First-principles investigation of the ferroelectric, piezoelectric and nonlinear optical properties of LiNbO3-type ZnTiO3

The newly synthesized LN-type ZnTiO3 (J. Am. Chem. Soc. 2014, 136, 2748) contains cations with the electronic configurations nd10 (Zn2+: 3d10) along with second-order Jahn-Teller (SOJT) nd0 (Ti4+: 3d0) cations. This is different from traditional ferroelectrics with the electric configurations of d0 transition metal ions or/and lone pair electrons of ns2. Using a first-principles approach based on density functional theory, we investigate the electronic structure, zone-center phonon modes, piezoelectric and nonlinear optical properties of the LiNbO3-type ZnTiO3. The electronic structure indicates that this compound is a wide direct-band-gap insulator. The results reveal that this compound is a good ferroelectric material with a large spontaneous polarization of 90.43μC/cm2. The Raman scattering peaks of A1 and E modes are assigned to their zone-center optical modes. Additionally, the large piezoelectric and nonlinear optical susceptibilities reveal that LiNbO3-type ZnTiO3 is a high-performance lead-free piezoelectric and nonlinear optical crystal.

structure and properties. First-principles calculations including the modern theory of polarization 25,26 , linear corresponding approaches, as well as the density functional perturbation theory (DFPT) [27][28][29] , can help us study the ferroelectric, piezoelectric, and nonlinear optical properties of materials. In addition to traditional ferroelectrics, first-principles calculations have made great progress in predicting and investigating new nonlinear optical crystal 30,31 , multiferroic materials 32,33 , two-dimensional ferroelectrics 34 and ferroelectric metal 35,36 , etc. In order to study the intrinsic correlation from the paraelectric to ferroelectric phase transition, we have investigated the zone-center phonon modes of the ilmenite, perovskite and LiNbO 3 -type ZnTiO 3 phases, respectively. Additionally, there have been no prior reports on the piezoelectric and nonlinear optical properties, including nonlinear optical susceptibilities and electro-optical coefficients, of LN-type ZnTiO 3 , and so in this paper we carry out first-principles calculations to investigate the origin of the ferroelectric behavior, and analyze the piezoelectric and nonlinear optical properties of LN-type ZnTiO 3 for the first time. Since the piezoelectric and nonlinear optical properties of this type of novel material with nd 10 and SOJT nd 0 cations have not been investigated previously, our calculations provide an important complement to experimental research.

Method
Our calculations are based on the framework of density functional theory (DFT). We mainly used the ABINIT package 37,38 . In order to calculate the electronic structure, spontaneous polarization, zone-center phonon modes, piezoelectric and nonlinear optical properties, we adopt norm-conserving pseudopotentials based on the local density approximation (LDA) as exchange-correlation potentials. An 8 × 8 × 8 Monkhorst-Pack k-point mesh is used with a plane wave cut-off of 45 hartrees. The Zn 3d, 4s and Ti 3d, 4s, as well as O 2s, 2p electrons are regarded as valence states to construct the pseudopotentials. The electronic structures are also calculated by adopting ultrasoft pseudopotentials 39 based on the Vienna ab initio simulation package (VASP) [40][41][42] , and the exchange-correlation potentials are based on LDA and the general gradient approximation (GGA) with Perdew-Burke-Ernzerhof (PBE) functional 43,44 , respectively. Additionally, due to DFT systematically underestimate band gaps with respect to experiment, the Heyd-Scuseria-Ernzerhof (HSE) screened hybrid functional (with the version HSE06) 45,46 is used to calculated the band structures.
