First Principles Calculation of Thermal Expansion of Carbon and Boron Nitrides Based on Quasi Harmonic Approximation

First Principles Calculation of Thermal Expansion of Carbon

and Boron Nitrides Based on Quasi-Harmonic Approximation

Tetsuya Tohei

1

, Hak-Sung Lee

1,2

and Yuichi Ikuhara

1,3

1_{Institute of Engineering Innovation, The University of Tokyo, Tokyo 113-8656, Japan}

2_{Materials Modeling and Characterization, Korea Institute of Materials Science, Changwon 641-831, Korea}

3_{Nanostructures Research Laboratory, Japan Fine Ceramics Center, Nagoya 456-8587, Japan}

We have performed theoretical analysis of thermal expansion of carbon and boron nitrides underfinite temperature based onfirst principles phonon state calculations. Volume dependence of phonon density of states and thermodynamic functions such as heat capacity and vibrational free energy were theoretically examined. Through the volume dependence of vibrational free energy, thermal expansion atfinite temperature is reproduced within quasi-harmonic approximation (QHA). Our calculation results have demonstrated that thermal expansion coefficients of typical ceramics materials (diamond, graphite,c-BN andh-BN) are reasonably well reproduced with thefirst principles approach employing QHA calculations. [doi:10.2320/matertrans.MA201574]

(Received March 9, 2015; Accepted April 6, 2015; Published June 19, 2015)

Keywords: first principles calculation, phonon, thermal expansion, quasi-harmonic approximation, carbon, boron nitride

1. Introduction

First principles calculations based on density functional theory (DFT)1,2)have been remarkably successful in predict-ing or reproducpredict-ing ground states properties of structures and energies for various materials. Nowadays the method is becoming a popular tool that is widely used in condensed matter theory and computational materials science. Recent advances in the computational technique further enable us to calculate full phonon dispersions fromfirst principles.3,4)_One of the most important applications of the phonon calculations is evaluation of thermodynamic functions of materials at

finite temperatures.5­8) _{Since phonon excitation constitutes} major contribution to thermodynamic functions at finite temperature, detailed information on the structure of phonon spectrum is significant for understanding thermodynamical properties of materials. With the quantitative knowledge of the phonon density of states (DOS), one can calculate thermodynamic functions such as heat capacity, vibrational entropy, or free energy according to formula of statistical thermodynamics. This allows us for instance discussions of relative stability of phases based upon free energy at finite temperature.5­7)_{Another important application of the phonon} calculation is the evaluation of thermal expansion based on quasi-harmonic approximation (QHA).7­9) _{In this work, we} have investigated thermal expansion behavior of carbon allotropes (diamond and graphite) and boron nitride poly-morphs (c-BN and h-BN) under finite temperature based on

first principles phonon state calculations. Thermodynamical properties of carbon and boron nitride polymorphs have been investigated in several previous works by first principles calculations.5,8,10­12)However, detailed examination of ther-modynamic functions of the compounds as a function of temperature and volume and thorough evaluation of thermal expansion behavior underfinite temperature are still lacking. In this paper, first we see how phonon states contribute to thermodynamic functions such as heat capacity and vibra-tional free energy under finite temperature. Next we will show how QHA works to reproduce the behavior of thermal expansion, taking diamond as an example. Then, calculation

results of thermal expansion coefficients for carbon and boron nitride polymorphs will be presented as typical example of ceramics materials, demonstrating the capability of the present approach based on first principles QHA calculations.

2. Calculation Method

First principles total energy calculations were performed using VASP code.13,14) _{The interaction between ions and} valence electrons was described by the projector augmented wave (PAW) method.15)_{The exchange and correlation effects} were treated by the local density approximation (LDA) parameterized by Perdew and Zunger.16) _{The k mesh was} sampled according to a Monkhorst-Pack scheme17) with a spacing of 0.4/¡. The cutoff energy in the plane wave expansion was 500 eV. Convergence of relative energies with respect to the k mesh and energy cut off was found to be better than 1 meV/atom.

Calculations of phonon dispersions were made by the direct method.4) Hellmann-Feynman forces exerted on all atoms in supercells by finite atomic displacements of every symmetrically nonequivalent atom were calculated. Super cells were composed of 64 atoms for diamond and c-BN (2©2©2 extension of the unit cell) and 128 atoms for graphite and h-BN (4©4©2 extension of the unit cell). Amount of displacements were «0.03¡ for diamond and c-BN and «0.05¡ for graphite and h-BN, respectively. Under harmonic approximation, force constant matrix and then the dynamical matrix were constructed. Phonon dispersion relations were obtained by the diagonalization of the dynamical matrix. The non-analytical behavior of the dynamical matrix in polar crystals (boron nitrides) has been treated by a conventional manner using the Born effective charge tensor and the electronic part of the dielectric constant.8,18)

From the calculated phonon density of states (DOS), thermodynamic functions were evaluated according to formula of statistical thermodynamics. Heat capacity at constant volumeCvwas calculated by following formula: Special Issue on Nanostructured Functional Materials and Their Applications

C_V¼rN_Ak_B

Z₁

0

d¯gð¯ÞWðh¯=k_BTÞ ð1Þ

where r is the number of degree of freedom in the unit cell,NAis Avogadro’s constant,g(¯) is phonon DOS andW is a weighting factor W(h¯/kBT)=(h¯/kBT)2exp(h¯/kBT)/ (exp(h¯/kBT)¹1)2.

