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First-principles study of the electronic structure, charge density, Fermi surface and optical properties of zintl phases compounds Sr2ZnA2 (A=P, As and Sb)
Journal of Magnetism and Magnetic Materials 345 (2013) 294–303
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First-principles study of the electronic structure, charge density, Fermi surface and optical properties of zintl phases compounds Sr2ZnA2 (A¼ P, As and Sb) A.H. Reshak a,b, Sikander Azam a,n a b
Institute of complex systems, FFPW, CENAKVA, South .Bohemia University in CB, Nove Hrady 37333, Czech Republic Center of Excellence Geopolymer and Green Technology, School of Material Engineering, University Malaysia Perlis, 01007 Kangar, Perlis, Malaysia
art ic l e i nf o
a b s t r a c t
Article history: Received 20 May 2013 Received in revised form 8 June 2013 Available online 5 July 2013
We present first-principles calculations of the electronic structure, Fermi surface, electronic charge density and optical properties of Sr2ZnA2 (A ¼P, As and Sb) based on density-functional theory using the local density approximation (LDA), generalized-gradient approximation (GGA) and the Engel–Vosko GGA formalism (EV-GGA). Additionally, modified Becke–Johnson (mBJ) is also used to improve the band splitting results. The calculated band structure and density of states show that Sr2ZnA2 compounds are metallic. The total DOS at Fermi level N(EF) is 72.92, 73.06 and 33.47 states/eV and the bare electronic specific heat coefficient (γ) is 12.64, 5.805 and 12.67 mJ/mol-K2 for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2, respectively. The Fermi surface of Sr2ZnA2 compounds is composed of two bands crossing along the Γ A direction of Brillouin zone. There exists a strong hybridization between Zn-p/s and Sb-d, Sb-p and Sr-d and also between Sr-s and Sr-p states. The bonding features are analyzed by using the electronic charge density contour in the (101) crystallographic plane. We found that Sr forms an ionic bond with Zn, whereas Zn forms a strong covalent interaction with P/As/Sb atoms. For further insight information about the electronic structure, the optical properties are derived and analyzed. & 2013 Elsevier B.V. All rights reserved.
Keywords: Zintl phases compound LDA GGA EVGGA mBJ: DFT
1. Introduction A motivating idea in the field of thermoelectric resources is the notion of PGEC (“phonon-glass, electron-crystal”) suggested by Slack [1]. PGEC resources should contain the low thermal conductivity of a glass, but keep the high electronic mobilities of crystalline semiconductors. It has been recommended that PGEC behavior would be finded in systems with heavy cations rattling inside oversized lattice cavities, to bring best conditions with regard to extensive thermal phonon scattering which leads to very low heat conductivity [2,3]. As energy PGEC resources, are considered the clathrate-type class of zintl compounds, in which their oversize cages are usually filled with large electropositive cation rattlers. Beneath these ingredients both the anionic frameworks are usually covalently bonded providing one medium with regard to heavy carrier range of motion. Indeed interest this class of ingredient has expanded currently [2]. The ingredient of the alkali and alkaline-earth materials with the early p-block resources are commonly understand as zintl
n Corresponding author at: Institute of complex systems, FFPW, CENAKVA, South Bohemia University in CB, Nove Hrady 37333, Czech Republic. Tel.: +420 775928620. E-mail addresses: [email protected], [email protected] (S. Azam).
0304-8853/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2013.06.021
phases [4,5]. Both the zintl stages containing heavy p-block elements generally screen a narrowed band difference. A precise acquaintance of the band gap and optical properties is very important for the design and study of various optoelectronic and photonic devices. It is most significant to understand the fundamental optical properties over a wide range of wavelengths [6]. The zintl stages which contains the p-block have remained widely examined due to their fascinating thermo electronic features, which includes electrical conductivity, thermo powder and also thermal conductivity [7–13]. The drawback of the above intermetallics is actually that they are very air-sensitive that significantly confines their own actual programs. The lanthanide analogs of the over phases, however, are much more air stable and therefore are more mean to be used as a new thermoelectric resources [12,13]. Subsequent the Zintl–Klemm formalism [14], the chemical bonding in this class of intermetallic stages can be rationalized if someone assumes the valence electrons are usually ascribed to the more electronegative metalloid element(s), which use them to achieve the electronic configuration(s) of a noble gas. An abundant examples of zintl phases with miscellaneous polyanionic fragments—clusters, chains, layers and frameworks—abound[15–37], yet the assortment of such novel compounds with intricate crystal and electronic structures appears far from being exhausted. Past efforts have been focused mostly on bismuthides and antimonides
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with zinc, manganese and cadmium, such as Ca2CdSb2 and Yb2CdSb2 [38], also on gallium pnictides e.g. A7Ga8Pn8 (A ¼Sr, Eu, Ba; Pn¼ Sb, Bi) [39], along with others. The synthesis of [Ba2Cd2Pn3 (Pn ¼As, Sb), BaGa2Pn2 (Pn ¼P, As) [40], and CaGa2Pn2 (Pn ¼P, As) and SrGa2As2 [41] resources has been made possible by the metal flux method [42], which permit the reactants to dissolve in a “solvent”, and attain higher diffusion rates at lower temperatures, by this means making it possible to get hold of even metastable phases. In the current paper we explain the results for the Zintl phase compounds i.e. Sr2ZnP2, Sr2ZnAs2 and A2ZnSb2. By using the LDA, GGA, EVGGA and mBJ techniques we calculated accurately the electronic band structure; density of states, charge density, Fermi surface and optical properties of these compounds. But here we will discuss the results obtained by modified Becke–Johnson (mBJ) exchange potential approximation with the reasons that mBJ gives better and more effective band splitting than that obtained using LDA, GGA and EVGGA. Because some other techniques, like optimized effective potentials (OEP), many body perturbation theory (MBPT), LDA+U, and GGA+U give excellent band gap, but these techniques are computationally expensive than mBJ approximation [43]. The rest of the paper has been divided in three parts. In Section 2, we briefly describe the computational method used in this study. The most relevant results obtained for the electronic and optical properties Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 are presented and discussed in Section 3. Finally, we summarize the main conclusions in Section 4.