The calculations of the polarization and piezoelectric properties are based on linear response functions, while the computations of nonlinear optical properties are based on nonlinear response functions. The linear and nonlinear response calculations are performed using density functional perturbation theory (DFPT). In order to calculate the above physical properties, the homogeneous electric fields, atomic displacements and stresses are regarded as perturbations. When stress is applied to a piezoelectric crystal, its interior will produce a polarization phenomenon where opposing positive and negative charges appear on its two opposite surfaces. This phenomenon is called the positive piezoelectric effect. On the contrary, when the electric field is applied in the polarization direction of the piezoelectric crystal, it will also deform. After the electric field is removed, the deformation of the piezoelectric crystal disappears. This phenomenon is called the reverse piezoelectric effect. In this work, we calculate the piezoelectric stress tensor from the zero-field derivative of polarization by using stress as a perturbation: where η and i represent second-rank stress tensor and the Cartesian coordinate direction. In the actual computation, the piezoelectric stress tensor comes from the response from fixed ions and relative displacement of the ions: where μ represents the displacement of the ions. For an insulator, the relation between the polarization and macroscopic electric field can be described as follows: where P i s represents spontaneous polarization in zero electric field; x ij (1) are linear dielectric constants, and x ijl (2) are second-order nonlinear optical (NLO) coefficients. The NLO and electro-optic (EO) coefficients can be calculated from third-order energy derivatives within the 2n + 1 theorem 47 . Like LiNbO 3 and LN-type ZnSnO 3 , we use the more convenient d tensor to represent NLO coefficients = ( ) (2) . When applying an electric field to some crystals, especially piezoelectric crystals, their refractive index changes. This characteristic of a crystal can be described by refractive index ellipsoids. The phenomenon is known as the linear electro-optic (EO) effect or Pockels Effect: where n and r ij represent refractive index and linear EO coefficients (i = 1~6, and j = 1~3), respectively. Pockels effect exists only in the NCS crystals. The electro-optical (EO) tensor arises from the sum of three contributions: the electronic part, ionic contribution and piezoelectric effects 48,49 .

Results and Discussions
electronic structure. Firstly, we perform the full geometry optimization to determine the crystal parameters in the paraelectric and ferroelectric phases. According to the optimized geometry (see Table 1), we have performed the electronic structure calculations of the LiNbO 3 -type ZnTiO 3 based on ultrasoft and norm-conserving pseudopotentials and HSE06, respectively. Figures 3 and 4 plots the energy band structure of this compound along the high symmetry points in the first Brillouin zone based on norm-conserving pseudopotentials within GGA and HSE06. When calculating the energy band based on HSE06, more high symmetry points in the first Brillouin zone are used. From Fig. 3, it is found that the top of the valence band (VB) and the bottom of the conduction band (CB) are both located at Γ(0.0, 0.0, 0.0), so this compound is a direct band gap insulator. The calculated energy gaps (e g ) are as follows: 3.054 eV (LDA) and 3.252 eV (GGA) based on norm-conserving pseudopotentials; 2.860 eV (LDA) and 2.956 eV (GGA) based on ultrasoft pseudopotentials. Therefore, the energy gap of norm-conserving pseudopotentials is slightly larger than that of ultrasoft pseudopotentials. From Fig. 4, it is found that the e g based on HSE06 (see in Fig. 4) are 4.25 eV. So, the band gap based on HSE06 is the largest Expt ref. 15   and is about 1 eV higher than that of GGA based on norm-conserving pseudopotentials. From Table 1, one can see that the lattice parameters of LDA and GGA are slightly lower and higher than the experimental values. The difference in the band gaps between GGA and LDA result from the difference in calculated lattice parameters. Compared with the results of LDA and GGA, the lattice parameters based on HSE06 are the lowest, however the band gap is the largest. Until now,as far as we know there is no experimental report on the band gap of LN-type ZnTiO 3 . Although the band gap of HSE06 is relatively larger than that of LDA and GGA, the agreement with the experimental band gap needs further verification. Figure 5 shows the total and partial density of states (DOSs) of LN-type ZnTiO 3 based on norm-conserving pseudopotentials within GGA. According to this figure, in the energy region from −18 to −16 eV, the total DOS mainly arises from O 2p electrons, with a small contribution from Zn 4s electrons and Ti 3p, 3d and 4s electrons.    Spontaneous polarization. The most basic characteristics of ferroelectric materials is the existence of spontaneous polarization over certain temperature ranges, and the fact that the polarization can be reversed by the external electric field. Using the Berry-phase approach proposed by R. D. King-Smith and D. Vanderbilt 25,26 , we have calculated the spontaneous electric polarization of LN-type ZnTiO 3 based on finite electric field calculations. The results show that the polarizations of P x and P y are almost equal to zero, so the polarization lies mainly along the z-axis direction. The total polarization is composed of the ionic (P ion ) and electronic (P ele ) contributions. The results based on LDA show that the total polarization is 90.43 μC/cm 2 , and the values of P ion and P ele are 89.07 μC/cm 2 and 1.36 μC/cm 2 , respectively. Therefore, the electric polarization mainly arises from the ionic contribution. Based on the result of our GGA calculation, the total polarization is 93.14 μC/cm 2 , the vales of P ion and electronic P ele are 91.73 μC/cm 2 and 1.41 μC/cm 2 . According to ref. 14 , the values of polarization obtained by Born effective charges and nominal charge are 88 μC/cm 2 and 75 μC/cm 2 . Our results are slightly larger than their values and consistent with their results based on Born effective charges.
