Prediction of Ferroelectric Behavior for PZT and Pb-Substituted Materials

In this study, first principles calculations based on density functional formalism were used to evaluate optimized structures, spontaneous polarizations, and effective charges for Sn(Zr0.5,Ti0.5)O3 (SZT), Ge(Zr0.5,Ti0.5)O3 (GZT), (Pb0.875,Sn0.125)(Zr0.5,Ti0.5)O3 (PSZT), and (Pb0.875,Ge0.125)(Zr0.5,Ti0.5)O3 (PGZT); the results were compared with corresponding values calculated for Pb(Zr,Ti)O3 (PZT). The stability of the calculated crystal structures differed depending on whether all Pb atoms were replaced (100% substituted materials, SZT and GZT) or whether only 12.5% of Pb atoms were replaced (12.5% substituted materials, PSZT and PGZT): the structures of 100% substituted materials were compressed and unstable, while the 12.5% substituted materials maintained stable structures. Both the 100% and 12.5% substituted materials had values of spontaneous polarization nearly as large as that for PZT. For 100% substitutions, atomic displacements increased and effective charges decreased relative to values for PZT. For 12.5% substitutions, atomic displacements also increased, and the effective charge along the macroscopic polarization axis decreased due to changes in covalent bonding. We evaluated the piezoelectricity of the materials by calculating the barriers to change in polarization axes. For 100% substituted materials, the barriers were larger than that for PZT, while for 12.5% substituted materials, the barriers were smaller. Therefore, 100% substituted materials have weaker piezoelectric properties than PZT, but 12.5% substituted materials have stronger piezoelectric properties. We conclude that 12.5% substituted materials maintain the crystal structure of PZT and probably have better piezoelectric behavior than PZT. [DOI: 10.1380/ejssnt.2014.40]


I. INTRODUCTION
One of the most famous Pb-based perovskite material is lead zirconate titanate (PZT), which has been extensively investigated for piezoelectric applications [1].Because of its remarkable piezoelectric properties, PZT is the most suitable material for industrial use [2].Unfortunately, PZT contains a high concentration of lead oxide, which is believed to be a threat to human health and the environment.Therefore, the development of lead-free piezoelectric materials is of considerable interest.There are many challenges in developing lead-free piezoelectric materials [3][4][5][6][7][8], and the best properties of recently studied candidates are not as good as those of PZT [9].
We know that to make a highly piezoelectric material the substance should have a high effective charge on the covalent bond between metal atoms and oxygen [10].When atoms are involved in covalent bonding, displacement of the atomic core induces electron movement.Accordingly, the Born effective charge consists not only of the atomic core contribution but also an electronic contribution.Normally, metal atoms and oxygen form ionic bonds; however, in PZT, the lead s-orbital and oxygen p-orbital form an effective covalent bond in addition to an ionic bond [11].These effective bonds make PZT a good piezoelectric material.Because we can identify those atoms having an s-orbital valence that can form a covalent bond with oxygen, we can also identify materials that should have good piezoelectric properties.For example, PbTiO 3 is a possible candidate, but it has the same drawbacks as PZT.
Some previous researchers have predicted that substituting Sn for Pb might be an effective way to decrease lead content [12], and this strategy has been tested experimentally.However, those experiments found that this substituted material assumes an ilmenite structure [13].Based on such previous work, we substitutionat 100% substitutions for Pb will change the crystal structure.To achieve crystal stability, some previous theoretical research has focused on increasing the number of different substituted elements.These efforts showed excellent results and some of the resulting materials are lead-free.However, these materials have complicated chemical formulae and crystal structures compared to PZT, so they are difficult to synthesize.The required crystalline stability depends on the radius of the Pb atom.Hence, we expect Sn would be a better candidate for substitution than Ge; however, Ge has an electronic structure similar to Pb.Therefore, we should investigate the possibility of substituting Ge for Pb.
In this research, we focus on simple chemical formulae and substitutions.In addition, substituted atoms with s and p valence orbitals have high effective charges and strong piezoelectric properties in the z-direction.However, these materials are unstable [13] and their properties do not change in the x-and y-directions.However, the field of applications is expanded if the material has superior properties in more than one direction.
In this study, we investigated the possibilities of making lead-free piezoelectric materials or, at least, materials containing decreased amounts of lead.First principles calculations were used to determine the structures, densities of states (DOS), and Born effective charges of the following tetragonal materials: PZT, Sn(Zr 0.5 ,Ti 0.5 )O 3 (SZT), Ge(Zr 0.5 ,Ti 0.5 )O 3 (GZT), (Pb 0.875 ,Sn 0.125 )(Zr 0.5 ,Ti 0.5 )O 3 (PSZT), and (Pb 0.875 ,Ge 0.125 )(Zr 0.5 ,Ti 0.5 )O 3 (PGZT).For materials in which 12.5% of the Pb was@replaced, we can expect that inversion symmetry was broken and property changes occurred in the x-and y-directions.