2. Calculation methodology The crystal structure of Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 are shown in Fig.1. The scrutinize compounds have the hexagonal symmetry with space group P63/mmc (No. 194). The lattice constant for Sr2ZnP2 are a ¼4.309(2) Å and c¼7.893(6) Å, for Sr2ZnAs2 a¼ 4.4185(8) Å and c¼ 8.042(2) Å and for Sr2ZnSb2 a ¼4.6234(13) Å and c¼ 8.345(3) Å. The atomic position in sequence for Sr2ZnP2, Sr2ZnAs2, Sr2ZnSb2 are given in Table 1. The electronic structure, Fermi surface, optical properties and charge density of Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 are obtained using the highly accurate full potential linearized augmented plane wave (FPLAPW) method, as implemented in theWien2k code [44]. This method is based on density functional theory (DFT) [45,46] within the local density approximation (LDA) [47], generalized gradient approximation (GGA) and GGA-PBE [48], and Engel– Vosko GGA [49] and the modified Becke–Johnson (mBJ) exchange correlation potential. Kohn–Sham wave functions were expanded
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in terms of spherical harmonic function inside the nonoverlapping muffin- tin (MT) spheres and with Fourier series in the interstitial region. The wave functions in the interstitial regions were expanded in the plane waves for the cut off of KMAXRMT ¼7 in order to achieve the convergence for energy eigenvalues. The valve for muffin-tin radii chosen for Sr2ZnP2: RMT (Sr) ¼ 2.50 a.u., RMT (Zn) ¼2.48 a.u. and RMT (P) ¼2.20 a.u., for Sr2ZnAs2 compound are chosen as RMT (Sr) ¼2.50 a.u., RMT (Zn) ¼2.50 a.u. and RMT (As)¼ 2.25 a.u., and for Sr2ZnSb2 compound are chosen as: RMT (Sr) ¼2.50 a.u., RMT (Zn) ¼2.50 a.u. and RMT (Sb)¼2.36 a.u. The dielectric function was calculated in the momentum representation, which requires matrix elements of the momentum p between occupied and unoccupied states. Thus, the components of the imaginary or the absorptive part of the dielectric function, εij2 ðωÞ was calculated using [50] the relation E D 4π 2 e2 ∑ knsjpi jkn′s kn′sjpj jkns Vm2 ω2 knn′s
εij2 ðωÞ ¼
f kn ð1f kn′ ÞsðEkn′ Ε kn ℏωÞ where e is the electron charge and m is the mass, ω is the frequency of the incoming electromagnetic radiation, V is the volume of the unit cell, (px, py, pz)¼p is the momentum operator jknsi the crystal wave function, corresponding to eigenvalue Ekn with crystal momentum k and spin s. Finally, f kn is the Fermi distribution function ensuring that only transitions from occupied to unoccupied states are counted, and sðΕ kn′ Ε kn ωÞ is the condition for total energy conservation. The real part ε1 ðωÞ can be obtained from the imaginary part ε2 ðωÞ using Kramer's Kronig
Table 1 Atomic positions of the Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds. Compound
Atom
x
y
z
Sr2ZnP2
Sr Zn P
0 1/3 1/3
0 2/3 2/3
0 3/4 1/4
Sr2ZnAs2
Sr Zn As
0 1/3 1/3
0 2/3 2/3
0 3/4 1/4
Sr2ZnSb2
Sr Zn Sb
0 1/3 1/3
0 2/3 2/3
0 3/4 1/4
Fig. 1. Unit cell structure for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds.
