Accepted Manuscript Structural, elastic, electronic and optical properties of novel antiferroelectric KNaX (X�=�S, Se, and Te) compounds: First principles study R. Belghit, H. Belkhir, M.T. Kadri, D. Heciri, M. Bououdina, R. Ahuja PII:
S0921-4526(18)30359-4
DOI:
10.1016/j.physb.2018.05.025
Reference:
PHYSB 310887
To appear in:
Physica B: Physics of Condensed Matter
Received Date: 20 April 2018 Revised Date:
15 May 2018
Accepted Date: 18 May 2018
Please cite this article as: R. Belghit, H. Belkhir, M.T. Kadri, D. Heciri, M. Bououdina, R. Ahuja, Structural, elastic, electronic and optical properties of novel antiferroelectric KNaX (X�=�S, Se, and Te) compounds: First principles study, Physica B: Physics of Condensed Matter (2018), doi: 10.1016/ j.physb.2018.05.025. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Structural, elastic, electronic and optical properties of novel antiferroelectric KNaX (X= S, Se, and Te) compounds: First principles study
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R. Belghit1, H. Belkhir1, M. T. Kadri1, D.Heciri1, M.Bououdina2,* , R. Ahuja3 Laboratory Studies of Surface and Interfaces of Solid Materials (LESIMS), Department of Physics, Faculty of Sciences, University Badji Mokhtar, P.O. Box 12, Annaba 23000, Algeria
Department of Physics and Astronomy, Uppsala University, 751 20 Uppsala, Sweden
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Department of Physics, College of Science, University of Bahrain, PO Box 32038, Kingdom of Bahrain
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*
Corresponding author: (M. Bououdina, PhD)
Tel: +973 1743 7585 Fax: +973 1744 9148
e-mail address:
[email protected]
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ACCEPTED MANUSCRIPT Abstract This work deals with the investigation of structural, elastic, electronic and optical properties of ternary compounds belonging to the inter-alkali metal chalcogenide family KNaS, KNaSe, and KNaTe by using density functional theory (DFT) based calculations.
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From the structural properties, it is found that the lattice parameters and the atomic coordinates of KNaX (X= S, Se, and Te) are in good agreement with the experimental results stated in the literature. The compounds are considered as soft materials since their bulk
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modulus values are small. In addition, the elastic properties indicate that KNaS possesses the largest parameters of elastic constant Cij compared with other compounds, thereby revealing
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that KNaX compounds exhibit an elastic anisotropy. Meanwhile, the bulk modulus, shear and Young moduli, and Poisson ratio are also calculated. It is observed that all compounds are brittle and that the ionic character is dominant. In order to confirm the anisotropic character of the mechanical properties, several parameters such as universal, bulk and shear anisotropic
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indexes are investigated. For instance, a novel fascinating approach has been proposed by using the 3D-surfaces. The electronic properties demonstrate their semiconductor nature with a direct wide band gap of 2.61, 2.25 and 2.00 eV for KNaS, KNaSe and KNaTe respectively,
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as well as confirm their ionic behavior by checking the charge density distributions.
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Furthermore, since the optical properties of our compounds have not been yet reported in previous works, the dielectric function, refraction index, extinction index, reflectivity, loss energy function, optical conductivity and absorption spectra have been studied in details. Keywords: KNaX, First principles, elastic properties, electronic properties, optical properties.
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ACCEPTED MANUSCRIPT 1. Introduction In early 1950s, the antiferroelectric behavior was first proposed by Kitell [1]. The newly discovered antiferroelectric materials (AFE) have shown very interesting properties like
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higher density of energy capacitance, high effective electrocaloric feedback, a large response of nonlinear strain and high efficiency of energy transfer, making them useful for wide range of industrial applications, such as energy storage and conversion devices, solid state cooling, and transducers [2–5]. The most important class of (AFE) materials that has attracted great
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attention is ABO3 perovskite oxides, with two complexes structures Pbcm (PbZrO3) and
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Pbam (NaNbO3)[6–12]. However, some researchers have been devoted to explore new simple structures, which can offer improved performance, satisfy the criteria of Kittel model [1] and show better compatibility with other materials.
In2013, Bunnette et al. [13,14] used first-principles method to predict a new class of
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antiferroelectric materials by proposing about 70 compounds that were classified as follow: 37 compounds already synthesized and included in ICSD data-base, 13 compounds chosen from Zhang et al. [15] research work and 20 compounds from another previous study [16].
