Journal of Alloys and Compounds 577 (2013) 587–599 Contents lists available at SciVerse ScienceDirect Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jalcom First-principles prediction of the structural, elastic, electronic and optical properties of the Zintl phases MIn2P2 (M = Ca, Sr) N. Guechi a, A. Bouhemadou b,⇑, A. Guechi c, M. Reffas d, L. Louail d, A. Bourzami a, M. Chegaar f, S. Bin-Omran e a Laboratoire d’Etudes des Surfaces et Interfaces des Matériaux Solides (LESIMS), Département de Physique, Faculté des Sciences, University de Sétif 1, 19000 Sétif, Algeria Laboratory for Developing New Materials and their Characterization, Department of Physics, Faculty of Science, University of Setif 1, 19000 Setif, Algeria Institute of Optics and Precision Mechanics, University of Setif 1, 19000 Setif, Algeria d Unité de Recherche Matériaux Emergents, University of Setif 1, 19000 Setif, Algeria e Department of Physics and Astronomy, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia f Department of Physics, Faculty of Science, University of Setif 1, 19000 Setif, Algeria b c a r t i c l e i n f o Article history: Received 27 May 2013 Received in revised form 28 June 2013 Accepted 1 July 2013 Available online 9 July 2013 Keywords: Zintl phases MIn2P2 First-principles calculations Elastic moduli Electronic properties Optical constants a b s t r a c t We have performed a detailed theoretical study of the structural, elastic, electronic and optical properties of two newly synthesized Zintl phases CaIn2P2 and SrIn2P2 by means of first-principles calculations based on density functional theory within the generalized gradient approximation of Wu and Cohen. The optimized lattice parameters, including the lattice constants and internal coordinates, are in good agreement with the existing experimental measurements. The relative changes of the structural parameters versus hydrostatic pressure have been investigated. The elastic properties of MIn2P2 have been examined by calculating all independent single-crystal elastic constants Cij using the static finite strain technique, and the polycrystalline isotropic elastic moduli, namely bulk modulus, shear modulus, Young’s modulus and Poisson’s coefficient, via the Voigt–Reuss–Hill approximations. The elastic wave velocities along some crystalline directions have been evaluated. The mechanical stability of the considered materials has been examined on the light of the pressure dependence of the elastic constants. The elastic anisotropy of the two phases has been studied using three different methods. The electronic properties have been studied throughout the calculations of the band structure, density of states, charge density distributions, charge transfers, and charge-carries masses. These two materials turn out to be narrow gap semiconductors. Finally, we have predicted the basic optical properties, such as the dielectric function, refractive index, extinction coefficient, reflectivity coefficient, absorption coefficient and loss function for polarized incident radiation with electrical vector E parallel to the crystalline axes a and c. A considerable anisotropy is observed in the frequency dependent optical spectra. Ó 2013 Elsevier B.V. All rights reserved. 1. Introduction In recent years, there have been numerous reports on the crystal chemistry and physical properties of ternary pnictides in the systems AE–Tr–Pn (AE = Ca, Sr, Ba, Eu, Yb; Tr = Al, Ga, In; and Pn = P, As, Sb), due to their fascinating structural variety and wide field of materials applications [1]. Such compounds can be classified as Zintl phases [2]. The classic Zintl phases constitute a class of intermetallic compounds, which are made of alkali- and alkaliearth metals in combination with the early post-transition metals and semimetals [3]. Zintl phases are intermetallic compounds with very diverse crystal structures, which could have a metallic/semimetallic or semiconductor nature. With the great expansion of ⇑ Corresponding author. Tel./fax: +213 36620136. E-mail address: a_bouhemadou@yahoo.fr (A. Bouhemadou). 0925-8388/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jallcom.2013.07.003 Zintl concept (formation of compensated valence intermetallic compounds based on a complete charge from an alkali or alkaline-earth element to a post-transition element), a large number of Zintl compounds are recently synthesized and these diverse structures bring abundant and interesting physical properties, such as semiconducting, superconductivity, colossal magnetoresistance, magnetic order, mixed-valence, thermoelectricity, and so on [4–10]. Two new isostructural Zintl compounds CaIn2P2 (calcium indium phosphide) and SrIn2P2 (strontium indium phosphide) have been synthesized and structurally characterized recently by Rauscher and co-workers [11]. CaIn2P2 and SrIn2P2 are isostructural with EuIn2P2 and crystallize in a hexagonal structure with the space group P63/mmc (No. 194 in the X-ray Tables). The alkaline earth cation M in the Zintl phases MIn2P2 is located at a site with  3m symmetry; In and P atoms are located at sites with 3m 588 N. Guechi et al. / Journal of Alloys and Compounds 577 (2013) 587–599 symmetry. To the author’s best knowledge, some fundamental physical properties of the herein considered materials, such as the elastic, electronic and optical properties, are not yet investigated theoretically or experimentally. It is well known that knowledge of the elastic properties of a material is important due to their closely relations with various physical fundamental properties. In particular, they provide information on the stability and stiffness of the material against externally applied strains. Knowledge of the pressure dependence of the elastic constants and lattice parameters are the most significant in many modern technologies [12]. For example, semiconductor layers are commonly subjected to large built-in strain since they are often grown on different substrates having considerable lattice mismatch within a difference in the thermal expansion coefficients between epitaxial layer and substrate [13,14]. Measurements of elastic and lattice parameters under pressure effect are generally very difficult, so the lack of experimental data can be compensated by theoretical simulation based on accurate ab initio theories. So, the first objective of the present work is to study the evolution of the structural and elastic properties with pressure. Knowledge of the electronic structure and optical properties of a material are the first required data for its eventual applications in the optoelectronic technology. Thus, the prediction of the electronic and optical properties of the newly discovered Zintl compounds CaIn2P2 and SrIn2P2 constitutes the second objective of the present work. In this paper, we have investigated the structural, elastic, electronic, and optical properties of the Zintl phases CaIn2P2 and SrIn2P2 by means of an ultrasoft pseudopotential plane-wave method in the first-principles density functional theory framework. The paper is organized as follows. In Section 2, we describe the computational methods and details. Section 3 is devoted to the presentation and discussion of the herein obtained results. We summarize our main results in Section 4. 