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Chinese Science Series G: Physics, Mechanics and Astronomy, Volume 38, Issue 2, 2008: 120~125 phys .scichina .com

Valence electronic structures of tantalum carbide (TaC) and tantalum nitride (TaN)

Fan Changzeng ① ② *, Sun Liling ②, Wei Zunjie ①, Ma Mingzhen ③, Liu Liping ③, Zeng Songyan ①, Wang Wenkui ③

① School of Materials Science and Engineering, Harbin University of Technology, Harbin 150001;

② State Key Laboratory of Superconductivity, Institute of Physics, Chinese Academy of Sciences, Beijing 100080;

③ State Key Laboratory of Metastable Material Preparation Technology and Science, Yanshan University, Qinhuangdao 066004

* E-mail:

Date of receipt: May 31, 2007; Acceptance date: July 4, 2007

Projects supported by the National Natural Science Foundation of China (Approval No.: 10702060, 50771090), the National Key Basic Research Development Plan (No.: 2005CB724404), the Yangtze River Scholars and Innovation Team Development Plan (No.: IRT0650)

The valence electronic structures of carbides and nitrides of transition metal tantalum have been calculated by using the empirical electronic theory of solids and molecules The results show that the chemical bonds of the two compounds contain not only covalent components, but also metallic and ionic components In order to quantitatively describe the strength of covalent bonds in two compounds, the calculation results of empirical electron theory are introduced into a valence bond theory model called PVL, and the ionic information of these compounds is obtained Both compounds have low ionic properties, indicating that they have strong covalence However, since the ionic property of TaC is lower than that of TaN, and the number of covalent electron pairs on the strongest bond of TaC is more than that of TaN, it can be inferred that the covalent bond of TaC is stronger than that of TaN The calculation results are consistent with the conclusions inferred by others from the first principle calculation results This is of great significance to systematically establish the relationship between the micro-electronic structure information and the macroscopic physical and mechanical properties of transition metal carbonitrides.

key word



Valence electron structure

Ionic property

The transition metal carbides and nitrides of the fourth and fifth groups have excellent properties such as high melting point, high hardness, metal conductivity and even superconductivity, and are widely studied [1-10] These excellent physical and chemical properties are generally believed to be caused by the coexistence of covalent bonds and metal bonds in these compounds [11,12] Recent research results have found that tantalum carbide (TaC) and tantalum nitride (TaN) are more likely to act as diffusion barriers in Cu coupling than other materials [13~16], so this research has also received extensive attention In theoretical research, many researchers have tried to establish the relationship between crystal structure and macroscopic properties For example, Lo - pez-de-la-Torre et al. [17] studied the elastic properties of TaC by using the first-principles method of plane wave-based pseudopotential scheme; Stampfl et al; Sahnoun et al

The relationship between the occupation state of valence electrons in TaC and TaN and the macroscopic physical properties shows that the covalent bond, ionic bond and metal bond coexist in the two compounds, but the covalent bond has the greatest impact on the macroscopic properties The valence electron analysis shows that the covalent bond of TaC is stronger than that of TaN However, so far, few quantitative analysis results have been published on the valence electronic structures of these two systems, which to some extent limits the establishment of the structure-performance relationship In this paper, we use the empirical electron theory of solids and molecules (EET) to calculate the detailed information of the valence electronic structures of TaC and TaN, and analyze the composition of the chemical bonds of the two compounds through the results of EET In addition, the ionic and covalent properties of the two systems were calculated by the parameters calculated by EET and the PVL valence bond theory model

1 Calculation method and results

The empirical electron theory of solids and molecules is a theory based on inductive principle, which combines the advantages of energy band theory, valence bond theory, Hume-Rothery's electron concentration theory, and a large number of experimental data (such as neutron scattering, electron diffraction, microwave, Mossbauer spectrum effect, spin resonance, positron annihilation technology, Compton scattering, etc.), For example, 78 elements (excluding rare metals) and thousands of compounds and molecules formed from them in the first six cycles, as well as alloy phase diagrams and a series of physical data, have been used to verify and systematize the empirical electron theory In the first approximation, this is a credible theory For the description of this theory, see literature [20] EET is mainly based on the following three basic assumptions and a bond distance difference analysis method [21]:

(i) Assumptions about atomic states in solids and molecules In solids and molecules, each atom is generally composed of two atomic states, namely h state and t state Both states have their own covalent electron number nc, lattice electron number nl and single bond half-distance R (l)

(ii) Assumption of discontinuity state hybridization Under certain conditions, state hybridization is discontinuous If Ct and Ch represent the components of t and h states in hybrid state respectively, in most structures, they can be given by the following formula:

