EVALUATION OF MECHANICAL AND THERMAL PROPERTIES OF CUBIC BORON NITRIDE BY AB-INITIO CALCULATION

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1 Materials Science Research International, Vol.4, No.1 pp (1998) General paper EVALUATION OF MECHANICAL AND THERMAL PROPERTIES OF CUBIC BORON NITRIDE BY AB-INITIO CALCULATION Yoshitada ISONO*, Hirokazu KISHIMOTO** and Takeshi TANAKA* *Department of Mechanical Engineering, Ritsumeikan University, 1-1-1, Nojihigashi Kusatsu-shi, Shiga , Japan. **Hirohata Works, Nippon Steel Corporation, 1, Fuji-cho Hirohata-ku Himeji-shi, Hyogo , Japan. Abstract: This paper describes the mechanical and thermal properties of a cubic boron nitride (cbn) by molecular orbital and molecular dynamics simulations. The interatomic potential of cbn used for the molecular dynamics simulation was proposed by an ab-initio molecular orbital calculation for a cbn cluster. The elastic stiffness and the bulk modulus of cbn were found to be close to those of diamond by the molecular simulation. The bulk modulus of cbn in the simulation agreed with that in experiment. The equilibrium molecular dynamics simulation estimated the effect of temperature on thermal conductivity and coefficient of thermal expansion of cbn. The thermal conductivity of cbn drastically decreased with increasing temperature above 150K. The coefficient of thermal expansion of cbn was independent of temperature at 50K-900K, but that of cbn increased above 900K with increasing temperature. Key words: Cubic boron nitride, Ab-initio calculation, Molecular dynamics, Elastic stiffness, Bulk modulus, Thermal conductivity, Coefficient of thermal expansion 1. INTRODUCTION Cubic boron nitride (cbn) is an abrasive material having diamond structure for grinding and polishing. CBN thin films have been used for machining tools and high temperature semiconductor devices due to its excellent hardness and thermal property [1-3]. Understanding of the mechanical and thermal properties of cbn is essential for improvement of its on the performance and life extension of machining tools and semiconductor devices. However, the mechanical and thermal properties of cbn abrasive and thin film, especially the elastic stiffness and thermal conductivity have not been well understood. Molecular orbital (MO) and molecular dynamics (MD) simulations are useful tools for studying the mechanical and thermal properties of materials. Many studies have been conducted on the calculation of the elastic stiffness and the bulk modulus of pure materials by MO and MD simulations. For example, Kugimiya et al. [4] calculated the elastic stiffness of graphite based on ab-initio MO calculations. Wang et al. [5] studied the elastic property of a gold under hydrostatic tension with the embedded-atom potential. However, few studies have been carried out for the elastic stiffness and the bulk modulus of the materials comprised with more than two kinds of atom [6]. Thermal conductivity and coefficient of thermal expansion (CTE) can be also estimated by the MD simulation. The former is physically understood as the propagation of the lattice vibration and the latter the variation of the lattice constant with temperature. Many researchers reported thermal property of silicon. Okada et al. [7] calculated the thermal conductivity of silicon by means of the MD simulation using the Tersoff threebody potential function, but did not mention the effect of temperature on thermal conductivity. Lee et al. [8] estimated the effect of temperature on the thermal conductivity of amorphous silicon using the Stillinger- Weber three-body potential function. However, the thermal property of cbn was scarcely reported. Lack of reliable potential function of cbn leads to no systematic research of cbn by molecular simulations. The objective of this paper is to study the elastic stiffness, bulk modulus, thermal conductivity and CTE of cbn by MO and MD simulations. An interatomic potential function of cbn was proposed and potential parameters of the function were determined by the abinitio MO calculation for a cbn cluster. Elastic stiffness and bulk modulus were calculated by the second derivative of the interatomic potential obtained in MO calculation. The effect of temperature on the thermal conductivity and CTE was computed by equilibrium MD simulation using the interatomic potential for canonical ensemble. Elastic stiffness and thermal conductivity of cbn were discussed by referring to those of diamond. 2. POTENTIAL FUNCTION AND POTENTIAL PARAMETERS OF cbn 2.1. Three-body Potential Function It is assumed in this paper that the interatomic potential of cbn is represented by the Tersoff threebody potential function [9], which is available to the material having covalent bonding. The Tersoff three- Received September 10,

Yoshitada ISONO, Hirokazu KISHIMOTO and Takeshi TANAKA body potential function is a sum of pairlike interactions, where the attractive term in the function includes threebody term.

