A for this work

required perspective is the thermodynamic model of free energy to model the

above mentioned effects of ferroelectrics without knowing the process on an

atomic level. To model a dielectric material six variables are needed for the

internal energy. 14

(1)

With U the internal

energy per unit volume, T the absolute temperature, S entropy, Xi

stress, xi strain, Ei electric field and Di

displacement. The small letter “i” show that this variables show be vectors. If

now conditions from the outside of the system can be changed like in the

following experiments we can define the free energy as follows:

(2)

with F the free energy

per unit volume. This term is called Gibbs free energy. Because the aim of this

work is not to measure strain or apply stresses, we assume this term as 0. This

leads to:

(3)

The experiments in

this work as will be described down will change the temperature and the polarization

of the system so the free energy should be modeled as follows:

(4)

The model will be

continued with the modeling of the system at the phase change from paraelectric

to ferroelectric phase. The Landau theory is the model normally used in these

cases. This theory asks to model the free energy as a power series. The effect

of polarization is the important effect and the origin is dielectric, so that

the free energy will be a function of the displacement:

(5)

with ?, ? and ? the Landau coefficients. The reason why the power

series is only with even powers is because the free energy cannot depend on the

sign of D. Furthermore, the power series is listed until the 6th power. The

reason is the influence of higher orders is lower and it simplifies the

treatment. The phase change takes place with falling temperature. So we assume

that the first coefficient ? is linearly

temperature dependent

, ? and ? are constant. Here ? is constant

and T0 the transition temperature. This is the Devonshire

approximation:

(6)

Furthermore, is the

displacement equal to the polarization of the material if no electric field is

applied and the problem will be modeled only in one dimension. So if E=0 we can

write:

(7)

With this model it is

possible to say something about the behavior of the crystal. The reader should

be reminded that the material tries to be in state with the lowest energy and

this state is the stable state of the material. If now the Landau coefficients ? and ? are bigger than 0 and

the parameter ? can be a positive or

negative value depends on the temperature, we can see if the temperature falls

under a value where ? gets negative the

function will have to minima at symmetric points from the energy axis where the

P is unequal to zero Ps =±Ps1,2 (see figure 1).

At these points the

material is polarizable because it can have one or the other state. The material

is ferroelectric because the polarization can be switched by an electric field.

If now the temperature rises ? at some point gets

positive and the function of the free energy will just have one minimum at the

point Ps =0 (see figure 1). The material is paraelectric. The phase

transition from ferroelectric to paraelectric is continuously and named as

second order transition.

Figure 2 The free energy as function of P

for different coefficients of ? 16

Now we focus on the behavior

of the function if ? is negative and ? remains positive and ? again depends on temperature. Here we have to

distinguish three cases. First is ? is smaller or equal

to zero what causes that the minimum is at Ps =±Ps1,2.

Second case is that ? is 0 < ? <
. This case causes that the minima of the function will be at Ps
=±Ps1,2 and Ps =0. This means that the shift from ±Ps1,2
to 0 is suddenly at one temperature. The third case ? ?
means Ps = 0. This is the
paraelectric phase again (see fig. 2). Because of the second case the phase
transition is discontinuously and named as first order transition.
Figure 3 The free energy as function of P
for different coefficients of ? with negative ? 16
We continue the model
to get an idea of the relation of electric field and displacement. We know that
the derivative of free energy as function of displacement is equal to the
electric field:
(8)
The derivative of the
sixed power was neglected. In the ferroelectric state with ? < 0 and ? > 0 the function

leads a third-degree polynomial with local extreme points in the second and

fourth quadrants in a Cartesian coordinate system. The inverse function would

describe the relation D=f(E). The inverse graph looks like figure 4 but it is

not unambiguously. This mean it cannot be described with a unique function. In

a physical sense the graph can be interpreted as follows: With the points

ABCDEF we see that the state of the material would be uncertain between the

points B and C. So the state of the material would chance suddenly from B to E

and from C to F forming the typical D-E hysteresis loop of a ferroelectric

material.

Figure 4 The inverse function of E vs. D

16

Furthermore, we can

see that the function D=f(E) can be modeled as a power series of E near to the

point of E = 0:

(9)

with

(n=1,2,3,…) the first, second and

third dielectric nonlinearities and ?0 electric field

constant. With the derivative we can determine the nonlinear permittivities on

the point E =0 as follows:

(10)

and with the

derivative of the function E = f(D) we can see:

(11)

This is giving a

direct connection between the nonlinear permittivities and the Landau

coefficients. From this relationship Ploss 13 launched the following

relationship between Ps and the nonlinear permittivities:

(12)

Furthermore, this

relationship was simplified by Ploss 13 to:

(13)

with m an constant

proportionality factor.