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dispersion relation [51]. Z 1 2 ω′ε2 ðω′Þ ε1 ðωÞ ¼ 1 þ Ρ dω′ π ω′2 ω2 0
2.1. Band structure and density of states The calculated electronic band structures of Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds are given in Fig. 2. The exposed figures of the electronic band structure show that the three compounds
exhibit metallic characters. Up to our knowledge there is no experimental data on the band structure of these three compounds are available in the literatures to be contrast with our results, by due to successful application of FP-LAPW methods we can discuss the performance of the energy band structure for these exacting material under the present study. Following Fig. 2, the bands situated around the Fermi levels are due to the admixture of Sr-s/p, Zn-s/p and P-p states. To further illuminate the nature of the electronic bands structure, we calculated the total and partial density of states as shown in Figs. 3 and 4. As we replace P2 by As2, the total structures
Fig. 2. (a–d): Calculated Band structures for LDA, GGA, EVGGA and mBJ for Sr2ZnP2/As2/Sb2 compounds.
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Fig. 2. (continued)
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Fig. 2. (continued)
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Fig. 3. Calculated combined total density of state for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds (State/eV unit cell).
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are shifted towards lower energies by around 0.3 eV. While when we replace As2 by Sb2, the total structures are shifted towards higher energies by around 1.3 eV. The peaks between 12.0 eV and 8.0 eV are due to Zn-d and P/As/Sb-s states and from 8.0 eV and 2.0 eV are due to P/As/ Sb-p and Zn-s/p states. While the peaks around the Fermi level are due to Zn-s/p and Sr-d. The peaks situated between 2.0 eV and 14 eV are due to the contribution of Sr-s/p/d, Zn-s/p and Sb-d states. There exists a strong hybridization between Zn-p/s and Sbd, Sb-p and Sr-d in the energy range 11.0 eV and 8.0 eV, whereas there is strong hybridization between Sr-s and Sr-p states in the energy range between 2.0 eV and 14 eV.
Fig. 4. (a–o): Calculated total and partial density of state for Sr2ZnP2/As2/Sb2 compound (State/eV unit cell).
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Fig. 5. Calculated Fermi surface for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds.
The total DOS at Fermi level N(EF) is 72.92, 73.06 and 33.47 states/eV for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2, respectively. The metallic character of all the three inter metallic is clearly seen from the finite DOS at the Fermi level. The electronic specific heat coefficient (γ), which is a function of the density of states, can be calculated using the expression γ¼
1 2 2 π ΝðΕ F ÞkB 3
where N(EF) is the DOS at the Fermi energy EF and kB is the Boltzmann constant. The calculated density of states at the Fermi energy N(EF) enables us to calculate the bare electronic specific heat coefficient, which is 12.64, 5.805 and 12.67 mJ/mol K2 for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2, respectively Fig. 4. 2.2. Fermi surface The occupation of electronic states at the Fermi level specifies the metallic measures of the compounds. For this reason, it is vital to establish the shape of the Fermi surface. The accepted way to find out the Fermi surface is to measure energy distribution curves (EDC) for distinct k-points of the Brilliouin zone and to ascertain the k-locations where bands traverse the Fermi energy. The electrons adjacent to the Fermi level are responsible for conductivity; the electronic structure of any metallic material is understandable from the Fermi surface (FS). Fermi surfaces can also be inspected by using de Haas van Alphen (dHvA) experiments, the
exact determination of the shape of the FS depends on models using easy geometries to define the magnetic field angle dependency of the measured frequencies, but the accurate shape of the FS can of course be much further complex. Therefore we figured the FS of Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 using the FPLAPW method outlined above. The results are shown in Fig. 5. The calculations for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2, exposes that there are two bands crossing along the Γ K direction. The Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 FS of compound consist of a set of holes and electronic sheets, where the empty areas contain holes and shaded areas electrons. In the three compounds there is the differences in FS topology, due to changes in inter-atomic distances, bonding angles and as well as to the degree of band filling. For the compounds, a merged structure is shown in the Fig. 5. In the Fermi surface the color changes occur due to the change in electron velocity. The red and violet color shows fast and the slow velocity of electrons while the remaining colors have the intermediate velocity of electron. The structure of the three compounds are considerably the same but the changes occur in the color which resulting the change in velocity of electron. 2.3. Electron charge density Electron charge density denotes the nature of the bond character. In order to predict the chemical bonding and also the charge transfer in Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds, the charge-density behaviors in 2D are calculated in the (101) plane
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and displayed in Fig. 6. To discuss the electronic charge density we calculated the bond lengths in unit lattices of Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds whose results are given in Table 2.
Fig. 6. Calculated electron charge density for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds.
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Table 2 Bond length. Sr2ZnP2
Sr2ZnAs2
Sr2ZnSb2
P3–Zn2 ¼2.488 P3–Sr1 ¼3.175 Zn2–Sr1 ¼ 3.175
As3–Zn2 ¼ 2.551 As3–Sr1 ¼ 3.248 Zn2–Sr1 ¼3.248
Sb3–Zn2 ¼ 2.669 Sb3–Sr1 ¼ 3.388 Zn2–Sr1 ¼ 3.388
Fig. 7. (a–c): Calculated average value of imaginary part of dielectric tenser component for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds ðε2 ðωÞÞ.