2 Yoshitada ISONO, Hirokazu KISHIMOTO and Takeshi TANAKA body potential function is a sum of pairlike interactions, where the attractive term in the function includes threebody term. The form of the potential energy, E, of the atomic system is Force acting on ƒ atom is the sum of the force from and ƒá atoms, so that it is equated as, 2.2, Interatomic Potential of cbn based on Ab-Initio MO Calculation Geometry optimized ab-initio MO calculations were performed to determine the suitable basis set and MO theory for cbn. Bond length, bond angle and potential energy were calculated by using Gaussian94 [10] for the BNH6 atom cluster shown in Fig. 1 [11]. Basis set and theory used in MO calculation is listed in Table 1. The results of the analysis for the BNH6 atom cluster are listed in Table 2. Errors in this table represent the difference between analytical and experimental results. Fig. 1. BNH6 atom cluster. where rƒ ƒà is the atomic distance between ƒ and ƒà atoms and ƒæƒ ƒàƒá the bond angle between vectors r and rƒ ƒá EfR represents a repulsive pair potential function and fa an attractive pair potential function associated with bonding energy between ƒ and ƒà atoms. fc is a cutoff function which limits the effective distance of the Table 1. Basis set and theory used in geometry optimization MO calculation. potential function. The term, bƒ ƒà, represents a measure of bond order and it decreases with increasing the number of ƒá atoms included in the cutoff region. Parameters A, B, ƒé, ƒê, ƒë, n, c, d, h, ƒô, R and S are constants. Table 2. Comparison of thebond length and bond angle of BNH6 atom cluster in MO analysis.

MECHANICAL AND THERMAL PROPERTIES OF cbn Table 3. Potential parameters included in the Tersoff model potential function for cbn. Fig.3. Fig.2. B4N4H18 atom cluster.

3 MECHANICAL AND THERMAL PROPERTIES OF cbn Table 3. Potential parameters included in the Tersoff model potential function for cbn. Fig.3. Fig.2. B4N4H18 atom cluster. Energy surface and energy counter map of B4N4H18 atom cluster. to B and N atoms for the periodicity of atoms in cbn. This paper calculated the potential energy of the cluster, changing the bond length between B and N atoms and the bond angles in Fig. 2. Figure 3 shows the variation of the total energy of the cluster with the bond length and bond angle. The total energy takes the minimum value at the bond length of 0.156nm and bond angle of The bond length and bond angle measured in experiments were nm and , respectively. The results in MO analysis agree with the experimental results. The difference between the analysis and experiment is about 0.3% and 1.8%, respectively. The difference is small, so that MO calculation accurately simulates the actual bonding behavior. Figure 4 shows the valence electron density of the cluster at the bond length of nm and bond angle of The contour map shows that electrons around B and N atoms are evenly distributed in the outer region but they are partly concentrated around the nitrogen atom in the inner region. The electron density in Fig. 4 shows that the bond between B and N atoms is covalent bonding. Potential parameters in Eq. (1) for cbn were determined by least-squares method so that the equation approximates the total energy in Fig. 3. Table 3 shows the parameters obtained by this fitting. 3. EVALUATION OF INTERATOMIC POTENTIAL AND ELASTIC CONSTANTS OF cbn Total energy of a cubic cell with 108 boron and 108 nitrogen atoms shown in Fig. 5 was calculated by Eq. Fig.4. Valence of electron density at the bond length of nm and the bond angle of The difference in the B-N bond length between analysis and experiment is more than 0.79% except that in 6-31G*/MP2. The difference between analysis of 6-31G*/MP2 and experiment is only 0.41%. The B-N bond length calculated by 6-31G*/MP2 well agrees with that in experiment [11], but those calculated by the other basis sets do not well agree with the experimental results. This paper uses 6-31G* basis set and MP2 theory in MO calculations for a cbn cluster. Potential parameters for cbn included in Eq. (1) were determined by the total energy obtained in MO calculation. Figure 2 shows the B4N4H18 atom cluster used in MO calculation. Hydrogen atoms were attached (1). In calculating the total energy, the distance between B and N atoms was changed from 0.130nm to 0.182nm. Figure 6 shows the relationship between the total energy and the lattice constant. Solid plots in Fig. 6 show the total energy of a diamond consisting of 216 carbon atoms, calculated by the Tersoff potential function [9]. The energy curve of cbn has a similar trend to that of diamond, but the minimum energy of cbn is larger than that of diamond. The value of the former material is about -600eV while that of the latter material is -1000eV. The lattice constant, which gives the minimum value of the total energy, is nm for cbn and nm for diamond. These values agree well with the experimental results reported by the articles [12, 13]. The difference is only 0.8% for cbn and 0.2% for diamond. These results show that the potential parameters of cbn based on ab-initio MO calculations are available to the analysis of crystal structure of cbn. The elastic stiffness and bulk modulus of the atomic model shown in Fig. 5 can be evaluated by the 41

