- Email: [email protected]
Compton profile study and electronic properties of tantalum diboride
Applied Radiation and Isotopes 77 (2013) 38–43
Contents lists available at SciVerse ScienceDirect
Applied Radiation and Isotopes journal homepage: www.elsevier.com/locate/apradiso
Compton profile study and electronic properties of tantalum diboride Veera Raykar, K.C. Bhamu, B.L. Ahuja n Department of Physics, ML Sukhadia University, Udaipur 313001, Rajasthan, India
H I G H L I G H T S c c c c c
Reported first-ever experimental Compton profile (CP) of TaB2. Interpreted experimental CP using theoretical CP within density functional theory. Analyzed equal-valence-electron-density experimental CPs of TaB2 and NbB2. Established metallic character by taking Fourier transform of experimental CP. Reported energy bands, DOS and Fermi surface of TaB2 using LCAO and FP-LAPW.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 5 January 2013 Accepted 21 February 2013 Available online 1 March 2013
We have reported the first-ever experimental Compton profile (CP) of TaB2 using 20 Ci137Cs Compton spectrometer. To compare the experimental data, we have also computed the theoretical CPs using density functional theory (DFT) and hybridization of DFT and Hartree–Fock (HF) within linear combination of the atomic orbitals (LCAO) method. In addition, we have reported energy bands and density of states of TaB2 using LCAO and full potential-linearized augmented plane wave (FP-LAPW) methods. A real space analysis of CP of TaB2 confirms its metallic character which is in tune with the cross-overs of Fermi level by energy bands and Fermi surface topology. A comparison of equal-valenceelectron-density (EVED) experimental profiles of isoelectronic TaB2 and NbB2 show more covalent (or less ionic) character of TaB2 than that of NbB2 which is in agreement with available ionicity data. & 2013 Elsevier Ltd. All rights reserved.
Keywords: X-ray scattering Band structure calculations Density functional theory
1. Introduction
is governed by the relation
Since last three decades, the Compton scattering technique has been applied to a variety of materials to test the accuracy of electronic structure calculations for the ground state (Cooper et al., 2004; Ahuja, 2010). The Compton profile (CP), which is spectrum of the Compton scattered photons, is very sensitive to the momentum distribution of valence electrons and therefore offers a straight forward way to explore the electronic structure of variety of materials. Mathematically, the CP, J(pz), is defined as Z Z Jðpz Þ ¼ nðpx ,py ,pz Þ dpx dpy , ð1Þ
pz ¼
px py
where pz is the component of the electron momentum along the scattering vector (chosen as the z-axis of Cartesian coordinate system) and the n(px,py,pz) is the ground state electron momentum density. It is worth mentioning that an electron with rest mass m and momentum pz shifts the scattered photon energy from o1 (incident) to o2 (scattered) at a scattering angle y. The pz n
Corresponding author. Tel.: þ91 941 4317048; fax: þ91 294 2411950. E-mail address: [email protected] (B.L. Ahuja).
0969-8043/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apradiso.2013.02.020
mcfo2 o1 þ o1 o2 ð1cos yÞ=mc2 g ðo21 þ o22 o1 o2 cos yÞ1=2
:
ð2Þ
Transition metal diborides have a unique combination of properties. Particularly, tantalum diboride (TaB2) possesses high hardness, chemical stability, high melting point, good electrical and thermal conductivity (Aizawa et al., 2005; Silvestroni et al., 2012). Due to these peculiar properties, TaB2 has promising applications in cutting tools, high temperature crucibles and thermal protection components, etc. Regarding earlier work on TaB2, several theoretical attempts have been made to investigate the electronic structure of TaB2 and other related diborides. Recently, Zhao and Wang (2009) have studied the structural, mechanical and electrical properties of TaB2 and other related borides using density functional theory (DFT) within the projector-augmented-wave method. Cohesive and thermal properties of a number of transition metal diborides have been investigated by Kaur et al. (2009) using a rigid ion model. Band structure calculations of MgB2 and related compounds have been presented by Ivanovskii (2003) using the self-consistent full-potential linear muffin-tin orbital (LMTO) method. Electronic structure of the
V. Raykar et al. / Applied Radiation and Isotopes 77 (2013) 38–43
AlB2-type diborides have been investigated by Paduani (2003) using the molecular cluster discrete variational method. Shein and Ivanovskii, 2002 have used the full potential LMTO method for computation of band structure calculations. Vajeeston et al. (2001) have investigated electronic structure, bonding and ground state electronic properties of metal diborides by using the tight binding LMTO method. Bond ionicity in MB2 (M¼Mg, Ti, V, Cr, Mn, Zr, Hf, Ta, Al, and Y) have been reported by Chen et al. (2001) using the complex chemical bond theory based on a generalization of the Phillips-Van-Vechten-Levine scheme. To the author’s knowledge, theoretical and experimental CPs of TaB2 have not been reported so far. In the present work, we report the first ever CP of TaB2 using 661.65 keV g-rays emitted by 20 Ci 137Cs source. The experimental CP is compared with the theoretical momentum densities, computed using DFT approximations and Hartree–Fock (HF) scheme within linear combination of atomic orbitals (LCAO). In addition, we have computed the energy bands, density of states (DOS), and Fermi surfaces, etc. of TaB2 using LCAO and full potential-linearized augmented plane wave (FP-LAPW) methods. The experimental CP is interpreted in terms of localization of valence d electrons in TaB2 and NbB2. The real space analysis of CP is also undertaken in the present paper.
