Structural phase transition, mechanical and ...

Computational Materials Science xxx (2013) xxx–xxx

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Structural phase transition, mechanical and optoelectronic properties of the tetragonal NaZnP: Ab-initio study A. Djied a, H. Khachai b, T. Seddik a, R. Khenata a,⇑, A. Bouhemadou c, N. Guechi c, G. Murtaza e, S. Bin-Omran d, Z.A. Alahmed d, M. Ameri f a

Laboratoire de Physique Quantique et de Modélisation Mathématique, Université de Mascara, 29000 Mascara, Algeria Physics Department, Djillali Liabes University of Sidi Bel-Abbes, Algeria Laboratory for Developing New Materials and their Characterization, Department of Physics, Faculty of Science, University Setif 1, 19000 Setif, Algeria d Department of Physics and Astronomy, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia e Modeling Laboratory, Department of Physics, Islamia College Peshawar, Pakistan f Physics and Chemistry of Advanced Materials Laboratory, Djillali Liabès University, BP: 89, Sidi Bel-Abbès 22000, Algeria b c

a r t i c l e

i n f o

Article history: Received 18 September 2013 Received in revised form 18 November 2013 Accepted 23 November 2013 Available online xxxx Keywords: NaZnP FP-APW + lo Phase transition Elastic constants Electronic properties Optical properties

a b s t r a c t Ab-initio full potential augmented plane wave plus local orbitals method has been used to investigate the structural phase transition, mechanical and optoelectronic properties of the Nowotny–Juza filled-tetrahedral compound NaZnP. The exchange-correlation potential was treated within the generalized gradient approximation of Perdew–Burke and Ernzerhof (GGA-PBE) and the modified Becke–Johnson potential (TB-mBJ) to improve the accuracy of the electronic band structure. Total-energy and geometry optimizations have been carried out for all structural phases of NaZnP. The following sequence of pressure-driven structural transitions has been found: Cu2Sb-type ? b-phase ? a-phase. The single-crystal elastic constants of NaZnP in the Cu2Sb-type structure have been calculated using total-energy versus strain method and their corresponding elastic moduli of polycrystalline aggregate, including Young’s modulus, shear modulus and Poisson’s ratio, have been derived. From the elastic parameters, it is inferred that this compound is brittle in nature. The elastic anisotropy was studied in detail using three different indexes; especially the 3D direction dependence of the Young’s modulus was visually described. Furthermore, calculated electronic band structure shows that NaZnP in the Cu2Sb-type phase has a direct energy band gap (C–C). The TB-mBJ approximation yields larger fundamental band gaps compared to those of PBEGGA. The examined charge density distributions for the Cu2Sb-type structure show a covalent character for Zn–P bond and ionic nature for Na–P bond. Additionally, real and imaginary parts of the dielectric function, reflectivity and energy loss function spectra have been calculated for radiation up to 30.0 eV with an incident radiation polarized parallel to both [1 0 0] and [0 0 1] crystalline directions. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction The Nowotny–Juza filled-tetrahedral compounds (FTC) [1,2] with the general formula AIBIICV (AI = Li, Na, Cu, Ag; BII = Mg, Zn and CV = N, P, As, Sb and Bi) have received considerable attention because of their promising properties for technological applications [3–31]. Most of these compounds usually crystallize in a cubic structure which can be obtained from the zinc-blende DIIICV by transmuting the DIII to the isovalent AI + BII atom pair. The zincblende structure (space group F43m) can be characterized by four lattice sites in the conventional cell, namely s1 = (0, 0, 0), s2 = (0.25, 0.25, 0.25), s3 = (0.5, 0.5, 0.5), and s4 = (0.75, 0.75, 0.75). For zinc⇑ Corresponding author. Address: LPQ3M-Laboratory, Faculty of Science and Technology, Mascara University, 29000 Mascara, Algeria. Tel./fax: +213 45802923. E-mail addresses: [email protected] (R. Khenata), [email protected] (G. Murtaza).

blende-like structure DIIICV compound the DIII and CV atoms occupy the s1 = (0, 0, 0) and s2 = (0.25, 0.25, 0.25) sites, respectively, whereas s3 = (0.5, 0.5, 0.5) and s4 = (0.75, 0.75, 0.75) sites are empty. The crystal structure of the cubic Nowotny–Juza AIBIICV can be described as follows: the BII and CV atoms form a regular zinc-blende crystal, and AI atom can occupy either s3 = (0.5, 0.5, 0.5), and forms the a-phase or the s4 = (0.75, 0.75, 0.75) site to form the b-phase. The c-phase can be obtained when the two sites are occupied. The NaZnX (X = P, As, Sb) compounds, belonging to the FTC family, have been less investigated compared to the other members of this family [3–31]. These materials form a special case; unlike the other FTC family members, they crystallize in tetragonal Cu2Sbtype structure (space group P4/nmm) [32]. NaZnAs is also found to crystallize in the MgAgAs-type structure [33]. By means of tight-binding linear muffin–tin orbital (TB-LMTO) method,

0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.11.041

Please cite this article in press as: A. Djied et al., Comput. Mater. Sci. (2013), http://dx.doi.org/10.1016/j.commatsci.2013.11.041