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These materials were placed in eight groups, with ananti-PbCl2crystal structure of MgSrSitype unknown as antiferroelectric material. The investigated physical parameters such as
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structural parameters, band gap, nonpolar distortion and the energy differences between AFE and FE phases, indicated that KNaS, KNaSe and KNaTe, among six materials are the most promising antiferroelectric materials which display a semiconductor behavior. KNaX (X=S, Se and Te) belong to inter-alkali metals chalcogenides, which has a formula of ABC, where A and B represent the elements of group II from periodic table (Li, Na, K, Rb and Cs), while C indicates a chalcogen element(O, S, Se and Te). The first synthesized ternary compounds are KNaO and RbNaO in 1981 [17], and have been
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ACCEPTED MANUSCRIPT extensively studied [18–27], but just their colors and crystal structures have been characterized. After that, Sabrowsky et al. [28] made an empirical rule to foresee whether that class of materials can crystallize in the anti-PbCl2 (orthorhombic) or in the anti-PbFCl (tetragonal) type structures. Among this range of materials, they found that KNaS, KNaSe
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and KNaTe compounds crystallize in the anti-PbCl2 structure type (Pnma, N°=62) [19–21]. To the best knowledge of the authors, no reports regarding the physical properties of these materials, except the experimental work on the structural properties and the theoretical work
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of Bunnette et al. [13] that partly deals with the electronic properties, have been reported so far in the literature. More importantly, there is no research work reporting on optical and
fundamental properties in-depth.
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elastic properties for the above mentioned materials, which motivate us to investigate their
This research work consists on a detailed investigation of structural, electronic, elastic and optical properties of KNaS, KNaSe and KNaTe compounds within the orthorhombic
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structure. The obtained results are discussed in terms of structural properties, elastic parameters and their related mechanical properties with the elastic anisotropy, the electronic properties such as structure bands energy, density of states and distributions of charge density,
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and finally prediction of optical properties such as dielectric constants, reflection and
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extinction indexes, reflectivity. A comparison with literature is presented. 2. Calculation details The investigation of the physical properties of KNaX ternary compounds was carried
out within the frame work of the density functional theory (DFT) [29] by using a method implemented in WIEN2K-code known as (FP-LAPW) [30]. Indeed, the exchange-correlation potential has been treated by the approximation of Wu and Cohen (WC-GGA) [31]. A degree of satisfactory convergence is reached by considering a number of FP-LAPW basic functions at a cuttof of RMTKmax= 7 (RMT is the smallest radius of the muffin-tin spheres and Kmax is the 4
ACCEPTED MANUSCRIPT largest value of the wave vector modulus of the mesh of the reciprocal lattice). When the total energy is stable at 0.1 mRy, the self-consistent calculations converge. For the integration, a mesh of 8×14×7k-points in the first Brillouin zone is used. In this study, 4s1 for K, 3s1 for Na, 3s2 3p4 for S, 4s2 4p4 for Se and 5s2 5p4 for Te are taken as valence states, and the rest are
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considered as core states.
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3. Results and discussion 3.1.Structural properties
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At ambient conditions, KNaX materials crystallize within the orthorhombic MgSrSi having space-group Pnma (N=62), where every primitive cell contains four formula units, as shown in Figure 1.
Before starting the calculations of any physical properties, it is important to determine the
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optimum volumes of the studied compounds. This optimization process is carried out as follow: after building up the first crystals based on the experimental parameters, the parameter (b/c) is optimized, then used to optimize the parameter (c/a). After that, both
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optimal values of (b/a) and (c/a) will be used to find the value of the optimum volume, and
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every step was accompanied with forces minimization to obtain the atomic positions. Table 1 shows all the calculated parameters including lattice constants, the positions of atoms, compression modulus and its first derivate of KNaS, KNaSe and KNaTe ternary compounds, alongside with the experimental and theoretical data reported in the literature for comparison. It is evident that for each structure, the lattice parameters with atomic positions are in accordance with the values obtained from experimental and simulation works. It is also observed that the largest error for the lattice parameters compared with the experimental values is about 0.7%, which confirms the reliability of our calculations, hence will be used
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ACCEPTED MANUSCRIPT for further calculations. On the other hand, it is noticed that the bulk modulus of KNaX compounds decreases according to the order KNaS>KNaSe>KNaTe, which is due to the larger volume of KNaTe>KNaSe>KNaS in accordance with the atomic radius of chalcogen
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element X, 0.37, 0.50 and 0.97 Å for S, Se and Te, respectively. 3.2.Elastic properties
The elastic constants are of great importance for the determination of physical properties because of their association with many other physical parameters [32].Since the
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studied compounds crystallize within an orthorhombic structure at ambient conditions, nine elastic constants are considered( , , , , , , , and ).The calculated
) for KNaX (X= S, Se and Te) given in Table 2 can be
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elastic constants ( and
considered as reference for future research works, since no previous values have been
reported in the literature so far. Primarily, all the computed elastic parameters for the investigated materials fulfill Born's mechanical stability conditions [33]:
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> 0, > 0, > 0, > 0, > 0, > 0 ( + − 2 )> 0, ( + − 2 )> 0, ( + − 2 )> 0 (01) ( + + + 2 + 2 + 2 )> 0
As shown in Table 2, The elastic parameters are found to decrease according to the
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order KNaS>KNaSe>KNaTe. It is common that, the elastic parameters , and are related to a, b and c-axes resistance to linear compression, respectively[34].Clearly, in Table
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2, the elastic constant is larger then and the parameter is larger than for all
compounds, which indicates that these ternary compounds KNaX are more compressible along c-axis than a-axis and more compressible along a-axis than b-axis.
