Electronic structure, elastic and optical properties of MnIn2S4

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The elastic constants and various optical properties of MnIn2S4 including the dielectric constant, absorption coefficient, electron energy loss function and reflectivity were calculated as a function of incident photon energy. Those results are discussed in this study and compared with available experimental results.

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JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2014, Vol. 59, No. 7, pp. 135-143 This paper is available online at http://stdb.hnue.edu.vn ELECTRONIC STRUCTURE, ELASTIC AND OPTICAL PROPERTIES OF MnIn2 S4 Nguyen Minh Thuy and Pham Van Hai Faculty of Physics, Hanoi National University of Education Abstract. The electronic, elastic, and optical properties of MnIn2 S4 were investigated using first-principle calculations based on density functional theory (DFT) with the plane wave basis set as implemented in the CASTEP code. Our study revealed that MnIn2 S4 has indirect allowed transitions for both DFT and DFT + U (U = 6 eV) with energy band gaps of 1.57 eV and 2.095 eV, respectively. The elastic constants and various optical properties of MnIn2 S4 including the dielectric constant, absorption coefficient, electron energy loss function and reflectivity were calculated as a function of incident photon energy. Those results are discussed in this study and compared with available experimental results. Keywords: Inorganic compounds, Ab initio calculations, electronic structure. 1. Introduction Recently, MnIn2 S4 which are ternary compounds of the AB2 X2 type have received much attention as materials which have potential for optoelectronic application and as magnetic semiconductors [1]. In the literature, physical properties of MnIn2 S4 have been reported [1, 2]. Recently, the optical absorption spectra of MnIn2 S2 single crystals have been measured and it was found that the fundamental absorption edge is formed by direct allowed transitions [3, 4]. However, Bodnar et al. showed that MnIn2 S4 has both direct and indirect transitions [5]. Therefore further calculations of MnIn2 S4 are needed to clarify the origin of its band gap structure. Density functional theory (DFT) has been the dominant method used when making electronic structure calculations in solid state physics. In this work we report on the band structure, optical and elastic properties of MnIn2 S4 using density functional theory. The calculated results can provide a good model for understanding and predicting other behaviors of this material. Received August 26, 2014. Accepted October 23, 2014. Contact Nguyen Minh Thuy, e-mail address: thuynm@hnue.edu.vn 135
Nguyen Minh Thuy and Pham Van Hai 2. Content 2.1. Calculation models and methods MnIn2 S4 is a spinel-type compound and crystallizes in the space group Fd3m with lattice parameters a = b = c = 10.722 A ˚ [4]. In this structure, the Mn atoms share the tetrahedral sites, while the In atoms share the octahedral sites, as shown in Figure 1. Figure 1. Crystal structure of cubic MnIn2 S4 First principle calculations were performed using the CASTEP module in Materials Studio 6.0 developed by Accelrys Software, Inc.. Electron-ion interactions were modeled using ultrasoft pseudopotentials. The wave functions of the valence electrons were expanded through a plane wave basis set and the cutoff energy was selected as 380 eV. The Monkhorst-Pack scheme k-points grid sampling was set at 8 × 8 × 8. The convergence threshold for self-consistent iterations was set at 2 × 10−6 eV/atom. In the optimization process, the energy change, maximum force, maximum stress and maximum displacement tolerances were set at 10−5 eV, 0.03 eV/A,˚ 0.05 GPa and 0.001 A,˚ respectively. 2.2. Results and discussion 2.2.1. Electronic structure We used density functional theory (DFT) to calculate the band structure and the density of states (DOS) of MnIn2 S4 . The generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) functional were used to describe the exchange-correlation effects. The core electrons were replaced by the ultrasoft core potentials. Electron configurations were 3p6 4s2 3d5 for Mn, 4d10 52 5p1 for In and 3s2 3p4 for S atoms. Both the lattice parameter and the atomic position are optimized. The optimized lattice constants calculated by GGA + PBE (10.854 A) ˚ show good 136
