- Email: [email protected]
Theoretical study of the thermodynamic properties of cubic Zr3N4 and Hf3N4 under high pressures
Accepted Manuscript Theoretical study of the thermodynamic properties of cubic Zr3N4 and Hf3N4 under high pressures Ji-Dong Zhang, Kun Yang PII: DOI: Reference:
S0925-8388(14)00933-5 http://dx.doi.org/10.1016/j.jallcom.2014.04.120 JALCOM 31085
To appear in: Received Date: Revised Date: Accepted Date:
9 March 2014 18 April 2014 18 April 2014
Please cite this article as: J-D. Zhang, K. Yang, Theoretical study of the thermodynamic properties of cubic Zr3N4 and Hf3N4 under high pressures, (2014), doi: http://dx.doi.org/10.1016/j.jallcom.2014.04.120
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Theoretical study of the thermodynamic properties of cubic Zr3N4 and Hf3N4 under high pressures Ji-Dong Zhang 1,2 , Kun Yang∗,1 1
Key Laboratory of Ecophysics and Department of Physics, School of Science, Shihezi University, Shihezi 832003
(PR China) 2
Xinjiang Laboratory of Phase Transitions and Microstructures in Condensed Matters, Yili Normal University, Yining
835000 (PR China)
Abstract The crystal structure and elastic properties of two nitrides c-Zr3N4 and c-Hf3N4 with Th3P4 structure are investigated using density functional theory within the local density approximation (LDA). The results at zero pressure are in good agreement with the available theoretical and experimental values. Through the quasi-harmonic Debye model, the variations of the Debye temperature, internal energies, entropy, heat capacity and the thermal expansion under pressure range from 0 to 100 GPa and temperature range from 0 to 600 K are successfully obtained and discussed.
Keywords: Zr3N4; Hf3N4; First-principles calculation; Thermodynamic property; High pressure
1. Introduction Recently, a great effort is focused on the synthesis and characterization of hard or superhard materials due to their important roles in fundamental science and technological applications [1-5]. Nitrides are an important family of hard materials with superior properties [6], and have been receiving increasing concerns from researchers. In recent years, many nitrides with superior properties have been synthesized one after another, such as cubic Zr3N4 (c-Zr3N4) and Hf3N4
(c-Hf3N4). The nitrides were obtained via chemical reactions of zirconium and hafnium with molecular nitrogen at pressures up to 18 GPa and temperatures up to 3000 K [7]. They have Th3P4 structure and exhibit high bulk modulus around 250 Gpa, which indicates high hardness [7]. And what’s more is that the nitrides exhibit interesting functional magnetic or superconducting properties ∗
Corresponding author: e-mail, [email protected].
[7]. Due to the attractive properties of the nitrides, they have been an interesting research subject, since they are synthesized. Kroll [8] characterized the structural, electronic, elastical and vibrational properties of the nitrides using first-principles calculations. Mattesini et al. [9] presented the ab initio calculations of the elastic properties for c-Zr3N4 and c-Hf3N4, and suggested the Vickers hardness value of about 20 GPa for the both compounds. Xu et al. [10] investigated the optical properties of c-Zr3N4 and c-Hf3N4, and found it suitable to be used in microelectronic devices. Guo et al. [11] studied the structural and electronic properties of c-Zr3N4 under high pressure. Lowther [12] investigated the elastic and electronic properties of c-Zr3 N4 with oxygen and carbon impurities. Chihi et al. [13] performed a systemic study of the structural, mechanic, electronic and optical properties of c-Zr3N4 and c-Hf3N4 using the pseudo-potential plane-wave (PP-PW) method. Despite the above investigations, the study on some fundamental properties of c-Zr3N4 and c-Hf3N4, such as the thermodynamic properties, is scarce. In this paper, we aim to investigate the thermodynamic properties of c-Zr3N4 and c-Hf3N4 under high pressure to provide some additional information to the existing data on the physical properties of the nitrides.
2. Model and Computational method Electronic structure calculations are performed based on the plane-wave pseudopotential density-functional theory (DFT) [14-15] as implement in CASTEP package [16]. We employ Vanderbilt ultrasoft pseudopotentials [17] to describe the electron-ion interactions in the calculation. Pseudoatomic calculations are performed for Zr 4d25s2, Hf 4f14 5d2 6s2 and N 2s22p3. The exchange correlation energy is described in the local density approximation (LDA) for the exchange correlation functional
by
Ceperley
and
Alder
[18-20].
The
structure
is
optimized
with
the
Broyden–Fletcher–Goldfarb–Shanno (BFGS) [21] method. In calculations, the electronic wavefunctions are expanded in a plane-wave basis set with energy cut-off 280.0 eV. As for the Brillouin-zone sampling, we use the Monkhorst–Pack mesh with 4 × 4 ×4 k-points. The thermodynamic properties are calculated by the quasi-harmonic model code (Gibbs) [22].
