Advances in Materials Science and Engineering

Volume 2017 (2017), Article ID 3981317, 13 pages

https://doi.org/10.1155/2017/3981317

## Consistent Atomic Geometries and Electronic Structure of Five Phases of Potassium Niobate from Density-Functional Theory

^{1}Department Physik, Universität Paderborn, 33095 Paderborn, Germany^{2}Dipartimento di Fisica e Astronomia, Universitá di Padova, 35131 Padova, Italy

Correspondence should be addressed to Arno Schindlmayr

Received 8 September 2016; Revised 11 November 2016; Accepted 24 November 2016; Published 30 January 2017

Academic Editor: Pascal Roussel

Copyright © 2017 Falko Schmidt et al. 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

We perform a comprehensive theoretical study of the structural and electronic properties of potassium niobate () in the cubic, tetragonal, orthorhombic, monoclinic, and rhombohedral phase, based on density-functional theory. The influence of different parametrizations of the exchange-correlation functional on the investigated properties is analyzed in detail, and the results are compared to available experimental data. We argue that the PBEsol and AM05 generalized gradient approximations as well as the RTPSS meta-generalized gradient approximation yield consistently accurate structural data for both the external and internal degrees of freedom and are overall superior to the local-density approximation or other conventional generalized gradient approximations for the structural characterization of . Band-structure calculations using a HSE-type hybrid functional further indicate significant near degeneracies of band-edge states in all phases which are expected to be relevant for the optical response of the material.

#### 1. Introduction

The study of perovskites, oxidic compounds with the chemical formula and an ideal or distorted cubic crystal structure, is of both academic and technological interest. On one hand, perovskite minerals are the most abundant components in the Earth’s mantle [1]. On the other hand, they exhibit a large variety of interesting physical properties [2, 3], which can be modified by composition-induced structural modifications.

Among the human-made perovskites, potassium niobate () belongs to the important class of perovskite-structure ferroelectrics [4]. It crystallizes at about 1325 K in a paraelectric cubic phase (space group ) and, similar to [5], undergoes three successive ferroelectric phase transitions [6]: to a tetragonal phase (space group ) at 691 K, from the tetragonal to an orthorhombic phase (space group ) at 498 K, and from the orthorhombic to a rhombohedral phase (space group ) at 263 K. In addition, a new monoclinic phase (space group ) was very recently discovered in nanomaterials synthesized via a hydrothermal method [7]. It has long been known that the room-temperature orthorhombic phase, in particular, is characterized by unusual dielectric properties and an exceptionally large nonlinear optical coefficient. For this reason, is widely used in nonlinear optics and holographic applications, where it represents a viable alternative to the more diffused .

Due to its technological importance, pure and defective has been subject to many experimental [8–12] and theoretical [13–19] studies. The previous theoretical investigations were typically aimed at the description of the ground-state structure and the identification of defect centers relevant for the optical properties. For the most part, they used density-functional theory [20, 21] with classical local or semilocal exchange-correlation functionals or even simpler semiempirical models. While these approaches are appropriate for atomic geometries and the mechanical properties of a material if not too high accuracy is required, they are much less accurate with regard to the electronic and optical properties, because important factors like quasiparticle renormalization of the electronic band gap, thermal effects due to the electron-phonon coupling, or excitonic effects in the linear and nonlinear optical absorption are missing. Furthermore, most of the published work so far concentrated on the structurally simple cubic and tetragonal phases, while less attention has been given to the remaining polymorphs. As a consequence, a thorough quantitative* ab initio* study of the electronic and optical properties of , especially of the more complex but technologically relevant room-temperature phase, with state-of-the-art methods that go beyond density-functional theory and take the above-mentioned effects into account, as recently carried out for its sister material [22–25], is still missing. The accurate theoretical description of the atomic structure of the relevant phases remains a prerequisite for all such studies, however.

