Structural, electronic and thermodynamic properties of Al- and Si-doped α-, γ-, and β-MgH2: Density functional and hybrid density functional calculations

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Structural, electronic and thermodynamic properties of Al- and Si-doped a-, g-, and b-MgH2: Density functional and hybrid density functional calculations Tuhina Adit Maark a,*, Tanveer Hussain a, Rajeev Ahuja a,b a

Condensed Matter Theory Group, Department of Physics and Astronomy, Box 516, Uppsala University, SE-751 20 Uppsala, Sweden Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden b

article info

abstract

Article history:

In this work, we present a detailed study of Al- and Si-doped a-, g-, and b-MgH2 phases

Received 17 November 2011

using the gradient corrected density functional GGA-PBE and the hybrid HartreeeFock

Received in revised form

density functionals PBE0 and HSE06 within the framework of generalized KohneSham

28 February 2012

density functional theory (DFT) using a plane-wave basis set. We investigate the structural,

Accepted 8 March 2012

electronic, and thermodynamical properties of these compounds with regard to their

Available online 13 April 2012

hydrogen storage effectiveness. PBE0 and HSE06 predict cell parameters and bond lengths that are in good agreement with the GGA-PBE calculations and previously known experi-

Keywords:

mental results. As expected smaller band gaps (Egs) are predicted by GGA-PBE for the pure

Hybrid density functionals

magnesium hydride phases. PBE0 overcomes the deficiencies of DFT in treating these

Magnesium hydride

materials better than HSE06 and yields Egs that compare even better with previous GW

Hydrogen storage

calculations. Both the hybrid functionals increase the Egs of the Al-doped magnesium

Density of states

hydrides by much less magnitudes than of the Si-doped phases. This difference is interpreted in terms of charge density distributions. Best H2 adsorption energies (DHads) are computed by HSE06 while GGA-PBE significantly overestimates them. Si-doped a- and bMgH2 exhibited the least negative DHads in close proximity to the H2 binding energy range of
0.21 to
0.41 eV ideal for practical H2 storage transportation applications. Copyright ª 2012, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.

1.

Introduction

For a material to be suitable for on-board storage of hydrogen (H2) in vehicles running on fuel cells, the required attributes include high H2 density, low operation temperatureepressure conditions, and rapid reversibility of H2 uptake/release. Magnesium hydride (MgH2) continues to be attractive in this regard because it can reversibly store large amounts of H2 (7.6 wt.%). But its slow reaction kinetics and high dissociation temperature associated with the H2 adsorption and desorption

processes limit its practical applications [1]. Thus, much of research in MgH2 is devoted to improving these two features. Numerous experimental efforts have been made to overcome the slow kinetics in MgH2 via particle size reduction achieved up on synthesizing it by mechanical alloying (MA) of Mg metal powders in a hydrogen atmosphere at room temperature. MA generates a high density of crystal defects which improve the mass transport and diffusion properties. Nanostructured alloys and intermetallics can also be synthesized by this method. It has been shown that in case of MgH2

* Corresponding author. E-mail address: [email protected] (T. Adit Maark). 0360-3199/$ e see front matter Copyright ª 2012, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijhydene.2012.03.038

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ball milling leads to the fabrication of nanocrystalline magnesium, which exhibits remarkable improvement of the H transport properties and H2 absorption/desorption kinetics [2,3]. It has been known for some time that when subjected to high-pressure and high-temperature conditions the rutile low-pressure form a-MgH2 transforms into an orthorhombic high-pressure modification g-MgH2 [4]. Theoretical findings [5] of the existence of other high-pressure phases (b, d and 3) led to a high-pressure synchrotron X-ray diffraction study [6] of the phase transitions associated with MgH2. Improvements in the H2 sorption properties of MgH2 can also be brought about by using transition metal (TM) and light metal atoms as additives [7e9]. The catalytic effect of TM ¼ Ti, V, Mn, Fe and Ni have been studied in ball milled MgH2eTM nanocomposites by Liang et al. [7] who showed that while desorption was most rapid for MgH2eV, composites containing Ti exhibited the fastest absorption kinetics. Though the formation enthalpy and entropy of magnesium hydride were unaltered by milling with TM, its activation energy of desorption significantly decreased. Shang et al. [10] carried out a systematic study of structural stability and dehydrogenation of (MgH2 þ Al, Nb) powder mixtures. Formation of a new bcc phase in the (MgH2 þ Nb) mixtures and a (Al, Mg) solid solution in the (MgH2 þ Al) mixture occurred. Thermogravimetry results displayed that the mechanically alloyed (MgH2 þ Nb) and (MgH2 þ Al) mixtures released w3.9 and 5.4 wt.% H2 at 300  C within 10 min, compared to only 1.5 mass% of the milled MgH2 powder. Several theoretical studies have also been carried out to develop a fundamental understanding of the effect of additives. A first-principles calculations of MgH2 and MgH2eX (X ¼ Al, Ti, Fe, Ni, Cu, or Nb) [11] revealed that the alloying elements destabilized the hydride in the decreasing order: Cu > Ni > Al > Nb > Fe > Ti. The destabilization was attributed to a weakening of the MgeH bonding. Recently Dai et al. [12,13] have analyzed the hydrogen desorption from metal (¼Al, Ti, Mn and Ni) doped MgH2 (110) and (001) surfaces. From density functional theory (DFT) calculations the site preference, namely, substitution of Mg versus occupation of interstitial site, for a dopant was determined. It was illustrated that each dopant improved the dehydrogenation properties of MgH2 by different mechanisms: Al by weakening the interactions between Mg and H atoms, Ti through its potential for generating a TiH2 phase, Mn via the formation of a MneH cluster and Ni due to the formation of thermodynamically less stable Mg2NiH4 phase. The effect of Al and Si substitution of Mg in MgH2 on the electronic structure, dehydrogenation reactions and H2 desorption kinetics of its a, g and b phases have been investigated via DFT-GGA calculations [14,15]. It was shown that compared to Ti, smaller fractions of Al or Si were needed to decrease the heat of formation of MgH2. The destabilization was correlated with the reduction in band gaps brought about by the two dopants. The zero band gaps predicted with Al doping and the small values (within 1 eV) obtained on Si doping suggest a shift from insulating to metallic and semiconducting behavior, respectively. However, such band gaps calculated from GGA eigenvalues in the KohneSham theory typically underestimate the true band gaps due to the

