Electronic structure, elastic and thermodynamic properties of α-phase Na3N under pressure from first principles

Physica B 407 (2012) 2272–2277

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Electronic structure, elastic and thermodynamic properties of a-phase Na3N under pressure from first principles Fanjie Kong a,n, Yanhua Liu b, Baolin Wang a, Yanzong Wang c, Yanfei Hu d, Lili Wang e, Lijuan Tang f a

Department of Physics, Yancheng Institute of Technology, Jiangsu 224051, China School of Information Engineering, Yancheng Institute of Technology, Jiangsu 224051, China c Department of Physics, Huaiyin Institute of Technology, Jiangsu 223003, China d School of Science, Sichuan University of Science and Engineering, Zigong 643000, China e Computer Application Institute of CAEP, Academy of Engineering Physics of China, Mianyang 621900, China f Yantai Research Institute of China Agricultural University, Yantai 264670, China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 November 2011 Received in revised form 1 March 2012 Accepted 5 March 2012 Available online 9 March 2012

The structural, electronic, elastic and thermodynamic properties of a-phase Na3N under pressure are investigated by performing first principles calculations within generalized gradient approximation. The elastic constants, bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio dependencies on pressure are also calculated. The thermodynamic properties of the a-phase Na3N are calculated using the quasi-harmonic Debye model. The dependencies of the heat capacity and the thermal expansion ¨ coefficient, as well as the Gruneisen parameter on pressure and temperature are investigated systematically in the ranges of 0–1 GPa and 0–100 K. Crown Copyright & 2012 Published by Elsevier B.V. All rights reserved.

Keywords: Na3N Elastic constant Thermodynamic properties

1. Introduction Recently, single crystalline and polycrystalline sodium nitride have been synthesized successfully on a laboratory scale by reaction of metallic sodium or a liquid Na–K alloy with plasmaactivated nitrogen [1]. It is found to be crystallized in cubic antiReO3-type structure at ambient pressure from x-ray diffraction data collected on powder and single crystals.a-phase Na3N is metastable with respect to the decomposition into elements at ambient pressure and temperature. It decomposes exothermally at around 100 1C. The open anti-ReO3-type structure transforms to hexagonal Li3N-type structure at 1.1 GPa. Two new phases with orthorhombic anti-YF3 structure and hexagonal Li3P structure appear simultaneously at 3.4 GPa. The Li3Bi-type is followed ¨ et al. [3] identified a number above 22.2 GPa [2]. Christian Schon of potential structural candidates for sodium nitride theoretically using the Hartree–Fock method. They calculated the volume– energy curves of Na3N with the Li3N, Li3P and Li3Bi types and concluded that the structure with the lowest energy at zero pressure is the Li3P-type. Vajenine et al. [2] investigated the structural phase transitions of Na3N at high pressure in the range up to 36 GPa. They used the angle dispersive synchrotron x-ray

n

Corresponding author. Tel.: þ86 515 88298271. E-mail address: [email protected] (F. Kong).

diffraction in combination with the diamond-anvil cell technique to identify four high pressure phases, namely, the Li3N, anti-YF3, Cu3P and Li3Bi types. They also utilized total energy calculations to compare with the obtained experimental high pressure phases. ¨ Bjorn Baumeier et al. [4] have systematically investigated the electronic structure of Na3N by density-functional calculations employing self-interaction-corrected pseudopotentials with the atom-centered Gaussian type orbitals (GTOs) which has been shown before to yield results in much better agreement with experiment than standard local-density calculations . The experimental high pressure study drives us to study the physical properties of sodium nitride under pressure theoretically. The elastic constants under pressure are very important to determine the response of the crystal to external forces, as characterized by the bulk and shear modulus, and they obviously play an important role in determining the strength and hardness of materials. Thermodynamic properties as a function of temperature and pressure may provide important information to understand the phase transitions and phase diagram. To our knowledge, there are few literatures reporting on the elastic and thermodynamic properties of Na3N under high pressure. A further discussion of physical properties of Na3N under pressure would be very meaningful at this point because of the lack of experimental data. Therefore, the aim of this work is to give a detailed description of the electronic, elastic and thermodynamic properties of a-phase Na3N under hydrostatic pressure using the plane

