Structural, Mechanical, Electronic, Elastic and Chemical Bonding Properties of the Complex
K 2 PtCl 6 -Structure Hydrides Sr 2 RuH 6 : First- Principles Study
BOUDRIFA OUASSILA
Research Center In Industrial Technologics CRTI, P.O. Box 64, Cheraga 16014
Algiers – Algeria o.boudhrifa@crti.dz
ABDELMADJID BOUHEADOU
Laboratory for Developing New Materials and their Characterization
Faculty of Science, University of Setif 1 Setif, Algeria
a_bouhemadou@yahoo.fr
Abstract— We report a systematic study of the structural, electronic and elastic properties of the ternary ruthenium-based hydrides Sr2RuH6 within two complementary first-principles approaches. We describe the properties of the Sr2RuH6systems looking for trends on different properties. Our results are in agreement with experimental ones when the latter are available. In particular, our theoretical lattice parameters obtained using the GGA-PBEsol to include the exchange-correlation functional are in good agreement with experiment. Analysis of the calculated electronic band structure diagrams suggests that these hydrides are wide nearly direct band semiconductors, with a very slight deviation from the ideal direct-band gap behaviour and they are expected to have a poor hole-type electrical conductivity. The TB-mBJ potential has been used to correct the deficiency of the standard GGA for predicting the optoelectronic properties. The calculated TB-mBJ fundamental band gap is about 2.99eV.
Calculated density of states spectra demonstrate that the topmost valence bands consist of d orbitals of the Ru atoms, classifying these materials as d-type hydrides. Analysis of charge density maps tells that these systems can be classified as mixed ionic-covalent bonding materials. The single-crystal and polycrystalline elastic
moduli and their related properties have been numerically estimated and analysed for the first time.
Keywords—Ruthenium-based hydrides; first-principles calculations; elastic constants; electronic structure
1. Introduction
The complex transition metal hydrides (CTMHs) such as theA2TH6hydrates, whereA=Mg, Ca, Sr, Ba, Eu, Sm;T=Fe, Ru, Os, which have the highest hydrogen density among the known condensed materials, are stable even at high temperature. Due to their high decomposition temperatureTd, which is generally greater than 550K, the CTMHs have been largely ignored for ambient temperature applications such as hydrogen solid-state storage in fuel cell vehicles [1]. However, the high stabilities and high hydrogen density of these hydrides make them potentially attractive for high-temperature applications, such as the thermochemical storage of heat for solar thermal plants or excess industrial heat for which hydrogen discharge temperature exceeding 700K are desired [2, 1, 3]. On other hand, the high temperature stabilities and semiconducting character of some of these materials led the researchers to the idea that these hydrides can be useful in semiconductor electronics [4, 5], photovoltaics [6] and optoelectronic devices [7]. The requested hydrides for the above-mentioned novel applications should be air- and
moisture-stable; their kinetics of
hydrogenation/dehydrogenation processes should be as slow
as possible; they should have open fundamental band gaps and be capable of conducting electrical current [8]. Consequently, in order to predict their properties, such as structural, elastic, and electronic properties, the characterizations of hydrides become highly required to verify the prerequisite for these novel applications. The studies on novel technological applications of hydrides have been classified into a new scientific field known as ‘hydride electronics’ [9,7], which is at the intersection of the ‘physics of semiconductors’ and
‘hydrogen storage materials’. Studies of hydrides are interesting from a fundamental standpoint and attractive in terms of technological applications. The present work is a part of a large effort aiming to understand the chemical and physical properties of this class of hydride materials. Accurate first-principles calculations based on the density functional theory (DFT) are the most used tools to study in detail each given hydride material without needing any experimental data.
The principal objective of the present work is to investigate the structural, elastic, electronic and optical properties of the ruthenium-based ternary hydride Sr2RuH6in order to improve the existing data and to provide database for the not yet investigated properties.
2. Results and discussions 2.1. Structural properties
The ternary complex transition metal hydride Sr2RuH6
crystalize in the cubic K2PtCl6-structure, space group Fm3m (No. 225). The Sr atoms are located at 8c(1/4, 1/4, 1/4), Ru at 4a(0, 0, 0) and H at 24e(xH, 0, 0) Wyckoff sites [10];the H atom is localized close to the (0.25, 0, 0) site. This structure is characterized by two structural parameters not fixed by the symmetry: the lattice parameter (a) and the internal x- coordinate of the H atom (xH), which can be determined by relaxing the lattice. Fig.1 shows the conventional-cell of Sr2RuH6 as a representative of the studied Sr2RuH6. These compounds can be viewed as RuH6octahedra, well separated by presumably inert Sr ions, whose role is to fill space and donate charge to the RuH6units.