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discontinuity of the exchange-correlation potentials. It is known that if metal hydrides are metallic they may be candidates for high-TC superconductivity [16e18]. In view of these facts accurate computation of electronic properties of MgH2 in pure and doped form is crucial. The abovementioned band gap problem in (semi)local functionals can be overcome by various methods including the DFT þ U method [19], self-interaction corrections (SIC) [20] and Green’s function based GW approximation [21]. The latter can be a particularly computationally expensive scheme when applied to complex systems. Recently, hybrid functionals which are a combination of an exact nonlocal orbitaldependent HartreeeFock (HF) exchange and a standard local exchange-correlation functional have been proposed. The manner in which the exact energy exchange is mixed in the energy functional determines the form of the hybrid functional. The PBE0 functional [22,23] was proposed by Ernzerhof and co-workers [24,25] and Adamo and Barone [23] as a “parameter-free” functional based on the PBE exchangecorrelation functional: 1 3 ¼ EHF þ EPBE þ EPBE EPBE0 x c 4 x 4 x

(1)

where EPBE and EPBE are the PBE exchange and correlation x c energy functionals, respectively and EHF x is the HF style exact exchange energy. Another attractive hybrid functional called HSE has been formulated by Heyd et al. [26] In this method the exchange interaction is described by a short-range (SR) and a long-range (LR) part. The resulting exchange-correlation functional is then given as follows: HF;SR ðmÞ þ ð1
aÞEPBE;SR ðmÞ þ EPBE;LR ðmÞ þ EPBE EHSE xc ¼ aEx x x c

(2)

In eq. (2) the mixing coefficient a is typically set to ¼ which is supported by perturbation theory, while m is called the screening parameter. To the best of our knowledge Arau´jo et al.’s [27] is the only attempt of its kind which goes beyond the standard LDA and GGA approaches to study the electronic and optical properties of a, g, and b phases of MgH2 by employing the all-electron PAW GW approximation. Unlike the GGA band gaps (3.9 eV14 and 4.2 eV5) obtained for a-MgH2 from other theoretical studies, the authors found that their indirect GW band gap of 5.58 eV showed excellent agreement with the experimental findings [28] (5.6  0.1 eV). However, no such accurate calculation has been carried out for doped magnesium hydride phases. In the present work we therefore, apply the PBE0 and HSE06 hybrid functionals to Al- and Si-doped a-, g- and b-MgH2 and examine how their structural, electronic and thermodynamic properties with regard to hydrogen storage differ from the standard gradient corrected PBE functional results.

2.

Computational aspects

To simulate the Al- and Si-doped magnesium hydrides we first generated their 2  2  2 supercells. By replacing one Mg atom out of a total of 16 Mg atoms in a-MgH2 and 32 Mg atoms in g-, and b-MgH2 by Al or Si atom we introduced fractions x ¼ 0.0625 and 0.03125 of these impurities, respectively. In the