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3. Optical properties The Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds have a hexagonal symmetry. Thus the consequent dielectric functions are εxx ðωÞ ¼εyy ðωÞ and εzz ðωÞ. The calculated dielectric functions contains both the energy eigenvalues and electron wave functions. The imaginary part of the frequency dependent dielectric function is obtained using the expressions given in the Ref. 52: x Z jP nm′ ðkÞj2 þ jP y nm′ ðkÞj2 dSk y ε2 xx ðωÞ ¼ ∑ ∇ωnm′ ðkÞ mω2 BZ nn′` ε2 zz ðωÞ ¼
12 mω2
Z
jP z nm′ ðkÞj2 dSk ∇ωnm′ ðkÞ BZ nn′` ∑
The above terms are written in atomic units with e2 ¼1/m ¼2 and ћ ¼1, where ω is the photon energy. P x nm′ ðkÞ and P z nm′ ðkÞ are the X and Z component of the dipolar matrix elements between initial jn′ki and final jn′ki states with their eigenvalues En(k) and Ε′n ðkÞ respectively. ωn′n ðkÞ is the energy difference, where ωn′n ðkÞ ¼En(k) Ε′n ðkÞ and Sk is a constant energy surface with Sk ¼ fk; ωnn′ ðkÞ ¼ ωg. Since the investigated compounds have metallic nature thus we have to include the intra band transitions (Drude term). Therefore both of the intra-band and inter-band transitions are contribute ε2 ðωÞ ¼ ε2inter ðωÞ þ ε2int ra ðωÞ where εxx 2int ra ðωÞ ¼
ωxx p τ ωzz p τ ; εzz 2int ra ðωÞ ¼ ωð1 þ ω2 τ2 Þ ωð1 þ ω2 τ2 Þ
where τ and ωp [53] is mean free time and mean frequency amongst the collisions. ωp 2 ¼
Fig. 8. (a–c): Calculated average value of real part of dielectric tenser component for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds ðε1 ðωÞÞ.
The result shows that the substitution of the P by As and As by Sb leads to redistribution of electron charge density. As an atom is interchanged by the other atom the VSCC (valence shell charge carrier) properties changes. By investigating the influence of replacing of P by As and As by Sb in Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds, it has been found that the charge density around “As” is greater than the both (P and Sb), as it is labeled by # 1 in Fig. 5 and also the bond length changes as listed in Table 2. From the calculated electron density it is clear that Sr forms an ionic bond with Zn, whereas Zn forms a strong covalent interaction with P/As/Sb and Zn atoms.
8π ∑ϑ2 δðε Þ 3 kn kn kn
As εkn is Ε n ðkÞΕ F and ϑkn is the velocity of electron. zz The calculated dielectric functions εxx 2 ðωÞ, and ε2 ðωÞ of the investigated compounds using mBJ are illustrated in Fig. 7. Following this figure we can notice that these compounds have a different optical spectrum, which attributed to the fact that the band structures for these compounds are different, which causes some changes in the optical transitions result in changing the peak positions and the peak heights. There is a considerable anisotropy zz between εxx 2 ðωÞ and ε2 ðωÞ components. We should emphasize that xx the ε2 ðωÞ component is the dominant in Sr2ZnAs2 and Sr2ZnSb2 spectrum while εzz 2 ðωÞ is the dominant one in Sr2ZnP2 spectra. We should highlights that subtracting P2 by As2 and As2 by Sb2, cause to change the peak heights, which shows that the Sr2ZnP2 compound more metallic with respect to Sr2ZnSb2 and Sr2ZnAs2 compound. From the imaginary part of dielectric functions εxx 2 ðωÞ, xx zz and εzz 2 ðωÞ the real parts ε1 ðωÞ, and ε1 ðωÞ are calculated using zz Kramer's–Kronig relations. The results of εxx 1 ðωÞ, and ε1 ðωÞ are shown in Fig. 8. It is clear from Fig. 8 that there is considerable anisotropy between these two components of the frequencydependent dielectric function.