2. Experiment The CP of TaB2 was measured using our indigenous 20 Ci 137Cs Compton spectrometer with an intermediate resolution of 0.34 a.u. (Ahuja et al., 2006, in press). The high purity (499.9%) powder of TaB2 (procured from Alfa Aesar) was kept in a circular ampoule of thickness 0.34 cm and diameter 2.03 cm. The sample was fixed vertically in a sample holder and collimated g-rays of energy 661.65 keV were allowed to scatter by the sample at a fixed scattering angle of 16070.61. The scattered radiations were detected by a high purity Ge solid-state detector (Canberra, GL0510P). The sample was exposed for 234 h to accumulate the Compton spectra and the stability of the system was checked from time-to-time using two weak calibrators, namely 57Co and 133 Ba g-emitting sources. To deduce the true experimental CP, the raw data were corrected for background, instrumental resolution (partial, limited to stripping off the low energy tail of the profile), detector efficiency, sample absorption, Compton cross-section, multiple scattering corrections, etc. (Felsteiner et al., 1974; Timms, 1989; Cooper et al., 2004; Ahuja, 2010). The corrected profile was then converted into momentum scale and normalized to free atom Compton profile area (Biggs et al., 1975) of 31.86 e– in the momentum range 0–7 a.u.
3. Computational details To compute the theoretical momentum densities, energy bands and DOS, we have used CRYSTAL09 code (Dovesi et al., 2005, 2009) which employs the wave functions generated from the LCAO method. This code includes various schemes, namely (i) HF, (ii) DFT with local density approximation (LDA) and generalized gradient approximation (GGA) and second order GGA (SOGGA) and (iii) hybridization of DFT and HF (so called B3LYP). In DFT–GGA, we have used the exchange and correlation potentials as prescribed by Becke (1988) and Perdew and Wang (1992), respectively. In the B3LYP (Becke three parameter) method, the correlation functional is a combination of functionals prescribed by Lee et al. (1988) and Vosko et al. (1980). The SO-GGA adopts the exchange functional as suggested by Zhao and Truhlar (2008) and the correlation functional of Perdew et al. (1996). These approximations in electronic structure calculations differ in the
39
form of monoelectronic Hamiltonian operators. In the HF calculations, the exact interaction between electrons is incorporated while the correlation effects are neglected. In the DFT–GGA/SOGGA calculations, both the exchange and correlation effects are treated approximately. The GGA accounts for variation in the density functional while the SO-GGA functional restores the gradient expansion for both exchange and correlation through second order (Zhao and Truhlar, 2008). In the absence of all-electron basis sets of Ta, we have used large core pseudopotential (PP) basis sets for Ta (Hay and Wadt, 1985). The all-electron basis sets for B were taken from http:// www.tcm.phy.cam.ac.uk/mdt26/basis_sets. The PP facilitates the expansion of electronic wave function using a much smaller number of plane wave basis sets. The basis sets were energy optimized, using the BILLY software (Dovesi et al., 2009) until the default tolerance values were achieved. The integration in the reciprocal space has been carried out on a grid of 845 k-points in the irreducible Brillouin zone (BZ). The lattice parameters for hexagonal structure were taken to be a ¼b¼3.062 and c¼ 3.301 A˚ (Paduani, 2003). Besides LCAO calculations, we have also derived the energy bands, DOS and Fermi surfaces of TaB2 using WIEN2k code (Blaha et al., 1990, 2011) which is based on the FP-LAPW method. In the FP-LAPW method, the unit cell is divided into two parts: atomic spheres centered on the atomic sites and interstitial regions. In the present calculations, we have used the latest and more accurate exchange and correlation potentials as suggested by Wu and Cohen (2006). The charge density, potential and wave functions were expanded in spherical harmonics with maximum radial expansion lmax ¼10. The convergence criterion for total energy was set to 0.01 m Ry with cut of charge density Gmax ¼12. The product of MT radii (2.42 a.u. for Ta and 1.69 a.u. for B) and Kmax (magnitude of the largest reciprocal lattice vector) which controls the convergence of basis sets was kept equal to 7.
4. Results and discussion In Figs. 1 and 2, the energy bands and DOS of TaB2 computed using LCAO–DFT–GGA and FP-LAPW calculations are presented. Since the energy bands and DOS computed using other prescriptions of DFT (like SO-GGA, B3LYP) as embodied in LCAO scheme show almost similar topology, only energy bands and DOS corresponding to DFT–GGA are shown. Except some fine structure and the energy values, the energy bands and DOS (Figs. 1 and 2) are found to be in reasonable agreement with reported calculations (Zhao and Wang, 2009; Ivanovskii, 2003; Paduani, 2003). The energy bands corresponding to B-2p and Ta-5d states cross the Fermi energy (EF) level at different branches of BZ, which confirm its metallic character. It is seen that the bands in the energy range from –10.15 to –2.45 eV (Fig. 1) are originated due to Ta-5d and B-2s states. The bands in energy range from –2.45 to 5.10 eV are due to the Ta-5d and B-2p states, with a small contribution of B-2s states. In the unoccupied region, the upper most group of bands in the energy range 5.10– 12.0 eV is due to strong hybridization between Ta-5d and B-2p states with a small contribution of B-2s states. In Fig. 3, differences between theoretical and experimental CPs of TaB2 are presented. Before taking the difference, all the theoretical profiles were convoluted with the experimental resolution function of 0.34 a.u. (Gaussian, full width at half maximum) and were normalized to the free atom CP area as discussed earlier. In the high momentum region (pz 44 a.u.), all the theoretical profiles are in good agreement with the experimental data. This is expected due to a major contribution of the core electrons in this region, which are well defined by the free atom wave functions. A reasonable agreement between the theoretical and experimental momentum densities confirms the accuracy of the data correction. In the low
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V. Raykar et al. / Applied Radiation and Isotopes 77 (2013) 38–43
Fig. 1. Band structure (E–k relation) and DOS of TaB2 along high symmetry directions of the first BZ using LCAO–DFT–GGA approach. The Fermi energy (EF) is taken at zero reference level. The positions of featured k-points are ! (0,0,0), M(1/2,0,0), K(1/3,1/3,0), A(0,0,1/2), L(1/2,0,1/2) and H(1/3,1/3,1/2).
Fig. 2. Same as Fig. 1 expect the theory, which is FP-LAPW.
Fig. 3. Difference between the isotropic experimental and the convoluted theoretical Compton profiles for TaB2 using HF, B3LYP and DFT–GGA and –SOGGA schemes within the LCAO model. The experimental error ( 7 s) is shown at few points.
momentum region (pz o3 a.u.), differences between theoretical and experimental CPs are evident (Fig. 3). It may also be noted that all the DFT based profiles and the hybrid calculations (B3LYP) show similar type of deviations from the experimental data. On the basis
of w2 fitting, it is seen that GGA based CP shows a better agreement with our measured profile than other theoretical profiles. In the vicinity of Compton peak, large differences between the HF based profile and the experimental CP show importance of inclusion of correlation effects in the electronic structure calculations. In the low momentum region, deviations between experimental and DFT based theoretical data may be attributed to incompleteness of basis sets, inapplicability of PP concept for heavier atom like Ta and noninclusion of relativistic effects in the present LCAO calculations. In Fig. 4, we have plotted the anisotropies in the unconvoluted theoretical CPs calculated for different principal directions within the LCAO based HF, DFT–GGA/SO-GGA and B3LYP approaches. It is seen that the general trend of oscillations in the anisotropies (J110–J001, J100–J001 and J100–J110) is almost similar. The anisotropies in the CPs can also be explained in terms of energy bands. For example, a negative amplitude in the anisotropies with respect to J001 (J110–J001 and J100–J001) near pz ¼0 a.u. arises due to higher momentum densities along GA[001] direction (because of large number of allowed states along the GA branch and degenerate states at A point near EF) in comparison to those along GK[110] and GM[100] directions. The positive oscillations in the anisotropies in J100–J110 near pz ¼0.3 a.u. is due to higher momentum densities after the GK/2 distances, which is attributed to cross-over of EF level by bands at the middle of GK distances and degeneracy at K point (close to EF). Other positive and negative oscillations in anisotropies can also be explained in terms of the degenerate and the allowed states of energy bands in the vicinity of EF.
V. Raykar et al. / Applied Radiation and Isotopes 77 (2013) 38–43
Fig. 4. Anisotropy in the unconvoluted theoretical directional Compton profiles of TaB2 derived from different schemes of LCAO–DFT. Solid lines are drawn to guide eyes.
To study the relative nature of bonding in isoelectronic TaB2 and NbB2, we have scaled our experimental profile on equalvalence-electron-density (EVED). The EVED profile, plotted as pF.J(pz) v/s pz/pF, is also known as profile on pz/pF scale (Podloucky and Redinger, 1984; Reed and Eisenberger, 1972). Such profiles are very useful in predicting the nature of bonding in isovalent compounds wherein the covalent bonding results from the sharing of electrons, which increases the localization of charge in the direction of bonding. Therefore, the materials with more covalent bonding show sharper EVED profile than those with less covalent character. In case of TaB2 and NbB2, the Fermi momenta (pF) were found to be 1.22 and 1.21 a.u., respectively. To obtain the EVED profile, convoluted free atom core profile (Biggs et al., 1975) of both the borides was subtracted from the corresponding absolute experimental profiles. The area of both the normalized EVED profiles was set equal to 5.5 e–, which is equal to area of CP of valence electrons. To deduce the EVED profile of NbB2, the experimental CP data were taken from Bhamu (2012). Our experimental EVED profiles, shown in Fig. 5, depict more sharpness of TaB2 data than that of NbB2. This leads to more localization of electrons (in momentum space) and more covalent nature of TaB2 than NbB2. Further, the sharpness in EVED profile of TaB2 shows that the 5d electrons of Ta in TaB2 are more localized (delocalized) in momentum (real) space than the 4d electrons of Nb in NbB2. Our conclusions based on EVED profiles are in tune with the theoretical band ionicity results (52.1% for NbB2 and 50.7 for TaB2) reported by Chen et al. (2001). Now, we discuss the real space CPs of TaB2 deduced from the experimental and free electron model profiles. The real space CP, so called the B(z) function, is the one dimensional Fourier transformation of CP (Mueller, 1977; Podloucky and Redinger, 1984). The B(z) function for the unit area of parabolic free electron profile can be given as 3 sinðpF zÞ BðzÞ ¼ 2 2 cosðpF zÞ : ð3Þ pF z pF z Eq. 3 dictates that when pFz ¼4.493 a.u., in the first instant, the value of B(z) becomes zero. From Fig. 6, we find that experimental B(z) curve cuts the pz-axis at z ¼3.7170.02 a.u. which shows the experimental value of pF as 1.2170.02 a.u. The present value of experimental pF is found to be very close to theoretical pF value (1.22 a.u.) obtained from the free electron hypothesis. Further, as evident from Fig. 6, the first cut in B(z) function of experiment is very close to that of free electron profile. A close agreement between the free electron based pF and that deduced from the
41
Fig. 5. Experimental equal-valence-electron-density (EVED) profiles of isoelectronic NbB2 and TaB2. The experimental EVED profile for NbB2 is derived from the CP data reported by Bhamu (2012).
Fig. 6. The real space Compton profiles, B(z) function, of TaB2 obtained from the valence experimental and free electron (FE) Compton profiles. The free electron profile consists of parabolic shape given by Jðpz Þ ¼ ap2z þ b for pz opF and 0 for pz Z pF, where constants a and b are the function of a number of electrons and pF.
experimental CP confirms a metal-like character of TaB2. The metallic character of TaB2 allows the mapping of Fermi surfaces that arises from electron ordering phenomena. The three dimensional Fermi surfaces of TaB2 within the framework of FP-LAPW are shown in Fig. 7. The main conclusions derived from Fig. 7 (b–e) are:
(i) Out of all the shown energy bands, the 10th band in the range from –2.07 to 2.33 eV (labeled in Fig. 2) leads to hollow doubly- and singly-necked out structures (Fig. 7b). (ii) The 11th band ranging from –1.25 to 4.69 eV (labeled in Fig. 2) produces the peculiar shape around M point and electron-occupancies near A point (Fig. 7c). (iii) The 12th band (Fig. 2), in the energy range almost similar to the 11th band, leads to bowl-like surface (Fig. 7d) at point A with base towards ! point.
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V. Raykar et al. / Applied Radiation and Isotopes 77 (2013) 38–43
Fig. 7. (a) Standard BZ for hexagonal structure (space group P6/mmm). Fermi surface (FS) of TaB2 arising from (b) 10th (c) 11th and (d) 12th bands using the FP-LAPW method. In (e), a complete mapping of FS from 10th–12th bands is shown.
(iv) An overall mapping of the Fermi surface, shown in Fig. 7e, clearly depicts the hole- and electron-like states which support metallic behavior of TaB2 as seen from the analysis of B(z) function.
5. Conclusions We have reported the isotropic electron momentum densities of TaB2 using 20 Ci 137Cs g-ray source. The experimental Compton profile shows a better agreement with the LCAO–DFT–GGA based profile than HF, SOGGA and B3LYP prescriptions. The band structure, density of states and Fermi surface topology for TaB2 as computed using LCAO–DFT and FP-LAPW schemes show a metallic character of TaB2. Equal-valence-electron-density profiles of TaB2 and NbB2 show more covalent character in TaB2 than that in NbB2. On the basis of EVED profiles, it is seen that 5d electrons of Ta in the TaB2 environment are more extended in real space than 4d electrons of Nb in NbB2.
Acknowledgments We are thankful to Prof. Dovesi and Prof. Blaha for providing the CRYSTAL09 and Wien2k codes, respectively. The authors are also grateful to BRNS (DAE), Mumbai for financial support.
References Ahuja, B.L., Sharma, M., Mathur, S., 2006. Anisotropy in the momentum density of tantalum. Nucl. Instrum. Methods B 244, 419–426. Ahuja, B.L., 2010. Recent Trends in Radiation Physics Research. Himanshu Publications, New Delhi, India. Ahuja, B.L., Joshi, R., Sahariya, J., Electronic properties of WC nanocompound: Compton spectroscopy and band structure calculations. J. Exp. Nanosci., in press. Aizawa, T., Suehara, S., Hishita, S., Otani, S., 2005. Surface core-level shift and electronic structure on transition-metal diboride (0001) surfaces. Phys. Rev. B 71, 165405–1–165405-9. Becke, A.D., 1988. Density functional exchange energy approximation with correct asymptotic behavior. Phys. Rev. B 38, 3098–3100. Bhamu, K.C., 2012. Study of Electronic Properties and Compton Profiles of Some Ceramic Materials. Ph.D. Thesis. Mohanlal Sukhadia University, Udaipur, India (unpublished). Biggs, F., Mendelsohn, L.B., Mann, J.B., 1975. Hartree-Fock Compton profiles. At. Data Nucl. Data Tables 16 (3), 201–308. Blaha, P., Schwarz, K., Sorantin, P., Rickey, S.B., 1990. Full-potential linearized augmented plane wave programs for crystalline systems. Comput. Phys. Commun. 59 (2), 399–415. Blaha, B., Schwarz, K., Luitz, J., 2011. Wien2k_11.1, An Augmented Plane Wave Plus Local Orbitals Program for Calculating Crystal Properties, Revised version. Vienna University of Technology, Vienna, Austria. Chen, X.L., Tu, Q.Y., He, M., Dai, L., Wu, L., 2001. The bond ionicity of MB2 (M ¼Mg, Ti, V, Cr, Mn, Zr, Hf, Ta, Al and Y). J. Phys.: Condens. Matter 13, L723–L727. Cooper, M.J., Mijnarends, P.E., Shiotani, N., Sakai, N., Bansil, A., 2004. X-ray Compton Scattering. Oxford Science Publications, Oxford University Press, New York (and references therein). Dovesi, R., Orlando, R., Civalleri, B., Roetti, C., Saunders, V.R., Zicovich-Wilson, C.M., 2005. Crystal: a computational tool for the ab-initio study of the electronic properties of crystal. Z. Kristallogr. 220 (5–6), 571–573.
