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Influence of different exchange correlation potentials on band structure and optical constant calculations of ZrGa2 and ZrGe2 single crystals
Computational Materials Science 78 (2013) 134–139
Contents lists available at SciVerse ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Influence of different exchange correlation potentials on band structure and optical constant calculations of ZrGa2 and ZrGe2 single crystals A.H. Reshak a,b,⇑, A.O. Fedorchuk c,g,⇑, G. Lakshminarayana d, Z.A. Alahmed f, H. Kamarudin b, S. Auluck e a
Institute of Complex Systems, FFPW, CENAKVA-South Bohemia University CB, Nove Hrady 37333, Czech Republic Center of Excellence Geopolymer and Green Technology, School of Material Engineering, University Malaysia Perlis, 01007 Kangar, Perlis, Malaysia c Lviv National University of Veterinary Medicine and Biotechnologies, Department of Inorganic and Organic Chemistry, Lviv, Ukraine d Materials Science and Technology Division (MST-7), Los Alamos National Laboratory, Los Alamos, NM 87545, USA e National Physical Laboratory, Dr. K S Krishnan Marg, New Delhi 110 012, Saudi Arabia f Department of Physics and Astronomy, King Saud University, Riyadh 11451, Saudi Arabia g P. Sagajdachnyj Miltary University, Lviv, Ukraine b
a r t i c l e
i n f o
Article history: Received 7 March 2013 Received in revised form 27 April 2013 Accepted 29 April 2013 Available online 19 June 2013 Keywords: Inorganic materials Crystal growth Electronic band structure
a b s t r a c t The all-electron full potential linearized augmented plane wave method was used to solve the Kohn Sham DFT equations. We have employed different approximations for the exchange correlation potentials, namely: LDA, GGA and EVGGA, and insignificant effect on the band structure and the density of states were found. Calculations show that there is a significant difference in the band dispersion with replacement of Ga by Ge that is attributed to the fact that in the ZrGe2 compound Zr atom is situated at 4c site and two Ge atoms are situated at 4c site. Whereas for ZrGa2 compound Zr is located at 4g site and the three Ga atoms are situated at 4h, 2c and 2a sites, respectively. There exists strong hybridization between the states. Moving from ZrGa2 to ZrGe2 has significant influence on the magnitudes and the peak positions of states. The optical properties of the two compounds were studied and analyzed. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction It is well known systematization of structural types with respect to the coordination of the atoms of the smallest sizes [1], which gives to classify structural types by classes, groups and to establish an affinity between structural types within the same class as well as between structural types, belong to different classes. Usually these approaches are different. In this work we try to look on the behavior of the corresponding energy band gaps. To study such relations the better way is to perform the calculations for the isostructural compounds. Among the binary compounds particular interest present halides, germanides, antimonides, arsenides, etc. Partially covalence bonds prevailingly form more electronegative atoms and atoms with less electronegativity usually occupy empty structural positions or occupy voids in cluster layers or polyhedra formed by anions. Very often these atoms are characterized by shorter chemical bonds to anions. This phenomenon may be explained similarly to
⇑ Corresponding authors. Address: Institute of Complex Systems, FFPW, CENAKVA-South Bohemia University CB, Nove Hrady 37333, Czech Republic (A.H. Reshak). Tel.: +420 777 729 583; fax: +420 386 361 219. E-mail addresses: [email protected], [email protected] (A.O. Fedorchuk). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.04.056
polyhedra Frank–Casper [2], i.e. like atomic agglomerates, where cationic electrons are shifted towards connections with anions. As a consequence the compound possessing the lowest electronegativity for compounds of different classes at fixed structural type should be changed. As a consequence, in the Ref. [3] an attempt has been made to perform a classification of structural types with respect to the atoms with maximal sizes. Following the reasons presented above in the present work we explore the band structure performed within a framework of DFT method with different exchange correlation potentials: LDA, GGA and EVGGA. For this reason we have chosen compounds ZrGa2 and ZrGe2 [4] possessing the same structural fragments (Fig. 1). At the same time their electronic configuration is different from each other. The choice of the different exchange correlations approach should add to help the influence of this screening on the band structure behaviors. The quantum chemical simulations should show how sensitive they will be to such different potential screenings. To the best of our knowledge no comprehensive work neither experimental data on the optical properties or first principles calculations on the structural, electronic, and optical properties of ZrGa2 and ZrGe2 compounds have appeared in the literature. Therefore as a natural extension to previous work on ZrGa2 and ZrGa3 compounds [5,6] a detailed depiction of the structural, electronic, and optical properties of ZrGa2 and ZrGe2 using full
A.H. Reshak et al. / Computational Materials Science 78 (2013) 134–139
135
Fig. 1. (a) The crystallochemical transformation of the structure for crystals ZrGa2 and ZrGe2; (b) crystal structure of ZrGa2 and ZrGe2; and (c) asymmetry unit of ZrGa2 and ZrGe2.
