EVALUATION OF MECHANICAL AND THERMAL PROPERTIES OF CUBIC …

Materials Science Research International, Vol.4, No.1 pp. 39-44 (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 525-77, Japan.

**Hirohata Works, Nippon Steel Corporation, 1, Fuji-cho Hirohata-ku Himeji-shi, Hyogo 671-11, 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 three-body 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 ab-initio 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 FunctionIt is assumed in this paper that the interatomic

potential of cBN is represented by the Tersoff three-body potential function [9], which is available to the material having covalent bonding. The Tersoff three-

Received September 10, 1997

39

Yoshitada ISONO, Hirokazu KISHIMOTO and Takeshi TANAKA

body potential function is a sum of pairlike interactions, where the attractive term in the function includes three-body term. The form of the potential energy, E, of the atomic system is

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

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.

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.

Table 1. Basis set and theory used in geometry optimization MO calculation.

Table 2. Comparison of thebond length and bond angle of BNH6 atom cluster in MO analysis.

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MECHANICAL AND THERMAL PROPERTIES OF cBN

Fig.2. B4N4H18 atom cluster.

Fig.3. Energy surface and energy counter map of B4N4H18 atom cluster.

Fig.4. Valence of electron density at the bond length of

0.1565nm and the bond angle of 109.47•‹.

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

Table 3. Potential parameters included in the Tersoff

model potential function for cBN.

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

107.5•‹. The bond length and bond angle measured in

experiments were 0.1565nm and 109.47•‹, 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 0.1565nm and bond angle

of 109.47•‹. 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.

(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 0.3578nm for cBN and 0.3561nm 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

Yoshitada ISONO, Hirokazu KISHIMOTO and Takeshi TANAKA

Fig. 5. Molecular dynamics simulation model of cBN.

Fig. 6. Relationship between total energy and lattice

constant.

Table 4. Elastic stiffness and bulk modulus of cBN and

diamond.

following equations based on the infinitesimal deformation theory [14].

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,

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

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.

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

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 200-300 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 ă is defined, following to

the Green-Kubo theory [8], asă

=1/kVT2•ç•‡0<ql(t)ql(0)>dt. (9)

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

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].

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

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.

4.2. Coefficient of Thermal Expansion (CTE) of cBN

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,

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 0.3566nm, 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

43

Yoshitada ISONO, Hirokazu KISHIMOTO and Takeshi TANAKA

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 three-time 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.

5. CONCULUSIONS

(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.

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