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Dovesi, R., Saunders, V.R., Roetti, C., Orlando, R., Zicovich-Wilson, C.M., Pascale, F., Civalleri, B., Doll, K., Harrison, N.M, Bush, I.J., D’Arco, Ph., Llunell, M., 2009. CRYSTAL09 Users’ Manual. University of Torino, Torino. Felsteiner, J., Pattison, P., Cooper, M.J., 1974. Effect of multiple scattering on experimental Compton profiles: a Monte Carlo calculation. Philos. Mag. 30 (3), 537–548. Hay, P.J., Wadt, W.R., 1985. Ab-initio effective core potentials for molecular calculations. Potentials for the transition metals Sc to Hg. J. Chem. Phys. 82, 270–283. Ivanovskii, A.L., 2003. Band structure and properties of superconducting MgB2 and related compounds (a reiew). Phys. Solid State 45, 1829–1859. Kaur, N., Mohan, R., Gaur, N.K., Singh, R.K., 2009. Cohesive and thermal properties of transition metal diborides. Phys. B 404, 1607–1610. Lee, C., Yang, W., Parr, R.G., 1988. Development of the Colle–Salvetti correlation energy formula into a functional of the electron density. Phys. Rev. B 37, 785–789. Mueller, F.M., 1977. Anisotropic momentum densities from Compton profiles: silicon. Phys. Rev. B 15 (6), 3039–3044. Paduani, C., 2003. Electronic structure of magnesium diborides and related compounds. Phys. Status Solidi (b) 240, 574–583. Perdew, J.P., Wang, Y., 1992. Accurate and simple analytical representation of the electron gas correlation energy. Phys. Rev. B 45, 13244–13249. Perdew, J.P., Burk, K., Ernzerhof, M., 1996. Generalized gradient approximation made simple. Phys. Rev. B 77, 3865–3868. Podloucky, R., Redinger, J., 1984. A theoretical study of Compton scattering for MgO. I. Momentum density, Compton profiles and B-functions. J. Phys. C: Solid State Phys. 16, 6955–6969.
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Reed, W.A., Eisenberger, P., 1972. Gamma-ray Compton profiles of diamond, silicon and germanium. Phys. Rev. B 6, 4596–4604. Shein, I.R., Ivanovskii, A.L., 2002. Band structure of ZrB2, VB2, NbB2 and TaB2 hexagonal diborides: comparison with superconducting MgB2. Phys. Solid State 44, 1833–1839. Silvestroni, L., Guicciardi, S., Melandri, C., Sciti, D., 2012. TaB2-based ceramics: microstructure mechanical properties and oxidation resistance. J. Eur. Ceram. Soc. 32, 97–105. Timms, D.N., 1989. Compton Scattering Studies of Spin and Momentum Densities. Ph.D. Thesis. University of Warwick, Coventry, UK (unpublished). Vajeeston, P., Ravindran, P., Ravi, C., Asokamani, R., 2001. Electronic structure, bonding and ground-state properties of AlB2-type transition-metal diborides. Phys. Rev. B 63, 045115-1–045115-12. Vosko, S.H., Wilk, L., Nusair, M., 1980. Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis. Can. J. Phys. 58, 1200–1211. Wu, Z., Cohen, R.E., 2006. More accurate generalized gradient approximation for solids. Phys. Rev. B 73, 235116-1–235116-6. Zhao, Y., Truhlar, D.G., 2008. Construction of a generalized gradient approximation by restoring the density-gradient expansion and enforcing a tight Lieb–Oxford bound. J. Chem. Phys. 128 (18), 184109-1–184109-8. Zhao, W.J., Wang, Y.X., 2009. Structural, mechanical and electronic properties of TaB2, TaB, IrB2 and IrB: first-principle calculations. J. Solid. State Chem. 182, 2880–2886.
Contents lists available at SciVerse ScienceDirect
Applied Radiation and Isotopes journal homepage: www.elsevier.com/locate/apradiso
Compton profile study and electronic properties of tantalum diboride Veera Raykar, K.C. Bhamu, B.L. Ahuja n Department of Physics, ML Sukhadia University, Udaipur 313001, Rajasthan, India
H I G H L I G H T S c c c c c
Reported first-ever experimental Compton profile (CP) of TaB2. Interpreted experimental CP using theoretical CP within density functional theory. Analyzed equal-valence-electron-density experimental CPs of TaB2 and NbB2. Established metallic character by taking Fourier transform of experimental CP. Reported energy bands, DOS and Fermi surface of TaB2 using LCAO and FP-LAPW.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 5 January 2013 Accepted 21 February 2013 Available online 1 March 2013
We have reported the first-ever experimental Compton profile (CP) of TaB2 using 20 Ci137Cs Compton spectrometer. To compare the experimental data, we have also computed the theoretical CPs using density functional theory (DFT) and hybridization of DFT and Hartree–Fock (HF) within linear combination of the atomic orbitals (LCAO) method. In addition, we have reported energy bands and density of states of TaB2 using LCAO and full potential-linearized augmented plane wave (FP-LAPW) methods. A real space analysis of CP of TaB2 confirms its metallic character which is in tune with the cross-overs of Fermi level by energy bands and Fermi surface topology. A comparison of equal-valenceelectron-density (EVED) experimental profiles of isoelectronic TaB2 and NbB2 show more covalent (or less ionic) character of TaB2 than that of NbB2 which is in agreement with available ionicity data. & 2013 Elsevier Ltd. All rights reserved.
Keywords: X-ray scattering Band structure calculations Density functional theory
1. Introduction
is governed by the relation
Since last three decades, the Compton scattering technique has been applied to a variety of materials to test the accuracy of electronic structure calculations for the ground state (Cooper et al., 2004; Ahuja, 2010). The Compton profile (CP), which is spectrum of the Compton scattered photons, is very sensitive to the momentum distribution of valence electrons and therefore offers a straight forward way to explore the electronic structure of variety of materials. Mathematically, the CP, J(pz), is defined as Z Z Jðpz Þ ¼ nðpx ,py ,pz Þ dpx dpy , ð1Þ
pz ¼
px py
where pz is the component of the electron momentum along the scattering vector (chosen as the z-axis of Cartesian coordinate system) and the n(px,py,pz) is the ground state electron momentum density. It is worth mentioning that an electron with rest mass m and momentum pz shifts the scattered photon energy from o1 (incident) to o2 (scattered) at a scattering angle y. The pz n
Corresponding author. Tel.: þ91 941 4317048; fax: þ91 294 2411950. E-mail address: [email protected] (B.L. Ahuja).