potential method is timely and would bring us important insights in understanding the origin of the band structure and densities of states. Hence it is very important to use a full potential method. Present study is aimed towards such calculation by using the full potential linear augmented plane wave (FP-LAPW) method which has proven to be one of the most accurate methods [7,8] for the computation of the electronic structure of solids within a framework density functional theory (DFT). Our calculations will demonstrate the effect of using a full-potential on the structural, electronic and optical properties of the single crystals of ZrGa2 and ZrGe2. In the Section 2 are given structural properties and principal computational details. The Section 3 present principal results concerning the band structure calculations and calculated optical
dispersions of the principal optical functions. Comparison of the results obtained by LDA, GGA and EVGGA is presented.
2. Structural properties and computational details For the calculations are chosen ZrGa2 and ZrGe2 single crystals. The crystal structures are listed in Tables 1 and 2 and illustrated at Fig. 1. Using the X-ray diffraction data (XRD) we have optimized the structure by minimization of the forces acting on the atoms. The structure is fully relaxed until the forces on the atoms reach values less than 1 mRy/a.u. Once the forces are minimized in this construction one can then find the self-consistent electron space density distribution at these positions by turning off the
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A.H. Reshak et al. / Computational Materials Science 78 (2013) 134–139
Table 1 Crystal data and the atomic coordinates. Formula sum Crystal system Space group Unit cell dimensions
ZrGe2 Ref. [3] Orthorhombic C m c m (No. 63) a = 3.7893 Å b = 14.9750 Å c = 3.7606 Å 213.39 Å3 4 oC12 NO2 c3
Cell volume Z Pearson code Formula type Wyckoff sequence Atomic coordinates Atom Ox.
Wyck.
x
y
z
Zr1 Ge1 Ge2
4c 4c 4c
0 0 0
0.10600 3/4 0.44100
1/4 1/4 1/4
+0 +0 +0
Table 2 Crystal data and the atomic coordinates. Formula sum Crystal system Space group Unit cell dimensions
ZrGa2 Ref. [4] Orthorhombic C m m m (No. 65) a = 12.8900 E b = 3.9940 E c = 4.1230 E 212.26 E3 4 oC12 NO2 hgca
Cell volume Z Pearson code Formula type Wyckoff sequence Atomic coordinates Atom Wyck.
x
y
z
Zr1 Ga1 Ga2 Ga3
0.35100 0.17600 1/2 0
0 0 0 0
0 1/2 1/2 0
4g 4h 2c 2a
relaxations and driving the system to self-consistency. From the obtained relaxed geometry the electronic structure and the chemical bonding can be determined and various spectroscopic features can be simulated and compared with existed experimental data. The all-electron full potential linearized augmented plane wave (FP-LAPW) method was used to solve the Kohn Sham DFT equa-
tions within the framework of the WIEN2K code [9]. This is an implementation of the DFT [10] with different possible approximation for the exchange correlation (XC) potentials. We have employed the local density approximation (LDA) by Ceperley–Alder (CA) [11], and the gradient approximation (GGA) [12], which are based on exchange–correlation energy optimization to calculate the total energy. In addition, Engel–Vosko generalized gradient approximation (EV-GGA) [13] also used to avoid the well-known LDA and GGA underestimation of the band splitting. Our calculations demonstrate the effect of the three different kinds of exchange–correlation potentials on the electronic structure and hence the band splitting. Thus we present the calculation using EVGGA. The unit cell was divided into two regions. The spherical harmonic expansion was applied inside the non-overlapping spheres of muffin-tin radius (Rmt) and the plane wave basis set was chosen in the interstitial region (IR) of the unit cell. The Rmt for Zr, Ga and Ge were chosen in such a way that the spheres did not overlap. In order to get the total energy convergence, the basis functions in the IR were expanded up to Rmt Kmax = 7.0 and inside the atomic spheres for the wave function. The maximum value of l were taken as lmax = 10, while the charge density is Fourier expanded up to Gmax = 20 (a.u)1. We have used 300 k-points in the irreducible Brillouin zone for structural optimization. For calculation of the electronic band structure dispersion in k-space along the high symmetry directions in the irreducible Brillouin zone, the total and the angular momentum decomposition of the atoms projected electronic density of states a denser meshes of 1000 kpoints were used. The convergence of the total energy in the selfconsistent calculations is taken with respect to the total charge of the system with a tolerance 0.0001 electron charges.
3. Results and discussion 3.1. Band structures and density of states The electronic band structure (BS) dispersion in k-space along the high symmetry directions in the irreducible Brillouin zone, the total and the angular momentum decomposition of the atoms projected electronic density of states (TDOS and PDOS) for ZrGa2 and ZrGe2 compounds are shown in Figs. 2 and 3. The BS, TDOS and PDOS confirm the metallic origin of the investigated compounds. We also emphasize that both compounds show continuous groups/structures along the entire energy region. The current calculations demonstrate the effect of using three different kinds
Fig. 2. Calculated electronic band structure of ZrGa2 and ZrGe2.
