Advances in Materials Science and Engineering

Volume 2018, Article ID 6416057, 9 pages

https://doi.org/10.1155/2018/6416057

## Electronic and Optical Properties of Sodium Niobate: A Density Functional Theory Study

Department of Chemistry, University of Bath, Claverton Down, Bath BA2 7AY, UK

Correspondence should be addressed to Daniel Fritsch; ku.ca.htab@hcstirf.d

Received 9 November 2017; Accepted 27 December 2017; Published 7 March 2018

Academic Editor: Pavel Lejcek

Copyright © 2018 Daniel Fritsch. 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

In recent years, much effort has been devoted to replace the most commonly used piezoelectric ceramic lead zirconate titanate Pb[Zr_{x}Ti_{1−x}]O_{3} (PZT) with a suitable lead-free alternative for memory or piezoelectric applications. One possible alternative to PZT is sodium niobate as it exhibits electrical and mechanical properties that make it an interesting material for technological applications. The high-temperature simple cubic perovskite structure undergoes a series of structural phase transitions with decreasing temperature. However, particularly the phases at room temperature and below are not yet fully characterised and understood. Here, we perform density functional theory calculations for the possible phases at room temperature and below and report on the structural, electronic, and optical properties of the different phases in comparison to experimental findings.

#### 1. Introduction

The most widely used piezoelectric ceramic to date is lead zirconate titanate Pb[Zr_{x}Ti_{1−x}]O_{3} (PZT), where the composition *x* is used to tailor specific properties for memory or piezoelectric devices. However, due to the toxicity of these lead-containing devices, much effort has been devoted in recent years to find suitable lead-free alternatives to PZT. One promising alternative materials system is the solid solution sodium potassium niobate (Na,K)NbO_{3} [1, 2]. While the structural and electronic properties of the one end member potassium niobate (KNbO_{3}) are relatively well known, this is much less the case for the ferroelectric (FE) perovskite sodium niobate (NaNbO_{3}).

Like many other perovskites, NaNbO_{3} exhibits a large range of structural phase transitions, accompanied by changes in the ferroelectric behaviour. A first comprehensive discussion of the different structural phase transitions in NaNbO_{3} was reported by Megaw [3]. According to Megaw [3], the high-temperature phase of NaNbO_{3} is paraelectric (PE) and crystallises in the simple cubic perovskite structure (), before it undergoes a phase transition to a PE tetragonal phase () at a transition temperature = 913 K. Next, there appear three distinct phase transitions into orthorhombic phases: to the PE phase () at = 848 K, to the PE *S* phase () at = 793 K, and to the antiferroelectric (AFE) *R* phase () at = 753 K, respectively. The orthorhombic AFE *R* phase () undergoes a phase transition into the orthorhombic AFE *P* phase () at = 633 K, which is the commonly assumed crystal structure at room temperature and stays stable over a wide temperature range down to = 173 K. Below , NaNbO_{3} crystallises in the rhombohedral FE *N* phase ().

However, while the structure of the high-temperature crystalline phases of NaNbO_{3} is commonly agreed on, there is still an ongoing discussion about the crystalline phases at room temperature and below. Darlington et al. [4] and Cheon et al. [5] reported on a possible admixture of a monoclinic phase () into the room-temperature orthorhombic AFE *P* phase () based on X-ray diffraction and neutron powder diffraction measurements. There are also reports about a room-temperature phase transition into a FE phase () induced by an applied electric field [6, 7], by nanoparticle growth [8] or by growth as a strained thin film [9], respectively. A full list of recent experimental data can be found in [2, 10–12].

While there is quite a lot information available on the structural phase transitions from experiment, this is much less the case for theoretical investigations. Diéguez et al. [13] reported a first-principle study of epitaxial strain in perovskites, including KNbO_{3} and NaNbO_{3}, while Li et al. [14] reported density functional theory (DFT) calculations for epitaxially strained KNbO_{3}/NaNbO_{3} superlattices, thereby including the unstrained simple cubic perovskite phase () as well. Finally, Machado et al. [15] reported on the relative phase stability and lattice dynamics of NaNbO_{3} from first principles. A rigorous assessment of the performance of different exchange-correlation functionals within DFT calculations and applied to the possible crystalline phases of NaNbO_{3} at room temperature and below is still missing to date.

