Lattice Parameter Calculation For Orthorhombic

[Stayla Casillas]

Piezoelectricceramic materials are central to a wide range of technical devices, such as sensors, actuators, crystal oscillators, and ultrasonic transducers. The most common piezoelectric ceramic used for commercial purposes is lead zirconate titanate (PbZr\(_x\)Ti\(_1-x\)O\(_3\), PZT), which is favored due to its large piezoelectric coefficient, physical strength, and relatively low manufacturing costs [1]. However, its toxicity has prompted an increasingly urgent search for safer, lead-free alternatives [2]. Among the possible substitutes, potassium sodium niobate (K\(_1-x\)Na\(_x\)NbO\(_3\), KNN) is regarded as particularly promising owing to its excellent piezoelectric properties that are comparable with PZT [3, 4] and its high Curie temperature, which is a prerequisite for many applications.

The solid solution KNN is part of the perovskite family and exhibits a multitude of different phases depending on composition and temperature [5,6,7,8]. Therefore, elucidating the complex phase diagram has been a central thrust of numerous experimental investigations. At room temperature, pure potassium niobate (KNbO\(_3\)), corresponding to \(x=0\), has a ferroelectric orthorhombic crystal structure that originates from a symmetry-lowering distortion of the cubic aristotype. This configuration remains stable if up to 52.5 mole % of K are replaced by Na, but successive phase transitions occur at \(x=0.525\), 0.675, and 0.825 [6]. The resulting structures, originally assumed to be orthorhombic [5], are now identified as ferroelectric monoclinic [7, 8]. Finally, above \(x=0.98\), it changes into the antiferroelectric orthorhombic structure of sodium niobate (NaNbO\(_3\)). As a function of temperature, KNN undergoes a polymorphic phase transition to one of several ferroelectric tetragonal structures at about 220 \(^\circ \)C and further to a paraelectric cubic structure at about 430 \(^\circ \)C [7]. These transition temperatures are essentially independent of x in pure KNN, except for very Na-rich samples, but they can be tuned to a great extent by the addition of other elements [9], thus opening a route to design KNN-based piezoceramics with improved functional properties [10].

Although the symmetry of the crystal structure changes at certain values of the compositional parameter x in KNN, it is generally believed that these phase transitions are not accompanied by major alterations of the external lattice parameters, but chiefly reflect internal rearrangements of the atomic positions inside the existent unit cells, such as different tilt patterns of the oxygen octahedra and cation displacements [8]. In particular, although some experimental studies identified small irregularities in the lattice parameters of KNN in a very narrow compositional region around the supposed phase boundary near \(x=0.525\) [11], which are indicative of a structural transformation, no systematic trends are visible on a larger scale. For example, Tellier et al. [12], who used X-ray diffraction to measure the lattice parameters of KNN in the range \(0.4 \le x \le 0.6\) in steps of 0.02, observed that the three lattice constants of the rectangular orthorhombic unit cell increase linearly with the potassium content, with no discernible anomalies. Consequently, the variation of the unit-cell volume is also linear. In the alternative rhombic metric, which utilizes two length parameters and one angle \(\beta > 90^\circ \) instead, the latter decreases towards a more symmetrical orthorhombic unit cell with increasing potassium content. The relation between \(\beta \) and x is nonlinear, featuring a shallower slope on the sodium-rich side of the investigated compositional range, but the absolute variation is extremely small with an overall drop from \(90.34^\circ \) at \(x=0.6\) to \(90.32^\circ \) at \(x=0.4\). Very similar results were obtained by Wu et al. [13], who measured the lattice parameters over an even larger compositional range and likewise observed a continuous, linear behavior for \(0.4 \le x \le 0.6\); in addition, they pointed out that anomalies outside this interval, especially for large x, correlate with deviations in the grain-size distribution of the analyzed nanoparticles, a factor that may also play a role in other studies.

As the apparent inconsistencies between some of these experimental studies are still debated and may be linked to technical details of the sample fabrication, the expectation grows that numerical simulations might provide answers. In this respect, recent first-principles calculations for KNN have challenged the picture of weakly, monotonically varying lattice parameters. In particular, Liu et al. [14] investigated the geometric structure, total energy, and electronic properties of three competing phases based on density-functional theory (DFT) in the full range \(0 \le x \le 1\) and claimed that the variation of the lattice parameters of the orthorhombic structure is highly nonlinear. Most surprisingly, their results suggest that the volume of the unit cell is minimal at \(x=0.5\), where the compositional disorder is largest, and significantly smaller than either at \(x=0\) or at \(x=1\). This is accompanied by a large distortion, as \(\beta \) peaks at \(94^\circ \) for \(x=0.5\), although both pure potassium niobate and pure sodium niobate have angles close to \(90^\circ \). Furthermore, the electronic bandgap derived within DFT reaches 2.7 eV at \(x=0.5\), compared to 2.2 eV for both end members of the solid solution.