A.H. Reshak et al. / Computational Materials Science 78 (2013) 134–139
Fig. 3. Calculated total and partial density of states of ZrGa2 and ZrGe2.
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of XC potentials on the electronic structure features. Since the EVGGA approach is able to reproduce better exchange potential at the expense of less agreement in the exchange energy and yields better band splitting compared to LDA and GGA. This is mainly due to the fact that LDA and GGA are based on simple model assumptions which are not sufficiently flexible to reproduce the exchange correlation energy and its charge space derivative. Thus we demonstrated the results using EVGGA only. Following Figs. 1 and 2; Tables 1 and 2, one can see that both compounds possess different structures. Thus there is difference in the band dispersion between ZrGa2 to ZrGe2. Fig. 3a shows the calculated TDOS of ZrGa2 and ZrGe2 with respect to each other. Generally both graphs showed quite similar features, except that the first structure of ZrGe2 is very much shifted towards lower energies with almost double heights. Fig. 3b–g shows Zr-s/p/d, Ga-s/p/d and Ge-s/p/d partial density of states (PDOS). From the PDOS we can identify the angular momentum character for the various structures and one can see that moving from ZrGa2 to ZrGe2 has significant influence on the magnitudes and the spectral peak positions. The magnitudes of Ge-d is double than that of Ga-d and is attributed to the fact that the two Ge atoms are situated at 4c site. Whereas the three Ga atoms are situated at 4 h, 2c and 2a sites, respectively. Deeply looking into the angular momentum decomposition of the atoms projected electronic density of states for ZrGa2 and ZrGe2 compounds one can see the strong hybridization between Zr-s and Zr-p states between 16.0 eV and 12.0 eV. Based on comparing the results of the calculated total densities of states and the angular momentum projected densities of states we can elucidate the origin of chemical bonding following the same method which was used in our previous work [14]. We have
established that the TDOS, and PDOS ranging from 8.0 eV up to 0.0 eV Fermi energy (EF) are larger for Zr-d states (0.7 electrons/ eV), Ga/Ge-p states (0.5 electrons/eV), Ga/Ge-s states (0.2 electrons/eV), Zr-p states (0.08 electrons/eV). This means some electrons from Zr and Ga/Ge atoms are transferred into valence bands and contribute to covalence interactions between Zr and Ga/Ge atoms. The interaction of charges between Zr and Ga/Ge atoms caused by strong hybridization, and the covalent bond arises due to the degree of hybridization. Hence, there exists a strong covalent bonding between these atoms. The angular momentum decomposition of the atoms projected density of states help to analyze the nature of the bonds according to a classical chemical concept. This concept is very useful to classify compounds into different categories with respect to different chemical and physical properties.
3.2. Optical properties ZrGa2 and ZrGe2 compounds are crystallizing in orthorhombic space group, thus it has three nonzero components exx ðxÞ; eyy ðxÞ and ezz ðxÞ. The imaginary part of the three principal complex tensor components completely defines the linear optical susceptibilities. The imaginary part of the three principal complex tensor components originates from inter-band transitions between valence and conduction band states. According to the dipolar selection rule only transitions changing the angular momentum quantum number l by unity are allowed. The imaginary parts of the optical function’s dispersion were calculated using the following expression which is taken from Ref. [15].