3. Results and discussion 3.1. Structural properties The ternary MIn2P2 compounds, where M is Ca or Sr, crystallize in EuIn2P2-type hexagonal structure, space group P63/mmc (No. 194 in the International X-ray Tables) [11]. The unit cell crystalline structure of the MIn2P2 (M = Ca, Sr) compounds is depicted in Fig. 1; it contains two chemical formula units. The atoms are positioned at the following Wyckoff positions: M (Ca, Sr): 2a (0, 0, 1/2), In: 4f (2/3, 1/3, zIn) and P: 4f (1/3, 2/3, zP), where zIn and zP are internal coordinates; the z coordinate of the In and P atoms, respectively. So, the unit cell is characterized by four structural parameters not fixed by the symmetry: two lattice constants (a and c) and two internal coordinates (zIn and zP). In order to explain the chemical bonding, all atoms in the unit cell have been numbered (Fig. 1). The optimized structural parameters at zero pressure, including the equilibrium lattice constants, a0 and c0, and the internal parameters, zIn0 and zP0, compared to the available experimental data are summarized in Table 1. As it can be seen from this table, our results are in very good agreement with the existing experimental data. The deviations of the calculated optimized lattice parameters values (a0, c0, zIn0 and zP0) from the measured ones are less than 0.40%, 0.35%, 0.46% and 0.43% (0.38%, 0.23%, 0.52% and 0.12%), respectively, in CaIn2P2 (SrIn2P2). This may be an indication of the capability of this chosen first-principles method to produce reliable and accurate results. The chemical and structural stabilities of the herein studied Zintl compounds MIn2P2 (M = Ca, Sr) have been estimated by means of the cohesive energy Ecoh, which is defined as the energy that is needed when the crystal decomposes into free atoms, using the following equation [22]: Ecoh ¼ 2. Computational methodology All herein reported results are obtained using a pseudopotential plane-wave (PP-PW) method in the framework of the density-functional theory (DFT) as implemented in the CASTEP code [15]. The generalized gradient approximation developed by Wu-Cohen (GGA-WC) [16] was used to treat the exchange–correlation energy. In all done electronic total energy calculations, the Vanderbilt-type ultrasoft pseudopotential was used to model the potential seen by the valence electrons because of the nucleus and the frozen core electrons. The Ca 3s23p64s2, Sr 4s24p65s2, P 3s23p3 and In 4d105s25p1 electron states are explicitly treated as valence electron states. The basis set for all performed calculations were defined by a plane-wave cut-off energy of 380 eV, and the Brillouin zone (BZ) integration is performed over the 7  7  2 grid sizes using Monkhorst-Pack scheme [17] for hexagonal structure. This set of parameters assures the maximum ionic Hellmann–Feynman force within of 0.01 eV Å1, maximum stress of 0.02 GPa, maximum displacement of 5  104 Å and self consistent convergence of the total energy of 5  106 eV/Atom. The optimized structural parameters were determined using the Broyden– Fletcher–Goldfarb–Shanno (BFGS) minimization technique [18], which provides a fast way to find the lowest energy structure. The elastic constants were determined by applying a set of given homogeneous deformation with a finite value and calculating the resulting stress with respect to optimizing the internal atomic freedoms [19]. Atoms were allowed to relax to their equilibrium positions when the energy change on each atom between successive steps was less than 1  106 eV/atom, the force on each atom was less than 0.002 eV/Å1, the stress on each atom was less than 0.02 GPa and the displacement was less than 1  106 Å. The optical properties were obtained from the complex dielectric function e(x) = e1(x) + ie2(x), which is mainly connected with the electronic structures. The imaginary part of the dielectric function e2(x) is calculated from the momentum matrix elements between the occupied and unoccupied wave functions [20,21]. Other macroscopic optical constants such as the refractive index n(x), extinction coefficient k(x), optical reflectivity R(x), absorption coefficient a(x) and energy-loss spectrum L(x) can be computed from the complex dielectric function e(x). For the optical properties computing, the numerical integration of Brillouin zone was performed using a 29  29  6 Monkhorst–Pack k-point sampling procedure [17]. h  i 1 MðatomÞ InðatomÞ PðatomÞ EMIn2 P2  NM ETot þ NIn ETot þ NP ETot NM þ NIn þ NP Tot Fig. 1. Crystal structure of the hexagonal Zintl compounds MIn2P2 (M = Ca, Sr) with P63/mmc space group. 589 N. Guechi et al. / Journal of Alloys and Compounds 577 (2013) 587–599 Table 1 0 Calculated optimized structural parameters for the Zintl compounds CaIn2P2 and SrIn2P2 at zero pressure: lattice constants (a0 and c0, in A Å), equilibrium unit cell volume (V0, in 0 3 A ) internal coordinates (zIn0 and zP0), bulk modulus B0 (in GPa) and pressure derivative of the bulk modulus B’, and cohesive energy (Ecoh, in eV/atom). B0 and B0 are derived from Å the Birch EOS. Available experimental results are reported for comparison. a0 c0 (c/a)0 V0 ZIn0 ZP0 B0 B0 Ecoh CaIn2P2 Present work Expt. [11] 4.0382 4.0220 17.3471 17.4080 4.2957 4.3282 244.98 243.87 0.3304 0.3290 0.3986 0.3970 58.44 – 4.27 – 4.486 SrIn2P2 Present work Expt. [11] 4.1099 4.0945 17.7709 17.8120 4.3239 4.3502 259.96 258.61 0.3284 0.3280 0.3924 0.3910 55.80 – 4.34 – 4.440 MðatomÞ InðatomÞ PðatomÞ 2 P2 where EMIn ; ETot ; ETot and ETot stand to the total energy Tot of the primitive cell of MIn2P2 compound and total energies of the isolated M, In and P atoms, respectively. The energy of the