As a first step towards the prediction of the physical properties of the Sr2RuH6 material, its structural parameters, including the lattice parameter a0and internal hydrogen atom coordinate xH, were optimized at the ambient pressure using the above-given calculation settings. Initial structural input were taken from Ref. [11]. The equilibrium lattice parameter a0 and the internal hydrogen atom coordinate xH (bond- lengths) of all considered compounds are collected in Table 1 (Table 2) together with their corresponding experimental values and previous theoretical results when they are available. Owing to the fact that the experimental data were measured at room temperature whereas our reported results are predicted at zero temperature, one will expect that correct calculated values will be slightly smaller than their corresponding measured ones. From Table 1, one can observe that the GGA96-calculated lattice parameters of all examined materials are somewhat larger than the measured ones and the
LDA-ones are somewhat smaller than the measured ones. This stems from the well-known fact that the LDA tends to slightly underestimate the interatomic distances and the GGA96 tends to slightly overestimate them. As it can be noted, the agreement with experiment is improved on passing to the GGA08; the GGA08-calculated lattice parameters of all examined systems are slightly smaller than their corresponding experimental ones as expected. The GGA- PBEsol [12] has been developed specifically to improve the description of the exchange-correlation effects in solids. The relative difference between the GGA08-calculated lattice parameters and the measured onesd(%):
d (%) = (Calculated value - Measured value)×100) / (Measured value) is -1%. This excellent concordance demonstrates the high accuracy and reliability of the obtained results. Owing to this excellent agreement, the optimized structural parameters obtained through the GGA-PBEsol will be adopted as input for the calculations of the electronic and optical properties. From Table 1, one can appreciate the good agreement between our obtained results within the LDA and GGA96 and previously reported ones using the same functionals 11, 13, 14, 15, 16, 17].
To ensure the thermodynamic and chemical stability of the examined material Sr2RuH6, we calculated the formation enthalpy ΔHfand cohesive energy Ecoh of each hydride. The formation enthalpies of the A2RuH6 compounds were computed using the following equation:
2 6 ( ) ( ) ( )
( 2 6) Sr RuH (2 Sr solid Ru solid 6 H gas )
H Sr RuHf ETot ETot ETot ETot
(1)
Here, ESr RuHTot2 6,ETotSr(solid) , ERu(solid)Tot and EH(gas)Tot are the total energy of the unit-cell of the Sr2RuH6 compound, the total energy per atom of the solid states of the pure elementsA and Ru, and the total energy per atom of the gas state of the hydrogen, respectively. The cohesive energy Ecohis the energy that is required for the crystal to decompose into free atoms.
The cohesive energies Ecoh of the A2RuH6 compounds were calculated using the following expression:
2 6 ( ) ( ) ( )
( 2 6) Sr RuH (2 Sr atom Ru atom 6 H atom) Ecoh Sr RuH ETot ETot ETot ETot (2) The obtained cohesive energies and formation enthalpies are listed in Table 1. From Table 1, we can see that the cohesive energies and formation enthalpies of all investigated compounds are negative, indicating that these materials are energetically stable.
In order to gain insight into the influence of pressure on the structural parameters of the considered materials, their geometries have been fully optimized at some fixed pressures in a 0 to 15 GPa range with a step of 3 GPa. Fig. 2 summarizes
Table 1:Optimized lattice parameter (a0, in Å), internal coordinate of the atom of hydrogen (xH), bulk modulus (B, in GPa) and its pressure derivative (B’), enthalpy of formation (ΔH, in eV) and energy of cohesion (Ecoh, in eV) at zero pressure for the Sr2RuH6material. Available experimental and theoretical data are presented for comparison. TheBandB’ were derived from the Birch, Murnaghan, Birch-Murnaghan and Vinet equations of states.
a B B’ xH ΔHf Ecoh
Present 7.5244A,7.6205B
7.3361C 51.591, 52.112
51.573,51.924 3.871,3.572
3.9073,3.8474 0.2319A, 0.2301B
0.2336C -6.27A -20.20A
Expt. 7.6088[18],7.60[10] 0.223[10]
Others 7.6292D[11],7.600E[16]
7.6536F[14],7.5218D[15],7.627D[13] 49.7D[13] 4.29D[13] 0.2268D[11],0.2254E[16]
0.2255F[14],0.2303D[15],0.228D[13]
aPresent work using the GGA08,bPresent work using the GGA96,cPresent work using the LDA,DPrevious calculations using the GGA96,EPrevious calculations using the LDA,FPrevious calculations using the Perdew-Wang GGA
1From the BirchEOS;2from the MurnaghanEOS,3from the Birch-MurnaghanEOS,4From the VinetEOS
the resulting theoretical pressure dependences of the Ru-H and Sr-H bonds. One can observe that the Sr-H bond-length decreases faster than the Ru-H bond length. The Ru-H bond- length is characterized by a weak pressure dependence. This indicates that the volume of the unit-cell decreases faster than the volume of the RuH6octahedron with increasing pressure.
Fig. 3 shows that the x-coordinate of H atom (xH) increases with increasing pressure. The decrease of the lattice parameter a with increasing pressure leads to a decrease of the Ru-H bond-length while the increase of thexHcoordinate leads to the increase of the Ru-H bond-length which explain the weak pressure dependence of the Ru-H bond-length. Variations of the considered chemical bond-lengths can be well approximated by a second-order polynomial expression (d/d0=1+BP+CP2, where d stands to the bond-length at a pressure P and d0 its corresponding value at zero pressure).
The obtained results are as follows:
= 1 - 6.67089 x 10-4P +1.36363 x 10-5P2
= 1 - 0.00599P+ 1.13016x 10-4P2
Fig. 1:Band dispersions for the Sr2RuH6crystal, calculated using the FP- LAPW method within both the GGA08 and TB-mBJ functionals.
To determine the bulk modulusB, which is a measure of the resistance of a solid to volume change under an applied hydrostatic pressure, the full optimized unit-cell volumes (V) at some fixed applied hydrostatic pressures (P) in the 0 – 15GPa range and the corresponding total energies (E) were fitted to different versions of the P-V and E-V equations of states (EOSs) [19-21]. The bulk modulus B and its pressure derivativeB’ values derived from the fits of the calculatedP-V andE-Vdata to the P-VBirch and MurnaghanEOSsandE-V Birch-Murnaghan and VinetEOSs are summarized in Table 1 and compared to available theoretical results. First, one can note the excellent agreement between the values ofBderived from different EOS forms for all considered materials. This yields a second proof of the reliability of the present calculations.