Fig. 4. (a) Calculated e2av erage ðxÞ using EVGGA for ZrGa2 (dark solid curve-black color online); ZrGe2 (light dashed curve-red color online); (b) calculated e1av erage ðxÞ using yy EVGGA for ZrGa2 (dark solid curve-black color online); ZrGe2 (light dashed curve-red color online); (c) calculated exx 2 ðxÞ (dark solid curve-black color online), e2 ðxÞ (light xx dashed curve-red color online) and ezz 2 ðxÞ (light dotted dashed curve-blue color online) spectra for ZrGa2 and ZrGe2 using EVGGA; (d) calculated e1 ðxÞ (dark solid curvezz black color online), eyy 1 ðxÞ (light dashed curve-red color online) and e1 ðxÞ (light dotted dashed curve-blue color online) spectra for ZrGa2 and ZrGe2 using EVGGA. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
A.H. Reshak et al. / Computational Materials Science 78 (2013) 134–139 2
eij2 ðxÞ ¼
8p2 h e2 XX picv ðkÞpjv c ðkÞ ðf f Þ d½Ec ðkÞ Ev ðkÞ hx c v m2 V E2v c k cv
where m, e and ⁄ are the electron mass, charge and Planck’s constant, respectively. fc and fv represent the Fermi distributions of the conduction and valence bands, respectively. The term picv ðkÞ denotes the momentum matrix element transition from the energy level c of the conduction band to the level v of the valence band at certain k-point in the BZ and V is the unit cell volume. To demonstrate the influence of replacing Ga by Ge on exx 2 ðxÞ, av erage zz eyy ð x Þ and e ð x Þ the e ð x Þ calculated using the expression gi2 2 2 exx þeyy þezz
ven in Ref. [16], ea2v erage ðxÞ ¼ 2 32 2 . The ea2v erage ðxÞ of ZrGa2 and ZrGe2 compounds are illustrated at Fig. 4a. The sharp rise below 1.0 eV is due to intra-band transitions (Drude term). Above 1.0 eV both compounds show different spectral dispersion that is attributed to the fact that both compounds possess different structures. The real part ea1v erage ðxÞ of the corresponding principal complex tensor components are obtained from the imaginary part of these principal complex tensor components by means of Kramers–Kronig transformation [17] as shown in Fig. 4b. yy Fig. 4c and d show the three nonzero components exx 2 ðxÞ; e2 ðxÞ and ezz ð x Þ of ZrGa and ZrGe compounds. It is clear that there is a 2 2 2 considerable anisotropy between the three principal complex tensor components. At low energies below 2.0 eV both compounds show that exx 2 ðxÞ is the dominant one. For the other two compozz nents, ZrGa2 show them as eyy 2 ðxÞ and e2 ðxÞ; whereas ZrGe2 show yy the ezz ð x Þ is dominated on e ð x Þ; that is due to the different struc2 2 tures of ZrGa2 and ZrGe2 compounds. Generally from the presented data it is clear that different approximation for the exchange correlation potentials, namely: LDA, GGA and EVGGA cause insignificant effect on the band structure and the density of states by using these three types of exchange correlation potentials. Usually such anisotropy reflects the anisotropy of the chemical bonds [18–21] where the dispersion of the band structure defines the local charge density distribution. It is necessary to add that some contribution may be given by localized states [22] and imperfections [23]. The peculiarities in the band structure serves as a bridge to understand the origin of their crystallochemistry properties. 4. Conclusions Using the X-ray diffraction data (XRD) of ZrGa2 and ZrGe2 compounds as starting for the all-electron full potential linearized augmented plane wave calculations to solve the Kohn Sham DFT equations within the framework of the WIEN2K code. We have optimized the structure by minimization of the forces acting on the atoms. The structure is fully relaxed until the forces on the atoms reach values less than 1 mRy/a.u. We have employed different approximation for the exchange correlation potentials, namely: LDA, GGA and EVGGA. We found
139
that there is insignificant effect on the band structure and the density of states by using these three types of exchange correlation potentials. The calculations show that there is significant differences in the band dispersion when replace Ga by Ge, that is attributed to the fact that in the ZrGe2 compound Zr atom is situated at 4c site, and the two Ge atoms are situated at 4c site. Whereas for ZrGa2 compound Zr is located at 4g site and the three Ga atoms are situated at 4h, 2c and 2a sites, respectively. Deeply looking into the angular momentum decomposition of the atoms projected electronic density of states for ZrGa2 and ZrGe2 compounds one can see the strong hybridization between; Zr-s and Zr-p states between 16.0 eV and 12.0 eV. Moving from ZrGa2 to ZrGe2 has significant influence on the magnitudes and the peak positions of states. The optical properties of the two compounds were calculated and analyzed.
Acknowledgments This work was supported from the institutional research concept of the Project CENAKVA (No. CZ.1.05/2.1.00/01.0024). School of Material Engineering, Malaysia University of Perlis, Malaysia.