4 Yoshitada ISONO, Hirokazu KISHIMOTO and Takeshi TANAKA where, V is the volume of atomic system and ƒó the potential function. Equation (3) shows the local elastic constant, which dose not include the effect of inner displacement between B and N on the elastic property. Cijkl has 21 independent values for the anisotropic material with no symmetry, but has only three independent values for the complete isotropic material. This paper denotes these three values as c11, c12 and c44 following to the Voigt notion as, Fig. 5. Molecular dynamics simulation model of cbn. 11=C1111, c12=c1122, c44=c1212. (5) Table 4 shows the elastic stiffness and bulk modulus evaluated by Eqs. (3)-(5) for cbn and diamond. The Fig. 6. Relationship between total energy and lattice constant. Table 4. Elastic stiffness and bulk modulus of cbn and diamond. calculated elastic stiffness and bulk modulus of diamond are in good agreement with those in experiments and the difference is less than 9%. This result indicates that Eqs. (3)-(5) are useful for evaluating the elastic stiffness and bulk modulus of materials. The bulk modulus of cbn in the analysis also agrees well with the experimental result [12], where the difference is only 3%. The elastic stiffness of cbn in the analysis could not be compared with experimental result since no experimental results were available. The elastic stiffness is presumably estimated properly, considering the accuracy of the calculation for elastic stiffness of diamond. The elastic stiffness of cbn calculated was c11=824.4gpa, c12=264.0gpa and c44=412.2gpa. These values are reliable enough to estimate the strength of machining tools of cbn. The elastic stiffness of cbn is close to that of diamond so that cbn thin film is an effective material for the protection of machining tools. 4. THERMAL PROPERTY OF cbn 4.1. Calculation of Thermal Conductivity of cbn The thermal conductivity of cbn was calculated by equilibrium MD simulation for NPT (N: number, P: pressure, T: temperature) ensemble corresponding to constant-pressure and temperature. In the MD simulation of this paper, the following Lagrangian equation proposed by Andersen [15] was used. following equations based on the infinitesimal deformation theory [14]. where m is the mass of atoms, E the potential energy, V the volume of atomic system, M a constant and PE the external pressure. Dots over the function stand for the derivative with respect to time. Lagrangian equations of motion are also expressed as, 42

MECHANICAL AND THERMAL PROPERTIES OF cbn The thermal conductivity of diamond was estimated in MD simulation using the Tersoff three-body potential.

