Compton scattering study and electronic properties of vanadium carbide: A validation of hybrid functional

Physica B 406 (2011) 2007–2012

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Compton scattering study and electronic properties of vanadium carbide: A validation of hybrid functional Ritu Joshi, Jagrati Sahariya, B.L. Ahuja n Department of Physics, University College of Science, M.L. Sukhadia University, Udaipur 313001, Rajasthan, India

a r t i c l e i n f o

abstract

Article history: Received 31 January 2011 Received in revised form 4 March 2011 Accepted 5 March 2011 Available online 15 March 2011

In this paper, we have reported the isotropic Compton profile of VC measured using high energy (661.65 keV) g-radiations from a 137Cs isotope. To compare the experimental momentum densities, we have also employed the linear combination of atomic orbitals (LCAO). In addition, energy bands, density of states and Fermi surface topology of VC have been computed using FP-LAPW and LCAO methods. It is seen that the LCAO with hybridization of density functional theory and Hartree–Fock (so called B3LYP) gives a better agreement with the present Compton profile experiment. This shows validation of an exact exchange part in hybrid density functional. On the basis of energy bands, we have discussed the microscopic origin for the anomalous behavior of hardness of VC. The relative nature of bonding in VC and NbC is also discussed in terms of valence charge densities and Mulliken0 s population analysis. To establish the role of Compton profiles in computation of cohesive properties of refractory materials, we have also calculated for the first time the cohesive energy using the present experimental Compton profile and compared it with the existing data. & 2011 Elsevier B.V. All rights reserved.

Keywords: X-ray scattering Transition metal compounds Density functional theory

1. Introduction The Compton scattering is an inelastic scattering in which an energetic photon collides with an electron and transfers a part of its energy to the electron. This technique has been employed to probe the ground-state electron momentum densities of a variety of materials. Within impulse approximation (IA), the Compton profile is related to the Doppler broadening of scattered radiations by the motion of electrons in the target [1,2]. In the Compton profile experiment, the double differential scattering cross-section is given as 

d2 s dO dE0



¼ C ðE,E0 , FÞJðpz Þ:

ð1Þ

Here C(E, E0 , F) is a conversion factor that depends on the experimental parameters namely the incident and scattered energy of photons (E and E0 , respectively), scattering angle (F) and z-component of linear momentum of electron (pz). In Eq. (1), dO is the solid angle element in the direction of scattering vector k. Theoretically, the Compton profile J(pz) is given as Jðpz Þ ¼

n

ZZ

nðpÞdpx dpy :

Corresponding author. Tel.: þ91 294 2423322; fax: þ 91 294 2411950. E-mail address: [email protected] (B.L. Ahuja).

0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.03.007

ð2Þ

The n(p) is the ground-state electron momentum density, which is written as Z 2  1 X   , nðpÞ ¼ C ðrÞexpðip:rÞdr ð3Þ   ð2pÞ3 where cðrÞ represents the electron wave function. The summation index is overall the occupied states. In such studies atomic units (a.u.) are used, where e¼m¼_¼1, c¼137.036 and 1 a.u. of momentum¼1.9929  10  24 kg m s  1. Transition metal carbides (TMC) in rock salt structure (space group Fm3m) share several basic properties, like high hardness at room temperature, high melting point, low diffusion coefficient and conductivity. Most of these properties justify the name metallic ceramics or refractory hard metals [3–5]. The possibilities of mixed covalent, ionic and metallic bonding lead to curious behavior of these metal carbides. Therefore, a rigorous understanding of the electronic structure of transition metal carbides like VC is very promising and is helpful to unravel the bonding character of refractory carbide materials. Regarding earlier important studies, few electronic structure calculations have been reported for VC with a view to study its band structure, density of states (DOS) and related properties. The self-consistent augmented plane wave (APW) method [6] and the APW method with Xa exchange approximation have been applied to VC [7], to explore its electronic behavior. Experimental Compton profile of polycrystalline VC has been reported by Deb and Chatterjee [8] at a poor resolution (0.60 a.u., Gaussian full width