The Born effective charges of the LN-type ZnTiO 3 are presented in Table 2. The nominal charges of Zn, Ti and O are +2, +4 and −2, respectively. By comparing the Born effective charges of these ions and their nominal values, it is found that charge numbers Z * of Zn, Ti and O (especially for Z xx * and Z yy * in Ti atom, Z yy * in O1, and Z xx * in O2 atoms) are obvious anomalous, which is an important characteristic in ferroelectric materials. From Table 2, one can see that Z * of Zn and Ti atoms mainly occur on the diagonal elements, and the values of the non-diagonal elements are small or equal to zero. This is in consistent with the spontaneous polarization being mainly along the z-axis direction. Compared with Z * of Zn and Ti atoms, the Z * tensor of O atoms displays strong anisotropy, and this should be attributed to the structural distortions caused by the Zn 3d-O 2p and Ti 3d-O 2p orbital hybridizations.
Zone-center phonon modes. In ferroelectric crystals that have undergone the paraelectric to ferroelectric transition, the relative displacements of anions and cations will lower the symmetry and lead to spontaneous polarization. This phenomenon can be explained by the theory of soft zone-center phonon modes. Firstly, we perform the symmetry analyses in ZnTiO 3 and divide its zone-center phonon modes into irreducible representations. The two paraelectric phases of ZnTiO 3 are ilmenite (IL)-type (hexagonal space group R3) and Perovskite (Pv)-type (orthorhombic space group Pnma) structures, and the ferroelectric phase is LiNbO 3 -type structure (hexagonal space group R3c). In this compound, there exists a continuous phase transition from ilmenite-type structure to perovskite-type, and then to LiNbO 3 -type structure.
In high-symmetry IL-type ZnTiO 3 (space group R3), there are 10 atoms in the primitive cell. All the 30 modes (including 3 acoustic modes) are composed of 4 irreducible representations (irreps).
In these modes, the three acoustic modes are composed of one A u and two E u * modes. In the optical modes, A g and E g * are Raman active, while A u and E u are infrared (IR) active. The corresponding zone-center optical phonon modes are presented in Table 3. In its perovskite-type paraelectric phase (space group Pnma), there are 20 atoms in the primitive cell. All the 60 phonon modes at Г point (including three acoustic modes) are composed of 8 irreps.  the corresponding zone-center optical phonon modes in Table 4. From this table, it is found that there are two imaginary frequencies in B 2u (124.12i) and B 3u (101.57i), respectively. However, in its high-symmetry IL-type ZnTiO 3 , according to Table 3, it is found that there are no imaginary frequencies in the zone-center phonon optical modes; additionally, we also calculated its phonon dispersion spectrum, and it is found that there are no imaginary frequencies in these phonon modes. In order to verify the stability of the R3c and R-3 structures, we also carried out the phonon dispersion relationship based on GGA. Figures 6 and 7      www.nature.com/scientificreports www.nature.com/scientificreports/ −8321.0790 (R3c) eV within GGA. The results show that the energy of its IL-type paraelectric phase is the lowest, the energy of Pv-type paraelectric phase is the highest, and the energy of LN-type ferroelectric phase lies in the middle. In order to verify whether it is due to the inaccuracy of the exchange correlation, we have carried out total energy calculation based on PBEsol, which is intended to improve on PBE for equilibrium properties such as bond lengths and lattice parameters. The calculated lattice parameters of PBEsol are also presented in Table 1. The results show that total energies are −8302.8974 (R3), −8302.3911 (pnma) and −8302.5810 (R3c) eV based on PBEsol. This is consistent with the above conclusion E(R3) < E(R3c) < E(pnma). This should be attributed to the synthesis of LN-type ZnTiO3 under high pressure.