3. Results and Discussion 3.1 Structural and elastic properties at zero pressure c-Zr3N4 and c-Hf3 N4 have a cubic Th3P4 structure with the space group I43d (No. 220) and have 14 atoms in unit cell. To obtain the equilibrium structure, we fully optimized both the lattice and internal coordinates of the nitrides using BFGS method [21] without any restrictions. The initial cell
parameters are taken from the experiment values [7]. In Table 1, we present the calculated equilibrium lattice constants, elastic constants, bulk modulus, shear modulus and Vickers hardness of c-Zr3N4 and c-Hf3N4 under zero pressure. The bulk modulus and shear modulus are calculated using the Voigt-Reuss-Hill averaging scheme, which is viewed as the best estimate for the theoretical value of polycrystalline elastic modulus [23]. The Vickers hardness can be obtained by an empirical model proposed by Chen et al. [24]:
H v = 2( k 2G ) 0.585 − 3
(1)
where Hv is Vickers hardness, k the Pugh’s modulus ratio (k=G/B) and G is the shear modulus. The available experimental data and others’ works are also shown in Table 1 to confirm the precision of our calculations. It is clear to see that the equilibrium lattice constants, elastic constants, bulk modulus, and shear modulus are in good agreement with the experimental [7] and previous theoretical values [8,9,11,12]. Our results exhibit a large deviation with the results reported by Xu et al. [10] and Chihi et al. [13], because they calculate the values by using GGA. Moreover, the calculated Vickers hardness basically agrees with the values reported by Mattesini et al. [9], and the difference may mainly due to the different calculation model we chose. 3.2 Thermodynamic properties In this study, the quasi-harmonic Debye model [22] is used to obtain the thermodynamic properties of the nitrides, in which the non-equilibrium Gibbs function G*(V; P, T) is expressed as:
G * (V ; P, T ) = E (V ) + PV + AVib (Θ (V );T )
(2)
Here E(V) is the total energy per unit cell for c-Zr3N4 and c-Hf3N4, which is obtained by using the first principles calculations. Θ(V) is the Debye temperature, and the vibrational Helmholtz free energy AVib can be written as Ref. [25, 26]
9 Θ Θ AVib (Θ;T ) = nkT + 3 ln(1 − e − Θ / T ) − D T 8 T
(3)
where D(Θ/T) represents the Debye integral, and n the number of atoms per formula unit. For an isotropic solid,
Θ=
B [6π 2V 1 / 2 n]1 / 3 f (σ ) s , K M
(4)
where M is the molecular mass per formula unit, Bs the adiabatic bulk modulus. The non-equilibrium
Gibbs function G*(V; P, T) can be minimized with respect to volume V as follows:
∂G * (V ; P,T ) = 0, ∂V P ,T
(5)
The thermal properties such as internal energies, entropy, heat capacity at constant volume CV, and thermal expansion α are, respectively, taken as
9 Θ U = nkT + 3 D(Θ / T ) 8 T
[
(6)
]
S = nk 4 D(Θ / T ) − 3 ln (1 − e− Θ / T )
Θ 3Θ / T CV = 3nk 4 D − Θ / T , − 1 T e
α=
γCV BT V
,
(7) (8)
(9)
Here γ is the Grüneisen parameter, and
γ = −(d ln Θ(V ) / d ln V ) ,
(10)
By using the above formulas (2-10), we calculated some thermodynamic properties of c-Zr3 N4 and c-Hf3N4. The Debye temperature of solid is an important physical quantity for a solid. In Fig. 1a and b, we present the obtained pressure-dependence of Debye temperature under different temperatures for c-Zr3N4 and c-Hf3N4. The Debye temperature of c-Zr3N4 at 0 K and 0 GPa calculated using quasi-harmonic Debye model is 673 K, which is in good agreement with the result reported by Chihi et al. (651.7 K) [13] derived from the averaged sound velocity. However, the calculated Debye temperature of c-Hf3N4 at 0 K and 0 GPa in this study is 550 K, which is obviously higher than the value reported by Chihi et al. (81.4 K) [13] using the same method with the calculation for c-Zr3N4. In order to evaluate the accuracy of our calculated results, we calculated the values again by using the averaged sound velocity. The obtained Debye temperatures are 674 K and 548 K for c-Zr3N4 and c-Hf3N4 respectively, which agree well with the results we calculated using quasi-harmonic Debye model. In Fig. 1, we plotted the Debye temperature Θ at the temperature of 0, 200, 400, and 600 K, respectively. We observe that the Debye temperature Θ tends to similar change for c-Zr3 N4 and c-Hf3N4. The Debye temperature decreases and increases with increasing temperature and pressure, respectively. This is because that the Debye temperature is related to volume V and the adiabatic bulk