In another computational study of the orthorhombic and tetragonal phases in the interval \(0.540 \le x \le 0.570\), Yang et al. [15] found large, nonmonotonic changes of up to 0.1 in the lattice parameters between x values separated by merely 0.005, and overall variations of 0.05 eV in the bandgap across this narrow compositional range. This suggests an extreme sensitivity with respect to the K:Na ratio that not only contradicts the experimental evidence [12, 13] but also the slow, albeit large, variation predicted by Liu et al. [14] using very similar theoretical methods. These discrepancies remain as yet unresolved.

In general, DFT is regarded as a highly accurate computational scheme for predicting crystal structures from first principles, i.e., without any empirical parameters, and although it underestimates the absolute size of electronic bandgaps [16], trends with respect to structural or compositional variations are typically obtained reliably. However, the modeling of solid solutions like KNN, where the potassium and sodium atoms are distributed randomly, poses a special challenge, because most implementations require three-dimensional periodic boundary conditions. In this situation, DFT studies of disordered systems often resort to the virtual-crystal approximation (VCA), which was also employed in Refs. [14] and [15]. In this approach, the two distinct atomic species are replaced by merely one virtual atom type created by mixing the two pseudopotentials of the original elements in the required ratio [17]. A single primitive unit cell then suffices to represent the solid solution, making the calculations extremely efficient. While the VCA is generally trusted and often describes the properties of disordered solids, including perovskites [17, 18], correctly, it ignores the true local interactions that act between the real atoms, which may lead to deviations from supercell calculations [19]. The electronic properties, in particular, are very sensitive to this. The most serious possible pitfall of the VCA, however, is that the automatic mixing of pseudopotentials is prone to produce so-called ghost states [20], unphysical extra bound states, even if the underlying pure pseudopotentials are well behaved. This leads to incorrect orbital occupancies and hence to seemingly converged but false results. Therefore, the VCA must always be applied with caution and carefully validated.

To examine the VCA as a possible explanation for the discrepancy between the theoretical results by Liu et al. [14] as well as Yang et al. [15] on the one hand and the experimental measurements [12, 13] on the other hand, we avoid the VCA in this work and instead choose the supercell approach to calculate the lattice parameters and the electronic bandgap of KNN. We focus specifically on K\(_0.5\)Na\(_0.5\)NbO\(_3\), denoted as KNN50 in the following, which has an outstanding piezoelectric coefficient [5] due to its proximity to the morphotrophic phase boundary at \(x=0.525\) and is, therefore, of principal interest for technical applications. In addition, the discrepancy between the previously published experimental and theoretical results is largest at this point.

Compared to the VCA, the supercell approach is computationally more demanding, because it requires larger simulation cells containing the proper ratio of K and Na atoms, whose spatial distribution should ideally be as close as possible to the statistical average over all possible random structures [21]. To avoid an excessive computational cost, we follow a different, less expensive route in this work by selecting a set of six configurations that possess a high degree of periodic ordering and can be embedded in relatively small supercells but nevertheless span a wide variety of local chemical environments. This approach is validated a posteriori by the fact that our results for KNN50 turn out to be insensitive to the actual atomic arrangement and agree well for all configurations.

This paper is organized as follows. In Sect. 2, we give an overview of our computational method, including the selected supercell configurations that we use for KNN50. In Sect. 3, we then present our results and compare the calculated lattice parameters and electronic bandgaps with the available theoretical and experimental data. Finally, we summarize our conclusions in Sect. 4.

The solid solution K\(_0.5\)Na\(_0.5\)NbO\(_3\) crystallizes in the same perovskite structure as KNbO\(_3\) [7], which is shown in Fig. 1a. The perfect cubic aristotype is realized above 430 \(^\circ \)C and undergoes a succession of symmetry-lowering deformations as the temperature decreases, first to a tetragonal phase and then to the room-temperature orthorhombic phase. The orthorhombic geometry refers to a rectangular unit cell containing two formula units (10 atoms), which is characterized by three distinct edge lengths oriented along the symmetry axes of the crystal structure. Alternatively, it can also be described in terms of the primitive unit cell, which contains one formula unit (5 atoms) and exhibits a rhombic symmetry [12, 22]. It has the shape of a right prism with a rhombus as base and is characterized by two length parameters and one obtuse angle. Further adding to the confusion, different notation systems are in common use: if the axis perpendicular to the other two is designated as b, the space group is labeled Bmm2. This convention was adopted, e.g., in Ref. [12]. If this axis is chosen as a instead, the space group becomes Amm2. In this work, however, we follow yet another notation established in Refs. [14, 15] to facilitate a direct comparison. In this system, the axis perpendicular to the other two is c, the edges of the rhombic base are \(a=b\), and the angle between the a and b axes is \(\beta > 90^\circ \), as illustrated in Fig. 1a.

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