Fig. 2:Variations of the relative bond-lengths (d/d0) of the bonds H-Ru, H-Sr and Sr-Ru versus pressure (P); ‘‘d’’ stands for the interatomic distance at a pressure P, whereas ‘‘d0’’ is the same distance at zero pressure. First- principles calculated values are represented by symbols and their quadratic polynomial fits are represented by solid line curves
Fig. 3: Pressure dependence of the internal coordinate of the atom of hydrogen (xH) of the Sr2RuH6hydride
Table 2: Calculated bond-lengths (H-Ru, A-Ru and H-A, in Å) at zero pressure for the Sr2RuH6
dH-Ru dA-Ru dH-A
Present 1.6999A,1.7031B
1.6970c 2.8744A, 2.8961B
2.8522D 2.3473A,2.3649B 2.3294C Others 1.70D[15] 2.88D[15]
aPresent work using the GGA08,bPresent work using the GGA96,cPresent work using the LDA,DPrevious calculations using the GGA96,EPrevious calculations using the LDA.
The present results are in good agreement with available theoretical results. There are no available experimental data to be compared with our results.
Second, the relatively lower values of B of the considered compounds indicate that these materials have a high compressibility. Third, the calculated bulk moduli at this step will be used to check the consistency and viability of the present calculations by comparing them to the corresponding ones, which will be achieved from the elastic constants later.
2.2. Electronic structure and chemical bonding
The energy band structures of the valence and low energy conduction electronic states determine the energy band gap and other important properties of materials. So, after geometry optimization adopting the GGA08 exchange-correlation functional, the electronic band structure along some high symmetry directions of the FCC Brillouin zone and their corresponding densities of states for the cubic hydride Sr2RuH6is calculated using the FP-LAPW method within the GGA08 and TB-mBJ functionals. The obtained energy band dispersions of the considered materials within both the GGA08 and TB-mBJ approaches are depicted in Fig. 1 for the sake of comparison. From a quick inspection of Fig. 1, some general features can be drawn. First, the bottommost conduction bands (CBs) and topmost valence bands (VBs) in k-space are not at the same point in BZ, revealing that the
Fig. 4:Total density of states (TDOS, presented in terms of States/eV/unit- cell) and partial density of states (PDOS, presented in terms of States/eV/atom) for the ternary hydrides Sr2RuH6
considered crystals are indirect band gap. The valence band maximum (VBMa) appears on the Γ and the conduction band minimum (CBMi) is located at the X-point. It is interesting to mention here that there is divergence about the position of the VBMa in BZ in the scientific literature. This discrepancy could be attributed to the following reasons: there is a very small energy difference between these different points (X, Γ) (see Table 3) on one side and on the other side the energy band dispersion is very sensitive to the crystal parameters and the used calculational method. Thus, experiment measurements of the band structure are required to confirm the nature of the band gap. Second, the general features of the electronic band dispersions obtained using GGA08 and TB- mBJ are qualitatively almost similar with a small difference in the dispersion at some regions of the Brillouin zone. As a general result, the TB-mBJ causes a rigid displacement of the conduction bands towards higher energies with respect to the top of the valence bands. The calculated TB-mBJ fundamental band gaps is about 2.99eV for Sr2RuH6. The corresponding values obtained within the GGA08 is 2.07eV. One can appreciate the important quantitative differences between the TB-mBJ band gaps and the GGA-PBEsol ones. Experimental determinations of the energy gaps of the examined crystals are not available in the literature to check the accuracy of the present results, however it is interesting to note the following:
(i) the general features of our electronic band dispersions are qualitatively similar to those reported in the literature [11,14,15,16,22,23].
(ii) Table 3 shows that our calculated fundamental band gaps through first-principle FP-LAPW method within the GGA08 functional agree well with earlier theoretical determinations of the energy gaps using the FP-LAPW within the GGA96 and LDA [15,16], projected augmented plane-wave (PAW) approach within the GGA96 and LDA [11,14,15] and the augmented-plane-wave (APW) method in its non-self- consistent version within the Xα-Slater method for the exchange term of the potential (APW-Xα-Slater) [22,23].
(iii) It is well known that the calculated band gaps using DFT within the common LDA and GGA approximations are likely to be approximately 30% ˗50% smaller than the experimental values [24,25] because of the well-known deficiency of the DFT within these standard approximations. Therefore, the real band gaps of the considered materials in this work are expected to be larger than those obtained using the common GGA-PBEsol, GGA-PBE and LDA, listed in Table 3.
Assuming that real band gaps of the studied hydrides are 1.3- 1.5 larger than the calculated ones using GGA-PBEsol (according to Refs. 43-40), the expected energy gaps will be in the ranges 2.76-3.18eV Sr2RuH6. Therefore, according to that, the obtained energy gaps using TB-mBJ are in the expected range of band gaps. One can appreciate the fact that the TB- mBJ functional significantly improves the band gaps of Sr2RuH6compared to the common GGA-PBEsol and the TB- mBJ band gaps can be expected to be in reasonable agreement with their corresponding experimental ones. It is seen from Fig. 1 that there are quasi-flat bands at the top of the valence bands (VBs) and relatively dispersive bands at the bottom of the conduction bands (CBs), which reveals that the second derivative from the eigenenegy on the k-vectors (
2E k ( ) /
2k
), which is inversely proportional to the effective mass m*, would be very small at the topmost VBs.Therefore, one can expect that the effective masses of the holes (charge carriers) would be heavy and their mobility should be small. This feature indicates that the d-t2g-Sr that form the topmost VBs are tightly bound to their atoms and make the valence band holes less mobile. As the top of the valence bands is more flat than the bottom of the conduction bands, it is expected that the mobility of holes will be lower than that of electrons. Consequently, these materials could possess n-type electrical conductivity, while the p-type electrical conductivity would be smaller even if the concentration of acceptors were larger than that of donors.