In order to pave the way for future investigations along these lines, here we perform a comprehensive study of the structural and electronic properties of all the five known phases of , including the newly discovered monoclinic phase, based on density-functional theory. The influence of different parametrizations of the exchange-correlation functional on the investigated properties is discussed in detail, and the results are compared to available experimental data. Such a systematic approach is especially important for because the energetic separation and the structural differences between the various phases are small and therefore potentially sensitive to details of the computational procedure. Indeed, our results show that several recently introduced exchange-correlation functionals give, overall, a better description of the atomic geometries throughout the various phases than common standard functionals as used in earlier studies. Band-structure calculations are additionally performed with a nonlocal hybrid functional to correct the well-known underestimation of the fundamental band gap incurred with conventional local or semilocal functionals, which stems from the absence of a derivative discontinuity with respect to the particle number as in the exact functional [26, 27].

This paper is organized as follows. In Section 2, we give an overview of our computational method. In Section 3, we present our results for the optimized atomic geometries of the five considered phases of and compare the performances of different exchange-correlation functionals in relation to experimental data. Furthermore, we discuss the effects of the differently broken symmetries on the electronic band structure through the successive phase transitions. We finally summarize our conclusions in Section 4.

#### 2. Computational Method

All calculations of this study are performed with the Vienna* Ab initio* Simulation Package (VASP) [28, 29], a plane-wave implementation of density-functional theory (DFT). The electron-ion interaction is described by the projector-augmented-wave (PAW) scheme [29, 30], where the K and the Nb electrons are treated explicitly as valence states. For the exchange and correlation between the electrons, we test several different schemes in order to compare their performance, namely, the local-density approximation (LDA), semilocal generalized gradient approximations (GGA), a semilocal meta-GGA, and a nonlocal hybrid functional. In detail, we use the LDA as parametrized by Perdew and Zunger [31]. The conventional Perdew-Burke-Ernzerhof (PBE) parametrization [32] and the more recently introduced PBEsol [33] and AM05 [34] functionals are considered on the GGA side; the latter two are chosen because they are known to yield accurate lattice parameters comparable to hybrid-functional DFT calculations at a moderate computational cost [35]. Furthermore, the revised Tao-Perdew-Staroverov-Scuseria (RTPSS) meta-GGA [36] is employed. Band-structure calculations are additionally performed for the relaxed PBEsol geometries with the Heyd-Scuseria-Ernzerhof (HSE) screened-Coulomb-potential hybrid functional [37, 38]. The fraction of the exact exchange from Hartree-Fock theory, 25% in the conventional HSE06 functional, is increased to 30% to match the experimentally observed band gap of the cubic phase. Nonlocal HSE-type hybrid functionals offer a reliable way to approximate more rigorous band-structure calculation schemes within the framework of quasiparticle theory and thus effectively overcome the band gap problem of conventional DFT calculations with local or semilocal exchange-correlation functionals at a lower numerical cost [39, 40].

The electronic wave functions are expanded into plane waves up to a kinetic energy of 600 eV for the lattice-parameter optimization. All structural degrees of freedom are relaxed until the forces on each atom are below 0.001 meV/Å. The cutoff energy is reduced to 400 eV in the hybrid-functional band-structure calculations. The Brillouin-zone integrations are performed using a shifted Monkhorst-Pack 6 × 6 × 6 -point mesh for all phases. In calculations with the HSE hybrid functional, a regular -centered 6 × 6 × 6 -point mesh is used instead.

#### 3. Results and Discussion

Figure 1 illustrates the five different phases of considered here with exaggerated deformations from the cubic configuration (Figure 1(a)). Starting from the cubic phase, the tetragonal phase is reached by elongation along axis and compression along the other two axes (Figure 1(b)). Further distortions of the lattice parameters transform the tetragonal phase into an orthorhombic polymorph with distinct lattice parameters , , and (Figure 1(c), left). The crystal structure of orthorhombic , whose rectangular unit cell contains two formula units, that is, ten atoms, can alternatively be characterized by a smaller rhombic unit cell with lattice parameters and which contains only one formula unit (Figure 1(c), right); it is this nonrectangular primitive unit cell that we employ in all structure optimizations and electronic-structure calculations. The relation between the orthorhombic and the rhombic unit cells, including the ion displacements, is further illustrated in Figure 2. The orthorhombic phase transforms into the monoclinic phase by choosing mutually distinct lattice parameters (Figure 1(d)). Alternatively, the rhombohedral phase is obtained by choosing equal lattice parameters together with equal but acute angles (Figure 1(e)).