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Crystal orbital schemes for solids. Titanium and vanadium monocarbides and mononitrides
Solid State Communications, Vol. 94, No. 10, pp. 861-866, 1995 Elsevier Science Ltd Printed in Great Britain 0038-1098/95 $9.50+.00 00381098(95)00175-l
TITANIllI\II
CRYSTAL ORBITAL SCHEMES FOR SOLIDS. AND VANADIUM MONOCARBIDES AND M~NONITRIDES iubomir
Benco
Institute of Inorganic Chemistry, Slovak Academy of Sciences, Dibravski cesta 9, SK-84236 Bratislava, Slovakia
(Received 22 December 1994 by A. Zuwadowski) Crystal orbital (CO) schemes are proposed for presenting the chemical bonding in solids. The CO scheme, based on the total DOS curve, shows interactions controlling main features of the electronic structure of a solid compound in a compact form. Its construction must be preceded by a detailed analysis of orbital interactions and bonding proclivities of states resulting from the band structure calculation. The schemes are demonstated for Although the chemical bonding in simple cubic transition metal carbides and nitrides. these simple compounds is well understood the analysis not only shows new features not described so far, but easily proves erroneous conclusions of several recent interpretations.
of the electronic structure calculation‘j, and in books explaining the chemical bonding in solids’ps. MO schemes are either presented in an abstact form or supported by energy level arrangement obtained for r point of the kspace. Both above mentioned approaches, however, have their own drawbacks. The former one is rather arbitrary and the latter one is an extrema.l case not representative of the overall bandstructure. As will be demonstrated below, the genuine interaction scheme for a solid originates from the detailed analysis of states spread over the energy scale (the DOS curve). The crystal orbital method - analog of the MO method developed for periodic systems9 - provides the description of a system in the term of crystal orbitalsl-‘. Instead of a set of energy levels (as in the MO method) we have the DOS curve as a result of the electronic band structure calculation. The states of the system are usually collected into bands (not considering the case of metallic systems with extremely wide bands). After analyzing the band composition, the CO scheme can be constructed which - by a.nalogy with the MO scheme - represents the interaction scheme indicating the splitting of atomic levels into band levels. In graphical representation these are displayed using band maxima or centers of mass. It is useful to realize at this point that the degree of interaction in the CO scheme is not only expressed by the value of the separation of the bonding and antibonding component, but also by the width of the band itself. Compared to the MO diagrams in the DOS representation the bandwidth is a new feature. However, besides the useful information it can also cause problems in interpreting too wide and overlapped bands. The transition metal monocarbides and mononitrides Tic, TiN, VC and VN are good candidates for the in-
1. INTRODUCTION During the last decade an extensive development of methods of theoretical chemistri has been noticeable furnishing the electronic structure of solid state on various levels of sophistication 1- 4. Since the electronic properties of a solid depend on energies of the bands and widths and the gaps between them the band structure represents the classical way of presenting results. Though valuable for physicists, the bandstructure along some chosen path in Brillouin’s zone, is extremely unappealing and discouraging for chemists. Density-of-states (DOS) curves are much more useful in evaluatiois of the chemical bonding. The total DOS curve expresses the partition of electronic states of the system into bands, it does not, however, give the information about the chemical bonding. On t,he other hand the projection of the total DOS onto the elements of the basis set used (atomic orbitals) is a. powerful tool showing whether certain atomic orbital (AO) takes place in orbital interactions, whether it creates bonding states thus contributing to the overall stability of the system, whether in the process of creating bonds it plays an antibonding or a nonbonding role. The method of molecular orbitals (MO) is well established in describing the chemical bonding of molecules. The basis of localized A0 used to describe the system in the form of the linear combination (LCAO) complies well with the way chemists imagine the directional character of covalent bonds. The results outlined in the form of the MO scheme are well understood and widespread. As well, refering lo the MO scheme is a common practice in the investigation of the chemical bonding in solids. Authors use it to interprete experimental dataS, results
861
CRYSTAL
862
ORBITAL
troduction of the CO scheme. Their NaCl-type structure is relatively simple, having a high degree of symmetry. Every metal atom is octahedrally surrounded by six nonmetal atoms and vice versa. Moreover, due to the unusual combination ol’ their physical properties, the transition melal ca.rbidcs and nitrides (TMCN) are compounds of great practica15*‘0-‘5 and theoretical interests~*6-3* . Having rocksalt slructure typical for ionic compounds they exhibit not, only metallic properties (conductivity or even superconductivity), but properties typical for strong covalent bonding (ultrahardness, high melting temperatures), as well. The overlap of the p and d band guarantees a non-zero density- of-states at the Fermi level situated within them, thus causing the metallic character of these ma.terials. As pointed out by several authors 27*20,33lhc properlies typica.I for covalent compounds are closely tied with empty places in the nonmetal sublattice. The vacancies enable the local strengthening of d-d connections via multicentred bonds, thus giving rise to the increasing stability of the system. Random oriented inlercollllecl,iolis of the multicentred d-d bonds were compared lo the effect of the whisker causing the increased fracture toughness of these materials33. The interesting physical properties of TMCN have attracted the interest of theoreticians for decades. Applying the large variety of methods of the theoretical physics and chemistry they have reported a huge amount of results for this relatively small group of compounds6J6-32. Within classical solid state concepts the most of stoichiometric systems arc covered and the attention has been currently focusing on nonstoichiometric systems31-33 (structu~_~ containing vacancies) and more complex structures 3’. In this paper we turn back to summarize the chemical bonding in the stoichio-
IO
TITANIUM
T
NITRIDE
TITANIUM
SCHEMES
Vol. 94, No. 10
metric first row TMCN. Constructing the CO schemes we compare and contrast their electronic structures in a concise and compact form. On one hand the schemes unify results obtained so far OII the same base, on the other hand they can serve as a good starting point for investigation of more complex structures ol this important class of compounds. In conclusion we point at several misinterpretations appearing recently in the literature thus demonstrating the usefulness of the global look at the chemical bonding in solids via the crystal orbital schemes.