0921-4526/$ - see front matter Crown Copyright & 2012 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2012.03.013

F. Kong et al. / Physica B 407 (2012) 2272–2277

wave method. In Section 2, we give a brief description of the theoretical method. The results and discussions are present in Section 3. Conclusions derived from our calculations are drawn in Section 4.

2. Method of calculation

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elastic material. They can be calculated in terms of the computed data using the following relations [13] " # 1 Bð2=3ÞG  ð6Þ n¼  2 B þ ð1=3ÞG Y¼

9GB G þ3B

ð7Þ

2.1. The electronic structure method The present results have been obtained using the CASTEP code [5], which is based on pseudo-potentials and plane waves. The self-consistent norm-conserving pseudo-potentials were used to replace the interactions between the core electrons and the valence electrons [6]. The electronic states were 2s22p3 for N, and 2s22p63s1 for Na. Plane waves were used as basis set for the electronic wave functions, the exchange-correlation effects were taken into account within the Perdew–Burke–Ernzerhof (PBE) scheme in the generalized gradient approximation (GGA) pseudopotential [7]. We have got a good convergence for the bulk totalenergy calculation with the choice of cut-off energies at 770 eV using the 10  10  10 Monkhorst-Pack mesh grid [8]. For accurate description of the mechanical properties of a-phase Na3N, a much denser 14  14  14 Monkhorst-Pack mesh grid is used to calculate the elastic constant of a-phase Na3N. 2.2. The elastic constants For a cubic lattice, there are three independent elastic constant components, C11, C12, and C44. They can be calculated from the stress variations by applying strains to the equilibrium configuration [9,10]. The elastic constant is defined as followings cijkl ¼

@sij ði,j,k,l ¼ x,y,zÞ @ekl

d 0

The derivatives in Eq. (1) were evaluated numerically by making a series of calculations with d ¼{ 70.00170.003} and fitting the calculated sij to quadratic polynomial in ekl, the linear terms of which provide the elastic constants. The elastic anisotropy of the cube crystal can be characterized by the Zener anisotropy factor A, which represents the ratio of the two extreme elastic-shear coefficients [11]. A¼

2C 44 C 11 C 12

Debye temperature (YD) is a fundamental physical property and used to distinguish between high- and low-temperature regions for a solid. If T4 YD we expect all modes to have energy kBT, and if To YD one expects high-frequency modes to be frozen [14]. In the present case, Debye temperature (YD) is estimated for Na3N, using the calculated elastic constant data, in terms of the following classical relations [15]    h 3n NA r 1=3 YD ¼ vm ð8Þ k 4p M where vm is the average wave velocity, and is approximately given by " !#1=3 1 2 1 þ ð9Þ vm ¼ 3 v3t v3l where vl and vt are the longitudinal and transverse elastic wave velocity, respectively, which are obtained from Navier’s equation sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3B þ4G vl ¼ ð10Þ 3r

ð1Þ

where sij is the ij component of the stress tensor and ekl is the kl component of the strain tensor. Following the conventional symbol C11 ¼cxxxx, C12 ¼cxxyy, C44 ¼cxyxy, we use the following strain tensor to calculate the elastic constant. 0 1 d 0 0 B C ð2Þ @0 0 dA 0

2.3. The Debye temperature

ð3Þ

For comparison with the macroscopic mechanical parameters, we calculated the bulk modulus B and the shear modulus G and they can be given by [12] B ¼ ðC 11 þ 2C 12 Þ=3