2

A. Djied et al. / Computational Materials Science xxx (2013) xxx–xxx

Jaiganesh et al. [34] studied the structural and electronic properties of NaZnX, in the tetragonal Cu2Sb-type structure, and cubic a-, band c-phases. They predicted structural phase transition from the Cu2Sb-type structure to the cubic a or b-phase under pressure effect. Moreover, they pointed out a semiconducting character with a direct band gap for NaZnP in the tetragonal Cu2Sb-type and cubic a- and b-phases and a metallic character in the cubic c-phase. On the other side, they reported that NaZnAs has a narrow band gap, NaZnSb was found to be metallic in the Cu2Sb-type structure, and both NaZnAs and NaZnSb were found to be semi-metallic in the a- and b-phases and metallic in the c-phase. From above, one can note that generally information about some fundamental physical properties of the NaZnX (X = P, As, Sb) compounds is scarce. There is no knowledge about the mechanical properties of these materials, either from experiments or theoretical models. There is no information about their optical properties. Therefore, to reveal the mechanical properties of these phases and to provide more optoelectronic properties data in order to understand the physical properties of these compounds in depth for satisfying scientific curiosity and for eventual technological application, it is necessary to further study their fundamental physical properties. The main objective of the present work is to investigate in details the structural, elastic, electronic and optical properties of NaZnP in the Cu2Sb-type structure. For this purpose, we use the full potential linear/augmented plane wave plus local orbitals (FP-L/ APW + lo) in conjunction with the generalized gradient approximation and modified Becke–Johnson potential approximation. The remainder of the paper is organized as follows. Section 2 briefly describes the computational techniques used in this study. Section 3 is devoted to the presentation and discussion of the obtained results. Finally, Section 4 summarizes the main conclusions of this work. 2. Computational details All herein done calculations were performed using the full potential augmented plane wave plus local orbitals (FP-APW + lo) method [35,36] in the framework of the density functional theory (DFT) as is implemented in the Wien2K code [37]. The exchangecorrelation potential was treated within the generalized gradient approximation (GGA) as is parameterized by Perdew et al. [38]; the so-called GGA-PBE. Further, we used the Tran–Blaha recipe, when applying a modified version of Becke and Johnson exchange potential [39]; the so-called (TB-mBJ), for the calculation of electronic and optical properties. This new potential improves the accuracy of the band gap energy of semiconductors and insulators with an orbital independent exchange–correlation potential which depends solely on semilocal quantities [39]. The calculations were performed with RKmax = 10 (R is the smallest muffin–tin radius and Kmax is the cut-off for the plane wave) for the convergence parameter for which the calculation stabilize and convergence in terms of the energy are achieved. We have used an appropriate set of kpoints, 10 10 10 Monkhorst-Pack sampling, to compute the total energy. The muffin–tin sphere radii RMT for Na, Zn and P were chosen to be 1.6, 2.1 and 2.0 atomic units (a.u.), respectively. Both plane wave cut-off and number of k-points were varied to ensure total energy convergence.

formula units, where Na, Zn and P atoms are positioned at 2c (0, 0, zNa), 2a (0, 0, 0) and 2c (0, 0.5, zP), respectively, where zNa = 0.3590 and zP = 0.7873 [34]. As the first step, the calculated total energy (ETot) versus primitive cell volume (V) were fitted to the Murnaghan’s equation of state [40] (Fig. 1) to determine the equilibrium lattice constants, bulk modulus and pressure derivative of the bulk modulus for the four polymorphs of the NaZnP compound: Cu2Sb-type, a-, b- and c-phases. It is worth to note here that the lattice constants, a and c, and the internal coordinate, z, are not fixed by the symmetry so we should relax the c/a ratio and z for each volume in order to obtain the optimized crystalline structure that minimizes the total energy. For each volume, we change the c/ a ratio with keeping the volume constant and calculating the total energy corresponding to each ratio and the best ratio is which minimizes the total energy. In principle, the internal coordinate z should also relaxed and the good one that minimizes the total energy. For the internal coordinate z, we have found that the value of the relaxed coordinate is practically equal to the measured one, so all calculations have been done using the measured value. The obtained results for all considered NaZnP phases are listed in Table 1 together with the existing theoretical results and experimental findings for the sake of comparison. Our results are in very good agreement with the available experiments [41]. Our computed value for the lattice constants a and c and the c/a ratio deviates from the measured value by only 0.02% and 1.1% and 0.5%, respectively, which confirms validity and reliability of the present performed calculations. Our obtained lattice constant values are also in very good agreement with the previously calculated ones [34,42–44]. Our calculated bulk modulus value for the a-phase is in good agreement with the reported one by Ref. [43], but it is smaller compared to that one obtained using the TB-LMTO calculation [34]. From Fig. 1, representing ETot versus V, one can see that the Cu2Sb-type structure is the most stable at zero pressure, and the E(V) curve of this phase crosses firstly that of b-phase and secondly that of a-phase, which indicates that under pressure effects the Cu2Sbtype phase undergoes phase transformation to the b-phase then to a-phase. The same sequence of phase transformations has been observed in NaZnSb [9]. The stability of a particular structure is decided by the minima of the Gibbs free energy given by: G = E0 + PV + TS. Since the herein theoretical calculations are performed at 0 K, the Gibbs free energy becomes equal to the enthalpy H: H = E0 + PV. At a given pressure, a stable structure is one for which enthalpy has its lowest value and the transition is calculated at which the enthalpies for the two phases are equal. Fig. 2 shows

3. Results and discussions 3.1. Structural properties and phase transition At ambient conditions, NaZnP crystallizes in the tetragonal Cu2Sb-type structure, with the space group P4/nmm (No. 129 in the international X-ray Tables). The conventional cell contains 2

Fig. 1. Energy versus volume curves of the Cu2Sb, a, b and c-phases of the NaZnP compound.