4. Conclusion In summary, we calculate the band structure, density of states, Fermi surface, charge density and optical properties of Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds by means of DFT within LDA, GGA, EVGGA and mBJ. The investigated compounds possess metallic nature, the values of DOS at EF (N(EF)) are 72.92, 73.06 and 33.47 states/eV and the bare electronic specific heat coefficient is 12.64,
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5.805 and 12.67 mJ/mol-K2 for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2, respectively. The Fermi surface of Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 is composed of two bands crossing along the Γ A direction of BZ. The bonding features are analyzed by using the electronic charge density contour in the (101) crystallographic plane. We found that Sr forms an ionic bond with Zn, whereas Zn forms a strong covalent interaction with P/As/Sb and Zn atoms. There exists a strong hybridization between Zn-p/s and Sb-d, Sb-p and Sr-d in the energy range between 11.0 eV and 8.0 eV and also between Sr-s and Sr-p states in the energy range between 2.0 eV and 14 eV. We also present the dielectric function, of Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 and the imaginary part ε2 ðωÞ of the dielectric function is discussed in detail. Acknowledgment This work was supported from the project CENAKVA (No. CZ.1.05/2.1.00/01.0024), the Grant no. 134/2013/Z/104020 of the Grant Agency of the University of South Bohemia. School of Material Engineering, Malaysia University of Perlis, P.O Box 77, d/a Pejabat Pos Besar, 01007 Kangar, Perlis, Malaysia. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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First-principles study of the electronic structure, charge density, Fermi surface and optical properties of zintl phases compounds Sr2ZnA2 (A¼ P, As and Sb) A.H. Reshak a,b, Sikander Azam a,n a b
Institute of complex systems, FFPW, CENAKVA, South .Bohemia University in CB, Nove Hrady 37333, Czech Republic Center of Excellence Geopolymer and Green Technology, School of Material Engineering, University Malaysia Perlis, 01007 Kangar, Perlis, Malaysia
art ic l e i nf o
a b s t r a c t
Article history: Received 20 May 2013 Received in revised form 8 June 2013 Available online 5 July 2013
We present first-principles calculations of the electronic structure, Fermi surface, electronic charge density and optical properties of Sr2ZnA2 (A ¼P, As and Sb) based on density-functional theory using the local density approximation (LDA), generalized-gradient approximation (GGA) and the Engel–Vosko GGA formalism (EV-GGA). Additionally, modified Becke–Johnson (mBJ) is also used to improve the band splitting results. The calculated band structure and density of states show that Sr2ZnA2 compounds are metallic. The total DOS at Fermi level N(EF) is 72.92, 73.06 and 33.47 states/eV and the bare electronic specific heat coefficient (γ) is 12.64, 5.805 and 12.67 mJ/mol-K2 for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2, respectively. The Fermi surface of Sr2ZnA2 compounds is composed of two bands crossing along the Γ A direction of Brillouin zone. There exists a strong hybridization between Zn-p/s and Sb-d, Sb-p and Sr-d and also between Sr-s and Sr-p states. The bonding features are analyzed by using the electronic charge density contour in the (101) crystallographic plane. We found that Sr forms an ionic bond with Zn, whereas Zn forms a strong covalent interaction with P/As/Sb atoms. For further insight information about the electronic structure, the optical properties are derived and analyzed. & 2013 Elsevier B.V. All rights reserved.
Keywords: Zintl phases compound LDA GGA EVGGA mBJ: DFT
1. Introduction A motivating idea in the field of thermoelectric resources is the notion of PGEC (“phonon-glass, electron-crystal”) suggested by Slack [1]. PGEC resources should contain the low thermal conductivity of a glass, but keep the high electronic mobilities of crystalline semiconductors. It has been recommended that PGEC behavior would be finded in systems with heavy cations rattling inside oversized lattice cavities, to bring best conditions with regard to extensive thermal phonon scattering which leads to very low heat conductivity [2,3]. As energy PGEC resources, are considered the clathrate-type class of zintl compounds, in which their oversize cages are usually filled with large electropositive cation rattlers. Beneath these ingredients both the anionic frameworks are usually covalently bonded providing one medium with regard to heavy carrier range of motion. Indeed interest this class of ingredient has expanded currently [2]. The ingredient of the alkali and alkaline-earth materials with the early p-block resources are commonly understand as zintl
n Corresponding author at: Institute of complex systems, FFPW, CENAKVA, South Bohemia University in CB, Nove Hrady 37333, Czech Republic. Tel.: +420 775928620. E-mail addresses: [email protected], [email protected] (S. Azam).
0304-8853/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2013.06.021
phases [4,5]. Both the zintl stages containing heavy p-block elements generally screen a narrowed band difference. A precise acquaintance of the band gap and optical properties is very important for the design and study of various optoelectronic and photonic devices. It is most significant to understand the fundamental optical properties over a wide range of wavelengths [6]. The zintl stages which contains the p-block have remained widely examined due to their fascinating thermo electronic features, which includes electrical conductivity, thermo powder and also thermal conductivity [7–13]. The drawback of the above intermetallics is actually that they are very air-sensitive that significantly confines their own actual programs. The lanthanide analogs of the over phases, however, are much more air stable and therefore are more mean to be used as a new thermoelectric resources [12,13]. Subsequent the Zintl–Klemm formalism [14], the chemical bonding in this class of intermetallic stages can be rationalized if someone assumes the valence electrons are usually ascribed to the more electronegative metalloid element(s), which use them to achieve the electronic configuration(s) of a noble gas. An abundant examples of zintl phases with miscellaneous polyanionic fragments—clusters, chains, layers and frameworks—abound[15–37], yet the assortment of such novel compounds with intricate crystal and electronic structures appears far from being exhausted. Past efforts have been focused mostly on bismuthides and antimonides
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with zinc, manganese and cadmium, such as Ca2CdSb2 and Yb2CdSb2 [38], also on gallium pnictides e.g. A7Ga8Pn8 (A ¼Sr, Eu, Ba; Pn¼ Sb, Bi) [39], along with others. The synthesis of [Ba2Cd2Pn3 (Pn ¼As, Sb), BaGa2Pn2 (Pn ¼P, As) [40], and CaGa2Pn2 (Pn ¼P, As) and SrGa2As2 [41] resources has been made possible by the metal flux method [42], which permit the reactants to dissolve in a “solvent”, and attain higher diffusion rates at lower temperatures, by this means making it possible to get hold of even metastable phases. In the current paper we explain the results for the Zintl phase compounds i.e. Sr2ZnP2, Sr2ZnAs2 and A2ZnSb2. By using the LDA, GGA, EVGGA and mBJ techniques we calculated accurately the electronic band structure; density of states, charge density, Fermi surface and optical properties of these compounds. But here we will discuss the results obtained by modified Becke–Johnson (mBJ) exchange potential approximation with the reasons that mBJ gives better and more effective band splitting than that obtained using LDA, GGA and EVGGA. Because some other techniques, like optimized effective potentials (OEP), many body perturbation theory (MBPT), LDA+U, and GGA+U give excellent band gap, but these techniques are computationally expensive than mBJ approximation [43]. The rest of the paper has been divided in three parts. In Section 2, we briefly describe the computational method used in this study. The most relevant results obtained for the electronic and optical properties Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 are presented and discussed in Section 3. Finally, we summarize the main conclusions in Section 4.