Vibrational contribution to the free energy was calculated as following formula:

F_vib¼rN_Ak_BT

Z ₁

0

d¯gð¯Þln½2 sinhðh¯=2k_BTÞ ð2Þ

To evaluate thermal expansion behavior under finite temperature, we have employed quasi-harmonic approxima-tion (QHA).9)_{Phonon DOS and vibrational free energy were} calculated at several different cell volumes. Free energy of a crystal at a given volume is obtained as a sum of the static total energy and vibrational free energy. Equilibrium volume at a given temperature was obtained from minima of free energy versus volume curve. Thus, we can obtain temper-ature change of equilibrium volume and thermal expansion coefficients atfinite temperature.

3. Results and Discussion

Calculation results of phonon DOS of two carbon allotropes, diamond and graphite are shown in Fig. 1. The calculations were made for cells with equilibrium volume at 0 K. The figure shows that diamond and graphite have very different phonon spectrum from each other, which reflects difference in crystal structure and chemical bonding of the different materials. In diamond, a prominent peak is observed at about 37 THz. This peak corresponds to high frequency optical modes associated with bending of rigid sp3 bonds in diamond. In graphite, on the other hand, there are more phonon states in low to middle frequency region. These low frequency bands correspond to layer-shearing, layer-breathing, and layer-bending modes. In the highest frequency region there is another band of vibra-tional states. These states correspond to in-plane bond stretching modes and their frequencies are even higher than the maximum frequency of diamond. These features of phonon DOS are in good agreement with experimentally observed phonon dispersions.19­24) _{In Fig. 1, the weighting} factor W(h¯/kBT) in the heat capacity formula are also shown for several temperatures. The weighting factor is a function of frequency and temperature and represents how many phonon states are excited and contribute to lattice heat capacity. The weighting factor decreases with frequency monotonically. With increasing temperature, the weighting factor increases at higher frequency region. This means excitation of higher frequency (energy) phonon states becomes increasingly important at higher temper-ature. By integrating the product of phonon DOS and the weighting factor, heat capacity at a given temperature is obtained. Calculated results of temperature dependence of heat capacity for diamond and graphite are shown in Fig. 2. Lines show the calculation values of Cv and symbols show heat capacity by experiments (Cp).25­28) Reasonably good agreement is found between the

calcu-lated Cv and the experimental values. Also important observation here is that graphite shows larger heat capacity than diamond at low temperature. This is because graphite is ‘softer’ and has more phonon states in low frequency region than diamond that contribute to heat capacity at low temperature.

Based upon above observations, next we will see the principles of the QHA method. In the method phonon dispersion and DOS are calculated for several different cell volumes. Figure 3 shows the change of phonon DOS in diamond calculated with different cell volumes. Solid line shows phonon DOS for the equilibrium volume at 0 K (with lattice constanta0=3.536¡). Dotted and dashed lines show phonon DOS calculated with a contracted cell (a=0.95a0) and an expanded cell (a=1.05a0), respectively. From the

figure we see that increasing cell volume lowers overall frequency of phonon DOS. This is due to the softening of lattice in an expanded cell. This lowering in phonon frequency should lead to larger heat capacity and larger vibrational entropy in expanded cells. Figure 4 shows calculated results of heat capacity and vibrational free energy of diamond for different cell volumes. It is observed that heat capacity is larger for the cell with larger volume (Fig. 4(a)). Also vibrational free energy becomes more negative for in the expanded cell, due to larger contribution of vibrational

0 10 20 30 40 50

0.00 0.05 0.10 0.15

0.0 0.5 1.0

1000 K 300 K

100 K

Phonon DOS,

g

(

ν

)/THz

-1

Frequency, ν/THz

W

eighting factor

,

W

(

h

ν

/k

B

T

)

graphite

diamond

Fig. 1 Calculation results of phonon DOS of diamond and graphite, and weighting factorW(h¯/kBT) in the heat capacity formula.