II. MODEL AND METHODS
We performed first principles calculations for Pb-based perovskite-type crystal structures (ABO 3 ) based on density functional theory using VASP and the Perdew-Burke-Ernzerhof generalized gradient approximation revised for solid the for the exchange correlation [14,15].We employed Blochl's projector augmented wave (PAW) method as implemented by Kresse and Joubert [16].The planewave cutoff energy was 600 eV.Spontaneous polarization and Born effective charges were calculated using the Berry phase method [17,18].The calculations were performed using grids of 7×7×7 and 7×7×17 for optimization and the Berry phase calculation, respectively.Figure 1 shows the unit cell used in this study.

A. Optimized structures
Results from structural optimization of tetragonal materials and displacements are summarized in Tables I and  II.Values for optimized structures and lattice parameters are satisfactory with respect to experiment data and previous first principles calculations [19,20].We found that displaced positions were more stable than inversion symmetry positions and that these materials had spontaneous polarization.For 100% substitutions, the lattice constant c along the z-axis increases with decreasing radius of the substituted atom, while the lattice constants a and b along the x-and y-axes decrease with decreasing ra-TABLE II: Optimized displacements and effective charges along the z-axis in 100% substitution materials (SZT and GZT) and 12.5% substitution materials (PSZT and PGZT).O1 is the oxygen atom under Ti (Zr) in the z-direction, and O2 is the oxygen atom under Ti (Zr) along the x-and y-directions.dius of substituted atoms (Table V).The displacements of metal and oxygen atoms are summarized in Table II.Distances between the A-site metal atom (Pb, Sn, Ge) and oxygen atom do not depend on the A-site atom.However, distances between the B-site metal atoms (Zr, Ti) and oxygen atom do depend on the A-site atom; these distances increase with increasing atomic radius of the Asite atom in the order Pb > Sn > Ge (Table V).This difference in tendency disturbs the octahedron composed of the B-site metal and six oxygen atoms, limiting movement in the narrow space around the B-site and making these crystals unstable.In addition to these restrictions in movement, small A-site atoms can move into an octahedron structure.Such movements contribute to the lack of stability of the perovskite material.In some experiments, some substituted atoms have been found to lead to pyrochore structures [3].We consider that our results explain this crystal transition from perovskite to pyrochore.The pyrochore phase in PZT shows paraelectric property and is extremely detrimental to dielectric and piezoelectric materials [21,22].Based on this lack of perovskite stability and pyrochore paraelectric property, we conclude that we cannot make these 100% substituted ferroelectric materials.Even if these crystals could be made, we could not easily control the ionic core positions using an external electric field.For 12.5% substitutions, the lattice constants a and c do not depend on the A-site substitution, and the size of the super cell is the same.Our optimization calculations show that the distance between the A-site metal and the oxygen atom do not change.In addition to this, the c/a  does not depend on the A site 12.5% substation.This means that ionic core movement does not break the octahedral structure and that the crystal structures of 12.5% substitutions are more stable than those for 100% substitution [23].

B. Electronic properties
Figure 2 shows the calculated density of states for PZT and PGZT.The densities of states show that these materials are insulators and covalent bonds form between oxygen p-orbitals and Ti (Zr) d-orbitals around −3.0 eV.In addition, these materials have a covalent bond between the oxygen p-orbital and s, p-orbitals of the A-site metal around −3.0 eV.Results for the Born effective charge and spontaneous polarization computed using the Berry phase method are shown in Tables II and III.Spontaneous polarization is proportional to atomic displacement multiplied by the effective charge, Here, k is an index over atoms, P sp is the spontaneous polarization, e is the elementary charge, c is the lattice constant along the z-axis, V is the volume of the unit cell, Z k zz is the k-th Born effective charge, u k is the k-th atom position, and u G is the center of mass.Normally, the Pb (Sn, Ge) valance is +2, but the effective charge is higher.This high effective charge originates from covalent bonds [24] between the metal s, p valence orbitals and the oxygen p-orbital.For 100% substitution, as the radius of a substituted atom decreases, the A-site metal moves closer to oxygen (Table V).According to the difference in electron negativity, metal electrons involved in covalent bonds are displaced toward oxygen: the metal atom loses electrons and the oxygen atom gains them.In other words, metal atoms and oxygen atoms become more ionic because of this movement of electrons.For this reason, the displacement increases and the effective charge decreases.Spontaneous polarization does not depend on A-site atom substitution because it is proportional to displacement multiplied by effective charge.Since the xand y-axes inversion symmetry is not broken, the oxygen http://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) e-Journal of Surface Science and Nanotechnology p x -and p y -orbitals do not form strong covalent bonds with metal atoms in the x-and y-directions; the valence electron concentrates along the z-axis and creates a high effective charge in this direction.From macroscopic and microscopic points of view, 100% substituted materials do not have spontaneous polarization along the x-and y-axes.
For 12.5% substituted materials, the size of the unit cell does not change, and the Ti (Zr) effective charge remains high.However, the effective charge of the substituted metal (Sn, Ge) decreases along the z-axis.This decrease is caused by a change in covalent bonding.For 12.5% substitution, the x-and y-axes inversion symmetry is broken (Table IV).Because of the change in A-site atomic radius, a small substituted atom leaves an empty space along the x-and y-directions around it (Table V).A small substituted atom gathers oxygen atoms around it and forms a covalent bond with these atoms in the xand y-directions.From a microscopic point of view, a dipole moment forms that involves the substituted metal (Sn, Ge) positive ion and oxygen negative ion; the macroscopic polarization along the x-and y-axes is equal to zero because the oxygen atoms move symmetrically (center of symmetric movement is the substituted atom).Because of this displacement and the changes in covalent bonding, the number of electrons involved in covalent bonds along the z-axis decreases.In fact, we can count the number of electrons from the density of states.The sum of Pb s, p z orbitals is 0.62 electrons between −5.00 eV and 0.00 eV (Fermi energy) and the sum of p x orbitals is 0.11 electrons in PZT.In contrast, in PGZT, the sum of Ge s, p z orbitals is 0.52 electrons between −5.00 eV and 0.00 eV (Fermi energy) and the sum of p x orbitals is 0.22 electrons.This low effective charge along the z-axis leads to weak spontaneous polarization.

