Ferroelectric superlattices have been widely studied due to the unusual and controllable properties which give them much potential in micro-electro-mechanical system (MEMS) domain for sensing or energy harvesting.^[1,2] In particular, perovskite structures containing PbTiO₃ component, which shows better piezoelectric response, mainly include PZT,^[3,4] PMN-PT,^[5] PIN-PMN-PT,^[6] and some PbTiO₃-based superlattices.^[7–9] In recent years, Niobate ceramics^[10,11] such as NaNbO₃ and KNbO₃, have been studied as promising Pb-free piezoelectric ceramics. Lanthanum aluminate (LaAlO₃),^[12,13] extensively used as the substrate material for the growth of functional thin films, is a popular target of fundamental investigation, since it exhibits a wide variety of physical and mechanical properties when the applied conditions are changed.

The ferroelectric oxide superlattices present an exciting opportunity for developing novel materials with extraordinary properties; also they are instrumental in studying the fundamental physics of ferroelectric materials. In the studies of complex solid–solution perovskite materials, cation compositional change and cation ordering,^[14,15] are the primary tool for designing these materials and a firm understanding of how individual cations contribute to the overall piezoelectric response would help the rational design of high-performance piezoelectric materials. Therefore, studying the contributions of the individual cations to the total piezoelectric response of such superlattices is very important.

In this paper, we change out-plane lattice constant to apply axial strains, noted as Δc, to investigate the polarization and piezoelectric behaviors of the tetragonal PbTiO₃/KNbO₃ and PbTiO₃/LaAlO₃ superlattices. The associated atomic displacements and Born effective charges are also calculated to explain the origin of the polarization and piezoelectric behaviors. The investigation in this paper can provide theoretical value for the comprehensive understanding piezoelectric response of PbTiO₃-based superlattices and research of the high-performance piezoelectric materials.

We perform first-principles calculations, using pseudopotential framework of density functional theory (DFT)^[16,17] for the tetragonal PbTiO₃/KNbO₃ and PbTiO₃/LaAlO₃ superlattices. All calculations use projector augmented wave (PAW) potential^[18] with the Vienna ab initio simulation package (VASP),^[19–21] and choose the local density approximation (LDA) for the exchange correlation functional. The positions of the ions were relaxed toward equilibrium until the Hellmann–Feynman forces became less than 0.001 eV/Å. A 6 × 6 × 3 Monkhorst–Pack k-point mesh^[22] and 500-eV plane wave cutoff were given to achieve good convergence of computed ground state properties.

As shown in Fig. 1, these are tetragonal structures of PbTiO₃/KNbO₃ and PbTiO₃/LaAlO₃ superlattices at the ground state, which consist of B cations at the centers of the oxygen coordinate octahedra and the A cations in the spaces (coordination 12) between the octahedra. The equilibrium structures of the tetragonal PbTiO₃/KNbO₃ and PbTiO₃/LaAlO₃ superlattices at the ground state are obtained through the constrained structural optimization. In order to study polarization and piezoelectricity behaviors of the individual cations in the tetragonal PbTiO₃/KNbO₃ and PbTiO₃/LaAlO₃ superlattices accurately, a relatively small Δc (−0.15 Å~ + 0.15 Å) with an interval of 0.025 Å is used; and when Δc changes from −0.15 Å to + 0.15 Å, representing out-plane strain varying from compressive state to tensile state. By applying the strain along the c axis from compressive strain to tensile strain, we conduct the constrained structural optimization until the stress tensor (i.e., σ₁₁ and σ₂₂) are all smaller than 0.01 GPa. Then, the corresponding polarization and Born effective charges were calculated by the Berry phase method^[23] and density functional perturbation theory, respectively.

Fig. 1.

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	Fig. 1. (color online) The tetragonal structure of (a) PbTiO3/KNbO3 and (b) PbTiO3/LaAlO3 superlattices in the ground state.

The A-site and B-site polarization contributions are calculated by formulas and , where and are the Born effective charges calculated by the density functional perturbation theory, D(A) and D(B) are the off-center displacement of cations from the center of the corresponding O₁₂ cage and O₈ cage, and V is the volume of the superlattice. Then the contributions of A-site and B-site atoms to the total piezoelectricity d₃₃ can be calculated as d₃₃(A) = ∂P(A)/∂δ₃₃ and d₃₃(B) = δP(B)/δδ₃₃,^[24] where P denotes atomic polarization and δ₃₃ is the stress along the c axis of the superlattice.

