Structure and composition

Figure 2-7 shows the basic perovskite unit cell of PZT, where Pb²⁺ on the A site is coordinated by 12 oxygen anions and Zr⁴⁺ or Ti⁴⁺ on B site is coordinated by 6 oxygen anions. Both Zr⁴⁺

and Ti⁴⁺ cations have similar ionic radii and are interchangeable within the structure.

Figure 2-7 A cubic perovskite unit cell of PZT illustrating a) 6-fold coordination of Zr⁴⁺/Ti⁴⁺ and b) 12-fold coordination of Pb²⁺ (reproduced from [34]).

Stability of the perovskite structure is described by the Goldschmidt tolerance factor tG [35], which is calculated from the average ionic radii of the constituting ions (rA, rB, rO) as follows:

𝑡_G 𝑟_A 𝑟_O

√2 𝑟_B 𝑟_O . (2-16)

When tG = 1, the perovskite unit cell assumes a stable cubic structure. In PZT, the cubic structure shown in Figure 2-7 is stable above the Curie temperature TC (approximately 240−480 °C depending on the Zr/Ti ratio). Ionic displacements are observed, when the tolerance factor deviates from 1. The ionic radii of Pb²⁺(XII), Zr⁴⁺(VI) and Ti⁴⁺(VI) ions are 1.49, 0.72 and 0.61∙10^-10 m respectively (numbers in brackets signify their coordination) [36].

The tG of PbTiO3 is >1, whereas the tG of PbZrO3 is <1, therefore, both perovskites favour different crystallographic structures below the Curie temperature. As illustrated by the PbZrO3 -PbTiO3 phase diagram shown in Figure 2-8, in Ti-rich PZT, the cations displace along the [100]

direction relative to the oxygen atoms, creating a tetragonal unit cell. In Zr-rich compositions, either an orthorhombic or rhombohedral phase is formed. The rhombohedral phase results from the shift of cations along the [111] direction accompanied by the tilt of oxygen octahedra. In the orthorhombic phase, cations assume an antiparallel arrangement along [110]/[ 110 ] directions, while the oxygen octahedra rotate around the [110] axis [37, 38]. The highest piezoelectric activity of Pb(Z^xTi1−x)O3 is achieved within the compositional range of x ≈ 0.52−0.55, near the morphotropic boundary (MPB) between the rhombohedral and tetragonal phases, as illustrated in Figure 2-9 [3]. Further discussions will be limited mostly to this range, as it contains the most of commercially used PZT compositions.

Figure 2-8 Phase diagram of Pb(Zr,Ti)O3 reproduced from the work of Zhang et al. [39]. TC = Curie temperature; MA, MB = two modifications of a monoclinic phase found in minor

concentrations in indicated areas.

Figure 2-9 Dependence of relative permittivity (εr) and planar coupling factor (kp) on the Zr/Ti ratio showing a maximum at the morphotropic phase boundary (reproduced from [26]).

Both tetragonal and rhombohedral unit cells are non-centrosymmetric and possess a spontaneous dipole, which can be permanently switched (i.e. both phases are ferroelectric). The cubic cell does not possess a dipole and is therefore paraelectric (shows no piezoelectric response). The dipole is formed by the displacement of ions after crossing the Curie temperature.

The paraelectric-ferroelectric phase transition can be described by the Landau theory, which characterises phase transitions by an order parameter, in this case polarisation PE. The Landau free energy density can be expressed as follows:

𝑓 𝑇, 𝑃_E 𝑓 𝑇 1

2a_f 𝑇 𝑇 𝑃_E 1

4𝑏_f 𝑇 𝑃_E 1

6𝑐_f 𝑇 𝑃_E , (2-17) where f0 is the free energy when PE = 0 (paraelectric state), bf and cf are temperature dependent expansion coefficients, af0 is a constant and T0 is the phase transition temperature. The equation is valid for homogeneous uniaxial ferroelectric systems in the absence of external electric and mechanical fields [40].