2. Calculation methodology The crystal structure of Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 are shown in Fig.1. The scrutinize compounds have the hexagonal symmetry with space group P63/mmc (No. 194). The lattice constant for Sr2ZnP2 are a ¼4.309(2) Å and c¼7.893(6) Å, for Sr2ZnAs2 a¼ 4.4185(8) Å and c¼ 8.042(2) Å and for Sr2ZnSb2 a ¼4.6234(13) Å and c¼ 8.345(3) Å. The atomic position in sequence for Sr2ZnP2, Sr2ZnAs2, Sr2ZnSb2 are given in Table 1. The electronic structure, Fermi surface, optical properties and charge density of Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 are obtained using the highly accurate full potential linearized augmented plane wave (FPLAPW) method, as implemented in theWien2k code [44]. This method is based on density functional theory (DFT) [45,46] within the local density approximation (LDA) [47], generalized gradient approximation (GGA) and GGA-PBE [48], and Engel– Vosko GGA [49] and the modified Becke–Johnson (mBJ) exchange correlation potential. Kohn–Sham wave functions were expanded
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in terms of spherical harmonic function inside the nonoverlapping muffin- tin (MT) spheres and with Fourier series in the interstitial region. The wave functions in the interstitial regions were expanded in the plane waves for the cut off of KMAXRMT ¼7 in order to achieve the convergence for energy eigenvalues. The valve for muffin-tin radii chosen for Sr2ZnP2: RMT (Sr) ¼ 2.50 a.u., RMT (Zn) ¼2.48 a.u. and RMT (P) ¼2.20 a.u., for Sr2ZnAs2 compound are chosen as RMT (Sr) ¼2.50 a.u., RMT (Zn) ¼2.50 a.u. and RMT (As)¼ 2.25 a.u., and for Sr2ZnSb2 compound are chosen as: RMT (Sr) ¼2.50 a.u., RMT (Zn) ¼2.50 a.u. and RMT (Sb)¼2.36 a.u. The dielectric function was calculated in the momentum representation, which requires matrix elements of the momentum p between occupied and unoccupied states. Thus, the components of the imaginary or the absorptive part of the dielectric function, εij2 ðωÞ was calculated using [50] the relation E D 4π 2 e2 ∑ knsjpi jkn′s kn′sjpj jkns Vm2 ω2 knn′s
εij2 ðωÞ ¼
f kn ð1f kn′ ÞsðEkn′ Ε kn ℏωÞ where e is the electron charge and m is the mass, ω is the frequency of the incoming electromagnetic radiation, V is the volume of the unit cell, (px, py, pz)¼p is the momentum operator jknsi the crystal wave function, corresponding to eigenvalue Ekn with crystal momentum k and spin s. Finally, f kn is the Fermi distribution function ensuring that only transitions from occupied to unoccupied states are counted, and sðΕ kn′ Ε kn ωÞ is the condition for total energy conservation. The real part ε1 ðωÞ can be obtained from the imaginary part ε2 ðωÞ using Kramer's Kronig
Table 1 Atomic positions of the Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds. Compound
Atom
x
y
z
Sr2ZnP2
Sr Zn P
0 1/3 1/3
0 2/3 2/3
0 3/4 1/4
Sr2ZnAs2
Sr Zn As
0 1/3 1/3
0 2/3 2/3
0 3/4 1/4
Sr2ZnSb2
Sr Zn Sb
0 1/3 1/3
0 2/3 2/3
0 3/4 1/4
Fig. 1. Unit cell structure for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds.
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dispersion relation [51]. Z 1 2 ω′ε2 ðω′Þ ε1 ðωÞ ¼ 1 þ Ρ dω′ π ω′2 ω2 0
2.1. Band structure and density of states The calculated electronic band structures of Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds are given in Fig. 2. The exposed figures of the electronic band structure show that the three compounds
exhibit metallic characters. Up to our knowledge there is no experimental data on the band structure of these three compounds are available in the literatures to be contrast with our results, by due to successful application of FP-LAPW methods we can discuss the performance of the energy band structure for these exacting material under the present study. Following Fig. 2, the bands situated around the Fermi levels are due to the admixture of Sr-s/p, Zn-s/p and P-p states. To further illuminate the nature of the electronic bands structure, we calculated the total and partial density of states as shown in Figs. 3 and 4. As we replace P2 by As2, the total structures
Fig. 2. (a–d): Calculated Band structures for LDA, GGA, EVGGA and mBJ for Sr2ZnP2/As2/Sb2 compounds.