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at half maximum). The authors have also computed Compton profiles using linear combination of Gaussian orbitals (LCGO). Band structure and cohesive properties of some 3d carbides, ¨ including VC, have been reported by Haglund et al. [9]. The effect of vacancies on TMC and nitrides is studied by Jhi et al. [10]. Wu et al. [11] have reported the elasticity in carbides using ab initio density functional perturbation theory. Zhang et al. [12] have reported a comparative study of chemical bonding in 3d TMC, ˜ es et al. [13] have studied the electronic structure of while Vin bulk and (0 0 1) surface of a few TMC. Using plane wave pseudopotentials, Vojvodic and Ruberto [14] have undertaken a study of electronic structure and bonding of TMC. The purpose of present work was multifold, namely (a) to measure the accurate Compton profiles of VC using a 20 Ci 137Cs Compton spectrometer, (b) to derive the first ever theoretical Compton profiles using linear combination of atomic orbitals (LCAO) with density functional theory (DFT) calculations and compare them with the experimental data, (c) to compare the nature of bonding between VC and its isoelectronic compound, namely NbC, (d) to derive energy bands, density of states (DOS) and Fermi surface topology using LCAO and also the full potential linearized augmented plane wave (FP-LAPW) calculations and (e) to establish Compton profile as a tool to compute the cohesive energy of refractory materials. Since large size single crystals of VC (diameter 15 mm, thickness 3 mm) were not available, we have reported the isotropic profiles.

2. Methodology 2.1. Experiment The measurements were performed on the 137Cs Compton spectrometer sited at the Compton profile laboratory, Udaipur. The instrument was originally described in Ref. [15]. Some salient features of the instrument are as follows. The incident beam consisting g-rays of 661.65 keV was scattered through a mean angle of 16070.61 from the pellets of high purity (99.9þ%) polycrystalline sample (diameter 2.56 cm and thickness 0.43 cm). To collect the Compton spectra, the sample was exposed for 146.8 h. The momentum resolution (Gaussian, full width at half maximum), which had been characterized in previous studies (see, for example Ref. [15]), was 0.38 a.u. The raw Compton spectra were corrected for a string of corrections, like background, instrumental resolution, Comptonscattering cross-section, multiple scattering, etc. as mentioned in Refs. [1, 16]. The instrumental resolution correction was restrained to stripping off the low energy tail from the data, leaving the experimental Compton spectrum convoluted with a Gaussian of the instrumental resolution. In such partial deconvolution, comparison of experiment and theory must be made with the theoretical profiles convoluted with the Gaussian resolution (FWHM of 0.38 a.u. in the present measurement). Finally, the experimental profile was normalized to the corresponding free atom Compton profiles area, viz. 13.26e  in the momentum range 0–7 a.u. [17]. The counts at Compton peak (at channel width 0.061 eV/channel) were about 70  103, leading to statistical error of 70.015e  /a.u. at pz ¼0. 2.2. Theory 2.2.1. LCAO In the present LCAO calculations we have used the quantum mechanical package CRYSTAL03 code [18] developed by the group of Dovesi. In this approach crystalline orbitals are expanded over the basis sets of the atomic orbitals. This code includes various schemes, namely Hartree–Fock (HF), DFT with local density and generalized gradient approximation (LDA and GGA), hybridization

of DFT and HF (B3LYP), etc. These prescriptions differ in the form of monoelectronic Hamiltonian operators. In the case of HF the correlation effect is neglected, while an exact exchange is taken into account. In the DFT approach, the exchange-correlation potential operator (VXC) is derived from the exchange-correlation energy per particle in a uniform electron gas. Mathematically, VXC ðrÞ ¼

@EXC ½r : @rðrÞ

In the LDA and GGA schemes, EXC is defined as Z   drrðrÞeXC rðrÞ , ELDA XC ðrÞ ¼

ð4Þ

ð5Þ

unitcell

EGGA XC ðrÞ ¼

Z unitcell

   drrðrÞeXC rðrÞ, rrðrÞ :

ð6Þ

The B3LYP (Becke0 s three-parameter hybrid functional) is a refined form of the hybrid DFT þHF method in which EXC ¼ ð1pÞEXLDA þ pEXHF þ q DEXB88 þrECLYP þð1rÞECVWN