3

A. Djied et al. / Computational Materials Science xxx (2013) xxx–xxx Table 1 Calculated equilibrium lattice constants (a, c, in Å), bulk modulus (B, in GPa) and its pressure derivative (B0 ), obtained from Murnaghan EOS for NaZnP in the Cu2Sb-type, a-, b- and c-phases. Phase

a

C

c/a

B

0

B

Hook’s law is applicable and the elastic energy DE is a quadratic function of strains:

DE ¼ V 0

6 X 1 i;j¼1

Cu2Sbtype

a-Phase

b-Phase c-Phase a b c e f

Ref. Ref. Ref. Ref. Ref.

Present

4.067

6.929 a

Expt. Other

4.066 4.012b 4.0673c

Present Other

6.146 5.7501b 6.146e 6.149f, 6.162f 6.068 6.29

Present Present

1.704 a

6.893 6.8011b 6.9068 c

55.48

C ij di dj

ð1Þ

4.21

a

1.695 1.6952b

2

124.85b 48.69 122.62b 50.76e

4.42 3.89e

51.08 34.63

4.57 4.51

[41]. [34]. [42]. [43]. [44].

where Cij are the elastic constants, V0 the equilibrium cell volume and di are the six components of strain tensor. Symmetry can be used to design an optimal set of deformations under two sometimes competing requirements: (i) use as few deformations as possible and (ii) use deformations that keep the highest possible symmetry on the deformed cell. For the tetragonal structure, there are six independent elastic constants, namely C11, C12, C13, C33, C44 and C66. Six different strain patterns (Table 2) were imposed on the crystal cell of the tetragonal NaZnP in order to calculate its six independent elastic constants. Three positive and three negative amplitudes were used for each strain component with a maximum strain value e = 0.02, and then the elastic constants were determined from a quadratic fit of the calculated energy (DE) as a function of strain. The calculated elastic constants Cij for the tetragonal NaZnP are summarized in Table 3. To the best of our knowledge, the present work is the first attempt to calculate the single-crystal elastic constants Cij of NaZnP; that is why comparison with other results is not possible. From Table 3 one can note that C11 is higher than C33 and C66 is higher than C44, so this material is anisotropic from an elastic point of view. The elastic constants C11 and C33 reflect the stiffness-to-uniaxial strains along the crystallographic a and c axes. The obtained results show that C11 > C33, indicating that the material is stiffer for strains along the a and b axes than along the c axis. C66 is larger than C44, implying that the [1 0 0] (0 0 1) shear is easier than the [1 0 0] (0 1 0) shear. Calculated elastic constants Cij for NaZnP satisfy the mechanical stability criteria for a tetragonal structure [46]:

C 11 > 0; C 33 > 0; C 44 > 0; C 66 > 0; ðC 11 C 12 Þ > 0; ðC 11 þ C 33 2C 13 Þ > 0; ð2C 11 þ C 33 þ 2C 12 þ 4C 13 Þ >0

Fig. 2. Variation of enthalpies per formula units as a function of hydrostatic pressure for the NaZnP compound in Cu2Sb, a and b-phases. The arrow marks the calculated transition pressure Pt.

the evolution of the enthalpies of the Cu2Sb-type, b- and a-phases of NaZnP with pressure. We have found that firstly the Cu2Sb-type undergoes a phase transition to the b-phase at pressure Pt1 = 71.8 GPa and secondly the b-phase undergoes a phase transition to the a-phase at pressure Pt2 = 99.4 GPa. It is worth to mention here that our obtained structural phase transition sequence ðCu2 Sb type ! b phase ! a phaseÞ is disagreeing with the reported results by Ref. [34] (Cu2 Sb type ! a phase ! b phase). There is no evident explanation for this discrepancy apart we have used a method that is different from that one used in Ref. [34]. Future experimental measurements will judge which one is accurate.

ð2Þ

The above independent elastic constants Cij values, summarized in Table 3, are calculated using energy-strain calculations in the framework of an ab-initio FP-L/APW + lo method for the NaZnP single-crystal. However, the majority of synthesized materials are prepared as polycrystalline, i.e., in the form of aggregated mixtures of monocrystalline with a random orientation. Thus, in this case, instead of measuring Cij, the couple bulk modulus B, which is a

Table 2 Parameterizations of the six strains used to calculate the six elastic constants of the NaZnP compound in the Cu2Sb-type structure. di (i = 1, 2, 3, 4, 5, 6) are the components of the strain tensor. DE = ETot(V, e) ETot(V, e = 0). Unlisted di are set to zero. Deformation

