Insight For The Construction of R-T Phase Boundary in KNN Piezoceramics From The View of Energy Band Structure

The crystal structures and associated energy band structures of classic lead-based PZT (PbZr 0.5 T i0.5 O 3 ) and lead-free KNN (K 0.5 Na 0.5 NbO 3 ) piezoelectric ceramics prepared by a solid-state method are systematically studied based on the First-principles calculations. For better understanding relations between energy band structures and the phase structures in the PZT and KNN, the components PT (PbT i O 3 )/PZ (PbZrO 3 ) and KN (KNbO 3 )/NN (NaNbO 3 ) are also calculated. The results suggest that the vital contribution to R-T morphotropic phase boundary (MPB) in PZT comes from the hybridization of A-site and B-site ions, which causes stretching and tilting of oxygen octahedron and therefore leads to the formation of R and T phases. The KNN behaves differently. Moreover, it’s concluded that the band structure highly inherited from tetragonal PT plays an important role in the construction of the R-T phase boundary according to the comparison of PZT and KNN. In general, the work inspires an idea from the view of energy band structure to realize R-T phase coexistence as well as improved piezoelectric properties in KNN-based lead-free piezoceramics.


Introduction
Lead zirconate titanate is a widely used piezoelectric ceramic that can transfer mechanical and electrical energy. With an increasing awareness of sustainable development, lead-free piezoceramics, especially KNN with high Curie temperature (T c ), has attracted more and more attention [1][2][3]. However, the electrical properties of KNN piezoceramics are inferior to PZT-based ones. Through dopant modi cation and advanced synthesis methods, the properties of KNN piezoceramics are enhanced dramatically and even reach comparable values to commercial PZT-5H and PZT-5A [4,5]. The polymorphic phase boundary (PPB) and nano ferroelectric domains are regarded as key factors to property enhancement, and they are closely related to the dopants [6]. Nevertheless, the effects of the dopants on phase structure as well as property enhancement are rarely mentioned from the view of energy bands. It's worth noting that both the PZT and KNN consist of ferroelectric materials (i.e. PT and KN) and anti-ferroelectric materials (i.e. PZ and NN), but their phase structures and electrical properties are different [7,8]. It's well known that the lattice deformation and associated phase transition are closely related to electrons' states and their interactions. Therefore, research on the energy band structures and the density of states for PZT and KNN probably opens up a new horizon to explain the differences between PZT and KNN, and makes a good start to help us understand the effects of dopants on phase structure and property enhancement [9,10].
Density functional theory (DFT) is an important method for studying the ground-state theory of multiparticle systems.
The basic idea is that the physical properties of the ground state of atoms, molecules and solids can be described by the particle number density function, and the ground state of the system is the state in which the energy functional takes the minimum value under the condition of constant particle number. [11,12] This not only provides a theoretical basis for simplifying the multi-electron problem into a single electron problem but also does not need to rely on any empirical parameters. The atomic and electronic structures and related physical properties of materials can be obtained by solving the intrinsic quantum mechanical equations. It provides a direct and effective method for studying the relationship between microstructure and macroscopic properties of materials [13].
By using the super-soft pseudopotential method, this pseudopotential introduces an overlap operator to plane wave, which reduces the plane-wave basis function. To solve the problem of large modulus conservation, this method is employed [14]. Generally speaking, the calculation of the pseudopotential method is small, in which the inner electronic state is eliminated, and an approximation is introduced more than that of the all-electron method [15].
In recent years, the generalized gradient approximation (GGA) has attracted increasing attention. Compared with the traditional LDA approximation, GGA further considers the in uence of the nearby charge density on the exchangecorrelation energy, i.e. the in uence of the rst-order gradient of the charge density [16]. Generally, the GGA method can give more accurate energy and structure, which can avoid underestimating exchange energy and overestimating correlation energy when LDA is used and applied to analyze the relation between PZT and KNN [17][18][19].
Therefore, in the work, the energy band structure and associated electron density of PZT and KNN piezoceramics prepared by a solid-state method are systematically studied based on the First-principles calculations [20]. The Perdew Burke Ernzerhofer (PBE) is a simpli ed form without experimental parameters to deal with the exchangecorrelation energy of the electron-electron interaction [21]. The PBE functional of general gradient approximation (GGA) was established by using the exchange-correlation energy function of electron and electron interaction. The electronic wave function was expanded based on the plane wave. The GGA with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation function was used in this work. The GGA and Perdew Burke Ernzerhof (PBE) were used to deal with the exchange-correlation potential [22,23]. The formula of the exchange-correlation energy function is written by: In the formula, E xc is the exchange-correlation energy and ρ(r) is the local electron density for triple integration.
Besides, the band structures of PT/PZ and KN/NN are also calculated to better investigate the origins of different phase structures in the PZT and KNN. Then the partial density of states (PDOS) of each matter is calculated to study the degrees of orbital hybridization and electron localization, which account for the lattice deformation as well as a phase transition. In General, the work inspires a thought to construct the R-T polymorphic phase boundary of KNNbased lead-free piezoceramics through the addition of the dopant with appropriate energy band structures.