                                             Ch  = 1 Ct ,

Where l, m, n and l ', m', n 'are the covalent and lattice electron numbers of s, p, d of h and t states, respectively When s electron is lattice electron τ= 0, otherwise take τ= 1 . When the valence electrons of the h state are all lattice electrons, formula (2) is not applicable. In this case, we should use:

(iii) Assumptions about bond distance Except for special cases, there are always covalent electron pairs between two similar atoms u and v in the structure The number of covalent electron pairs is expressed by n α The distance between the two atoms is called the covalent bond distance, expressed by Dv In EET, the results of Pauling are directly used to deal with Dv and Ru (l), Rv (l), n α Relationship between [22]:

                                                          Dn(u)v  = Ru (l) + Rv (l) −  βlg nα,      

among β Is the coefficient, determined according to the following conditions:

Where n is the largest n α . 0< ε< 0.050. If the system is metallic, it is generally used first β= 0.0600 nm test calculation, and then according to the calculation results, the final determination is made according to formula (5) β Value; If the system is non-metallic, take β= 0.0710 nm.

(iv) Key distance difference (BLD) method The bond distance difference method is one of the EET methods for calculating the valence electronic structures of solids and molecules This method can calculate the valence electronic structures of crystals and molecules from the known crystal and molecular structures The basic theoretical tool used in BLD calculation is Equation (4) In the same system (molecule and solid), the bond distance between all atoms connected by covalent bonds follows the bond distance formula For any group of atoms, it is one-to-one corresponding to the bond distance of all other atoms with which it forms covalent bonds, and the number of corresponding covalent electron pairs assigned to each bond From this, the ratio of the number of covalent electron pairs on each corresponding bond can be derived according to the difference of each bond distance In addition, considering that the basic structural unit of molecule or crystal is electrically neutral, all covalent electrons contributed by each atom in a structural unit in the system should be completely distributed on all covalent bonds in the structural unit The combination of the two can give the covalent electron pairs on each covalent bond By comparing the experimental values of all covalent bond distances in a structural unit measured by experiments with the corresponding theoretical bond distances calculated by the specified atomic states, we can determine whether the given atomic states of the composition are in line with the objective reality The correct state of atoms in the system can be determined by purposeful exploration It is generally believed that the absolute value of the difference between the experimental bond distance and the theoretical bond distance is less than 0.0050 nm Only non-negligible keys are considered in the specific calculation

Since its publication, this theory has been widely used in different fields and can give reasonable results. For transition metals that are not easy to be treated by other methods, this method can also be easily treated [23~27] So we think it is feasible and credible to study the valence electronic structure of transition metal tantalum carbonitride

When applying EET to a system, we should first know its crystal structure Both tantalum carbide and tantalum nitride belong to NaCl (B1) structure (space group: 225), and their lattice constants a are 0.4450 [28] and 0.4427 nm [29], respectively A cubic structural unit with a volume of 1/8a3 and containing 0.5 Ta atoms and 0.5 C or N atoms is used in the calculation EET calculation is divided into the following three processes:

(i) Calculation of experimental bond distance and equivalent bond number For the above crystal structure and selected structural unit, the analysis results are shown in Table 1

Table 1 Experimental bond distance and equivalent bond of TaC and TaN

D      =   a = 0.22250 nm

D       =      a = 0.31466 nm

D     =      a = 0.31466 nm

D      =     a = 0.38538 nm

D      =   a = 0.22135 nm

D       =      a = 0.31304 nm

D      =      a = 0.31304 nm

D      =     a = 0.38339 nm

IA = 0.5 ×6 ×2 = 6

IB = 0.5 × 12 × 1 = 6

IC = 0.5 × 12 × 1 = 6

ID = 0.5 × 8 ×2 = 8

(ii) According to the experimental bond distance and formula (4), r α Equation:

lg rB  = lg(nB / nA ) = (DnA − DnB ) /  β+ [RTa (l) − Rc(N) (l)] /  β,

lg rC  = lg(nC / nA ) = (DnA − DnC ) /  β− [RTa (l) − Rc(N) (l)] /  β,

lg rD  = lg(nD / nA ) = (DnA − DnD ) /  β.

