J. Cent. South Univ. Technol. (2011) 18: 1773-1779

DOI: 10.1007/s11771-011-0901-5

First-principles calculation of structural and

thermodynamic properties of titanium boride

LI Yan-feng(李燕峰)^{1, 2}, XU Hui(徐慧)^{1, 2}, XIA Qing-lin(夏庆林)¹, LIU Xiao-liang(刘小良)¹

1. School of Physical Science and Technology, Central South University, Changsha 410083, China;

2. School of Materials Science and Engineering, Central South University, Changsha 410083, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2011

Abstract: The equilibrium lattice parameters, electronic structure, bulk modulus, Debye temperature, heat capacity and Gibbs energy of TiB and TiB₂ were investigated using the pseudopotential plane-wave method based on density functional theory (DFT) and the improved quasi-harmonic Debye method. The results show that the total density of states (DOS) of TiB₂ is mainly provided by the orbit hybridization of Ti-3d and B-2p states, and the total DOS of TiB is mainly provided by the hybrids bond of Ti-3d and B-2p below the Fermi level and Ti—Ti bond up to the Fermi level. The Ti—B hybrid bond in TiB₂ is stronger than that in TiB. Finally, the enthalpy of formation at 0 K, heat capacity and Gibbs free energy of formation at various temperatures were determined. The calculated results are in excellent agreement with the available experimental data.

Key words: electronic structure; Debye model; thermodynamic properties; density functional theory; titanium boride

1 Introduction

Titanium borides, including TiB and TiB₂, are important members of the large family of transition metal–metalloid compounds possessing many unexpected properties and have been investigated extensively in many aspects [1]. Owing to their high strength and durability, the traditional applications of titanium borides are mainly limited to areas such as impact resistant armor, cutting tools and particulate reinforced steel [2]. However, since titanium borides can resist electron migration and prevent diffusion due to the strong interatomic bonding in them, there has been an upwelling of recent interest in further applications. Therefore, an increasing number of applications in thin-film interconnect and diffusion barriers in very large-scale integrated circuits [3] have been found at present. For example, in an important application recently evolved, titanium diboride has been used as a coating on particles in a bioactive glass-ceramic matrix designed for surgical implant materials [4].

The space groups of TiB and TiB₂ are Pnma [5] and P6/mmm [6], respectively. As for their properties such as elastic modules, the result of ultrasonic pulse-echo measurements provides four elastic coefficients and an estimate for the value of C₁₃. While the study based on resonant ultrasound spectroscopy [7] gives a very different set of elastic coefficients, with off-diagonal terms being one order of magnitude smaller than those previously reported, and the data are in noticeably better agreement with the results for polycrystals [8]. In a more recent theoretical study, PEROTTONI et al [9] evaluated the mechanical properties of TiB₂ using the periodic Hartree–Fock (HF) method, but their results disagreed with the experimental data. MILMAN and WARREN [10] even thought that there were no reliable data either on static or shock compression of titanium diboride.

Besides many studies on the elastic properties of titanium boride, there are many studies on the electronic structure of titanium boride. HAN et al [1] studied the transition-metal diborides (TMDB) surfaces by first principle calculation and found that strong covalent    B—B bonds exist between B atoms in the same layer and the Ti—B bonds form between the neighboring layers, which were observed in both theoretical and experimental works in bulk TiB₂ [11-12]. There are two conflicting opinions on the electron transfer between B and Ti atoms, which are closely related to the chemical bonding between Ti and B layers [12-13]. Even in the field of thermodynamic properties, such as the entropy, the heat capacity and the enthalpy of titanium boride remain highly controversial [14].

The aim of this work is to give a comparative and complementary investigation on structural, thermo- dynamic and electronic properties of the Ti-B compounds using the plane-wave pseudopotential based on DFT method, and to understand the reaction mechanisms and thermo-dynamic properties of the material.