2. CRYSTAL
ORBITAL
SCHEMES
Figs. I and 2 show the CO schemes for stoichiometric TX, TiN, VC, and VN on the same energy scale. To construct the schemes besides the total DOS three other quantities are necessary: positions of interacting atomic levels, projected DOS, and over1a.p populations. The total DOS of the EHT quality35 are obtained using parameters collected by AlvarezX for experimental interatomic separations (a = 4.327, 4.242, 4.160, and 4.140A, respectively) sampling 5G k-points regularly spaced within the irreducible wedge of the Brillouin’s zone. r Ionization energies are used to indicate positions of interacting atomic levels3G. The projected DOS (not reported because of the wealth of data available in the literature6@-32) prove the dominant role of nonmetal p and metal dorbitals in the chemical bonding. This is because of similar atomic energy levels (Figs. 1 and 2) and size of relevant orbitals ensuring effective overlaps. Four types of orbital interactions occur: &, pd,, dd,, and d&. Contributions of all other orbitals to the valence region are negligible. The overtap populations, defined a~ C cicjSij, are used to characterize bonds between two atoms. c+ and cj are LCAO coefficients and Sij represents the matrix element of the overlap between two
CARBIDE
5
10
.I%
w
FOR SOLIDS
VANADIUM
0
T t -5
-10
-15
-20
Fig. 1. The crystal orbital schemes for TiC and TiN. Atomic levels are taken from Ref. 36. Full horizontal lines within DOS indicate the Fermi level positions. The center of mass of broad upper band is estimated. The doubled d band line for TIN symbolizes two indistinguished bands.
NITRIDE
VANADIUM
CARBIDE
5
Id
0
-5
-10
-15
-20
Fig. 2. The crystal orbital schemes for VC and VN. Two components within the main d band of VN are indicated by the doubled band line.
CRYSTAL
Vol. 94, No. 10
ORBlTAL
SCHEMEIS FOR SOLIDS
863
lr* ____ f\ n
*
/
\
+
-20
-15
-10
-5
0
5
E (ev)
Fig. 3. Crystal orbital overlap populations for TX. TiN (full line) COOP curve shows bonding characteristics of the pd interaction between nearest neighbours. Ti-Ti (dotted line) curve characterizes metal-to-metal interaction. The vertical line shows the Fermi level position (EF).
atomic orbitals i and j. They can be evaluated either or in numerical form3’. Fig. 3 shows in graphicaWzs an example of COOP curves for TiC. The COOP quantity (Crystal Orbital Overlap Population) introduced by Hoffmann et &a*26 is an overlap population-weighted density of states. It represents a useful tooI indicating bonding proclivities of states comprised in investigated bands. In Fig. 3 two COOP curves are displayed, TCN and Ti-Ti. The first one characterizes the pd interaction of the nearest neighbours (all contributions except the nonmetal p and the metal d admixtures to the DOS are negligible) and the second one represents the dd bonding between two transition met& The COOP curves are positive for the bonding and negative for the antibonding energy regions. For nonbonding states the’ COOP curve approaches zeros. Pig. 3 shows bonding characteristics of the TIC valence states demonstrating e.g. clear pd nonbonding and dd antibonding character of states within the subband situated at e-9 eV. The evaluation of the COOP curves for both, carbides and nitrides indicates that two interaction schemes are taking place at the same time, 1~11 and dd (Fig. 4). In fact the distribution of the energy states is controlled by the pd interaction because of the smaller metal-to-nonmetal interatomic distace (larger overlap of AO’s) compared to the dd interaction. Therefore only interaction lines of the pd interaction are indicated in the CO schemes. As shown in Fig. 4 the main bonding band (p band) consists of both, u and z pd bonding states. The next couple of bands is usually referred to as “d band consiating of nonbonding d states”. The annlysis shows that there are in fact two bands. The first one comprise pd nonbouding states, i.e. both t2, aucl e, symmetry components, but no p states a.dmixture. The second band exhibits strong pd antibonding character. Consisting of only the nonmetal p and the metal ts9 states it represents pd, antibonding states. Though rather overlapped and
Fig. 4. Interaction schemes for the first row TMCN. Only nonmetal p and metal d orbitals participate in the chemical bonding. Dashed lines symbolize band energies (band maxima or centers of mass).
