Electronic structure and bonding properties for Laves-phase RV2 (R=Ti, Nb, Hf, and Ta) compounds

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Physica B 403 (2008) 2088–2092 www.elsevier.com/locate/physb

Electronic structure and bonding properties for Laves-phase RV2 (R ¼ Ti, Nb, Hf, and Ta) compounds Chang-wen Zhang School of Science, University of Jinan, Jinan 250022, China Received 14 April 2007; received in revised form 19 November 2007; accepted 22 November 2007

Abstract The electronic structure and bonding properties of Laves-phase compounds RV2 (R ¼ Ti, Nb, Hf, and Ta) with C15 structure have been investigated using the full-potential linearized augmented plane-wave method. The results show that the chemical bonding is metallic–ionic–covalent in nature in these compounds, and the covalent bonding between V and V atoms strengthens with the atomic number, increasing among the RV2 (R ¼ Ti, Nb, Hf, and Ta) compounds. The density of states (DOS), equilibrium volume, and elastic properties are discussed, which is important for understanding the physical properties of RV2 (R ¼ Ti, Nb, Hf, and Ta) and may inspire future experimental research. r 2007 Elsevier B.V. All rights reserved. PACS: 75.50.y; 71.15.Nc Keywords: Laves phase; Bonding properties; Electronic structure

1. Introduction The ground-state properties of Laves-phase RB2 (R stands for transition metals or rare earth elements, and B stands for transition metals) compounds are of considerable scientific and technological interests, and have been the subject of many investigations during the last decade [1]. These phases have quite high melting temperatures, low densities, and high oxidation resistances, which are necessary for high-temperature structural applications. Rather recently, interests have been focused on the development of hydrogen-storage materials, and the class of Laves-phase compounds with C15 crystal structure is a promising candidate [2]. However, their potential has not been exploited unfortunately, largely because their lowtemperature brittleness adversely affects the fabrication and use of these materials. Consequently, it is highly desirable to find ways to improve the low-temperature E-mail address: [email protected] 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.11.033

ductility without compromising much on the attractive high temperature properties. The RV2 (R ¼ Ti, Nb, Hf, and Ta) compounds with C15 structure have been studied for high-temperature structural applications [3–7] in the past few years. The results show the high melting point, high strength at elevated temperatures, and reasonably good oxidation resistance. However, very few of the theoretical research work for Laves-phase RV2 (R=Hf, Ti, Nb, and Ta) compounds are reported to date [8,9]. Here, in order to further understand the physical and chemical properties of these compounds, a thorough and detailed investigation of the electronic structure and bonding characteristic of RV2 (R=Ti, Nb, Hf, and Ta) compounds seems to be necessary. In our present paper, we report the calculated results of full-potential augmented plane-wave method for RV2 (R=Ti, Hf, Nb, and Ta) compounds with C15 structure. We have computed various ground-state properties, such as the structural parameters, bulk modulus, density of states (DOS), and band structures, which are helpful for a general understanding

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of the physical and chemical properties for RV2 (R=Ti, Hf, Nb, and Ta) compounds.

valence states are treated within the scalar relativistic approximation. For the case of description of d and f states, extra local orbitals are used in these phases. In our present calculations, self-consistency is achieved by demanding the convergence of the integrated charge difference between the last two iterations to be smaller than 105 electron, since it ensures better stability of the calculated values than the corresponding energy criterion.

2. Details of the calculation The electronic structure and bonding properties of Laves-phase RV2 (R=Ti, Hf, Nb, and Ta) compounds with C15 structure are investigated using the full-potential linearized augmented plane-wave method (FLAPW) with the local density approximation (LDA) exchange and correlation potential [10]. The FLAPW method solves the local-density-functional equation without any shape approximation to the potential or charge density. In this method, the unit cell is divided into non-overlapping muffin-tin (MT) spheres around atoms and an interstitial region. In the spheres, wave functions, charge density, and potential are expanded in spherical harmonics, while in the interstitial region they are expanded in plane waves. The C15 Laves phase is an FCC-based structure containing six atoms in the primitive unit cell. It belongs to the space group Fd3m-number 227 and Pearson symbol cF24 [11]. The MT radii of 2.3 and 2.15 a.u. were used for R and V atoms, respectively. The Brillouin zone integrations within the self-consistency cycles are performed via a tetrahedron method using 84 k-points in the irreducible wedge of the Brillouin zone for the C15 structure. The cutoff parameter RmtKmax for limiting the number of plane waves is set to 8, where Rmt is the smallest value of all atomic sphere radii and Kmax the largest reciprocal lattice vector used in the plane-wave expansion. The maximum l value for the waves used inside the atomic spheres is set to lmax ¼ 10, whereas the charge density is Fourier expanded up to Gmax ¼ 14. The core states are treated fully relativistically, while the