Strains

DE/V

1 2

d1 = d2 = e d1 = d2 = e

ðC 11 þ C 12 Þe2 þ 0½e3 ðC 11 þ C 12 þ 2C 33 4C 13 Þe2 þ 0½e3

d3 ¼ eð2þeÞ 2 ð1þeÞ

3

3.2. Mechanical properties Elastic constants are one of the important fundamental physical properties of solids whose knowledge is essential to the understanding of many of their properties. In particular, elastic constants determine the elasticity and mechanical stability of crystals. Determination of the elastic constants requires knowledge of the curvature of the energy curves as a function of strain of selected deformation of the unit cell [45]. For a small strain on a solid,

d3 = e h i1=2 ð1þeÞ 1 d1 ¼ ð1eÞ h i1=2 ð1eÞ 1 d2 ¼ ð1þeÞ

C 33 e2 2

5

d4 ¼ d5 ¼ e

C 44 e2 þ 0½e4

6

d3 ¼ e2 =4 d6 = e

4

þ 0½e3

ðC 11 C 12 Þe2 þ 0½e4

C 66 e2 =2 ¼ 0½e4 2

1=2

2

1=2

d1 ¼ ð1 þ e4 Þ d2 ¼ ð1 þ e4 Þ

1 1


4


Table 3 Calculated bulk modulus (B, in GPa), shear modulus (G, in GPa), Poisson’s ratio (m), Young’s modulus (E, in GPa) obtained from calculated independent elastic constants (Cij, in GPa), for the NaZnP compound in the Cu2Sb-type structure. The subscripts V, R and H denote the Voigt, Reuss and Hill approximations. System

C11

C12

C13

C33

C44

C66

Present

128.5 BV

24.4 BR

35.9 BH

71.3 GV

39.7 GR

104.1 GH

Present

57.8 BH/GH

54.8 E

56.3

52.2

41.3

46.7

r

1.20

109.9

0.175

Present

measure of the resistance to the volume change of the material by an applied pressure, and isotropic shear modulus G, which is a measure of the resistance to reversible deformations upon the shear strain, may be determined experimentally on the polycrystalline sample to characterize its mechanical properties. Theoretically, B and G of the polycrystalline phase of a material can be obtained by a special averaging of the independent elastic constants Cij of its monocrystalline phase. The orientation-averaged elastic moduli B and G are calculated using The Reuss–Voigt–Hill approximations [47–49,30–32]. In Voigt model the bulk and shear (BV and GV) are estimated as:

BV ¼

2 1 ðC 11 þ C 12 þ 2C 13 þ C 33 Þ 9 2

ð3Þ

BR ¼

1 ðC 11 þ C 12 þ 2C 33 4C 13 þ 12C 55 þ 12C 66 Þ 30

ð4Þ

In Reuss approximation the bulk and shear (BR and GR) are estimated as:

BR ¼

ðC 11 þ C 12 ÞC 33 2C 212 C 11 þ C 12 þ 2C 33 4C 13

ð5Þ

BR ¼

5 ½ðC 11 þ C 12 ÞC 33 2C 212 C 55 C 66 2 3BV C 55 C 66 þ ½ðC 11 þ C 12 ÞC 33 2C 212 ðC 55 þ C 66 Þ

ð6Þ

Voigt (BV, GV) and Reuss (BR, GR) averages limits values for B and G, and Hill recommended that the arithmetic mean of these two limits is used as effective moduli in practice for polycrystalline samples:

BH ¼ ðBV þ BR Þ=2; GH ¼ ðGV þ GR Þ=2

ð7Þ

With the values of B and G, we define the orientation-averaged Young’s modulus (E) and Poisson’s ration (r):

E¼

9BG 3B 2G ;r ¼ 3B þ G 6B þ 2G

ð8Þ

The obtained values of the above-mentioned isotropic elastic parameters are summarized in Table 3, which allow us to make the following conclusions: (i) From Tables 1 and 3, one can see that the bulk modulus value of NaZnP deduced from the single-crystal elastic constants Cij is in good agreement with its value derived from Murnaghan E(V) EOS, depicted in Fig. 1. This might be an estimate of the reliability and accuracy of this theoretical estimation of the elastic constants for the tetragonal NaZnP in the Cu2Sb-type structure. (ii) The bulk modulus of the considered material is quite small (lower than 60 GPa) and so this material should be classified as a relatively soft material with high compressibility (higher than 0.02). In addition, Young’s modulus, defined as the ratio of linear stress and linear strain, can give