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Fig. 2. (continued)
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Fig. 2. (continued)
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Fig. 3. Calculated combined total density of state for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds (State/eV unit cell).
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are shifted towards lower energies by around 0.3 eV. While when we replace As2 by Sb2, the total structures are shifted towards higher energies by around 1.3 eV. The peaks between 12.0 eV and 8.0 eV are due to Zn-d and P/As/Sb-s states and from 8.0 eV and 2.0 eV are due to P/As/ Sb-p and Zn-s/p states. While the peaks around the Fermi level are due to Zn-s/p and Sr-d. The peaks situated between 2.0 eV and 14 eV are due to the contribution of Sr-s/p/d, Zn-s/p and Sb-d states. There exists a strong hybridization between Zn-p/s and Sbd, Sb-p and Sr-d in the energy range 11.0 eV and 8.0 eV, whereas there is strong hybridization between Sr-s and Sr-p states in the energy range between 2.0 eV and 14 eV.
Fig. 4. (a–o): Calculated total and partial density of state for Sr2ZnP2/As2/Sb2 compound (State/eV unit cell).
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Fig. 5. Calculated Fermi surface for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds.
The total DOS at Fermi level N(EF) is 72.92, 73.06 and 33.47 states/eV for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2, respectively. The metallic character of all the three inter metallic is clearly seen from the finite DOS at the Fermi level. The electronic specific heat coefficient (γ), which is a function of the density of states, can be calculated using the expression γ¼
1 2 2 π ΝðΕ F ÞkB 3
where N(EF) is the DOS at the Fermi energy EF and kB is the Boltzmann constant. The calculated density of states at the Fermi energy N(EF) enables us to calculate the bare electronic specific heat coefficient, which is 12.64, 5.805 and 12.67 mJ/mol K2 for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2, respectively Fig. 4. 2.2. Fermi surface The occupation of electronic states at the Fermi level specifies the metallic measures of the compounds. For this reason, it is vital to establish the shape of the Fermi surface. The accepted way to find out the Fermi surface is to measure energy distribution curves (EDC) for distinct k-points of the Brilliouin zone and to ascertain the k-locations where bands traverse the Fermi energy. The electrons adjacent to the Fermi level are responsible for conductivity; the electronic structure of any metallic material is understandable from the Fermi surface (FS). Fermi surfaces can also be inspected by using de Haas van Alphen (dHvA) experiments, the
exact determination of the shape of the FS depends on models using easy geometries to define the magnetic field angle dependency of the measured frequencies, but the accurate shape of the FS can of course be much further complex. Therefore we figured the FS of Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 using the FPLAPW method outlined above. The results are shown in Fig. 5. The calculations for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2, exposes that there are two bands crossing along the Γ K direction. The Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 FS of compound consist of a set of holes and electronic sheets, where the empty areas contain holes and shaded areas electrons. In the three compounds there is the differences in FS topology, due to changes in inter-atomic distances, bonding angles and as well as to the degree of band filling. For the compounds, a merged structure is shown in the Fig. 5. In the Fermi surface the color changes occur due to the change in electron velocity. The red and violet color shows fast and the slow velocity of electrons while the remaining colors have the intermediate velocity of electron. The structure of the three compounds are considerably the same but the changes occur in the color which resulting the change in velocity of electron. 2.3. Electron charge density Electron charge density denotes the nature of the bond character. In order to predict the chemical bonding and also the charge transfer in Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds, the charge-density behaviors in 2D are calculated in the (101) plane
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and displayed in Fig. 6. To discuss the electronic charge density we calculated the bond lengths in unit lattices of Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds whose results are given in Table 2.
Fig. 6. Calculated electron charge density for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds.
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Table 2 Bond length. Sr2ZnP2
Sr2ZnAs2
Sr2ZnSb2
P3–Zn2 ¼2.488 P3–Sr1 ¼3.175 Zn2–Sr1 ¼ 3.175
As3–Zn2 ¼ 2.551 As3–Sr1 ¼ 3.248 Zn2–Sr1 ¼3.248
Sb3–Zn2 ¼ 2.669 Sb3–Sr1 ¼ 3.388 Zn2–Sr1 ¼ 3.388
Fig. 7. (a–c): Calculated average value of imaginary part of dielectric tenser component for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds ðε2 ðωÞÞ.