The focus of the present work is on the reported possible crystalline phases of NaNbO_{3} at room temperature and below, especially on the coexistence of the rhombohedral FE *N* phase () with the monoclinic AFE *P* phase () and the orthorhombic AFE *P* phase (). Since the delicate interplay of structural and electronic properties determines properties like the spontaneous polarisation, an improved description would help demystify the crystalline phases of this material at room temperature and below and to tailor it better for technical applications. Here, we present results of DFT calculations for the structural, electronic, and optical properties of the crystalline phases of NaNbO_{3} at room temperature and below, with a special emphasis on the performance of different flavours of the generalised gradient approximation (GGA) to the unknown exchange-correlation potential. The results include calculations based on the conventional PBE parametrisation of Perdew et al. [16], the AM05 parametrisation [17], and the PBE parametrisation revised for solids (PBEsol) [18]. In addition, we also perform benchmark calculations for the simple cubic PE perovskite phase () utilising the hybrid functional PBE0 [19], where a quarter of the exchange potential is replaced by Hartree–Fock exact-exchange to better account for electronic correlation effects [20]. In accordance with similar investigations for the other end member KNbO_{3} [21] of the solid solution sodium potassium niobate (Na,K)NbO_{3}, we find that the improved GGA approximations of AM05 and PBEsol perform better for the structural, electronic, and optical properties compared to the conventional PBE approximation. The results will be beneficial for future theoretical works concerning strain influences from underlying substrates or calculations of the spontaneous polarisation.

This paper is organised as follows. Section 2 introduces the necessary theoretical background and details of the calculations. Section 3 is devoted to the discussion of the obtained structural properties in comparison to available experimental data, the electronic properties, and finally, the optical properties. The final section provides a summary of the presented results and their main conclusion.

#### 2. Materials and Methods

##### 2.1. Computational Details

The results of the present work have been obtained by DFT calculations employing the Vienna ab initio simulation package (VASP 5) [22–24] together with the projector-augmented wave (PAW) formalism [25]. For the latter, standard PAW potentials supplied with VASP were used, providing 9 valence electrons for Na atoms (223), 13 valence electrons for Nb atoms (4454), and 6 valence electrons for O atoms (22), respectively.

Structural relaxations have been performed within a scalar-relativistic approximation with a plane wave energy cutoff of 500 eV. -centred *k* point meshes have been used to sample the Brillouin zone and amounted to for the simple cubic perovskite phase, for the monoclinic phase, for the orthorhombic phase, and for the rhombohedral phase, respectively.

To evaluate the performance of different exchange-correlation functionals, the structural and electronic properties have been calculated employing the GGA in the conventional PBE parametrisation of Perdew et al. [16], the AM05 parametrisation [17], and the PBE parametrisation revised for solids (PBEsol) [18]. Both AM05 and PBEsol have been developed to increase accuracy in structural properties for crystalline solids [18, 26]. For the smallest unit cell of the simple cubic perovskite phase of NaNbO_{3}, the results have additionally been benchmarked against hybrid functional calculations using the PBE0 functional [19] to better account for electronic correlation effects [20].

The obtained relaxed ground state structures served as a starting point for subsequent calculations of the electronic band structures and the real and imaginary parts of the dielectric functions. Thereby, the imaginary part of the dielectric tensor (in VASP) is determined by a summation over empty states usingwhere *c* and *v* denote the conduction and valence band states, respectively, and is the cell periodic part of the orbitals at . In order to ensure converged results, the number of empty bands in the calculations has been increased by a factor of three. The real part of the dielectric tensor is obtained via a Kramers–Kronig transformation:where *P* denotes the principal value of the integral. Details of the method can be found in [27]. The real and imaginary parts of the dielectric functions for the noncubic phases have been obtained by diagonalising the dielectric tensors for every energy point and averaging over the resulting main diagonal values, respectively.

##### 2.2. Crystalline Phases at Room Temperature and Below

The present work focuses on the following crystalline phases at room temperature and below: the rhombohedral FE *N* phase (, SG 161, *Z* = 6), the monoclinic AFE *P* phase (, SG 6, *Z* = 8), and the orthorhombic AFE *P* phase (, SG 57, *Z* = 8). In addition, benchmark calculations have been carried out for the high-temperature simple cubic perovskite phase (, SG 221, *Z* = 1). All four phases have been initially set up using experimental data reported by Jiang et al. [28] for the phase, by Cheon et al. [5] for the phase, by Johnston et al. [6] for the phase, and by Darlington and Megaw [4] for the phase, respectively. For all four phases, we performed a full geometry optimisation for several unit-cell volumes centred around the experimentally reported ones. The geometries have been fully optimised employing three different GGA functionals (PBE, PBEsol, and AM05), until the forces on each atom were smaller than 0.001 eVÅ^{−1}. In addition, for the simple cubic perovskite phase, we also employed the PBE0 hybrid functional. Together with the plane wave energy cutoff and the *k* point meshes reported in Section 2.1, this ensured well-converged structural and electronic properties. Exemplary, the PBEsol relaxed structures for the four different phases are shown in Figure 1, and its CIF files can be found in the Supplemental Material (available here). The volume dependence of the total energies for the four different phases gives access to the bulk modulus , defined aswhere is the total energy and is the equilibrium bulk volume. For cubic crystals, the bulk modulus can also be expressed in terms of the elastic moduli and [29–31]: