^{†}Corresponding author. Email: ruizhang ccmst@hit.edu.cn
^{‡}Corresponding author. Email: ltq@hit.edu.cn
^{*}Project supported by the National Basic Research Program of China (Grant No. 2013CB632900).
The piezoelectric properties of K_{1− x}Na_{ x}NbO_{3} are studied by using firstprinciples calculations within virtual crystal approximation. To understand the critical factors for the high piezoelectric response in K_{1− x}Na_{ x}NbO_{3}, the total energy, piezoelectric coefficient, elastic property, density of state, Born effective charge, and energy barrier on polarization rotation paths are systematically investigated. The morphotropic phase boundary in K_{1− x}Na_{ x}NbO_{3} is predicted to occur at x = 0.521, which is in good agreement with the available experimental data. At the morphotropic phase boundary, the longitudinal piezoelectric coefficient d_{33} of orthorhombic K_{0.5}Na_{0.5}NbO_{3} reaches a maximum value. The rotated maximum of
Leadbased piezoelectric materials have been widely used in sensors, actuators, and accelerometers.^{[1]} However, due to Pb’ s toxicity, there is an increasing attention in the development of highperformance leadfree piezoelectric materials. Environmentally friendly (K, Na)NbO_{3} (KNN) and its solid solutions show the superior piezoelectric properties at a K/Na ratio of around 50/50, and are the potential substitutes for leadbased piezoelectrics.^{[2– 7]} It is well known that the piezoelectric response is enhanced for piezoelectrics at the morphotropic phase boundary (MPB).^{[1, 8]} Therefore, a great deal of experimental effort has been devoted to search the MPB composition in the K_{1− x}Na_{x}NbO_{3} system. Studies^{[9– 12]} on the phase transition process in the KNN system suggested that an MPB lies near the composition with 50 mol.% NaNbO_{3}. In those studies, the Na content intervals are too large to verify the accurate K/Na ratio at the MPB. The small intervals in K_{1− x}Na_{x}NbO_{3} ceramics have been derived by Dai et al.^{[13]} Their results suggested that a typical morphotropic phase boundary exists at x = 0.52– 0.525. Although the accurate MPB in K_{1− x}Na_{x}NbO_{3} has been confirmed experimentally, some fundamental problems still need to be solved, such as the dominant factor for the high piezoelectric response in K_{1− x}Na_{x}NbO_{3}, the effect of K– O (or Na– O) and Nb– O bonding on the piezoelectric performance, and the effect of Na substitution on the polarization rotation. Firstprinciples calculations can predict various properties well, ^{[14– 18]} such as the phase transition and thermodynamic properties, which have been adopted for the purpose of this report.
In this paper, we use firstprinciples density functional theory within the virtual crystal approximation to predict the MPB in K_{1− x}Na_{x}NbO_{3} and evaluate its piezoelectric response behaviors. Moreover, the mechanism of a high piezoelectric response and the effect of the domainengineering method on polarization rotation in K_{1− x}Na_{x}NbO_{3} near the MPB are also investigated. To the best of our best knowledge, it is the first time that the MPB of KNN has been obtained theoretically. This paper is organized as follows. Theoretical methods are described in Section 2. The results for the structural parameter, bulk modulus, band gap, Born dynamical charge, and piezoelectric property are discussed in Section 3. Finally, the conclusions are given in Section 4.
