Advances in Condensed Matter Physics

Volume 2019, Article ID 3176148, 12 pages

https://doi.org/10.1155/2019/3176148

## Structural, Electronic, Lattice Dynamic, and Elastic Properties of SnTiO_{3} and PbTiO_{3} Using Density Functional Theory

Correspondence should be addressed to Shiferaw Kuma; moc.liamg@asidagfihs and Menberu Mengesha Woldemariam; te.ude.uj@ahsegnem.urebnem

Received 18 March 2019; Revised 5 August 2019; Accepted 27 August 2019; Published 29 September 2019

Academic Editor: Sergio E. Ulloa

Copyright © 2019 Shiferaw Kuma and Menberu Mengesha Woldemariam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The structural, electronic, and elastic properties of tetragonal phase of SnTiO_{3} and PbTiO_{3} are investigated using first principle calculations. The unknown exchange-correlation functional is approximated with generalized gradient approximation (GGA) as implemented in pseudopotential plane wave approach. The convergence test of total energy with respect to energy cutoff and *k*-point sampling is preformed to ensure the accuracy of the calculations. The structural properties such as equilibrium lattice constant, equilibrium unit cell volume, bulk modulus, and its derivative are in reasonable agreement with the previous experimental and theoretical works. From elastic constants, mechanical parameters such as anisotropy factor *A*, shear modulus *G*, bulk modulus *B*, Young’s modulus *E*, and Poison’s ratio *n* are determined by using Voigt–Reuss–Hill average approximation. In addition, Debye temperature and longitudinal and transversal sound velocities are predicted from elastic constants. The electronic band structure and density of states of both compounds are obtained and compared with the available experimental as well as theoretical data. Born effective charge (BEC), phonon dispersion curve, and density of states are computed from functional perturbation theory (DFPT). Lastly, the spontaneous polarization is determined from the modern theory of polarization, and they are in agreement with the previous findings.

#### 1. Introduction

ABO_{3} perovskites are important for a variety of high technology applications as a result of their diverse physical properties [1]. Ferroelectric perovskite oxides are important for many emerging industrial applications including high capacity memory cells, catalysis, optical wave guides, integrated optics applications, and substrates for high-Tc cuprate superconductor growth [2]. Lead titanate (PbTiO_{3}) is one of the interesting and more studied perovskites possessing a ferroelectric phase under ambient conditions [3]. Due to its high spontaneous polarization and wide temperature stability of ferroelectric phase, the compound has got a strong interest. At room temperature, PbTiO_{3} compound has a tetragonal phase (space group *P4mm*) with ferroelectric property, while for the temperature above 763 K, it shows cubic phase (space group *Pm3m*) with paraelectric characteristics [4]. For a long time, there have been some efforts towards determining the electronic and optical properties of the cubic and tetragonal state of PbTiO_{3} from either first principles calculations or by experiment [5, 6]. But, from a theoretical point of view, a proper description of its electronic properties is still an area of active research. Theoretical computations have had difficulty in predicting the correct band gap energy and other related electronic properties of PbTiO_{3} from first principle.

In spite of great physical importance, the most widely used ferroelectric ceramics based on the PbTiO_{3} and PbZrO_{3} solid solution are generically called PZT. The PZT is composed of about 60 percent of lead, which raises ecological concerns; thus, some countries have legislated to replace this material by lead-free ceramics [7] since lead is a toxic element that affects the human health and the environment. As a result, recent studies have extensively focused on identifying new and more environmentally friendly ferroelectric materials and other alternative compounds [8]. The total replacement of Pb-based materials in technological devices remains almost improbable because of the unsatisfactory performance of other materials.

However, modification efforts to reduce the consumption of toxic Pb^{2+}, such as by substitution or doping techniques, remain necessary. SnTiO_{3} is one of the promising Pb-free ferroelectric materials, which is theoretically having a high dielectric constant and ferroelectric polarization [9]. Recently, Sn^{2+} is widely used to design a novel piezoelectric of free Pb-based material using the first principle study. However, most of the theoretical reports with regard to the SnTiO_{3} materials are merely focused on their physical properties and high polarization effect in the ferroelectric phase [10]. According to our knowledge, the elastic properties, Poison’s ratio, anisotropic index, Debye temperature, born effective charge, phonon dispersion, and spontaneous polarization of tetragonal phase of SnTiO_{3} which is, expected to replace PbTiO_{3} is, not well studied and needs more investigation. In general, to predict a specific device application and improvements, a deeper and fundamental understanding of the properties of the ferroelectric material are necessary. Therefore, studying the structural, elastic, lattice dynamics, and electronic properties as well as understanding the overall characteristics of the system is utmost important.

#### 2. Computational Method

Electronic and structural computations are performed by using density functional theory as implemented in the Quantum ESPRESSO (QE) open-source package. Plane wave self-consistent calculation (PWscf) is a first principle energy code that uses norm-conserving pseudopotentials (PP) and ultrasoft pseudopotentials (US-PP). It is well known that the exchange-correlation functional is the challenging term to approximate in first principle calculations. Today, the hybrid exchange functional like B3LYP and B3PW allows obtaining a band gap which is in good agreement with the experimental value [11, 12]. However, GGA-PBE [13] is used in this study to approximate the exchange-correlation functional as we have no pseudopotential generator for hybrid functional in Quantum ESPRESSO package currently. The *k*-point sampling of the Brillion zone is constructed using Monk Horst Pack Mesh scheme [14] with 6 × 6 × 6 grids in primitive cells of compounds. The Born effective charges, optical dielectric constants, and phonon dispersion curve of the ferroelectric materials are calculated from density functional perturbation theory (DFPT) using local density approximation (LDA).

#### 3. Results and Discussion

##### 3.1. Total Energy with respect to Energy Cutoff and k-point Sampling

The convergence test for total minimum energy as a function of cutoff energy is performed with an increment of 10 Ry in the range of 20 to 130 Ry. While varying the energy cutoff, the other parameters in the input file are fixed. For good total minimum energy convergence, we have used the criteria that the change in energy () from the minimum energy at the reference point (130 Ry) to be approximately equal to Ry per atom. In our calculation, per cell ( Ry per atom) for PbTiO_{3} at 80 Ry energy cutoffs and convergence is achieved. Moreover, per cell ( Ry per atom) for SnTiO_{3} at 80 Ry energy cutoffs and the energy is converged. Similarly, the convergence test for total minimum energy versus *k*-point sampling with an increment of in the range from to is performed by fixing the other parameters constant. Based on the criteria, the convergence is achieved at point grid. The energy is monotonically decreasing with increasing cutoff energy due to a direct result of the variational principle. Moreover, it is possible to argue the energy is monotonically decreasing with respect to *k*-point grid size in this calculation. However, this does not necessarily happen all the time. The systematic trend cannot be predicted just by increasing the sampling point for the approximation to the integral Figure 1.