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paper the pure hydrides are referred to as a-Mg16H32, gMg32H64 and b-Mg32H64 and the doped hydrides are represented as a-Mg15H32X, g-Mg31H64X and b-Mg31H64X, where X ¼ Al or Si, respectively. All calculations have been performed using the DFT [29] based Vienna ab initio simulations package (VASP) [30,31]. The following approximations have been applied to treat exchange and correlation: (a) the PBE approach utilizing a standard GGA scheme (referred as GGA-PBE) according to the parameterization of Ref. [32], and (b) two types of hybrid functionals, namely, the PBE0 approach [22e25] and the HSE method [26] as implemented in VASP [33,34]. As can be seen from eqs. (1) and (2) the PBE0 and HSE approaches are similar as they mix 25% of the exact HF exchange with 75% of the PBE exchange functional. However, their treatment of the longrange part of exact exchange interaction is different. Only the short-range component is preserved in the nonlocal part in the HSE functional. The cut-off of the long-range part helps in overcoming the convergence problem associated with the slow decay of the Coulomb 1/r kernel. The screening param˚
1 for the nonlocal and local eter (m) was chosen to be 0.2 A part, which is identical to the implementation of the HSE06 functional [35]. Projector augment waves (PAWs) [36,37] and plane-wave (PW) basis sets as provided with VASP were employed. The geometries were optimized by force and stress minimization while relaxing the unit cell shape and volume. Relaxation was performed until the force on each atom became less than ˚ . A cut-off energy of 440 eV was chosen for the PW 0.005 eV/A basis set. Previous experience [14,15] has shown that standard GGA total energies of MgH2 in its different phases converge well using a 6  6  6 Monkhorst-Pack [38] k-point mesh and is therefore, employed when calculating hydrogen adsorption energies (DHads). The corresponding formulae by which DHads

are computed are discussed in Section 3.3. The necessity of performing geometry optimization with hybrid functionals was tested by comparing calculated structural parameters against results obtained using the GGA-PBE functional in order to determine the optimal compromise between accuracy and computing time. The quality of density of states was also analyzed by generating plots both at 2  2  2 and 6  6  6 kpoints.

3.

Results

3.1.

Structural properties

We first examine the structural properties as represented by lattice parameters of the pure MgH2 phases. In Table 1 the a, b, and c values calculated from GGA-PBE, PBE0 and HSE06 at 2  2  2 and 6  6  6 k-point meshes are reported along with the corresponding percentage errors relative to experiment [6]. It is to be noted that in the table the optimized lattice parameters of the supercells have been halved to have a better comparison with the experimental results for MgH2 unit cells. We see from Table 1 that with respect to experiment the lattice parameters of a and g phases are slightly underestimated within 0.46e0.83% and of the b phase are overestimated by GGA-PBE by 2.75%. The general effect of the two hybrid functionals is to reduce the lattice parameters from the GGA-PBE values. This could be a result of the enhanced localization of electrons which favors tight interatomic bonding. Compared to GGA-PBE with these hybrid functionals the percentage error in computation of lattice parameters nearly doubles (
1.05% to
1.42%) for a-Mg16H32 and (
0.99% to
1.02%) for g-Mg32H64 and decreases to 2.1e2.2% for b-Mg32H64. In general the magnitude of relative error (1.01e2.13%) at

Table 1 e Lattice parameters and relative percentage error with respect to experiment in ( ) of a-, g-, b-MgH2 as calculated by GGA-PBE, PBE0 and HSE06 approaches at 2 3 2 3 2 and 6 3 6 3 6 k-point mesh. System a-Mg16H32

Method Experiment GGA-PBE

k-points a

PBE0 HSE06 g-Mg32H64

Experimenta GGA-PBE PBE0 HSE06

b-Mg32H64

Experimenta GGA-PBE PBE0 HSE06

a Ref [6].

222 666 222 666 222 666 222 666 222 666 222 666 2 6 2 6 2 6

2 6 2 6 2 6

2 6 2 6 2 6

˚) a (A

˚) b (A

˚) c (A

4.5176 4.4965 (
0.47) 4.4942 (
0.52) 4.4632 (
1.20) 4.4687 (
1.08) 4.4712 (
1.03) 4.4695 (
1.06) 4.5246 4.4894 (
0.78) 4.4869 (
0.83) 4.4623 (
1.38) 4.4644 (
1.33) 4.4606 (
1.41) 4.4648 (
1.32) 4.6655 4.7939 (2.75) 4.7934 (2.74) 4.7652 (2.14) 4.7689 (2.22) 4.7651 (2.13) 4.7682 (2.20)

4.5176 4.4969 (
0.46) 4.4945 (
0.51) 4.4636 (
1.19) 4.4685 (
1.09) 4.4716 (
1.02) 4.4699 (
1.06) 5.4442 5.4185 (
0.47) 5.4150 (
0.54) 5.3746 (
1.28) 5.3854 (
1.08) 5.3817 (
1.15) 5.3856 (
1.08) 4.6655 4.7939 (2.75) 4.7934 (2.74) 4.7652 (2.14) 4.7689 (2.22) 4.7651 (2.13) 4.7682 (2.20)

3.0206 3.0010 (
0.65) 3.0027 (
0.59) 2.9849 (
1.18) 2.9888 (
1.05) 2.9781 (
1.41) 2.9881 (
1.08) 4.9285 4.9047 (
0.48) 4.9055 (
0.47) 4.8785 (
1.01) 4.8798 (
0.99) 4.8784 (
1.02) 4.8795 (
0.99) 4.6655 4.7939 (2.75) 4.7934 (2.74) 4.7652 (2.14) 4.7689 (2.22) 4.7651 (2.13) 4.7682 (2.20)

