First-principles study of structural, elastic, electronic and optical properties of orthorhombic NaAlF4

Computational Materials Science 50 (2011) 2822–2827

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First-principles study of structural, elastic, electronic and optical properties of orthorhombic NaAlF4 Qi-Jun Liu ⇑, Zheng-Tang Liu, Li-Ping Feng, Hao Tian State Key Lab of Solidification Processing, College of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 8 December 2010 Received in revised form 19 April 2011 Accepted 26 April 2011 Available online 14 May 2011

a b s t r a c t We have performed the ab initio total energy calculations using the plane-wave ultrasoft pseudopotential technique based on the first-principles density-functional theory (DFT) to study the structural parameters, elastic, electronic, chemical bonding and optical properties of orthorhombic NaAlF4. The calculated lattice parameters are in good agreement with experimental work. The bulk, shear and Young’s modulus, Poisson’s coefficient, compressibility and Lamé’s constants are firstly obtained using Voigt–Reuss–Hill method and the Debye temperature is estimated using Debye-Grüneisen model. Band structure shows a direct band gap at C point. Density of states and charge density have been studied, which show the bonding between Na and F is mainly ionic as well as that between Al and F. In order to clarify the mechanism of optical transitions of orthorhombic NaAlF4, the complex dielectric function, refractive index, extinction coefficient, reflectivity, absorption efficient, loss function and complex conductivity function are calculated. The optical properties and origins of the structure have been analysed. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction The binary system NaFAAlF3 used in aluminum smelting has been extensively investigated [1–3]. Other systems such as NaFACaF2AAlF3 [4], Na3AlF6AAlF3ALiFACaF2 [5], NaFAAlF3AAl2O3 [6], Na3AlF6AAlF3AAl2O3ACaF2ALiFANaCl [7] for the aluminum production have been reported. Among them, the system NaFAAlF3 is a fundamental electrolyte composition, which contains three compounds named mineral cryolite Na3AlF6, mineral chiolite Na5Al3F14 and sodium tetrafluoroaluminate NaAlF4 [8,9]. The well-known cryolite and chiolite have been largely discussed, and this compound NaAlF4 which was first to be found by condensing the vapor of NaFAAlF3 mixture by Howard [10] has also gained wide interests due to its industrial applications such as the excellent substitute for fluoride compounds used as flux in secondary aluminum, the deoxidizing agent during the deoxidation process, etc. [11–20]. Recently, this compound NaAlF4 has been obtained, and its crystal structure and thermal stability have been investigated by X-ray powder diffraction method, thermal analysis and high temperature X-ray diffraction, which show that the compound NaAlF4 has an orthorhombic structure and the area of the thermodynamic stability is between 690 and 730 °C [15]. However, there is a lack of fundamental understanding at the microscopic level on NaAlF4 due to its rather short temperature interval of stability. These problems can be resolved using computational modellings such as the MD method for the structure of molten system ⇑ Corresponding author. Tel.: +86 029 88488013. E-mail address: [email protected] (Q.-J. Liu). 0927-0256/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2011.04.037

(NaF)x(AlF3)1x (x = 0.8, 0.75, 0.67, 0.5) at 1323 K [2], firstprinciples calculations for some molecular properties and thermochemical quantities of Na3AlF6, Na2AlF5, NaAlF4, AlF3, NaF, (NaF)2, (NaAlF4)2 system [12], etc. As far as we know, there is no detailed theoretical work to study its physical properties, so it is necessary to investigate the structural, elastic, electronic and optical properties of NaAlF4 to understand its physical properties for the future technological applications. In this paper, we present a systematic study of structural, elastic, thermodynamic, electronic, chemical bonding and optical properties of orthorhombic NaAlF4 using the plane-wave ultrasoft pseudopotential technique based on the first-principles densityfunctional theory. The paper is organized in the following way: Section 2 gives a brief description of the calculation method, Section 3 is devoted to the results and discussion, including the geometry and structural properties, elastic and mechanical properties, electronic and chemical bonding, optical properties, Section 4 gives our conclusions. 2. Computational methodology In our paper, we use the CASTEP code [21] based on the planewave ultrasoft pseudopotential using the generalized gradient approximation (GGA) with the Perdew–Wang 1991 (PW91) functional [22] within the framework of density functional theory. The ionic cores are represented by ultrasoft pseudopotentials for Na, Al and F atoms. The Na 2s2, 2p6, 3s1 electrons, Al 3s2, 3p1 electrons and F 2s2, 2p5 electrons are explicitly treated as valence electrons. The plane-wave cutoff energy is 410 eV and the Brillouin-zone