information about the stiffness of a material. The Young’s modulus of NaZnP is found to be about 102 GPa; thus, this compound will show a rather small stiffness. (iii) Poisson’s ratio r is connected with the volume change and the nature of interatomic forces. If r is 0.5, no volume change occurs, lower than 0.5 means that large volume change is associated with elastic deformation [50]. We have found that r = 0.175, which means that a considerable volume change can be associated with the elastic deformation in NaZnP. According to Frantsevich rule [51], the Poisson’s ratio r can be used as a criterion for ductility or brittleness behavior of materials. The critical value that separates the ductile and brittle nature of material is 0.26: if r < 0.26, a material demonstrates brittleness and if r > 0.26, a material behaves in a ductile manner. In our case, according to this indicator (r 0.18), NaZnP will behave as a brittle material. The Poisson’s ratio can also provide information about the bonding nature. The values of the Poisson’s ratio r for covalent materials are small (r 0.1), for systems with predominantly central interatomic interactions (i.e., ionic crystals), the values of r is usually close to 0.25, whereas for metallic materials is typically 0.33 [52], thus according to this, NaZnP has a certain ionic bonding character. (iv) The bulk and shear moduli provide information about the brittle-ductile nature of materials through the Pugh’s B/G empirical criterion [53]. According to this criterion, a B/ G > 1.75 (B/G < 1.75) indicates the ductile (brittle) nature of the material. For NaZnP, the calculated value is B/G = 1.20, i.e., indicating a brittle nature of the material. The brittleness properties of NaZnP based on B/G ratio criterion is in good agreement with the result estimated according to the Poisson’s ratio index. Another interesting physical parameter regarding the elastic properties of solids is the so-called elastic anisotropy. The elastic anisotropy is related to different bonding nature in different crystallographic directions, and correlates with the possibility to induce microcracks in materials. Hence, it is important to estimate elastic anisotropy in order to improve its mechanical durability. Today, a set of approaches is proposed to estimate the elastic anisotropy numerically. Here we used three different indexes to estimate the elastic anisotropy of NaZnP. (i) One measure of the elastic anisotropy is the percentage of anisotropy in the compression AComp and shear AShear which are defined as: AComp ¼ ðBV BR Þ=ðBV þ BR Þ and AShear ¼ ðGV GR Þ=ðGV þ GR Þ, where B and G denote the bulk and shear moduli, and the subscripts V and R stand to the Voigt and Reuss approximations. For an isotropic crystal the factors AComp and AShear must equal to zero, while a value of 100% is associated with the largest anisotropy [54]. The percentage of the bulk and shear anisotropies for NaZnP are equal to 2% and 11%, implying a weak anisotropy in the compression and a noticeable anisotropy in shear. (ii) Another index allows us to estimate the elastic anisotropy is the so-called universal anisotropy index AU [55], given by the following relation: AU ¼ 5GV =GR þ BV =BR 6. As a matter of fact, for isotropic crystals, AU = 0, and deviations of AU from zero define the extent of elastic anisotropy. The obtained value of AU for the NaZnP compound is equal to 1.37, which points to noticeable elastic anisotropy in this material. (iii) An illustrative way of describing the elastic anisotropy is a three-dimensional (3D) surface representation showing the variation of elastic modulus with crystallographic direction. In 3D representation, an isotropic system would exhibit a spherical shape and a deviation from spherical shape



indicates a degree of anisotropy. To visualize the elastic anisotropy of the herein considered material, we have plotted a three-dimensional dependence of the Young’s modulus E on crystallographic directions using the equation given in Ref. [56] for a tetragonal structure. In Fig. 3, we present the visualization of the Young’s moduli surfaces and their cross-sections in the (1 0 0), (0 1 0) and (0 0 1) planes for NaZnP. The lowest value, Emin, is about 35% of the highest value of the Young’s modulus, Emax, indicating a strong anisotropic elastic behavior.

3.3. Electronic properties It is well-known that the major drawback of the DFT formalism in conjunction with the common local density approximation (LDA) and generalized gradient approximation (GGA), is the improper interpretation of the excited state properties, such as the severe underestimation of the band gap value or the overestimation of the electron delocalization in systems with localized d and f electrons [57,58]. Actually, some approximations beyond the LDA and GGA, such as GW, hybrid functional, LDA + U and LDA + DMFT, are developed in order to describe correctly the electronic structure of semiconductors and insulators. However, most of these methods are computationally expensive or not satisfactory in all cases [59,60]. Fortunately, the recently proposed modified Becke–Johnson (mBJ) potential approximation [39] is an alternative way to have a band gap close to the experimental value [61,62] but computationally cheaper than the other mentioned methods.

5

The calculated band structures along high-symmetry lines in the Brillouin zone for the tetragonal NaZnP within GGA-PBE and TBmBJ approaches are depicted in Fig. 4. The overall behavior of the band structures calculated using these two exchange-correlation functionals is very similar except for the values of their band gaps, which are higher within the TB-mBJ (Table 4). The mBJ generally improves the electronic band structure, but especially improves the band gap. As a general result the TB-mBJ potential causes a rigid displacement of the conduction bands toward higher energy, with small differences in the dispersion at some regions of the Brillouin zone. Both valence band maximum (VBM) and conduction band minimum (CBM) of the tetragonal NaZnP are located at the C point. Thus, NaZnP in the Cu2Sb-type structure is a direct band gap (C–C) semiconductor, and its value is 1.86 eV, using the TB-mBJ. Comparing this value with those obtained by us using the common GGAPBE (0.9 eV) and by Jaiganesh et al. [34] using the TB-LMTO in conjunction with the LDA (0.6 eV). No experimental data on the NaZnP band gaps were found in the literature for comparison with the above given theoretical estimations (Table 4). Nevertheless, we know that generally the band gaps calculated using DFT with the common LDA and GGA approximations are likely to be approximately 30–50% smaller than experimental values [62]. According to this, the real gap Eg of LiZnN is expected to be approximately between 1.3 eV and 1.8 eV. So it is clear that the TB-mBJ approximation considerably improved the band gap value. In order to get an insight into the electronic nature and chemical bonding of the tetragonal NaZnP, the total and site-projected l-decomposed densities of states (TDOS and PDOS) diagrams of the tetragonal NaZnP were depicted in Fig. 5 and analyzed in

Fig. 3. Visualizations of the Young’s modulus surface (left panel) and its cross-section in the (1 0 0), (0 1 0) and (0 0 1) planes (right panel) for the tetragonal NaZnP compound. The axes units are in GPa.