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3. Optical properties The Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds have a hexagonal symmetry. Thus the consequent dielectric functions are εxx ðωÞ ¼εyy ðωÞ and εzz ðωÞ. The calculated dielectric functions contains both the energy eigenvalues and electron wave functions. The imaginary part of the frequency dependent dielectric function is obtained using the expressions given in the Ref. 52: x Z jP nm′ ðkÞj2 þ jP y nm′ ðkÞj2 dSk y ε2 xx ðωÞ ¼ ∑ ∇ωnm′ ðkÞ mω2 BZ nn′` ε2 zz ðωÞ ¼
12 mω2
Z
jP z nm′ ðkÞj2 dSk ∇ωnm′ ðkÞ BZ nn′` ∑
The above terms are written in atomic units with e2 ¼1/m ¼2 and ћ ¼1, where ω is the photon energy. P x nm′ ðkÞ and P z nm′ ðkÞ are the X and Z component of the dipolar matrix elements between initial jn′ki and final jn′ki states with their eigenvalues En(k) and Ε′n ðkÞ respectively. ωn′n ðkÞ is the energy difference, where ωn′n ðkÞ ¼En(k) Ε′n ðkÞ and Sk is a constant energy surface with Sk ¼ fk; ωnn′ ðkÞ ¼ ωg. Since the investigated compounds have metallic nature thus we have to include the intra band transitions (Drude term). Therefore both of the intra-band and inter-band transitions are contribute ε2 ðωÞ ¼ ε2inter ðωÞ þ ε2int ra ðωÞ where εxx 2int ra ðωÞ ¼
ωxx p τ ωzz p τ ; εzz 2int ra ðωÞ ¼ ωð1 þ ω2 τ2 Þ ωð1 þ ω2 τ2 Þ
where τ and ωp [53] is mean free time and mean frequency amongst the collisions. ωp 2 ¼
Fig. 8. (a–c): Calculated average value of real part of dielectric tenser component for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds ðε1 ðωÞÞ.
The result shows that the substitution of the P by As and As by Sb leads to redistribution of electron charge density. As an atom is interchanged by the other atom the VSCC (valence shell charge carrier) properties changes. By investigating the influence of replacing of P by As and As by Sb in Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds, it has been found that the charge density around “As” is greater than the both (P and Sb), as it is labeled by # 1 in Fig. 5 and also the bond length changes as listed in Table 2. From the calculated electron density it is clear that Sr forms an ionic bond with Zn, whereas Zn forms a strong covalent interaction with P/As/Sb and Zn atoms.
8π ∑ϑ2 δðε Þ 3 kn kn kn
As εkn is Ε n ðkÞΕ F and ϑkn is the velocity of electron. zz The calculated dielectric functions εxx 2 ðωÞ, and ε2 ðωÞ of the investigated compounds using mBJ are illustrated in Fig. 7. Following this figure we can notice that these compounds have a different optical spectrum, which attributed to the fact that the band structures for these compounds are different, which causes some changes in the optical transitions result in changing the peak positions and the peak heights. There is a considerable anisotropy zz between εxx 2 ðωÞ and ε2 ðωÞ components. We should emphasize that xx the ε2 ðωÞ component is the dominant in Sr2ZnAs2 and Sr2ZnSb2 spectrum while εzz 2 ðωÞ is the dominant one in Sr2ZnP2 spectra. We should highlights that subtracting P2 by As2 and As2 by Sb2, cause to change the peak heights, which shows that the Sr2ZnP2 compound more metallic with respect to Sr2ZnSb2 and Sr2ZnAs2 compound. From the imaginary part of dielectric functions εxx 2 ðωÞ, xx zz and εzz 2 ðωÞ the real parts ε1 ðωÞ, and ε1 ðωÞ are calculated using zz Kramer's–Kronig relations. The results of εxx 1 ðωÞ, and ε1 ðωÞ are shown in Fig. 8. It is clear from Fig. 8 that there is considerable anisotropy between these two components of the frequencydependent dielectric function.
4. Conclusion In summary, we calculate the band structure, density of states, Fermi surface, charge density and optical properties of Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 compounds by means of DFT within LDA, GGA, EVGGA and mBJ. The investigated compounds possess metallic nature, the values of DOS at EF (N(EF)) are 72.92, 73.06 and 33.47 states/eV and the bare electronic specific heat coefficient is 12.64,
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5.805 and 12.67 mJ/mol-K2 for Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2, respectively. The Fermi surface of Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 is composed of two bands crossing along the Γ A direction of BZ. The bonding features are analyzed by using the electronic charge density contour in the (101) crystallographic plane. We found that Sr forms an ionic bond with Zn, whereas Zn forms a strong covalent interaction with P/As/Sb and Zn atoms. There exists a strong hybridization between Zn-p/s and Sb-d, Sb-p and Sr-d in the energy range between 11.0 eV and 8.0 eV and also between Sr-s and Sr-p states in the energy range between 2.0 eV and 14 eV. We also present the dielectric function, of Sr2ZnP2, Sr2ZnAs2 and Sr2ZnSb2 and the imaginary part ε2 ðωÞ of the dielectric function is discussed in detail. Acknowledgment This work was supported from the project CENAKVA (No. CZ.1.05/2.1.00/01.0024), the Grant no. 134/2013/Z/104020 of the Grant Agency of the University of South Bohemia. School of Material Engineering, Malaysia University of Perlis, P.O Box 77, d/a Pejabat Pos Besar, 01007 Kangar, Perlis, Malaysia. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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