ð7Þ

where p, q and r are 0.2, 0.72 and 0.81, respectively [18]. In the case of DFT–GGA, we have adopted the exchange potential of Dirac–Slater [18] and the correlation potential of Perdew–Zunger [19]. In DFT–LDA, we have used the exchange and correlation potentials as prescribed by Becke [20] and Perdew and Wang [21], respectively. In the B3LYP approach, the correlation functional is a combination of functionals due to Lee–Yang–Parr (LYP) [22] and Vosko–Wilk–Nusair (VWN) [23]. The all-electron Gaussian basis sets for V and C have been used from the website www.tcm.phy.cam.ac.uk. Energy optimization of these basis sets was undertaken using the BILLY software [18]. Self-consistent calculations have been performed at 256 k points in the irreducible Brillouin zone (IBZ). These k values correspond to oblique coordinates in the units of the shrinking factor (20, 0, 20) for crystalline environment. The first shrinking factor IS (20) is according to Pack–Monkhorst while another shrinking factor ISP (20) corresponds to Gilat net.

2.2.2. FP-LAPW The FP-LAPW method within DFT scheme as embodied in WIEN2K code [24] has also been implemented to compute the energy bands, DOS and Fermi surface topology of VC. This method is known as full potential because no shape approximation to the potential is required. The crystal potential is expanded into two parts: (a) Spherical harmonics within the muffin-tin (MT) atomic sphere and (b) Plane wave outside the MT sphere. All the calculations were performed for the first time using the latest version of gradient corrected exchange-correlation potential as suggested by Wu and Cohen [25]. The convergence criterion was set to 10  5Ry with cut-off charge density Gmax ¼12. The radii of MT spheres of V and C were 2.23 and 1.85 a.u., respectively. To include the crystalline effects, 256k points in the IBZ {9000 (20  20  20)} were considered. The convergence of basis set was controlled by a cut-off parameter RMTKmax ¼7, where (RMT is the radius of MT sphere and Kmax is the magnitude of the largest k vector in the combined basis set of LAPWs). This code does not include the computation of the momentum densities.

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3. Results and discussion 3.1. Energy bands, DOS and Fermi surface Energy bands of VC along with total and partial DOS obtained using the FP-LAPW and LCAO–B3LYP schemes are shown in Figs. 1 and 2, respectively. Except for some fine structures the energy bands and DOS computed from the LCAO–LDA and GGA are similar to those in Fig. 2; therefore these are not shown here. From the energy bands of VC (Figs. 1 and 2) it is evident that several bands cross the Fermi level (EF) in different branches. It is seen that the lowest valence band (Fig. 1) disperses within the energy range from 14.05 to  10.14 eV. This dispersion results from the mixing of C 2s and V 3d states. Since these bands lie much below the valence state, their contribution in the formation of bonds is negligible. The next three bands, which have

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degeneracy at G0 25 in VC, originate mainly from C 2p and V 3d states. The 9th and 10th bands of VC, which degenerate at G12, cross the EF level. The bands arising due to the strong hybridization of 2p states of C and 3d states of V atoms are responsible for the metallic behavior. Moreover, this strong overlap interaction below the EF leads to covalent bonding and hardness of VC. The three-dimensional Fermi surface (FS) structures, shown for the first time in Fig. 3, are mainly formed by these bands. In Fig. 3(a), the standard BZ for cubic structure marked with symmetry directions is shown. In the case of VC the 9th band, which disperses in the energy range 1.73 to 1.22 eV, gives a multiply connected ‘‘jungle-gym’’ like structure that touches the surfaces of the BZ around the X point (and extends towards W). This structure is shown in Fig. 3(b). A deformed sphere Fig. 3(c) is formed at G point due to the 10th band. An overall structure arising from 9th and 10th bands is shown in Fig. 3(d).

Fig. 1. (a) Selected energy bands (E–k relation) of VC along high symmetry directions of the first Brillouin zone using the FP-LAPW method. Here G(0, 0, 0), X(1/2, 0, 1/2), W(1/2, 1/4, 3/4), K(3/8, 3/8, 3/4) and L(1/2, 1/2, 1/2) are featured k points in the Brillouin zone. (b) Partial and total DOS of VC computed using FP-LAPW.