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3. Results and discussion

mechanically stable. To the best of our knowledge, this is the first reported values of the elastic constants which await experimental confirmation. According to the Voigt approximation which proposed the averaging of relations expressing the stress in terms of the given strain [30], we can obtain the bulk modulus and shear modulus using the following equations:

3.1. Geometry and structure properties

BVoigt ¼

integration is performed over the 7  7  5 grid sizes using the Monkorst–Pack method for orthorhombic structure optimization. This set of parameters assures the maximum force of 0.01 eV/Å, the maximum stress of 0.02 GPa and the maximum displacement of 5.0  10–4 Å.

The structure of orthorhombic NaAlF4 presented in Fig. 1 belongs to the space group Cmcm and the local symmetry D17 2h . Firstly, the ground state properties of orthorhombic NaAlF4 are studied. The total energy versus volume is fitted to the third-order BirchMumaghan equation of state [23] to obtain the equilibrium lattice constants and other ground state properties. The experimental [15] and computed results are shown in Table 1 which shows the estimations of a, b and c using the GGA exchange–correlation functional are larger than the experimental data of 1.5%, 2.3% and 1.5%, respectively. The difference between our calculated values and experimental data is due to the use of an approximate DFT and the obtained values using the GGA are slightly overestimated to the experimental data. Though there is a difference, the agreement with experimental data can be considered to be very good. The average pressure on the unit cell after the GGA calculation is 0.0034 GPa, and the components of the symmetrised stress ten0 1 0:000417 0 0 A. sor are: @ 0 0:007972 0 0 0 0:002762 3.2. Elastic and mechanical properties

1 ðC 11 þ C 22 þ C 33  C 12  C 13  C 23 Þ 15 1 þ ðC 44 þ C 55 þ C 66 Þ 5

ð2Þ

ð3Þ

According to the Reuss approximation which proposed the averaging of the relations expressing the strain in terms of the given stress [31], the bulk modulus and shear modulus can be gained using the following equations:

BReuss ¼ ½ðS11 þ S22 þ S33 Þ þ 2ðS12 þ S13 þ S23 Þ1

ð4Þ

GReuss ¼ 15½4ðS11 þ S22 þ S33 Þ  4ðS12 þ S13 þ S23 Þ þ 3ðS44 þ S55 þ S66 Þ1

ð5Þ

where Sij is the inverse matrix of the elastic constants matrix C ij , which is given by [24]:

S11 ¼ ðC 22 C 33  C 223 Þ=ðC 11 C 22 C 33 þ 2C 12 C 13 C 23  C 11 C 223  C 22 C 213  C 33 C 212 Þ

ð6Þ

S12 ¼ ðC 13 C 23  C 12 C 33 Þ=ðC 11 C 22 C 33 þ 2C 12 C 13 C 23  C 11 C 223

The elastic constants determine the stiffness of a crystal against an externally applied strain, which are important to provide information of the structural stability and the strength of materials. Elastic constants are defined by means of a Taylor expansion of the total energy, namely the derivative of energy as a function of a lattice strain [24–26]. The calculated nine independent components of the elastic stiffness constants are presented in Table 2. According to the mechanical stability criteria of the elastic constants in orthorhombic structure [27–29]:

ðC 11 þ C 22  2C 12 Þ > 0;

ðC 11 þ C 33  2C 13 Þ > 0;

ðC 22 þ C 33  2C 23 Þ > 0;

C 11 > 0;

C 22 > 0;

GVoigt ¼

1 2 ðC 11 þ C 22 þ C 33 Þ þ ðC 12 þ C 13 þ C 23 Þ 9 9

ð7Þ

S13 ¼ ðC 12 C 23  C 22 C 13 Þ=ðC 11 C 22 C 33 þ 2C 12 C 13 C 23  C 11 C 223  C 22 C 213  C 33 C 212 Þ

ð8Þ

S22 ¼ ðC 11 C 33  C 213 Þ=ðC 11 C 22 C 33 þ 2C 12 C 13 C 23  C 11 C 223  C 22 C 213  C 33 C 212 Þ

ð9Þ

S23 ¼ ðC 12 C 13  C 11 C 23 Þ=ðC 11 C 22 C 33 þ 2C 12 C 13 C 23  C 11 C 223  C 22 C 213  C 33 C 212 Þ

C 33 > 0;