In order to identify the angular momentum character of the electronic states that composed the band structures of the examined hydrides, the total and atomic-resolved l-projected densities of states (TDOSs and PDOSs) diagrams were computed. This is depicted in Fig. 4, which shows the orbital and site projected density of states (PDOS) of each element that constitutes the considered compounds in the present work.
In semiconductors, it is frequently to classify materials depending on the origin of the topmost VBs, so there ares-,p- ord-type materials. As demonstrated in Fig. 4, the dominant contribution to the topmost VBs comes from the d-t2g- electrons of the Ru atoms, so, using this criterion, such
hydrides can be classified as d-type hydrides. Fig. 4 shows that the valence band group located below the Fermi level in the energy range from about ~ -6eV up to Fermi level in Sr2RuH6consists of four regions of bands that labelled V1, V2, V3 and V4. These four subbands are clearly separated by energy gaps. At the bottom of the energy scale, the first structure, termed V4, consists of the hybridized (Ru-s/ Sr-p/H- s) states. The second structure, labelled V3, predominantly originates from the Rudegand Hs states. The V2 is formed from the H-sstates slightly hybridized with the Sr-pd states in the case of Sr2RuH6. This material is characterized by a covalent bonding between the Rudegand H-sstates. The occupied non-bonding Rudt2g states are separated by an indirect band gap from the empty antibonding Rudeg and Sr -dstates in Sr2RuH6.
To visualize the chemical bonding character between the atoms constituting the considered ternary ruthenium-based hydrides, we have calculated the charge density distribution in the (110) plane, which is depicted in Fig. 5. Charge density maps serve as a complementary tool for achieving more understanding of the electronic structure of materials. Fig. 5 shows a sharing of charge between the neighboring Ru and H atoms due to the Ru-4dand H-1shybridization, indicating the presence of a covalent bonding between the ruthenium and hydrogen elements. The near spherical charge distribution around the site of the alkaline-earth atom Sr indicates that the chemical bonding between the Sr atom and RuH6 unit is mainly ionic. So, one can conclude that the chemical bonding in the ternary ruthenium-based hydrides Sr2RuH6 is characterized by a mixture of covalent and ionic characters;
mainly covalent bonding between Ru and H atoms inside the RuH6unit and ionic bonding between the alkaline-earth atom and the RuH6unit.
Fig. 5:Valence charge density distribution maps in (110) plane for the ternary ruthenium-based hydrides Sr2RuH6
Table 3: Calculated fundamental energy gaps (Eg, in eV) for Sr2RuH6system along with available theoretical results GGA08
(PP-PW) GGA96
(PP-PW) LDA
(PP-PW) TB-mBJ
(FP-LAPW) GGA08
(FP-LAPW) Other 2.28123 (Γ-X)
2.35211 (X-X) 2.46510 (Γ -X)
2.51141 (X-X) 2.20191 (Γ -X)
2.29670 (X-X) 2.98699 (Γ-X)
3.05667 (X-X) 2.07001(Γ-X)
2.14511 (X-X) 2.09[13],2.06(Γ-X)[28],2.19(X-X) [26],2.58(X-X),[27],2.22 (X-X) [27], 3.27 (~Γ-X) [29]
2.3. Elastic properties
Three independent elastic constants Cijs, namely C11, C12and C44, are required to describe the behaviour of the cubic crystals under compressional and/or shear strains. They were evaluated for the cubic hydride Sr2RuH6 at the optimized lattice parameters. The obtained results using three different functionals for treating the exchange-correlation potential, namely GGA08, GGA96 and LDA, are listed in Table 4. The elastic properties of the title materials have not yet been discussed in the scientific literature, so we are not aware of any experimental or theoretical data for the elastic constants Cijs in order to be compared with our findings. In view of Table 4, one can make the following conclusions:
(i) All computed Cijs (i, j = 1, 1; 1, 2; 4, 4) for the four considered crystals satisfy the well-known requirements of
mechanical stability for cubic crystals [30]:
C112C12
0,C44 0,
C11C12
0. This indicates that the investigated ternary hydrides are mechanically stable.(ii) A comparison between the CijLDA(the calculated Cijwithin LDA) and the CijGGA96 (the calculated Cij within GGA96) demonstrates that the CijLDA are relatively higher than the
96 ijGGA
C as a result of the aforementioned systematic overestimation of the equilibrium lattice parameter obtained within the GGA96 compared to that obtained within the LDA.
From Table 4, one can appreciate that the arithmetic mean of these two limits of each elastic constant, i.e.,
( LDA GGA96) / 2
ij ij ij
C C C is approximately equal to the corresponding one obtained using the GGA08 for all studied crystals. According to the fact that the LDA overestimates the elastic constants and the GGA96 underestimates them, one can expect that the elastic constants CijGGA08would be in a good agreement with the expected experimental ones.
(iii) The unidirectional elastic constant C11is larger than C44, indicating that these compounds present a relatively weaker resistance to the pure shear deformations compared to their resistance to the compressional deformations.