free atom has been calculated using a cubic box with a large lattice constant that contains the considered atom. The obtained cohesive energies are listed in Table 1. From Table 1, we can see that the cohesive energies of the herein studied Zintl compounds are negative which means that these materials are energetically stable. In Fig. 2, we illustrate the pressure dependence of the relative changes X/X0 of the lattice parameters, namely a/a0, c/c0, zIn/zIn0, zP/zP0 and V/V0 for MIn2P2 (M = Ca, Sr), where X is the lattice parameter at the considered pressure and X0 its value at zero pressure. The obtained results for a/a0, c/c0, zIn/zIn0, zP/zP0 in the considered range of pressure were well fitted to a third-order polynomials: P XðPÞ=X 0 ¼ 1 þ bX P þ 3n¼2 K n P n . The following relations were obtained from these calculations:  CaIn2 P2 a ¼ 1  0:00535P þ 1:43  104 P2  2:12  106 P3 a0  CaIn2 P2 c ¼ 1  0:00571P þ 1:43  104 P2  2:07  106 P 3 c0  CaIn2 P2 V ¼ 1  0:01627P þ 4:76  104 P2  7:18  106 P3 V0  CaIn2 P2 z ¼ 1 þ 7:62  104 P  1:68  105 P2 þ 2:35  107 P3 z0 P  CaIn2 P2 z ¼ 1  1:32  104 P þ 1:14  105 P2  2:34  107 P3 z0 In  SrIn2 P2 a ¼ 1  0:00563P þ 1:56  104 P2  2:32  106 P3 a0  SrIn2 P2 c ¼ 1  0:00586P þ 1:53  104 P2  2:33  106 P3 c0  SrIn2 P2 V ¼ 1  0:01696P þ 5:13  104 P2  7:88  106 P3 V0  SrIn2 P2 z ¼ 1 þ 5:88  104 P  1:40  105 P2 þ 2:24  107 P3 z0 P  SrIn2 P2 z ¼ 1  1:24  104 P þ 1:07  105 P 2  2:05  107 P3 z0 In As the applied pressure increases from 0 to 25, the ratio c/c0 decreases more quickly than a/a0, indicating that both considered materials are much more compressible along the c-axis than along the a-axis. This means that CaIn2P2 and SrIn2P2 materials are stiffer for strains along the a axis than along c axis. The z coordinate of the P atom is more sensible to pressure than that of the In atom; which practically do not change its position with the pressure changes. The calculated unit cell volumes at fixed values of applied hydrostatic pressure in the range 0–30 GPa were fitted to the third-order Birch equation of state [23], depicted in Fig. 3(a). The obtained bulk modulus B and its pressure derivative B’ from this fitting are listed in Table 1. To the best of the authors’ knowledge, no experimental or theoretical data on the bulk modulus of the herein investigated materials, so at this time it is not possible to check out the obtained values. However, an indirect check for consistency of the above-given estimations can be made by direct calculations of the elastic constants. In order to fully characterize the pressure dependence of the structural parameters, we turn our attention to the pressure dependence of the interatomic distances (Fig. 3(b)). The bondlengths at zero pressure of the considered bonds are listed in Table 2 along with the existing experimental data [11]. Our results are in good agreement with the reported experimental results. Analysis of the relative variations of the considered bond-lengths shown in Fig. 3(b) allows us to draw some conclusions: (i) P–Ca (Sr) bonds are more compressible than the other bonds, namely P–In, Ca–In and In–In; (ii) P–In bonds are the less compressible; so they are the softest ones; (iii) P–In and Ca(Sr)–In bonds are less compressible than the In–In bond, which explains the pressure dependencies of the lattice constants a and c, shown in Fig. 2 and discussed above; and (iii) variations of all chemical bonds can be well approximated by a third-order polynomial: L/L0 = 1 + BP + CP2 + DP3, where L stands to the bond-length at a pressure P and L0 its corresponding value at zero pressure.  CaIn2 P2 L ¼ 1  0:00439P þ 1:10  104 P2  1:56  106 P3 L0 P—In  CaIn2 P2 L ¼ 1  0:00624P þ 1:90  104 P2  3:03  106 P3 L0 In—In  CaIn2 P2 L ¼ 1  0:00656P þ 1:72  104 P2  2:51  106 P3 L0 P—Ca  CaIn2 P2 L ¼ 1  0:00542P þ 1:30  104 P2  1:80  106 P3 L0 Ca—In  SrIn2 P2 L ¼ 1  0:00489P þ 1:28  104 P2  1:84  106 P3 L0 P—In  SrIn2 P2 L ¼ 1  0:00637P þ 1:98  104 P2  3:20  106 P3 L0 In—In  SrIn2 P2 L ¼ 1  0:00656P þ 1:78  104 P2  2:68  106 P3 L0 P—Sr  SrIn2 P2 L ¼ 1  0:0563P þ 1:42  104 P2  2:07  106 P3 L0 Sr—In 3.2. Electronic structure and chemical bonding The electronic structure of the valence band and the low energy conduction states determine the band gap and other properties of materials. The calculated band structures at the highly symmetric points of the Brillouin zone and along directions between them for the CaIn2P2 and SrIn2P2 compounds are shown in Fig. 4. For the CaIn2P2 compound, our ab initio calculation within the GGA-WC 590 N. Guechi et al. / Journal of Alloys and Compounds 577 (2013) 587–599 Fig. 2. Pressure dependence of the relative variations of the lattice constants, a and c, z coordinate of P and In atoms, ZP and ZIn, and unit cell volume, V, for the CaIn2P2 and SrIn2P2 materials. The ‘‘0’’ subscript denotes the parameter value at zero pressure. Pressure (GPa) 25 25 CaIn2P2 20 20 15 15 10 10 5 5 0 (a) 0.75 SrIn2P2 (a) 0 0.80 0.85 0.90 0.95 1.00 0.75 0.80 0.85 Relative bond-lengh, d/d0 1.00 CaIn2P2 0.98 0.96 1.00 P-In In-In P-Ca Ca-In 0.95 SrIn 2P2 1.00 P-In In-In P-Sr Sr-In 0.98 0.96 0.94 0.94 0.92 0.90 0.90 V/V0 V/V0 0.92 (b) 0 (b) 0.90 5 10 15 20 25 Pressure (GPa) 0 5 10 15 20 25 Pressure (GPa) Fig. 3. (a) Computed pressure versus primitive-cell volume data for the CaIn2P2 and SrIn2P2 materials. The symbols are the calculated results and the continuous lines are Birch EOS fits. (b) Pressure dependence of the relative bond-lengths for the CaIn2P2 and SrIn2P2 materials. The calculated values are shown by symbols and the quadratic approximations are shown by solid line. ‘‘d’’ stands for the interatomic distance at a pressure P, whereas ‘‘d0’’ is the same distance at zero pressure. at the optimized lattice parameters at zero pressure yielded an indirect band gap of 0.39 eV between the valence band maximum (VBMa) at the C point and the conduction band minimum (CBMi) at M point. For the SrIn2P2 compound, our calculation yielded a direct band gap C–C of 0.28 eV. It is worth to note that the substitution of the Ca atom by the Sr atom leads to narrowing the energy band gap. The top of the valence bands is mainly derived from the P-3p and In-5p states, while the bottom of the conduction bands is mainly made up of the M-d and In-5s + 5p states and a small contribution from P-3p states. We have gathered in Table 3 some direct and indirect transitions of interest for the herein studied materials. It should be