Furthermore, the polycrystalline moduli are estimated by utilizing Voigt, Reuss, and Hill approximations [35–37]; compression and shear moduli can be written as follow [38]: =
1 ( + + − − − + 3 + 3 + 3 ) (02) 15 6
=
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1 ( + + + 2 + 2 + 2 ) (03) 9
1 = ( + + ) + 2 ( + + ) (04)
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1 4 3 ( + + − − − ) + ( + + ) (05) = 15 15
here, V and R refer to Voigt and Reuss approaches. By using the Hill approximation, the mechanical moduli can be estimated as follow [37]:
!=
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=
( + ) (06) 2
( + ) (07) 2
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=
9 (08) 3 +
#=
3 − 2 (09) 2(3 + )
%=
! (10) 2(1 + #)
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#! (11) (1 + #)(1 − 2#)
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&=
The Lamé's constants µ and λareobtained from E and ν as follow[39]:
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The mechanical properties of KNaX given in Table 3, indicate that the bulk moduli of
KNaS, KNaSe and KNaTe are 24.88, 22.29 and 19.62 GPa, respectevely, as obtained from the fit of Britch-Murnaghan equation. Moreover, sincethe values of bulk modulus are small, these ternary compounds can be consideredas soft materials. It's obvious that the shear modulus of 15.98 for KNaS is larger compared with that of KNaSe and KNaTe, which fundamentally originates from its higher value.The stiffness is associated with Young's
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As mentioned by Pugh's rule [40], the value of ⁄ is considered as a criterion to
foresee the brittleness and ductility of materials. In our case, the values of ⁄ are lower
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than 1.75 for all studied compounds, which denotes their brittle nature. Frabtsevich [41]
proposed the utilization of Poisson's ratio as a criterion to classify compounds as follow: with ν< 0.26 as brittle and with ν> 0.26 as ductile. As stated in Table 3, the values of Poisson's ratio for KNaS, KNaSe and KNaTe are smaller than 0.26, indicating once again and
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confirming their brittle nature, in accordance with the prediction by usingB⁄G.
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The Poisson's ratio is also used to inform about the character of the liaison forces. For
covalent character, the Poisson’s ratio should be small (# = 0.1), while for ionic character the
value of # is 0.25 [42]. The computed Poisson’s ratios are 0.232, 0.233and 0.24 for KNaS,
KNaSe and KNaTe, respectively. Consequently, it can be concluded that the ionic character
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is dominant for all compounds.
the hardness of a material is considered an essential property for its industrial applications. Numerous models [43] for calculating the hardness have been reported in the
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literature. In this study, two models (* = 2(+ ),.. − 3
and * = 0.151, where
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+ = /) have been adopted. The obtained values presented in Table 3, can be considered as
perspective data for future research work, because no experimental hardness values for KNaX (X= S, Se and Te) ternary compounds can be found in the literature. In this research work, the obtained elastic parameters suggest that our studied
compounds are elasticity anisotropic. Thereby, it is important to represent the elastic anisotropy by utilizing a common indexes such as universal anisotropic (01 ), bulk anisotropy
and shear anisotropy (02 and 03 ), as well as the factors of shear anisotropic (0 , 0 and 0 ).
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0 = 0 =
4 (15) ( + − 2 )
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0 =
( − ) (14) ( + )
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03 =
( − ) (13) ( + )
4 (16) ( + − 2 ) 4 (17) ( + − 2 )
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02 =
+ − 6 ≥ 0 (12)
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01 = 5
It should be highlighted that the total elastic anisotropy produces the zero value while the value of the greatest elastic anisotropy corresponds to100%. From Table 4, it can be
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noticed that the obtained 02 is lower than 03 for the three KNaS, KNaSe and KNaTe
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compounds, indicating that the bulk modulus has less directional dependence than shear modulus.