difficult to distinguish in nitrides these two bands are well separated in carbides (Figs. 1 and 2). In numerous papers reporting DOS diagrams for carbides6*31Js*40 this separation is not described, because it becomes apparent only when complete interaction scheme is constructed. The last’band is again pd antibonding. It consists of only the nonmetal p and the metal e, states thus representing pd,’ antibonding states. The dd interaction scheme (Fig. 4, right) shares most features of the pd scheme because d orbitals participate in both, pd and dd interactious at the same time, positions of bands being fixed by much stronger pd interaction. Fig. 5 (left) shows an example of the cf& orbital interaction where the pd# interaction plays the dominant role. The only exception is the dd, interaction (Fig. 5, right) in which the nonmetal p orbitals cannot participate for symmetry reasons. The states originating from this interaction are not concentrated giving rise to their own peak. The appearance of the “net dd states” is indicated by a rectangular in the interaction scheme, and by a hatched area in the CO schemes. Note that no net dd states are indicated for carbides (Figs. 1 and 2)! The CO schemes display all relevant features influencing the distribution of states in solids. Supplemented with few additional data (A0 overlaps) they represent the complete picture of the chemical bonding. The example of TMCN shows the interaction pattern which rather differs for carbides and nitrides, thus demonstrating nonvalidity of the rigid band model for TM compounds with different nonmetal atoms. Carbides remind a homonuclear system. The strong pd interaction results in large stabilization shift of levels, the large bandwidth of the p band (=4eV), the d band splitting into the nonbonding and antibonding component, and pd versus dd competition shifted towards the former one. Contrary, nitrides represent typical heteronuclear system. The pd interaction is much weaker than in carbides. The bandwidth of the main bonding band is only =3eV, and the components of the d band cannot be discriminated. The weaker pd interaction favours strengthening of the dd interaction resulting into the appearence of the net dd states. These states have been shown both, to play an experimentally12*13 and theoretically28J1-ss*ss
864
CRYSTAL ORBITAL SCHEMES FOR SOLIDS
important role in rocksalt structures containing vacancies, and also in complex structures of TM compounds34. Because the understanding of the chemical bonding of this class of compounds is only being developed the construction of the CO schemes for such systems would help the process considerably. Recently Hiiglund et al. have developed a theory of bonding in transition-metal carbides and nitrides40. Within the application of the rigid band model they have evaluated bonding energies A&.,,d of 3d-, 4d-, and Sd-transition-metal carbides and nitrides. Based on this quantity plotted versus the position of 114 in the d series their conclusion is that gradual population of bonding, antibonding, and nonbonding electronic states occurs. Striking is the fact tl1a.t this interpretation differs. from the usual picture of the chemical bonding in which bonding states are stabilized (lowered), antibonding states are destabilized (shifted to higher energies) and nonbonding states are situated within splitted bonding and antibonding counterparts. Hgglund et al. do not realize that as soon as more than one pair of interacting bodies exists the using of the global terms “bonding, antibonding, and nonbonding” becomes inadequate. The analysis above shows that two kinds of interaction, pd and dd, play the role in the chemical bonding of TMCN. This means that the bonding properties of a particular state can be described within both interaction schemes. The example of the subband at x-9 eV in TiC (Fig. 3) demonstrates that the same state can reveal different bonding characteristics for different interactions. The states described by Higlund et aI.‘O as “bonding states” are therefore better described as pd and dd bonding. Following arc the states which exhibit the pd nonbonding and dd antibonding properties (c/ Fig. 3), contrary to the “antibondiug states” according to Higlund’s interpretation. And finally the upper states of the valence region which are filled in corn; pounds in the end of the TM series we describe as pd antibonding and dd antibonding, while based on calculated bonding energies theese have been previously characterized as “nonbonding statesnqo. It must be stressed that this interaction scheme is valid only for the first row TMCN. Because of the non negligible role of the TM s and p orbitals the interaction scheme for the second and third series of TM compounds differs from the first one4r. New bonding states which appear in the valence region causes that bonding energies for the second and the third TMCN can increase a.gain iu the end of series (Ref. 40, Fig. 7). In the first series of TMCN, contrary, all bonding states are filled in TiC and ScN and (supposing that the lattice parameter is kept constant) no reason exists for the bonding energy increase when going from CrN to MnN, as presented in Fig. 7 of Ref.40. The bonding energy curve should smoothly decrease throughout the whole series of nitrides. The in-
Vol. 94, No. 10
Fig. 5. An example of dd interactions in rocksalt structures showing net dd, interaction (right) where nonmetal p orbitals cannot participate for symmetry reasons. d orbitals pointing towards nonmetal atoms allow both, dd, and pd, interactions (left). Full circles indicate nonmetal atom positions.