Fig. 2. Same as Fig. 1 except the methodology, which is LCAO–B3LYP.

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3.2. Compton profiles

profiles as

To deduce the anisotropies in momentum densities, we have calculated the difference, DJ(pz), between different directional

DJðpz Þ ¼ Jhkl ðpz ÞJh0 k0 l0 ðpz Þ

ð8Þ

In Eq. (8), [h k l] and [h0 k0 l0 ] are the low indexed crystallographic directions. The DJ(pz) in the theoretical Compton profiles of VC is depicted in Fig. 4. It is seen that the anisotropies in the momentum densities derived within the DFT–LDA and GGA scheme are almost similar. In the low momentum region the B3LYP (hybrid HFþDFT) differs from the DFT–LDA/GGA in predictions. The origin of these anisotropies (Fig. 4) can be explained on the basis of topology of energy bands and DOS shown in Fig. 2. The negative amplitude of anisotropy J111–J110 in the vicinity of Compton peak (pz ¼0) is due to more degenerate states at point G in (1 1 0) or say G–X ( length 0.56 a.u.) branch and existence of energy band near EF, in comparison to G–L(1 1 1) branch. In a similar way, Table 1 Isotropic experimental and unconvoluted theoretical (spherically averaged) Compton profiles, J(pz), using DFT–GGA, DFT–LDA and B3LYP of VC. Also included here is the LCGO data taken from Ref. [8]. pz (a.u)

Fig. 3. (a) Standard BZ for rock salt (Fm3m) structure with high symmetry directions. Fermi surface of VC from (b) 9th and (c) 10th bands using the FPLAPW scheme. In (d) the overall mapping resulting from both bands is shown.

Fig. 4. Anisotropies in the unconvoluted theoretical Compton profiles of VC calculated using different density functional schemes, namely local density and generalized gradient approximations (LDA and GGA, respectively) and the hybridization of DFT and HF (B3LYP).

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 6.0 7.0

J(pz) (e/a.u.) DFT–GGA

DFT–LDA

B3LYP

LCGO

Expt.

7.311 7.263 7.167 7.024 6.832 6.592 6.304 5.975 5.611 4.822 4.028 3.314 2.729 2.284 1.953 1.075 0.716 0.520 0.386 0.301

7.325 7.276 7.180 7.038 6.846 6.606 6.318 5.987 5.620 4.823 4.023 3.304 2.718 2.275 1.948 1.0737 0.715 0.520 0.386 0.300

7.283 7.235 7.138 6.992 6.798 6.557 6.272 5.949 5.593 4.826 4.052 3.346 2.760 2.307 1.967 1.073 0.714 0.520 0.386 0.304

8.231 8.034 7.875 7.647 7.326 6.909 6.427 5.926 5.435 4.515 3.732 3.109 2.613 2.224 1.878 1.024 0.712 0.523 0.375 0.278

7.0337 0.015 7.042 6.942 6.790 6.630 6.406 6.153 5.889 5.541 4.845 70.012 4.096 3.396 2.851 2.405 2.0547 0.006 1.127 70.004 0.7197 0.003 0.5117 0.002 0.3727 0.001 0.2767 0.001

Fig. 5. Differences between the isotropic experimental Compton profile and the convoluted theoretical profiles using DFT–GGA, DFT–LDA and B3LYP schemes for VC. Also included here is the LCGO data taken from Ref. [8].

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Fig. 6. Two-dimensional valence charge density (VCD) for (a) VC and (b) NbC in the (1 1 1) plane, involving the coordinates (1, 0, 0), (1, 1, 0) and (1, 1, 1).

