All crystals can be classified into one of the 32 crystal classes. Of these classes, 21 lack an inversion centre. Of these 21 classes, 20 are piezoelectric and display an electric polarity when subject to external stress (and vice-versa). Of the 20 piezoelectric classes, 10 are polar and display a spontaneous polarisation which is also temperature dependent, making crystals within these classes pyroelectric. To be ferroelectric, a material requires not only to be polar but also to exhibit at least two possible polarisation states in the absence of external electric field and it must be possible to switch between the spontaneous polarisation states repeatedly with an electric field of the proper magnitude and orientation. Because of these symmetry constraints, ferroelectric materials are also pyroelectric and piezoelectric as illustrated in figure2.1, making them a technologically relevant class of materials. Thanks to their high piezoelectric, pyroelectric and dielectric constants, they are materials of choice for transducers and actuators, infrared detectors and capacitors.
Ferroelectric materials exhibit a hysteresis of the polarisation upon cycling of an external electric field, an example of which is shown for a real sample in figure2.2a. The polarisation at zero field (or bias) is called the remanent polarisation,Pr. When the voltage is increased and reaches the coercive voltageVc, the polarisation of domains not oriented parallel to the field switches and aligns with the external field. If the field is further
Piezoelectric
Pyroelectric
Ferroelectric
Figure 2.1:Ferroelectric materials are also pyroelectric and piezoelectric.
increased, the measured polarisation can increase as a result of dielectric charging of the material. During polarisation switching, a current can also be measured between the electrodes connected to the sample, as shown in figure 2.2b, due to changes in the surface bound charges caused by the reversal of the polarisation. Integration of these currents can be used to extract the polarisation of the material. As discussed in chapter3, polarisation reversal occurs in discrete events called jerks, which can be studied by carefully measuring these switching currents.
Figure 2.2:(a) Typical ferroelectric polarisation hysteresis loop. (b) Corresponding switching current peaks corresponding to the coercive voltage of the sample. From [1].
At the microscopic scale, different mechanisms can give rise to the polar-isation states. In order-disorder ferroelectrics such as polyvinylidene fluoride (PVDF), ferroelectricity arises as an ordering of initially randomly oriented electric dipoles upon cooling under a critical temperature. In displacive ferroelectrics, the dipolar moments are caused by ionic displacements within the lattice spontaneously appearing below a critical temperature. Although perovskite ferroelectrics were first considered as displacive, the exact mecha-nism for ferroelectricity in some of the most common perovskite ferroelectrics such as BaTiO3and PbTiO3 is under debate [2], and a mixed character is
2.1 General properties
possible.
Some of the most studied and technologically relevant ferroelectric mate-rials have a perovskite structure. These matemate-rials have a chemical formula ABX3, where A and B are cations and X is usually oxygen. The ideal perovskite lattice is cubic with the A atoms located at the corners of the cube, B in the centre and the oxygen (X) atoms are located at the centres of the faces of the cube and form an octahedron around the central atom.
An ideal cubic realisation of such a structure is shown in figure2.3a. The perovskite structure is highly robust to changes in the atoms in the A and B sites, leading to a wide variety of materials adopting this structure and rich physical properties [3], as well as the relative ease of preparing solid solutions of different materials within a perovskite structure. Furthermore, materi-als with a perovskite structure can show a variety of symmetry-lowering distortions such as distortions of the octahedral shape and symmetry, rota-tions of the octahedron or off-centering of the carota-tions. These distorrota-tions can lead to significant changes in the material properties such as the colossal magnetoresistance in LaMnO3[4], control of the metal-insulator transition in rare-earth nickelates [5] and the emergence of ferroelectricity in a wide variety of perovskite oxides [6].
The presence of these distortions can often be rationalised by the ionic radiirof the A and B cations and oxygen anions, which control the optimal packing of atoms within the lattice. Ideally, the lattice parameteraand the ionic radii should be related bya=√
2(rA+rO) = 2(rB+rO), leading to a cubic lattice. This relation is often not satisfied, leading to deviations from the cubic structure which can be estimated from the Goldschmidt tolerance factor [7].
t= rA+rO
√2(rB+rO) (2.1)
Materials witht <1 have an A atom that is small, leaving ample space between oxygen octahedra. These materials are often not ferroelectric and for 0.7 < t < 0.9, rotations of the oxygen octahedra are common while for lower values of t, the perovskite structure is unstable and the lattice adopts a different structure. In materials witht >1, the central cation has enough space to move and polar distortions are often favoured, giving rise to ferroelectricity, as in BaTiO3 and PbTiO3 where the tolerance factors are 1.07 and 1.03 respectively [8]. The stability of ferroelectricity in PbTiO3
is further assisted by the smaller size of the A cation, Pb, along with its hybridisation with the O ions. Both lead to the stabilisation of a tetragonal structure. The Ti ion also bonds with the closer oxygen ion, further pulling the Ti off-centre. Simultaneously, the Ti and oxygen octahedron are displaced in the same direction with respect to the Pb ion, albeit with a higher shift amplitude for the oxygen octahedron, shifting the centres of charge and further contributing to the dipole moment [9]. The resulting structure is
shown in figure 2.3b-c for polarisations pointing up and down respectively.
Figure 2.3:(a) Perovskite in the cubic phase. (b, c) Atomic positions of PbTiO3
(A=Pb, B=Ti) in the ferroelectric tetragonal phase for up and down oriented polari-sations. The Ti ion and oxygen octahedron are shifted in the same direction but the magnitude of the shift is greater for the oxygen octahedron.
PbTiO3is a widely used and studied material and is commonly grown in solid solutions by replacing some of the central Ti atoms with other elements such as Zr, where it forms Pb(Zr1-xTix)O3.