modulus Bs. Although volume V increases and decreases with increasing temperature and pressure respectively, the adiabatic bulk modulus Bs is on the contrary. From Formula 4, it is easy to find the adiabatic bulk modulus Bs makes more contribution to the Debye temperature than volume V. The effect of the temperature on the Debye temperature Θ is obviously smaller than that of the pressure. Moreover, with the increasing of pressure, the temperature exhibits smaller and smarter effect on the Debye temperature Θ, which is same to that of HfB2 [27]. The calculated internal energies for c-Zr3N4 and c-Hf3N4 as a function of pressure under different temperatures are depicted in Fig. 2a, b, respectively. It can be seen that the total internal energies increase with pressure and temperature. It is more sensitive to the temperature than the pressure. For example, at 0 GPa pressure, the internal energies increase 151% and 202% for c-Zr3N4
and from c-Hf3N4 respectively, when the temperature changes from 0 K to 600 K. While, at the temperature of 0 K, the internal energies increase 56% and 53% for c-Zr3N4 and c-Hf3N4
respectively, when the pressure changes from 0 GPa to 100 GPa. Moreover, the higher the temperature is, the slower the internal energies increase with pressure. At the temperature of 600 K, the internal energies increase 8.7% and 5.5% for c-Zr3N4 and c-Hf3N4 respectively, when the pressure changes from 0 GPa to 100 GPa. The pressure-dependence of entropy for c-Zr3N4 and c-Hf3N4 under different temperatures are depicted in Fig. 3a, b, respectively. The values of the entropy of c-Zr3N4 are slighty smaller than that of c-Hf3N4.The entropies decrease with pressure (except at 0 GPa) and increase with temperature. Obviously, the entropy is more sensitive to the temperature than the pressure. For example, at 0 GPa pressure, the entropy increases 265% and 201% for c-Zr3N4 and c-Hf3N4 respectively, when the temperature changes from 200 K to 600 K. While, at the temperature of 200 K, the entropy decreases 59% and 52% for c-Zr3N4 and c-Hf3N4 respectively, when the pressure changes from 0 GPa to 100 GPa. When the temperature increases, the decrease of the entropy with pressure is slower and slower. At the temperature of 600 K, the entropy decreases 36% and 29% for c-Zr3N4 and c-Hf3N4 respectively, when the pressure changes from 0 GPa to 100 GPa. The heat capacity is an important property of crystal, which not only provides an essential insight into its vibrational properties, but also possesses many applications. The variations of heat capacity at constant volume C V with pressure are depicted in Fig. 4a and b for c-Zr3N4 and c-Hf3N4 , respectively. The change tendencies of CV are almost same for c-Zr3N4 and c-Hf3N4. CV increases
with temperature and decreases with pressure. It is more sensitive to the temperature than the pressure, especially at low temperature. With the increasing of the temperature, the effect of the pressure on CV decreases gradually, which can be viewed as the general features of the CV of a solid. At the temperature of 600 K, the calculated CV at 0 GPa for c-Zr3N4 and c-Hf3N4 are 165.5 and 168.1 Jmol-1 K-1 respectively, which is close to Dulong-Petit limit 174.5 Jmol-1K-1. The thermal expansion α of c-Zr3N4 and c-Hf3N4 are also plotted in Fig. 5a and b, respectively. The values of the thermal expansion α of c-Zr3N4 are slight higher than that of c-Hf3N4. As shown in Fig. 5a and b, the thermal expansion α decreases with pressure; the higher the temperature is, the faster the thermal expansion α decreases with pressure; and furthermore, the decrease becomes slower as pressure increases. Moreover, the thermal expansion α increases with temperature, and the higher the temperature is, the smaller the effect of temperature on the thermal expansion α is. 4. Conclusions In this paper, we have investigated the thermodynamic properties of two nitrides c-Zr3N4 and c-Hf3N4 using density functional theory within the local density approximation (LDA) and the quasi-harmonic Debye model. The pressure dependence of Debye temperature, internal energies, entropy, heat capacity and thermal expansion under different temperatures are obtained. The main results are as follows: (1) The obtained lattice parameters, elastic constants, bulk modulus, shear modulus and Vickers hardness at ambient pressure are in agreement with the experimental and theoretical values. (2) The calculated Debye temperature of c-Zr3N4 and c-Hf3N4 at 0 K and 0 GPa are 673 K and 550 K, respectively. The Debye temperature decreases and increases with increasing temperature and pressure, respectively. (3) The total internal energies of c-Zr3 N4 and c-Hf3N4 increase with pressure and temperature. (4) The entropies decrease with pressure and increase with temperature, so do the heat capacity at constant volume CV and the thermal expansion α.
