# A D. Furthermore, the power series is listed until

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)

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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.