Electronic structure, elastic and optical properties of MnIn2 S4 agreement with experimental details 10.722 A ˚ [4] in which the difference value is about 3 - 5 percent. As is well known, the GGA structural results are somewhat overestimated in comparison with experimental values. Calculated band structures of MnIn2 S4 are shown in Figure 2a. Coordinates of the special points of the Brillouin zone area are as follow (in terms of unit vectors of the reciprocal lattice): W (0.5, 0.25, 0.75), L (0.5, 0.5, 0.5), G (0, 0, 0), X (0.5, 0, 0.5) and K (0.375, 0.375, 0.750). The calculated band gap Eg 1.57 eV by GGA is smaller than that derived by experiment data, 1.97 eV [4], due to the well-known underestimation of conduction band state energies in DFT calculations. One can seen that in MnIn2 S4 the top of the valence band and the bottom of the conduction band are simply realized at different points of the Brillouin zone. Determination of an appropriate effective Hubbard U parameter is necessary in DFT + U calculation to correctly interpret the intra-atomic electron correlation. Here, the effective on-site Coulomb interaction is U = 6.0 eV and the calculated band gap of spinel MnIn2 S4 is 2.095 eV (see Figure 2b). The compound has indirect band gap, which is in agreement with previous data [5]. Since the energy gap is indirect, the phonon contribution to the absorption processes should be important. Composition of the calculated energy bands can be resolved with the help of projected density of states (PDOS) and a total density of states (DOS) diagram. Figure 3 describes the total and projected density of states of MnIn2 S4 . Figure 2a. Calculated band structure of MnIn2 S4 with GGA In Figure 2a Fermi level is set as zero of energy and is shown by the dashed line. Coordinates of the special points in the Brillouin zone are in units of the reciprocal lattice unit vectors. 137
Nguyen Minh Thuy and Pham Van Hai Figure 2b. Calculated band structure of MnIn2 S4 with GGA + U, U = 6 eV From these diagrams one can seen that the conduction band is about 5 eV wide and is formed by the Mn 4s and 3d states, which are hybridized with the S 3p states and the In 4s and 3p states. The valence band is wider by about 7 eV and consists of two sub-bands that are clearly seen in the band structure as well; the upper one (between -5 and 0 eV) is a mixture of the S 3p states and Mn 3d states and the lower one is narrow (between -7 and -5 eV) due to the In 5s states. Another band between -10 and -15 eV consists of two sub-bands created by the In 4d states (between -15 and -13 eV) and the S 4s states (between -12 and -10 eV). Figure 3. Calculated total DOS (bottom) and partial density of states PDOS for In, Mn (middle) and S (top) 138
Electronic structure, elastic and optical properties of MnIn2 S4 2.2.2. Elastic properties and bulk modulus Elastic properties of single cubic crystal can be described using the independent elastic moduli C11 , C12 and C44 . For the cubic crystal, its mechanical stability requires Born’s stability criteria: [6, 7]. (C11 − C12 ) > 0, C11 > 0, C44 > 0, (C11 + 2C12 ) > 0 (2.1) These conditions also lead to a restriction on the magnitude of the bulk modulus B [7]: C12 < B < C11 (2.2) These conditions are satisfied by the calculated elastic constants at zero external pressure in Table 1. This ensures the elastic stability of the compound and the accuracy (2C44 + C12 ) of the calculated elastic modulus. The anisotropy factor A = = 1.45 shows C11 that MnIn2 S4 can be regarded as elastically