Fig. 4. Electronic energy dispersion curves for the tetragonal NaZnP along some high symmetry directions of the Brillouin zone within TB-mBJ and GGA-PBE.


6


Table 4 Comparison of some calculated direct band gaps for NaZnP in the Cu2Sb-type using two different exchange-correlation energy methods: GGA-PBE and TB-mBJ. (All energies are in eV).

Present

GGA-PBE TB-mBJ

Other [9]

LDA

R–R

C– C

X–X

M–M

5.72053 6.03139

0.9 1.86

3.21819 3.98595

4.32737 5.05945

0.64

Fig. 5. Calculated total and partial densities of states for the tetragonal NaZnP.

comparison with the available theoretical data. Our results are consistent with those obtained in previous theoretical studies [34]. The lower in energy structure between about 11.24 eV and 9.57 eV consists mainly of the P-3s states with a minor contribution coming from the Zn-3d states. The DOS between about 7.47 and 6.34 is due to a mixture of the Zn-3d states with the P 3s + 3p states. In the energy region from about 4.88 eV to the Fermi level, the so-called upper valence band (UVB), the DOS are attributed to the overlapping of the Zn-4s + 4p + 3d and P-3s + 3p states, which suggests a covalent bonding character of the Zn–P bond. To visualize the nature of the bonding characteristics in NaZnP compound, the charge density distributions for the Gu2Sb-type structure is examined. The contour map of the charge density in (1 0 0) plane is shown in Fig. 6. Deformation of charge distribution of the Zn and P atoms indicates the existence of a directional bonding between these two atoms due the hybridization of their upper valence band. Since there is no obvious electron density accumulation in the regions of Na–P and because of the different electronegativity between sodium and phosphor, the bonding between the Na and P atoms has ionic nature. Therefore, the bonding in Cu2Sb–NaZnP may be expressed as a combination of covalent-ionic behavior.

Fig. 6. Calculated valence charge density plots (2D) for the tetragonal NaZnP in the (1 0 0) plane.

3.4. Optical properties On the basis of calculated electronic structure of the tetragonal NaZnP material using the mBJ exchange-correlation potential, we computed the imaginary part of the dielectric function Imðeab Þ ¼ ea2b ðxÞ for incident photon of energy ⁄x up to 30 eV. The real part of the dielectric tensor Reðeab Þ ¼ ea1b ðxÞ is then

Fig. 7. Calculated real part e1(x) and imaginary e2(x) part of the complex dielectric function e(x) for the tetragonal NaZnP for two different polarizations of incident radiations.



7

Fig. 8. (Right panel) Calculated optical constants for tetragonal NaZnP for two different polarizations of incident radiations; (a) refractive index n(x), (b) reflectivity R(x) and (c) energy-loss function L(x). (Left panel) The average of the (a) refractive index n(x), (b) reflectivity R(x) and (c) energy-loss function L(x).

determined using Kramers–Kronig relation. The rest of the optical constants, such as the reflectivity R(x) and the loss function L(x), can be calculated from the dielectric function e(x) = e1(x) + je2(x). Due to the tetragonal symmetry of the herein studied compound, all the optical spectra are plotted for electrical field ! vector E of the incident radiation polarized both perpendicular ! ! ( E ? c) and parallel ( E ==c) to the tetragonal c-axis of the crystal. Fig. 7 displays the calculated real (e001 ðxÞ; e? 1 ðxÞ) and imaginary (e002 ðxÞ; e? 2 ðxÞ) parts of the complex dielectric function of the ! ! tetragonal NaZnP for E ==c and E ? c; respectively. As usually elaborated materials are polycrystalline, we present also in Fig. 7 an average imaginary part, eA2v er ðxÞ ¼ ðe002 ðxÞ þ 2e? 2 ðxÞÞ=3, and an average real part, eA1v er ¼ ðe001 ðxÞ þ 2e? 1 ðxÞÞ=3. From this figure, one can notice a considerable anisotropy between the parallel and perpendicular components of the frequency dependent dielectric function; the absorption edge is slightly shifted to the lower ! energy for the E ==c. The optical absorption edge starts at about 1.84 eV (Fig. 7a); this gives the threshold for direct optical transition between the highest valence band and the lowest conduction band (CV–CC transition). Beyond this point, e2 ðxÞ curve increases rapidly due to the abruptly increase in the number of transitions contributing toward e2(x). Structures of the e2 ðxÞ spectrum represent photon-absorption caused by electronic transitions from the occupied valence band (Vi) to the conduction empty band (Ci). The e? 2 ðxÞ spectrum presents two sharp main peaks at about 4.25 and 4.80 eV, which are probably originated from direct transition from the V4 to C2 at C-point and from V3 to C4 at C-point, respectively, whereas the two sharp main peaks of the e002 ðxÞ spectrum are located at 4.80 and 5.75 eV and they are due probably to the direct transitions: V3 to C4 at C-point and V1 to C2 at X-point, respectively. The real part of the complex dielectric function e1 ðxÞ which indicates how electromagnetic energy is dispersed when it penetrates in a medium is depicted in Fig. 7(b). The calculated static 00 dielectric constant e? 1 ð0Þ (e1 ), which represents the dielectric response of a material to a static electric field, is found to be about 00 7.17 (6.46). The e? 1 ðxÞðe1 ðxÞ) presents two main sharp peaks at about 3.41 and 3.91 eV (3.83 and 4.56 eV) with magnitude of about 14.18 and 13.55 (13.32 and 9.72) respectively and a negative valley at about 6.57 eV.