C 44 > 0; C 55 > 0; C 66 > 0; ðC 11 þ C 22 þ C 33 þ 2C 12 þ 2C 13 þ 2C 23 Þ > 0;

 C 22 C 213  C 33 C 212 Þ

ð1Þ

we can find that the elastic constants in Table 2 satisfy all of these stability conditions, which indicates orthorhombic NaAlF4 is

ð10Þ

S33 ¼ ðC 11 C 22  C 212 Þ=ðC 11 C 22 C 33 þ 2C 12 C 13 C 23  C 11 C 223  C 22 C 213  C 33 C 212 Þ

ð11Þ

S44 ¼ 1=C 44

ð12Þ

S55 ¼ 1=C 55

ð13Þ

S66 ¼ 1=C 66

ð14Þ

Al

Hill [32] have proved that the Voigt and Reuss equations represent upper and lower bounds of elastic constants, respectively. Then, he took an arithmetic mean values of the two approaches given by:

Na

F3

F1

F2

Fig. 1. The structure of orthorhombic NaAlF4.

BHill ¼

 1 BReuss þ BVoigt 2

ð15Þ

GHill ¼

 1 GReuss þ GVoigt 2

ð16Þ

From the calculated results using Voigt–Reuss–Hill approach, we can obtain Young’s modulus (E), Poisson’s coefficient (m), Lamé’s constants (l, k), shear sound velocity (ts ), longitudinal

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Table 1 Calculated lattice parameters a, b, c (in Å), volume (in Å3), atomic coordinates x, y, z (in fractional units of cell parameters) and bulk modulus B (in GPa) compared with available experimental data [15] for orthorhombic NaAlF4 (the atomic coordinates presented below the lattice parameters). Atom

a

b

c

V

B

15.2835 0.0839 0.1972 0.1681 0.0800 0

5.3387 1/4 3/4 0.0146 1/4 0

299.21

75.67

Al Na F1 F2 F3

3.6671 1/2 0 1/2 0 1/2 3.6124(1) 1/2 0 1/2 0 1/2

14.9469(7) 0.0836(2) 0.1917(2) 0.1691(1) 0.0820(4) 0

5.2617(3) 1/4 3/4 0.0115(5) 1/4 0

284.10(2)

–

Al Na F1 F2 F3

This work CASTEP GGA-PW91

Expt. [15]

Table 2 Calculated elastic constants of orthorhombic NaAlF4 (in GPa).

GGA

C11

C12

C13

C22

C23

C33

C44

C55

C66

185.73

28.99

37.41

112.68

30.83

117.37

41.96

45.74

38.42

sound velocity (tl ), average sound velocity (tm ) and Debye temperature (H) from [25,33,34]

EX ¼

9BX GX GX þ 3BX

ð17Þ

mX ¼

  1 BX  ð2=3ÞGX 2 BX þ ð1=3ÞGX

ð18Þ

lX ¼ EX =½2ð1 þ mX Þ

ð19Þ

kX ¼ mX EX =½ð1 þ mX Þð1  2mX Þ

ð20Þ

ts ¼

pffiffiffiffiffiffiffiffiffiffiffiffi GX =q

ð21Þ

tl ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðBX þ 4GX =3Þ=q

ð22Þ

tm ¼ ½ð2=t3s þ 1=t3l Þ=31=3

ð23Þ

The calculated mechanical and thermo-acoustics properties of orthorhombic NaAlF4 are summarized in Table 3. We can see BX > GX which shows that the parameter limiting the stability of orthorhombic NaAlF4 is the shear modulus [35]. Additionally, in order to evaluate material ductility or brittleness, a parameter BX/GX has been defined that the shear modulus is relative with the resistance to plastic deformation and the bulk modulus is associated with the resistance to fracture and Pugh proposed that if the BX/ GX is more than the critical value (1.75), the material would be ductile [36]. The values of BX/GX shown in Table 3 indicate orthorhombic NaAlF4 is brittle. The lower calculated values of mX presented in Table 3 mean that an increasing in the volume is associated with the uniaxial tensile deformation. The calculated results of the shear, longitudinal, average sound velocity and Debye temperature are shown in Table 3. Unfortunately, as far as we know, there are no data in the literature about these parameters for comparison. Hence, these values still await experimental confirmation.

3.3. Electronic band structure, density of states and chemical bonding

 1 h 3nN q 3 H¼ tm k 4p M

ð24Þ

where X is Voigt, Reuss or Hill, h is Planck’s constant, k Boltzmann’s constant, n the number of atoms in unit cell, N Avogadro’s number, q the density, M molecular mass.

