**Introduction**

**What is PZT?**

Pb(Zr_{1-X}Ti_{X})O_{3}, PZT, a perovskite material that shows a marked piezoelectric effect. Among all the compositions of PZT, Pb(Zr_{0.52}Ti_{0.48})O_{3}, which is near the morphotropic phase boundary (MPB), is the most important one because of its excellent ferroelectric and electro-optic properties. Due to its excellent properties, PZT-based compounds are used in the sensors and actuator, as well as high-value ceramic capacitors and DRAMs. In this term paper, I would like to introduce how to calculate the piezoelectric coefficient (e_{ij}) by the first-principles^{[1]}.

**Piezoelectricity**

Piezoelectric materials exhibit an effect whereby they expand or contract in the presence of an applied electric field. This “induced strain” or change in length occurs as electrical dipoles in the material rotate to align with an orientation that more closely aligns with the direction of the applied electric field. The change in length is generally proportional to the field strength as applied via the device actuation voltage. The relations for piezoelectric coefficients are

(1)Where D is electric charge density displacement, S is strain, and T is stress. Here, we focus on the piezoelectric tensor element e_{ij}^{[1]}.

Fig 1. PZT crystal structure

Fig 2. Phase diagram of the solid solution system PbZrO3-PbTiO3 ^{[2]}

**Paper review**

**Case 1. First-principles study of Pielectricity in PbTiO3 ^{[3]}**

First, let’s start from a easier system (PbTiO_{3}: an end member of PZT) to calculate the piezoelectric coefficient of Pb(Zr_{1-X}Ti_{X})O_{3}. This letter is by R. E. Cohen et al ^{[3]}. And PbTiO_{3} is one of the simplest ferroelectric oxides (ABO_{3} tetragonal 4mm) at low T, could be the starting point for understanding piezoelectricity in ferroelectrics.

In this Letter, the complete piezoelectric response and dynamical charge tensors are computed from first-principles within the density functional theory using the all-electron full-potential linearized augmented plane wave (LAPW) ^{[6]} method, and is the first application of modern polarization theory using the Berry’s phase ^{[4,5]}approach to compute piezoelectric response in a ferroelectric.

**How to calculate piezoelectric tensor by first-principles ^{[3]}**

1.

To understand how to calculate the piezoelectric tensor element (e_{ij}). First, we should know the relation between the macro polarization (P^{T}_{i}) of a strained sample. It can be expressed as

Where P_{i}^{S} is the spontaneous polarization of the unstrained sample, $\epsilon_{v}$ is the strain tensor element, and e_{iv} defines the piezoelectric tensor elements in Voigt notation. The three independent piezoelectric tensor components are e_{31} =e_{32} and e_{33}, which describe the zero field polarization induced along the z axis when the crystal is uniformly strained in the basal xy plane or along the z axis, respectively, and e_{15} =e_{24} which measures the change of polarization perpendicular to the z axis induced by shear strain.

The change in total macroscopic polarization, containing both **electronic** and **rigid ionic core** contributions, is a well defined bulk property and could be determined experimentally under shorted boundary conditions. **Therefore the total piezoelectric constant can be calculated from finite differences of polarizations (ΔP ^{T}) between crystals of different shapes and volumes.**

2.

The electronic part of the polarization was determined using the Berry’s phase approach^{[4,5]}, a quantum mechanical theorem dealing with a system coupled to a slowly changing environment. One can calculate the polarization difference between two states of the same solid, under the necessary condition that the crystal. Common origins to determine electronic and core parts were arbitrarily assigned along the crystallographic axes.

3.

The elements of the macroscopic piezoelectric tensor can be further separated into two parts: a clamped-ion or homogeneous strain contribution evaluated at vanishing internal strain u, and a term that is due to an internal microscopic strain,

and can be rewritten in terms of the clamped-ion part and the Born (transverse) effective charge tensor as

(8)where Ω is the volume, ai is the lattice parameter, the clamped-ion term es0d is the first term of Eq. (8), , and subscript k corresponds to the atomic sublattices. $Z^{*} _{k,iv}$ is the transverse (Born) effective charge:

The first term in eq (8) can be evaluated from polarization differences as a function of strain, with the internal parameters kept fixed at their values corresponding to zero strain, in other words, once we calculate the polarization difference with Berry’s phase approach, the first term can be solved. The second term, which arises from internal microscopic relaxation can be calculated after determining the elements of the dynamical transverse charge tensors and variations of internal coordinates ui as a function of strain. Therefore, we could obtain the piezoelectric tensor e_{iv}.