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2  2  2 k-points is not significantly large and it can be said that PBE0 and HSE06 perform reasonably well. It is also noticed that a, b and c parameters computed for the three hydrides at 2  2  2 and 6  6  6 k-points are in close proximity. Thus, for the proceeding analysis only the 2  2  2 k-point optimized geometries of the doped phases are considered. In Table 2 the lattice parameters (halved from optimized values) and equilibrium cell volumes (V0) of Al- and Si-doped a-, g-, and b-MgH2 phases calculated by the different DFT functionals are reported. For all the cases both hybrid functionals yield smaller cell parameters and volumes than GGAPBE. However, the difference (d) in magnitudes is not large: ˚ for the former and between 0.48 and d is within 0.01e0.04 A ˚ 3/n (where n ¼ 16 for a phase and n ¼ 32 for g and 0.66 A b phases) for cell volume. This showcases that optimizing geometries of these systems with PBE0 and HSE06 does not offer any distinct advantage over GGA-PBE. Thus, using GGAPBE optimized geometries for performing static calculations (i.e. only electronic relaxation) with PBE0 and HSE06 at a suitable k-point mesh would be a computationally inexpensive yet effective strategy for calculating accurate hydrogen adsorption energies. The effect of Al and Si doping can be understood by looking at the percentage change in V0 of the doped hydrides relative to V0 of their corresponding pure form calculated at the same method. These are listed in ( ) in Table 2 and indicate that at the small doping fractions x ¼ 0.0625 and 0.03125 considered herein the effect of Al doping is to decrease the cell volume and of Si doping is to increase it. Furthermore, for each case the percentage volume increase caused by Si is more than the percentage decrease due to Al. Interestingly the cell parameters of b-Mg31AlH64 are not much different from those of bMg32H64. Regardless of the DFT functional used and the

Table 2 e Lattice parameters and equilibrium volumes (V0) of Al- and Si-doped a-, g-, b-MgH2 as calculated by GGA-PBE, PBE0 and HSE06 approaches using a 2 3 2 3 2 kpoint mesh. In ( ) the percentage change in V0 relative to the corresponding pure phases for the same method are given. System

Method

˚) a (A

˚) b (A

˚) c (A

˚ 3/n)a V0 (A

a-Mg15H32Al

GGA-PBE PBE0 HSE06 GGA-PBE PBE0 HSE06 GGA-PBE PBE0 HSE06 GGA-PBE PBE0 HSE06 GGA-PBE PBE0 HSE06 GGA-PBE PBE0 HSE06

4.4855 4.4490 4.4556 4.4844 4.4435 4.4532 4.4956 4.4681 4.4636 4.5004 4.4802 4.4758 4.7931 4.7644 4.7641 4.7972 4.7691 4.7691

4.4855 4.4490 4.5566 4.4844 4.4435 4.4532 5.3637 5.3205 5.3288 5.3995 5.3602 5.3695 4.7931 4.7644 4.7641 4.7972 4.7691 4.7691

3.0096 2.9951 2.9894 3.0339 3.0232 3.0159 4.9286 4.9010 4.9033 4.9249 4.8963 4.8974 4.7931 4.7644 4.7641 4.7972 4.7691 4.7691

30.27 (
0.23) 29.64 (
0.30) 29.67 (
0.34) 30.50 (0.53) 29.84 (0.27) 29.90 (0.44) 29.71 (
0.30) 29.13 (
0.41) 29.16 (
0.44) 29.91 (0.37) 29.40 (0.51) 29.42 (0.44) 27.53 (
0.00) 27.04 (
0.05) 27.03 (
0.07) 27.60 (0.22) 27.12 (0.26) 27.12 (0.26)

a-Mg15H32Si

g-Mg31H64Al

g-Mg31H64Si

b-Mg31H64Al

b-Mg31H64Si

a n ¼ 16 for a phases and n ¼ 32 for g and b phases.