The calculated energy band structure of orthorhombic NaAlF4 along with the high-symmetry points of the Brillouin zone is shown in Fig. 2. The top of the valence band is taken as the zero of energy. Table 4 shows the band gap energies together with the valence-to-conduction band transition for orthorhombic NaAlF4.

Table 3 Calculated bulk modulus (B), shear modulus (G), Young’s modulus (E) (in GPa), BX/GX, Poisson coefficient (m), compressibility (b) (in GPa1), Lamé constants (l, k) (in GPa), shear sound velocity (ts ), longitudinal sound velocity (tl ), average sound velocity (tm ) (in m/s) and Debye temperature (H) (in K). BX BR

BV

BH

GX (lX) GR

GV

GH

BX/GX X=R

X=V

X=H

EX ER

EV

EH

64.64

67.80

66.22

44.69

46.46

45.57

1.45

1.46

1.45

108.96

113.46

111.20

mX

b

mR

mV

mH

0.219

0.221

0.220

tXs X=R 3998

kX

0.0155

tXl X=V 4076

X=H 4037

X=R 6665

X=V 6812

kR

kV

34.85

36.83

tXm X=H 6739

X=R 4423

X=V 4510

kH 35.84

HX X=H 4467

X=R 568

X=V 579

X=H 574

Q.-J. Liu et al. / Computational Materials Science 50 (2011) 2822–2827

Fig. 2. Band structure of orthorhombic NaAlF4 along with the high-symmetry points of the Brillouin zone.

Table 4 Band gap energies Eg (eV) and symmetry of the valenceto-conduction band transitions of orthorhombic NaAlF4. Valence-to-conduction band transition

Band gap energies

C? C C?Z C?T C?Y C?S C?R

6.726 8.586 9.235 7.708 10.255 11.048 9.094 9.855 7.787 10.281 11.458

Z?Z T?T Y?Y S?S R?R

2825

In order to elucidate the nature of the electronic structure, the total and partial density of states are presented in Fig. 3. We can see that the bands of the lowest energy from 48.21 eV to 46.38 eV are due to Na-s states. There are three peak values at 20.86 eV, 20.04 eV and 19.07 eV in the middle valence bands between 22.65 eV and 17.99 eV. The first peak corresponds to F2-2s and F3-2s states, the second peak corresponds to F1-2s staes and the third peak is due to Na-2p ststes. Additionally, Al-3s and Al-3p states are a small quantity of contribution in the middle valence bands. The upper valence bands are essentially dominated by F-2p states, with a minor admixture from Al-3s and Al-3p states, which induces a covalent bonding contribution in orthorhombic NaAlF4 though these bondings are weak. The conduction bands are composed mostly of Al-3s and Al-3p. The s, p states of Na and F are also contributing to the conduction bands, but the values are quite small. Contours maps of the charge density of orthorhombic NaAlF4 in the (1 0 0), (2 0 0) and (4 0 0) planes of primitive cell are shown in Fig. 4 to understand of the electronic structure and chemical bonding of orthorhombic NaAlF4. We can see that the NaAF bonds are mainly ionic characters and AlAF bonds are mainly ionic characters as well as partially covalent characters

It can be see that the direct band gap appears at C point with the energy of 6.726 eV. Though there is no reported experimental data of band gap energy, the calculated value is smaller than the experimental data due to the well-known underestimation of conduction band state energies in DFT calculations [37].

Fig. 3. The total and partial density of states of orthorhombic NaAlF4.

Fig. 4. Charge densities in the (a) (1 0 0), (b) (2 0 0) and (c) (4 0 0) planes of orthorhombic NaAlF4.