**DFT Method and parameters**

Total energy calculations were performed within the **general gradient approximation (GGA)** using the **fullpotential ab initio LAPW method with local orbital (LO) extension**. **The Perdew-Burke-Ernzerhof exchange-correlation parametrization** was used in the calculations. The value of RKmax was set to 8.3, LAPW sphere radii of 2.0, 1.7, and 1.6 a.u. were used for Pb, Ti, and O, respectively. Pb 5d, 6s, 6p, Ti 3s, 3p, 3d, 4s, and O 2s and 2p orbitals were treated as valence orbitals. Atomic core states were calculated relativistically, ignoring spin-orbit coupling, while valence states were treated semirelativistically. **The special points method was applied for Brillouin-zone samplings with a 4 X 4 X 4 k-point mesh.** **The k-space integrations in the Berry’s phase calculations were made on a uniform 4 X 4 X 20 k-point mesh**. The results of the calculations were checked for convergence with respect to the number of k points and the plane wave cutoff energy.

**Result**

There are 2 different kind homogeneous contributions, it may looks a little bit confusing, but the only difference is that the proper homogeneous constants only in materials with nonzero polarization in the unstrained crystal. Which means that it excluded the effect of the rotation or dilation of the spontaneous polarization. And the results show that the computed intrinsic piezoelectric moduli are found to be in relatively good agreement with experimental data measured on single crystal material. The large piezoelectric response in this material is mainly due to the large relative displacement of cationic and anionic sublattices induced by the macroscopic strain. This effect is further amplified in tetragonal solid PbTiO_{3} by the nomalously large elements of Born dynamic charge tensors.

**Case 2: First-principles study of piezoelectricity in tetragonal PbTiO3 and PbZrTiO3 ^{[7]}**

This paper is by the same group as case 1. Basically, they use the same method and the approach. The only difference is the B-site cation ordering. Chemically ordered PbZr_{1/2}Ti_{1/2}O_{3} phases investigated in this study are the computationally simplest systems with stoichiometry close to that of the morphotropic-phase boundary. Here, two chemically ordered PbZr_{1/2}Ti_{1/2}O_{3} (PZT 50/50) phases, with B-site cations ordered along [001] (P4mm), and [111] (I4mm) directions are considered.

Fig 3. (a) B-site cations ordered along [001] (P4mm), and (b) along [111] (I4mm) directions

The authors ignore the B-site ordering effect in PZT system. Because chemical ordering has not been considered as a possibly important feature of PZT, mainly because long-range chemical ordering has not been observed experimentally in pure PZT. Besides, the degree of order is determined by the size and charge difference between the two B-site ions. Since there are no obvious differences in sizes and charge between Zr and Ti. Therefore, the energy difference between ordered and disordered PZT phases is expected to be small.

**Approach method and parameters**

Basically, the approach is the same as the case 1. All properties presented formerly and in this paper were computed within the general gradient approximation using the full-potential ab initio LAPW method with local-orbital (LO) extension. And the electronic part of the polarization was determined using the Berry’s phase approach.

The table compares the parameters between case 1 and case 2.

Result

The table below shows the structural parameters and Z_{33}^{*} values of tetragonal PZT 50/50. Internal coordinates (u) are given in terms of the lattice constants of the P4mm unit cell. This shows that a highly strained oxygen octahedron was found around the Zr atom in tetragonal PZT with P4mm symmetry. Calculated O-O distances are 2.82, 2.85 and 3.13 Å compare to the ideal 3.00 Å. It leads to a larger internal strain.

And this table shows e_{33} piezoelectric stress tensor elements (C/m^{2}) of tetragonal PbTiO3 and ordered PZT 50/50. Here, we can see that the PTO result is close the experimental data measured on single crystal material. But the for the both PZT 50/50 sytems, the calculated results are very different with the experimental results. The first measurements give the value of e_{33}=27.0 C/m^{2} in the ceramic material at room temperature and the second measurement is measured at low temperature. The measurements at RT do include external contributions to the piezoelectric modulus, such as domain-wall and thermal-defect motions. However, using the low-temperature data of standard piezoelectric resonance measurements obtained for poled, pure, ceramic PZT 50/50, the measured value of e_{33}=11.9 C/m^{2} is still more than twice as big as the theoretical value.

