Mathematical and Physical Sci., 2014, Vol. 59, No. 7, pp. 135-143 This paper is available onUne at http://stdb.lmue.edu.vn
ELECTRONIC STRUCTURE, ELASTIC AND OPTICAL PROPERTIES OF MnIn2S4
N g u y e n M i n h T h u y a n d P h a m V a n H a i Faculty of Physics. Hanoi Natioruil University of Education
Abstract. The electtonic, elastic, and optical properties of MnIn2S4 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 MnIn2S4 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 MnIn2S4 including the dielecttic constant, absorption coefficient, electton 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, electtonic structure.
1. Introduction
Recentiy, MnIn2S4 which are ternary compounds of the AB2X2 type have received much attention as materials which have potential for optoelecttonic application and as magnetic semiconductors [1]. In the Uterature, physical properties of MnIn2S4 have been reported [1, 2]. Recently, the optical absorption spectta of Mnln2S2 single crystals have been measured and it was found that the fundamental absorption edge is formed by direct allowed ttansitions [3,4]. However, Bodnar er a/, showed that MnIn2S4 has both direct and indirect ttansitions [5], Therefore further calculations of MnIn2S4 are needed to clarify the origin of its band gap sttucture.
Density functional theory (DFT) has been the dominant method used when making electtonic sttucture calculations in solid state physics. In this work we report on the band sttucture, optical and elastic properties of MnIn2S4 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. [email protected]. vn
135
Nguyen Minh Thuy and Pham Van Hai 2. Content
2.1. Calculation models and methods
MnIn2S4 is a spinel-type compound and crystalhzes in the space group Fd3m with lattice parameters a = b = c = 10.722 A [4]. In this stiructure, the Mn atoms share the tettahedral sites, while the In atoms share the octahedral sites, as shown in Figine 1.
Figure 1. Crystal structure of cubic MnIn2S^
First principle calculations were performed using the CASTEP module in Materials Studio 6.0 developed by Accelrys Software, Inc.. Electton-ion interactions were modeled using ulttasoft pseudopotentials. The wave functions of the valence electtons 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 x 8 x 8. The convergence threshold for self-consistent iterations was set at 2 x 10~^ eV/atom. In the optimization process, the energy change, maxunum force, maximum sttess and maximum displacement tolerances were set at 10"^ 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 sttucture and die density of states (DOS) of MnIn2S4. The generalized gradient approximation (GGA) witii the Perdew-Burke-Ernzerhof (PBE) functional were used to describe the exchange-correlation effects. The core electtons were replaced by the ulttasoft core potentials. Electton configurations were 3p^4s23d^ for Mn, 4d^°525p^ for In and 3s^3p^
for S atoms. Bodi the lattice parameter and the atomic position are optimized.
The optimized lattice constants calculated by GGA + PBE (10.854 A) show good 136
agreement with experimental details 10.722 A [4] in which tiie 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 stiuctures of MnIn2S4 are shown in Figure 2a. Coordinates of the special points of die 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 MnIn2S4 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 intta-atomic electton correlation. Here, the effective on-site Coulomb interaction is U = 6.0 eV and the calculated band gap of spinel Mnln2S4 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 conttibution 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 MiiIn2S4.
a) GGA
Figure 2a. Calculated band structure ofMnln2S^ with GGA In Figure 2a Fermi level is set as zero of energy and is shown by the dashed hne.
Coordinates of die special points in the Brillouin zone are in units of the reciprocal lattice unit vectors.
137
Nguyen Minh Thuy and Pham Van Hai b) GGA+U
4.S 4.0 3.5 3.0 2.5 2,0 1.5 1.0 o.s 0.0 -0.5 1.0 -1.5 -2.0 -2.5
H > ^ -.^^SS-^^^*™
-
I S - .095 eV
>
iS
Figure 2b. Calculated band structure ofMnIn2Si 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 sttucture 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).
4 5 - 3 0 - 1 5 - 4 5 - 3 0 - 1 5 - 4 5 - 3 0 - 1 5 - 4 5 - 3 0 - 1 5 -
n-
s - states
• ' — p - states
• - N . , - - ^ — - - ^ ' s-States'
11 s - states / I p - states
/ \ -'-"":
l\ •
/ L ^ /..-^^^^
/ w ^ ^ / ^ - ^ ' >
E S
' /\ "".
'
In Total
^—/ •" ':i^ ^J
-^
Energy (eV)
Figure 3. Calculated total DOS (bottom) and partial density of states PDOS for In, Mn (middle) and S (top)
2.2.2. Elastic properties and bulk modulus
Elastic properties of single cubic crystal can be described using the independent elastic moduli Cn, C12 and C44. For the cubic crystal, its mechanical stability requires Born's stability criteria: [6, 7].