Using the calculated imaginary and real parts of frequency dependent dielectric function, we have calculated the refractive index n(x), reflectivity coefficient R(x) and energy loss-function L(x), which are displayed in Fig. 8. At zero frequency, the refractive index n(0) of the tetragonal NaZnP is about 2.62. The refractive index increases from the static value to attain a maximum of 3.97 at 3.85 eV then decreases rapidly to its minimum value which is smaller than 1. From Fig. 8(b), one can say that the tetragonal NaZnP shows small reflectivity at low energies then a rapid increase of the reflectivity occurs in the energy range 4–14 eV. A reflectivity maximum occurs between 5.5 and 7.0 eV which coincides with the lower negative values of e1(x). The energy-loss function L(x) is a physical parameter describing the energy-loss of a fast electron traversing a material. Its main peak occurs at the energy ⁄xP, where xP is called screened plasma frequency [63,64]. From Fig. 8(c), we can see that the main sharp structure of L(x) is located at about 15 eV, corresponding to a rapid decrease of reflectance.

4. Conclusion This paper reports a detailed investigation of the structural, phase transition, mechanical and optoelectronic properties of the NaZnP compound using ab-initio L/APW + lo calculations within the GGA and TB-mBJ approaches. The optimized lattice parameters are in good accordance with the experimental findings. Total energy versus hydrostatic pressure calculations revealed that NaZnP undergoes a structural phase transition in the following sequence: CU2Sb-type phase ? a-phase ? b-phase. Independent single crystal elastic constants Cij have predicted and the polycrystalline elastic parameters have been deduced using the Voigt–Reuss–Hill approximations. Calculated elastic constants show that the tetragonal NaZnP compound is elastically stable and deduced computed Poisson’s and B/G ratios clearly indicated that this compound can be classified as brittle materials. Elastic anisotropy of the considered material was investigated according to three different indexes. Three and two dimensional visualization of the Young’s modulus clearly showed its variation with the crystallographic directions indicating its elastic anisotropy. Analyses of the band structure and density of states revealed that the Cu2Sb-type phase


8


of NaZnP is a direct band gap semiconductor. The TB-mBJ approximation yields larger fundamental band gaps compared to those of PBE-GGA and LDA. Density of states and charge distribution spectra showed obvious Zn–P covalent bond and the ionic nature of the Na–P bond. Therefore, the bonding in Cu2Sb–NaZnP may be expressed as a combination of covalent-ionic behavior. Additionally, we have calculated the optical properties, namely, the real and the imaginary parts of the dielectric function, reflectivity and energy loss function for radiation up to 30.0 eV for two different polarizations of the incident radiations.

[27] [28] [29] [30] [31] [32]

[33] [34] [35]

Acknowledgements Authors (Khenata, Bouhemadou, Bin-Omran and Alahmed) acknowledge the financial support by the Deanship of Scientific Research at King Saud University for funding the work through the research group project No. RPG-VPP-088. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

H. Nowotny, K. Bachmayer, Monatsh. Chem. 81 (1950) 488. R. Juza, F. Hund, Z. Anorg, Chem. 257 (1948) 1. R. Bacewicz, T.F. Ciszek, Appl. Phys. Lett. 52 (1988) 1150. K.M. Wong, Jpn. J. Appl. Phys. 48 (2009) 085002. K.M. Wong, W.K. Chim, J.Q. Huang, L. Zhu, J. Appl. Phys. 103 (2008) 054505. A. Mellouki, L. Kalarasse, B. Bennecer, F. Kalarasse, Comp. Mat. Sci. 42 (2008) 579. F. Kalarasse, B. Bennecer, A. Mellouki, L. Kalarasse, Comp. Mat. Sci. 43 (2008) 791. D.M. Wood, A. Zunger, R. de Groot, Phys. Rev. B 31 (1985) 2570. A.E. Carlsson, A. Zunger, D.M. Wood, Phys. Rev. B 32 (1985) R1386. S.-H. Wei, A. Zunger, Phys. Rev. Lett. 56 (1986) 528. K. Kuriyama, F. Nakamura, Phys. Rev. B 36 (1987) 4439. K. Kuriyama, T. Katoh, Phys. Rev. B 37 (1988) R7140. K. Kuriyama, T. Kato, T. Tanaka, Phys. Rev. B 49 (1994) 4511. K. Kuriyama, T. Kato, K. Kawada, Phys. Rev. B 49 (1994) 11452. K. Kuriyama, T. Kato, T. Kato, h. Matsuno, J. Cryst. Growth 166 (1996) 631. K. Kuriyama, K. Kushida, R. Taguchi, Solid State Commun. 108 (1998) 429. K. Kuriyama, R. Taguchi, K. Kushida, K. Ushiyama, J. Cryst. Growth 198 (199) (1999) 802. K. Kuriyama, K. Nagasawa, K. Kushida, J. Cryst. Growth 237 (2002) 2019. K. Kuriyama, K. Kushida, J. Appl. Phys. 87 (2000) 2303. K. Kuriyama, K. Kushida, J. Appl. Phys. 87 (2000) 3168. K. Kushida, Y. Kaneko, K. Kuriyama, Phys. Rev. B 70 (2004) 233303. L.H. Yu, K.L. Yao, Z.L. Liu, Physica B 353 (2004) 278. F. Kalarasse, B. Bennecer, J. Phys. Chem. Solids 67 (2006) 846. K. Kuriyama, T. Ichikawa, Phys. Rev. B 72 (2005) 233201. L.H. Yu, K.L. Yao, Z.L. Liu, Solid State Commun. 135 (2005) 124. F. Kalarasse, B. Bennecer, J. Phys.: Condens. Matter 18 (2006) 7237.

