In the late thirties [4], Landau captured the non-trivial concepts of phase transitions in a symmetry-based model. The mainansatz in the Landau model is that in the vicinity of the phase transition, the Helmoltz free energy F can be written as a polynomial expansion of an order param-eter. The order parameter is a physical quantity either macroscopic or microscopic that goes from zero to a non-zero value through the transi-tion. From symmetry arguments, the expansion ofF has to be valid for temperatures above and below the transition. The terms that are allowed in the expansion are determined from the phase of the highest symmetry.
The phenomenological approach of the theory comes from the fact that it takes as inputs parameters that are either measured or calculated by first-principle calculations. (Hence, it is important to point out that such macroscopic model can only be as good as the given parameters are.) It is based on the assumption that the fluctuations of the order parameter are small and hence it is a useful model for long range interactions sys-tems such as ferroelectrics. It is in 1949 [3, 5] that A.F. Devonshire was the first in implementing Laudau theory to ferroelectrics. It is important to mention that the Landau-Devonshire model is valid for a bulk system with uniform polarization. In order to introduce domains i.e. spatial varia-tion of the polarizavaria-tion and study the thin films with boundary condivaria-tions,
2.1 Phase transition:
A phenomenological approach Landau-Ginzburg theory is required. (Laudau theory is valid for displacive materials since by definition of the construction of the Landau theory, the order parameter vanishes uniformly everywhere above the critical transi-tion temperature)
Focusing now on ferroelectric materials with the ABO3chemical struc-ture, it feels natural to use the ferroelectric polarization as the main order parameter. Such structural phase transition comes along with a lattice distortion which should also be taken into account as a secondary order parameter. Nevertheless, for the rest of the discussion we will neglect it for simplicity.
The paraelectric case is centro-symmetric therefore only the even terms are allowed in the expansion. The reason for that, is that the energy should not depend on the orientation of the polarization.
F = α 2P2+γ
4P4+δ
6P6 (2.3)
To find the polarization state, one should minimizeF with respect to P
∂F
∂P =E (2.4)
giving rise to a simple expression of the electric field as a function of the polarization
E=αP +γP3+δP5. (2.5)
In this theory, and close to the phase transition one assumes that α lin-early depends on the temperature, changes sign atT0 and that the other coefficients are temperature independent;
α=β(T −T0) (2.6)
withβ >0andT0the transition temperature. To ensure the stability of the ferroelectric phase, it is required that eitherγorδare positive. As we will see bellow, the sign of the coefficientγ strongly changes the evolution of the order parameter as a function of the temperature. Ifγis negative, the transition will be called first order (discontinuous) and theδterm should be taken into account and positive. For positiveγ, it is a second order (continuous) phase transition and theδterm can be neglected.
First order phase transition
The first order phase transition appears whenγ <0in Eq. (2.3), whileδ is positive. The Curie temperatureTC corresponds to the temperature at which the system abruptly develops a polarization and is defined by two conditions: F = 0and∂F/∂P = 0forP 6= 0. Such a discontinuity in the order parameter is the signature of a first order phase transition. We then have the following withE= 0:
As sketched in Fig. 2.3, forT > TC, three different wells appear inF with only one being a minimum atP = 0. ForT =TC, the three wells appear as three local minima withF = 0: one state hasP = 0 and the two others haveP 6= 0. ForT0< T < TC the state atP = 0is still a local minimum of the free energy. AtT =T0, only two solutions with non zero polarization are stable and those states correspond to the spontaneous ferroelectric polarization. It is hence possible to look at the temperature dependence of the spontaneous polarizationPsfrom Eq.2.8.
Ps2(T) = −γ±p
γ2−4δβ(T −T0)
2δ . (2.9)
AtTC, the three possible polarization states are degenerate. Whether the system will be inP = 0or P 6= 0 state will depend whether TC is ap-proached from a higher (TC+) or a lower (TC−) temperature. If we are cooling, atTC the system will stay at zero polarization and upon further cooling, the system will jump to eitherPs+orPs−; this jump being a signa-ture of a first order phase transition.
Ps+=
Upon heating, atTC, the system will be in either of the two polarizations Ps+ orPs−depending on its history.
2.1 Phase transition:
In the case of a second order phase transition sinceγ is positive, higher order terms can be neglected. As a starting point we shall write with E= 0:
∂F
∂P = 0 =β(T−T0)P+γP3. (2.12) Forα >0, only one solution of polarization is found to Eq. 2.12: P = 0.
For α < 0, two solutions of polarization are valid for Eq.2.12. The fer-roelectric transition occurscontinuously whenα= 0with a spontaneous polarization appearing atT =T0which by definition is the Curie temper-ature and therefore for a second order phase transitionT0 =TC. Hence the solution of Eq. 2.12 forP 6= 0givesPs:
Ps2= β
γ(TC−T). (2.13)
At zero field and for T < TC, the free energy landscape displays two minima withP =±Ps. The energy barrier between the wells determines the minimum electric field required to switch all the dipoles together.
Dielectric stiffness
Let’s now look at the dielectric stiffness:
κ= 1 Hence depending on whether we are above or belowTCand using Eq. 2.13 we get:
ForT > TC, κ/0=β(T−TC).
ForT < TC, κ/0= 2β(TC−T). (2.15)
T0<T<Tc
Figure 2.3: The top part of the figure describes the first order phase transi-tion and the bottom is for the second order. For both types the free energy as a function of the polarization is displayed for different temperatures across the phase transition. The dielectric susceptibility is displayed as a function of temperature showing a clear difference between the two types of phase transitions. Finally, the behavior of the polarization across tem-peratures shows the continuous increase of P for the second order and the sudden jump for the first order atTC.
The dielectric susceptibility which is defined as χ = 1/κ ∝ T−T1
C re-flects that a singularity is present atTC. This behavior is called the Curie-Weiss Law. Note the fact that theαterm is directly linked to the dielectric constant. From Eq. 2.14 we obtain:
ForT > TC, P = 0,→χ= 1
2.1 Phase transition:
A phenomenological approach