The calculated lattice constants, Born effective charge, polarization and piezoelectricity d₃₃ of the tetragonal PbTiO₃/KNbO₃ and PbTiO₃/LaAlO₃ superlattices at the ground state are listed in Table 1. The in-plane lattice constant of the tetragonal superlattices is between the two individuals.^[25–27] The Born effective charge tensor characterizes the influence of long-range Coulomb interactions on the vibrational and optical properties.^[28] It is clear that of the A- and B-site atoms are relatively larger than their corresponding nominal ionic value, revealing the charge redistribution caused by the dynamic modification of orbital hybridizations.^[29,30] According to Table 1, equilibrium lattice constant a of tetragonal PbTiO₃/KNbO₃ and PbTiO₃/LaAlO₃ superlattices are 3.941 Å and 3.817 Å, so PbTiO₃ layer in two superlattices suffers from different strains, inducing different charge transfers between Ti and O atoms in PbTiO₃/KNbO₃ and PbTiO₃/LaAlO₃ superlattices, and there is different Z*(Ti) in these two superlattices. In PbTiO₃/KNbO₃, the charges of the B-site atoms (Ti, Nb) are surprisingly large in the sense that they reach about twice the value they would have in a pure ionic picture, indicating the strong covalent hybridization between O and the transition metal (Ti, Nb).^[31,32] Comparing to PbTiO₃/KNbO₃, (Pb) in PbTiO₃/LaAlO₃ increase a little, and (Ti) decreases a lot, showing that the strong covalent hybridization between O 2p and Ti 3d orbitals is weakened and the charge transfers between Pb and O increase. Besides, PbTiO₃/KNbO₃ shows better polarization and piezoelectric response than PbTiO₃/LaAlO₃.

Table 1.

Table 1.

Table 1. Lattice constants (in unit Å), effective charges (in unit e), polarization (in units C/m2), and d33 (in units pC/N) of the ground state of the tetragonal PbTiO3/KNbO3 and PbTiO3/LaAlO3 superlattices. . PbTiO3/KNbO3 PbTiO3/LaAlO3 a/Å 3.941[25,26] 3.817[27] c/Å 8.026 7.669 (A1)/e 2.612 3.068 (A2)/e 1.309 4.233 (B1)/e 8.165 4.872 (B2)/e 8.932 3.507 P 0.532 0.202 d33 44.594 13.944

Table 1. Lattice constants (in unit Å), effective charges (in unit e), polarization (in units C/m2), and d33 (in units pC/N) of the ground state of the tetragonal PbTiO3/KNbO3 and PbTiO3/LaAlO3 superlattices. .

As shown in Fig. 2, Born effective charges of the A- and B-site atoms versus c-axis length change (Δc) in superlattices are given. In PbTiO₃/KNbO₃ superlattice, of the B-site atoms (Ti and Nb atom) declines gradually and the of A-site atoms (Pb and K) changes insignificantly as Δc is varied from −0.15 Å to + 0.15 Å, which indicates strain induces charge transfer from B-site atoms to O atom and little charge transfer from A-site atoms to O atom. The effect of strain on charge transfer of A-site atoms and B-site atoms of PbTiO₃ layer is identical to that of KNbO₃ layer in PbTiO₃/KNbO₃ superlattice. In PbTiO₃/LaAlO₃ superlattice, with Δc changing from −0.15 Å to + 0.15 Å, (Ti) of B-site atoms and (Pb) of A-site atoms decrease in PbTiO₃ layer, while (La) of B-site atoms increases a little and (Al) of A-site atoms has little change in LaAlO₃ layer; showing that strain leads to charge transfer from A-site atom and B-site atom to O atom in PbTiO₃ layer, and charge transfer from A-atom to O atom in LaAlO₃ layer. The effect of strain on charge transfer in PbTiO₃ layer is different from that in LaAlO₃ layer.

Fig. 2.

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	Fig. 2. (color online) The Born effective charges in (a) PbTiO3/KNbO3 and (b) PbTiO3/LaAlO3 versus c-axis length change.