The 3D shape of mechanical properties represents a good approach to determine the
anisotropy in mechanical properties. For an orthorhombic structure, the equations of mechanical moduli surfaces can be expressed as [45]:
1 = ( + + )5 + ( + + )5 + ( + + )5 (18)
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1 = 5 + 5 + 5 + 25 5 + 225 5 + 25 5 + 5 5 + 5 5 ! + 5 5 (19)
In these equations 5 , 5 and 5 are the spherical coordinates flow the x, y and z axes,
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are the matrices of compliance constants. Figure 2 plots the three dimensional surface of
bulk, shear and Young's moduli for KNaX (X=S, Se and Te). To obtain the isotropic structure, the 3D-figure should possess a spherical form. If the form of the 3D-figure deviates
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from the shape of a sphere, this expresses the degree of anisotropy. As Figure 2 demonstrates, bulk modulus shows less anisotropic degree than that illustrated in both shear and Young's moduli for the studied compounds, which confirms the results given in Table 4.
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Debye temperature (67 ) is known as an essential parameter in thermodynamic properties of materials because of its relation to many physical constants such as thermal expansion, specific heat and elastic constants, and can be written in the form [46]:
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ℎ 3: => ? 67 = 9 < AB #C (20) +2 4; @
where ℎ represents Plank's constant and +2 is the constant of Boltzmann, : is the total atoms
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number per unit cell, => is the number of Avogadro, @ is the molecular mass. The average sound velocity #C can be written as:
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1 1 2 #C = D E + HI 3 5F 5G
J
(21)
4 + 3 #F = K L (22) ?
#G = < A (23) ? 10
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The results of both parameters(ΘM and #C ) for the studied compounds are given in
Table 5. The Debye temperature is well correlated with thermal conductivity; i.e. high Debye temperature means a large thermal conductivity. According to Table 5, KNaS has the largest thermal conductivity whereas KNaTe has the smallest. Once again, no experimental or
the ternary compounds are availablefor comparison.
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theoretical data reported in the literature for Debye temperature values and sound velocity for
The calculated values for the variation of elastic wave velocities along different axes
the deferent directions can be described as follow: Q100R#F = T ⁄? ;
Q010R#G = T ⁄? ; Q010R#F = T ⁄? ;
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Q100R SQ001R#G = T ⁄? ; (24)
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for KNaX, is also considered. According to Christoffel relation [47], the sound velocity on
Q010R SQ100R#G = T ⁄? ; (25)
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Q001R#G = T ⁄? ; Q001R#F = T ⁄? ;
Q001R SQ100R#G = T ⁄? ; (26)
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Q010R#G = T ⁄? ;
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where#F and #G are the sound velocity of longitudinal and transverse waves, respectively. From the obtained results for anisotropic sound velocity given in Table 6, it can be
noticed that KNaS possessed the smallest density ? and larger elastic constants resulting in
higher sound velocities. The elastic parameters , and are used for computing the
sound velocities of the longitudinal wave along x, y and z directions, subsequently. Thus, it is clear that the longitudinal waves of all compounds are fastest along y direction, while the
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The electronic properties, including band structure, density of states and distributions of charge density have been investigated. Figure 3 shows the band structure beside the total density of states for KNaX (X=S, Se and Te) ternary compounds. It can be noticed that the
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three compounds are semiconductors with a direct band gap, due to the fact that both maximum value of valence band and minimum value of conduction band are situated at Г
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point for all compounds. Furthermore, the less dispersive character (almost flat) of the valence band energy, can be interpreted by the localization of electrons. Thus, it is expected that the dominant bonding nature is the ionic character for all compounds. The values of band gap energy using the WC-GGA approximation as listed in Table 7, are similar to the values
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predicted in refs. [13,14] for both KNaS and KNaSe, while the value of band gap energy for KNaTe is found to be smaller than that reported in refs. [13,14], therefore future experimental research works are needed for confirmation.
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For more understanding of the electronic band structure, the partial density of states
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are plotted in Figure 4, where the Fermi level is taken as origin of energies. The profile of the density of states of the three compounds is similar, slight qualitative differences can be observed. The following remarks can be highlighted: The diagram of energy bands can be divided into four groups. The deeper region, is located around -12.7 eV for KNaS, -12.8 eV for KNaSe and 13.2 eV for KNaTe and is dominated mainly by the contribution of the p-orbital of K. The second region is situated at -10 eV and originates from the s orbital of chalcogen elements. 12
ACCEPTED MANUSCRIPT The top valence band originates from the p states of chalcogen elements with small contribution of the s-orbital of Na. The conduction band originates mainly from d states of K and a small contribution
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from s states of Na and p states of chalcogen elements. In order to obtain more information about the attraction between atoms (chemical bonding), the distributions of charge density for KNaS, KNaSe and KNaTe, have been plotted in Figure 5 as prototypes. No difference between the contour maps of the investigated
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materials can be observed. It is clear that the ionic character appears in the region between
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chalcogen elements(S-, Se- and Te-) and Na+, and that refers to the inequality of electronegativity, at the same time, the ionic character appears with lower degree between (S-, Se- and Te-) and K+ (K+(1.38 Å)>Na+ (1.02 Å)). Therefore, the ionic character is dominant, which confirms the above obtained results from the Poisson's ratios and band structures.