[36] [37]

[38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51]

[52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64]

A. Bouhemadou, R. Khenata, F. Zerarga, Solid State Commun. 141 (2007) 288. A. Bouhemadou, R. Khenata, J. Phys. Chem. Solids 68 (2007) 549. A. Bouhemadou, R. Khenata, Semicond. Sci. Technol. 23 (2009) 105024. A. Bouhemadou, Mater. Sci. Semicond. Process. 12 (2009) 198. A. Bouhemadou, R. Khenata, D. Rached, B. Amrani, Comput. Mater. Sci. 49 (2010) 64. P. Villars, L.D. Calverts, Pearson’s Handbook of Crystallographic Data for Intermetallic Phases, 2nd ed., Vol. 1 and 4. ASM international, The materials international society, Materials Park, OH, p. 1188, 4508, 4520. F.S. Galasso, Structure and Properties of Inorganic Solids, 1st ed., Pergamon, Oxford, 1970. p. 95. G. Jaiganesh, T. Merita Anto Britto, R.D. Eithiraj, G. Kalpana, J. Phys. Condens. Matter 20 (2008) 085220. K.M. Wong, S.M. Alay-e-Abbas, A. Shaukat, Y. Fang, Y. Lei, J. Appl. Phys. 113 (2013) 014304. K.M. Wong, S.M. Alay-e-Abbas, Y. Fang, A. Shaukat, Y. Lei, J. Appl. Phys. 114 (2013) 034901. P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, WIEN2k, An Augmented Plane Wave Plus Local Orbitals Program for Calculating Crystal Properties, Vienna University of Technology, Austria, 2001. J.P. Perdew, S. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. F. Tran, P. Blaha, Phys. Rev. Lett. 102 (2009) 226401. F.D. Murnaghan, Proc. Nat. Acad. Sci. USA 30 (1944) 244. B. Eisenman, M. Somer, Z. Naturforsch. 40 (1985) 1419. H. Kabbour, L. Cario, Mater. Chem. 15 (2005) 3525. H. Mehnane, B. Bekkouche, S. Kacimi, A. Hallouche, M. Djermouni, A. Zaoui, Superlattices Microstruct. 51 (2012) 772. D. Kieven, R. Klenk, S. Naghavi, C. Felser, T. Gruhn, Phys. Rev. B 81 (2010) 075208. M.J. Mehl, Phys. Rev. B 47 (1993) 2493. Z.J. Wu, E.J. Xiang, H.P. Hao, X.F. Liu, J. Meng, Phys. Rev. B 76 (2007) 054115. W. Voigt, Lehrbuch der Kristallphysik, Taubner, Leipzig, 1928. A. Reuss, Z. Angew, Math. Mech. 9 (1929) 55. R. Hill, Proc. Phys. Soc., London, Sect. A 65 (1952) 349. S. Appalakondaiah, G. Vaitheeswarm, S. Lebègue, N.E. Christensen, A. Svane, Phys. Rev. B 86 (2012) 035105. I.N. Frantsevich, F.F. Voronov, S.A. Bokuta, Elastic Constants and Elastic Moduli of Metals and Insulators, in: I.N. Frantsevich (Ed.), Naukova Dumka, Kiev, 1983, pp. 60–180. J. Haines, J.M. Leger, G. Bocquillon, Annu. Rev. Matter Res. 31 (2001) 1. S.F. Pugh, Philos. Mag. 45 (1954) 823. D.H. Chung, W.R. Buessem, in: F.W. Vahldiek, S.A. Mersol (Eds.), Anisotropy in Single Crystal Refractory Compound, Vol. 2, Plenum, New York, 1968, p. 217. S.I. Ranganathan, M. Ostoja-Starzewski, Phys. Rev. Lett. 101 (2008) 055504. A. Cazzani, M. Rovati, Int. J. Solids Struct. 42 (2005) 5057. F. Zhou, C.A. Marianetti, M. Cococcioni, D. Morgan, G. Ceder, Phys. Rev. B. 69 (2004) 201101. M.V. Ganduglia-Pirovano, A. Hofmann, J. Sauer, Surf. Sci. Rep. 62 (2007) 219. D. Koller, F. Tran, P. Blaha, Phys. Rev. B 83 (2011) 195134. I.V. Anisimov, J. Zaanen, K.O. Andersen, Phys. Rev. B 44 (1991) 943. D.A. Becke, R.E. Johnson, J. Chem. Phys. 124 (2006) 221101. F. Tran, P. Blaha, K. Schwarz, J. Phys. Condens. Matter 19 (2007) 196208. S.Zh. Karazhanov, P. Ravindran, H. Fjellvag, B.G. Svensson, J. Appl. Phys. 106 (2009) 123701. M. Fox, Optical Properties of Solids, Academic Press, New York, 1972.