2 Theoretical methods

2.1 Electronic structure calculation

The pseudopotential plane-wave method based on DFT was applied to doing all the calculations. Broyden- Fcetcher-Goldfarb-Shanne (BFGS) algorithm was used to relax the crystal structures. In the electronic structure calculations, the ultra-soft pseudopotentials introduced by VANDERBILT [15] were employed with the generalized gradient approximation (GGA) proposed by PERDEW et al (PW91) [16] as an exchange-correlation function. A plane-wave basis set with an energy cut-off of 280 eV was applied. Pseudo-atomic calculations were performed for Ti 4s²3d² and B 2s²2p¹. For the Brillouin zone sampling, a 6×6×6 Monkhorst–Pack mesh was used in the calculation of ground state properties, while a 12×12×12 Monkhorst–Pack mesh was used in the band structure calculation. The self-consistent convergence of the total energy is 10^-6 eV/atom. All these total-energy electronic structure calculations were carried out using the CASTEP code.

2.2 Thermodynamic properties

The quasi-harmonic Debye model was employed to calculate the equation of state (EOS) and the thermodynamic properties which are constructed from the Helmholtz free energy at the temperature below melting point in the quasi-harmonic approximation. This method was adapted for the investigation of thermo- dynamic properties [17], in which the non-equilibrium Gibbs function of the crystal phase G*(V, P, T) can be written in the form of [18]

G*(V, P, T)=E(V)+PV+A_vib(V, T)                                 (1)

where E(V) is the total energy per unit cell; PV corresponds to the constant hydrostatic pressure condition; A_vib is the vibrational Helmholtz free energy, which can be written as

                           (2)

where k is a constant; Θ_D is the Debye temperature; N is the number of atoms per formula unit; and D(y) is the Debye integral defined as

                                             (3)

The Debye characteristic temperature of the solid, Θ_D in Eq.(2) is related to an average sound velocity (v_D), because in Debye’s theory the vibrations of the solid are considered as elastic waves. For an isotropic solid, Θ_D can be computed as follows with Poisson ratio  0.25 [19]:

(4)

where k_B, B_s, V and r are the Boltzman constant, the elastic modulus, volume of primitive cell and density, respectively.

In order to perform static optimizations for Eq.(1), an appropriate approximate algorithm was applied using the G_st(V, P)=H_st(V, P)=E(V)+PV. Then, after the static calculation, Eq.(1) can be rewritten as

                           (5)

where E_vib(T) and S_vib(T) are the vibrational internal energy and vibrational entropy, respectively; N_A is the Avogadro constant. According to the quasi-harmonic Debye model, the heat capacity (c_V) and vibrational entropy (S_vib) can be obtained by

                  (6)

       (7)

here, in the approximate formula at low temperature,   c_{V, low} is given as follows:

                                      (8)

The calculation methods above are not computationally expensive, while the Debye temperature Θ_D and B_s are all equal to the values of standard equilibrium state. Using the methods mentioned above, the elastic and thermodynamic properties of TiB_x (x=1, 2) can be successfully obtained.

3 Results and discussion

3.1 Geometry, elastic constants and modulus

The lattice constants and mechanical constants for TiB after optimization are shown in Table 1. As listed in Table 1, the calculated lattice parameters for TiB are in excellent agreement with the previous experimental results and theoretical estimates.

For TiB₂, it can be seen that previous theoretical estimates of the bulk modulus, based on the DFT approach [10] and experiment [7], are close to the results obtained in the present studies. However, a TB–LMTO study with the LDA predicts much larger cell parameters [11] and a correspondingly lower bulk modulus than any other theoretical work (Table 1). The error of the LMTO calculation is especially large, which reveals that the LMTO results are unreliable. It is thus unclear why the LDA results of the all-electron TB–LMTO calculations overestimate the cell volume to such a degree. The most likely reason is the general shortcoming of the LMTO techniques that does not contain a full-potential treatment.