Acknowledgment This work is supported by outstanding young talent cultivation plan of science and technology of shihezi university, China (No.:2012ZRKXYQ08)
References [1] R.A. Varin, C. Chiu, S. Li, D. Wexler, J. Alloys Compd. 370 (2004) 230. [2] L.A. Shi, Y. Gu, L.Y. Chen, Z.H. Yang, Y.T. Qian, Mater. Lett. 58 (2004) 23. [3] R.B. Kaner, J.J. Gilman, S.H. Tolbert, Science 308 (2005) 1268. [4] J.A. Montoya, A.D. Hernandez, E. Gregoryanz, S. Scandolo, Appl. Phys. Lett. 90 (2007) 011909. [5] C. Jiang, Z.J. Lin, Y.S. Zhao, Phys. Rev. Lett. 103 (2009) 185501. [6] C. Jiang, Z.J. Lin, Y.S. Zhao, Scripta Materi 63 (2010) 532. [7] A. Zerr, G. Miehe, R. Riedel, Nat. Mater. 2 (2003) 185. [8] P. Kroll, Phys. Rev. Lett. 90 (2003) 125501. [9] M. Mattesini, R. Ahuja, and B. Johansson, Phys. Rev. B 68 (2003) 184108. [10] M. Xu, S.Y. Wang, G. Yin, J. Li, Y.X. Zheng and L.Y. Chen, Appl. Phys. Lett. 89 (2006) 151908. [11] Q.X. Guo, W.K. Kwan, X.L. Cheng, and H. Zhang, Phys. Status Solidi B 247 (2010) 67. [12] J.E. Lowther, Solid. State Commun. 148 (2008) 563. [13] T. Chihi, M. Fatmi, B. Ghebouli, M. Guemmaz, Solid State Sci. 13 (2011) 1414. [14] P. Hohenberg, W. Kohn, Phys. Rev. B 136 (1964) 864. [15] W. Kohn, L.J. Sham, Phys. Rev. A 140 (1965) 1133. [16] V. Milman, B. Winkler, J.A. White, C.J. Packard, M.C. Payne, E.V. Akhmatskaya, R.H. Nobes, Int. J. Quantum Chem. 77 (2000) 895. [17] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892. [18] P. Hohenberg and W. Kohn, Phys. Rev. B 136 (1964) 864. [19] W. Kohn, and L.J. Sham, Phys. Rev. A 140 (1965) 1133. [20] D.M. Ceperley, and B.J. Alder, Phys. Rev. L 45 (1980) 566. [21] T.H. Fischer, J. Almlof, J. Phys. Chem. 96 (1992) 9768. [22] M.A. Blanco, E. Francisco, and V. Luana, Comput. Phys. Commun. 158 (2004) 57. [23] L.H. Li, W.L. Wang, L. Hu, B.B. Wei, Intermetallics 46 (2014) 211. [24] X.Q. Chen, H. Niu, D. Li, Y. Li. Intermetallics 19 (2011) 1275. [25] M.A. Blanco, A.M. Penda´s, E. Francisco, J.M. Recio, and R. Franco, J. Mol. Struct. (Theochem) 368 (1996) 245. [26] M. Flo´rez, J.M. Recio, E. Francisco, M.A. Blanco, and A.M. Penda´s, Phys. Rev. B 66 (2002) 144112. [27] J.D. Zhang, X.L. Cheng and D.H. Li, J. Alloys Compd. 509 (2011) 9577.
Table Captions Table 1. The calculated structural data, elastic constants Cij (GPa), bulk modulus B (GPa), shear modulus G (GPa) and Vickers hardness H (GPa) of c-Zr3 N4 and c-Hf3N4 under zero pressure and related experimental and theoretical data. c-Zr3N 4 Structure
Present
a
Experiment
c-Hf3N4 Other work
Present
work a (Å)
6.66
a
Experiment
work 6.74
b
c
6.69
d
6.69
6.79
6.76
6.70
6.67e 6.69f 6.84g c11 (GPa)
458
-
c
d
f
454.4 325.7 491
540
-
310.1g c12 (GPa)
161
-
165.3c 102d 160f
187
-
-
260
250
d
f
d
f
145.9 119.4 152
170
-
c
148
-
3.127e
6.58f
6.58g
c
443.9d
167.3c
156.5d
b
521
261.7 177.2 270 145.4c 158f 120.3g
174b 163b
304
260
Hv (GPa)
16.2
a
Reference 7, experiment.
b
Reference 8, LDA.
c
Reference 9, LDA.
d
Reference 10, GGA.
e
Reference 11, LDA.
f
Reference 12, LDA.
g
Reference 13, GGA.