References [1] W. Blasé, G. Cordier, Z. Naturforsch. B: Anorg. Chem. Org. Chem. 44 (1989) 1479–1482. [2] A.-K. Larsson, L. Stenberg, S. Lidin, Acta Cryst. B50 (1994) 636–643. [3] J.F. Smith, D.M. Bailey, Acta Crystallogr. 10 (1957) 341–342. [4] M. Potzschke, K. Schubert, ZEMTAE, CRYSTMET Ver. 2.0, vol. 53, 1962, 474– 488. [5] A.H. Reshak, G. Lakshminarayana, J. Ebothe, A.O. Fedorchuk, M.F. Fedyna, H. Kamarudin, P. Mandracci, S. Auluck, J. Alloys Comp. 556 (2013) 259–265. [6] A.H. Reshak, I.V. Kityk, J. Ebothe, A.O. Fedorchuk, M.F. Fedyna, H. Kamarudin, S. Auluck, J. Alloys Comp. 546 (2013) 14–19. [7] S. Gao, Comput. Phys. Commun. 153 (2003) 190–198. [8] K. Schwarz, J. Solid State Chem. 176 (2003) 319–328. [9] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, An augmented plane wave + local orbitals program for calculating crystal properties, in: WIEN2K, Karlheinz Schwarz, Techn. Universitat, Wien, Austria, 2001, ISBN 3-95010311-2. [10] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) 864–871. [11] D.M. Ceperley, B.I. Ader, Phys. Rev. B 8 (1973) 4822–4832. [12] J.P. Perdew, S. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865–3868. [13] E. Engel, S.H. Vosko, Phys. Rev. B 47 (1993) 13164–13174. [14] A.H. Reshak, D. Stys, S. Auluck, I.V. Kityk, H. Kamarudin, Mater. Chem. Phys. 130 (2011) 458–465. [15] F. Bassani, G.P Parravicini, Electronic States and Optical Transitions In Solids, Pergamon Press Ltd., Oxford, 1975. [16] A. H. Reshak, Ph.D. thesis, Indian Institute of Technology-Rookee, India, 2005. [17] H. Tributsch, Z. Naturforsch. 32A (1977) 972. [18] M.I. Kolinko, I.V. Kityk, A.S. Krochuk, J. Phys. Chem. Solids 53 (1992) 1315– 1320. [19] I.V. Kityk, Phys. Solid State 33 (1991) 1026–1030. [20] I.V. Kityk, A. Kassiba, K.J. Plucinski, J. Berdowski, Phys. Lett. A 265 (2000) 403– 410. [21] I.V. Kityk, P. Smok, J. Berdowski, T. Łukasiewicz, A. Majchrowski, Phys. Lett. A 280 (2001) 70–76. [22] I.V. Kityk, Semicond. Sci. Technol. 18 (2003) 1001–1009. [23] I.V. Kityk, Mater. Sci. 27 (2003) 342–350.
Contents lists available at SciVerse ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Influence of different exchange correlation potentials on band structure and optical constant calculations of ZrGa2 and ZrGe2 single crystals A.H. Reshak a,b,⇑, A.O. Fedorchuk c,g,⇑, G. Lakshminarayana d, Z.A. Alahmed f, H. Kamarudin b, S. Auluck e a
Institute of Complex Systems, FFPW, CENAKVA-South Bohemia University CB, Nove Hrady 37333, Czech Republic Center of Excellence Geopolymer and Green Technology, School of Material Engineering, University Malaysia Perlis, 01007 Kangar, Perlis, Malaysia c Lviv National University of Veterinary Medicine and Biotechnologies, Department of Inorganic and Organic Chemistry, Lviv, Ukraine d Materials Science and Technology Division (MST-7), Los Alamos National Laboratory, Los Alamos, NM 87545, USA e National Physical Laboratory, Dr. K S Krishnan Marg, New Delhi 110 012, Saudi Arabia f Department of Physics and Astronomy, King Saud University, Riyadh 11451, Saudi Arabia g P. Sagajdachnyj Miltary University, Lviv, Ukraine b
a r t i c l e
i n f o
Article history: Received 7 March 2013 Received in revised form 27 April 2013 Accepted 29 April 2013 Available online 19 June 2013 Keywords: Inorganic materials Crystal growth Electronic band structure
a b s t r a c t The all-electron full potential linearized augmented plane wave method was used to solve the Kohn Sham DFT equations. We have employed different approximations for the exchange correlation potentials, namely: LDA, GGA and EVGGA, and insignificant effect on the band structure and the density of states were found. Calculations show that there is a significant difference in the band dispersion with replacement of Ga by Ge that is attributed to the fact that in the ZrGe2 compound Zr atom is situated at 4c site and two Ge atoms are situated at 4c site. Whereas for ZrGa2 compound Zr is located at 4g site and the three Ga atoms are situated at 4h, 2c and 2a sites, respectively. There exists strong hybridization between the states. Moving from ZrGa2 to ZrGe2 has significant influence on the magnitudes and the peak positions of states. The optical properties of the two compounds were studied and analyzed. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction It is well known systematization of structural types with respect to the coordination of the atoms of the smallest sizes [1], which gives to classify structural types by classes, groups and to establish an affinity between structural types within the same class as well as between structural types, belong to different classes. Usually these approaches are different. In this work we try to look on the behavior of the corresponding energy band gaps. To study such relations the better way is to perform the calculations for the isostructural compounds. Among the binary compounds particular interest present halides, germanides, antimonides, arsenides, etc. Partially covalence bonds prevailingly form more electronegative atoms and atoms with less electronegativity usually occupy empty structural positions or occupy voids in cluster layers or polyhedra formed by anions. Very often these atoms are characterized by shorter chemical bonds to anions. This phenomenon may be explained similarly to
⇑ Corresponding authors. Address: Institute of Complex Systems, FFPW, CENAKVA-South Bohemia University CB, Nove Hrady 37333, Czech Republic (A.H. Reshak). Tel.: +420 777 729 583; fax: +420 386 361 219. E-mail addresses: [email protected], [email protected] (A.O. Fedorchuk). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.04.056