ð4Þ

G ¼ ðGV þ GR Þ=2

ð5Þ

GV is the Voigt shear modulus corresponding to the upper bound of G values and GR is the Reuss shear modulus corresponding to the lower bound of G values and can be written as GV ¼(2Cs þ3C44)/5, GR ¼15(6/Cs þ 9/C44)  1, Cs ¼(C11–C12)/2, Cs is the shear modulus. The Poisson ratio v reflects the stability of a crystal against shear and the Young’s modulus is a measure of the stiffness of an

vt ¼

sffiffiffiffi G

r

ð11Þ

2.4. Thermodynamic properties The quasi-harmonic Debye model is used to investigate the thermodynamic properties of Na3N. The method has been successfully utilized to predict the thermodynamic properties of material [16]. In the quasi-harmonic Debye model [17], the non-equilibrium Gibbs function Gn(V,P,T) can be written as Gn ðV,P,TÞ ¼ EðVÞ þ PV þAVib ðYðVÞ; TÞ

ð12Þ

where E(V) is the total energy, PV corresponds to the constant hydrostatic pressure condition, Y(V) is the Debye temperature, and the vibrational contribution AVib can be written as [18,19]   9Y þ3lnð1eY=T ÞDðY=TÞ AVib ðY; TÞ ¼ nkB T ð13Þ 8T where D(Y/T) represents the Debye integral, n is the number of atoms per formula unit. The non-equilibrium Gibbs function G*(V,P,T) as a function of (V,P,T) can be minimized with respect to volume V. By solving Eq. (10), one can get the thermal equation of state (EOS) V (P,T). The isothermal bulk modulus BT, the heat capacity CV, and the thermal expansion coefficient a are given by [17] ! @2 Gn ðV,P,TÞ ð14Þ BT ðP,VÞ ¼ V @V 2 P,T   3Y=T C V ¼ 3nkB 4DðY=TÞ Y=T 1 e

ð15Þ

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a¼

F. Kong et al. / Physica B 407 (2012) 2272–2277

gC V BT V

ð16Þ

¨ where g is the Gruneisen parameter and it is defined as

g¼

dlnYðVÞ dlnV

ð17Þ

3. Results and discussion 3.1. Structure properties We calculated the total energy as a function of the unit-cell volume around the equilibrium cell volume V0. The calculated total energies are fitted to the Birch–Murnaghan equation of state [20] to obtain an analytical interpolation of our computed points from which to calculate derived structural properties. ! 0 B0 V ðV 0 =VÞB0 B0 V 0 EðVÞ ¼ E0 þ 0 þ 1  0 ð18Þ B00 1 B0 1 B0 where E0 is the equilibrium energy, B0 is the bulk modulus, B00 is the first derivative of B0 with pressure. The calculated lattice constant a, bulk modulus B0 and its pressure derivative B00 are presented in Table 1, together with the available experimental and theoretical results. It can be seen that the obtained lattice constant for a-phase ˚ which deviates 0.78% from the experimental value. Na3N is 4.77 A, The bulk modulus and the pressure derivative of bulk modulus of a-phase Na3N are close to the experimental and calculated values using the full potential and linear combinations of the augmented plane wave method (FP-LAPW). Taking into the contributions from lattice vibration of Eq. (13) account, the variational minimum of the non-equilibrium Gibbs function corresponds to the thermal equilibrium EOS V (P,T). The isothermal EOS of a-phase Na3N is shown in Fig. 1(a), it can be seen that the volume decreases linearly with pressure at different temperatures. The dependencies of BT and BT’ on pressure are plotted in Fig. 1(b) and (c), respectively. The bulk modulus BT increases and the first derivative of the bulk modulus B0T decreases linearly with pressure. The linear dependencies of volume, bulk modulus and the first derivative of the bulk modulus on pressure can be attributed to the nature of volumeGibbs energy papa curve. 3.2. Electronic structure The band structure and the density of states for a-phase Na3N are presented in Fig. 2. It can be seen that the calculated band structure is characterized with the Fermi level across the energy band showing metallic electronic structure. The calculated band structure is contradicted with the experimental observed semiconductor nature of the a-phase Na3N with band gap of 1.6 eV [1].

