AB-INITIO STUDY OF BULK
MODULUS AND CHARGE DENSITY OF
CUBIC SrMO
3
PEROVSKITES (M = Ti,
Zr, Mo, Rh, Ru)
AVINASH DAGA
Research Scholar, Govt. Dungar College, Bikaner, Rajasthan India 334003
SMITA SHARMA
Lecturer. in Physics, Govt. Dungar College, Bikaner, Rajasthan India 334003
K. S. SHARMA
Professor of Physics, The IIS University, Mansarovar,
Jaipur, Rajasthan, India [email protected]
Abstract:
Bulk modulus & charge density of cubic SrMO3 perovskites (M = Ti, Zr, Mo, Rh & Ru) have been investigated systematically using the first principle density functional calculations. Local density approximation (LDA) method has been used to compute the two quantities for five perovskites. It is found that the calculated bulk modulus for all the transition metal oxides are in good agreement with the available experimental data and with other theoretical results previously reported in the literature. ABINIT computer code is used to carry out all the calculations. Charge density plots for all the five cubic SrMO3 perovskites have been drawn using MATLAB. The maximum and minimum values of charge density along with the corresponding reduced coordinates are reported for all the perovskites.
Keywords: Ab-initio method; perovskite; LDA approximation; Density Functional Theory; MATLAB; ABINIT.
1. Introduction
Perovskite-type oxide materials have received considerable interest due to their various functional properties, such as insulating, semiconducting, metallic, superconducting, and ferroelectric as well as a spectrum of important potential applications. Several members of this family have been carefully studied and some of them are widely used in electronic devices. SrMO3 perovskites have attracted much attention from material scientists because of the unusual combination of their magnetic, electronic and transport properties [1, 2]. Strontium Titanate, SrTiO3 (STO) is widely studied material because of its cubic crystal structure at room temperature and high dielectric constant. SrTiO3 is a generic representative of transition metal oxides [3]. It is a prototype of room temperature cubic perovskite. It becomes tetragonal and ferroelectric at temperatures T > 30 K.
material for various uses incorporating a number of perovskite-based ferroelectric, dielectric, and conductive films. Yet, numerous studies have shown that this is no longer the case for some promising TMO such as ruthenates and molybdates. These oxides are observed to exhibit unconventional superconductivity [8] and research is going on to understand the mechanism [9] of this new quantum order due to strong electron-electron interaction. In addition to this, non-Fermi liquid behavior has been observed in SrRuO3 [10]. Pseudo gap
formation [11], metal-insulator transitions [12, 13] and high-voltage applications are the other significant properties, which draw a considerable attention to 4d TMO.
The density functional theory (DFT) is an extremely successful approach for the description of ground state properties of metals, semiconductors, and insulators. The computer implementations of DFT have been found to be a highly productive tool for studying complex solid state systems. Among these, the plane wave pseudopotential technique which is based on the total energy of the system has become popular as the method of choice for computational solid state physics. The success of this technique lies in that it has allowed us to better understand materials and processes. It has also enabled us to make deepened interpretation of experimental findings. It has offered new possibilities to design novel materials and devices by giving precise quantitative physical predictions.
Numerical modeling has become an important tool to understand the behavior of materials and ab initio
calculations result in an ideal method to understand the quantum origin of the properties observed in the macroscopic world. Experimental methods can tell us that something is happening but ab initio calculations can
explain why?
In the present study we deal with the bulk modulus and charge density of SrMO3-type perovskites,
where M stands for Ti, Zr, Mo, Rh and Ru respectively. We have chosen perovskite crystals of simple, cubic Pm3m symmetry, in order to keep all computational conditions simple and identical.
In order to perform first-principles DFT calculations we used the ABINIT computer code. This code is open source ab initio electronic structure calculation software and has been under continuous development.
2. Calculation Method
The simplest and easiest way to implement density functional approximation is a local one, in which the function is a simple integral over a function of the density at each point in space.
In this approximation,
E
XC
r
is a sum of contributions from each point in space depending only upon the density at each point independent of other points and given byE
XC
r
XC
r
r
d
r
(1)
where XC is the exchange-correlation energy per electron. The above relation represents what is called
the local density approximation (LDA). In the present study we have used pseudopotential method based on density functional theory in the local density approximation (LDA).
The ABINIT code, based on the DFT using plane waves and pseudopotentials, has been utilized for the computation of bulk modulus. Plane waves are used as a basis set for the electronic wave functions. In order to solve the Kohn-Sham equations, conjugate gradients minimization method is employed as implemented by the ABINIT code. Many types of pseudopotentials can be utilized by ABINIT; we used Hartwigsen-Goedecker-Hutter pseudopotential for the present calculations.