-
19.7
g
152.4c
290b
133.1d
275.9c
251.6d
275f 153.7g 173
-
167b 150.7
c
493.2
152f 148.5g
153.7g G (GPa)
6.84d
167 51.0 c
122.4g B (GPa)
6.59c
f
75.5 148
6.59b
493f 358.9g
g
c44 (GPa)
Other work
18.0
-
21.3c
156.6c g
156f
Figures (Color online only)
Figure1. Variation of the Debye temperature of c-Zr 3N4 (a) and c-Hf3N 4 (b) as a function of pressure. Figure2. Variation of the internal energies for c-Zr3N 4 (a) and c-Hf3N4 (b) as a function of pressure. Figure3. Variation of the entropies for c-Zr3N4 (a) and c-Hf3N4 (b) as a function of pressure. Figure4. Variation of the heat capacities for c-Zr3N 4 (a) and c-Hf3N4 (b) as a function of pressure. Figure5. Variation of the thermal expansions for c-Zr 3N4 (a) and c-Hf3N 4 (b) as a function of pressure.
Figure1a
Figure1b
Figure2a
Figure2b
Figure3a
Figure3b
Figure4a
Figure4b
Figure5a
Figure5b
The thermodynamic properties of two nitrides c-Zr3N4 and c-Hf3 N4 with Th3P4 structure are investigated firstly. The Vickers hardness of c-Zr3 N4 and c-Hf3 N4 are calculated. The variations of the Debye temperature, internal energies, entropy, heat capacity and the thermal expansion with pressure and temperature are successfully obtained and discussed.
S0925-8388(14)00933-5 http://dx.doi.org/10.1016/j.jallcom.2014.04.120 JALCOM 31085
To appear in: Received Date: Revised Date: Accepted Date:
9 March 2014 18 April 2014 18 April 2014
Please cite this article as: J-D. Zhang, K. Yang, Theoretical study of the thermodynamic properties of cubic Zr3N4 and Hf3N4 under high pressures, (2014), doi: http://dx.doi.org/10.1016/j.jallcom.2014.04.120
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Theoretical study of the thermodynamic properties of cubic Zr3N4 and Hf3N4 under high pressures Ji-Dong Zhang 1,2 , Kun Yang∗,1 1
Key Laboratory of Ecophysics and Department of Physics, School of Science, Shihezi University, Shihezi 832003
(PR China) 2
Xinjiang Laboratory of Phase Transitions and Microstructures in Condensed Matters, Yili Normal University, Yining
835000 (PR China)
Abstract The crystal structure and elastic properties of two nitrides c-Zr3N4 and c-Hf3N4 with Th3P4 structure are investigated using density functional theory within the local density approximation (LDA). The results at zero pressure are in good agreement with the available theoretical and experimental values. Through the quasi-harmonic Debye model, the variations of the Debye temperature, internal energies, entropy, heat capacity and the thermal expansion under pressure range from 0 to 100 GPa and temperature range from 0 to 600 K are successfully obtained and discussed.
Keywords: Zr3N4; Hf3N4; First-principles calculation; Thermodynamic property; High pressure
1. Introduction Recently, a great effort is focused on the synthesis and characterization of hard or superhard materials due to their important roles in fundamental science and technological applications [1-5]. Nitrides are an important family of hard materials with superior properties [6], and have been receiving increasing concerns from researchers. In recent years, many nitrides with superior properties have been synthesized one after another, such as cubic Zr3N4 (c-Zr3N4) and Hf3N4
(c-Hf3N4). The nitrides were obtained via chemical reactions of zirconium and hafnium with molecular nitrogen at pressures up to 18 GPa and temperatures up to 3000 K [7]. They have Th3P4 structure and exhibit high bulk modulus around 250 Gpa, which indicates high hardness [7]. And what’s more is that the nitrides exhibit interesting functional magnetic or superconducting properties ∗
Corresponding author: e-mail, [email protected].