In Fig. 3, the displacements of the cations from the corresponding O₈ and O₁₂ cage center versus c-axis length change (Δc) were calculated in tetragonal PbTiO₃/KNbO₃ and PbTiO₃/LaAlO₃ superlattices. According to Fig. 3, it is found that, when strain changes from compressive state to tensile state, all atomic displacements from the corresponding O₈ and O₁₂ cage center increase gradually, and atomic displacement response to strain in PbTiO₃/KNbO₃ superlattices is stronger than that in PbTiO₃/LaAlO₃ superlattices. In PbTiO₃/KNbO₃ superlattices, compared with K and Nb atoms in KNbO₃ layer, Pb and Ti atoms in PbTiO₃ layer have larger displacement response to strain because of excellent piezoelectricity in PbTiO₃. In PbTiO₃/LaAlO₃ superlattices, atomic displacement response to strain of La and Al atoms in LaAlO₃ layer is smaller than that of Pb and Ti atoms in PbTiO₃ layer. In both superlattices, PbTiO₃ layer has the largest atomic displacement response to strain; KNbO₃ layer is middle, LaAlO₃ layer is smallest.

Fig. 3.

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	Fig. 3. (color online) The displacement from the center of the corresponding O12 cage and O8 cage in (a) PbTiO3/KNbO3 and (b) PbTiO3/LaAlO3 versus c-axis length change.

The polarization of each cation in PbTiO₃/KNbO₃ and PbTiO₃/LaAlO₃ superlattices is indicated in Fig. 4. The atomic polarization is calculated by formulas , where , D, and V are Born effective charge, the atomic displacement from the corresponding O₈ and O₁₂ cage in the axial direction, and volume V of the unit cell, respectively. As indicated in Fig. 4, the polarization of the PbTiO₃/KNbO₃ superlattice is larger than that of the PbTiO₃/LaAlO₃ superlattice, illustrating that the PbTiO₃/KNbO₃ superlattice may have better piezoelectricity than that of the PbTiO₃/LaAlO₃ superlattice. With Δc varying from −0.15 Å to + 0.15 Å, both total polarization of superlattices and polarization of each cations increase gradually, and polarization of PbTiO₃/KNbO₃ and PbTiO₃/LaAlO₃ superlattices raise from 0.42 C/m² to 0.63 C/m² and from 0.15 C/m² to 0.27 C/m², respectively. Polarization from B-site is significant because of the strong hybridization between Ti 3d (Nb 4d) orbitals and O 2p orbital, while the A-site atoms (Pb and K) have a relatively lower Born effective charge. Pb atom contributes more to the total polarization because displacement and Born effective charge of the Pb atom is larger than those of the K atom. The weak polarization contribution of K atom ascribes to its small displacement and Born effective charge. Besides, the rate of change of P(Ti) and P(Nb) is bigger than that of P(Pb)and P(K) in response to the c-axis strain. For the tetragonal PbTiO₃/KTaO₃ and PbTiO₃/LaAlO₃ superlattices, P(Pb,Ti) in the PbTiO₃ layer has larger contribution to the total polarization than P(K,Nb) in the KNbO₃ layer and P(La,Al) in the LaAlO₃ layer.

Fig. 4.

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	Fig. 4. (color online) The polarization distributions in (a) PbTiO3/KNbO3 and (b) PbTiO3/LaAlO3 versus c-axis length change.

Lastly, the axial stress σ₃₃ of PbTiO₃/KNbO₃ and PbTiO₃/LaAlO₃ versus c-axis length change (Δc) are calculated, as indicated in Fig. 5. It is shown that the variation trend of σ₃₃ versus c-axis strain is very similar. The value of σ₃₃ decreases slowly to around zero, then increases inversely. But for the same c-axis length change Δc, σ₃₃ in the PbTiO₃/KNbO₃ superlattice is smaller than that in the PbTiO₃/LaAlO₃ superlattice.

Fig. 5.

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	Fig. 5. (color online) The σ33 of (a) PbTiO3/KNbO3 and (b) PbTiO3/LaAlO3 versus c-axis length change.