polyhedra Frank–Casper [2], i.e. like atomic agglomerates, where cationic electrons are shifted towards connections with anions. As a consequence the compound possessing the lowest electronegativity for compounds of different classes at fixed structural type should be changed. As a consequence, in the Ref. [3] an attempt has been made to perform a classification of structural types with respect to the atoms with maximal sizes. Following the reasons presented above in the present work we explore the band structure performed within a framework of DFT method with different exchange correlation potentials: LDA, GGA and EVGGA. For this reason we have chosen compounds ZrGa2 and ZrGe2 [4] possessing the same structural fragments (Fig. 1). At the same time their electronic configuration is different from each other. The choice of the different exchange correlations approach should add to help the influence of this screening on the band structure behaviors. The quantum chemical simulations should show how sensitive they will be to such different potential screenings. To the best of our knowledge no comprehensive work neither experimental data on the optical properties or first principles calculations on the structural, electronic, and optical properties of ZrGa2 and ZrGe2 compounds have appeared in the literature. Therefore as a natural extension to previous work on ZrGa2 and ZrGa3 compounds [5,6] a detailed depiction of the structural, electronic, and optical properties of ZrGa2 and ZrGe2 using full
A.H. Reshak et al. / Computational Materials Science 78 (2013) 134–139
135
Fig. 1. (a) The crystallochemical transformation of the structure for crystals ZrGa2 and ZrGe2; (b) crystal structure of ZrGa2 and ZrGe2; and (c) asymmetry unit of ZrGa2 and ZrGe2.
potential method is timely and would bring us important insights in understanding the origin of the band structure and densities of states. Hence it is very important to use a full potential method. Present study is aimed towards such calculation by using the full potential linear augmented plane wave (FP-LAPW) method which has proven to be one of the most accurate methods [7,8] for the computation of the electronic structure of solids within a framework density functional theory (DFT). Our calculations will demonstrate the effect of using a full-potential on the structural, electronic and optical properties of the single crystals of ZrGa2 and ZrGe2. In the Section 2 are given structural properties and principal computational details. The Section 3 present principal results concerning the band structure calculations and calculated optical
dispersions of the principal optical functions. Comparison of the results obtained by LDA, GGA and EVGGA is presented.
2. Structural properties and computational details For the calculations are chosen ZrGa2 and ZrGe2 single crystals. The crystal structures are listed in Tables 1 and 2 and illustrated at Fig. 1. Using the X-ray diffraction data (XRD) we have optimized the structure by minimization of the forces acting on the atoms. The structure is fully relaxed until the forces on the atoms reach values less than 1 mRy/a.u. Once the forces are minimized in this construction one can then find the self-consistent electron space density distribution at these positions by turning off the
136
A.H. Reshak et al. / Computational Materials Science 78 (2013) 134–139
Table 1 Crystal data and the atomic coordinates. Formula sum Crystal system Space group Unit cell dimensions
ZrGe2 Ref. [3] Orthorhombic C m c m (No. 63) a = 3.7893 Å b = 14.9750 Å c = 3.7606 Å 213.39 Å3 4 oC12 NO2 c3
Cell volume Z Pearson code Formula type Wyckoff sequence Atomic coordinates Atom Ox.
Wyck.
x
y
z
Zr1 Ge1 Ge2
4c 4c 4c
0 0 0
0.10600 3/4 0.44100
1/4 1/4 1/4
+0 +0 +0
Table 2 Crystal data and the atomic coordinates. Formula sum Crystal system Space group Unit cell dimensions
ZrGa2 Ref. [4] Orthorhombic C m m m (No. 65) a = 12.8900 E b = 3.9940 E c = 4.1230 E 212.26 E3 4 oC12 NO2 hgca
Cell volume Z Pearson code Formula type Wyckoff sequence Atomic coordinates Atom Wyck.
x
y
z
Zr1 Ga1 Ga2 Ga3
0.35100 0.17600 1/2 0
0 0 0 0
0 1/2 1/2 0
4g 4h 2c 2a
relaxations and driving the system to self-consistency. From the obtained relaxed geometry the electronic structure and the chemical bonding can be determined and various spectroscopic features can be simulated and compared with existed experimental data. The all-electron full potential linearized augmented plane wave (FP-LAPW) method was used to solve the Kohn Sham DFT equa-
tions within the framework of the WIEN2K code [9]. This is an implementation of the DFT [10] with different possible approximation for the exchange correlation (XC) potentials. We have employed the local density approximation (LDA) by Ceperley–Alder (CA) [11], and the gradient approximation (GGA) [12], which are based on exchange–correlation energy optimization to calculate the total energy. In addition, Engel–Vosko generalized gradient approximation (EV-GGA) [13] also used to avoid the well-known LDA and GGA underestimation of the band splitting. Our calculations demonstrate the effect of the three different kinds of exchange–correlation potentials on the electronic structure and hence the band splitting. Thus we present the calculation using EVGGA. The unit cell was divided into two regions. The spherical harmonic expansion was applied inside the non-overlapping spheres of muffin-tin radius (Rmt) and the plane wave basis set was chosen in the interstitial region (IR) of the unit cell. The Rmt for Zr, Ga and Ge were chosen in such a way that the spheres did not overlap. In order to get the total energy convergence, the basis functions in the IR were expanded up to Rmt Kmax = 7.0 and inside the atomic spheres for the wave function. The maximum value of l were taken as lmax = 10, while the charge density is Fourier expanded up to Gmax = 20 (a.u)1. We have used 300 k-points in the irreducible Brillouin zone for structural optimization. For calculation of the electronic band structure dispersion in k-space along the high symmetry directions in the irreducible Brillouin zone, the total and the angular momentum decomposition of the atoms projected electronic density of states a denser meshes of 1000 kpoints were used. The convergence of the total energy in the selfconsistent calculations is taken with respect to the total charge of the system with a tolerance 0.0001 electron charges.