0969-8043/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apradiso.2013.02.020
mcfo2 o1 þ o1 o2 ð1cos yÞ=mc2 g ðo21 þ o22 o1 o2 cos yÞ1=2
:
ð2Þ
Transition metal diborides have a unique combination of properties. Particularly, tantalum diboride (TaB2) possesses high hardness, chemical stability, high melting point, good electrical and thermal conductivity (Aizawa et al., 2005; Silvestroni et al., 2012). Due to these peculiar properties, TaB2 has promising applications in cutting tools, high temperature crucibles and thermal protection components, etc. Regarding earlier work on TaB2, several theoretical attempts have been made to investigate the electronic structure of TaB2 and other related diborides. Recently, Zhao and Wang (2009) have studied the structural, mechanical and electrical properties of TaB2 and other related borides using density functional theory (DFT) within the projector-augmented-wave method. Cohesive and thermal properties of a number of transition metal diborides have been investigated by Kaur et al. (2009) using a rigid ion model. Band structure calculations of MgB2 and related compounds have been presented by Ivanovskii (2003) using the self-consistent full-potential linear muffin-tin orbital (LMTO) method. Electronic structure of the
V. Raykar et al. / Applied Radiation and Isotopes 77 (2013) 38–43
AlB2-type diborides have been investigated by Paduani (2003) using the molecular cluster discrete variational method. Shein and Ivanovskii, 2002 have used the full potential LMTO method for computation of band structure calculations. Vajeeston et al. (2001) have investigated electronic structure, bonding and ground state electronic properties of metal diborides by using the tight binding LMTO method. Bond ionicity in MB2 (M¼Mg, Ti, V, Cr, Mn, Zr, Hf, Ta, Al, and Y) have been reported by Chen et al. (2001) using the complex chemical bond theory based on a generalization of the Phillips-Van-Vechten-Levine scheme. To the author’s knowledge, theoretical and experimental CPs of TaB2 have not been reported so far. In the present work, we report the first ever CP of TaB2 using 661.65 keV g-rays emitted by 20 Ci 137Cs source. The experimental CP is compared with the theoretical momentum densities, computed using DFT approximations and Hartree–Fock (HF) scheme within linear combination of atomic orbitals (LCAO). In addition, we have computed the energy bands, density of states (DOS), and Fermi surfaces, etc. of TaB2 using LCAO and full potential-linearized augmented plane wave (FP-LAPW) methods. The experimental CP is interpreted in terms of localization of valence d electrons in TaB2 and NbB2. The real space analysis of CP is also undertaken in the present paper.
2. Experiment The CP of TaB2 was measured using our indigenous 20 Ci 137Cs Compton spectrometer with an intermediate resolution of 0.34 a.u. (Ahuja et al., 2006, in press). The high purity (499.9%) powder of TaB2 (procured from Alfa Aesar) was kept in a circular ampoule of thickness 0.34 cm and diameter 2.03 cm. The sample was fixed vertically in a sample holder and collimated g-rays of energy 661.65 keV were allowed to scatter by the sample at a fixed scattering angle of 16070.61. The scattered radiations were detected by a high purity Ge solid-state detector (Canberra, GL0510P). The sample was exposed for 234 h to accumulate the Compton spectra and the stability of the system was checked from time-to-time using two weak calibrators, namely 57Co and 133 Ba g-emitting sources. To deduce the true experimental CP, the raw data were corrected for background, instrumental resolution (partial, limited to stripping off the low energy tail of the profile), detector efficiency, sample absorption, Compton cross-section, multiple scattering corrections, etc. (Felsteiner et al., 1974; Timms, 1989; Cooper et al., 2004; Ahuja, 2010). The corrected profile was then converted into momentum scale and normalized to free atom Compton profile area (Biggs et al., 1975) of 31.86 e– in the momentum range 0–7 a.u.