pointed out here, that the calculated 591 N. Guechi et al. / Journal of Alloys and Compounds 577 (2013) 587–599 Table 2 Calculated bond-lengths (in Å) for the bonds having bond-length less than 4 Å and their multiplicity (given between brackets) for the Zintl compounds CaIn2P2 and SrIn2P2. Existing data are quoted for comparison. Bond CaIn2P2 P-In (4) In-In (2) P-M (4) M-In (4) (1  1), (1  4), (1  1), (1  1), (2  2), (3  3), (4  4) (2  3) (2  2), (3  1), (4  2) (2  2), (1  3), (2  4) SrIn2P2 Present work Expt. [11] Present work Expt. [11] 2.61498 2.78746 2.92000 3.75433 2.6021 2.7625 2.936 – 2.63188 2.78540 3.04692 3.86427 2.6216 2.7620 3.0572 – Fig. 4. Electronic band dispersion curves along the high symmetry directions in the Brillouin zone for the MIn2P2 (M = Ca, Sr) compounds. band gaps obtained using ab initio methods are as a rule somewhat underestimated, so our reported value can serve as a lower estimate of the band gap, which can be checked experimentally. All states of conduction and valence bands exhibit strong dispersion, which would manifest itself in high electrons and holes mobility, especially in the vicinity of the C point, the center of the Brillouin zone. Effective charge-carrier mass is one of the main factors determining the transport properties, Seebeck coefficient and electrical conductivity of materials. Here, the effective charge-carrier mass m⁄ have been evaluated by fitting the E–k diagram around the valence band maximum (VBMa) and conduction band minimum (CBMi) by a paraboloid, then the effective mass m⁄ at a given point along the direction given by ~ k is: 1 1 @ 2 EðkÞ ¼ 2 2 m h  @ k The calculated effective masses for electrons and holes from the electronic band energy dispersions at the points of interest in the Brillouin zone for the Zintl compounds MIn2P2 (M = Ca, Sr) (all in units of free electron mass) are summarized in Table 3. The effective electron mass is indicated by the under script ‘‘e’’ (me ) and the hole mass by ‘‘h’’ (mh ). Our calculations show that charge-carriers effective masses of CaIn2P2 are larger than that of SrIn2P2; this predicts that the mobility of charge-carriers in CaIn2P2 is smaller than in SrIn2P2. One can note that the electron effective masses at the bot- tom of the conduction band along the M ? L and M ? C directions in the BZ of CaIn2P2 are different, indicating an anisotropy of the electron effective masses in this compound at the M point, whereas the effective masses at the bottom of the conduction band along the C ? M and C ? K directions in the BZ of SrIn2P2 are quite equal, indicating an isotropy of the electron effective masses in this compound. The calculated hole effective masses at the top of the valence band along the C ? M and C ? K directions in the BZ of the studied materials are also quite equal, indicating also an isotropy of the hole effective masses in the herein investigated Zintl phases. Calculated effective masses of holes are larger than that of electrons, so the electrons would be the main resources of chargecarriers in these two materials. Small values of the effective masses of both electrons and holes indicate that these materials would be characterized by high charge-carriers mobility. The hydrostatic pressure affects significantly the value and nature of the energy band gaps of both herein studied compounds, as illustrated in Fig. 5. At about 15.87 GPa the bottom of the conduction band of CaIn2P2 moves with increasing pressure from the M point at zero pressure to the K point in the Brillouin zone, whereas the top of the valence band remains at the C point, regardless of the applied pressure. At about 3.60 GPa the band gap character of SrIn2P2 changes from direct band gap C–C to an indirect band gap C–M; the bottom of the conduction band moves with increasing pressure from the C point at zero pressure to the M point in the Brillouin zone, whereas the top of the valence band remains at the C point, regardless of the applied pressure. The variations of the C– C and C–K band-gap-energies with the increasing pressure P are approximated by the following polynomials: CaIn2 P2 : SrIn2 P2 : ( ( ECg —K ¼ 0:93369  0:01973P þ 2:34525  104 P2 ECg —C ¼ 0:57659 þ 0:09163P  0:00131  104 P2 EgC—K ¼ 1:22284  0:01672P þ 3:37462  104 P2 ECg —C ¼ 0:28119 þ 0:08971P  0:00116  104 P 2 We note here that the pressure dependence of the C–M band gap energy do not accept a second-order polynomial fit; this is due to the fact that two different conduction bands compete the point M. To further elucidate the nature of the electronic band structure, we have calculated the total (TDOS) and related partial (PDOS) densities of states diagrams. The valence bands of the two herein studied materials can be divided in five groups named V1, V2, V3, V4, and V5, which are separated by gaps; V3, V4 and V5 are not shown in Fig. 6 for more clarity of the shown diagrams. The lowest sharp structure V5 located at about 41 eV (35 eV) is Table 3 Calculated fundamental energy band gap (Eg, in eV), electron and hole effective masses (me and mh , respectively; in unit of free electron mass) for the CaIn2P2 and SrIn2P2 compounds. System Eg (C–C) Eg (C–M) Eg (C–K) me (M–L) me (M–C) me (C–M) me (C–K) mh (C–M) mh (C–K) CaIn2P2 SrIn2P2 0.561 0.263 0.390 0.521 0.937 1.226 0.012 – 0.0254 – – 0.0070 – 0.0070 0.0633 0.0371 0.0632 0.0371 592 N. Guechi et al. / Journal of Alloys and Compounds 577 (2013) 587–599 Fig. 5. Pressure dependence of the calculated band gaps: C–K, C–C and C–M, for the MIn2P2 (M = Ca, Sr) compounds. The calculated values are shown by symbols and the fits by solid lines. Fig. 6. Total (TDOS) and partial (PDOS) densities of states diagrams for the Zintl compounds MIn2P2 (M = Ca, Sr). Table 4 Mulliken’s and Hirshfeld’s atomic charges of the CaIn2P2 and SrIn2P2 materials. Atom species CaIn2P2 Mulliken’s Ca Sr In P SrIn2P2 Hirshfeld’s 1.04 0.12 – – 0.27 0.79 0.18 0.24 Mulliken’s Hirshfeld’s – – 0.91 0.26 0.72 0.16 0.17 0.25 due to the Ca-4s (Sr-5s) states. Another sharp structure, V4, centered at about 22 eV (17 eV) consists of the Ca-3p (Sr-4p) states. In-4d states form the structure V4 peaked at about 14 eV in both compounds. V5, V4 and V3 with energies much below the Fermi level ensure that these bands almost retain an atomic character with small band width. The structure V2 extended from about 11.5 to 9 eV is mainly made of the occupied P-3s states with small contributions from the In-5s + 5p + 4d states. The