The calculated elastic constants for TiB and TiB₂ are presented in Table 2, with values from previous theoretical and experimental works. The GGA calculation agrees well with this set of data except for C₁₂ and C₆₆ components. The error of the FLAPW calculation is especially larger than that of PBE method [21] in TiB and experimental data [7] in TiB₂ for C₁₂ and C₆₆ components. The results of the present work thus strengthen the conclusion that the values of FLAPW [22] are erroneous. The consistency between the present calculations demonstrates that the bulk modulus calculated from the elastic constants in Table 2 agrees with the values derived from the previous theoretical and experimental data.

3.2 Electronic structure

The calculated electronic structures for the two compounds are displayed in Fig.1. It is clear that TiB and TiB₂ exhibit metallic behavior. The curves of them (see Figs.1(a) and (c)) show clearly that there has a valley region (pseudo-gap) which can distinguish the high- energy anti-bonding states and low-energy bonding states, especially for TiB₂. This pseudo-gap which manifests a typical stability effect was observed earlier by ROSNER et al [25]. The pseudo-gap in binary alloys is believed to be of ionic origin or come from the covalent hybridization [11]. However, ionicity does not contribute much to bonding here. The most popular opinion is that the pseudo-gap is due to the p—d σ bonding between B and Ti atoms. According to VAJEESTON et al [11], the pseudo-gap originates from strong covalent bonding between B atoms, i.e. from the B(p)—B(p) covalent contribution. ROSNER et al [25] pointed out that the pseudo-gap is due to the relatively large velocities in the region of Fermi level (E_F) rather than any semi-metallic overlapping of bands.

Table 1 Lattice parameters and bulk modulus for TiB and TiB₂

Table 2 Elastic constants C_ij for TiB and TiB₂ (GPa)

As can be seen from Fig.1, the DOSes of TiB and TiB₂ consist of the hybrid bond of Ti-3d and B-2p below the Fermi level. In the vicinity of the Fermi level, the DOSes of TiB and TiB₂ are mainly provided by the Ti-3d orbit. In the high-energy region above the Fermi level, the DOS of TiB consists of Ti—Ti bond, but the DOS of TiB₂ is more complex which includes Ti—Ti, Ti—B and B—B bonds. Meanwhile, strong hybrid bond of Ti-3d and B-2p appears in the region above the Fermi level, which indicates that the Ti—B hybrid bond in TiB₂ is greater than that of TiB. The conclusion is consistent with the views of PANDA and CHANDRAN [22, 26].

In addition, the band stretching of TiB₂ (Fig.1(b)) is significantly wider than that of TiB (Fig.1(d)), and the valence band of TiB₂ shows obvious parabolic shape and the feature of sp-like band. This indicates that the non-local level of the electrons as well as the covalence in TiB₂ is greater. So, the differences in electronic structure determine that the elastic modulus of TiB₂ is higher than that of TiB.

The present results of DOSes for TiB and TiB₂ in Table 3 are almost consistent with the data from experiments and previous calculation. But the DOS at the Fermi level shows a remarkable discrepancy with the atomic sphere approximation (LMTO-ASA) method. This discrepancy appears to be caused by the lack of consideration of the electron-phonon interaction in the calculation. In the vicinity of the Fermi level, the DOS of TiB₂ is lower than that of TiB, which indicates that TiB₂ is poor metal, in contrast with the semiconducting property of the pyrite phase.
 

Fig.1 Electronic structure of TiB and TiB₂: (a) Density of states of TiB₂; (b) Band structure of TiB₂; (c) Density of states of TiB;   (d) Band structure of TiB

Table 3 Comparisons of present results with previous works

Table 4 Values of Debye temperature Θ_D and enthalpy of formation ΔH_f at 0 K

3.3 Thermodynamic properties of Ti-B compound

The Ti-B system has been assessed by MURRAY  et al [29] and MA et al [14]. The thermodynamic data of TiB [30] and TiB₂ [29] have already been given by many authors. But there are many disagreements and contradictories among these literatures on the thermodynamic data, such as melting temperature and compound solution phases [14].