These aforementioned elastic moduli can be obtained from the Cijs. Two well-known methods are generally used for evaluating these moduli from the Cijs. These are the Voigt [31]
and Reuss [32] methods, which yield the maximum and minimum bounds of these elastic moduli, respectively. For a
cubic structure, the bulk modulus is given by the same formula in both Voigt and Reuss approximations:
11 12
BV BR C 2C / 3 (3)
In the Voigt and Reuss methods, the shear modulusGis given by the following relationships:
11 12 44
( 3 ) / 5
GV C C C , (4)
44 11 12 44 11 12
5 (C C ) / (4 3( ))
GR C C C C (5)
The bulk (B) and shear (G) moduli that characterize the mechanical behaviour of the polycrystalline phase of materials can be calculated from the elastic moduli of its mono- crystalline phase using the Voigt-Reuss-Hill [31-33]
procedure:
V R
2B B B / , G
GVGR
/ 2 (6) The Young’s modulus E, Poisson’s ratio σ and Lamé’s coefficients (μ, λ) can be derived from the moduli B and G through the following relationships:(iv) It is important to evaluate the sound velocities in a crystal because they are related to some physical properties of the material such as its thermal conductivity. Acoustic wave velocities in a crystal can be obtained from the resolution of the Christoffel equation [68]. The calculated sound wave velocities propagating in the [100], [110] and [111] directions are listed in Table 5.
(iv) It is important to evaluate the sound velocities in a crystal because they are related to some physical properties of material such as its thermal conductivity. Acoustic wave velocities in a crystal can be obtained from the resolution of
Fig. 6:Variation of elastic constants C11, C12, C44and C’ with the la pressure for Sr2RuH6.
C11 C12 C44 AE AZ
GGA08 106.5 23.2 30.9 0.286 0.740
GGA96 101.9 21.5 28.0 0.331 0.696
LDA 112.1 27.6 33.6 0.222 0.794
Table 4:Calculated single-crystal elastic constants (C11, C12, C44, in GPa), Zener anisotropy (AZ, dimensionless) and Every anisotropy (AE, dimensionless) for the Sr2RuH6.
the Christoffel equation [34]. The calculated sound wave velocities propagating in the [100], [110] and [111] directions are listed in Table 5.
(v) In addition to the independent elastic constants Cijs, other elastic parameters, such as the bulk (B), shear (G) and Young’s (E) moduli, Poisson’s ratio (σ), Lamé’s coefficients (μ, λ) and compliance components Sijs, have been widely used to characterize the elastic properties of solids.
9 / (3 )
E BG B G , (3B2 ) / (2(3G B G )) (7) / (2(1 ))
E , E/ (2(1)(1 2 )) (8) Our calculated values of the above-mentioned elastic moduli for the considered materials are quoted in Table 10 and allow us to make the following conclusions:
a) From Tables 1 and 6, one can appreciate the good agreement between the calculated values of the bulk modulus from the elastic constants and that from theEOSs fits for each examined compound. This good concordance between results obtained through two different methods suggests that our calculated elastic moduli are reliable.
b) The Young’s modulus, defined as the ratio of the linear stress to the linear strain, can give information about the stiffness of materials. The relatively small values of the Young’s moduli of the examined hydrides suggest that this material will show a rather small stiffness.
) Brittle/ductile behaviour is one of the important mechanical characteristics of materials, which is closely related to their reversible compressive deformation and fracture ability [37].
An empirical relationship proposed by Pugh [35] is widely used to predict the brittle/ductile character of solids.
According to Pugh’s criterion, the material behaves in a ductile manner if the ratio B/G is greater than 1.75 (B G/ 1.75); otherwise, the material behaves in a brittle manner. The obtained value of the Pugh’s B/G ratio for Sr2RuH6is smaller than the critical value 1.75, suggesting that these compounds can be classified as brittle materials. Brittle material is not resistant to large thermal shocks because it cannot efficiently dissipate thermal stress via plastic deformation; its mechanical properties decrease quickly with increasing temperature.
d) The obtained set of isotropic elastic moduli, such as the bulk (B) and shear (G) moduli can be used for semi-empirical estimations of a number of important physical parameters of the polycrystalline phase of materials. For example, the transverse (Vt) and longitudinal (Vl) sound wave velocities for polycrystalline materials can be calculated from the Band G elastic parameters via the known relationships:
3 4
3Vl B G , Vt G
(9)Here,ρis the mass density of the material. The average sound wave velocity (Vm) can be estimated from the Vt and Vl
through the following formula:
2 3 1 3
3 1/3m t l
V V V (10)
An important fundamental parameter closely related to many physical properties such as the elastic constants, specific heat and melting temperature, namely the Debye temperature
D, could be estimated from Vm using the following expression [36]:3 1/3
4 A
D m
B
N
h n V
k M
(11)
Here,
h
is the Planck constant,k
Bis the Boltzmann constant, n is the number of atoms in the formula unit, NA is the Avogadro number and M is the molecular weight. The predicted values for the aforementioned parameters are quoted in Table 6.(vii) The elastic anisotropy of crystals has an important implication in engineering science because it is highly correlated with the possibility to induce microcracks in the materials [38]. Recent research [39] demonstrates that the elastic anisotropy has a significant influence on the nanoscale precursor textures in alloys. Thus, it becomes necessary and significant to estimate the elastic anisotropy of materials for a better understanding of this property and hopefully find mechanisms that will help to improve its durability and resistivity to microcracks. Therefore, numerous approaches were developed for evaluating the elastic anisotropy in solids.
Four different indexes were used in the present work to explore the elastic anisotropy of the Sr2RuH6hydride.