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due to F-2p and Al-3s (3p) hybridization, which are in agreement with our results of population analysis presented in Table 5. 3.4. Optical properties In this section, we investigate the optical response functions which are shown in Figs. 5–7. The imaginary parts e2 ðxÞ and the real parts e1 ðxÞ of the complex dielectric function as a function of the photon energy are presented in Fig. 5. The absorptive parts of the imaginary parts exhibit three structures labeled A, B and C. Structure A (two peaks at 10.573 eV and 11.848 eV from (1 0 0) direction, one peak at 10.755 eV from (010) direction and one peak at 10.682 eV from (0 0 1) direction) corresponds mainly to the transitions F-2pp to the lower conduction bands. Structure B (one peak at 16.109 eV from (1 0 0) direction, one peak at 15.152 eV from (0 1 0) direction and one peak at 15.854 eV from (0 0 1) direction) corresponds mainly to the transitions F-2p to the middle conduction bands as well as p bonding from F-2p and Al-3p to the middle conduction bands. Structure C (one peak at 19.023 eV from (1 0 0) direction, one peak at 18.404 eV from (0 1 0) direction and one peak at 18.768 eV from (0 0 1) direction) corresponds mainly to the transitions F-2p to the upper conduction bands as well as sr bonding from F-2p and Al-3s (3p) to the upper conduction bands. At the moment, the calculated positions of the structures are not available due to the GGA underestimating the band gap of orthorhombic NaAlF4. However, these results are good prediction of the electrons transitions in the orthorhombic NaAlF4. The real parts of the complex dielectric function have been obtained using the Kramers-Kroning transformation [38,39], which are shown in Fig. 5. The calculated static dielectric constants are

Fig. 6. The calculated refractive index and extinction coefficient of orthorhombic NaAlF4 from (1 0 0), (0 1 0) and (0 0 1).

Table 5 Calculated results of population analysis. Atom

s

p

Total

Charge (e)

F1 F2 F3 Na Al

1.95 1.93 1.93 2.13 0.40

5.73 5.76 5.75 6.04 0.70

7.68 7.69 7.68 8.17 1.10

0.68 0.69 0.68 0.83 1.90

Fig. 7. The calculated reflectivity, absorption coefficient, loss function and complex conductivity function of orthorhombic NaAlF4 from (1 0 0), (0 1 0) and (0 0 1).

Fig. 5. The calculated dielectric function from (1 0 0), (0 1 0) and (0 0 1) of orthorhombic NaAlF4.

1.667, 1.662 and 1.657 from (1 0 0), (0 1 0) and (0 0 1) directions, respectively. The calculated refractive index and the extinction coefficient are shown in Fig. 6. The static refractive index are 1.291, 1.289 and 1.287 from (1 0 0), (0 1 0) and (0 0 1). We can see that the refractive index increase with energy reaching three peaks at 8.060 eV, 14.834 eV, 17.967 eV from (1 0 0) direction, 7.950 eV, 13.086 eV, 17.311 eV from (0 1 0) direction and 7.987 eV, 13.341 eV, 17.821 eV from (0 0 1) direction due to probably interband transitions and these curves then decrease to the minimum levels at 23.649 eV, 23.685 eV and 23.357 eV from (1 0 0), (0 1 0) and (0 0 1) directions. The origin of the structures of the extinction coefficient are the same to the imaginary parts of the complex dielectric function. Fig. 7 shows the calculated results on the reflectivity,

Q.-J. Liu et al. / Computational Materials Science 50 (2011) 2822–2827

absorption coefficient, loss function and complex conductivity function from polarization vectors (1 0 0), (0 1 0) and (0 0 1) for orthorhombic NaAlF4. However, there is no experimental results for the optical properties for orthorhombic NaAlF4. We hope the calculated values can motivate experimental work to study this compound and offer a theoretical basis for the experiments and applications. 4. Conclusion We have performed first-principles computations on orthorhombic NaAlF4, including structural parameters, elastic, electronic, bonding and optical properties. It has been shown that the structural parameters obtained after relaxation are in favorable agreement with the experimental work. We have calculated the elastic constants and derived the bulk, shear and Young’s modulus, Poisson’s coefficients, compressibility, Lamé’s constants and Debye temperature. The results show orthorhombic NaAlF4 is mechanically stable and behaves in a brittle manner. The electronic structure presents that orthorhombic NaAlF4 has a direct band gap with 6.726 eV. From the band structure, the charge density is obtained and the chemical bonding is analyzed, which show the bonding between Na and F is mainly ionic while that between Al and F is mainly ionic as well as partially covalent. The calculated optical properties show an optical anisotropy from (1 0 0), (0 1 0) and (0 0 1) directions for orthorhombic NaAlF4. Acknowledgements This work was financially supported by the Scholarship Award for Excellent Doctoral Student granted by Ministry of Education, China, the Doctorate Foundation of Northwestern Polytechnical University (Contract no. cx201005), the 111 Project (Contract No. B08040) and theResearchFundoftheStateKeyLaboratoryofSolidificationProcessing (NWPU),China (Contract No. 58-TZ-2011). References [1] Q. Xu, Y.M. Ma, Z.X. Qiu, Acta Metall. Sinica 13 (2000) 1125–1130 (English Letters).

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