In the ferroelectric phase (space group R3c) in ZnTiO 3 , the primitive cell contains 10 atoms, so there are 30 phonon modes at the Г point. The three zero-frequency acoustic modes comprise one in A 1 and two in E. The zone-center phonon optical modes are as follows: 1 2 In these modes, A 1 and E optical modes are both Raman and infrared (IR) active, while A 2 modes are silent. The optical modes of the LN-type ZnTiO 3 (see in Table 5) infer that the imaginary frequencies disappear from perovskite-type paraelectric phase to LiNO 3 -type ferroelectric phase. Due to A1 and E optical modes being IR active, these frequencies are splitting into LO and TO modes described by the Lyddane-Sachs-Teller (LST) relation 50 . Generally, the large Bern effective charge corresponds to a large LO-TO splitting. From    www.nature.com/scientificreports www.nature.com/scientificreports/ that C 66 = (C 11 − C 12 )/2. In Table 7, we give the corresponding elastic stiffness coefficients (relaxed ion) of LN-type ZnTiO 3 . The point group of LN-type ZnTiO 3 belongs to the triangle crystal. As discussed in refs. 51,52 , the triangle crystals need to satisfy the following Born mechanical stability criterion. The results presented in Table 7 reveal that the elastic constants of the LN-type ZnTiO 3 satisfy the constraints of the Born stability condition. Therefore, the structure of LN-type ZnTiO 3 is stable. By using homogeneous strains and electric fields as perturbations, we can calculate the piezoelectric properties of LN-type ZnTiO 3 . The obtained piezoelectric tensor of this compound has the four independent elements (Voigt notations) e 11  Obviously, this tensor is the same as that of LN-type ZnSnO 3 and ZnGeO 3 , however it is a little different from that of LiNbO 3 and LiTaO 3 since independent elements of the piezoelectric tensor in LiNbO 3 and LiTaO 3 are e 22 , e 15 , e 31 , and e 33 . In Table 8   www.nature.com/scientificreports www.nature.com/scientificreports/ together with LiNbO 3 and LiTaO 3 30,53 . From this table, the piezoelectric constants e 11 , e 15 , e 31 , e 33 are −0.93, 1.00, 1.01 and 2.51 C/m 2 , respectively. It is well known that lead titanate (PbTiO 3 ) piezoelectric ceramics have excellent piezoelectric properties 54,55 . By comparing the piezoelectric coefficients of LN-type ZnTiO and that of PbTiO 3 55 calculated by density functional pertubation theory and finite strain method (see Table 8), it is found that values of e 31 , e 33 of LN-type ZnTiO 3 are close to and exceeds one-half of that of PbTiO 3, and the value of e 15 is relatively small compared to that of PbTiO 3 . The large piezoelectric constants reveal that LN-type ZnTiO 3 is a promising candidate piezoelectric material. nonlinear optical properties. As mentioned above, the zone-center phonon optical modes of ferroelectric phase ZnTiO 3 can be divided into 4A 1 ⊕ 5A 2 ⊕ 9E, and A 1 and E modes are both Raman and infrared active. The Raman scattering efficiencies of the phonon modes can be calculated using the following formula 56,57 : where α m , c,  and n m are the Raman susceptibility, the speed of light in vacuum, the Planck constant and Bose-Einstein factor, respectively; ω 0 and ω m are the frequencies of an incoming photon and the mth zone-center phonon mode, the frequency of scattered outgoing photon is (ω 0 − ω m ), e s and e 0 indicate incoming and outgoing polarizations within an angle Ω. According to the structural symmetry, the Raman tensors of A 1 mode (along z axis) and E mode (in the x-y plane) of LN-type ZnTiO 3 are similar to that of LiNbO 3 and described as follows: Figure 8 displays the calculated Raman spectrum of LiNbO 3 -type ZnTiO 3 for the x(zz)y scattering configuration. In this configuration, as in the case of LiNbO 3 or PbTiO 3 , only the transverse A 1 modes can be detected. In order to see the peaks of TO1 and TO3 clearly, we have made two enlarged small illustrations in this figure. The results show that TO2 and TO4 modes have a very strong Raman scattering efficiency. In comparison, the Raman scattering efficiencies of TO1 and TO3 are very weak and are much smaller than that of the TO2 and TO4 modes. As presented in Tables 5 and Table 6, notice that TO2 and TO4 of A 1 modes have large LO-TO splittings and oscillator strengths, and the two frequencies just correspond to large Raman peaks. Figure 9 presents the calculated Raman spectrum of LiNbO 3 -type ZnTiO 3 for the x(yz)y configuration. In this configuration, the same as LiNbO 3 , only the TO and LO of E modes can be detected. From this figure, it is found that five transverse modes (TO3,  TO4, TO6 and TO7 and TO8)   www.nature.com/scientificreports www.nature.com/scientificreports/ The