3. Computational details To compute the theoretical momentum densities, energy bands and DOS, we have used CRYSTAL09 code (Dovesi et al., 2005, 2009) which employs the wave functions generated from the LCAO method. This code includes various schemes, namely (i) HF, (ii) DFT with local density approximation (LDA) and generalized gradient approximation (GGA) and second order GGA (SOGGA) and (iii) hybridization of DFT and HF (so called B3LYP). In DFT–GGA, we have used the exchange and correlation potentials as prescribed by Becke (1988) and Perdew and Wang (1992), respectively. In the B3LYP (Becke three parameter) method, the correlation functional is a combination of functionals prescribed by Lee et al. (1988) and Vosko et al. (1980). The SO-GGA adopts the exchange functional as suggested by Zhao and Truhlar (2008) and the correlation functional of Perdew et al. (1996). These approximations in electronic structure calculations differ in the
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form of monoelectronic Hamiltonian operators. In the HF calculations, the exact interaction between electrons is incorporated while the correlation effects are neglected. In the DFT–GGA/SOGGA calculations, both the exchange and correlation effects are treated approximately. The GGA accounts for variation in the density functional while the SO-GGA functional restores the gradient expansion for both exchange and correlation through second order (Zhao and Truhlar, 2008). In the absence of all-electron basis sets of Ta, we have used large core pseudopotential (PP) basis sets for Ta (Hay and Wadt, 1985). The all-electron basis sets for B were taken from http:// www.tcm.phy.cam.ac.uk/mdt26/basis_sets. The PP facilitates the expansion of electronic wave function using a much smaller number of plane wave basis sets. The basis sets were energy optimized, using the BILLY software (Dovesi et al., 2009) until the default tolerance values were achieved. The integration in the reciprocal space has been carried out on a grid of 845 k-points in the irreducible Brillouin zone (BZ). The lattice parameters for hexagonal structure were taken to be a ¼b¼3.062 and c¼ 3.301 A˚ (Paduani, 2003). Besides LCAO calculations, we have also derived the energy bands, DOS and Fermi surfaces of TaB2 using WIEN2k code (Blaha et al., 1990, 2011) which is based on the FP-LAPW method. In the FP-LAPW method, the unit cell is divided into two parts: atomic spheres centered on the atomic sites and interstitial regions. In the present calculations, we have used the latest and more accurate exchange and correlation potentials as suggested by Wu and Cohen (2006). The charge density, potential and wave functions were expanded in spherical harmonics with maximum radial expansion lmax ¼10. The convergence criterion for total energy was set to 0.01 m Ry with cut of charge density Gmax ¼12. The product of MT radii (2.42 a.u. for Ta and 1.69 a.u. for B) and Kmax (magnitude of the largest reciprocal lattice vector) which controls the convergence of basis sets was kept equal to 7.
4. Results and discussion In Figs. 1 and 2, the energy bands and DOS of TaB2 computed using LCAO–DFT–GGA and FP-LAPW calculations are presented. Since the energy bands and DOS computed using other prescriptions of DFT (like SO-GGA, B3LYP) as embodied in LCAO scheme show almost similar topology, only energy bands and DOS corresponding to DFT–GGA are shown. Except some fine structure and the energy values, the energy bands and DOS (Figs. 1 and 2) are found to be in reasonable agreement with reported calculations (Zhao and Wang, 2009; Ivanovskii, 2003; Paduani, 2003). The energy bands corresponding to B-2p and Ta-5d states cross the Fermi energy (EF) level at different branches of BZ, which confirm its metallic character. It is seen that the bands in the energy range from –10.15 to –2.45 eV (Fig. 1) are originated due to Ta-5d and B-2s states. The bands in energy range from –2.45 to 5.10 eV are due to the Ta-5d and B-2p states, with a small contribution of B-2s states. In the unoccupied region, the upper most group of bands in the energy range 5.10– 12.0 eV is due to strong hybridization between Ta-5d and B-2p states with a small contribution of B-2s states. In Fig. 3, differences between theoretical and experimental CPs of TaB2 are presented. Before taking the difference, all the theoretical profiles were convoluted with the experimental resolution function of 0.34 a.u. (Gaussian, full width at half maximum) and were normalized to the free atom CP area as discussed earlier. In the high momentum region (pz 44 a.u.), all the theoretical profiles are in good agreement with the experimental data. This is expected due to a major contribution of the core electrons in this region, which are well defined by the free atom wave functions. A reasonable agreement between the theoretical and experimental momentum densities confirms the accuracy of the data correction. In the low
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Fig. 1. Band structure (E–k relation) and DOS of TaB2 along high symmetry directions of the first BZ using LCAO–DFT–GGA approach. The Fermi energy (EF) is taken at zero reference level. The positions of featured k-points are ! (0,0,0), M(1/2,0,0), K(1/3,1/3,0), A(0,0,1/2), L(1/2,0,1/2) and H(1/3,1/3,1/2).
Fig. 2. Same as Fig. 1 expect the theory, which is FP-LAPW.
Fig. 3. Difference between the isotropic experimental and the convoluted theoretical Compton profiles for TaB2 using HF, B3LYP and DFT–GGA and –SOGGA schemes within the LCAO model. The experimental error ( 7 s) is shown at few points.
momentum region (pz o3 a.u.), differences between theoretical and experimental CPs are evident (Fig. 3). It may also be noted that all the DFT based profiles and the hybrid calculations (B3LYP) show similar type of deviations from the experimental data. On the basis
of w2 fitting, it is seen that GGA based CP shows a better agreement with our measured profile than other theoretical profiles. In the vicinity of Compton peak, large differences between the HF based profile and the experimental CP show importance of inclusion of correlation effects in the electronic structure calculations. In the low momentum region, deviations between experimental and DFT based theoretical data may be attributed to incompleteness of basis sets, inapplicability of PP concept for heavier atom like Ta and noninclusion of relativistic effects in the present LCAO calculations. In Fig. 4, we have plotted the anisotropies in the unconvoluted theoretical CPs calculated for different principal directions within the LCAO based HF, DFT–GGA/SO-GGA and B3LYP approaches. It is seen that the general trend of oscillations in the anisotropies (J110–J001, J100–J001 and J100–J110) is almost similar. The anisotropies in the CPs can also be explained in terms of energy bands. For example, a negative amplitude in the anisotropies with respect to J001 (J110–J001 and J100–J001) near pz ¼0 a.u. arises due to higher momentum densities along GA[001] direction (because of large number of allowed states along the GA branch and degenerate states at A point near EF) in comparison to those along GK[110] and GM[100] directions. The positive oscillations in the anisotropies in J100–J110 near pz ¼0.3 a.u. is due to higher momentum densities after the GK/2 distances, which is attributed to cross-over of EF level by bands at the middle of GK distances and degeneracy at K point (close to EF). Other positive and negative oscillations in anisotropies can also be explained in terms of the degenerate and the allowed states of energy bands in the vicinity of EF.