3. Results and discussion 3.1. Band structures and density of states The electronic band structure (BS) dispersion in k-space along the high symmetry directions in the irreducible Brillouin zone, the total and the angular momentum decomposition of the atoms projected electronic density of states (TDOS and PDOS) for ZrGa2 and ZrGe2 compounds are shown in Figs. 2 and 3. The BS, TDOS and PDOS confirm the metallic origin of the investigated compounds. We also emphasize that both compounds show continuous groups/structures along the entire energy region. The current calculations demonstrate the effect of using three different kinds
Fig. 2. Calculated electronic band structure of ZrGa2 and ZrGe2.
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Fig. 3. Calculated total and partial density of states of ZrGa2 and ZrGe2.
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of XC potentials on the electronic structure features. Since the EVGGA approach is able to reproduce better exchange potential at the expense of less agreement in the exchange energy and yields better band splitting compared to LDA and GGA. This is mainly due to the fact that LDA and GGA are based on simple model assumptions which are not sufficiently flexible to reproduce the exchange correlation energy and its charge space derivative. Thus we demonstrated the results using EVGGA only. Following Figs. 1 and 2; Tables 1 and 2, one can see that both compounds possess different structures. Thus there is difference in the band dispersion between ZrGa2 to ZrGe2. Fig. 3a shows the calculated TDOS of ZrGa2 and ZrGe2 with respect to each other. Generally both graphs showed quite similar features, except that the first structure of ZrGe2 is very much shifted towards lower energies with almost double heights. Fig. 3b–g shows Zr-s/p/d, Ga-s/p/d and Ge-s/p/d partial density of states (PDOS). From the PDOS we can identify the angular momentum character for the various structures and one can see that moving from ZrGa2 to ZrGe2 has significant influence on the magnitudes and the spectral peak positions. The magnitudes of Ge-d is double than that of Ga-d and is attributed to the fact that the two Ge atoms are situated at 4c site. Whereas the three Ga atoms are situated at 4 h, 2c and 2a sites, respectively. Deeply looking into the angular momentum decomposition of the atoms projected electronic density of states for ZrGa2 and ZrGe2 compounds one can see the strong hybridization between Zr-s and Zr-p states between 16.0 eV and 12.0 eV. Based on comparing the results of the calculated total densities of states and the angular momentum projected densities of states we can elucidate the origin of chemical bonding following the same method which was used in our previous work [14]. We have
established that the TDOS, and PDOS ranging from 8.0 eV up to 0.0 eV Fermi energy (EF) are larger for Zr-d states (0.7 electrons/ eV), Ga/Ge-p states (0.5 electrons/eV), Ga/Ge-s states (0.2 electrons/eV), Zr-p states (0.08 electrons/eV). This means some electrons from Zr and Ga/Ge atoms are transferred into valence bands and contribute to covalence interactions between Zr and Ga/Ge atoms. The interaction of charges between Zr and Ga/Ge atoms caused by strong hybridization, and the covalent bond arises due to the degree of hybridization. Hence, there exists a strong covalent bonding between these atoms. The angular momentum decomposition of the atoms projected density of states help to analyze the nature of the bonds according to a classical chemical concept. This concept is very useful to classify compounds into different categories with respect to different chemical and physical properties.
3.2. Optical properties ZrGa2 and ZrGe2 compounds are crystallizing in orthorhombic space group, thus it has three nonzero components exx ðxÞ; eyy ðxÞ and ezz ðxÞ. The imaginary part of the three principal complex tensor components completely defines the linear optical susceptibilities. The imaginary part of the three principal complex tensor components originates from inter-band transitions between valence and conduction band states. According to the dipolar selection rule only transitions changing the angular momentum quantum number l by unity are allowed. The imaginary parts of the optical function’s dispersion were calculated using the following expression which is taken from Ref. [15].