LN-type ZnTiO 3 belongs to the 3 m point group, therefore this compound is a candidate promising nonlinear optical material. As discussed by Inaguma et al. 14 , the SHG response of LN-type ZnTiO 3 is 24 times greater than that of LN-type ZnSnO 3 and has about 5% of the SHG response of LiNbO 3 . By using electric fields and atomic displacements as perturbations based on DFPT, we have calculated the nonlinear optical (NLO) susceptibilities and electro-optic (EO) coefficients of LN-type ZnTiO 3 . According to reports by M. Veithen, et al. 48 , the NLO properties of various semiconductors, such as LiNbO 3 , PbTiO3, and AlAs, etc. were calculated by DFPT based on the ABINIT package. The comparation between the theoretical results of NLO and EO coefficients with the experimental data shows that the method is reliable. The calculated results show that the three independent elements of the NLO tensor of this compound are d 12 , d 15 and d 33 (Voigt notations). The NLO tensor form of LN-type ZnTiO 3 is the same as that of LN-type ZnSnO 3 and ZnGeO 3 , but it is not the same as that of LiNbO 3 and LiTaO 3 since the independent elements of NLO tensor of the latter are d 22 , d 31 , d 33 . The calculated NLO susceptibilities of LN-type ZnTiO 3 together with those of LN-type ZnGeO 3 43 , LN-type ZnSnO 3 31 , LiNbO 3 58 and LiTaO 3 59 are presented in Table 9. From this table, the NLO susceptibilities d 12 , d 15 and d 33 are 1.37, 1.46 and −20.18 Pm/V, respectively, and one can see that the biggest susceptibility (d 33 ) of this compound is larger than that of ZnSnO 3 , and lower than that of LiNbO 3 . For the electro-optic (EO) coefficients of LN-type ZnTiO 3 , the results reveal that its independent elements are γ 11 , γ 13 , γ 33 , and γ 51 .  www.nature.com/scientificreports www.nature.com/scientificreports/ As can be seen, the EO tensor is the same as that of LN-type ZnSnO 3 and ZnGeO 3, but is different from that of LiNbO 3 and LiTaO 3 since the independent elements of latter are γ 13 , γ 33 , γ 22 and γ 51 . The EO coefficients can be divided into three contributing parts: electronic, ionic and piezoelectric. In Table 10, we give the calculated EO coefficients of LN-type ZnTiO 3 . For ease of comparison, the EO coefficients of LN-type ZnGeO 3 , LN-type ZnSnO 3 , LiNbO 3 60 and LiTaO 3 61 are also listed in this table. Our obtained EO coefficients γ 11 , γ 13 , γ 33 , and γ 51 for LN-type ZnTiO 3 are 0.46, 3.71, 17.17 and 1.62 Pm/V, respectively. Obviously, the coefficients γ 13 and γ 33 are much larger than that of LN-type ZnSnO 3 and ZnGeO 3 . The large NLO susceptibilities and EO coefficients reveal that the LN-type ZnTiO 3 is a high performance lead-free nonlinear optical crystal.

conclusions
In this work, the electronic structure, zone-center phonon modes, piezoelectric and nonlinear optical properties of LN-type ZnTiO 3 are investigated by first-principles calculations based on DFT. The electronic structure shows that this compound is a wide direct-band-gap insulator. By investigating the zone-center phonon modes of the paraelectric and ferroelectric phases, it is found that there are two imaginary frequencies in B 2u and B 3u in the perovskite paraelectric phase, and subsequently the imaginary frequencies disappear in the ferroelectric phase. The calculated spontaneous polarizations are 90.43 μC/cm 2 and 93.14 μC/cm 2 based on LDA and GGA, respectively. Our results are in good agreement with the experimental results, and the large spontaneous polarization reveals that this compound is a good ferroelectric material.
The elastic constants of LN-type ZnTiO 3 satisfy the constraints of the Born stability condition, and therefore this compound has a stable structure. The obtained piezoelectric tensor has four independent elements e 11 , e 15 , e 31 , e 33, with values −0.93, 1.00, 1.01 and 2.51 C/m 2 . This shows that this compound is a promising piezoelectric crystal. Like LiNbO 3 , the calculated Raman spectrum for the x(zz)y and x(yz)y configurations correspond to A 1 and E modes, and the Raman scattering peaks of A 1 and E modes are assigned to their zone-center optical modes, respectively. The independent nonlinear optical susceptibilities of this compound are d 12 , d 15 and d 33 with values 1.37, 1.46 and −20.18 Pm/V, respectively. For the EO coefficients, the independent elements are γ 11 , γ 13 , γ 33 , and γ 51 with values 0.46, 3.71, 17.17 and 1.62 Pm/V, respectively. The results show that LN-type ZnTiO 3 exhibits better nonlinear optical properties than LN-type ZnSnO 3 . The large piezoelectric and nonlinear optical susceptibilities reveal that this compound is a high-performance lead-free piezoelectric and nonlinear optical crystal.