creased value of the bonding energy for MnN is probably due to a cumulative effect of the va.rious errors inherent in electron-structure calculations. Didziulis et aL5 have recently investigated the electronic structure and bonding in TiC and TiN exploring valence level photoelectron spectroscopies using both, X-rays and synchrotron radiation Fitting the band intensity changes in the valence region they have shown in Fig. 5 (Ref. 5) increased intensities for peaks A and C (s band and the lower part of the p band of Tic). Similar data are presented for TiN. lu conclusion Didziulis at al. have stressed that this is due to the participation of Ti 4s and 4p levels in covalent bouds remarkiug that many calculations ignore 4s and 4p bonding interactions (no citation is given). In spite of the wealth of results available in literature, during last two decades gained for TiC and TiN by reliable metl~ods6~20~3*~32, they support their interpretation by the MO scheme dated 1970. Making no restrictions on intera.ctions in the valence region modern solid state theories1-4 furnish the picture of the chemical bonding in TiC and TiN where the admixture of Ti 4s and 4p states into the p band is negligible and there is practically no admixture into the s band. The CO schemes presented above provide the idea how the energy separation of interacting levels influences the chemical bonding (compare schemes for TiC and TiN where the p-d separation reads 0.2 and 2.2 eV, respectively). Another important circumstance is that as soon as one interaction is strong (p-d in Tic) others are negligible. Therefore not only the d-d interaction is insignificant, but to much more extent the Ip(C)&(Ti) interaction with the energy separation of the value 2.5 eV. And what is to be said about the 2s(C)-4s(Ti) interaction where the energy difference is 12.5 eV? The authors should better find out other reasons for increased band intensities with increasing photon energy. Acknowledgement - This work has been supported in part by a research grant No. 4340 of the Grant Agency for Science of the Slovak Republic.
CRYSTAL
Vol. 94, No. 10
ORBITAL
SCHEMES
FOR SOLIDS
REFERENCES 1. M.-H. Whangbo, M. Evain, T. Hungbanks, M. Kertesz, S. Wijeyesekera, C. Wilker, C. Zheng, and R. Hoffmann, Extended Hiickel Molecular, Crystal and Properties Package, QCPE Program No.571, Indiana University (198s). 2. R. Dovesi, C. Pisani, C. Roetti, and M. CausB, CRYSTAL 88: An AL Initio All-Electron LCAO Hartree-Fock Program for Periodic Systems, QCPE Program No.577, Indiana University (1988); R. Dovesi, V.R. Saunders, C. Roetti, CRYSTAL 92, An Ab-Initio Hartree-Fock LCAO Program for Periodic Systems, University of Turin and SERC Daresbury Laboratory (1992). 3. M. Methfessel, Phys. Rev. B38, 1537 (1988); M. Methfessel, C. 0. Rodriguez, and 0. K. Andersen, Pfiys. Rev. B40, 2009 (19S9). 4. P. Blaha, I<. Schwarz, P.I. Sorantin, S.B. Trickey, 59, 399 (1990); P. Comput. Phys. Commun. Blaha, I<. Schwarz, and R. Augustyn, WIEN93, A Full Potential Linearized Augmented Plane Wave Package for Calculating Crystal Properties, Technical University of Vienna (1993). (Improved and updated Unix version of the original copyrighted WIEN-code, which was published by P. Blaha, K. Schwarz, P. Sorantin, and S. B. Trickey, in Comput. Phys. Commun. 59, 399 (1990)). 5. S. V. Didziulis, J. R. Lince, and T. B. Stewart, Inorg. Chem. 33, 1979 (1994). 6. I<. Schwarz, CRC Critical Reviews in Solid State and MateriaJs Sciences 13, 211 (1987); 7. P. A. Cox, The Electronic Structure and Chemistry ofSolids, Oxford University, Oxford, (1987). 8. R. Hoffmann, Solids and Surfaces: A Chemist’+ View ofBonding in Extended Structures, VCH Publishers, Inc., New York, (1988). 9. J. -M. Andre,
J. &em.
Phys.
50, 1536 (1969).
10. L. E. Toth, Transition Metal Carbides trides, Academic, London (1971). 11. V. Schier, H. nius J. Anal. H.-J. Michel, Chem., in the
and
Ni-
-J. Michel, and J. Halbritter, FreseChem. 346, 227 (1993); B. Hornetz, and J. Halbritter, Fresenius J. Anal. press.
12. L. Porte, L. Roux, and J. Hanus, Phys. Rev. B28, 3214 (1993). P. Steiner, 13. H. Hcchst, R. D. Bringans, Wolf, Phys. Rev. B25, 71S3 (19S2).
and T.
14. A. Dunand, H.D. Flack, and K. Yvon, Pitys. Rev. B31, 2299 (1985); F. Kubel, W. Lengauer, I<. Yvon, I<. Knorr, and A. Junod, I%ys. Izev. 1338, 12908 (19SS). 15. J. Rundgren, Y. JoIy, and 4445 (1992); M. Hammar, 10769 (1993).