(Cn - C12) > 0, On > 0, C44 > 0, (Cu + 2C12) > 0 (2.1)
B[7]:
These conditions also lead to a resttiction on the magnitude of the bulk modulus
Cn<B< Cu (2.2)
These conditions are satisfied by the calculated elastic constants at zero external pressure in Table 1. This ensures the elastic stabihty of the compound and the accuracy
(1C -t- C \
of the calculated elastic modulus. The anisottopy factor A = — = 1.45 shows Cn
that MnIn2S4 can be regarded as elastically anisottopic [8]. The value of the B/G ratio of MnIn2S4 is 2.96 (where G is the isottopic shear modulus), which is larger than the critical value 2.75 in Ref. [9], separating the ductile and brittle materials, indicating that MnIn2S4 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
^ (3J5 + G) (2.3)
1 r S - ( 2 / 3 ) G
2 [B-H(1/3)G (2.4)
Tcd}le 1. Elastic constants C^, bulk modulus B, shear modulus G, Young's modulus Y (all in GPa), Poisson's ratio 7 at zero pressure and anisotropy factor A C n
95 C,2
67 C44
35 B 77(1)
G 26
B/G 2.96
7 0.35
A 1.45
Y 70(1) The numbers in paranlheses are the estimated errors of the mean in the last decimal place.
e.g., 77(1) = 77 ^h or 3.2(1) = 3.2 ± 0.1 It is known that the values of tiie 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 conttibution in intta-bonding.
Comparing the bulk modulus and its pressitte derivate witii the above calculations, we calculated die optimized geometty 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 MnIn2S4 maintains a spinel-type crystal sttucture 139
Nguyen Minh Thuy and Pham Van Hai
itil a pressure of up to around 7 GPa. Figiue 3 presents the dependence of the relative .'olume change V/VQ on pressure P for MnIn2S4. The calculated results shown by squares m Figiue 3 were fitted to the Bkch-Murnaghan equation of state (EOS):
PiV): 3Bo V,
m'
1 + J (Bi - 4) (2.5)Clakulalcil l-illcd from K)S
where Bo and B'o are the bulk modulus and its pressure derivative, respectively.
8 • 'S 6 • a- ST 4
0
0.90 0.92 0.94 0.96 0.9S 1.00
W o
Figure 4. Dependence of V/VQ volume ratio on pressure The least-squares fits to Eq. (5) are shown in Figure 4 by solid lines. From these approximations, die values of BQ andB'o are 66 ± 1 GPa and 4.4 ±0.1 GPa, respectively Table 2 shows the bulk moduli BQ values obtained using different methods. The plot value exttacted from the bulk moduli BQ (fitted EOS) is smaller than those obtained as die 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 MnIn2S4
Parameters
Bulk modulus (GPa) Bo Bulk modulus pressure derivative B'o
Exp.
[3]
78(4) 3.2(1)
Exp.
[3]
73(2) 2.8(6)
Calculations Theor.
[10]
80(2) 3.9(3)
Fitted from Birch-Murnaghai
EOS 66(1) 4.4(1)
Calculated 1 from elastic
constants 77(1)
140
2.2.3, Optical properties
The optical properties of MnIn2S4 are detemuned by the frequency dependent dielecttic function e (w) = si (w) +ie2 (w) that describes die response of the system in the presence of electtomagnetic radiation and governs the propagation behavior of radiation in a medium. The imaginary part of the dielecttic constant EI (w) can be calculated from the momentum mattix elements between the occupied and unoccupied electtonic 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 tiie refractive index n(cj), absorption coefficient a (ijj), reflectivity R(u;) and electton energy-loss function L(a;), can be deduced from £i (w) and €2 (w).
Figure 5 shows the imaginary part £2 (.^) and the real part ei (w) of the dielecttic function for MnIn2S4. Here we have calculated the dielecttic constant within GGA and a scissors operator 0.9 eV is used to correct the theoretical and experimental band gap.
Experimental dielecttic functions measured for single crystals of MnIn2S4 using variable angle spectioscopic ellipsometty [12] are taken for comparison. Very good agreement with experiment data is obtained for the dielecttic functions in both components. The static dielecttic constants at a; —> 0 are £1 = 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 £2 (w) is different from zero can be related to the absorption specttum and originate predominantiy from the ttansitions of 01 2p and 02 2p electtons into the Mn 5d and In 3d conduction band.
b 10
8 6 4 2 0- -2 -4-
/ \ A ^^^^1
/ \ \ -~^2
/ \ \ ^
0 2 4 6 8 10 12 14 16 18 20 22 24 Energy (eV)
Figure 5. Calculated dielectric function ofMnln2Si
Nguyen Minh Thuy and Pham Van Hai
The optical parameters of interest, namely, die 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 MnIn2S4 single crystals [4, 11].
Electton energy-loss function (ELF) is an important optical parameter, indicating the energy-loss of a fast electton ttaversing the material. The prominent peak in the specttum is identified as the energy of plasmon oscillation, signaling the collective excitations of the electtonic charge density in the material. For MnIn2S4 (Figure 7), this energy is found to be approximately 13 eV.
E"«'^J'<«V) Energy (eV)
Figure 6. Calculated optical properties ofMnIn2Si
8 10 12 14 16 18 20 22 Energy (eV)
Figure 7. Electron energy loss function for MnIn2Si
142
3. Conclusion
In summary, DFT and DFT + U approaches are used to study the electronic structure and the optical and elastic propeties of MnIn2S4 bulk crystal in the present paper. The band structure reveals that MnIn2S4 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 AB2X4 compounds.
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