25

20

15

10

5

0

0 200 400 600 800 1000 1200 1400

Cv

,

Cp

/J

K

-1mol

-1

Temperature, T/K

diamond graphite

[image:2.595.318.533.74.211.2] [image:2.595.318.532.259.417.2]

entropy (Fig. 4(b)). This dependence of vibrational free energy on volume is the key for QHA. Figure 5 shows volume dependence of free energy at finite temperature and its original components of static total energy (energy at 0 K) and vibrational free energy. The static energy shows minima at an equilibrium volume at 0 K (Fig. 5(a)). On the other hand, vibrational free energy decreases monotonically with increasing cell volume (Fig. 5(b)). The dependence of vibrational free energy on volume becomes stronger in higher temperatures. By adding these two components of static energy and vibrational free energy, free energy versus volume curves at finite temperature are obtained (Fig. 5(c)). Equilibrium volume at a given temperature is determined

from minima of a free energy-volume curve. We see from the

figure that equilibrium volume increases with increasing temperature. In this way, thermal expansion behavior is reproduced by QHA. It should be noted that the volume dependence of vibrational free energy is essential for the approximation.

0 500 1000 1500 2000

0 5 10 15 20 25

Cv

/J

mol

-1K

-1

Temperature, T/K

a= 0.95a0

a=a0

a= 1.05a0

(a)

Fvib

/eV/f.u.

2000

0 500 1000 1500

0.2

0

-0.2

-0.4

Temperature, T/K

(b)

a= 0.95a0

a=a0

a= 1.05a0

Fig. 4 Thermodynamic functions of diamond calculated at different cell volumes. (a) heat capacityCv, (b) vibrational free energyFvib.

0 0.05 0.10 0.15 0.20 0.25

0 10 20 30 40 50

Frequency

,

ν

/THz

Phonon DOS,

g

(

ν

)/THz

-1

a= 0.95a0

a=a0

a= 1.05a0

Fig. 3 Calculated phonon DOS of diamond with different cell volumes.

Es

/eV/f.u.

Volume, V/Å3

-10.09

-10.10

-10.11

-10.12

-10.13

-10.14

5.3 5.4 5.5 5.6 5.7 5.8 5.9

Fvib

/eV/f.u.

1500K 1000K 500K 0K

0.2

0.15

0.1

0.05

0

-0.05

5.3 5.4 5.5 5.6 5.7 5.8 5.9

Volume, V/Å3

(a)

(b)

Free energy

,

F

/eV/f.u.

Volume, V/Å3

1500K 1000K 500K 0K -9.9

-9.95

-10

-10.05

-10.1

-10.15

5.3 5.4 5.5 5.6 5.7 5.8 5.9 (c)

[image:3.595.325.527.162.462.2] [image:3.595.68.275.250.469.2] [image:3.595.95.505.513.760.2]

Finally, we will present our calculation results of thermal expansion based on QHA for several different materials. Figure 6 shows calculation results of volume thermal expansion coefficients of diamond and graphite, with comparing experimental values.29,30) _{We see that the} calculation shows generally good agreements with experi-ments, reproducing different behavior of thermal expansion in different materials. Thermal expansion coefficients of diamond and graphite were calculated as 3.35 (3.75) and 19.8 (21.9) at 300 K, and 12.8 (13.2) and 18.6 (29.5) at 1000 K, respectively (experimental values in parentheses29,30)_{). For} graphite at high temperature, some difference was observed between the calculations and experiments. The discrepancy may be attributed the poor description of the inter-plane interaction in graphite by the currently used energy functional of LDA. Calculation results of thermal expansion coefficients for boron nitride polymorphs (c-BN andh-BN) are shown in Fig. 7. Again, we see reasonable agreements between the calculations and the experiments.30,31) _{Thermal expansion} coefficients of c-BN and h-BN were calculated as 5.33 (5.4) and 29.8 (30.1) at 300 K, and 16.2 (17.7) and 24.4 at 1000 K (experimental values in parentheses30,31)_{). Thermal}

expansion behavior ofc-BN andh-BN is well reproduced by the present calculations except for high temperature behavior in h-BN where similar deviation to the graphite case was observed. Overall, the present results on carbons and boron nitrides show that different behavior of thermal expansion in different materials is reasonably well reproduced by the present approach based on the first principles QHA calculations.

4. Summary and Conclusions

In the present study, we have performed theoretical analysis of thermal expansion behavior of several crystalline materials based onfirst principles phonon state calculations. Volume dependence of phonon DOS and thermodynamic functions such as heat capacity and vibrational free energy were examined and shown to be the key for the quasi-harmonic approximation. Thermal expansion coefficients of several different materials including carbon and boron nitride polymorphs were calculated and compared with experimental values. The calculation results show generally good agree-ment with experiagree-ments, with reproducing different behavior of thermal expansion in different materials. The present results demonstrate the capability of the present approach based on first principles QHA calculations in reproducing or predicting thermal expansion behavior of crystalline materials.

Acknowledgements

This work was supported by Grant-in-Aids for Scientific Research on Innovative Areas (25106003), Young Scientists (B) (24760533), and Elements Strategy Initiative for Structural Materials (ESISM) from The Ministry of Educa-tion, Culture, Sports, Science and Technology (MEXT), Japan.

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Temperature, T/K

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