The firstprinciples calculations are conducted using the density functional theory (DFT) and plane wave pseudopotential method implemented in the ABINIT software package.^{[19– 22]} In order to quantitatively investigate the MPB and related phenomena, the virtual crystal approximation (VCA)^{[23]} is adopted because it has been successfully applied to some perovskite solutions (e.g., PZT, ^{[24, 25]} NBTBT, ^{[26]} PSN, ^{[27]} and LNTO^{[28]}). The exchange correlation potential is described by the localdensity approximation within the Hartwigsen– Goedecker– Hutter (HGH) pseudopotentials.^{[29]} We treat 9 valence electrons for K (3s^{2}3p^{6}4s^{1}) and Na (2s^{2}2p^{6}3s^{1}), 6 for O (2s^{2}2p^{4}), and 13 for Nb (4s^{2}4p^{6}4d^{4}5s^{1}). The Brillouin zone (BZ) is sampled by a 5 × 5 × 5 Monkhorst– Pack mesh of k points for cubic, rhombohedra, orthorhombic, and tetragonal K_{1− x}Na_{x}NbO_{3}. To achieve high accuracy in the total energy calculation and ionic relaxation, we use a very high planewave energy cutoff of 3591 eV, and the convergence for the maximum tolerance on the wave function squared residual is set to 10^{− 18}. Firstly, the appropriate cell parameters of K_{1− x}Na_{x}NbO_{3} are obtained after optimizing the lattice constants and internal atom coordinates based on the maximal force tolerance (0.0257 eV/Å ). Secondly, the total energy, piezoelectric constant, Born effective charge, and elastic property calculations for K_{1− x}Na_{x}NbO_{3} are performed by the density functional perturbation theory (DFPT).^{[30]}
Above the Curie temperature (around 624 K), KNbO_{3} (KNO) is a typical perovskite structure with the cubic symmetry. Cooling KNO from the Curie temperature leads to a continuous phase transition in a cubictetragonalorthorhombicrhombohedral sequence. The calculated lattice parameter, bulk modulus, and band gap for these phases are presented in Table 1. The lattice constant a and the cell volume V of the cubic phase are calculated to be 3.968 Å and 62.46 Å ^{3}, respectively, which are in good agreement with the experimental values (4.021 Å and 65.01 Å ^{3}).^{[32]} The calculated c/a ratio for the tetragonal KNO is 1.034, which agrees with the available experimental result (1.017).^{[36]} According to the present DFT calculations, the unitcell parameters of a = 5.629 Å , b = 3.959 Å , and c = 5.633 Å are predicted for the orthorhombic KNO. The rhombohedral KNO with the space group R3m only exists below room temperature.
Compared with the reported experimental data, ^{[32, 36, 37]} our calculated cell parameters listed in Table 1 have an accuracy with a relative error of ± 1.6%. The predicted bulk modulus for the cubic and orthorhombic KNO are 132 GPa and 185 GPa, respectively, which are consistent with the experimental data (138 GPa and 172 GPa).^{[34, 38]} The calculated piezoelectric coefficients d_{33} are also in reasonable agreement with the experimental and theoretical data.^{[33, 39]}
At the MPB, the piezoelectric materials would show the high piezoelectric properties. In order to further verify our prediction, we calculate the piezoelectric coefficients (d_{33} and d_{15}). Figure 2 only shows d_{33} of the orthorhombic K_{1− x}Na_{x}NbO_{3} in the range of x = 0.0– 0.7, because the elastic constant calculations suggest that the orthorhombic K_{1− x}Na_{x}NbO_{3} becomes mechanically instable in the range of x > 0.7. The d_{33} of the orthorhombic structure shows a maximum value at x = 0.5. This result is in agreement with the experimental measurements.^{[9– 12]} The reason for the peak value at x = 0.2 is unclear yet. There may be another MPB or phase boundary. Similarly, the d_{33} also shows a maximum at x = 0.2 in both BiFe_{1− x}Co_{x}O and (NBT)_{1− x}(K_{1/2}BiTiO_{3})_{x}, which has been thoroughly verified by the experimental^{[41– 43]} and theoretical^{[44]} studies.
The shear piezoelectric coefficient d_{15} is also very important for piezoelectrics. Therefore, we calculate d_{15} as a function of x for the orthorhombic K_{1− x}Na_{x}NbO_{3}. As shown in Fig. 3, there are three peaks located at x = 0.2, 0.464, and 0.7. Although d_{15} at x = 0.7 is the highest among the three peaks, the corresponding d_{33} is the lowest. At x = 0.464, d_{15} is calculated to be 153.52 pC/N, which further identifies the MPB in K_{1− x}Na_{x}NbO_{3}.