1) The anisotropy factors AE, introduced by Every [40], and AZ, introduced by Zener [75], given by the following expressions:AE (C11C122C44)/ (C11C44)
andAZ 2C44/
C11C12
are usually used to evaluate the elastic anisotropy of cubic crystals. For a completely isotropic material AE is equal to zero (AZ is equal to one) and any deviation ofAE(AZ) from zero (from one) is an indication of the presence of elastic anisotropy. The magnitude of the deviation from zero forAEand one forAZis a measure of the degree of the elastic anisotropy possessed by the considered crystal. The computed values for AE and AZ for the studied hydrides are given in Table 4. It is obvious that Sr2RuH6 is practically elastically anisotropic. The AZfactor is lower than one, which suggests that these materials are more rigid along the [100] crystallographic direction [41].VL(100) VT(100) VL(110) VT(110) VL(111) VT(111)
GGA08 4920 2647 4663 3078 4574 2941
GGA96 4904 2569 4601 3080 4495 2920
LDA 4978 2723 4781 3057 4713 2950
Table 5:Acoustic wave velocities (in m/s) propagating along the [100], [110]
and [111] directions in the cubic Sr2RuH6compound. L and T stand for the longitudinal and transverse polarizations, respectively, regardless the elastic wave propagation direction.
G GV GR B AG AU B/G E σ λ Vl Vt Vm θD
GGA08 34.81 35.19 34.44 50.95 50.95 0.110 1.4641 85.07 0.222 27.74 4703 2812 3112 407
GGA96 32.36 32.87 31.85 48.32 48.32 0.160 1.4932 79.37 0.226 26.74 4646 2763 3060 395
LDA 36.80 37.03 36.57 55.77 55.77 0.064 1.5163 90.50 0.230 31.24 4814 2852 3159 416
Table 6:Calculated elastic moduli: bulk modulus (B, in GPa), Reuss and Voigt shear modulus (GRand GV, in GPa), Young’s modulus (E, in GPa), Poisson’s ratio (σ), longitudinal, transverse and average sound velocity (Vl, Vtand Vm, in m/s), Debye temperatures (θD, in K), anisotropy in shear (AG) and anisotropy universal index (AU) for the Sr2RuH6.
2) The difference between Voigt and Reuss values of the shear modulus, GV and GR, can be used to characterize the elastic anisotropy in solids. It is useful to define a measurement of the
elastic anisotropy in shearing as
follows:AG(GV GR) / (GV GR). AGis null for isotropic crystals. The percentage of shear anisotropy of the studied materials are listed in Table 10. These materials show a weak elastic anisotropy in shearing.
3) A universal index AU, defined as follows [42]:
AU=5GV/GR+BV/BR-6, is a quantitative measure of the elastic anisotropy accounting for both bulk and shear contributions.
For an isotropic crystal, theAUis equal to zero and deviations of AU from zero define the extent of elastic anisotropy. The numerical estimations ofAUfrom the calculated values ofGV
andGRfor the considered materials are shown in Table 6.
4) Three-dimensional (3D) surface representation of elastic moduli is an effective method for visualizing the elastic anisotropy character of a crystal. An isotropic crystal would exhibit a 3D-surface representation with spherical shape. Any deviation of the 3D-surface representation from the spherical shape indicates the presence of elastic anisotropy and its extent is measured by the deviation degree of the 3D-surface representation from the spherical form. For a cubic crystal, the directional dependence of the Young’s modulus is expressed as [43]:
2 2 2 2 2 2
11 11 12 44 1 2 2 3 1 3
1/ E S 2( S S 0.5 )( S l l l l l l )
(12)TheSijare the elastic compliance constants and thel1,l2andl3
are the directional cosines with respect to the x-, y- and z- axes, respectively. The above-equation determines three- dimensional closed surface and the distance from the origin of system coordinate to this surface is equal to the Young’s modulus in a given direction. Fig. 7 illustrates the 3D-surface representations of the Young’s moduli versus crystallographic directions and the cross-sections of these closed surfaces in the ab- and ac-planes for the Sr2RuH6crystal. From Fig. 7, one can observe that the Sr2RuH6 hydride exhibit a noticeable anisotropic character in the Young’s modulus; the shape of the 3D surface deviates from the spherical form. The highest value of the Young’s modulus Emax is carried out for the external stress applied along the crystallographic directions [100], [010] and [001] and the lowest valueEminis carried out for the external stress applied along the [111] crystallographic direction. The lowest Emin and the highestEmaxvalues of the Young’s modulus for the considered materials are given in Table 7. For an isotropic material, theEmin/Emaxratio should be equal to unity and the degree of elastic anisotropy is measured by the deviation of this ratio from one.
3. Conclusions
Using two complementary DFT-based first-principles methods, systematic detailed calculations of the structural, electronic, optical and elastic properties for the ternary ruthenium-based hydrides Sr2RuH6 is performed. These materials offer a singular opportunity to investigate the role of the alkaline-earth element in an isostructural series of ternary ruthenium-based hydrides. Our main findings are as follows:
(i) The best-optimized structural parameters compared to the measured ones were obtained using the GGA-PBEsol for the exchange-correlation potential. The negative values of the cohesion and formation energies conform the chemical stability of the Sr2RuH6systems.
(ii) The TB-mBJ method yielded improved band gap value, which would be consistent with the expected experimental ones. The fundamental band gap of the considered hydride is in the ultraviolet range of the solar energy spectrum.
(iii) Our analysis of the energy band dispersions reveals that the hole-type electrical conductivity of these hydrides is expected to be very small.