Fig. 1. Isothermalequation of state (a), dependencies of bulk modulus (b) and its pressure derivative (c) of a-phase Na3N on pressure at different temperature.

The discrepancy is due to self-interaction errors and lack of an integer discontinuity of the exchange correlation energy and potential upon changing the number of electrons in the conventional frame work of density functional theory, which leads to the underestimation of the energy band gap [21]. The mixture of 2p orbital of N atom and 3s orbital of Na contributes to energy range from  1.89 eV to the Fermi level. The energy bands from the Fermi level to 10 eV are mainly composed by the 3s and 2p orbitals of Na and 2p orbital of N. The mixture of the 2p orbital of N atom and 3s orbital of Na atom around and above the Fermi level leads to the calculated metallic band structure of a-phase Na3N similar to the band structure calculation based on LDA [4]. However, the inclusion of the self-interaction corrections significantly reduces the mixture of N 2p and Na 3s states around the Fermi level so that a gap opens [4]. 3.3. Elastic constants and the Debye temperature

Table 1 Lattice constant, bulk modulus and the pressure derivative of bulk modulus of a-phase Na3N.

Present work Exp GTOs-LDA GTOs-SIC FP-LAPW a b

Reference [1]. Reference [4].

˚ a0 (A)

B0 (Mbar)

B00

4.77 4.7330a 4.57b 4.56b 4.7485a

0.21 0.2042a 0.27b 0.28b 0.2192a

4.17

4a

The dependences of the elastic constants Cij and the bulk modulus B on pressure for the a-phase Na3N within 0–1 GPa are given in Table 2. The dependencies of elastic constants Cij, bulk modulus B and shear modulus CS on pressure for the a-phase Na3N are shown in Fig. 3(a). It is shown that the elastic constants C11, C12, bulk modulus B and shear modulus CS increase linearly with increasing pressure, but with different slopes. The elastic constant C11 shows the largest slope which means much stronger resistance to unixal stress along [100] direction for a-phase Na3N under pressure and the elastic constant of C12 remains unchanged. However, the elastic constant C44 decreases slightly with increasing pressure.

F. Kong et al. / Physica B 407 (2012) 2272–2277

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Fig. 2. Band structure (a), density of states (b), Na partial density of states (c) and N partial density of states (d) of a-phase Na3N.

The calculated Zener anisotropy factor A, Poisson ratio n, Young’s modulus Y, shear modulus G for a-phase Na3N are given in Table 3. The dependencies of bulk, Young’s and shear modulus on pressure are depicted in Fig. 3(b), they all increase with pressure indicating the resistance to uniform compression, linear stress and shear is enhanced under pressure. The dependencies of

the Zener anisotropy factor A and Poisson ratio n on pressure for a-phase Na3N are presented in Fig. 4, the result shows the Zener anisotropy factor decreases with pressure indicating that elastic anisotropy decreases under high pressure. The Poisson ratio for a-phase Na3N increases slightly with pressure, ranging from 0.27 to 0.30 between 0 and 1 GPa.

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F. Kong et al. / Physica B 407 (2012) 2272–2277

Using the calculated elastic constant and Eqs. (9)–(11), the Debye temperature along with the longitude, transverse, average elastic wave velocity is presented in Table 4.

3.4. Thermodynamic properties The specific volume heat capacity CV can be calculated by Eq. (16) and the dependence of heat capacity on temperature at different pressures is illustrated in Fig. 5(a), the heat capacity at constant volume follows the Debye model in low temperature and it illustrates that the heat capacity increases with temperature at the same pressure and decreases with the pressure at the same temperature, the influences of the temperature on the heat capacity are much more significant than that of the pressure on it. Within the quasi-harmonic approximation, the anharmonicity is restricted to the thermal expansion. The dependence of the thermal expansion coefficient a on temperature between 0 and 1 GPa is plotted in Fig. 5(b). The calculated thermal expansion coefficient is 9.33601  10  5 K  1 at 100 K and 0 GP. The effect of pressure on the thermal expansion coefficient is small in low temperature and the effect is increasingly obvious as the temperature increases. As pressure increases, the thermal expansion coefficient decreases slowly.