5 MECHANICAL AND THERMAL PROPERTIES OF cbn The thermal conductivity of diamond was estimated in MD simulation using the Tersoff three-body potential. The thermal conductivity of diamond in the analysis decreases with increasing temperature. The MD results of diamond closely agree with the experimental results. This result indicates that Eqs. (6)-(10) as well as the Fig.7. Effect of temperature on the thermal conductivity of cbn and diamond. where Fƒ is the force acting on ƒ atom and P the internal pressure. Double dot in Eqs. (7) and (8) stands for the second derivative with respect to time. Equation (8) means that the external vibration is superimposed on the lattice vibration in the simulation cell. However, the external vibration did not influence the amplitude of lattice vibration since the period of external vibration was about times larger than that of lattice vibration. So, Eq. (8) did not influence the thermal conductivity. MD analyses were performed for the cbn cubic cell in Fig. 5, imposing the periodic boundary condition to the outer surfaces in the three directions of x, y and z. Time integration of motion was used by discrete Verlet's method [16] at every 0.5fs. Thermal conductivity ƒé is defined, following to the Green-Kubo theory [8], asƒé =1/kVT2 ç 0<ql(t)ql(0)>dt. (9) parameters in these equations are useful for estimating the thermal conductivity of diamond. The thermal conductivity of cbn in the analysis increases with increasing temperature at the temperature range of 50K-150K, but it turns to decrease at 150K with increasing temperature. This results from the difference in specific heat and phonon mean free pass between low and high temperatures. The main carrier of heat in an insulator as cbn is phonons, and the specific heat and the phonon mean free pass determine the thermal conductivity. The thermal conductivity can be approximated as ƒé `ClƒË/3, where C is the specific heat, l the phonon mean free pass and ƒë the sound velocity. At low temperatures, C increases in proportion to T3 but l is considered to be constant due to the small interaction of phonons [17]. Thus, the thermal conductivity increases in proportion to T3 at low temperatures. At high temperatures, C is regarded as a constant value, whereas l decreases in proportion to T-1 owing to the heavy interaction of phonons [17] and then the thermal conductivity decreases in proportion to T-1. The thermal conductivity of cbn in MD analysis is smaller than that of diamond in all the temperature range examined. This results from the difference in mean free pass of phonons between cbn and diamond. The phonon mean free pass of cbn is smaller than that of diamond since atoms in the material consisting of more than two different kinds of atoms scatter more than that consisting of only one kind of atoms. The lower thermal conductivity of cbn at high temperatures prevents the heat flux to machining tool in a machining process so that cbn is a suitable material for protecting machining tool. The protecting capability is comparable to diamond Coefficient of Thermal Expansion (CTE) of cbn ql is a heat flux vector in l-direction, <ql(t)ql(0)> the correlation function of the heat flux, V the volume of the atomic system, T temperature and k the Boltzmann constant. Heat flux vector q is defined as CTE, the thermal expansion of unit lattice length per temperature, was discussed by calculating the expansion of the cbn cubic cell in Fig. 5 in MD simulation. CTE is defined as the expansion ratio per temperature, so it is equated as, q=ƒ ƒ Eƒ ƒëƒ +1/2ƒ ƒ,ƒàrƒ ƒà(ƒëƒ EFƒ ƒà). (10) Eƒ and ƒëƒ are the total energy and velocity vector of ƒ atom, respectively. Fƒ is the interatomic force between ƒ and ƒà atoms and rƒ ƒà is a vector sensing from ƒà to ƒ atoms. MD simulation based on the equilibrium NPT ensemble calculated the heat flux. The correlation function of the heat flux was integrated up to 2 ~10 steps. Figure 7 shows the variation of the thermal conductivity of cbn and diamond with temperature together with the experimental results of diamond [13]. where a is the lattice constant and T temperature of the atomic system. Figure 8 shows the relationship between the lattice constant of cbn and temperature. The lattice constant in MD simulation at room temperature is nm, which is smaller than the experimental result [12]. However, the difference is only 1.3% and is small. The increase ratio of lattice constant in the analysis is small in temperature range between 50K and 900K. The small

Yoshitada ISONO, Hirokazu KISHIMOTO and Takeshi TANAKA 5. CONCULUSIONS Fig.8. Relationship between lattice constant and temperature for cbn. Fig.9. Effect of temperature on the CTE of cbn.