[7]. Due to the attractive properties of the nitrides, they have been an interesting research subject, since they are synthesized. Kroll [8] characterized the structural, electronic, elastical and vibrational properties of the nitrides using first-principles calculations. Mattesini et al. [9] presented the ab initio calculations of the elastic properties for c-Zr3N4 and c-Hf3N4, and suggested the Vickers hardness value of about 20 GPa for the both compounds. Xu et al. [10] investigated the optical properties of c-Zr3N4 and c-Hf3N4, and found it suitable to be used in microelectronic devices. Guo et al. [11] studied the structural and electronic properties of c-Zr3N4 under high pressure. Lowther [12] investigated the elastic and electronic properties of c-Zr3 N4 with oxygen and carbon impurities. Chihi et al. [13] performed a systemic study of the structural, mechanic, electronic and optical properties of c-Zr3N4 and c-Hf3N4 using the pseudo-potential plane-wave (PP-PW) method. Despite the above investigations, the study on some fundamental properties of c-Zr3N4 and c-Hf3N4, such as the thermodynamic properties, is scarce. In this paper, we aim to investigate the thermodynamic properties of c-Zr3N4 and c-Hf3N4 under high pressure to provide some additional information to the existing data on the physical properties of the nitrides.
2. Model and Computational method Electronic structure calculations are performed based on the plane-wave pseudopotential density-functional theory (DFT) [14-15] as implement in CASTEP package [16]. We employ Vanderbilt ultrasoft pseudopotentials [17] to describe the electron-ion interactions in the calculation. Pseudoatomic calculations are performed for Zr 4d25s2, Hf 4f14 5d2 6s2 and N 2s22p3. The exchange correlation energy is described in the local density approximation (LDA) for the exchange correlation functional
by
Ceperley
and
Alder
[18-20].
The
structure
is
optimized
with
the
Broyden–Fletcher–Goldfarb–Shanno (BFGS) [21] method. In calculations, the electronic wavefunctions are expanded in a plane-wave basis set with energy cut-off 280.0 eV. As for the Brillouin-zone sampling, we use the Monkhorst–Pack mesh with 4 × 4 ×4 k-points. The thermodynamic properties are calculated by the quasi-harmonic model code (Gibbs) [22].
3. Results and Discussion 3.1 Structural and elastic properties at zero pressure c-Zr3N4 and c-Hf3 N4 have a cubic Th3P4 structure with the space group I43d (No. 220) and have 14 atoms in unit cell. To obtain the equilibrium structure, we fully optimized both the lattice and internal coordinates of the nitrides using BFGS method [21] without any restrictions. The initial cell
parameters are taken from the experiment values [7]. In Table 1, we present the calculated equilibrium lattice constants, elastic constants, bulk modulus, shear modulus and Vickers hardness of c-Zr3N4 and c-Hf3N4 under zero pressure. The bulk modulus and shear modulus are calculated using the Voigt-Reuss-Hill averaging scheme, which is viewed as the best estimate for the theoretical value of polycrystalline elastic modulus [23]. The Vickers hardness can be obtained by an empirical model proposed by Chen et al. [24]:
H v = 2( k 2G ) 0.585 − 3
(1)
where Hv is Vickers hardness, k the Pugh’s modulus ratio (k=G/B) and G is the shear modulus. The available experimental data and others’ works are also shown in Table 1 to confirm the precision of our calculations. It is clear to see that the equilibrium lattice constants, elastic constants, bulk modulus, and shear modulus are in good agreement with the experimental [7] and previous theoretical values [8,9,11,12]. Our results exhibit a large deviation with the results reported by Xu et al. [10] and Chihi et al. [13], because they calculate the values by using GGA. Moreover, the calculated Vickers hardness basically agrees with the values reported by Mattesini et al. [9], and the difference may mainly due to the different calculation model we chose. 3.2 Thermodynamic properties In this study, the quasi-harmonic Debye model [22] is used to obtain the thermodynamic properties of the nitrides, in which the non-equilibrium Gibbs function G*(V; P, T) is expressed as:
G * (V ; P, T ) = E (V ) + PV + AVib (Θ (V );T )
(2)
Here E(V) is the total energy per unit cell for c-Zr3N4 and c-Hf3N4, which is obtained by using the first principles calculations. Θ(V) is the Debye temperature, and the vibrational Helmholtz free energy AVib can be written as Ref. [25, 26]
9 Θ Θ AVib (Θ;T ) = nkT + 3 ln(1 − e − Θ / T ) − D T 8 T
(3)
where D(Θ/T) represents the Debye integral, and n the number of atoms per formula unit. For an isotropic solid,