The considered SrMO3 perovskites are assumed to have ideal cubic structure (e.g., Pm3m) where
atomic positions in the elementary cell are M: 1a (0, 0, 0); O: 3d (0, 0, 1/2); and Sr: 1b (1/2, 1/2, 1/2). The electronic configurations are taken Ar3d24s2 for Ti, Kr4d25s2 for Zr, Kr4d54s1 for Mo, Kr4d75s2 for Rh, Kr4d65s2
for Ru, Kr5s2 for Sr, and He2s22p4 for O. Here, the noble gas cores are distinguished from the sub-shells of
valence electrons.
Fig. 1. Cubic perovskite single unit cell
The equilibrium lattice parameter of a crystal is that value of the lattice parameter which minimizes the total energy and from the curvature of the Φ (ρ) curve near the minimum we obtain the bulk modulus of the crystal. The bulk modulus B is defined by B= -V ∂P/ ∂V, P= -∂Φ / ∂V and B= [V ∂2Φ / ∂V2]V=Vo.
Then Φ(V) curve is fitted by a quadratic function, i.e., Φ = c0 + c1V + c2V2. This can be done by the
MATLAB command polyfit and the result is c2 in units of eV/Å6. Then, with minimum volume V0, the bulk
modulus is given by B = 2V0C2 (160.2) GPa.
The computation of charge densities of the above mentioned transition metal perovskites has been done by choosing the appropriate variables and the analysis was done by using the post-processor, called “Cut 3D”. It analyzes the charge contained in an atomic sphere and performs the Hirshfeld computation of atomic charges. In order to visualize the charge density contours MATLAB is used.
3. Results & Discussions
Binding energy per atom versus atomic volume curves for all the SrMO3 perovskites
considered in this work are shown in figures 2-6. These curves have been used to calculate the bulk modulus for SrMO3 perovskites. A quadratic polynomial fitting of the curves have been obtained and
have been shown below in the equations 2-6.
Y = 0.00959x2 - 1.13769x - 2,829.99836 [SrTiO
3] (2)
Y = 0.00708x2 - 0.95735x - 2,568.78825 [SrZrO
3] (3)
Y = 0.00717x2 - 0.88849x - 1,512.47223 [SrMoO
3] (4)
Y = 0.00576x2 - 0.68105x- 1,897.57491 [SrRhO
3] (5)
Y = 0.00660x2 - 0.80765x- 1,746.22288 [SrRuO
3] (6)
From this polynomial, ‘c2’ which is the coefficient of x2 in the quadratic equation is obtained
separately for each perovskite. By using the atomic volume for minimum value of binding energy and the calculated value of c2, we obtained the bulk modulus for each of the perovskites. The computed
values of bulk modulus along with the available experimental data and other theoretical results are assembled in table 1.
The calculated bulk modulus for SrTiO3 is within 0.5% and for SrZrO3 is within 2.37% of the
respective experimental results. Thus the calculated values are in good agreement with the experimental results. In all cases comparison of the present results with the theoretical work of Mete et al. [14] suggests that the pseudopotentials used in the present work are reliable and perform better.
Table 1. Calculated & Experimental Values of Bulk Modulus (GPa) for SrMO3 (M=Ti, Zr, Mo, Rh, Ru)in units of GPa.
SrMO3 Calculated Experimental Others
SrTiO3 183.949 183 [15] 191 [14]
SrZrO3 153.562 150 [16] 171 [14]
SrMoO3 146.410 - 145 [14]
SrRhO3 112.166 - 111 [14]
SrRuO3 129.393 - 127 [14]
A at (0, 0, 0),
B at (1/2, 1/2, 1/2) a O at (1/2, 0, 0) a,
Fig. 2. B. E. versus Atomic Vol. for SrTiO3 Fig. 3. B. E. versus Atomic Vol. for SrZrO3 Fig. 4. B. E. versus Atomic Vol. for SrMoO3
Fig. 5. B. E. versus Atomic Vol. for SrRhO3 Fig. 6. B. E. versus Atomic Vol. for SrRuO3
The 3-D and 2-D charge density contour plots for the above mentioned perovskites have been shown in figures 7-11, in which (a) represent 2-D contours and (b) represent 3-D charge density plots. It is observed that the charge density distribution in all the perovskites considered in this work have almost similar 3-D form. However, it is observed that SrMO3-type metallic perovskites, where M stands for Mo, Rh and Ru, show charge
density regions near the corners also, which are absent in SrMO3-type insulating perovskites, where M stands
for Ti and Zr.