3. Results and discussion The ground-state structural parameters for RV2 with C15 crystal structure are obtained by a minimization of the total energy with respect to the volume with LDA approach. The lattice parameters, total energies, and bulk moduli are listed in Table 1 for Laves-phase RV2 (R ¼ Zr, Ti, Nb, Hf, and Ta) with C15 structure. We note that the equilibrium lattice constant is evaluated within 2.54–5.17% for RV2, slightly smaller than the available experimental data [6,7], and therefore the calculated lattice constants are used in our calculations. In general, the discrepancy was attributed to shortcomings of the local exchange-correlation approximations of LDA potential. Elastic properties of a solid are very important because they relate to various fundamental solid-state phenomena (e.g., mechanical properties, equations of state, and phonon spectra). Most importantly, elastic constants are essential for many practical applications related to the mechanical properties of a solid: thermoelastic stress, internal strain, sound velocity, and fracture toughness. The calculated elastic constants for RV2 compounds are derived from the total energies as a function of suitably selected distortions. The energies are fitted to third-order polynomials, where the elastic constants at the equilibrium constants shown (Table 1) are calculated. For comparison, it should be noted that the calculated elastic moduli for TaV2 is about 202.6 GPa, rather larger than that of other RV2 phases, which indicates the former is harder than the latter. This phenomenon probably results from the effects of Ta-4f states. To our knowledge, no experimental data for the elastic properties of RV2 have been published so far. Therefore, our calculated data shown in Table 1 are predictions and may inspire future experimental research. Table 2 lists the total and partial DOS at the Fermi energy, together with the charges Q within the MT spheres of RV2 (R ¼ Ti, Nb, Hf, and Ta) compounds. The

Table 1 Optimized structural parameters for RV2 compounds with the C15 structure Structure

a

V0

B

B0

Total energy

TiV2 HfV2 NbV2 ZrV2 TaV2

7.1219 7.3156 7.1656 7.3474 7.2215

45.03 48.91 45.70 49.66 47.13

147.1 140.5 190.4 134.7 202.6

3.55 4.23 3.34 3.39 4.89

11009.8786 70556.9549 22876.5347 21987.4345 70081.1675

2089

The lattice parameters are given in angstrom (A˚), equilibrium volume V0 in A˚3, bulk moduli in GPa, and total energies in Ryd per formula unit.

Table 2 Total DOS n(EF) (states/eV) and MT charges (electrons/atom) for RV2 compounds R Compounds

HfV2 NbV2 TaV2 TiV2

n

7.51 5.45 5.25 5.12

R

0.81 0.55 0.7 0.61

V

3.6 2.8 2.45 2.55

R-d

0.6 0.41 0.52 0.49

V

V-d

3.2 2.6 2.2 2.48

s

p

d

f

s

p

d

f

2.1847 2.1607 2.2310 2.1980

5.8875 5.9696 6.0068 6.1109

1.1521 2.2320 1.8307 1.7408

13.9440 0.0134 13.9809 0.0082

2.2497 2.2018 2.2026 2.2029

6.1166 6.0677 6.0698 6.0440

2.8529 2.5682 2.8526 2.6536

0.0117 0.0101 0.0111 0.0093

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350 NbV2 Nb V

300

DOS(States/eV)

