Strain-Coupling

In the previous section the Gibbs free energy density was only ex-panded relative to the polarisation. However, perovskite ferroelectrics also have a dominant feature of being extremely sensitive to elastic stress σij

[26, 32, 34]. It is, therefore, necessary to understand how applied stress will affect the properties of these materials. Landau-Devonshire theory can be extended to incorporate these effects by including terms from the Cauchy strain tensor ηij:

where ui is the displacement along i of a point in the material relative to position r_i.

Common industrial applications such as FeRAM or high sensitivity actuators take advantage of homogeneous biaxial strain though epitaxial growth of tetragonal (001) ferroelectrics onto a (001) substrate with differ-ent lattice parameters [35]. In such cases, the in-plane strains (η11 & η22) are fixed and defined by the lattice mismatch between the ferroelectric and substrate bulk lattice parameters (a & as, respectively):

η11= η22 = a− a^s

a_s (2.10)

Pertsev et al identified that, in addition to the fixed in-plane strain, the correct mechanical boundary conditions occur in the limit of vanishing shear strains with the out-of-plane normal and shear stresses free to fully relax [36, 37]. Consequently, they were able to show the functional form of

Strain-Coupling 13

the free energy density to be minimised to be [35–38]:

G(P, η) + EP = 1 where the first line is the original expansion with the classic W-potential (Eqn. 2.7). The terms in the second line relate the elastic constants Cij

to the strain deformations, giving the elastic energy contributions to the free energy density. The third line contains the terms corresponding to the

‘polarisation-strain coupling’ which acts to renormalise the quadratic term of the free energy [39]: This polarisation-strain coupling causes giant piezoelectric responses typically 100-fold greater than non-ferroelectric piezoelectrics such as quartz [40]. Moreover, these additional coupling terms can be used to tune the co-efficient of the quadratic term via the epitaxial strain conditions enabling so-called ‘strain-engineering’ of the ferroelectric instability. By modifying the strain, the coefficient can be made more negative, lowering the energy of the double well in the W-potential and increasing the stability of the ferroelectric phase as was shown experimentally by Choi et al for BaTiO₃ grown on single crystal substrates of DyScO3 and GaScO3 [41]. These epi-taxial films were found to increase the transition temperature by 500 K and the spontaneous polarisation by 250% relative to bulk BaTiO3. For non-ferroelectric materials, reductions in the quadratic coefficient can even induce a ferroelectric phase as has been predicted for rocksalt BaO and EuO [42]. SrTiO₃, an incipient ferroelectric below 37 K, has been experi-mentally shown to support in-plane polarisation at room temperature when

epitaxially grown onto DyScO3[43]. Alternatively, if strain conditions cause the quadratic term coefficient (2.12) to become less negative, the ferroelec-tric character of the material will be reduced or even suppressed entirely if it becomes positive.

If higher order strain terms are considered in the free energy expansion (2.11), then additional cofactors will enter into the coefficient for the quartic term [26]. As the sign of this coefficient leads to the order of the phase transition, external stresses from lattice mismatch, chemical substitutions or hydrostatic pressures can change the order of the transition. Notably, the phase transition of PbTiO3 has been shown to become second-order under pressures exceeding 12.1 GPa [44].

2.3 Phonons

Landau theory provides an excellent phenomenological introduction to ferroelectric phase transitions but makes no effort to describe the underlying physical origin. One of the most convincing microscopic models was derived by Cochran [45] who framed the problem in term of lattice dynamics. In this Section, the concept of lattice dynamics and soft modes are briefly introduced to aid the concluding sections of this Chapter. Greater detail on the fundamental physics of the ferroelectric phenomena including lattice dynamics are presented in Section 3.2.

Phonons are quasiparticles describing lattice vibrations, the collective excitations of the ions from their equilibrium Bravais sites. The adiabatic (Born-Oppenheimer) approximation (see §3.1) is applied wherein electrons are assumed to remain in their groundstate. If atomic displacements are small, the potential energy can be considered harmonic such that the restor-ing force actrestor-ing upon an ion κ is linear. In this case, the total energy E_i+e of small lattice vibrations can be expanded as:

Phonons 15

where Cκακ⁰α⁰ are the interatomic force constants for displacements of atom κ in cell a along direction α:

C_κακ0α⁰ = ∂²E_i+e

∂τκα^(a)∂τ_κ^(b)0α⁰

(2.14)

Assuming a periodic crystal, the classical equation of motion derives from Newton’s second law using the derivative of equation 2.13. One expects planewave solutions of the form:

where ˜Cκακ⁰α⁰ is the Fourier transform of the force constants matrix (eqn. 2.14), Mκis the mass of ion κ and ˜Dκακ⁰α⁰ is the Fourier transform of the Dynamic matrix. The eigenvectors of equation 2.16 describe the lattice vibrations (phonons) of the system.

The simple case of a 1-dimensional diatomic chain of atoms has two solutions to the eigenequation (2.16). The eigenvalues for the two branches w(q) are shown in Figure 2.3 and referred to as a dispersion curve. The subfigure shows the patterns of ionic displacements for optical and acoustic phonons of the same frequency. In three dimensions, solutions of eqn. 2.16

permit transverse and shear waves as well as the compressive waves consid-ered in 1D.

Figure 2.3: Phonon dispersion of the optical (purple) and acoustic (green) modes of a 1D diatomic chain. The sub-figure shows the pattern of ionic displacements for transverse optical and acoustic waves of the same frequency.

The seminal paper by Cochran related the zone centre (q=0) optical phonon frequency to theF² Landau coefficient [45]:

ω_m,q=0² ∝ 1

χ (2.18)

This relation holds remarkably well for many ferroelectric materials. Ex-ample prototypical perovskite ferroelectrics are introduced in the following Section.

Ferroelectric Perovskites 17