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dopant, amongst the three hydrides maximum change in cell volume is observed for the g phase. Substitution of Mg with Al and Si also resulted in slight variations in the arrangement of the atoms. The GGA-PBE optimized geometries of a-Mg15H32X, g-Mg31H64X, and bMg31H64X, where X ¼ Al or Si, are shown in Fig. 1(a)e(c). In order to simplify the picture we compare the GGA-PBE derived bond distances of the pure and doped a phase only. In the pure a-MgH2 phases the closest MgeMg distances are 3.003 and ˚ , while the closest MgeH distances are 1.935 and 3.514 A ˚ . The corresponding AleMg and AleH lengths in a1.949 A ˚ ) and 1.796 A ˚ , respectively. Mg15H32Al are (3.017 and 3.543 A The MgeMg and MgeH bonds of the atoms in the vicinity of the dopant Al atom are lengthened significantly to 3.536 and ˚ , respectively. For the constituent Mg and H 1.956e2.003 A atoms which are not neighboring Al the MgeMg distances are practically unaffected and only the shorter MgeH is decreased ˚ . These changes in bond lengths are similar to those to 1.915 A found by Dai et al. [12] in the Al-doped MgH2 (110) surface. When Si is the doping element the SieMg and SieH ˚ ) and 1.852 A ˚, distances in a-Mg15H32Si are (3.029 and 3.483 A ˚ respectively. The neighboring MgeH bonds are 1.950e1.996 A ˚ and MgeMg bonds are 3.029e3.552 A in length. In contrast the other MgeMg bonds in Si-doped a-MgH2 are the same as in the pure phase but the longer (shorter) MgeH bond is increased ˚ . Thus, even though the Al/SieH (decreased) to 1.960 (1.923) A distances are shorter than the MgeH distances the rest of the MeM and MeH bonds, where M ¼ Mg or Al/Si, are lengthened and more when Si is the impurity. Given the similar effect of Al and Si a direct correlation between the bonding interactions and the variations in cell volume may be debatable. Also it is noteworthy that in our geometry optimizations the symmetry and shape were not restricted. Consequently the increase or decrease in cell volume is not a result of only an increase or decrease of the three a, b, and c lattice parameters. For instance, both Al and Si doping decrease a and increase c lattice parameter of the a phase. But in case of a-Mg15H32Al the decrease in a exceeds the increase in c and so it exhibits a smaller cell volume than the pure magnesium hydride. For a-Mg15H32Si it is the opposite and the increase in c supercedes the decrease in a so that the cell volume increases. The lattice parameters derived for MgH2eX (X ¼ Al. Ti, Fe, Ni, Cu, or Nb) alloys by Song et al. [11] through first-principles investigations are such that they have smaller cell volumes than MgH2. Similar to the effect of Si, Cu and Nb doping increased the c lattice parameter while decreasing a and b values but overall the cell volume reduced.

3.2.

Electronic properties

3.2.1.

Quality of DOS with k-points

We first decipher the effect of k-points on the quality of total density of states (DOS) generated and thereby on the band gap of the systems under study. To this end we plotted DOS of aMgH2 at 2  2  2 and 6  6  6 k-points (Supplementary material Fig. S1). A comparison with a previously reported first-principles investigation at 1000 k-points by Song et al. [11] revealed that the 2  2  2 k-point total DOS plot was unable to exhibit some of the features correctly giving zero DOS even

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Fig. 1 e GGA-PBE optimized geometries of (a) a-Mg15H32X, (b) g-Mg31H64X, and (c) b-Mg31H64X (where X [ Al or Si) with the labeling scheme. Green, white and red spheres are Mg, H and X atoms. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

where it is reasonable. In contrast our 6  6  6 k-point total DOS plot showed a very good resemblance. Similarly we found that the quality of the PBE0 and HSE06 plots improved at the higher k-point grid and it was therefore chosen to perform all further DOS calculations. As shown in Section 3.1 geometry optimization with PBE0 and HSE06 yielded lattice parameters in reasonable agreement with GGA-PBE results. So for studying the electronic properties with hybrid functionals we have carried out only electronic relaxations utilizing the GGAPBE relaxed structures as input in order to save computational time. The results thereby obtained are discussed in Sections 3.2.2 and 3.2.3.

3.2.2.

Effect of doping

In order to understand how the dopants are affecting the electronic properties of MgH2, we closely inspect the density of states (DOS) of the pure phases with those of the doped counterparts. As the general features of the a- and g-MgH2 plots are similar herein we present the total and partial DOS of only the former in Fig. 2 calculated using GGA-PBE, PBE0 and HSE06. As the corresponding plots for b-MgH2 exhibit slight differences these are displayed in Supplementary information Fig. S2. In these figures the zero represents the Fermi level (EF). We notice sharp bonding peaks between EF and
2.0 eV in the total DOS of a-Mg16H32 and g-Mg32H64 and a broad bonding

Fig. 2 e Calculated total and partial density of states (DOS) for a-Mg16H32 using (a) GGA-PBE, (b) PBE0, and (c) HSE06 functionals. Fermi level (EF) is set to zero.

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peak centered at w2.0 eV in case of b-Mg32H64. Analysis of the partial DOS reveals that these arise due to the bonding between H s and Mg s and p electrons. This strong sep hybridization is responsible for the high formation energy associated with MgH2. The rest of the valence band (VB) region arises from bonding of H s and Mg s electrons. In all three hydrides the conduction band (CB) has a purely Mg character comprising of both s and p states. In b-MgH2 the intensity of DOS is low at EF as opposed to a- and g-MgH2, indicative of a relatively weaker interaction between Mg and H. Magnesium has two valence electrons but the substituting atoms Al and Si atom possess three and four valence electrons, respectively. These extra electrons of the dopant atoms can be used to increase the electrons in the CB. The substitution also disturbs the structural symmetry giving rise to inequivalent atoms displaying significantly different atom DOS. Atoms in the neighborhood of the dopant are denoted as type-I and those away as type-II. This notation has been depicted in Fig. 1. Irrespective of the hydride phase the effect of Al and Si doping on the DOS is similar and so only the GGAPBE, PBE0 and HSE06 total and partial DOS of a-Mg15H32Al and a-Mg15H32Si are presented in Figs. 3 and 4. For information the plots for b-Mg31H64Al and b-Mg31H64Si are shown in Supplementary information Figs. S3 and S4. The main dissimilarity between the total DOS of the doped and pure magnesium hydrides is noticeable in the generation of new bands below and above the original VB of the pure phases. The former in the Al (Si) doped phases is predominantly an Al (Si) s state with considerable (strong) contributions