higher valence bands structure, stretching from about 6.8 eV (6.6 eV), show several DOS peaks and is formed from the occupied P-3p and In-5s + 5p states. Careful look at the states with largest contribution in the higher part of V1, the close proximity of the Fermi level, shows that they are predominantly from the hybridized P-3p and In-5p orbitals, whereas the P-3p and In-5s states form the lower part of V1. This means that In-P bonding is most significant for configuring the top of the valence band. The bottom of the conduction band 593 N. Guechi et al. / Journal of Alloys and Compounds 577 (2013) 587–599 the ionic bonding character percentage. Thus, from Hirshfeld’s charge (Mulliken’s charge) values given in Table 4, we estimate that the ionic bonding character is about 7% (23%) in both studied compounds. Thus these two materials have more covalent properties. A further insight into the chemical bonding can be gained by considering the charge density distributions. Fig. 7 shows the 2D charge density maps in the plane (1 1 0). From Fig. 7, the obvious deformation of charge distribution of In and P atoms indicates the existence of a directional bonding between In atoms and between In and P atoms. The more spherical charge distribution around M (M = Ca, Sr) atom reveals an ionic bonding between M and P atoms with a somewhat covalent feature. The hybridization of the In-5p and P-3p states, which is clearly visible in the DOS spectra shown in Fig. 6, is responsible for the covalent In-P bonds. 3.3. Elastic properties Fig. 7. Charge density distribution maps in (1 1 0) plane for the Zintl compounds MIn2P2 (M = Ca, Sr). Electron density is high in the red regions and is low in the blue regions. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) C1 is mainly derived from hybridization between the unoccupied P-3p, In-5s + 5p and Ca-3d (Sr-4d) states. Distribution of valence electron charges in a crystal is an important aspect of the electronic structure since it reveals the bonding pattern of the crystal. Thus, we have calculated the effective charge Q⁄ on each atom of the studied crystals using two different calculation methods: Mulliken population analysis (MPA) [24] and Hirshfeld population analysis (HPA) [25]; the obtained results are summarized in Table 4. It is well known that the absolute values of the atomic charges yielded by the population analysis have little physical meaning, since they display a high degree of sensibility to the atomic basis set [26]. However, we still can find some useful information by considering the relative values of bond populations. MPA is less reliable, as it arbitrarily divides the overlap electron population [27]. For small charge transfers, Fukui function indices estimated using MAP are eventually unpredictable [28]. HPA, in contrast, is more accurate, leading to improved Fukui function indices capable of predicting reactivity trends within a molecule better than MPA [29]. From Table 4, it can be seen that Ca (Sr) and In atoms donate electrons, while P atoms accept electrons. The negatively charged P atom is bounded to two positive metal atoms: M (M = Ca and Sr) and In. The amounts of charge transfers readily reveal the presence of a certain degree of ionic bonding character in these materials. The effective ionic valences, defined as the difference between the formal ionic charge and the calculated charge on the ion species in the crystal, can be used as a measure of ionicity; a value of zero implies an ideal ionic bond, while a value greater than zero indicates the presence of covalent bonding character. Nominal oxidation states of the atoms of our considered materials are M+2 (M = Ca, Sr), In+3 and P3. Considerable difference between the formal and calculated charges for Ca/Sr, In and P atoms highlights strongly covalent chemical bonds between them. We can exploit also the ratio of the net charge of an atom in a material to the charge of its nominal oxidation state to evaluate Table 5 Calculated elastic constant Cij (in GPa) for the MIn2P2 (M = Ca, Sr) compounds. System C11 C12 C13 C33 C44 C66 CaIn2P2 SrIn2P2 112.62 106.79 30.26 29.41 31.55 31.00 106.64 103.41 32.81 32.60 41.18 38.69 3.3.1. single-crystal elastic constants and related properties The elastic properties of a hexagonal crystal are described by five independent elastic constants: C11, C33, C44, C12 and C13. Our calculated elastic constants Cij of the CaIn2P2 and SrIn2P2 materials at zero pressure and for the optimized crystal structures are collected in Table 5. To the best of our knowledge, there are no experimental or theoretical results available in the literature for the elastic constants Cij of the herein studied materials, which can be compared to our present theoretical estimation. From Table 5, it can be seen that C11, C33, C12 and C13 of CaIn2P2 are slightly larger than that of SrIn2P2, while they have practically the same value for C44. C11 is slightly larger than C33 and C12 is slightly smaller than C13, so these materials would have a weak anisotropy from an elastic point of view. Knowing that C11 and C33 reflect the stiffness-touniaxial strains along a ([1 0 0]) and c ([0 0 1]) directions, this shows that these materials are stiffer for strains along the a axis than along c axis; this is in agreement with the response of a and c under hydrostatic pressure, shown in Fig. 2 and discussed above in Section 2.1. In addition, C44 is smaller than both C11 and C33, which reflects the weak resistance to shear deformation compared to the compressional deformations. Acoustic wave velocities in a material can be obtained from the Christoffel equation [30]. In a hexagonal structure, the sound wave velocities propagating in the [1 0 0], [0 0 1] and [1 2 0] directions are given by the following relations: m½1L 0 0 ¼ m½1L 2 0 ¼ m½1T10 0 ¼ m½1T12 0 m½1T20 0 ¼ m½1T22 0  1=2 C 11 q ;  1=2 C 11  C 12 ¼ ; 2q  1=2  1=2 C 44 C 33 ½0 0 1 ¼ ; mL ¼ ; m½0T10 1 ¼ m½0T20 1 ¼ m½0T10 1 ¼ m½0T20 1 ¼ q  1=2 C 44 q  1=2 C 44 q ; m½0L 0 1 ¼  1=2 C 33 q ; q where q is the mass density, T and L stand for transverse and longitudinal polarization regardless the elastic wave propagation direction. The sound velocities extracted from the elastic constants Cij at zero pressure are given in Table 6. From Table 6, we can see that there is a small difference between the values of the longitudinal velocities along the a-axis ([1 0 0] directions) and c-axis ([0 0 1] directions). On the other side, going from CaIn2P2 to SrIn2P2 both longitudinal and transverse waves decrease in the same trend as the elastic constants Cij, since the sound velocities are proportional to the square root of the corresponding elastic constant. 