Y. Gauthier, R. Baudoings-Savois, L. I. Johansson, Phys. Rev. B45, K. L. Hgkansson, L. I. Johansson, and M. Giithelid, Phys. Rev. B47,
16. L. E. Toth, J. Zbasnik, Y. Sato, W. Gardner, in Anisotropy in Single-Cqstal Rejractory Compounds, (Edited by F. W. Valdhiek, and S. A. Mersol), Plenum, New York, (1968); L. E. Toth, M. Ishikawa, and Y. A. Chang, Acta Met. 16, 1183 (1968); J. Zbasnik, ‘and L. E. Toth, Phys. Rev. B8,452 (1973). 17. S. P. Denker, J. Phys. Chem. Solids 25, 1397 (1964); S. P. Denker, J. Less-Common Metals 14, l(196S). 18. J. B. Conklin Jr., D. J. Silversmith, ht. J. @canturn Chem. 2S, 243 (1968). 19. G. E. Hollox, J. Mater. Sci. Eng. 3, 121 (19681969). 20. A. Neckel, P. Rastl, K. Schwarz, and R. EiblerMechtler, Z. Naturforsch. 29a, 107 (1974); A. Neckel, P. Rastl, R. Eibler, P. Weinberger, and K. Schwarz, J. Phys. C: Solid Slate Phys. 9, 579 (1976); K. Schwarz, a.nd N. Riisch, J. Pjty.9. C: Solid State Phys. 9, L433 (1976). 21. J. ,L. Calais, Adv. Phys. 26, 847 (1977). 22. G. Ries, and I-I. Winter, J. Phys. F: Metal Phys. 10, 1 (1980). and L. 23. B. M. Klein, D. A. Papaconstantopoulos, L. Bqyer, Phys. Rev. B22, 1946 (1980). 24. J. Klima, Czech. J. Phys. B30, 905 (19SO). 25. A. Neckel, Int. J. Quantum Chem. XXII, 1317 (1983). and R. Hoffmann, Organome26. S. D. Wijeybekera, tallies 3, 949 (1984). 27. P. Blaha, J. Redinger, and K. Schwarz, Phys. Rev. B31,2316 (1985). P. Weinberger, A. Neckel, R. Zel2s. P. Marksteiner, ler, and P. H. Dederichs, Phys. Rev. B33, 812 (1986). 29. W. E. Pickett, B. M. Klein, and R. Zeller, Phys. Rev. B34, 2517 (19S6). 30. P. Bfaha, and I<. Schwarz, Phys. Rev. B36, 1420 (1987). 31. J. Redinger, R. Eibler, P. Herzig, A. Neckel, R. Podloucky, and E. Wimmer, J. Phys. Chem. Solids 46,383 (1985); 47, 387 (19S6). 32. P. Herzig, J. Redinger, R. Eibler, and A. Neckel, J. Solid State Chem. 70, 281 (1987). 33. i. Benco, J. Solid. State Chem. 110, 58 (1994). 34. R. Eibler, J. Phys.: Condens. Il4atter 5, 5621 (1993). 35. R. Hoffmann, J. Chem. Phys. 39, 1397 (1963); R. Hoffmann, and W. N. Lipscomb, J. Chem. Phys. 36, 2179 (1962); 2189 (1962). 36. S. Alvarez, Ta6le of Parameters for Eztended f’*ckel Calculations, Barcelona (1985), unpublished. 37. R. Dovesi, C. Roetti, C. Freyria-Fava, E. Aprl, V. R. Saunders, and N. M. Harrison, Phil. Trans. R. Sot. Lond. A341,203 (1992); P. D’Arco, G. Sandrone, R. Dovesi, R. Orlando, and V. R. Saunders, Phys. Chem. Minerals 20, 407 (1993).
865
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38. V. Ozoli& and J. H&+td, Phys. Rev. B48,5069 (1993). 39. i. Benco, J. Solid Stale Claem. 111, 440 (1994).
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40. J. Hkigiund, A. F.Guillermet, G. Grimvall, and M. Kkling, Phys. Rev. B48, 11685 (1993). 41. i. Benco, unpublished results.
TITANIllI\II
CRYSTAL ORBITAL SCHEMES FOR SOLIDS. AND VANADIUM MONOCARBIDES AND M~NONITRIDES iubomir
Benco
Institute of Inorganic Chemistry, Slovak Academy of Sciences, Dibravski cesta 9, SK-84236 Bratislava, Slovakia
(Received 22 December 1994 by A. Zuwadowski) Crystal orbital (CO) schemes are proposed for presenting the chemical bonding in solids. The CO scheme, based on the total DOS curve, shows interactions controlling main features of the electronic structure of a solid compound in a compact form. Its construction must be preceded by a detailed analysis of orbital interactions and bonding proclivities of states resulting from the band structure calculation. The schemes are demonstated for Although the chemical bonding in simple cubic transition metal carbides and nitrides. these simple compounds is well understood the analysis not only shows new features not described so far, but easily proves erroneous conclusions of several recent interpretations.
