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Electronic and optical properties of SrTiO3 perovskite using semi-local and hybrid first principles density functional theory

The lattice constants of Strontium titanate perovskite has been studied using the Density functional theory formalism at the PBE and the HSE levels of theory. The study has been done in all the 3 existing phases; namely cubic, orthorhombic and tetragonal, however, experimental details are available only for the cubic phase so far.


Introduction
Complex correlated oxides in general have several interesting properties such as high Tc superconductivity, multiferroicity, colossal magnetoresistance and ferroelectricity. Most interesting complex oxides have a perovskite structure [1][2][3][4]; hence SrTiO3 is considered to be an ideal starting point for their study. Strontium titanate (SrTiO3) is an important perovskite oxide that has been studied extensively over the past years due to it's technological importance. [5][6][7] Several modern concepts of condensed matter and physics of phase transition have been studied while investigating this unique material. Among it's wide range of applications, includes optical switches, grain boundary barrier layer capacitors, oxygen gas sensors, wide band gap semiconductivity, etc. [8][9][10][11][12][13][14][15][16] Over the past years, the structural, electronic, optical, and elastic properties of SrTiO3 have been investigated thoroughly both experimentally [17][18][19][20][21][22][23] and theoretically [24][25][26][27]. The theoretical description of this material is however still an active area of research. Conventionally, theoretical computation of solid materials face difficulty in predicting their correct band gap and the associated electronic properties from first principles. Density functional theory (DFT) additionally with the Coulomb interaction formalism (DFT+U) [28][29][30][31] has been successful in obtaining the correct energy band gaps, however only for correlated and localised electrons in transition and rare earth metal oxides. The underestimation of the energy band gaps and the associated electronic properties is typically owed to the inadequacies of the current density functional potentials for the description of ground state properties of semiconductors. In some cases, methods beyond DFT also have proven to be unsuccessful towards predicting the electronic properties of semiconducting materials [32][33][34]. In general, the perovskite materials exist in 3 phases dependent on the temperature. At low temperatures <100K, these exist in the orthorhombic phase, at temperatures between 100K and 300K, these exist in the tetragonal phase, and at high temperature above 300K, cubic phase of the perovskites are found to be the most stable. [35][36][37] The major motivation of this work is to study the crystallography predict the energy band gap of SrTiO3 in all these 3 phases using first principles DFT calculations. We are able to predict band gaps close to the experimental band gap by using the HSE functional that mixes certain amount of Hartree-Fock exchange to the standard DFT functional that is known to improve the band gaps of solids.

Computational Details
All calculations are done using plane-wave Density Functional Theory (DFT) simulations as implemented within the Vienna Ab-initio Software Package (VASP). The initial geometries of the perovskites in all the 3 phases (cubic, orthorhombic and tetragonal) are taken from Materials Project website and are given in Table 1. The geometries of the perovskites are optimised using the Perdew-Burke-Ernzerhof (PBE) functional using a cut off energy of 520 eV. Plane wave basis sets with ultrasoft pseudopotentials as implemented within the VASP package have been employed for Sr, Ti, and O. The sampling of the Brillouin zone has been done using a k-point mesh of 5x5x5. The energy band gaps are calculated using both the PBE and the hybrid HSE functionals. The parameters have been tested computationally for convergence.

Results and Discussions
The geometries of the SrTiO3 perovskites are optimised using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm as implemented within the VASP package. The optimised crystals of SrTiO3 perovskite is seen to be undistorted in all the 3 phases. For the cubic and the orthorhombic crystal phases, however, the lattice parameters are seen to decrease after optimisation, thus reducing the cell volume. For the tetragonal crystal structure, the lattice parameters increase by little amount post optimisation, thus increasing the cell volume by a small percentage. The initial and the final lattice parameters, lattice volume and the percentage differences are shown in table 1. The energy band gaps of the perovskite have been predicted by using both the PBE as well as the HSE functional in all the 3 crystal phases. PBE functional tends to underestimate the energy band gap of periodic solids. The band gap calculation of solids using the PBE functional comes with a phenomenon that is called self-interaction error, which arises in the occupied states in DFT and in the unoccupied states in Hartree-Fock calculations. These spurious self interaction error effects in the occupied states in semi local DFT delocalises these occupied states even further, thus shooting up the energy and reducing the band gap. Hartree-Fock theory has the same problem, however, in the unoccupied states; these states are over delocalised and hence pushed up, thus increasing the band gap. Empirically, it's found that mixing a semi-local DFT functional with a small amount of exact Hartree-Fock exchange equalises the self interaction error of the top occupied band and the lowest unoccupied band, thus making the band gap close to experimental band gap. The PBE and the HSE band gaps of the perovskite in all 3 phases are shown in

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The 0K phase stability of the perovskite can be determined approximately from the DFT total energies as shown in figure 2 . Conventionally, the room temperature phase stability of the perovskites can be estimated using the empirical Goldschmidt tolerance factor t. Goldschmidt tolerance factor is an empirical descriptor-based method used for predicting the stability of the perovskites. This method is based on the chemical composition of the perovskite ABX3 and the ionic radii of the A, B, and t X ions. t is calculated using the formula:

Conclusions
The crystal structure, electronic properties and crystallographic phase stabilities of SrTiO3 perovskite has been studied using first principles DFT calculations. From our findings, we conclude that this perovskite material can be used as a potential wide band gap semiconductor in all 3 possible crystallographic phases. Experimental data is available so far for the cubic crystal phase only. However, from our study, the other two phases are also predicted to be quite promising for applications as wide band gap semiconductors.