(iv) The calculated DOS diagrams show that the Ru-4d states give the main contributions to the upper valence band region (near the Fermi level), indicating that this material can be classified asd-type materials.
(v) Analysis of the calculated charge density maps reveals that the interatomic bonding is of a covalent character inside the [RuH6] units, whereas the bonding between the Sr atom and the [RuH6] units is of an ionic type.
(vi) The obtained optical spectra suggest that the title materials are promising candidate for the application as antireflection coatings.
(vii) The predicted elastic moduli suggest that the examined hydride is mechanically stable with rather moderate stiffness.
The Sr2RuH6 compound has a brittle character and demonstrate noticeable elastic anisotropy.
E ([1 00]) E ([1 11]) Emin/Emax
97.08 72.66 0.75
Table 7:Calculated values of the Young′s modulus (E, in GPa) in the [100]
and [111] directions. Emin and Emax are the lowest and highest values, respectively, of the Young′s modulus
Fig. 7: 3D-surface representation of the directional dependence of the Young’s modulus (in GPa) and its projection on the ab {(001)} and ac {(110)}
planes for the Sr2RuH6.
References
[1] K. M. Nicholson and D.S. Sholl, “First-principles screening of complex transition metal hydrides for high temperature applications”, Inorg. Chem. vol. 53, 2014, pp. 11833–11848.
[2] K. M. Nicholson and D.S. Sholl, “First-principles prediction of new complex transition metal hydrides for high temperature applications”, Inorg. Chem. vol. 53, 2014, pp. 11849–11860.
[3] B. Bogdanovic, A. Reiser, K. Schlichte, B. Spliethoff, and B. Tesche,
“Thermodynamics and dynamics of the Mg–Fe–H system and its potential for thermochemical thermal energy storage”, J. Alloys Compd. vol. 345, 2002, pp. 77–89.
[4] S. Z. Karazhanov, A.G. Ulyashin, P. Ravindran, and P. Vajeeston,
“Semiconducting hydrides”, 82, Phys. Lett. vol. 17006, 2008, pp. 1–5.
[5] S. Z. Karazhanov, P. Ravindran, P. Vajeeston, and A.G. Ulyashin,
“Hydrides as materials for semiconductor electronics”, Philos. Mag.
vol. 88, 2008, pp. 2461–2476.
[6] S. Z. Karazhanov and A.G. Ulyashin, “Similarity of electronic structure and optical properties of Mg2 NiH4 and Si”, Phys. Lett. vol.
82 2008, pp. 1–6.
[7] S. Z. Karazhanov, A.G. Ulyashin, P. Ravindran, and P. Vajeeston, “ Hydride electronics, Phys. Status Solidi (a)”, vol. 204, 2007, pp.
3538–3544.
[8] S. Z. Karazhanov, U. Sheripov, and A.G. Ulyashin, “Classification of hydrides according to features of band structure”, Philos. Mag. vol.
89, 2009, pp. 1111–1120.
[9] S. Bouras, B. Ghebouli, M. Benkerri, M.A. Ghebouli, H. Choutri, L.
Louail, T. Chihi, M. Fatmi, A. Bouhemadou, R. Khenata, and H.
Khachai, “ Theoretical characterization of quaternary iridium based hydrides NaAeIrH6 (Ae = Ca, Ba and Sr)”, Mater. Chem. Phys. vol.
149–150, 2015, pp. 87–93.
[10] R. O. Moyer Jr., C. Stanitski, and J. Tanaka, “Ternary hydrides of calcium and strontium with iridium, rhodium and ruthenium”, J. Solid State Chem. vol. 3, 1971, pp. 541–549.
[11] K. Kadir, D. Moser, M. Munzel, and D. Noreus, “Investigation of Counterion Influence on an octahedral IrH6- complex in the solid state hydrides AAeIrH6 (A = Na, K and Ae = Ca, Sr, Ba, and Eu) with a new structure type”, Inorg. Chem. vol. 50, 2011, pp. 11890–11895.
[12] A. Reuss, Berechnung der Fliesgrenze von Mischkristallen auf Grund der Plastizitatsbedingung fur Einkristalle “Calculation of the yield strength of solid solutions based on the plasticity condition of single crystals”, ZAMM – Zeitschrift fur Angewandte Mathematik und Mechanik. vol. 9, 1929, pp. 49–58.
[13] H. Hagemann, V. D’Anna, L.M. Lawson Daku, S. Gomes, G.
Renaudin, and K. Yvon, “Structural and vibrational properties of Ca2FeH6 and Sr2RuH6”, J. Phys. Chem. Solids. vol. 72, 2011, pp.
286–289.
[14] S.Z. Karazhanov and A.G. Ulyashin, “Similarity of optical properties of hydrides and semiconductors for antireflection coatings”, Philos.
Mag. vol. 90, 2010, pp. 2925–2937.
[15] E. Orgaz and A. Aburto, “Electronic structure of ternary ruthenium- based hydrides”, J. Phys. Chem. C. vol. 112, 2008, pp. 15586–15594.
[16] S. V. Halilov, D.J. Singh, M. Gupta, and R. Gupta, “Stability and electronic structure of the complex K2PtCl6-structure hydrides DMH6 (D = Mg, Ca, Sr; M = Fe, Ru, Os)”, Phys. Rev. B. vol. 70, 2004, pp.
1–6.
[17] [L .M. L. Daku and H. Hagemann, “First-principles study of the pressure dependence of the structural and vibrational properties of the ternary metal hydride Ca2RuH6”, Phys. Rev. B. vol. 76, 2007, pp. 1–
7.