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3.4. Optical properties
The optical properties of a material appear from its interactions with electromagnetic
waves which are fully defined by the dielectric function V(W) = V (W) + XV (W). The rest of
optical parameters, such as absorption coefficient Y(W), optical conductivity Z(W), refractive
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index :(W), extinction coefficient +(W), optical reflectivity [(W), and energy-loss spectrum
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\(W), can be computed from the imaginary part V2 (W) by the Kramers-Kronig transform [48]. As KNaX (X= S, Se, Te) compounds have orthorhombic crystal structure with Pnma space group, three tenser components (i. e., V]] (W), V^^ (W), V__ (W)) are necessary to perform calculations of their optical properties. These components are expressed in ref. [49].
Figure 6 presents the dielectric function (real and imaginary parts) of KNaX (X= S, Se, Te) compounds versus photon energy up to 30 eV for the three crystal directions. It is clear that the dielectric function shifts to the low energy region when moving from S to Te, 13
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density of states (DOS). According to Figure 6(a), a considerable anisotropy between V`` (W) bb and both Vaa (W)and V (W) components is shown at low energy region, i.e. V`` (W) is the
dominant component for the three compounds. The imaginary part of the dielectric constant
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has prominent peaks with some shoulders on the right situated in the energy regions 4.09 to 7.8 eV, 3.76 to 7.63 and 4.06 to 7.28 eV for KNaS, KNaSe and KNaTe respectively. Principal peak values are given in Table 8, which are associated with direct interband transitions from occupied states (VBs) to unoccupied states (CBs). Considering the calculated
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PDOS, the dominant peaks of V`` (W) centred at 5.12, 5.47 and 3.99 eV are attributed to the
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optical transition from the occupied S-3p/Se-4p/Te-5p valences states to the unoccupied K-3d conduction states. For KNaTe compound, the high absorbance in ultraviolet (UV) range represented by the strong absorption peak located around 3.99 eV suggests that this
compound may be used for the fabrication of specific UV optoelectronic devices. One can note also the appearance of the threshold energy of the dielectric function at 2.64, 2.29, and
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1.9 eV for KNaS, KNaSe and KNaTe respectively. The Real parts V (W)of the dielectric function presented in Figure 6(b) show a high anisotropy between the non-zero components
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in the energy range 2-10 eV, the main peaks occur at 4.60, 4.28 and 3.50 eV, for KNaS, KNaSe and KNaTe respectively. According to this figure, the studied compounds show a
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Drude-like behaviour, since the real part of dielectric function V (W) crosses zero in some energy ranges, which is found more pronounced in the case of KNaTe. The real part of the
dielectric function at the zero frequency limit V (0) represents the dielectric response of a
material to a static electric field. According to Penn model [50], which is V(0) = 1 +
cℏωe ⁄Ef g , the obtained results reveal an inversely proportional relation between band gap
bb and V (0)and values of Vaa (0), V (0) and V`` (0)are listed in Table 8.
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edges located at 2.57eV for KNaS, 2.24 eV for KNaSe and 1.88 eV for KNaTe, are found to be close to the calculated band gap. The absorption spectrum decreases rapidly in the high energy region, which is the typical characteristic of semiconductors.
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The photoconductivity is defined as the increase in the electrical conductivity
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resulting from the increase in the number of free carriers generated when photons are absorbed [51,52]. The shape of the optical conductivity presented in Figure 7(b) shares
similar characteristics with V (W) spectrum. It can be noted that the maximum photoconductivity of KNaX (X=S, Se and Te) compounds lies in the UV (4-8 eV) and
extreme-UV (17-24 eV) region. In UV range, the pronounced peaks arise at about 5.06
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(Z `` (W)), 4.71 (Z `` (W)) and 3.99 (Z `` (W)) eVof magnitude 0.64x10-4, 0.67x10-4 and 0.58x10-4 (Ohm.cm)-1 for KNaS, KNaSe and KNaTe, respectively. Whereas, for extreme-UV
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range, prominent peaks occur around 21.18 (Z bb (W)), 21.16 (Z bb (W)) and 21.67 (Z aa (W))
eV of magnitude 0.39x10-4, 0.38x10-4 and 0.29x10-4 (Ohm.cm)-1 for for KNaS, KNaSe and
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KNaTe, respectively.
