The Landau-Devonshire theory is valid for bulk mono-domain (poled) fer-roelectrics. Nevertheless the ferroelectric properties are known to strongly depend on strain; therefore we shall add a strain-dependent term to the LD theory in Eq. 2.3. This will be of great interest since later on we will study ferroelectric epitaxial films strained to substrates. The difference in lattice parameter will hence induce a strain state [6]. The polarization is now decomposed into its components with respect to the three crystallo-graphic axis i,j and k. A correspondence will be used between α/2, γ/4 andδ/6withαi,αijandαijk. The allowed terms in the energy expansion should imperatively be permitted by the highest symmetry structure. The free energy can be written as follows:
FLD=FLD(Pi) +Fstrain(Si) +Fstrain−polarization−coupling(Pi, Si) α1,α11andα12are the phenomenological LD coefficients being physically linked to the dielectric stiffness and higher order stiffness coefficient at constant strain. Sn defines the strain tensors with n=1,...,6 defined as:
S= (Sxx, Syy, Szz, Syz, Sxz, Sxy)≡(S1, S2, S3, S4, S5, S6).cnlare the elas-tic stiffnesses andgnl are the electrostrictive tensors, both at constant or-der parameter (here polarization). (A discussion on the use of Helmholtz or Gibbs energy can be found in Appendix A.)
For the specific case of a tetragonal film grown on a cubic substrate with an out-of-plane polarization, namelyP1 =P2 = 0,P3 =Pz 6= 0, we haveS4=S5=S6= 0due to the absence of any shear [7],S1=S2=Sx
andS3=Sz. Sx= asuba−a0
0 wherea0is the equivalent cubic in-plane lattice constant of the free standing film.
One can then rewrite Eq. 2.17 in a simpler way:
FLD =α1Pz2+α11Pz4+α111Pz6+1
2c11(2S2x+S2z) +c12(Sx2+ 2SxSz)
−g11SzPz2−2g12SxPz2.
(2.18) Let’s consider now the specific case of resulting stresses due to an in-plane clamping (and hence strain) that are such asσ1=σ2andσ3=σ4=
Putting this into Eq. 2.18 gives:
FLD =Sx2c211+c12c11−2c212
In other words, the strain renormalizes all the Landau coefficients and adds an extra coupling term. We will now call the renormalized termsα∗. Hence, A renormalization ofα1straightforwardly implies a renormalization ofTC
sinceα1∼(T−TC)implies thatα1∗∼(T−TC∗). From these results stems the influence of the strain on the ferroelectric transition temperature but also on the dielectric and piezoelectric properties for instance.
This macroscopic treatment of the strain-dependence in a ferroelectric film clamped on a substrate is a good starting point towards the under-standing of the observed physical properties which are usually different
2.1 Phase transition:
A phenomenological approach from the ones in the unstrained system. As an example, PbTiO3 as bulk is a ferroelectric for which the transition is first order. Once in thin film form (without considering the strain), theα11term becomes positive indi-cating a second order phase transition, while experimentally the transition appears as first order. Nevertheless once strain is taken into account, the α∗11is found to be negative, consistent with a first order phase transition.
In other words the first order “character” of the transition for a thin film material is totally due to the clamping to the substrate. (This is shown in the reference [8].)1
Hysteresis - the ferroelectric signature
When an electric field is applied to a piezoelectric (dielectric) material it gets polarized. 2 Within the approximation of smallE, one can express the polarizationP of a ferroelectric as follows
P =Ps+0χE (2.25)
wherePsis the spontaneous polarization,χthe electric susceptibility and 0the vacuum permittivity.
The definition of ferroelectricity enforces that the polarization should be switchable by the application of an electric field leading to an hys-teresis represented in Fig.2.4. Mathematically, such hysteretic behavior is obtained by solving the thermodynamic equationdF/dP =Efor P :
dFLD
dP |P6=0= 2α∗1Pz+ 4α∗11Pz3+ 6α111Pz5=E. (2.26) The electric field asymmetrizes the free energy and therefore one state is favored over the other leading to an hysteretical behavior like described in Fig. 2.4. This polarization loop shows that the sample will switch polar-ization for fields bigger than a threshold value called the coercive fieldEc. If the field increases further, the extra contribution to the polarization is the one due to the dielectric response (dielectric charging). Once the field is reduced to zero, all the dipoles stay aligned and the system retains its polarizationPr. Increasing the field in the opposite direction, the dipoles
1As a curiosity, it is important to notice that here we have stopped the expansion at the fourth order term. This comes again from the first order nature of the transition that we are interested in. Nevertheless, it has been mentioned by Kvasov and Tagantsev [9] that for strained thin films where the sixth order term is needed, higher order of the electromechan-ical coupling (P and strain coupling) are necessary.
2If the material already has a polarization, when a field is applied, the polarization will increase or decrease depending on the direction of the field.
Figure 2.4: Polarization versus electric field of a ferroelectric material.
Once the applied field is zero, the system is either in a state up or down with a remanent polarization Pr. Ps is an estimate of the spontaneous polarization defined as the extrapolation to zero field of the polarization at higher field. Experimentally, if the voltage cycle is slow enough for the charges to equilibrate, Pr=Ps.
will suddenly switch into the opposite polarization at the coercive value.
This is of course a theoretical view which is not the one happening in real systems due to inhomogeneity of the polarization, defects, domains, etc.
A more realistic scenario for switching is the growth and nucleation of regions with opposite polarizations [10].