of the electronic structure calculation‘j, and in books explaining the chemical bonding in solids’ps. MO schemes are either presented in an abstact form or supported by energy level arrangement obtained for r point of the kspace. Both above mentioned approaches, however, have their own drawbacks. The former one is rather arbitrary and the latter one is an extrema.l case not representative of the overall bandstructure. As will be demonstrated below, the genuine interaction scheme for a solid originates from the detailed analysis of states spread over the energy scale (the DOS curve). The crystal orbital method - analog of the MO method developed for periodic systems9 - provides the description of a system in the term of crystal orbitalsl-‘. Instead of a set of energy levels (as in the MO method) we have the DOS curve as a result of the electronic band structure calculation. The states of the system are usually collected into bands (not considering the case of metallic systems with extremely wide bands). After analyzing the band composition, the CO scheme can be constructed which - by a.nalogy with the MO scheme - represents the interaction scheme indicating the splitting of atomic levels into band levels. In graphical representation these are displayed using band maxima or centers of mass. It is useful to realize at this point that the degree of interaction in the CO scheme is not only expressed by the value of the separation of the bonding and antibonding component, but also by the width of the band itself. Compared to the MO diagrams in the DOS representation the bandwidth is a new feature. However, besides the useful information it can also cause problems in interpreting too wide and overlapped bands. The transition metal monocarbides and mononitrides Tic, TiN, VC and VN are good candidates for the in-
1. INTRODUCTION During the last decade an extensive development of methods of theoretical chemistri has been noticeable furnishing the electronic structure of solid state on various levels of sophistication 1- 4. Since the electronic properties of a solid depend on energies of the bands and widths and the gaps between them the band structure represents the classical way of presenting results. Though valuable for physicists, the bandstructure along some chosen path in Brillouin’s zone, is extremely unappealing and discouraging for chemists. Density-of-states (DOS) curves are much more useful in evaluatiois of the chemical bonding. The total DOS curve expresses the partition of electronic states of the system into bands, it does not, however, give the information about the chemical bonding. On t,he other hand the projection of the total DOS onto the elements of the basis set used (atomic orbitals) is a. powerful tool showing whether certain atomic orbital (AO) takes place in orbital interactions, whether it creates bonding states thus contributing to the overall stability of the system, whether in the process of creating bonds it plays an antibonding or a nonbonding role. The method of molecular orbitals (MO) is well established in describing the chemical bonding of molecules. The basis of localized A0 used to describe the system in the form of the linear combination (LCAO) complies well with the way chemists imagine the directional character of covalent bonds. The results outlined in the form of the MO scheme are well understood and widespread. As well, refering lo the MO scheme is a common practice in the investigation of the chemical bonding in solids. Authors use it to interprete experimental dataS, results
861
CRYSTAL
862
ORBITAL
troduction of the CO scheme. Their NaCl-type structure is relatively simple, having a high degree of symmetry. Every metal atom is octahedrally surrounded by six nonmetal atoms and vice versa. Moreover, due to the unusual combination ol’ their physical properties, the transition melal ca.rbidcs and nitrides (TMCN) are compounds of great practica15*‘0-‘5 and theoretical interests~*6-3* . Having rocksalt slructure typical for ionic compounds they exhibit not, only metallic properties (conductivity or even superconductivity), but properties typical for strong covalent bonding (ultrahardness, high melting temperatures), as well. The overlap of the p and d band guarantees a non-zero density- of-states at the Fermi level situated within them, thus causing the metallic character of these ma.terials. As pointed out by several authors 27*20,33lhc properlies typica.I for covalent compounds are closely tied with empty places in the nonmetal sublattice. The vacancies enable the local strengthening of d-d connections via multicentred bonds, thus giving rise to the increasing stability of the system. Random oriented inlercollllecl,iolis of the multicentred d-d bonds were compared lo the effect of the whisker causing the increased fracture toughness of these materials33. The interesting physical properties of TMCN have attracted the interest of theoreticians for decades. Applying the large variety of methods of the theoretical physics and chemistry they have reported a huge amount of results for this relatively small group of compounds6J6-32. Within classical solid state concepts the most of stoichiometric systems arc covered and the attention has been currently focusing on nonstoichiometric systems31-33 (structu~_~ containing vacancies) and more complex structures 3’. In this paper we turn back to summarize the chemical bonding in the stoichio-
IO
TITANIUM
T
NITRIDE
TITANIUM
SCHEMES
Vol. 94, No. 10
metric first row TMCN. Constructing the CO schemes we compare and contrast their electronic structures in a concise and compact form. On one hand the schemes unify results obtained so far OII the same base, on the other hand they can serve as a good starting point for investigation of more complex structures ol this important class of compounds. In conclusion we point at several misinterpretations appearing recently in the literature thus demonstrating the usefulness of the global look at the chemical bonding in solids via the crystal orbital schemes.