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from H-I s electrons, while the latter is primarily due to Al (Si) s and Mg-I s (Mg-I s and H-I s) states. The in between VB region of 6e8 eV width is a result of interaction between Mg, H and the dopant atoms. The CB is no longer a purely Mg state but also has contributions from the dopant s and p states. Depending up on the density functional used the new occupied band above the original VB is found to be positioned 0.0e0.5 eV and 0.5e2.6 eV below the CB minimum for the Aldoped and Si-doped hydrides, respectively. Consequently the band gaps become much reduced. Overall it can be judged that the Al/Si-Mg-I bonding is controlled by pep and ses interactions while that between Al/Si and H-I is sep interaction. It can also be seen that the atom DOS of Mg and H atoms is relatively reduced at the EF due to the presence of Al and Si impurities. This is indicative of weakening of the interaction between the two atoms. Thus, both on account of decreased band gaps and bond strengths an easier dissociation of MgeH bond and lower dehydrogenation energy are expected in the doped MgH2 hydrides.

3.2.3.

Band gaps

From an experimental study [28] on thin films a wide band gap of 5.6  0.1 eV has been previously determined for a-MgH2 by spectrophotometry and ellipsometry. In Table 3 our results of GGA-PBE, PBE0 and HSE06 band gaps (Eg) for the pure and Aland Si-doped MgH2 phases are presented. A comparison of our GGA-PBE DOS plot for a-Mg16H32 with the band structure reported in Ref. [14] suggests that our Eg corresponds to an indirect band gap. Therefore, in Table 3 we list for the previously

Fig. 3 e Calculated total and partial density of states (DOS) for a-Mg15H32Al using (a) GGA-PBE, (b) PBE0, and (c) HSE06 functionals. The zero is the Fermi level (EF).

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Fig. 4 e Calculated total and partial density of states (DOS) for a-Mg15H32Si using (a) GGA-PBE, (b) PBE0, and (c) HSE06 functionals. The zero is the Fermi level (EF).

known GW indirect (and not direct) band gaps and the predicted expected experimental Egs for the three pure phases from Ref. [27]. As discussed in the previous section the decrease in band gap in Al- and Si-doped a-MgH2 is due to generation of localized states belonging to the dopant atom. Consequently a question of direct or indirect band gap does not arise. The GGA-PBE Egs for a-Mg16H32, g-Mg32H64, and b-Mg32H64 are 3.47, 3.46 and 2.46 eV, respectively. Though GGA-PBE is

Table 3 e Results of GGA-PBE, PBE0 and HSE06 calculations of band gap compared with previous experimental [28] and theoretical values [27]. Values in ( ) are expected experimental band gaps predicted in Ref. [27]. System

Band gap (eV) GGA-PBE PBE0 HSE06

a-Mg16H32 a-Mg15H32Al a-Mg15H32Si g-Mg32H64 g-Mg31H64Al g-Mg31H64Si b-Mg32H64 b-Mg31H64Al b-Mg31H64Si a Ref. [27]. b Ref. [28].

3.47 0.00 0.53 3.46 0.00 1.29 2.46 0.00 0.91

5.31 0.46 1.78 5.24 0.32 2.61 4.12 0.09 2.13

4.58 0.00 1.05 4.59 0.00 1.93 3.47 0.00 1.54

GWa indirect

Expt.

5.58

5.6  0.1b (5.6)

5.24

(5.3)

3.90

(4.2)

able to predict the insulating nature of the three hydrides, it strongly underestimates the Egs relative to the GW and expected experimental values by 34e38%. HSE06 improves band gaps of pure a-, g-, and b-MgH2 and the percentage underestimation of Egs reduces to 11e18%. The best band gaps are obtained with PBE0: (i) the a-MgH2 Eg is underestimated only by 5%, (ii) the g phase value is equivalent to the GW indirect bang gap, and (iii) b-MgH2 Eg agrees even better than GW with the expected experimental result. This excellent correspondence is an indicator of the quality of the predictions with PBE0 and of its appropriateness for examination of the electronic properties of the doped magnesium hydride systems. So in the proceeding discussion we consider only the PBE0 band gaps. A metallic state is found with GGA-PBE for all the Al-doped MgH2 phases. In comparison Si-doped MgH2 band gap magnitudes with GGA-PBE (0.53e1.29 eV) suggest that they are semiconducting. The effect of PBE0 is to make these band gaps larger: nearly doubling them for the Si-doped hydrides and increasing them by much less (0.09e0.46 eV) for the corresponding Al-doped hydride phases showcasing that these systems are insulating and semiconducting, respectively. In order to account for the differing magnitudes of band gap enhancement for Al and Si doping we analyzed the charge density distributions. In Fig. 5 only the PBE0 charge distributions of g-Mg32H64, g-Mg31H64Al and g-Mg31H64Si plotted at 6  6  6 k-points on the (001) plane are displayed. As the color changes from blue to red the charge density increases from a minimum to a maximum value in the plots. Based on this scheme it is understood that in the pure gamma phase the