The simplified quasi-harmonic Debye model empowers us to obtain the enthalpy ΔH_f, Debye temperature Θ_D, heat capacity c_V and the Gibbs free energy of formation. The calculated Debye temperature Θ_D and enthalpy of formation at 0 K of the two intermetallic compounds are illustrated in Table 4. For pure boron, which is one of the most enigmatic elements [31], there are at least sixteen kinds of known polymorphs, and most of them are not yet experimentally established even at ambient conditions. Boron is known to exist in four pure phases at present, i.e. rhombohedral α-B12, β-B106, tetragonal T-B192 and orthorhombic γ-B28 [32], and γ-B28 is the newly synthesized high- pressure phase of boron.

The calculated value of Θ_D is a little smaller than Bloch-Gruneisun parameters [33]. For the enthalpy of formation, the four boron phases including the latest one [32] are considered and the calculated values are in conformity with the literatures.

The calculated relationship between heat capacities and temperatures at standard pressure is plotted in Fig.2. It is clear that when the temperature is below 300 K, the heat capacity is strongly dependent on temperature and increases rapidly with temperature increasing, which is due to the anharmonic approximation of the Debye approximation used here. However, the anharmonic effect on heat capacity is suppressed when the temperature is above 300 K. Heat capacity is very close to the Dulong–Petit limit, which is common to all solids at high temperatures. These results show the fact that the interactions between ions in TiB and TiB₂ have great effect on the heat capacity especially at low temperatures. Unfortunately, there are no theoretical results or experimental data to check the calculated results.

Fig.2 Relationship between heat capacities and temperatures at standard pressure

The zero-pressure Gibbs free energy of formation for the reaction (solid-solution phase) can also be calculated using the following relation:

Ti+B=TiB                                                                    (9)

Ti+2B=TiB₂                                                              (10)

The calculated Gibbs free energies of formation at zero-pressure together with the literature data [37] are given in Fig 3. It shows the comparison of the calculated phase diagram with the experimental data. Satisfactory agreement is obtained. The temperature differences of all the invariant reactions between the thermodynamic prediction and the practical measurement are within experimental error.

As can be seen from Fig.3, the reaction of TiB₂ is easier than that of TiB. The Gibbs free energies of formation of them are negative, and the data of TiB₂ are smaller than these of TiB, which indicates that TiB₂ is of priority to be generated in Ti-B reaction. One thing needs to be noted is that although the TiB₂ is easier to be generated than TiB in thermodynamic, the diffusion coefficient of boron in TiB₂ to the titanium matrix and the growth rate of TiB is very high, which causes the real product is TiB, rather than TiB₂.

Fig.3 Variations of ΔG with temperature for reactions

4 Conclusions

1) The calculated lattice parameters and mechanical constants for TiB are in excellent agreement with the previous experiment and theoretical estimates, except for C₁₂ and C₆₆ components. The errors of TB-LMTO-LDA in cell parameters and the FLAPW in elastic constants especially are large and the results are unreliable.

2) The DOSes of TiB and TiB₂ consist of the hybrid bonds of Ti-3d and B-2p below the Fermi level and    Ti—Ti 3d orbit in the vicinity of the Fermi level. In the high-energy region up to the Fermi level, the Ti—B hybrid bond in TiB₂ is stronger than that in TiB. The band stretching indicates that the non-local level of the electrons as well as the covalence in TiB₂ is greater.

3) The value of E_F indicates that the GGA method is appropriate to calculate the DOS of TiB. The calculated values of Debye temperature and the enthalpy of formation at 0 K are in good agreement with the experience considering the four boron phases. These estimated data are used to predict the Gibbs energy and the heat capacities. The values of Gibbs free energy of formation and heat capacities at zero-pressure in variation temperatures are calculated, the differences between calculated values and the experimental data are within experimental error.

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(Edited by HE Yun-bin)

Foundation item: Project(07JJ3102) supported by the Natural Science Foundation of Hunan Province, China; Project(k0902132-11) supported by the Changsha Municipal Science and Technology, China

Received date: 2010-09-16; Accepted date: 2011-04-11

Corresponding author: XU Hui, Professor, PhD; Tel: +86-731-88836142; E-mail: xuhui@mail.csu.edu.cn