2. CRYSTAL
ORBITAL
SCHEMES
Figs. I and 2 show the CO schemes for stoichiometric TX, TiN, VC, and VN on the same energy scale. To construct the schemes besides the total DOS three other quantities are necessary: positions of interacting atomic levels, projected DOS, and over1a.p populations. The total DOS of the EHT quality35 are obtained using parameters collected by AlvarezX for experimental interatomic separations (a = 4.327, 4.242, 4.160, and 4.140A, respectively) sampling 5G k-points regularly spaced within the irreducible wedge of the Brillouin’s zone. r Ionization energies are used to indicate positions of interacting atomic levels3G. The projected DOS (not reported because of the wealth of data available in the literature6@-32) prove the dominant role of nonmetal p and metal dorbitals in the chemical bonding. This is because of similar atomic energy levels (Figs. 1 and 2) and size of relevant orbitals ensuring effective overlaps. Four types of orbital interactions occur: &, pd,, dd,, and d&. Contributions of all other orbitals to the valence region are negligible. The overtap populations, defined a~ C cicjSij, are used to characterize bonds between two atoms. c+ and cj are LCAO coefficients and Sij represents the matrix element of the overlap between two
CARBIDE
5
10
.I%
w
FOR SOLIDS
VANADIUM
0
T t -5
-10
-15
-20
Fig. 1. The crystal orbital schemes for TiC and TiN. Atomic levels are taken from Ref. 36. Full horizontal lines within DOS indicate the Fermi level positions. The center of mass of broad upper band is estimated. The doubled d band line for TIN symbolizes two indistinguished bands.
NITRIDE
VANADIUM
CARBIDE
5
Id
0
-5
-10
-15
-20
Fig. 2. The crystal orbital schemes for VC and VN. Two components within the main d band of VN are indicated by the doubled band line.
CRYSTAL
Vol. 94, No. 10
ORBlTAL
SCHEMEIS FOR SOLIDS
863
lr* ____ f\ n
*
/
\
+
-20
-15
-10
-5
0
5
E (ev)
Fig. 3. Crystal orbital overlap populations for TX. TiN (full line) COOP curve shows bonding characteristics of the pd interaction between nearest neighbours. Ti-Ti (dotted line) curve characterizes metal-to-metal interaction. The vertical line shows the Fermi level position (EF).
atomic orbitals i and j. They can be evaluated either or in numerical form3’. Fig. 3 shows in graphicaWzs an example of COOP curves for TiC. The COOP quantity (Crystal Orbital Overlap Population) introduced by Hoffmann et &a*26 is an overlap population-weighted density of states. It represents a useful tooI indicating bonding proclivities of states comprised in investigated bands. In Fig. 3 two COOP curves are displayed, TCN and Ti-Ti. The first one characterizes the pd interaction of the nearest neighbours (all contributions except the nonmetal p and the metal d admixtures to the DOS are negligible) and the second one represents the dd bonding between two transition met& The COOP curves are positive for the bonding and negative for the antibonding energy regions. For nonbonding states the’ COOP curve approaches zeros. Pig. 3 shows bonding characteristics of the TIC valence states demonstrating e.g. clear pd nonbonding and dd antibonding character of states within the subband situated at e-9 eV. The evaluation of the COOP curves for both, carbides and nitrides indicates that two interaction schemes are taking place at the same time, 1~11 and dd (Fig. 4). In fact the distribution of the energy states is controlled by the pd interaction because of the smaller metal-to-nonmetal interatomic distace (larger overlap of AO’s) compared to the dd interaction. Therefore only interaction lines of the pd interaction are indicated in the CO schemes. As shown in Fig. 4 the main bonding band (p band) consists of both, u and z pd bonding states. The next couple of bands is usually referred to as “d band consiating of nonbonding d states”. The annlysis shows that there are in fact two bands. The first one comprise pd nonbouding states, i.e. both t2, aucl e, symmetry components, but no p states a.dmixture. The second band exhibits strong pd antibonding character. Consisting of only the nonmetal p and the metal ts9 states it represents pd, antibonding states. Though rather overlapped and
Fig. 4. Interaction schemes for the first row TMCN. Only nonmetal p and metal d orbitals participate in the chemical bonding. Dashed lines symbolize band energies (band maxima or centers of mass).