**Case 3: Compare to another group: Intrinsic Piezoelectric Response in Perovskite Alloys: PMN-PT versus PZT**

The reason to introduce this paper is that it use a different approach method to calculate the piezoelectric tensor. Compare to the former group, it perform local-density approximation (LDA) ^{[8]} calculations and using the Vanderbilt ultrasoft-pseudopotential scheme. The semicore shells of all of the metals are included in the valence, which leads to 44, 88, and 220 electrons per cell in PT, PZT, and PMN-PT, respectively. (6,6,6), (6,6,3), and (6,6,1) Monkhorst-Pack meshes are used for the PT, PZT, and PMN-PT supercells, respectively, in order to provide good convergence of the results.

The results shows that the e_{33} of PZT that calculated by LDA is smaller than the Cohen’s work. And for the both groups, the results differ significantly from the experimental values. This means that the calculation models may only describe the intrinsic properties in the complicated system (AB’B’’O_{3}), the enhancement of piezoelectricity experimentally found when going from the parent compound (PT) to the alloy (PZT) is entirely due to extrinsic contributions, such as the motion of ferroelectric domain walls, or effects associated with the ceramic microstructure of the experimental material, or the coexistence of different crystallographic phases in the vicinity of the morphotropic phase boundary.

**Further Investigation**

**Study the piezoelectric properties at finite temperature**

The three cases discussed above are all show the results at 0 K. In order to close the actual situation, it is very important to understand the properties above 0 K. Vanderbilt et al^{[9]} try to use another approach to build a model at finite temperature. Virtual crystal approximation (VCA) and effective Hamiltonian method were used to calculate the properties. It proof this method works to predict the piezoelectric tensor at low temperature (have a close result to former cases), but fail at predict correct Tc and the e_{33} (4.3 C/m^{2}) still too small compare to experimental data. Therefore, it may put more investigations on the properties of piezoelectric at finite temperature.

**Build a better model in the complex system**

The calculated results are shows that the large difference between the calculated results and experimental ones. Although the authors in the case 2 and case 3 explained that the difference may be due to the extrinsic contribution. But Krakauer et al[9] calculated a result that close to experimental one by using a monoclinic model (instead of tetragonal). They claims that Greatly enhanced piezoelectric coefficients are observed due to polarization rotation as a function of applied strain in the monoclinic phase with fully relaxed internal atomic coordinates. As the polarization rotates within the (11^{¯} 0) mirror plane between the [001] and [*vv*1] (pseudocubic) directions, values as large as e_{33}= 12.6 C/m^{2}, e_{15}= 10.9 C/m^{2}, e_{13}= -33 C/m^{2}, and e’_{11}=36 C/m^{2} are observed at _{v}= 1.27 , where e’11 is defined as 0.5(e_{11}+e_{12}). Such large values are consistent with the measured piezoelectric response in ceramic PZT.

But there are still lots of different opinions on the structure around the MPB region. Therefore, to build a better model for complicated AB’B’’O_{3} model may be could improve the result of calculation.

**Reference**

1. http://en.wikipedia.org/wiki/Piezoelectricity

2. L. E. Cross, Ferroelectric Ceramics-Tutorial Reviews, Theory, Processing and Applications, N. Setter, and E. L. Colla, ed., 1 (Birkhauser Verlag, Basel, 1993)

3. G. Saghi-Szabo, R. E. Cohen and H. Krakauer, Phys. Rev. lett. 80, 4321

4. R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47,1651 (1993)

5. R. Resta, Rev. Mod. Phys. 66, 899

6. D. J. Singh, Planewaves, Pseudopotentials and the LAPW Method (Kluwer Academic Publishers, Boston, (1994)

7. . Saghi-Szabo, R. E. Cohen and H. Krakauer, Phys. Rev. B. 59, 12771

8. P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964);

9. Z. Wu and H. Krakauer, Phys. Rev. B. 68, 014112