594 N. Guechi et al. / Journal of Alloys and Compounds 577 (2013) 587–599 Table 6 Acoustic wave velocities (in m/s) for different propagation directions in the MIn2P2 (M = Ca, Sr) compounds. System VL CaIn2P2 SrIn2P2 4999 4630 ½1 0 0 ½1 2 0 ¼ VL ½1 0 0 V T1 ½1 2 0 ½1 0 0 ¼ V T1 V T2 3018 2805 ½1 2 0 ½0 0 1 ¼ V T2 2682 2589 ½0 0 1 ½0 0 1 VL V T1 V T2 4846 4538 2682 2589 2682 2589 Fig. 8. Calculated pressure dependence of the elastic constants Cij and bulk modulus B for the CaIn2P2 and SrIn2P2 materials. The symbols are the calculated results and the continuous lines are the second-order polynomial fits to the results. Finally, we turn to the pressure dependence of the elastic constants Cij and bulk modulus B. Fig. 8 shows the pressure dependence of the elastic constants Cij and bulk modulus B for the herein studied materials. The dots show the first-principles results for the given pressures and the full lines were obtained by a simultaneous least squares fit of each of the six independent elastic constants (C11, C33, C44, C12 and C13) and the bulk modulus (B) to a second-order polynomial expression (Cij(P) = q + rP + tP2). The second-order polynomials describing the pressure dependence of the elastic constants and bulk modulus for CaIn2P2 and SrIn2P2 are given by the following expressions: CaIn2 P2 : SrIn2 P2 : 8 C 11 ¼ 112:54 þ 5:5929P  0:04591P 2 > > > > > C 33 ¼ 105:64 þ 4:61992P  0:02684P2 > > > > 2 > > < C 44 ¼ 32:57  0:07511P  0:0508P C 12 ¼ 30:30 þ 3:80594P  0:01149P2 > > 2 > > > C 13 ¼ 31:67 þ 3:63284P  0:00344P > > 2 > > > C 66 ¼ 41:12 þ 0:89348P  0:01721P > : B ¼ 56:96 þ 3:38499P  0:00369P2 8 C 11 ¼ 104:54 þ 5:58826P  0:05062P2 > > > > > C 33 ¼ 100:35 þ 5:17698P  0:05324P2 > > > > 2 > > < C 44 ¼ 33:34  0:13227P  0:05409P C 12 ¼ 27:25 þ 4:18524P  0:01689P2 > > > > C 13 ¼ 30:16 þ 3:80144P  0:00931P2 > > > > > C 66 ¼ 38:65 þ 0:70151P  0:01687P2 > > : B ¼ 53:33 þ 3:60889P  0:01241P2 The elastic constants C11 and C33 represent the elasticity in length; they change with the longitudinal strain. The elastic constants C12, C13 and C44 are related to the elasticity in shape. We can see quite different behaviors depending on the considered elastic constants. The elastic constant C11, C33, C12 and C13 increase with the increasing pressure. The exception is C44, which decreases with the increasing pressure, predicting a mechanical instability of these materials at high pressure. For a hexagonal crystal, the requirement of mechanical stability leads to the following restrictions of the elastic constants [31]: e 44 > 0; C e 11  j C e 12 j > 0; C e 33 ð C e 11 þ C e 12 Þ  2 C e 13 > 0, C where e aa ¼ C aa  P and C e ab ¼ C ab þ P. It is found that the elastic conC stants Cij for CaIn2P2 and SrIn2P2 satisfy these criteria only for pressure range 0–16.86 GPa and 0–16.48 GPa, respectively, so CaIn2P2 and SrIn2P2 are mechanically instable for hydrostatic pressures higher than about 16 GPa. 3.3.2. Polycrystalline elastic moduli and related properties Generally, large single-crystals are currently unavailable and consequently measurements of the individual elastic constants are impossible. The bulk modulus B and shear modulus G may be determined experimentally on the polycrystalline samples to characterize their mechanical properties. Theoretically, these two elastic parameters of the polycrystalline phase of a material can be obtained by a special averaging of the individual elastic constants of the monocrystalline phase. There are several different schemes of such homogenization method. The Reuss–Voigt–Hill homogenization method [32–34] is the most used one. Here, Voigt and Reuss approximations represent extreme values, and Hill recommended Table 7 Calculated polycrystalline elastic moduli: Reuss and Voigt bulk modulus (BR and BV, in GPa), Reuss and Voigt shear modulus (GR and GV, in GPa), Young’s modulus (E, in GPa) and Poisson’s ratio r for isotropic polycrystalline MIn2P2 (M = Ca, Sr) aggregates. System BR BV BH GR GV GH E r B/G CaIn2P2 SrIn2P2 57.45 53.77 57.48 53.77 57.47 53.77 36.49 35.05 36.91 35.23 36.70 35.14 90.78 86.57 0.24 0.23 1.57 1.53 595 N. Guechi et al. / Journal of Alloys and Compounds 577 (2013) 587–599 that the arithmetic mean of these two limits is used as effective moduli in practice for polycrystalline samples. Their definitions for hexagonal systems are as follows [35]:   2 1 C 11 þ C 12 þ C 33 þ 2C 13 þ C 33 ; 9 2 1 ð7C 11  5C 12 þ 12C 44 þ 2C 33  4C 13 Þ; GV ¼ 30 BR ¼ CðC 11 þ C 12 þ 2C 33  4C 13 Þ1 ; 5 GR ¼ fCC 44 C 66 gf3BV C 44 C 66 þ CðC 44 þ C 66 Þg1 ; 2 C ¼ ðC 11 þ C 12 ÞC 33  2C 213 ; BH ¼ ðBV þ BR Þ=2; GH ¼ ðGV þ GR Þ=2 BV ¼ Here (BV, GV), (BR, GR) and (BH, GH) are Voigt (V), Reuss (R) and Hill (H) bounds of the bulk modulus and shear modulus. Our calculated values of the above mentioned elastic moduli for the herein considered materials are quoted in Table 7. Using the calculated values of the bulk and shear moduli, we have evaluated the Young’s modulus E and Poisson’s coefficient r using the known relations [35], and the obtained results are summarized in Table 7, which allow us to make the following conclusions: (i) From Tables 1 and 7 one can see that the bulk modulus value evaluated for each compound from the single-crystal elastic constants is in good agreement with its value derived from the Birch equation of state P(V) depicted in Fig. 3. This might be an estimate of the reliability and accuracy of our calculated elastic constants for the CaIn2P2 and SrIn2P2 compounds. (ii) As the bulk modulus represents the resistance to volume change under hydrostatic pressures, this indicates that the average bond strength in CaIn2P2 is slightly higher than in SrIn2P2. The bulk moduli of this two considered materials are quite small (lower than 100 GPa) and so these materials should be classified as a relatively soft materials with high compressibility (higher than 0.01). 