–T direction. c. The transition v4–c1 at 5.78 along T–G direction. Other peaks and band to band transitions are listed in Table 2, and transition energy band structure is plotted in Fig. 6. To the best of our knowledge, no experimental optical spectrum is available for this compound. It seems for us, that the theoretical optical spectrum is presented in this study for the first time. 3.4. Standard enthalpy of formation The formation energy DH has been calculated according to the following reaction equations: NaH þ MgH2 -NaMgH3
ð4Þ
Naþ MgH2 þ 12H2 -NaMgH3
ð5Þ
NaH þ Mgþ H2 -NaMgH3
ð6Þ
Naþ Mg þ 32H2 -NaMgH3
ð7Þ
The enthalpy of the formation is calculated by taking the difference in total electronic energy of the products and the reactants. The total energies of NaH, Na and Mg have been computed for the ground-state structures, viz. in space group Fm3m for NaH, Im3m for Na, P63/mmc for Mg and P42/mnm for MgH2, with full geometry optimization. Calculated enthalpies for the reactions (4–7) are shown in Table 3. The calculated enthalpies are in good agreement with the theoretical findings [12,13,27]. Also, our calculated enthalpy using the reactions (6) and (7) is in good agreement with the experimental value [15,28,29]. For the NaMgH3 perovskite, the experimental standard enthalpy of formation Hf0 has been measured by Ikeda et al. [15,28] and by Bouamrane et al. [14]. However, the values reported by Ikeda et al. [15,28] differ markedly from the value reported by Bouamrane et al. [14]. This disagreement can be interpreted by the fact that calorimetric measurements [14] are more direct than the extraction of enthalpies from pressure– composition–temperature (PCT) isotherms [15,28].
Table 3 Calculated hydride formation energy (DH) according to Eqs. (4–7). Enthalpy of reactions (kJ/mol)
D H4 D H5 D H6 D H7 a
Ref. [13]. Ref. [27]. c Ref [12]. d Ref. [15]. e Ref. [28]. f Ref. [29]. b
This work
Calculated
10.88 54.53 71 151.83
a
Experiment b
c
11 , 10 , 10 56.01a 74.80a 179.70a, 147b
887 0.9d, 93.9e, 947 15f 145.05d, 147e
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Table 4 The calculated and experimental standard enthalpy of formation of NaMgH3.
Standard enthalpy of formation DHf of NaMgH3 in (kJ/mol) a b c
Calculated
Experimental
151.83
145.05a, 147b, 232c
Ref. [15]. Ref. [28]. Ref. [14].
In the following, we want to restrict ourselves to the concept of the standard enthalpy of formation. Following Ref. [30] it can be stated: the standard enthalpy of formation DHf is the energy exchange that takes place when 1 mole of a substance is formed from the basic elements in their natural state. For elements in their natural state the standard enthalpy of formation is considered to be zero. Corresponding to the above definition, reaction (7) determines directly this standard enthalpy for the formation of NaMgH3 since it is formed solely by the basic constituents Na, Mg and H. According to the data for the standard enthalpy mentioned separately in Table 4, the result from the measurements of Bouamrane et al. [14] disagrees with ours, and with the one obtained by Ikeda et al. [15,28]. It should be noted that our calculation using DFT is at T¼0 K and P¼0, and further studies might depend on temperature corrections such as zero-point energy, polaritons and vibration excitation. However, the influence of these corrections on the calculated results was previously found to be minor [31,32]. On the other hand, our results reproduce the experimental results of Ikeda et al. [15,28], which make us think that the experimental results of Bouamrane et al. [14] may contain errors. In fact, Bouamrane et al. [14] determined the standard enthalpy of formation of NaMgH3 via the reaction NaMgH3(cr) +3HCl(aq)-NaCl(aq)+ MgCl2(aq)+ 3H2(g)
(8)
Using the values of the other compounds’ formation enthalpies known in the literature, Bouamrane et al. [14] obtained the standard enthalpy of formation of NaMgH3, without adding corrections of solution energies of compounds in water. Additional experimental measurements would be useful in determining the correct value.
4. Conclusion In this paper, the crystal structure, electronic structure, optical and thermodynamic properties of NaMgH3 have been studied by first-principles calculations within DFT. The optimized structure is obtained from both LDA and GGA total-energy minimizations. From the single-particle wave functions, we calculate the dielectric functions.