The same trend is observed for the refractive index :(W)presented in Figure 7(a),
which closely follows V (W) spectrum. One can note also that :(W) shows a slight
anisotropic behaviour. Prominent peak shifts towards lower energy while going from S to Te, which is found at 4.58, 4.21 and 3.57 eV for KNaS, KNaSe and KNaTe, correspondingly.
The static refractive index :(0)for the three crystal directions : aa (0):bb (0) and : `` (0), is
found to be 2.03, 2.02 and 2.05 for KNaS, 2.00, 2.00 and 2.12 for KNaS, 2.21, 2.23 and
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2.23for KNaTe. Furthermore, the refractive index in the static limit :(W) is expressed by the
relation :(0) = V ⁄ (0). The obtained results of :(0) are in good agreement with the values
calculated by this relation, and indicating the reliability of the results.
The zero of the real part of the dielectric function V (W) corresponds to the local
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maxima of the extinction coefficient +(W), which is shown in Figure 8(b). The sharp edges in the extinction coefficient +(W) is typical characteristic of semiconductor, due to the fact that occupied states (VBs) to unoccupied states (CBs).
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the incident photon requires more energy to cause optical excitations to raise an electron from
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The reflectivity spectrum shows a sharp drop situated in the ranges 6~10 eV and 23~26 eV for the herein studied compounds (see Figure 8(c)). This reduction results from the zero value of the real part V (W). The loss function of electron energy L(W) is illustrated in Figure 8(d), which represents an essential factor interpreting the energy loss of a fast electron
traversing a material. The \(W)spectra peaks show the plasma resonance and the plasma
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frequency Wh [53]. However, the \(W)peaks correspond to the edges in the reflective index
4. Conclusion
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spectra, which are given in Table 8, and corresponding also to the reduction of [(W).
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In summary, KNaX represent an interesting class of inter-alkali metal chalcogenide with antiferroelectric behavior where several insightful details on structural, mechanical, electronic and optical properties were largely missing from previous experimental and theoretical studies. To address these, in this research work, numerous physical properties of KNaS, KNaSe, and KNaTe ternary compounds were investigated in details by using a theoretical method based on density functional theory and implemented within WIEN2k code. The obtained results can be summarized as follow:
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The determined structural constants and atomic positions are in good agreement with the values reported in the literature, meanwhile the bulk moduli increase with changing the chalcogen atom from S to Te, which is well correlated to the ionic radius of the corresponding atoms. From the elastic properties, it is found that KNaS possesses the largest elastic
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•
constants, meanwhile the elastic parameters of the three compounds reveal a degree of anisotropy in the mechanical properties. The mechanical properties of KNaX
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compounds indicate a brittle behavior for the three compounds in addition to the ionic character being dominant in the inter-atomic bonding. The obtained elastic properties
•
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of KNaX are the first quantitative study.
The electronic properties of KNaX compounds confirm their semiconductor behavior with a direct wide band gap. The TDOS and PDOS are also investigated. The charge distribution confirms the elastic properties about their ionic bonding.
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This study demonstrates the first calculation of the optical properties of KNaS, KNaSe, and KNaTe. The dielectric function (real and imaginary parts), the extinction and the
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refractive indexes, the reflectivity, the loss energy function, the optical conductivity and the absorption spectra were studied. From the wide absorption in the UV region, these
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compounds can be considered as suitable materials for specific applications in UV optoelectronic devices.
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ACCEPTED MANUSCRIPT References [1]
C. Kittel, Theory of antiferroelectric crystals, Phys. Rev. 82 (1951) 729–732. doi:10.1103/PhysRev.82.729. Y. Yu, J. Tu, R.N. Singh, Phase Stability and Ferroelectric Properties of Lead
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ACCEPTED MANUSCRIPT Table Captions Table 1. The calculated structural parameters (a, b and c in Å), atomic positions, bulk modulus
B (in GPa) and its first derivative B' of KNaS, KNaSe and KNaTe ternary
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compounds. Table 2. The calculated elastic constants (in GPa) and their related elastic compliance matrices of KNaS, KNaSe and KNaTe ternary compounds.
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Table 3. The Calculated mechanical properties of KNaS, KNaSe and KNaTe ternary
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compounds with orthorhombic structure ( B, G and E in GPa).
Table 4. The calculated anisotropic index for KNaS, KNaSe, KNaTe ternary compounds in orthorhombic structure.
Table 5. The calculated density (ρ), the sound velocity (in m/s) and Debye temperature of KNaS, KNaSe and KNaTe ternary compounds.
compounds.