Θ=
B [6π 2V 1 / 2 n]1 / 3 f (σ ) s , K M
(4)
where M is the molecular mass per formula unit, Bs the adiabatic bulk modulus. The non-equilibrium
Gibbs function G*(V; P, T) can be minimized with respect to volume V as follows:
∂G * (V ; P,T ) = 0, ∂V P ,T
(5)
The thermal properties such as internal energies, entropy, heat capacity at constant volume CV, and thermal expansion α are, respectively, taken as
9 Θ U = nkT + 3 D(Θ / T ) 8 T
[
(6)
]
S = nk 4 D(Θ / T ) − 3 ln (1 − e− Θ / T )
Θ 3Θ / T CV = 3nk 4 D − Θ / T , − 1 T e
α=
γCV BT V
,
(7) (8)
(9)
Here γ is the Grüneisen parameter, and
γ = −(d ln Θ(V ) / d ln V ) ,
(10)
By using the above formulas (2-10), we calculated some thermodynamic properties of c-Zr3 N4 and c-Hf3N4. The Debye temperature of solid is an important physical quantity for a solid. In Fig. 1a and b, we present the obtained pressure-dependence of Debye temperature under different temperatures for c-Zr3N4 and c-Hf3N4. The Debye temperature of c-Zr3N4 at 0 K and 0 GPa calculated using quasi-harmonic Debye model is 673 K, which is in good agreement with the result reported by Chihi et al. (651.7 K) [13] derived from the averaged sound velocity. However, the calculated Debye temperature of c-Hf3N4 at 0 K and 0 GPa in this study is 550 K, which is obviously higher than the value reported by Chihi et al. (81.4 K) [13] using the same method with the calculation for c-Zr3N4. In order to evaluate the accuracy of our calculated results, we calculated the values again by using the averaged sound velocity. The obtained Debye temperatures are 674 K and 548 K for c-Zr3N4 and c-Hf3N4 respectively, which agree well with the results we calculated using quasi-harmonic Debye model. In Fig. 1, we plotted the Debye temperature Θ at the temperature of 0, 200, 400, and 600 K, respectively. We observe that the Debye temperature Θ tends to similar change for c-Zr3 N4 and c-Hf3N4. The Debye temperature decreases and increases with increasing temperature and pressure, respectively. This is because that the Debye temperature is related to volume V and the adiabatic bulk
modulus Bs. Although volume V increases and decreases with increasing temperature and pressure respectively, the adiabatic bulk modulus Bs is on the contrary. From Formula 4, it is easy to find the adiabatic bulk modulus Bs makes more contribution to the Debye temperature than volume V. The effect of the temperature on the Debye temperature Θ is obviously smaller than that of the pressure. Moreover, with the increasing of pressure, the temperature exhibits smaller and smarter effect on the Debye temperature Θ, which is same to that of HfB2 [27]. The calculated internal energies for c-Zr3N4 and c-Hf3N4 as a function of pressure under different temperatures are depicted in Fig. 2a, b, respectively. It can be seen that the total internal energies increase with pressure and temperature. It is more sensitive to the temperature than the pressure. For example, at 0 GPa pressure, the internal energies increase 151% and 202% for c-Zr3N4
and from c-Hf3N4 respectively, when the temperature changes from 0 K to 600 K. While, at the temperature of 0 K, the internal energies increase 56% and 53% for c-Zr3N4 and c-Hf3N4
respectively, when the pressure changes from 0 GPa to 100 GPa. Moreover, the higher the temperature is, the slower the internal energies increase with pressure. At the temperature of 600 K, the internal energies increase 8.7% and 5.5% for c-Zr3N4 and c-Hf3N4 respectively, when the pressure changes from 0 GPa to 100 GPa. The pressure-dependence of entropy for c-Zr3N4 and c-Hf3N4 under different temperatures are depicted in Fig. 3a, b, respectively. The values of the entropy of c-Zr3N4 are slighty smaller than that of c-Hf3N4.The entropies decrease with pressure (except at 0 GPa) and increase with temperature. Obviously, the entropy is more sensitive to the temperature than the pressure. For example, at 0 GPa pressure, the entropy increases 265% and 201% for c-Zr3N4 and c-Hf3N4 respectively, when the temperature changes from 200 K to 600 K. While, at the temperature of 200 K, the entropy decreases 59% and 52% for c-Zr3N4 and c-Hf3N4 respectively, when the pressure changes from 0 GPa to 100 GPa. When the temperature increases, the decrease of the entropy with pressure is slower and slower. At the temperature of 600 K, the entropy decreases 36% and 29% for c-Zr3N4 and c-Hf3N4 respectively, when the pressure changes from 0 GPa to 100 GPa. The heat capacity is an important property of crystal, which not only provides an essential insight into its vibrational properties, but also possesses many applications. The variations of heat capacity at constant volume C V with pressure are depicted in Fig. 4a and b for c-Zr3N4 and c-Hf3N4 , respectively. The change tendencies of CV are almost same for c-Zr3N4 and c-Hf3N4. CV increases