[18] M. Kritikos and D. Noreus, “Synthesis and characterization of ternary alkaline-earth transition metal hydrides containing octahedral [Ru(II)H6]−4and [Os(II)H6]−4complexes”, J. Solid State Chem. vol.
93, 1991, pp. 256–262.
[19] F. D. Murnaghan, “The compressibility of media under extreme pressure”, Proc. Nat. Acad. Sci. vol. 30, 1944, pp. 244–247.
[20] F. Birch, “Finite elastic strain of cubic crystals”, Phys. Rev. vol. 71, 1947, pp. 1–16.
[21] P. Vinet, J.H. Rose, J. Ferrante, and J.R. Smith, “Compressibility of solids”, J. Geophys. Res. vol. 92, 1987, pp. 9319–9326.
[22] M. Gupta, “Electronic structure of intermetallic hydrides: Mg2FeH6
and Ca2RuH6”, J. Less Common. Met. vol. 103, 1984, pp. 325–335.
[23] E. Orgaz and M. Gupta, “The electronic properties of intermetallic hydrides with the K2PtC16, Structure”, J. Phys: Condens. Matter. vol.
5, 1993, pp. 6697–6718.
[24] C. S. Wang and W.T. Pickett, “Density-functional theory of excitation spectra of semiconductors: application to si”, Phys. Rev. Lett. vol. 51, 1983, pp. 597–600.
[25] D. F. R. Gilson and R. O. Moyer Jr., “Counterion influence on the vibrational wavenumbers in ternary and quaternary metal hydride salts, A2MH6 (A = Alkali Metal, Alkaline Earth, and Lanthanides; M
= Ir, Fe, Ru, Os, Pt, Mn)”, Inorg. Chem. vol. 51, 2012, pp. 1231–1232.
[26] B. Huang, K. Yvon, and P. Fischer, “LiMgERuH7, a new quaternary metal hydride containing octahedral, [Ru(II)H6]4– complex anions”, J. Alloys Compd. vol. 210, 1994, pp. 243–246.
[27] B. Huang, P. Fischer, and K. Yvon,” LiMg4Os2H13containing double layers of octahedral [Os(II)H6]4−complexes”, J. Alloys Compd. vol.
245, 1996, pp. L24–L27.
[28] R.O. Moyer Jr., D.F.R. Gilson, and B.H. Toby, “Neutron powder diffraction, and solid-state deuterium NMR analyses of Yb2RuD6 and spectroscopic vibrational analysis of Yb2RuD6and Yb2RuH6”, J. Solid State Chem, vol. 184, 2011, pp. 1895–1898.
[29] W. Bronger, K. Jansen, and G. Auffermann, “ Mg3RuH6, a complex hydride containing two types of hydrogen atoms differing in their bonding”, J. Alloys Compd. vol. 199, 1993, pp. 47–51.
[30] M. H. Born, “Dynamical theory of crystal lattices”, Oxford University Press, New York, NY, 1988.
[31] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz,
“WIEN2 k: An Augmented Plane wave + Local Orbitals Program for calculating Crystal Properties”, University Karlheinz Schwarz, Techn.
Universitat Wien, Wien, Austria. 2013, pp. 1–238, ISBN 3-9501031- 1-2.
[32] F. Tran and P. Blaha, “Accurate band gaps of semiconductors and insulators with a semi local exchange-correlation potential”, Phys.
Rev. Lett. vol.102, 2009, pp. 1–8.
[33] A. D. Becke and E. R. Johnson, “A simple effective potential for exchange, J. Chem. Phys. vol. 124, 2006”, pp. 1–4.
[34] L.D. Landau and E.M. Lifshitz, Theory of elasticity, course of theoretical physics, Pergamum Press, New York, NY, 1980.
[35] S. F. Pugh, Relations between the elastic moduli and plastic properties of polycrystalline pure metals, Philos. Mag. vol. 45, 1954, pp. 823–
843.
[36] O .L. Anderson, A simplified method for calculating the Debye temperature from elastic constants, J. Phys. Chem. Solids. vol. 24, 1963, pp. 909–917.
[37] I.R. Shein, S.L. Skornyakov, V.I. Anisimov, A.L. Ivanovskii,
“Structural, elastic and electronic properties of new layered superconductor HfCuGe2 in comparison with isostructural HfCuSi2, ZrCuGe2, and ZrCuSi2 from first-principles calculations”, Intermetallics, vol. 42, 2013, pp. 130–136.
[38] P. Ravindran, L. Fast, P.A. Korzhavyi, and B. Johansson, Density functional theory for calculation of elastic properties of orthorhombic crystals: Application to TiSi2, J. Appl. Phys. vol. 84, 1998, pp. 4891–
4904.
[39] P. Lloveras, T. Castan, M. Porta, A. Planes, and A. Saxena, Influence of elastic anisotropy on structural nanoscale textures, Phys. Rev. Lett.
vol. 100, 2008, pp. 1–4.
[40] [39] A.G. Every, General closed-form expressions for acoustic waves in elastically anisotropic solids, Phys. Rev. B. vol. 22, 1980, pp. 1746–
1760.
[41] R. E. Newnham, Properties of materials; Anisotropy, symmetry, structure, Oxford University Press, New York, NY, 2005.
[42] S. I. Ranganathan and M. Ostoja-Starzewski, Universal elastic anisotropy index, Phys. Rev. Lett. vol. 101, 2008, pp. 1–4.
[43] J. F. Nye, Properties of crystals, Oxford University Press, New York, NY, 1985.