we can explain the other positive and negative trends in the anisotropy. Now we compare the experimental isotropic Compton profile with our theoretical profiles and the available LCGO data [8]. The isotropic experimental Compton profiles of VC along with the corresponding unconvoluted theoretical values computed within the framework of the DFT (LDA and GGA) and B3LYP are listed in Table 1. In Fig. 5, we have shown the difference between our experiment and convoluted theories. To take into account the experimental resolution, all the theoretical profiles including LCGO data [8] have been smeared with the instrumental resolution of 0.38 a.u. (Gaussian, FWHM). In the high momentum side (pz Z4 a.u.), the theoretical values agree well with the experimental data. This is expected because the core electrons, which almost remain unaffected in bonding mechanism, mainly dominate in this region. Significant differences between the experimental and LCGO profiles show total failure of LCGO calculations in modeling the momentum densities and hence the electronic properties of VC. It is also clear from Fig. 5 that the B3LYP based Compton line is in better agreement with the present experiment than other theoretical profiles. This is also confirmed on the basis of v2 fitting. The differences between LCAO theory and present experiment, particularly in the low momentum region, may be due to non-inclusion of relativistic effects and the Lam–Platzman (LP) correlation [1] in LCAO computations. It may be noted that the LP electron correlation effect shifts the momentum density from below the Fermi momentum (pF) to above the pF. Therefore, an inclusion of this correction is expected to reduce the amplitude of theoretical profiles near pz ¼0. To check the relative nature of bonding between VC with its isoelectronic NbC, we have computed the valence charge densities (VCDs) of both TMC using FP-LAPW. It is worth noting that VCD provides a pictorial representation of the distribution of electrons in the specimen. In Fig. 6(a and b), two-dimensional VCDs of VC and NbC in (1 1 1) plane of the unit cell are shown. The relative nature of bonding in both the carbides can be explained by these maps. In both cases, VCDs show almost spherical charge densities around the C ions. In the case of NbC, besides a spherical charge density, additional bond charge is clearly visible as a cloud around the C ions. This additional charge represents more amount of charge transfer from Nb to C than that in VC, which depicts more ionic behavior of NbC than that of VC. This is also in agreement with our LCAO based Mulliken0 s population (MP) analysis which shows a charge transfer of 0.202e  from V to C, which is less than that of Nb to C (1.676e  ). The overlap population between nearest neighbors in case of VC is 0.174e  , while in NbC it is very small (0.056e  ). Now we discuss the cohesive energy, which is the difference between the bulk and atomic total energies. One can compute

cohesive energy from the Compton data using the following relation [1]; Z pmax Ecoh ¼ p2z ½J S ðpz ÞJ FA ðpz Þdpz , ð9Þ 0

FA

where J and JS are the Compton profiles in free atom and solid state phases, respectively. In Eq. (9), JS corresponds to our experiment on bulk VC and JFA is the free atom Compton profile taken from Biggs et al. [17]. In Eq. (9), we have taken pmax ¼1.5 a.u., because after this value there are small deviations between free atom and the experimental Compton profiles. The Ecoh deduced from the Compton experimental data comes out to be 13.7070.07 eV, which is close to the data reported by various workers viz. 13.87 eV [12], 13.88 eV [13] and 13.88 eV [14]. The large value of Ecoh in the case of VC is in tune with the strong covalent interaction between V and C, as evident by energy bands also.

4. Conclusions The energy bands and DOS of VC computed using LCAO and FP-LAPW calculations reveal metallic character, which is attributed to the strong hybridization between C 2p and V 3d states near the Fermi level. The LCAO–B3LYP (hybrid HFþDFT) based Compton profile for this refractory carbide is found to be in reasonable agreement with our experimental Compton data. Small deviations between experimental and theoretical profiles in the low momentum region may be attributed to non-inclusion of relativistic effects and electron–electron correlation effect in the LCAO theory. Using LCAO based Mulliken0 s population analysis and the FP-LAPW based valence charge densities, it is concluded that NbC is more ionic (or say less covalent) than VC. A reasonable value of cohesive energy calculated from the experimental Compton data highlights the applicability of momentum densities in predicting the cohesive properties of refractory materials. The covalent bonding, Compton profiles and cohesive energy are also explained in terms of energy bands of VC. Directional Compton profile measurements are required to compare the anisotropy in momentum densities.

Acknowledgements The authors are thankful to Prof. R. Dovesi and CRYSTAL support team for providing the CRYSTAL03 code. The authors would like to express their gratitude to Prof. P. Blaha for providing the WIEN2k code. This work is supported by UGC, New Delhi.

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