difficult to distinguish in nitrides these two bands are well separated in carbides (Figs. 1 and 2). In numerous papers reporting DOS diagrams for carbides6*31Js*40 this separation is not described, because it becomes apparent only when complete interaction scheme is constructed. The last’band is again pd antibonding. It consists of only the nonmetal p and the metal e, states thus representing pd,’ antibonding states. The dd interaction scheme (Fig. 4, right) shares most features of the pd scheme because d orbitals participate in both, pd and dd interactious at the same time, positions of bands being fixed by much stronger pd interaction. Fig. 5 (left) shows an example of the cf& orbital interaction where the pd# interaction plays the dominant role. The only exception is the dd, interaction (Fig. 5, right) in which the nonmetal p orbitals cannot participate for symmetry reasons. The states originating from this interaction are not concentrated giving rise to their own peak. The appearance of the “net dd states” is indicated by a rectangular in the interaction scheme, and by a hatched area in the CO schemes. Note that no net dd states are indicated for carbides (Figs. 1 and 2)! The CO schemes display all relevant features influencing the distribution of states in solids. Supplemented with few additional data (A0 overlaps) they represent the complete picture of the chemical bonding. The example of TMCN shows the interaction pattern which rather differs for carbides and nitrides, thus demonstrating nonvalidity of the rigid band model for TM compounds with different nonmetal atoms. Carbides remind a homonuclear system. The strong pd interaction results in large stabilization shift of levels, the large bandwidth of the p band (=4eV), the d band splitting into the nonbonding and antibonding component, and pd versus dd competition shifted towards the former one. Contrary, nitrides represent typical heteronuclear system. The pd interaction is much weaker than in carbides. The bandwidth of the main bonding band is only =3eV, and the components of the d band cannot be discriminated. The weaker pd interaction favours strengthening of the dd interaction resulting into the appearence of the net dd states. These states have been shown both, to play an experimentally12*13 and theoretically28J1-ss*ss
864
CRYSTAL ORBITAL SCHEMES FOR SOLIDS
important role in rocksalt structures containing vacancies, and also in complex structures of TM compounds34. Because the understanding of the chemical bonding of this class of compounds is only being developed the construction of the CO schemes for such systems would help the process considerably. Recently Hiiglund et al. have developed a theory of bonding in transition-metal carbides and nitrides40. Within the application of the rigid band model they have evaluated bonding energies A&.,,d of 3d-, 4d-, and Sd-transition-metal carbides and nitrides. Based on this quantity plotted versus the position of 114 in the d series their conclusion is that gradual population of bonding, antibonding, and nonbonding electronic states occurs. Striking is the fact tl1a.t this interpretation differs. from the usual picture of the chemical bonding in which bonding states are stabilized (lowered), antibonding states are destabilized (shifted to higher energies) and nonbonding states are situated within splitted bonding and antibonding counterparts. Hgglund et al. do not realize that as soon as more than one pair of interacting bodies exists the using of the global terms “bonding, antibonding, and nonbonding” becomes inadequate. The analysis above shows that two kinds of interaction, pd and dd, play the role in the chemical bonding of TMCN. This means that the bonding properties of a particular state can be described within both interaction schemes. The example of the subband at x-9 eV in TiC (Fig. 3) demonstrates that the same state can reveal different bonding characteristics for different interactions. The states described by Higlund et aI.‘O as “bonding states” are therefore better described as pd and dd bonding. Following arc the states which exhibit the pd nonbonding and dd antibonding properties (c/ Fig. 3), contrary to the “antibondiug states” according to Higlund’s interpretation. And finally the upper states of the valence region which are filled in corn; pounds in the end of the TM series we describe as pd antibonding and dd antibonding, while based on calculated bonding energies theese have been previously characterized as “nonbonding statesnqo. It must be stressed that this interaction scheme is valid only for the first row TMCN. Because of the non negligible role of the TM s and p orbitals the interaction scheme for the second and third series of TM compounds differs from the first one4r. New bonding states which appear in the valence region causes that bonding energies for the second and the third TMCN can increase a.gain iu the end of series (Ref. 40, Fig. 7). In the first series of TMCN, contrary, all bonding states are filled in TiC and ScN and (supposing that the lattice parameter is kept constant) no reason exists for the bonding energy increase when going from CrN to MnN, as presented in Fig. 7 of Ref.40. The bonding energy curve should smoothly decrease throughout the whole series of nitrides. The in-
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Fig. 5. An example of dd interactions in rocksalt structures showing net dd, interaction (right) where nonmetal p orbitals cannot participate for symmetry reasons. d orbitals pointing towards nonmetal atoms allow both, dd, and pd, interactions (left). Full circles indicate nonmetal atom positions.
creased value of the bonding energy for MnN is probably due to a cumulative effect of the va.rious errors inherent in electron-structure calculations. Didziulis et aL5 have recently investigated the electronic structure and bonding in TiC and TiN exploring valence level photoelectron spectroscopies using both, X-rays and synchrotron radiation Fitting the band intensity changes in the valence region they have shown in Fig. 5 (Ref. 5) increased intensities for peaks A and C (s band and the lower part of the p band of Tic). Similar data are presented for TiN. lu conclusion Didziulis at al. have stressed that this is due to the participation of Ti 4s and 4p levels in covalent bouds remarkiug that many calculations ignore 4s and 4p bonding interactions (no citation is given). In spite of the wealth of results available in literature, during last two decades gained for TiC and TiN by reliable metl~ods6~20~3*~32, they support their interpretation by the MO scheme dated 1970. Making no restrictions on intera.ctions in the valence region modern solid state theories1-4 furnish the picture of the chemical bonding in TiC and TiN where the admixture of Ti 4s and 4p states into the p band is negligible and there is practically no admixture into the s band. The CO schemes presented above provide the idea how the energy separation of interacting levels influences the chemical bonding (compare schemes for TiC and TiN where the p-d separation reads 0.2 and 2.2 eV, respectively). Another important circumstance is that as soon as one interaction is strong (p-d in Tic) others are negligible. Therefore not only the d-d interaction is insignificant, but to much more extent the Ip(C)&(Ti) interaction with the energy separation of the value 2.5 eV. And what is to be said about the 2s(C)-4s(Ti) interaction where the energy difference is 12.5 eV? The authors should better find out other reasons for increased band intensities with increasing photon energy. Acknowledgement - This work has been supported in part by a research grant No. 4340 of the Grant Agency for Science of the Slovak Republic.
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