Methods And Parameters
The investigated ceramics were prepared by the conventional solid-state reaction method. Firstly, weighed raw materials were ball-milled in a polyethylene jar using ZrO 2 balls and the anhydrous ethanol for 12 h and calcined after drying. Then the calcined powders were ball milled again for 12 h. After drying, the powders were mixed with 6% para n and then pressed into disc particles with a diameter of 10.0 mm and thickness of 1.2 mm under the pressure of 10 MPa. The preparation process for the above ceramics is listed in Table 1. Then, the phase of the sintered samples was detected by X-ray diffraction (XRD) with a Cu Kα radiation (Xpert-PRO, PANalytical Company, The Netherlands). The band structure was calculated using Materials Studio software (Version of CASTEP-19.11) and the Vienna ab initio simulation package (VASP) which is based on density functional theory (DFT) and the plane-wave pseudopotential methods.
The First-principles calculations were based on the crystal structure of the targeted material. All the calculations were carried out by the CASTEP package which was based on density functional theory (DFT) and the ultrasoft pseudopotential methods. The plane-wave cutoff energy was set at 300 eV and the convergence for energy is chosen as 2×10 −5 eV. The accuracy of interatomic force was 0.05 eV/Å, the accuracy of internal stress was 0.1 GPa, and the accuracy of displacement tolerance was 2×10 -4 nm. When the DFT theory of GGA was used, only the local electronic states of the outermost s and p orbits were considered, and the in uence of the inner electronic states was not taken into consideration, which inevitably leads to the smaller energy gap. Even so, the differences in the band structure, the density of states of various systems were still able to reveal the origins of the R-T phase boundary in PZT [19]. Consequently, this work inspires an idea from the view of energy band structure to realize R-T phase coexistence as well as improved piezoelectric properties in KNN-based lead-free piezoceramics.
Results And Discussion Figure 1 is the crystal structure, energy band structure and PDOS of NaNbO 3 (NN) piezoceramics. The space group of NN is Pmc21 (Orthorhombic phase), as shown in Fig. 1(a). Figure 1(b) gives the corresponding band structure. It's well known that the top of the valence bands (VB) and the bottom of the conduction bands (CB) are located at the same high symmetric point in the Brillouin region, suggesting that the material is a direct gap semiconductor and vice versa. Therefore, NN is a direct gap semiconductor and the forbidden bandgap (E g ) is 2.4 eV. Besides, the energy bands are compact and at, resulting from the high degree of electron localization. The results are consistent with the PDOS in which the peaks of O 2p and Nd 4d are detected, as presented in Fig. 1(c). According to the PDOS, it's found that the energy bands are mainly ascribed to covalent interaction of O 2p orbit and Nd 4d orbit over the energy range of 2.5 eV-5 eV (for calculation convenience, we de ne the Fermi level as 0 eV), and the overlapped sharp peaks imply the formation of quasi s-p hybridized orbit between O and Nb atoms. Furthermore, the energy bands lying in the energy range of -5 eV-0 eV are also mainly attributed to Nb 4d and O 2p orbits. The hybridized orbit facilitates the formation of strong binding between adjacent atoms, which is closely related to the crystal structure. More importantly, the A-site ions, i.e. Na, are not involved in the bonding. Fig. 2 is the crystal structure, energy band structure and PDOS of KN piezoceramics using the lattice parameters reproduced. As shown in Fig. 2(a), the space group of KN is Bmm2 (Orthorhombic phase). Fig. 2(b) is the band structure of KN. It's observed that KN is an indirect gap semiconductor since the bottom CB and the top VB lie at different positions. That means both the energy and the momentum change when the electron jumps between two energy states. Besides, as can be seen, the E g of KN is 2.11 eV, smaller than that of NN. However, unlike NN, the CB are loosened. Fig. 2 A(0) 3 ) since the hybrid orbits with speci c spatial orientations restrict the degree of freedom of lattice deformation. Furthermore, the bands become loosened, indicating that the electron localization of KN weakens compared to NN. Figure 3 is the crystal structure, energy band structure and PDOS of KNN piezoceramics using the lattice parameters reproduced. As shown in Fig. 3(a), the space group of KNN is Amm2 (O phase). It's universally acknowledged that KNN is composed of antiferroelectric NN and ferroelectric KN. However, the energy band structure of KNN seemingly inherits from NN, as shown in Fig. 3 PZT is extensively used which is also composed of antiferroelectric PZ and ferroelectric PT. However, the piezoelectric properties of PZT are superior to KNN. To explore the differences between PZT and KNN, the energy band structures and PDOSs of PZ, PT and PZT piezoceramics are calculated using the same method, as exhibited in Fig. 4. Figures 4(a1), 4(b1) and 4(c1) are the crystal structures of PZ, PT and PZT piezoceramics. It's observed that the space groups of PZ, PT and PZT are Pbam (Orthorhombic phase), P4mm (Tetragonal phase) and C1m1 (Centered monoclinic) respectively. The phase C1m1 was discovered at low temperatures along the R-T MPB according to the high-resolution x-ray diffraction study of PZT, and therefore, it's a transitional phase between the R and T phase [24][25][26]. Figures 4(a2), 4(b2) and 4(c2) give the energy band structures of PZ, PT and PZT ceramics. As presented in Fig. 4(a2), antiferroelectric PZ is an indirect gap semiconductor with E g~2 .4 eV. Similar to the NN, the at and compact energy band structures are detected. Accordingly, as exhibited in Fig. 4(a3), the hybridization of Pb 6p and Zr 4d orbits plays an important role in the formation of the quasi s-p hybridized orbit at the bottom of CB within the range of 2.7 eV-4.5 eV. Nevertheless, the top of VB (-4.4 eV-0 eV) is highly attributed to O 2p, Zr 4d while Pb 6s and 6p orbits are involved to some extent. The involvement of A-site Pb and featured orbit hybridization in the bottom of CB are regarded as the origins of the R phase of PZT. Figure 4(b2) is the band structure of ferroelectric PT. It's seen that the energy band structures become loosened for PT (indirect gap semiconductor and E g~2 .2 eV). However, unlike KN, the bottom of CB is mainly derived from the hybridization of Ti 3d and Pb 6p orbits. More importantly, compared to KN, at higher CB from 8.1 eV to 14 eV, an electron-forbidden area is detected.