According to the suggestion of Davis et al., ^{[45]} the largest longitudinal piezoelectric response would be found away from the polar direction when the ratio d_{15}/d_{33} is larger than 1.5, and the crystals can be categorized into “ rotator” ferroelectrics with d_{15}/d_{33} > 1.5 and “ extender” with d_{15}/d_{33} < 1.5. According to the predicted piezoelectric coefficients d_{33} and d_{15}, the d_{15}/d_{33} value is 7.29 for the orthorhombic K_{0.5}Na_{0.5}NbO_{3}. Thus, the orthorhombic K_{0.5}Na_{0.5}NbO_{3} belongs to “ rotator” ferroelectrics and is highly anisotropic. The large anisotropy is also observed in other singledomain relaxorPTbased crystals near their MPBs.^{[8]} It is necessary for investigating the crystal orientation dependence of the piezoelectric properties. From the calculated singledomain data of the orthorhombic K_{1− x}Na_{x}NbO_{3} with x = 0.5 and 0.0, the orientation dependence of the intrinsic piezoelectric coefficient
For the orthorhombic K_{1− x}Na_{x}NbO_{3} with x = 0.5, there are nine independent elastic constants (c_{11}, c_{22}, c_{33}, c_{44}, c_{55}, c_{66}, c_{12}, c_{13}, and c_{23}), which are obtained using DFPT. From the calculated elastic constants, both the bulk modulus (B) and shear modulus (G) can be derived using Voigt– Reuss– Hill (VRH) approximation, ^{[48]} and the results are presented in Fig. 5. The VRH bulk modulus is defined as the average B = (B_{V} + B_{R})/2 between the Voigt upper and Reuss lower bounds
The VRH shear modulus G = (G_{V} + G_{R})/2 is taken from the average between the Voigt upper G_{V} and Reuss lower G_{R} bounds
From Fig. 5, three minimum value of the bulk B and shear G modulus are located at x = 0.2, 0.464, and 0.7. The bulk B and shear G modulus at x = 0.7 are the smallest among the three minimum values. The corresponding d_{33} is also the smallest, as shown in Fig. 2. This implies that the small bulk and shear modulus would not necessarily give rise to a high piezoelectric response. Otherwise, the exceptionally high bulk and shear modulus are also not conducive to a high piezoelectric response. Therefore, the moderate bulk and shear modulus are better for the piezoelectric response.
The firstprinciples studies have suggested that the orbital hybridization between the Pb 6s state and O 2p states plays a crucial role for large ferroelectricity in tetragonal PbTiO_{3}.^{[49]} The prediction is later verified experimentally, ^{[50]} i.e., the Pb– O bonds in tetragonal PbTiO_{3} show rather strong covalency. In order to investigate the interaction between Nb– O and K– O bonds, we plot the partial density of state (PDOS) of the Nb, K, and O orbitals for the orthorhombic K_{1− x}Na_{x}NbO_{3} with x = 0.5. As shown in Fig. 6, the K 2s and 2p, O 2p orbits are mainly located in the range from − 5.6 to 0 eV, indicating that the K– O bonds are mainly ionic. The calculated Born effective charge of K is 1.13, which is close to its oxidation state, further confirming the characteristic of ionic interaction between K and O ions. The Nb 4d orbits mixed with O 2p orbits are located in the valence band and conduction band, suggesting that the Nb– O bonds are both ionic and covalent. The calculated average Born effective charge of Nb is 8.79, larger than its oxidation state (+ 5), indicating that the Nb– O bonds are not purely ionic. Thus, the high piezoelectric for orthorhombic K_{1− x}Na_{x}NbO_{3} with x = 0.5 is related to the Nb– O covalency. This agrees with the recent theoretical study.^{[51]}
To understand the physical mechanism of the high piezoelectric response for the orthorhombic K_{1− x}Na_{x}NbO_{3}, we calculate the energy barrier on polarization rotation paths along the [001] direction through a simple approach: E(θ ) = E_{0}(θ ) − E_{0}(0), where E(θ ) is the energy barrier with θ being the angle of polarization rotation, E_{0}(θ ) is the total energy of the orthorhombic K_{1− x}Na_{x}NbO_{3}, and E_{0}(0) denotes the total energy with θ = 0° . The adopted model for the energy barrier calculations is illustrated in Fig. 7, and the results are depicted in Fig. 8. As shown in Fig. 8, the energy barrier of the orthorhombic K_{1− x}Na_{x}NbO_{3} with x = 0.0 and x = 0.5 increases with the increase of the angle θ . However, the energy barrier of the orthorhombic K_{1− x}Na_{x}NbO_{3} with x = 0.5 increases more slowly than the one with x = 0.0, indicating that at MPB, the phase instability of the orthorhombic K_{1− x}Na_{x}NbO_{3} with x = 0.5 is enhanced and the polarization rotation becomes easier than the pure KNO, i.e., the flat free energy profile is the most important factor for inducing a high piezoelectric response.
Firstprinciples density functional theory within the virtual crystal approximation is applied to investigate the morphotropic phase boundary and piezoelectric properties of K_{1− x}Na_{x}NbO_{3}. The structure parameter, bulk modulus, band gap, and piezoelectric coefficients have been calculated. The MPB in K_{1− x}Na_{x}NbO_{3} is predicted to be located at x = 0.521, which is in good agreement with the experimental data. The dependence of the longitudinal piezoelectric coefficient on the polarization orientation proves that the highest value of the piezoelectric coefficient
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