(6) There are four unknowns in formula~(8), which requires an independent equation to solve. This is the so-called nA equation in EET. Its core idea is that all valence electrons of all atoms in a single cell should be distributed on all covalent bonds:

nIαrα = ncj  = 0.5 × (nC(cN) + nc(Ta) ),

J refers to different kinds of atoms in a structural unit

(iii) In formula (5) β 0.06 nm, scan all possible hybrid state combinations of Ta and C or N in literature [25] through a self-written program, and select Δ Dn α ≤ The combination of 0.0005 nm conditions is the possible ground state The results show that there are 7 and 3 cases of TaC and TaN respectively In the following analysis, we think that the minimum Δ Dn α The hybrid state of TaC and TaN is the actual atomic state (of course, other cases cannot be strictly excluded). The hybrid state combinations of TaC and TaN are shown in Table 2 and Table 3 respectively

Table 2 Hybrid ground state parameters of TaC

lattice constant

a = b = c = 0.4450 nm























Table 3 Hybrid ground state parameters of TaN

lattice constant

a = b = c = 0.4427 nm























According to the parameters listed in Tables 2 and 3 and referring to the composition of h and t states in the hybrid table of literature [25], it can be easily concluded that the valence electron configuration of TaC and TaN are Ta 6s 1.3147, 6p 1.3147, 5d 2.3706 (including lattice electrons), C 2s 1.832, 2p 2.1681 (including lattice electrons) and Ta 6s 1.0, 6p 1.8738, 5d 2.1262 (including lattice electrons), N 2s 1.0, 2p 2.0, respectively Both compounds have lattice electrons, which is the reason why they conduct electricity and show metallicity; 5d electrons of metallic elements and 2p electrons of non-metallic elements constitute the main components of covalent electrons. Their interaction makes chemical bonds covalent; In addition, due to the different number of valence electrons provided by metal Ta and nonmetallic elements when forming chemical bonds, there is bound to be electron transfer or shift between them, which means that the chemical bonds show certain ionic properties In short, the chemical bonds of transition metal carbides TaC and nitrides TaN are composed of complex components, including both the dominant covalent components and metal and ionic components

It is generally believed that the excellent physical and mechanical properties of transition metal nitrides and carbides are mainly attributed to the strong covalent bonds between transition metals and non-metallic elements In order to quantitatively describe the strength of the covalent bond of TaC and TaN, we substituted the parameters of EET into a theoretical model of valence bond called PVL [30], and studied the ionic properties of their chemical bonds According to PVL theory, the ionic property of chemical bonds in crystals can be expressed as

fi = 1 − Eh2/Eg2, (10)

Where fi is ionic; Eg is the average energy gap between the bonding orbital and the anti-bonding orbital, and Eh is the homopolar energy gap Eg is divided into two parts: homopolar Eh and heteropolar C, and

Eg2 = Eh2 + C2 ,                                                                 

Eh = 39.74/d 2.48 ,                                                     

        C = 14.4bexp(ksr0)[ZA/rZB/rB],   

Where d is the key length; B is a constant related to the crystal structure; ZA and ZB are the number of covalent electrons on atoms A and B that form the chemical bond; RA and rB are the covalent radii of A and B atoms, respectively; Exp (− ksr0) is the Thomas Fermi shielding factor; R0 is the half key length; Ks can be calculated by Fermi wave vector kF, and the latter can be calculated by electron density Ne:

                                                                                                                       ks = (4kF/aB)1/2,

                                                                                                                       kF3 = 3π2Ne .

According to the definition of parameters in the PVL model, ZA or ZB is the covalent electron number nc of the corresponding atom in the EET theory, and rA or rB is the single bond half-distance R (l) of the corresponding atom in the EET theory In this way, the ionic properties of TaC and TaN covalent bonds can be calculated by some parameters of EET Some parameters and results in the calculation process are listed in Table 4 The results show that both of them have little ionic property, and TaC has less ionic property and stronger covalence In addition, the covalent electron logarithm of the shortest bond in TaC, that is, the strongest bond, nA=0.4643, while that of the strongest bond in TaN=0.4536 All these indicate that the covalent bond in TaC is stronger than that in TaN This conclusion is consistent with that obtained by first-principle calculation [19]

Table 4 Some parameters used to calculate TaC and TaN







E /eV




















2 Conclusion

In this paper, the valence electronic structures of carbides and nitrides of transition metal tantalum have been obtained by using the empirical electronic theory of solids and molecules The results show that the chemical bonds of these two compounds mainly contain covalent components, as well as some metallic and ionic components At the same time, the ionic properties of the two compounds are calculated based on the parameters obtained by EET and the definition of ionic properties in the PVL model of valence bond theory The results show that both of them have low ionic properties, which indicates that the chemical bonds that form them have strong covalence In comparison, the ionic property of TaC is lower than that of TaN, and since the number of covalent electrons on the strongest bond of the former is higher than that of the latter, it can be inferred that the covalent bond of TaC is stronger

Thank you to Professor Fei Weidong of Harbin Institute of Technology for his introduction of empirical electronic theory and Guo Yong of the General Institute of Iron and Steel Research

A useful discussion by researcher Quan


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