Concerning PZT, in the rst place, the E g (2.03eV) decreases compared to PZ and PT. As depicted in Fig. 4(c2), the synergistic effect of PZ and PT gives rise to comparably at and compact band structures alike to KNN, since the calculated supercell is smaller than KNN. Furthermore, the band structures of PZT are highly similar to PT, as presented in Fig. 4(b2). However, as exhibited in Fig. 4(c3), the higher CB within 8.2 eV-14.3 eV, disappear for PZT compared to PT, indicating that the effects of Ti 4s and Ti 3p vanish when the PZT is constituted. Figure 5(a) gives the electron densities of KNN and PZT. The typical quasi s-p hybrid orbit is observed for KNN according to the isosurface scale. The positive isosurface represents the accumulation of charge while the negative isosurface suggests the reduction of the charge. Therefore, the linear type s-p hybrid orbit appears around O ions and orientates along Nb and O ions. On the contrary, the involvement of the A-site K ion for the hybridization is absent. For PZT, the hybridization of B-site Zr and Ti and O ions is investigated. However, the degree of such hybridization decreases according to the weakened color of the positive isosurface. Moreover, it's detected that the proximate region of A-site Pb ions presents the positive isosurface for accumulation of charge. The circular region implies s orbit exists around Pb ions. Consequently, the process of lattice deformation as well as phase transition is deduced, as exhibited in Fig. 5(b). The s-p hybrid orbit results in the symmetric stretching of oxygen octahedron and hence the c increases, resulting in the formation of the T phase. Meanwhile, the Pb ions with like charges repel each other, leading to the tilting of oxygen octahedron, and therefore the R phase forms [27][28][29]. For the convenient observation of the tilting, only the bendings in (000) and (001) planes are presented. The bendings in other planes are alike to the situation of (000) and (001) planes.
Therefore, the construction of the R-T phase boundary in KNN can take a cue from the PZT, that is the strengthening of the A-site ion's effect and the weakening of the B-site ion's effect. The key factors are easy to understand. Firstly, the involvement of the A-site ion gives birth to the tilting of the B-site ion. Secondly, the deformation of oxygen octahedron would be restricted to the stretching of c if the hybridization ability of the ion in the B site is relatively strong. Consequently, only the T phase is constructed.
Above all, a feasible strategy is proposed to obtain R-T phase boundary as well as enhanced the piezoelectric properties of KNN piezoceramics, the end members composed of the A-site or B-site dopants with appropriate hybridization ability. The hybridization ability can be re ected partly by the electronegativity which is also representative of the ability to bind the electrons. Namely, the electronegativity of dopants can be utilized to estimate the feasibility of dopant modi cation.

Conclusions
To understand why the phase structure of KNN differs from PZT, even though they are both constituted by the antiferroelectric and ferroelectric, the First-principles calculations of KNN and PZT as well as their components-KN and NN, PT and PZ, are conducted. The results suggest that the R-T phase boundary which is demonstrated to facilitate the property enhancement of perovskite piezoceramic is generated by the hybridization of A-site ion and relatively weakened hybridization of B-site ion. Besides, the electronegativity represents the ability of hybridization to some extent. Therefore, the electronegativity is an important consideration for the options of dopants or end members.
Besides, the contents and other characteristics of dopants, such as ionic radius and valence, are also taken into consideration to construct the R-T phase boundary and obtain promoted piezoelectric properties of KNN.    The electron densities of (a1) KNN and (a2) PZT; (b) the schematic of lattice deformation as well as a phase transition of PZT