with temperature and decreases with pressure. It is more sensitive to the temperature than the pressure, especially at low temperature. With the increasing of the temperature, the effect of the pressure on CV decreases gradually, which can be viewed as the general features of the CV of a solid. At the temperature of 600 K, the calculated CV at 0 GPa for c-Zr3N4 and c-Hf3N4 are 165.5 and 168.1 Jmol-1 K-1 respectively, which is close to Dulong-Petit limit 174.5 Jmol-1K-1. The thermal expansion α of c-Zr3N4 and c-Hf3N4 are also plotted in Fig. 5a and b, respectively. The values of the thermal expansion α of c-Zr3N4 are slight higher than that of c-Hf3N4. As shown in Fig. 5a and b, the thermal expansion α decreases with pressure; the higher the temperature is, the faster the thermal expansion α decreases with pressure; and furthermore, the decrease becomes slower as pressure increases. Moreover, the thermal expansion α increases with temperature, and the higher the temperature is, the smaller the effect of temperature on the thermal expansion α is. 4. Conclusions In this paper, we have investigated the thermodynamic properties of two nitrides c-Zr3N4 and c-Hf3N4 using density functional theory within the local density approximation (LDA) and the quasi-harmonic Debye model. The pressure dependence of Debye temperature, internal energies, entropy, heat capacity and thermal expansion under different temperatures are obtained. The main results are as follows: (1) The obtained lattice parameters, elastic constants, bulk modulus, shear modulus and Vickers hardness at ambient pressure are in agreement with the experimental and theoretical values. (2) The calculated Debye temperature of c-Zr3N4 and c-Hf3N4 at 0 K and 0 GPa are 673 K and 550 K, respectively. The Debye temperature decreases and increases with increasing temperature and pressure, respectively. (3) The total internal energies of c-Zr3 N4 and c-Hf3N4 increase with pressure and temperature. (4) The entropies decrease with pressure and increase with temperature, so do the heat capacity at constant volume CV and the thermal expansion α.
Acknowledgment This work is supported by outstanding young talent cultivation plan of science and technology of shihezi university, China (No.:2012ZRKXYQ08)
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Table Captions Table 1. The calculated structural data, elastic constants Cij (GPa), bulk modulus B (GPa), shear modulus G (GPa) and Vickers hardness H (GPa) of c-Zr3 N4 and c-Hf3N4 under zero pressure and related experimental and theoretical data. c-Zr3N 4 Structure
Present
a
Experiment
c-Hf3N4 Other work
Present
work a (Å)
6.66
a
Experiment
work 6.74
b
c
6.69
d
6.69
6.79
6.76
6.70
6.67e 6.69f 6.84g c11 (GPa)
458
-
c
d
f
454.4 325.7 491
540
-
310.1g c12 (GPa)
161
-
165.3c 102d 160f
187
-
-
260
250
d
f
d
f
145.9 119.4 152
170
-
c
148
-
3.127e
6.58f
6.58g
c
443.9d
167.3c
156.5d
b
521
261.7 177.2 270 145.4c 158f 120.3g
174b 163b
304
260
Hv (GPa)
16.2
a
Reference 7, experiment.
b
Reference 8, LDA.
c
Reference 9, LDA.
d
Reference 10, GGA.
e
Reference 11, LDA.
f
Reference 12, LDA.
g
Reference 13, GGA.
-
19.7
g
152.4c
290b
133.1d
275.9c
251.6d
275f 153.7g 173
-
167b 150.7
c
493.2
152f 148.5g
153.7g G (GPa)
6.84d
167 51.0 c
122.4g B (GPa)
6.59c
f
75.5 148
6.59b
493f 358.9g
g
c44 (GPa)
Other work
18.0
-
21.3c
156.6c g
156f
Figures (Color online only)
Figure1. Variation of the Debye temperature of c-Zr 3N4 (a) and c-Hf3N 4 (b) as a function of pressure. Figure2. Variation of the internal energies for c-Zr3N 4 (a) and c-Hf3N4 (b) as a function of pressure. Figure3. Variation of the entropies for c-Zr3N4 (a) and c-Hf3N4 (b) as a function of pressure. Figure4. Variation of the heat capacities for c-Zr3N 4 (a) and c-Hf3N4 (b) as a function of pressure. Figure5. Variation of the thermal expansions for c-Zr 3N4 (a) and c-Hf3N 4 (b) as a function of pressure.
Figure1a
Figure1b
Figure2a
Figure2b
Figure3a
Figure3b
Figure4a
Figure4b
Figure5a
Figure5b
The thermodynamic properties of two nitrides c-Zr3N4 and c-Hf3 N4 with Th3P4 structure are investigated firstly. The Vickers hardness of c-Zr3 N4 and c-Hf3 N4 are calculated. The variations of the Debye temperature, internal energies, entropy, heat capacity and the thermal expansion with pressure and temperature are successfully obtained and discussed.