¨ Gruneisen constant, which describes the effect that changing the volume of a crystal lattice has on its vibrational properties, and, as a consequence, the effect that changing temperature has on the size or dynamics of the lattice, can correctly predict the anharmonic properties of a material, such as the thermal expansion coefficient, the temperature dependence of phonon frequen¨ cies and linewidths. The dependence of Gruneisen constant on ¨ pressure is shown in Fig. 5(c). It can be seen that the Gruneisen constant g decreases significantly with pressure but the temperature effect becomes less pronounced. Table 3 Calculated Zener anisotropy factor (A), Poisson ratio (v), Young’s modulus (Y), shear modulus (G) for a-phase Na3N. A

n

Y (GPa)

G (GPa)

0.27

0.27

29.02

11.39

Table 2 Elastic constant Cij, the shear modulus Cs, the bulk modulus B of a-phase Na3N at different pressure. Pressure (GPa)

C11 (GPa)

C12 (GPa)

C44 (GPa)

Cs (GPa)

B (GPa)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

53.27 54.23 55.14 56.08 56.69 57.64 58.71 59.60 60.54 61.68 62.46

5.36 5.54 5.62 5.85 5.69 5.94 6.28 6.48 6.61 7.02 7.07

6.57 6.55 6.55 6.50 6.41 6.37 6.32 6.22 6.18 6.15 6.11

23.95 24.38 24.76 25.11 25.50 25.85 26.21 26.56 26.96 27.33 27.69

21.33 21.77 22.12 22.60 22.69 23.18 23.76 24.19 24.59 25.24 25.53

Fig. 4. Dependencies of Zener anisotropy factor and Poisson ratio on pressure.

Fig. 3. Dependencies of elastic constant (a) and elastic modulus (b) of a-phase Na3N on pressure.

F. Kong et al. / Physica B 407 (2012) 2272–2277

Table 4 Longitude, transverse, mean elastic wave velocity and Debye temperature for a-phase Na3N. vl (Km/s)

vt (Km/s)

vm (Km/s)

YD (K)

5.36

2.99

3.33

338.51

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experimental ones. The dependence of the elastic properties on pressure is investigated. It is found that the elastic constants and the bulk modulus vary linearly with elevated pressure. The shear modulus, Young’s modulus, Poisson’s ratio at high-pressure as well as the sound velocity and Debye temperature are calculated. The thermodynamic properties are calculated using the quasiharmonic Debye model. The dependencies of the heat capacity, ¨ the thermal expansion coefficient and the Gruneisen parameter on pressure and temperature are obtained systematically in the ranges of 0–1 GPa and 0–100 K.

Acknowledgments The project was supported by Natural Science Foundation of China (Grant nos. 10874052 and 11174242), the Foundation of Yancheng Institute of Technology, the research fund of Key Laboratory for Advanced Technology in Environmental Protection of Jiangsu Province (Grant no. AE201034), the Education Department of Sichuan Province (Grant no. 11ZB099) and the Third batch of Key Technology Research Fund of Zigong Science and Technology Bureau in 2011 (Grant no. 3).

References

Fig. 5. Dependencies of the heat capacity (a), the thermal expansion coefficient ¨ a (b) and Gruneisen constant (c) on temperature at different pressure.

4. Conclusion In summary, the electronic structure, elastic and thermodynamic properties of a-phase Na3N under high pressure have been investigated using the first principles calculations. The structural parameters have been calculated. It is found that the calculated lattice constant and the EOS parameters are in agreement with the

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