Newly formed dipoles create charges near interfaces, surfaces and defects, which gives rise to a depolarising field. To minimise this field and the energy of the system, the piezoelectric crystal is separated into regions possessing certain directions of polarisation, called domains.

The direction of dipoles is homogeneous within each domain and the domains are separated by domain walls. In the case of a depolarising field, the directions of domains alternate by 180°, making them the 180° domains or 180° domain walls [27, 40].

In polycrystalline ceramics, each crystal is clamped and the distortion accompanying the phase transition gives rise to internal stresses. The stress can be minimised by formation of non-180°

domains (non-180° domain walls). The tetragonal phase exhibits 90° domain walls, whereas the rhombohedral phase exhibits 71° and 109°domain walls [27].

In a polycrystalline material, although the domains have precise crystallographic directions within each crystallite, as the crystallites are randomly oriented, a polycrystalline ceramic body has zero net polarisation and shows no response to applied mechanical stress or electric field.

In order to make the body piezoelectric, domains must be aligned by application of an external electric field of sufficient strength. This process is also known as poling.

A piezoelectric response originating from atomic displacements within the individual unit cells is called intrinsic. Non-180° domain wall motion is the most important extrinsic contribution to the piezoelectric response in ferroelectrics [27].

As stated previously, the piezoelectric response of PZT is much higher than that of most other ferroelectric ceramics. This phenomenon is closely related the structure of the material near the MPB and despite several decades of research, there is still an ongoing debate about the exact mechanisms behind it. Both tetragonal and rhombohedral phases coexist near the MPB and it has been shown by Rossetti et al. [41] that the “MPB” is thermodynamically required to be a two-phase field. The presence of 2 phases increases the number of available polarisation-switching angles and increases the overall polarisation after poling. In addition to tetragonal and rhombohedral phases, Noheda et al. [42] also detected a monoclinic phase using the high-resolution synchrotron X-Ray powder diffraction. A Rietveld study by Zhang et al. [39] found long- and short-range monoclinic regions present throughout the entire rhombohedral phase region in 2 different structures – MA and MB. Their proposed phase diagram is shown in Figure 2-8. The presence of a monoclinic phase was also shown as thermodynamically possible [43, 44]. Further research showed that the potential barrier between tetragonal and rhombohedral phases is shallow and both phases can be reversibly transformed between one another using electric field [45-48]. The transition is facilitated by polarisation rotation, as illustrated in Figure

2-10, and all the intermediate stages of rotation can be considered as monoclinic, which explains the presence of this phase as well as the abnormally high piezoelectric response of PZT.

Figure 2-10 A cubic cell with directions in which it distorts to achieve tetragonal (T), rhombohedral (R) or orthorhombic (O) symmetry. MA, MB and MC distortions illustrate possible

intermediate monoclinic phases. (reproduced from [44])

Distortion of the tetragonal unit cell cT/aT where aT and cT are lattice parameters, decreases in a linear fashion as the Zr/Ti ratio approaches the MPB, which is shown in Figure 2-11. The rhombohedral unit cell distortion (90 𝛼R, where αR is the rhombohedral angle) is significantly lower and shows a little change with the Zr/Ti ratio [26].

Figure 2-11 The dependence of tetragonal and rhombohedral unit cell distortion on the Zr/Ti ratio (reproduced from [26]).

It has been demonstrated, that the maximum unidirectional strain of PZT induced by an electric field is not achieved exactly at the MPB, but rather on the tetragonal side of it. Hoffmann and Kungl [49] found the highest strain response at the tetragonal phase volume fraction of 60−70 %.

The authors speculated that the higher lattice distortion but reduced non-180° domain wall motion in the tetragonal phase are more effective at contributing to the strain response than the higher non-180° domain switching activity but smaller lattice distortion of the rhombohedral phase.