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Fig. 5 e Charge density distribution of (a) g-Mg32H64, (b) g-Mg31H64Al, and (c) g-Mg31H64Si, respectively, plotted in the range ˚ 3 on the (001) plane at an interplanar spacing of 0.35. Contours are drawn in linear between 0.003 (min) to 0.058 (max) e/A 3 ˚ scale with increment of 0.005 e/A .

charge density is localized around H atoms only while the Mg atoms are bare. This is in agreement with the well-known ionic bonding scenario. The picture is maintained for the interaction between Mg and H atoms in the doped hydride phase as well. In comparison one can see the development of a slight covalent character in the AleH bonds as depicted by the increased elliptical red and green regions. However, there is a much stronger directional charge density along the SieH bonds indicative of pure covalent bonding. Thus, the SieH bonds are likely to be sp3 type and the corresponding bands in the DOS are expected to be stabilized than the valence bands due to Al and H interaction. Furthermore, even though both the dopants are n-type donors and lead to generation of filled states closer to the conduction band in MgH2, Si (3s2 3p2) has one more electron in its p orbital than Al (3s2 3p1). Consequently, even based on electronic configuration the localized states due to Si will be energetically more stable than those due to Al and would be further away from the CB. Thus, we believe that the considerably smaller increase in band gap predicted by PBE0 for Al doping compared to by Si doping is not a manifestation of the functional used but a real effect. Overall the three functionals Egs reflect the same trends: (a) b-Mg32H64 < a-Mg16H32 w g-Mg32H64, (b) Al-doped MgH2 < Sidoped MgH2 < pure MgH2 (c) Al-doped Egs w 0.0 eV, and (d) aMg15SiH32 < b-Mg31SiH64 < g-Mg31SiH64. Recently Zhao and coworkers [39] have studied several high-density hydrogen storage materials (MgH2, LiBH4, LiNH2 and NaAlH4) and their alloys from a DFT based first-principles plane-wave pseudopotential method. The authors summarize that though alloying reduces the stability of these hydrides it is the width of band gap which characterizes the bond strength. Wider the Eg harder it is to break the bond and higher is the hydrogen desorption temperature (Tdes). Thus, from the abovementioned Eg orders it can be anticipated that amongst the pure phases b-MgH2 will have the lowest Tdes and that Al doping will decrease Tdes more than Si doping. To understand how the hybrid functionals lead to improved band gaps we compare the (a), (b), and (c) panels of the total DOS in Figs. 2e4 and in Supplementary material Figs. S2eS4. PBE0 and HSE06 leave the spectral shape of the total DOS plots of a-Mg16H32 unchanged but increase the VB

bandwidth by w1 eV. A wide DOS is due to a steep band in the band structure, which in turn implies a stronger interaction. Accordingly, we can expect the hybrid functionals to yield increased hydrogen desorption energies. Comparing with the GGA-PBE calculations, the position of VB is unaffected with respect to the Fermi level when employing PBE0 and HSE06 but the CB minimum is shifted upwards by w1.7 and 1.1 eV, respectively. It is this shift of CB which is responsible for the bringing the calculated band gaps close to the experimental values. PBE0 and HSE06 do not alter the position of the CB minimum relative to EF in the Al-doped magnesium hydrides and the band gaps remain close to zero. Similar to the case of pure phases the increase in Si-doped magnesium hydride band gaps with the hybrid functionals is a result of a shift of the CB minimum to higher energies relative to EF.

3.3.

Hydrogen adsorption energies

In this section, we focus on the hydrogen adsorption energies (DHads) for the systems by considering the following reaction: ð1
xÞMg þ H2 þ xX/Mg1
x H2 Xx

(3)

where X ¼ Al or Si, x ¼ 0 and 0.0625 for a-MgH2 and x ¼ 0 and 0.03125 for g- and b-MgH2. DHads for these reactions are then computed by using the formula:

DHads Mg1
x H2 Xx ¼Etot Mg1
x H2 Xx
ð1
xÞEtot ðMgÞ
xEtot ðXÞ
Etot ðH2 Þ

(4)

The GGA-PBE total energies (Etot) for all except H2 were obtained by geometry (cell shape, volume and ionic) relaxations at 6  6  6 k-points. For evaluating the properties of H2 ˚ cell parameter was used and molecule a cubic cell with a 15 A only ionic relaxations were performed at 6  6  6 k-points. For Mg, Al and Si their solid-state unit cell structures are used. As in case of DOS plots, for obtaining the PBE0 and HSE06 Etot static electronic relaxations were carried out at 6  6  6 kpoints using the GGA-PBE optimized structures as input. The heats of adsorption of H2 by the different pure and Aland Si-doped magnesium hydrides using the three density functionals considered herein are listed in Table 4. Some of