Through formulas d₃₃(A) = d P(A)/d σ₃₃ and d₃₃(B) = d P(B)/d σ₃₃, we gain the piezoelectric response d₃₃ of each cation in PbTiO₃/KNbO₃ and PbTiO₃/LaAlO₃ superlattices in Fig. 6. As the c-axis length change (Δc) changes from −0.15 Å to + 0.15 Å, in two superlattices, both total d₃₃ and contributions from all cations increase continually, and the trend is more significant in the tensile state. In the PbTiO₃/KNbO₃ superlattice, there is a significant gap between the d₃₃(A) and d₃₃(B), and the total d₃₃ increases from 37.07 pC/N to 75.64 pC/N, and the piezoelectric contributions from B-site atoms (Ti, Nb) is larger than that from A-site atoms (Pb, K). In the PbTiO₃/LaAlO₃ superlattice, the total d₃₃ increases from 7.86 pC/N to 25.76 pC/N, and the piezoelectric contribution almost comes from Pb and Ti in the PbTiO₃ component, and the difference between d₃₃ (La) and d₃₃ (Al) is very minor. In particular, in the compressive c-axis condition, d₃₃ (Pb) contributes most to the total d₃₃. It is concluded that there is the largest piezoelectric contribution of PbTiO₃layer to the total d₃₃ and the smallest piezoelectric contribution of LaAlO₃ layer to the total d₃₃, which indicate that the total d₃₃ of PbTiO₃/KNbO₃ superlattice is bigger than that of the PbTiO₃/LaAlO₃ superlattice.

Fig. 6.

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	Fig. 6. (color online) The d33 piezoelectric distributions in (a) PbTiO3/KNbO3 and (b) PbTiO3/LaAlO3 versus c-axis length change.

In order to investigate the influence of the component PbTiO₃ to the piezoelectric response of PbTiO₃/KNbO₃ and PbTiO₃/LaAlO₃ superlattices, the proportion of the piezoelectric contributions from PbTiO₃ component in the superlattices is calculated, as shown in Fig. 7. The proportion is around 60% at the ground state of the tetragonal superlattices, and the proportion shows different responses to the c-axis strain but still with a value larger than 50%. In the PbTiO₃/KNbO₃ superlattice, as strains change from compressive state to tensile state, the proportion of PbTiO₃ layer first increases and then decreases, and reaches maximum near the equilibrium state. However, in PbTiO₃/LaAlO₃ superlattices, the c-axis strain increases the proportion, especially in the compressive state. In two superlattices, the proportion of the piezoelectric contributions from PbTiO₃ component in the PbTiO₃/LaAlO₃ superlattice is larger than that of the PbTiO₃/KNbO₃ superlattice, because of small piezoelectric contribution of La and Al atoms.

Fig. 7.

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	Fig. 7. (color online) The proportion of the piezoelectric contributions from PbTiO3 component in PbTiO3/KNbO3 and PbTiO3/LaAlO3 versus c-axis length change.

In this paper, we performed a first-principles study on the polarization and piezoelectric behaviors of the tetragonal PbTiO₃/KNbO₃ and PbTiO₃/LaAlO₃ as a function of c-axis length change. The contributions of the A-site and B-site atoms to polarization and the total d₃₃ are discussed. In PbTiO₃/KNbO₃ superlattices, the B-site atoms make relatively larger contributions to polarization and the total d₃₃ than the A-site atoms because of the strong covalent interactions between the transition metal (Ti, Nb) and the oxygen atoms, which is demonstrated by the large dynamic charge transfers between Nb 4d, Ti 3d orbitals, and O 2p orbitals. The A-site atom Pb has a relatively larger polarization and piezoelectric contribution than K due to more intensive interaction between Pb and O atoms and the huge offset displacement. Meanwhile in PbTiO₃/LaAlO₃ superlattices, contributions from the B-site atoms are weakened because of the weak contribution from Al atom, highlighting the influence from A-site atom Pb which has a huge offset displacement. By analyzing the proportion of the piezoelectric contributions from PbTiO₃ component in the superlattices, the influence from PbTiO₃ component is weakened in PbTiO₃/KNbO₃ as the c-axis strain is applied. However, in the PbTiO₃/LaAlO₃ superlattice, the PbTiO₃ component plays a leading role to the total d₃₃, especially in the compressive strains, and the proportion decreases as the tensile strains increase.