Fig. 4. (a) Calculated e2av erage ðxÞ using EVGGA for ZrGa2 (dark solid curve-black color online); ZrGe2 (light dashed curve-red color online); (b) calculated e1av erage ðxÞ using yy EVGGA for ZrGa2 (dark solid curve-black color online); ZrGe2 (light dashed curve-red color online); (c) calculated exx 2 ðxÞ (dark solid curve-black color online), e2 ðxÞ (light xx dashed curve-red color online) and ezz 2 ðxÞ (light dotted dashed curve-blue color online) spectra for ZrGa2 and ZrGe2 using EVGGA; (d) calculated e1 ðxÞ (dark solid curvezz black color online), eyy 1 ðxÞ (light dashed curve-red color online) and e1 ðxÞ (light dotted dashed curve-blue color online) spectra for ZrGa2 and ZrGe2 using EVGGA. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
A.H. Reshak et al. / Computational Materials Science 78 (2013) 134–139 2
eij2 ðxÞ ¼
8p2 h e2 XX picv ðkÞpjv c ðkÞ ðf f Þ d½Ec ðkÞ Ev ðkÞ hx c v m2 V E2v c k cv
where m, e and ⁄ are the electron mass, charge and Planck’s constant, respectively. fc and fv represent the Fermi distributions of the conduction and valence bands, respectively. The term picv ðkÞ denotes the momentum matrix element transition from the energy level c of the conduction band to the level v of the valence band at certain k-point in the BZ and V is the unit cell volume. To demonstrate the influence of replacing Ga by Ge on exx 2 ðxÞ, av erage zz eyy ð x Þ and e ð x Þ the e ð x Þ calculated using the expression gi2 2 2 exx þeyy þezz
ven in Ref. [16], ea2v erage ðxÞ ¼ 2 32 2 . The ea2v erage ðxÞ of ZrGa2 and ZrGe2 compounds are illustrated at Fig. 4a. The sharp rise below 1.0 eV is due to intra-band transitions (Drude term). Above 1.0 eV both compounds show different spectral dispersion that is attributed to the fact that both compounds possess different structures. The real part ea1v erage ðxÞ of the corresponding principal complex tensor components are obtained from the imaginary part of these principal complex tensor components by means of Kramers–Kronig transformation [17] as shown in Fig. 4b. yy Fig. 4c and d show the three nonzero components exx 2 ðxÞ; e2 ðxÞ and ezz ð x Þ of ZrGa and ZrGe compounds. It is clear that there is a 2 2 2 considerable anisotropy between the three principal complex tensor components. At low energies below 2.0 eV both compounds show that exx 2 ðxÞ is the dominant one. For the other two compozz nents, ZrGa2 show them as eyy 2 ðxÞ and e2 ðxÞ; whereas ZrGe2 show yy the ezz ð x Þ is dominated on e ð x Þ; that is due to the different struc2 2 tures of ZrGa2 and ZrGe2 compounds. Generally from the presented data it is clear that different approximation for the exchange correlation potentials, namely: LDA, GGA and EVGGA cause insignificant effect on the band structure and the density of states by using these three types of exchange correlation potentials. Usually such anisotropy reflects the anisotropy of the chemical bonds [18–21] where the dispersion of the band structure defines the local charge density distribution. It is necessary to add that some contribution may be given by localized states [22] and imperfections [23]. The peculiarities in the band structure serves as a bridge to understand the origin of their crystallochemistry properties. 4. Conclusions Using the X-ray diffraction data (XRD) of ZrGa2 and ZrGe2 compounds as starting for the all-electron full potential linearized augmented plane wave calculations to solve the Kohn Sham DFT equations within the framework of the WIEN2K code. We have optimized the structure by minimization of the forces acting on the atoms. The structure is fully relaxed until the forces on the atoms reach values less than 1 mRy/a.u. We have employed different approximation for the exchange correlation potentials, namely: LDA, GGA and EVGGA. We found
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that there is insignificant effect on the band structure and the density of states by using these three types of exchange correlation potentials. The calculations show that there is significant differences in the band dispersion when replace Ga by Ge, that is attributed to the fact that in the ZrGe2 compound Zr atom is situated at 4c site, and the two Ge atoms are situated at 4c site. Whereas for ZrGa2 compound Zr is located at 4g site and the three Ga atoms are situated at 4h, 2c and 2a sites, respectively. Deeply looking into the angular momentum decomposition of the atoms projected electronic density of states for ZrGa2 and ZrGe2 compounds one can see the strong hybridization between; Zr-s and Zr-p states between 16.0 eV and 12.0 eV. Moving from ZrGa2 to ZrGe2 has significant influence on the magnitudes and the peak positions of states. The optical properties of the two compounds were calculated and analyzed.
Acknowledgments This work was supported from the institutional research concept of the Project CENAKVA (No. CZ.1.05/2.1.00/01.0024). School of Material Engineering, Malaysia University of Perlis, Malaysia.
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