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Table 4 e Calculated heats of adsorption (in eV/H2) of pure and Al- and Si-doped a-, g-, and b-MgH2 with GGA-PBE, PBE0 and HSE06 functionals compared with previously reported theoretical and experimental values. System

Pure a-Mg16H32 g-Mg32H64 b-Mg32H64 Al doping a-Mg15H32Al g-Mg31H64Al b-Mg31H64Al Si doping a-Mg15H32Si g-Mg31H64Si b-Mg31H64Si

DHads (this work)

Fraction x

DHads (previous work)

GGA-PBE

PBE0

HSE06

PW91a

Expt.b

0.0 0.0 0.0


0.577
0.576
0.483


0.660
0.656
0.562


0.676
0.674
0.579


0.657


0.682 to
0.772

0.0625 0.03125 0.03125


0.459
0.513
0.424


0.509
0.575
0.483


0.533
0.598
0.505

0.0625 0.03125 0.03125


0.409
0.492
0.404


0.453
0.555
0.463


0.475
0.576
0.484

a Ref. [40]: Calculations at 12  12  16 k-points. b Ref [41,42].

the previously reported theoretical [40] and experimental results [41,42] are also presented in the table. We first discuss the general trends observed irrespective of the density functional used. We firstly notice that pure a- and g-MgH2 exhibit very similar DHads but these are more negative that the value for b-MgH2. The effect of fractional doping by Al and Si is to increase the hydrogen adsorption energies (i.e. to make them less negative) of the three phases. For the same fraction x of impurity and the same phase, a greater increase in DHads is brought about by Si than Al. Whatever the dopant X at the same amount of doping, b-Mg(1
x)H2Xx displays the least negative DHads. A less negative heat of adsorption is implies lesser stability and more ease of decomposition into the elements. Thus, we infer that amongst all the systems analyzed at the same fractional doping Si-doped b-MgH2 will be associated with the lowest dehydrogenation temperature. Based on simple entropic arguments Lochan and HeadGordon [43] have made rough estimates of the ideal H2 binding energy at a series of temperature. Within the operation temperature range of
30  C to 50  C for a hydrogen storage system in practical transportation applications this ideal binding energy is
20 to
40 kJ/mol or
0.21 to
0.41 eV/ H2. Thus, accurate computation of DHads is crucial. Experimental values of DHads of magenisum hydride reportedly range from
0.682 to
0.772 eV/H2. Er and coworkers [40] in a first-principles study calculated a DHads of
0.657 eV for pure a-MgH2 using the GGA-PW91 density functional with a high k-point grid of 12  12  16. Our GGAPBE result (
0.577 eV) for a-Mg16H32 significantly overestimates these values. In comparison our PBE0 and HSE06 DHads are much improved and show an even better agreement with the experimental result than the PW91 value. Furthermore between the two hybrid functionals HSE06 performs better than PBE0. Amongst all the cases x ¼ 0.0625 Al- and Sidoped a-MgH2 and x ¼ 0.03125 Al and Si-doped b-MgH2 have HSE06 DHads that are nearest to the upper end of the ideal binding energy range. It can be inferred that using a higher doping fraction would bring these energies within
0.21 to
0.41 eV/H2. Finally, a comparison of the amount of impurity required to bring about a similar increase in DHads showcases

that it would be more advantageous to use Si as a dopant than Al for releasing H2 more readily than from the pure MgH2 phases.

4.

Conclusions

We have presented a detailed study of structural, electronic and thermodynamic properties of pure, Al-doped, and Sidoped a-, g-, and b-MgH2 using PBE, PBE0 and HSE06 density functionals. All approaches have been applied consistently within the PAW-DFT framework. The hybrid functionals are able to provide a good description of the structural parameters in agreement up to 2.2% with experimental values. Though standard GGA-PBE gives better a, b, and c lengths it underestimates the band gaps (Eg) of the three pure phases by 30e40% compared to previously known experimental and GW calculations’ based band gaps. However, HSE06 and even more so PBE0 gives Egs which match exceedingly well. Analysis of density of states revealed that the improved Egs by hybrid functionals are a result of an increased shift in position of the conduction band minimum relative to the Fermi level. Only a slight increase in Eg (0.1e0.3 eV) is predicted by PBE0 for the Al-doped magnesium hydrides. The increase for the corresponding Si-doped phases is much higher such that the hybrid functionals favor an insulating character over the semiconducting nature according to GGA-PBE. Furthermore HSE06 and PBE0 also yield the best heats of adsorptions as compared to experiment while GGA-PBE overestimates them strongly. Our results therefore, ascertain the good performance of these hybrid functionals for studying various properties of the MgH2 phases and their doped counterparts.

Acknowledgements TAM would like to acknowledge FUTURA for her postdoctoral scholarship. RA would like to thank FORMAS for funding. TH is thankful to the Higher Education Commission of Pakistan for

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the doctoral fellowship. The authors acknowledge SNIC and UPPMAX for the computing time.

Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.ijhydene.2012.03.038.

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