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Fig. 4. Anisotropy in the unconvoluted theoretical directional Compton profiles of TaB2 derived from different schemes of LCAO–DFT. Solid lines are drawn to guide eyes.
To study the relative nature of bonding in isoelectronic TaB2 and NbB2, we have scaled our experimental profile on equalvalence-electron-density (EVED). The EVED profile, plotted as pF.J(pz) v/s pz/pF, is also known as profile on pz/pF scale (Podloucky and Redinger, 1984; Reed and Eisenberger, 1972). Such profiles are very useful in predicting the nature of bonding in isovalent compounds wherein the covalent bonding results from the sharing of electrons, which increases the localization of charge in the direction of bonding. Therefore, the materials with more covalent bonding show sharper EVED profile than those with less covalent character. In case of TaB2 and NbB2, the Fermi momenta (pF) were found to be 1.22 and 1.21 a.u., respectively. To obtain the EVED profile, convoluted free atom core profile (Biggs et al., 1975) of both the borides was subtracted from the corresponding absolute experimental profiles. The area of both the normalized EVED profiles was set equal to 5.5 e–, which is equal to area of CP of valence electrons. To deduce the EVED profile of NbB2, the experimental CP data were taken from Bhamu (2012). Our experimental EVED profiles, shown in Fig. 5, depict more sharpness of TaB2 data than that of NbB2. This leads to more localization of electrons (in momentum space) and more covalent nature of TaB2 than NbB2. Further, the sharpness in EVED profile of TaB2 shows that the 5d electrons of Ta in TaB2 are more localized (delocalized) in momentum (real) space than the 4d electrons of Nb in NbB2. Our conclusions based on EVED profiles are in tune with the theoretical band ionicity results (52.1% for NbB2 and 50.7 for TaB2) reported by Chen et al. (2001). Now, we discuss the real space CPs of TaB2 deduced from the experimental and free electron model profiles. The real space CP, so called the B(z) function, is the one dimensional Fourier transformation of CP (Mueller, 1977; Podloucky and Redinger, 1984). The B(z) function for the unit area of parabolic free electron profile can be given as 3 sinðpF zÞ BðzÞ ¼ 2 2 cosðpF zÞ : ð3Þ pF z pF z Eq. 3 dictates that when pFz ¼4.493 a.u., in the first instant, the value of B(z) becomes zero. From Fig. 6, we find that experimental B(z) curve cuts the pz-axis at z ¼3.7170.02 a.u. which shows the experimental value of pF as 1.2170.02 a.u. The present value of experimental pF is found to be very close to theoretical pF value (1.22 a.u.) obtained from the free electron hypothesis. Further, as evident from Fig. 6, the first cut in B(z) function of experiment is very close to that of free electron profile. A close agreement between the free electron based pF and that deduced from the
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Fig. 5. Experimental equal-valence-electron-density (EVED) profiles of isoelectronic NbB2 and TaB2. The experimental EVED profile for NbB2 is derived from the CP data reported by Bhamu (2012).
Fig. 6. The real space Compton profiles, B(z) function, of TaB2 obtained from the valence experimental and free electron (FE) Compton profiles. The free electron profile consists of parabolic shape given by Jðpz Þ ¼ ap2z þ b for pz opF and 0 for pz Z pF, where constants a and b are the function of a number of electrons and pF.
experimental CP confirms a metal-like character of TaB2. The metallic character of TaB2 allows the mapping of Fermi surfaces that arises from electron ordering phenomena. The three dimensional Fermi surfaces of TaB2 within the framework of FP-LAPW are shown in Fig. 7. The main conclusions derived from Fig. 7 (b–e) are:
(i) Out of all the shown energy bands, the 10th band in the range from –2.07 to 2.33 eV (labeled in Fig. 2) leads to hollow doubly- and singly-necked out structures (Fig. 7b). (ii) The 11th band ranging from –1.25 to 4.69 eV (labeled in Fig. 2) produces the peculiar shape around M point and electron-occupancies near A point (Fig. 7c). (iii) The 12th band (Fig. 2), in the energy range almost similar to the 11th band, leads to bowl-like surface (Fig. 7d) at point A with base towards ! point.
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Fig. 7. (a) Standard BZ for hexagonal structure (space group P6/mmm). Fermi surface (FS) of TaB2 arising from (b) 10th (c) 11th and (d) 12th bands using the FP-LAPW method. In (e), a complete mapping of FS from 10th–12th bands is shown.
(iv) An overall mapping of the Fermi surface, shown in Fig. 7e, clearly depicts the hole- and electron-like states which support metallic behavior of TaB2 as seen from the analysis of B(z) function.
5. Conclusions We have reported the isotropic electron momentum densities of TaB2 using 20 Ci 137Cs g-ray source. The experimental Compton profile shows a better agreement with the LCAO–DFT–GGA based profile than HF, SOGGA and B3LYP prescriptions. The band structure, density of states and Fermi surface topology for TaB2 as computed using LCAO–DFT and FP-LAPW schemes show a metallic character of TaB2. Equal-valence-electron-density profiles of TaB2 and NbB2 show more covalent character in TaB2 than that in NbB2. On the basis of EVED profiles, it is seen that 5d electrons of Ta in the TaB2 environment are more extended in real space than 4d electrons of Nb in NbB2.
Acknowledgments We are thankful to Prof. Dovesi and Prof. Blaha for providing the CRYSTAL09 and Wien2k codes, respectively. The authors are also grateful to BRNS (DAE), Mumbai for financial support.
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