anisotropic [8]. The value of the B/G ratio of MnIn2 S4 is 2.96 (where G is the isotropic shear modulus), which is larger than the critical value 2.75 in Ref. [9], separating the ductile and brittle materials, indicating that MnIn2 S4 behaves in a ductile manner. Young’s modulus and Poisson’s ratio are major elasticity related characteristic properties for a material and are calculated using the following relations [10]: 9GB Y = (2.3) (3B + G) [ ] 1 B − (2/3) G γ= (2.4) 2 B + (1/3) G Table 1. Elastic constants Cij , bulk modulus B, shear modulus G, Young’s modulus Y (all in GPa), Poisson’s ratio γ at zero pressure and anisotropy factor A C11 C12 C44 B G B/G γ A Y 95 67 35 77(1) 26 2.96 0.35 1.45 70(1) The numbers in parantheses are the estimated errors of the mean in the last decimal place, e.g., 77(1) = 77 ± 1, or 3.2(1) = 3.2 ± 0.1 It is known that the values of the Poisson ratio are minimal for covalent materials and increase for ionic systems. In our case, the calculated Poisson ratio is 0.35, which means a sizable ionic contribution in intra-bonding. Comparing the bulk modulus and its pressure derivate with the above calculations, we calculated the optimized geometry for different values of pressure in the range from 0 to 8 GPa, which corresponds to typical range of pressure experiments [10, 11]. Experimental studies have shown that MnIn2 S4 maintains a spinel-type crystal structure 139
Nguyen Minh Thuy and Pham Van Hai until a pressure of up to around 7 GPa. Figure 3 presents the dependence of the relative volume change V/V0 on pressure P for MnIn2 S4 . The calculated results shown by squares in Figure 3 were fitted to the Birch-Murnaghan equation of state (EOS): [( ) 73 ( ) 53 ] { [( ) 32 ]} 3B0 V0 V0 3 V0 P (V ) = − 1 + (B0′ − 4) −1 (2.5) 2 V V 4 V where B0 and B’0 are the bulk modulus and its pressure derivative, respectively. Figure 4. Dependence of V/V0 volume ratio on pressure The least-squares fits to Eq. (5) are shown in Figure 4 by solid lines. From these approximations, the values of B0 and B’0 are 66 ± 1 GPa and 4.4 ± 0.1 GPa, respectively. Table 2 shows the bulk moduli B0 values obtained using different methods. The plot value extracted from the bulk moduli B0 (fitted EOS) is smaller than those obtained as the results of the elastic constants calculations (Table 1) and experiments in Ref. [10], indicating that elastic constant calculations provide better results. Table 2. Summary of elastic parameters for MnIn2 S4 Calculations Fitted from Calculated Exp. Exp. Theor. Parameters Birch-Murnaghan from elastic [3] [3] [10] EOS constants Bulk modulus (GPa) B0 78(4) 73(2) 80(2) 66(1) 77(1) Bulk modulus pressure 3.2(1) 2.8(6) 3.9(3) 4.4(1) derivative B’0 140
Electronic structure, elastic and optical properties of MnIn2 S4 2.2.3. Optical properties The optical properties of MnIn2 S4 are determined by the frequency dependent dielectric function ε (ω) = ε1 (ω)+iε2 (ω) that describes the response of the system in the presence of electromagnetic radiation and governs the propagation behavior of radiation in a medium. The imaginary part of the dielectric constant ε1 (ω) can be calculated from the momentum matrix elements between the occupied and unoccupied electronic states within the selection rule, and its real part can be derived from the Kramer–Kronig relationship. All of the other optical constants, such as the refractive index n(ω), absorption coefficient α (ω), reflectivity R(ω) and electron energy-loss function L(ω), can be deduced from ε1 (ω) and ε2 (ω). Figure 5 shows the imaginary part ε2 (ω) and the real part ε1 (ω) of the dielectric function for MnIn2 S4 . Here we have calculated the dielectric constant within GGA and a scissors operator 0.9 eV is used to correct the theoretical and experimental band gap. Experimental dielectric functions measured for single crystals of MnIn2 S4 using variable angle spectroscopic ellipsometry [12] are taken for comparison. Very good agreement with experiment data is obtained for the dielectric functions in both components. The static dielectric constants at ω → 0 are ε1 = 6.21, which show consistent agreement with an experimental value of 6.24 [4], suggesting that the choice of parameters is reasonable. The regions in which the imaginary part ε2 (ω) is different from zero can be related to the absorption spectrum and originate predominantly from the transitions of O1 2p and O2 2p electrons into the Mn 5d and In 3d conduction band. Figure 5. Calculated dielectric function of MnIn2 S4 141