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 moduli of CaIn2P2 and SrIn2P2 are found to be 91 and 87 GPa, respectively; thus, these compounds will show a rather small stiffness. (iii) Pugh’s B/G ratio empirical criterion [36] is one of the widely used to provide information about brittle (ductile) nature of materials. If B/G > 1.75, a ductile behavior is predicted; otherwise, the material behaves in a brittle manner. According to the calculation results shown in Table 7, the B/G ratios of the studied compounds are smaller than 1.75, i.e., indicating a brittle nature of these materials and thus they will not be resistant to thermal shocks; their mechanic properties decrease quickly with increasing temperature. Debye temperature, which determines the thermal characteristics of a material, can be estimated from the average sound velocity in an isotropic material [37]. The obtained values of the longitudinal, transverse and average sound velocities (Vl, Vt and Vm, respectively), and the Debye temperature TD for polycrystalline CaIn2P2 and SrIn2P2 are gathered in Table 8. The value of TD decreases from Table 8 Calculated density of mass q, longitudinal, transverse and average sound velocity (Vl, Vt and Vm, respectively), and the Debye temperatures TD for the MIn2P2 (M = Ca, Sr) compounds. System q (g/cm3) Vl (m/s) Vt (m/s) Vm (m/s) TD (K) CaIn2P2 SrIn2P2 4.50 4.84 4865 4558 2857 2693 3167 2984 325 314 Table 9 Calculated anisotropy in the compression, AB, anisotropy in the shear, AG, anisotropy factors, A1 = A2 and A3, and the anisotropy universal index AU for the MIn2P2 (M = Ca, Sr) compounds. System AB (%) AG (%) AU A1 A2 A3 CaIn2P2 SrIn2P2 0.027 0.002 0.58 0.26 0.06 0.03 0.84 0.91 0.84 0.91 1 1 CaIn2P2 to SrIn2P2; this result is expected due to the fact that the stiffness decreases in the same sequence. 3.3.3. Elastic anisotropy Practically, all known crystals are elastically anisotropic, and a proper description of such anisotropic behavior has an important implication in engineering science as well as in crystal physics since the elastic anisotropy could introduce microcracks in materials [38,39]. Moreover, recent research demonstrates that the elastic anisotropy of crystals has a significant influence on the nanoscale precursor textures in alloys [40,41]. Therefore, several criteria have been developed to investigate the elastic anisotropy. The elastic anisotropy behavior of a crystal can be described by three shear factors: A1, A2 and A3 [42]. The shear anisotropic factors provide a measure of the degree of anisotropy in the bonding between atoms in different planes. For a hexagonal structure A1, A2 and A3 can be expressed as: A1 = A2 = 4C44/(C11 + C33  2C13), for the (1 0 0) and (0 1 0) planes; A3 = 4C66/(C11 + C22  2C12), for the (0 0 1) plane (C66 = (C11  C12)/2). For an isotropic crystal the factors A1, A2 and A3 must be one, while any value smaller or greater than unity is a measure of the degree of elastic anisotropy possessed by the crystal. The shear anisotropic factors obtained from our theoretical studies are given in Table 9. From Table 9, it can be seen that values of A3 are equal to 1 for both studied materials, indicating that the (0 0 1) shear plane exhibits isotropy. These two materials exhibit a weak anisotropy for the (1 0 0) and (0 1 0) shear planes; SrIn2P2 is less anisotropic than CaIn2P2. Another way to evaluate the elastic anisotropy is by introducing the concept of percent elastic anisotropy which is a measure of elastic anisotropy possessed by the crystal [43]. The percentage anisotropy in compressibility and shear are defined as: AB = (BV  BR)/(BV + BR); AB = (GV  GR)/(GV + GR), respectively, where B and G are the bulk and shear moduli, and the subscripts V and R represent the Voigt and Reuss bounds. For these two expressions, a value of zero represents elastic isotropy and a value of 1 (100%) is the largest possible anisotropy. The percentage of bulk and shear anisotropies are listed in Table 9. Clearly, the computed elastic constants of the herein studied compounds exhibit a weak anisotropy. The above elastic anisotropy criteria quantify the anisotropy degree from single bulk or shear contribution [41]. In order to quantify the extent of the anisotropy accurately, a new and more universal index AU has been proposed by Ranganathan and Ostoja-Starzewski [44] to measure the single-crystal anisotropy accounting for both bulk and shear contributions, where the AU is defined as follows: AU = 5GV/GR + BV/BR  6. For isotropic crystals, the universal index is equal to zero (AU = 0); the deviations of AU from zero definite the extent of crystal anisotropy. The obtained results shown in Table 9 reveal that both studied materials are characterized by a weak anisotropy and SrIn2P2 is less anisotropic than CaIn2P2. Three-dimensional (3D) surface representation of the elastic moduli is an effective method to visualize the elastic anisotropy of a material along its crystallographic directions. In 3D representation, an isotropic system would exhibit a spherical shape, and a deviation from spherical shape indicates the degree of anisotropy. 596 N. Guechi et al. / Journal of Alloys and Compounds 577 (2013) 587–599 Fig. 9. 3D directional dependence of the Young’s modulus (in GPa) and its projection on the ab {(0 0 1)} and ac {(0 1 0)} planes for the CaIn2P2 and SrIn2P2 materials. Fig. 10. Calculated imaginary part e2(x) and real part e1(x) of the dielectric function e(x) for the hexagonal Zintl compounds MIn2P2 (M = Ca, Sr). So, for a deep look into the peculiar features of the elastic anisotropy of our herein studied materials we have plotted in Fig. 9 their direction-dependent Young’s modulus surface using the following relation [45]: E ¼ S ½a n2 þb1 ðn2 þn2 Þn2 , where a2 = S11  S13, b2 = 2S11 11 2 3 2 1 2 3  2S13  S44, the Sij are the elastic compliance constants that can be obtained through an inversion of the elastic constant matrix, and n1, n2 and n3 are the directional cosines with respect to the x-, y- and z-axes, respectively. We have plotted the cross-sections of these surfaces in the (ab){(0 0 1)} and (ac){(0 1 0)} coordinate planes. From the 2D plane projections, one can see that the Young’s modulus of the (ab) plane is isotropic but that of the (ac) plane has a weak anisotropic character; the elastic anisotropy of SrIn2P2 is less weak than that of CaIn2P2. In the two considered compounds the highest value of the Young’s modulus Emax is realized for the external stress applied along the crystallographic axes a and b N. Guechi et al. / Journal of Alloys and Compounds 577 (2013) 587–599 597 Fig. 11. Calculated refractive index n(x) and extinction coefficient k(x) for the Zintl phases MIn2P2 (M = Ca, Sr). ([1 0 0] and [0 1 0] directions), and the lowest value Emin is for the stress along approximately any bisector direction in the (ac) and (bc) coordinate planes. The lowest value, Emin (about 84.41 GPa in CaIn2P2 and 82.97 GPa in SrIn2P2), is about 87% of the highest value of the Young’s modulus, Emax, in CaIn2P2 and about 91% in SrIn2P2, indicating a weak elastic anisotropy behavior of these materials and SrIn2P2 is less anisotropic than CaIn2P2. 