6 Yoshitada ISONO, Hirokazu KISHIMOTO and Takeshi TANAKA 5. CONCULUSIONS Fig.8. Relationship between lattice constant and temperature for cbn. Fig.9. Effect of temperature on the CTE of cbn. increase ratio of lattice constant below 900K can be explained by the shape of potential curve. The potential curve of cbn shown in Fig. 6 is more concave than that of metal bonding and ionic bonding materials. So, the expansion of lattice constant of cbn with increasing the potential energy is smaller than that of metal bonding and ionic bonding materials. The lattice constant in the analysis shows a sharp increase with increasing temperature above 900K. The sharp increase in the lattice constant is considered to be due to decreasing covalent bonding force between B and N atoms above 900K. Figure 9 shows the effect of temperature on CTE calculated by Eq. (11) together with the experimental results [12]. CTE in MD analysis takes almost the constant value at the temperature range of 50K-900K. At the temperatures higher than 900K, however, CTE increases with increasing temperature, having the threetime larger value at 1200K in comparison with that at room temperature. Comparing analytical results with experimental results, the temperature dependence of CTE in MD analysis agrees with that in experiment, but CTE in experiment is larger than that in MD analyses at the temperature range of 700K-900K. The low CTE in the analysis is attributed to the strong attractive force in the long range between B and N atoms. The potential function of cbn used in the analysis estimated slightly stronger covalent bonding force since the cbn cluster size in the ab-initio calculation was small, so that the long-range Coulomb interaction must be taken account for the better accuracy in the long-range between B and N atoms. (1) Potential parameters in Tersoff potential function for cbn were proposed based on the by ab-initio MO calculation using B4H4H18 atom cluster. The calculated elastic stiffness and bulk modulus of cbn using these parameters were c11= 824.4GPa, c12=264gpa c44= 412.2GPa, and B=450GPa. The elastic property of cbn is close to that of diamond. (2) The thermal conductivity of cbn in MD analysis increased with increasing temperature at the temperature range of 50K-150K, but it decreased with increasing temperature above 150K. The thermal conductivity of cbn in MD analysis was smaller than that of diamond in all the temperature range examined, the cause of which was discussed in connection with the phonon scattering. (3) The coefficient of thermal expansion of cbn above 900K increased with increasing temperature in experiments and MD analyses. The temperature dependence of CTE in MD analysis agreed with that in experiment, but CTE in experiment was larger than that in MD analyses at the temperature range of 700K-900K. The low CTE in the analysis was attributed to the strong attractive force in the long range between B and N atoms. Acknowledgement -The authors express their gratitude to Prof. Sakane of Ritsumeikan University for the extensive and detailed discussion on this paper. REFERENCES 1. H. Erhardt, Surface Coated Technology, 74/75 (1995) G. Demanzeau, Diamond and Related Materials, 2 (1993) H. Sachdev, R. Haubner, H. Noth and B. Lux, Diamond and Related Materials, 6 (1997) T. Kugimiya, Y. Shibutani and Y. Tomita, Proc. of Molecular Dynamics Symp. JSMS, (1996) 72 (in Japanese). 5. J. Wang, J. Li and S. Yip, Phys. Rev. B, 52 (1995) H. Kitagawa, Y. Shibutani and S. Ogata, Modelling Simul. Mater. Sci. Eng., 3 (1995) T.K. Okada, S. Kambayashi, M. Yabuki, Y. Tsunashima, Y. Mikata and S. Onga, Mater. Res. Soc. Symp. Proc., 283 (1993) Y.H. Lee, it Biswas, C.M. Soukoulis, C.Z. Wang, C.T. Chan and K.M. Ho, Phys. Rev. B, 43 (1991) J. Tersoff, Phys. Rev. Let., 61 (1988) Gaussian, Inc., Gaussian 94 Reference Manual, (1994). 11. L.R. Thorne, R.D. Suenram and F.J. Lovas, J. Chem. Phys. 78 (1983) L. Vel, G. Demazeau and G., J. Etourneau, Mater. Sci. and Eng. B, 10 (1991) R. Berman, Physical Properties of Diamond, Oxford University Press, Oxford, (1965) p J.W. Martin, J. Phys. C, 8 (1975) H.C. Andersen, J. Chem. Phys. 72 (1980) L. Verlet, Phys. Rev., 159 (1967) C. Kittel, Introduction to Solid State Physics, John Wiley & Sons, Inc., New York, (1956) p

Potentials, periodicity

Potentials, periodicity Lecture 2 1/23/18 1 Survey responses 2 Topic requests DFT (10), Molecular dynamics (7), Monte Carlo (5) Machine Learning (4), High-throughput, Databases (4) NEB, phonons, Non-equilibrium