250 200 150 100 50 0 -5

-4

-3

-2

-1

0

1

2

3

4

Energy(eV) 400 TiV2 Ti V

300

DOS(States/eV)

construction of the MT spheres is not unique, and the charges contained within by no means correspond to the actual ionic charge. Also, the charges in the interstitial region cannot be assigned to any particular atomic species. Still, analyzing the charges confined in the MT spheres can be useful to get an overall picture about the possible charge transfer between constituent atoms. We note that the MT charges per V atom is equal to about 2.6 electrons for Nb and Ti compounds, and about 2.8 electrons for Ta and Hf compounds, indicating that the V-d states for Ta and Hf are more spatially confined than those of R (R ¼ Nb, Ti). Ohba et al. [12] obtained the structure factors of MgZn2 and MyCu2 by the single-crystal X-ray diffraction method. They showed that charge transfers occurred between the constituent atoms by the population analysis, and localized electrons were seen in the center of the tetrahedral formed by the small atoms in the difference Fourier maps. Here, we account for the different MT radii for all atoms in the Laves-phase compounds, and a similar phenomenon is observed for all RV2 compounds in which some electrons are transferred from R atoms to V atoms. Furthermore, by comparing our calculated valence charges with the starting free atoms and examining the charge changes, we see that the trends of charge transfer for all atoms are rather similar to results discussed above. These phenomena are an indication that the bonding in these metallic compounds could also have a partly ionic character. Fig. 1 shows the calculated total and partial DOS for RV2 (R ¼ Ti, Nb, and Ta) compounds. The Fermi energy is similar for Ti, Nb, and Ta compounds, but moves to a lower level (EF) by about 0.5 eV, in comparison to the HfV2 phase [13] for which EF crosses at a relatively high local minimum. The DOS at the Fermi energy, n(EF), is found to be 5.12–5.45 states/eV for TiV2, NbV2, and TaV2, whereas for HfV2 it reaches up to 7.51 states/eV per formula unit. Also note that the DOS relative to EF is dominated by the V-d states, and n(EF) reaches up to 2.2–2.64 states/eV, which is comparable to the value (2.17 states/eV) for the superconducting vanadium metal. On the other hand, the most important feature of these DOS for these compounds is the overlapping of d states from R and V in the energy range of 3 to 4 eV, which implies hybridization between R-d and V-d states on formation of the RV2 (R ¼ Ti, Nb, and Ta) compounds. R–V hybridizations result in the transfer of electrons between R and V atoms, in good agreement with the corresponding results as shown in Table 1. Fig. 2 shows the band structure of RV2 (R ¼ Nb, Ti, Ta) phases. Note that the peaks arise from the almost dispersionless bands near the Fermi energy, which is similar with HfV2 [13] and ZrV2 [8]. In the region 2.5 to 2 eV near the Fermi energy, the V-d states for these compounds are dominant, with important contributions from R-d states and others as well. The R-d states appear to be confined to this energy range, while the V-d states are seen to have significant additional contributions in the range of 2 eV near EF. By systematically analyzing the

200

100

0 -5

-4

-3

-2

-1

0

1

2

3

4

1

2

3

4

Energy(eV) 350 TaV2 Ta V

300 250 DOS(States/eV)

2090

200 150 100 50 0 -5

-4

-3

-2

-1

0

Energy(eV)

Fig. 1. Total and partial density of states for (a) NbV2, (b) TiV2, and (c) TaV2.

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Fig. 2. Band structure for (a) NbV2, (b) TiV2, and (c) TaV2.

Fig. 3. Contour plot of charge density in the (1 1 0) plane for (a) NbV2, (b) TiV2, and (c) TaV2. The contour lines are drawn from 0 to 0.6e A˚3 with 0.02e A˚3 intervals.

band characteristic for RV2, one also finds that the R-d and V-d hybridization is mainly located in the energy range of 3 to 0 eV, while R-sp and V-pd hybridization is in the range of 0–3 eV for Nb- and Ti-based compounds. For Taand Hf-based compounds [13], however, the R-d and V-d hybridization is mainly located in the energy range of 1 to 0.5 eV, while R-sp and V-pd hybridization is in the range of 3 to 3.5 eV. Moreover, the R-4f band, located in the 9.3 to 12 eV range, is rather localized, and therefore there is no hybridization with V-spd bands. Therefore, we can conclude that the hybridization between the V-d states and R-d states is strongest in the region around 1.5 to 0 eV. In general, the hybridizations between Ta or Hf and V atoms result in a bigger shift of empty states to the Fermi energy for the Hf (Ta)-d states upon formation of Hf (Ta)V2 compared to transition intermetallic RV2 (R ¼ Ti, Zr, Nb) compounds.