3.4. Optical properties The calculated imaginary part e2(x) and real part e1(x) of the dielectric function e(x) of the hexagonal Zintl phases MIn2P2 (M = Ca, Sr) in the energy range from 0 to 15 eV are displayed in Fig. 10. Their calculations were performed for two light polariza- tions along [1 0 0] and [0 0 1] directions (the [1 0 0] and [0 1 0] directions are identical due to the hexagonal structure of our studied materials). The behavior of the dielectric function is rather similar for the two studied materials with some differences in details, which is attributed to the fact that the band structures of these materials are similar with minor differences causing insignificant changes in the structures of e(x). A common feature of the calculated dielectric functions spectra is that both the absorptive parts (e2(x)) and dispersive parts (e1(x)) of the complex dielectric functions are smaller for the [0 0 1] polarization, which indicates that the anisotropy of the optical properties of these materials is noticeable. Absorption edge is slightly shifted to the lower energy for the [0 0 1] polarization. The imaginary part of the complex dielectric function determines the absorption properties of a material. From Fig. 10, we Fig. 12. Calculated optical constants for the Zintl phases MIn2P2 (M = Ca, Sr): (a) absorption, (b) reflectivity and (c) energy-loss spectrum. 598 N. Guechi et al. / Journal of Alloys and Compounds 577 (2013) 587–599 can see that there are three T1, T2 and T3 (four: T1, T2, T3 and T 03 ) intensive peaks of absorption, centered at about 1.83 (1.67), 2.45 (2.37) and 3.12 (2.89, 3.43) eV in CaIn2P2 (SrIn2P2) for the [1 0 0] polarization. These peaks are due to the electronic transitions from the valence bands (V1) states to the conduction bands (C1) states. Consideration of the PDOS diagrams illustrated in Fig. 6 helps in assigning the peaks T1, T2, T3 and T4 to the transitions from the occupied P-3p and In-5p valences states to the unoccupied P-3p, In-5s + 5p and Ca-3d (Sr-4d) conduction states. The values of real part of the dielectric function at the zero frequency limit e1(0) are 20.62 (20.61) and 14.20 (13.20) for incident radiations polarized along the [1 0 0] and [0 0 1] polarizations, for CaIn2P2 (SrIn2P2), correspondingly. The calculated refractive index n(x) and extinction coefficient k(x) spectra for the two studied materials in the energy range from 0 to 15 eV for two incident radiation polarizations along [1 0 0] and [0 0 1] are presented in Fig. 11. It is clear from Fig. 11 that the refractive index and extinction coefficient exhibit a noticeable anisotropy. From the dispersion curves of refractive index, the static refractive index n(0) values are equal to 4.56 (4.54) and 3.78 (3.63) for incident radiation polarized along the [1 0 0] and [0 0 1] polarizations, for CaIn2P2 (SrIn2P2), correspondingly. When the photon energy increases, the refractive index n(x) displays a broad maximum with three humps in the energy range from 1.2 eV to 3.0 eV in the transparency region for the polarized incident radiation along [1 0 0] direction. Then it decays abruptly to its minimum level. The refractive index n(x) values are small than one for photon energy higher than 5.8 eV. The local maxima of the extinction coefficient k(x) correspond to the zero of e1(x). The origin of the structures in the imaginary part of the dielectric function e2(x) also explains the structures in the refractive index n(x). The absorption coefficient a(x) is a parameter, which characterize the decay of light intensity spreading in unit distance in medium. From Fig. 12(a), we can see that the absorption coefficient a(x) increases rapidly when the photon energy is higher than the absorption edge, which is the typical characteristic of semiconductors and insulators. These two materials exhibit a noticeable absorption in the visible and far-ultraviolet range so they are not a transparent crystal. The reflectivity coefficient R(x) is depicted in Fig. 12(b). These materials are characterized by a reflectivity coefficient R(x) value higher than 40% for a wide energy range from 0 to 13 eV. The reflectivity R(x) reaches a maximum value of about 70% for photon energies 3.6 and 6.0 eV (6.03 and 7.03 eV) for CaIn2P2 (SrIn2P2). The loss energy function L(x), an important factor describing the energy loss of a fast electron traversing in a material, is displayed in Fig. 12(c). For radiation polarized in [1 0 0] direction, this function has a main peak, the so called plasmon frequency (xp), at 14.84 eV for CaIn2P2 (14.42 eV for SrIn2P2), which corresponds to the abrupt reduction of reflectivity. 4. Conclusion In summary, by means of pseudopotential plane-wave method in the framework of density functional theory calculations we studied in details the structural, elastic, electronic, and optical properties of the hexagonal Zintl phases CaIn2P2 and SrIn2P2. The theoretically predicted lattice parameters for CaIn2P2 and SrIn2P2 are in good agreement with existing experimental measurements. Calculated dependencies of the unit cell on pressure were fitted by the Birch equation of state, and the relative changes of the lattice parameters and bond-lengths were approximated by third-order polynomials. Numerical estimates of the single-crystal and polycrystalline elastic moduli and their related properties were performed for the first time. Calculated dependencies of the single-crystal elastic constants on pressure shows that these considered materials are mechanically stable at zero pressure but they become mechanically instable for external hydrostatic pressure higher than about 16 GPa. Our analysis of the predicted elastic moduli shows that CaIn2P2 and SrIn2P2 are relatively soft materials with high compressibility, behave in a brittle manner, and characterized by a weak elastic anisotropy. The performed investigation of the electronic properties of CaIn2P2 and SrIn2P2 shows that they are narrow gap semiconductors possessing charge-carriers high mobility. 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