The NaMgH3 is a large gap insulator with direct band gap (! -! ) 3.4 eV. The valence band is dominated by hydrogen atoms. The bonding within the NaMgH3 involves mixing between Mg and H orbitals but shows a strong ionic character. The theoretical optical spectrum is presented in this study, and some peaks in the linear optical spectrum have been identified from the band structure. To the best of our knowledge, no experimental optical spectrum is available for this compound. The theoretical optical spectrum has been presented in this study for the first time. Formation energy for NaMgH3 is calculated for different possible reaction pathways. Our result of the standard enthalpy of formation is in good agreement with the experiment. Reference [1] Y. Bouhadda, A. Rabehi, Y. Boudouma, N. Fenineche, S. Drablia, H. Meradji, Int. J. Hydrogen Energy 34 (2009) 4997. [2] J. Zhang, D.W. Zhou, L.P. He, P. Peng, J.S. Liu, J. Phys. Chem. Solids 70 (2009) 32. [3] P. Dantzer, Topics in Applied Physics, Hydrogen in Metals III, vol. 73, Springer, Berlin, 1997, p. 279. [4] M. Gupta, L. Schlapbach, Electronic properties of metal hydrides, in: L. Shlapbach (Ed.), Topics in Applied Physics, Hydrogen in Intermetallic Compounds, vol. 63, Springer, Berlin, 1988, pp. 139–217. [5] L. Schlapbach, A. Zuttel, Nature 414 (2001) 358. [6] B. Sakintuna, F. Lamari-Darkrim, M. Hirscher, J. Int., Hydrogen Energy 32 (2007) 1121. ¨ [7] A. Zaluska, L. Zaluski, J.O. Strom-Olsen, J. Alloys Compd. 307 (2000) 157. [8] F. Gingl, T. Vogt, E. Akiba, K. Yvon, J. Alloys Compd. 282 (1999) 125. [9] A. Bouamrane, J.P. Laval, J.P. Soulie, J.P. Bastide, Mater. Res. Bull. 35 (2000) 545. ¨ [10] B. Bertheville, T. Herrmannsdorfer, K. Yvon, J. Alloys Compd. 325 (2001) L13. [11] Y. Li, B.K. Rao, T. McMullen, P. Jena, P.K. Khowash, Phys. Rev. B 44 (1991) 6030. [12] M. Fornari, A. Subedi, D.J. Singh, Phys. Rev. B 76 (2007) 214118. [13] P. Vajeeston, P. Ravindran, A. Kjekshus, H. Fjellvag, J. Alloys Compd. 450 (2008) 327. [14] A. Bouamrane, C. De Brauer, J.P. Soulie, J.M. Letoffe, J.P. Bastide, Thermochim. Acta 326 (1999) 37. [15] K. Ikeda, Y. Kogure, Y. Nakamori, S. Orimo, Scr. Mater. 53 (2005) 319. [16] X. Gonze, J.M. Beuken, R. Caracas, F. Detraux, M. Fuchs, G.M. Rignanese, L. Sindic, M. Verstraete, G. Zerah, F. Jollet, M. Torrent, A. Roy, M. Mikami, Ph. Ghosez, J.Y. Raty, D.C. Allan, Comput. Mater. Sci. 25 (2002) 478. [17] W. Kohn, L.J. Sham, Phys. Rev. A 140 (1965) 1133. [18] S. Goedecker, SIAM J. Sci. Comput. 18 (1997) 1605. [19] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, J.D. Joannopoulos, Rev. Mod. Phys. 64 (1992) 1045. [20] X. Gonze, Phys. Rev. B 54 (1996) 4383. [21] J.P. Perdew, k Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [22] M. Fuchs, M. Scheffle, Comput. Phys. Commun. 119 (1999) 67. [23] S. Goedecker, M. Teter, J. Hutter, Phys. Rev B 54 (1996) 1703. ¨ [24] E. Ronnebro, D. Nore´us, K. Kadir, A. Reiser, B. Bogdanovic, J. Alloys Compd. 299 (2000) 101. [25] P.K. Khowash, B.K. Rao, T. McMullen, P. Jena, Phys. Rev. B 55 (1997) 1454. [26] C. Ambrosch-Draxl, J.O. Sofo, Comput. Phys. Commun. 175 (2006) 1. [27] A. Klaveness, O. Swang, H. Fjellvag, Europhys. Lett. 76 (2006) 285. [28] K. Ikeda, S. Kato, Y. Shinzato, N. Okuda, Y. Nakamori, A. Kitano, H. Yukawa, M. Morinaga, S. Orimo, J. Alloys Compd. 446–447 (2007) 162. [29] K. Komiya, N. Morisaku, R. Rong, Y. Takahashi, Y. Shinzato, H. Yukawa, M. Morinaga, J. Alloys Compd. 453 (2008) 157. [30] R.E. Sonntag, G.J. Van Wylen, C. Borgnakke, Fundamentals of Thermodynamics, fifth ed., John Wiley & Son, Inc., New York, 1998. [31] X. Ke, I. Tanaka, Phys. Rev. B 71 (2005) 024117. [32] A. Klaveness, H. Fjellvag, A. Kjekshus, P. Ravindran, O. Swang, J. Alloys Compd. 469 (1–2) (2009) 617–622.