Nguyen Minh Thuy and Pham Van Hai The optical parameters of interest, namely, the complex refractive index, n, the normal incidence reflectivity and the absorption coefficient, have been computed using well known mathematical expressions (Figure 6). The values obtained are in good agreement with those estimated using optical absorption measurements performed on MnIn2 S4 single crystals [4, 11]. Electron energy-loss function (ELF) is an important optical parameter, indicating the energy-loss of a fast electron traversing the material. The prominent peak in the spectrum is identified as the energy of plasmon oscillation, signaling the collective excitations of the electronic charge density in the material. For MnIn2 S4 (Figure 7), this energy is found to be approximately 13 eV. Figure 6. Calculated optical properties of MnIn2 S4 Figure 7. Electron energy loss function for MnIn2 S4 142
Electronic structure, elastic and optical properties of MnIn2 S4 3. Conclusion In summary, DFT and DFT + U approaches are used to study the electronic structure and the optical and elastic propeties of MnIn2 S4 bulk crystal in the present paper. The band structure reveals that MnIn2 S4 has a K-G indirect band transition in the Brillouin zone. The top valance band consists mainly of a mixture of the S 3p and Mn 3d states whereas the bottom of the conduction band is formed by Mn 4s and Mn 3d states. An effective Hubbard parameter U = 6 eV was added to the Mn d-d interaction in order to correct the energy band gap using experimental values. The obtained values of lattice constant, elastic constants and optical parameters are in very good agreement with other studies. Therefore, this model can be useful to investigate different properties of AB2 X4 compounds. REFERENCES [1] N. N. Niftiev, 1994. Solid State Communications. 92 (9), pp. 781-783. [2] V. Sagredo, M. C. Moron, L. Betancourt and G. E. Delgado, 2007. Journal of Magnetism and Magnetic Materials 312 (2), pp. 294-297. [3] F. J. Manjon, A. Segura, M. Amboage, J. Pellicer-Porres, J. F. Sanchez-Royo, J. P. Itie, A. M. Flank, P. Lagarde, A. Polian, V. V. Ursaki and I. M. Tiginyanu, 2007. Physica Status Solidi B-Basic, Solid State Physics 244 (1), pp. 229-233. [4] M. Leon, S. Levcenko, I. Bodnar, R. Serna, J. M. Merino, M. Guc, E. J. Friedrich and E. Arushanov, 2012. Journal of Physics and Chemistry of Solids 73 (6), pp. 720-723. [5] I. V. Bodnar, V. Y. Rud and Y. V. Rud, 2009. Semiconductors 43 (11), pp. 1506-1509. [6] M. D. Segall, P. J. D. Lindan, M. J. Probert, C. J. Pickard, P. J. Hasnip, S. J. Clark and M. C. Payne, 2002. Journal of Physics-Condensed Matter 14 (11), 2717-2744. [7] J. J. Wang, F. Y. Meng, X. Q. Ma, M. X. Xu and L. Q. Chen, 2010. Journal of Applied Physics 108 (3), 034107-034106. [8] A. M. Hao, X. C. Yang, X. M. Wang, Y. Zhu, X. Liu and R. P. Liu, 2010. Journal of Applied Physics 108 (6). [9] G. Vaitheeswaran, V. Kanchana, R. S. Kumar, A. L. Cornelius, M. F. Nicol, A. Svane, A. Delin and B. Johansson, 2007. Physical Review B 76 (1), 014107. [10] D. Santamaría-Pérez, M. Amboage, F. J. Manjón, D. Errandonea, A. Mu˜noz, P. Rodríguez-Hernández, A. Mújica, S. Radescu, V. V. Ursaki and I. M. Tiginyanu, 2012. The Journal of Physical Chemistry C 116 (26), pp. 14078-14087. [11] J. Ruiz-Fuertes, D. Errandonea, F. J. Manjon, D. Martinez-Garcia, A. Segura, V. V. Ursaki and I. M. Tiginyanu, 2008. Journal of Applied Physics 103 (6), 063710-063715. 143