C. Piezoelectric properties
To evaluate the ferroelectric and piezoelectric properties [25], we calculated the activation barrier to change in the spontaneous polarization axis from [001] to [00 1] and from [001] to [010].Polarization changes cause lattice constants c/a change and transformation and the crystal transformation will product mechanical force.In other words, a small activation barrier means that we can get mechanical force with a small electric field.Changes in the ratio of lattice constants c/a indicate crystal transformation.Every point in Fig. 3 and Fig. 4 was optimized about atom positions with the different ratio of lattice constants c/a and the same unit cell volume.The calculated values are shown in Figs. 3 and 4. The big c/a value corresponds the big polarization value.The most unstable point means a tetragonal structure when the c/a is equal to 1. Figures 3 and 4 show line symmetry.It comes from that the cases of c/a and a/c are the same unit cells, just axes changed.The result shows that activation barriers from [001] to [00 1] increase for 100% substitution.This is caused by the magnitude of the ratio of lattice constants c/a (Table V).For 12.5% substitution, activation barriers decrease with decreasing radius of substituted atoms.The reason for this trend is the weak covalent bonds.The substituted atom can move more easily than Pb because of the weak constraint that comes from the long distances between oxygen atoms.Oxygen atoms have the same trend for the same reason.The activation barriers show the same trend for the transition from [001] to [010].For 100% substitution, activation barriers increase with decreasing atomic radius because of the c/a trend (Table V).In contrast, for 12.5% substitution, activation barriers decrease with decreasing atomic radius (Table V).In addition to weak covalent bonds, small substituted atoms leave enough empty space that other atoms can occupy.This mechanism is the same as that which causes high ionic conductivity with a dopant.A small activation barrier means that we can change the polarization with a small electric field.These small activation barriers are necessary to achieve good piezoelectricity, so we can say that 12.5% substitution produces a better piezoelectric material than PZT.

IV. CONCLUSIONS
We have used first principles calculations to obtain lattice parameters and spontaneous polarizations for PZT and Pb-substituted materials.The values for lattice parameters and atomic positions indicate that 100% substitution of Pb leads to unstable crystals while materials with 12.5% substitution remain stable.For the materials considered in this study, spontaneous polarization does not strongly depend on substitution.This is because changes in effective charge and in displacement are opposite and tend to compensate for one another.Partial replacement appears to be a good method for maintaining piezoelectric behavior while decreasing lead content.In particular, 12.5% substitution sustains piezoelectricity because of the formation of weak covalent bonds between metal atoms and oxygen as well as because empty space becomes available around small substituted metal atoms.

FIG. 1 :
FIG. 1: Unit cell used in the first principles calculations.Large black spheres represent Pb and small purple ones indicate Sn or Ge.Also, red is O, blue is Zr, and green is Ti.

FIG. 2 :
FIG. 2: Partial density of states (PDOS) for (a) PZT and (b9 PGZT.Horizontal axis shows energy with the Fermi energy (EF) used as the zero point (eV).Vertical axis shows the number of states per unit energy (/eV).

FIG. 3 :
FIG. 3: Activation barriers for changes in the spontaneous polarization axis from [001] to [00 1].Horizontal axis contains the ratio of lattice constants c/a.Vertical axis shows the energy (eV) of the activation barrier.In the calculation for each point, the volume of the unit cell was fixed.

TABLE III :
Calculated values of spontaneous polarization for PZT, 100% substituted, and 12.5% substituted materials.

TABLE IV :
Displacement of oxygen atoms (in fractional coordinates) about Sn in PSZT and about Ge in PGZT (12.5% substituted materials).O1 denotes the same oxygen atoms as in TableII.

TABLE V :
Electron negativity and van der Waals radii about each element.