Fig. 7a. Charge Density 2-D Contour plot for SrTiO3 Fig. 7b. Charge Density 3-D plot for SrTiO3 -2863.740
-2863.730
59.8 60.2 60.6 61.0 61.4
B
indi
Atomic Volume V (Angstrom3)
-2,601.166740 -2,601.166739
67.69567.69667.69767.69867.699
B
indi
Atomic Volume V (Angstrom3)
-1917.70 -1917.50
59.7060.7061.7062.7063.7064.70
Bind
ing
Energ
y
per A
tom
(eV
)
Atomic Volume V …
SrRhO3
-1,770.92585 -1,770.92580
61.18561.19061.19561.20061.205
Bind
ing
Energ
y
per A
tom
(eV
)
Atomic Volume V (Angst
SrRuO3
5 10 15 20 25 30 35 40 45
5 10 15 20 25 30 35 40 45
10 20
30 40
Fig. 8a. Charge Density 2-D Contour plot for SrZrO3 Fig. 8b. Charge Density 3-D plot for SrZrO3
Fig. 9a. Charge Density 2-D Contour plot for SrMoO3 Fig. 9b. Charge Density 3-D plot for SrMoO3
Fig. 10a. Charge Density 2-D Contour plot for SrRhO3 Fig. 10b. Charge Density 3-D plot for SrRhO3
5 10 15 20 25 30 35 40 45 50
5 10 15 20 25 30 35 40 45 50
10 20
30 40
50
10 20 30 40 50 10 20 30 40 50
5 10 15 20 25 30 35 40 45 50 55 60 5
10 15 20 25 30 35 40 45 50 55 60
10 20
30 40
50 60
10 20 30 40 50 60 10 20 30 40 50 60
5 10 15 20 25 30 35 40 45 50 55 60
5 10 15 20 25 30 35 40 45 50 55 60
10 20
30 40
50 60
Fig. 11a. Charge Density 2-D Contour plot for SrRuO3 Fig. 11b. Charge Density 3-D plot for SrRuO3
The maximum and minimum values of charge density and their occurrence however differ considerably in all the perovskites. The maximum and minimum values of charge density for different perovskites along with their reduced coordinates have been assembled in table 2. It is observed that the first maximum value obtained is highest in SrTiO3, while it is lowest in SrZrO3. It is also seen from table 2 that the
minimum value of charge density is lowest in SrRhO3 and highest in SrRuO3. It is also observed that the value
of second maximum of charge density is same as that of the first maximum in all the perovskites, whereas the value of second minimum differs from the value of first minimum in all the perovskites, except for SrTiO3.
Table 2. Maximum & Minimum Values of Charge Density Along With Their Reduced Coordinates for SrMO3 (M=Ti, Zr, Mo, Rh, Ru)
4. Conclusion
The bulk modulus of five transition metal oxides with cubic perovskite structure, viz., SrZrO3, SrTiO3,
SrMoO3, SrRhO3, and SrRuO3, has been studied by employing an ab-initio pseudopotential method using
ABINIT code. The Bulk modulus values of all the perovskites are found to compare well with the available experimental data and other theoretical results, showing the relevance of the present approach. Charge density 2D contour plots and 3D plots for the transition metal perovskites under consideration are found to be nearly similar; the insulating perovskites showing slightly different distribution as compared to the metallic perovskites. The maximum and minimum values of charge density are found to be different in different
10 20 30 40 50 60 70
10 20 30 40 20 40 60 20 40 60 10 20 30 40
SrMO3 Max. I
[el/Bohr^3] Red. Coord. Max. II [el/Bohr^3] Red. Coord. Min. I [el/Bohr^3] Red. Coord. Min. II [el/Bohr^3] Red. Coord.
SrTiO3 1.2557 0.0000
0.0000 0.9111
1.2557 0.0000
0.0000 0.0889
8.2582E-06 0.4889 0.4889 0.4889
8.2582E-06 0.5111 0.4889 0.4889
SrZrO3 1.0010 0.0000
0.5600 0.9800
1.0010 0.0000
0.4400 0.9800
3.0945E-06 0.5000 0.5000 0.5000
6.2742E-06 0.5000 0.5000 0.4800
SrMoO3 1.0219 0.5000
0.9500 0.9667
1.0219 0.9500
0.5000 0.9667
8.0708E-07 0.5000 0.5000 0.5000
1.7630E-06 0.0000 0.0000 0.0000
SrRhO3 1.0941 0.5000
0.9500 0.9667
1.0941 0.9500
0.5000 0.9667
9.7724E-07 0.0000 0.0000 0.0000
2.3743E-06 0.5000 0.5000 0.5000
SrRuO3 1.0960 0.5000
0.9444 0.9861
1.0960 0.9444
0.5000 0.9861
1.1043E-06 0.5000 0.5000 0.5000
Acknowledgments
The authors (Avinash Daga and Smita Sharma) are grateful to the Principal, Dungar College Bikaner for providing necessary facilities for this work. They are also grateful to Paridhi Jain, IIIT New Delhi for rendering help in MATLAB programming. K.S. Sharma is grateful to the Vice Chancellor, The IIS University for the similar facilities provided for this work.
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