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Table 6. The anisotropic sound velocities (in km/s) of KNaS, KNaSe and KnaTe ternary
compounds.
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Table 7. The calculated energy band gaps (in eV) of KNaS, KNaSe and KnaTe ternary bb Table 8. Calculated maximum peak values Vaa (W), V (W) and V`` (W), static dielectric
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bb constantVaa (0), V (0) and V`` (0), dominant peak values of \aa (W),\bb (W) and \`` (W) and
zero frequency limit reflectivity [ aa (0)[ bb (0)[ `` (0) for KNaS, KNaSe and KNaTe ternary
compounds.
25
ACCEPTED MANUSCRIPT Figure Captions Figure 1. Crystal structure of KNaX (X= S, Se and Te) ternary compounds. X represents the chalcogen atoms.
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Figure 2. The 3D surfaces of bulk, shear and Youngs moduli for KNaX (X= S, Se and Te) ternary compounds
Figure 3. The band structure and total density of states (TDOS) for KNaX (X= S, Se and Te)
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ternary compounds.
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Figure 4. The partial density of states (PDOS) of KNaX (X= S, Se and Te) ternary compounds.
Figure 5. The charges destrubition contours of KNaX (X= S, Se and Te) ternary compounds. Figure 6. The real part of dielectric function (a) and the imagenry part of dielectric function
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for KNaX (X= S, Se and Te) ternary compounds. Figure 7. The optical conductivity (a) and the absorption spectra (b) for KNaX (X= S, Se and
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Te) ternary compounds.
Figure 8. The refraction index (a), extinxction index (b), refrelectivity (c) and loss energy
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function (d) for KNaX (X= S, Se and Te) ternary compounds.
26
ACCEPTED MANUSCRIPT Table 1. The calculated structural parameters (a, b and c (Å), Atomic positions, the bulk modulus B (GPa) and its first derivative B' of KNaS, KNaSe and KNaTe compounds. b
c
xK
zK
xNa
zNa
xCh
zCh
B
B'
Calc.
7.73
4.61
8.32
0.012
0.684
0.146
0.075
0.279
0.396
26.09
4.69
Othersd.e.
7.59
4.51
8.31
0.017
0.682
0.146
0.074
0.276
0.391
-
-
Expta
7.70
4.60
8.29
0.013
0.685
0.147
0.076
0.279
0.396
-
-
Calc.
8.10
4.85
8.70
0.013
0.684
0.146
0.074
Othersd.e
7.88
4.70
8.65
0.017
0.683
0.147
0.073
Exptb
8.06
4.82
8.65
0.013
0.685
0.147
0.075
Calc.
8.55
5.17
9.28
0.013
0.685
0.146
Othersd.e
8.35
5.00
9.26
0.017
0.686
0.147
Exptc
8.52
5.13
9.26
0.013
0.685
0.147
KNaSe
KNaTe
a
Ref.[19] Ref.[21] c Ref.[20] d Ref.[13] e Ref.[14] b
0.282
0.392
22.54
4.42
0.274
0.391
-
-
0.279
0.396
-
-
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KNaS
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a
0.074
0.281
0.391
18.34
4.45
0.072
0.272
0.390
-
-
0.075
0.279
0.393
-
-
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KNaX
C11 49.38 45.48 39.31
C12 11.59 10.40 9.52
S11
S12
KNaS KNaSe KNaTe
0.022673 0.024399 0.028221
-0.003125 -0.003121 -0.004501
C13 13.44 11.44 9.87
C22 51.75 48.81 39.82
C23 16.29 14.99 13.31
C33 40.88 34.37 32.48
C44 18.79 16.23 13.74
C55 13.37 11.94 9.44
C66 16.13 14.45 11.47
S13
S22
S23
S33
S44
S55
S66
-0.006212 -0.006760 -0.006732
0.022525 0.024055 0.029823
-0.007947 -0.009454 -0.010857
0.029672 0.035465 0.037283
0.053206 0.061599 0.072783
0.074742 0.083691 0.105934
0.061987 0.069191 0.08718
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KNaX KNaS KNaSe KNaTe
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Table 2. The calculated elastic constants (in GPa) and their related elastic compliance matrices of KNaS, KNaSe and KNaTe compounds.