In order to further explore the bonding characteristics in the RV2 (R ¼ Ti, Nb, Hf, and Ta) compounds, the charge density maps of the (1 1 0) plane are theoretically calculated as shown in Fig. 3, where the contour lines of the higher density regions are omitted. In the C15 structure of RV2 compounds, R atoms occupy the diamond-like lattice and V atoms are located at tetrahedral sites. The (1 1 0) plane bisects the V tetrahedral in RV2 compounds. Fig. 3 shows that R (R ¼ Ti, Nb, and Ta) atoms are almost spherical as shown in HfV2 [13], whereas V atoms in TiV2 are rather more anisotropic in comparison to RV2 (R ¼ Nb, Hf, and Ta). We also note that the contour line of low charge density around R atoms moves obviously toward V atoms, which indicates the directional d-bonding characteristic in RV2, probably due to the electron transfer from the R to the V atoms. Moreover, the overlap of electron densities between V and V atoms implies a covalent bonding

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between V and V atoms due to hybridization of the d orbitals. Also note that the height of the charge density at the bond midpoint between V and V increases from 0.34 to 0.64e A˚3 for TiV2 in HfV2, with increasing atomic number of R, whereas the height of the charge density at the bond midpoint between R and V is almost the same, i.e. 0.28e A˚3 for NbV2, TiV2, and TaV2, except for 0.38e A˚3 in HfV2, probably because the lattice parameter in HfV2 is larger than that of other RV2. Therefore, the hybridization of V–V is stronger than R–V hybridization in RV2 compounds and even stronger than V–V hybridization in HCP-type V metal. In the interatomic region, the electrons distribute evenly like a metallic bond whose nature is well explained by the nearly free electron model. This could be explained from the point of view of the C15 Laves-phase structure. In comparison, it could be said that the hybridization between V and V atoms in HfV2 is rather stronger than in other RV2. Therefore the strength of covalent bonding for HfV2 is the strongest compared with those in RV2 (R ¼ Ti, Nb, and Ta) compounds, which are helpful in increasing the phase stability and Young’s moduli of these compounds. In order to obtain the enthalpies of formation of RV2 (R ¼ Ti, Nb, Hf, and Ta) compounds, the total energies of pure R and V metals were also calculated using the optimized lattice constants. The equilibrium enthalpies of formation are defined as the energy differences between the total energy of compound and the constituents in proportion to composition, i.e., DH ¼ Etot(ER+2EV). Etot is the total energy per formula unit. ER and EV are the total energies of the pure R and V metals, respectively, which we evaluated using the FCC structure, respectively. Note that the contribution to enthalpies of formation comes from not only the R–V hybridization, but also from the rather strong V–V hybridization upon the formation of RV2 compounds. The difference in enthalpies of formation for these compounds may be explained by the different degrees of covalent bonding existing in these RV2 compounds. Therefore, the stronger covalent bonding between atoms, in particular between V and V atoms, seems to correspond to the higher magnitude of enthalpies of formation in the RV2 compounds. The calculated enthalpy of formation shows that HfV2 is the largest compared to those of RV2 (R ¼ Ti, Nb, and Ta) compounds, in good agreement with the results above. Furthermore, the relatively large bulk moduli are found to be 202.6 GPa for HfV2 compared with those in RV2 compounds, which indicates the strong bonding revealed in the electronic DOS.

4. Conclusions The total energy, density-of-states (DOS), and band structure of Laves-phase RV2 (R ¼ Ti, Nb, Hf, and Ta) with C15-type structure have been investigated using the FLAPW method. The results revealed the overlap of charge densities between the neighboring V–V atoms, resulting in a rather strong covalent bonding between V and V atoms. It was also found that there is a metallic bonding and an ionic bonding between R and V atoms. In comparison, the covalent bonding between V and V atoms is found to strengthen with increasing atomic number in the C15 structure RV2 (R ¼ Ti, Nb, Hf, and Ta) compounds. The DOS, equilibrium volume, and elastic properties are also discussed, which is important for understanding the physical properties of RV2 and may inspire future experimental research.

Acknowledgments This work was supported partly by the National Nature Science Foundation of China (Grant no. 5057205), and the Doctoral Foundation of University of Jinan of China (Grant no. B0632).

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