Table 3. The Calculated mechanical properties of KNaS, KNaSe and KNaTe with orthorhombic structure ( B, G and E in (GPa). KNaX KNaS KNaSe KNaTe
BV
B BR
BH
GV
GR
G GH
24.96 22.48 19.67
24.81 22.10 19.56
24.88 22.29 19.62
16.37 14.64 12.19
15.98 14.19 11.84
16.17 14.42 11.79
HV E
ν
ܤ/ܩ
µ
λ
2ሺ݇ ଶ ܩሻ.ହ଼ଷ −3
KNaS KNaSe KNaTe
39.89 35.59 28.99
0.232 0.233 0.245
1.53 1.54 1.66
16.19 14.43 11.30
14.01 12.59 12.05
3.15 2.75 1.66
0.151ܩ 2.44 2.17 1.78
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Table 4. The calculated anisotropic index for KNaS, KNaSe, KNaTe in Orthorhombic Structure. AU
AB
AG
A1
A2
A3
0.1280 0.1757 0.2642
0.0030 0.0085 0,0028
0.0120 0.0156 0.0252
1.1858 1.1395 1.0456
0.8905 0.8977 0.7474
0.8277 0.7865 0.6303
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KNaX KNaS KNaSe KNaTe
Table 5. The calculated density (ρ), the sound velocity (The unit of velocity is m/s) and Debye temperature of KNaS, KNaSe and KNaTe compounds.
2.10 2.80 3.11
vl
vm
2772.76 2269.26 1905.62
4698.81 3850.29 3338.04
3072.22 2514.62 2117.66
ϴD 313.791 246.915 195.156
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KNaS KNaSe KNaTe
vt
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Sound velocity
ρ
KNaX
Table 6. The anisotropic sound velocities of KNaS, KNaSe and KnaTe ternary compounds. (The unit of velocity is km/s) KNaX
[100] [100]ߥ 5.108 4.157 3.554
KNaS KNaSe KNaTe
[010]ߥ௧ଵ 3.091 2.267 1.920
[001]ߥ௧ଶ 2.884 2.061 1.741
[011] [010]ߥ 5.232 4.167 3.577
[100]ߥ௧ଵ 3.091 2.267 1.920
[001]ߥ௧ଶ 3.163 2.403 2.101
[001] [001]ߥ 4.540 3.754 3.231
[100]ߥ௧ଵ 2.884 2.061 1.741
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Table 7. The calculated energy band gaps of KNaS, KNaSe and KNaTe. (The unit of velocity is km/s) The calculated Energy band gaps (eV) KNaX
Other works
Г-S
Г-X
Г-Y
Г-R
Г-T
Г-Z
Г-U
KNaS
2.61
3.93
4.91
4.47
4.59
4.27
3.91
4.54
2.59a,b
KNaSe
2.25
3.59
4.51
4.05
4.18
3.93
3.66
4.17
2.21a,b
KNaTe
2.00
3.12
3.81
3.45
3.45
3.27
3.23
3.59
2.49a,b
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Г-Г
a
Ref.[13]
b
Ref.[14]
[010]ߥ௧ଶ 3.163 2.403 2.101
ACCEPTED MANUSCRIPT ௬௬ Table 8. Calculated maximum peakvaluesߝଶ௫௫ ሺ߱ሻ, ߝଶ ሺ߱ሻ and ߝଶ௭௭ ሺ߱ሻ, static dielectric ௬௬ constant ߝଵ௫௫ ሺ0ሻ, ߝଵ ሺ0ሻ and ߝଵ௭௭ ሺ0ሻ, dominant peak values of ܮ௫௫ ሺ߱ሻ,ܮ௬௬ ሺ߱ሻ and ܮ௭௭ ሺ߱ሻ and zero frequency limit reflectivity ܴ ௫௫ ሺ0ሻܴ௬௬ ሺ0ሻܴ ௭௭ ሺ0ሻ for KNaS, KNaSe and KNaTe compounds.
KNaSe 4.77 4.74 4.74 4.41 4.41 4.53
KNaTe 4.52 3.99 3.99 4.90 4.96 5.00
ܮ௫௫ ሺ߱ሻ ܮ௬௬ ሺ߱ሻ ܮ௭௭ ሺ߱ሻ ܴ ௫௫ ሺ0ሻ ܴ௬௬ ሺ0ሻ ܴ ௭௭ ሺ0ሻ
KNaS 13.56 13.33 14.04 0.110 0.111 0.114
24.47 24.44 24.92
KNaSe 13.16 12.77 13.42 0.126 0.126 0.130
24.61 24.21 24.64
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KNaTe 3.87 3.66 3.86 0.142 0.144 0.146
7.72 7.66 7.66
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KNaS 5.37 5.07 5.12 4.00 3.97 4.08
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ߝଶ௫௫ ሺ߱ሻ ௬௬ ߝଶ ሺ߱ሻ ߝଶ௭௭ ሺ߱ሻ ߝଵ௫௫ ሺ0ሻ ௬௬ ߝଵ ሺ0ሻ ߝଵ௭௭ ሺ0ሻ
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