1. investigations in bi and multilayer samples of PVDF

1.  
Introduction:

Ferroelectric materials possess pyroelectric properties and spontaneous
polarisation. All ferroelectric materials are pyroelectric, however, not all
pyroelectric materials are ferroelectric. Below a transition temperature called
the Curie temperature ferroelectric and pyroelectric materials are polar and
possess a spontaneous polarization or electric dipole moment. Inhomogeneous
polarization is the typical property of ferroelectric materials, particularly
at the surface region. Different mechanisms can cause decrease of polarization
near the surface.1

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Example:         High electric
fields

                        Structural
variation of the sample near the surface

 

Applications for Ferroelectric Materials:

 

Capacitors, Non-volatile memory, Piezo electrics for ultrasound imaging
and actuators, Electro-optic materials for data storage applications,
Thermistors, Switches, Oscillators and filters, Light deflectors, modulators
and display.

Here we are working on PVDF and P (VDF-TrFE) materials. The used P
(VDF-TrFE) compositions are P (VDF-TrFE) 56/44 mol% and P (VDF-TrFE) 70/30
mol%. Bi and multilayer films of P (VDF-TrFE) and PVDF are produced and
measured the polarization distributions using LIMM method.

 

To measure the charge and polarization distributions, various
experimental techniques are available, which are based on the piezoelectric or
on the pyroelectric effect. Pyroelectric effect methods are implemented in the
time or in the frequency domain. The time domain is the thermal pulse method,
while the LIMM uses thermal waves in the frequency domain. Thermal methods are
good enough to achieve high resolution near the sample surface. With LIMM we
can achieve 0.5µm resolution. The results of polarization investigations in bi
and multilayer samples of PVDF and P (VDF-TrFE) are presented.2

1.1 Principle of operation:

 

Charge and polarization distribution in the sample is measured based on
the pyroelectric response to a non uniform variation of temperature. Sample is
prepared in the form of round shape and the film is covered with electrodes on
both sides of surface. Laser light is focused on one side of the surface of the
film and that surface gets heated due to absorption of laser light. Heat goes
inside the sample through the surface it causes change in the temperature
inside the sample.1

 

 

1.2 LIMM (Laser Intensity Modulation
Method):

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1: Laser Intensity Modulation
Method (LIMM)

 

 

 

 

For the Measurement of pyroelectric current a current to voltage
converter, lock in amplifier has been used in computer controlled equipment.
The pyroelectric specimen and the reference photo diode both are connected to
the input of the current to voltage converter. For the measurement of
pyroelectric spectrum laser light is incident on the specimen, for the
measurement of reference spectrum laser light is incident on the photo diode.
Through a fast operational amplifier with a gain band width product of 1.7GHZ
is used in the I-U converter, a significant amount of phase shift can be
avoided in MHZ.1

 

 

 

 

 

 

 

 

 

1.  Theory

2.1
Ferroelectric effects

Solid materials are
classified as ferroelectric if they show two or more orientation states of
spontaneous polarization Ps without
a permanent electric field forcing the crystal in a polarization. Polarization
in this sense is the fixed separation of charges. These charges are arranged
through chemical bonding or through motions of sub lattices 14.

To be ferroelectric
the spontaneous polarization must be switchable in two or more different states
and the states must be stable in zero field. The value of the polarization
measurable at zero field is called remnant polarization Pr. Normally the polarization will be compensated through free
charges in the crystal and the surrounding media. The arrangements of charges
which are polarized in the opposite direction are also compensating the
spontaneous Polarization. This is possible because the whole crystal does not polarize
in the same direction if it’s not been forced through an electric field. Only
certain areas polarize in the same direction this area is called domains 14.

 Furthermore, the spontaneous polarization is
temperature dependent and disappears continually with higher temperature or
disappears suddenly above a certain temperature. This is called pyroelectric
effect. The temperature where the spontaneous polarization disappears
discontinuously is called the Curie temperature TC. Under the TC point the material is ferroelectric. Some
materials also show that the ferroelectric effect disappear under a certain
temperature, so the material is ferroelectric in certain temperature range 14.

Another property is
piezoelectric effect. This means that the material shows a spontaneous
polarization under mechanical stress and if an electric field is applied to
polarize the material, it shows a strain. The effect is linear 14.

In addition a
ferroelectric material shows nonlinear optic effects. It should be noted that
nonlinear optic effects can appear in more than ferroelectric materials. The
effects appearing in ferroelectrics are the spontaneous Kerr effect (or
electro-optic effect) which leads to birefringence and piezo-optic effect which
also occurs in any crystal material. With the change of the polarization in
ferroelectrics the birefringence changes as well 14.

2.2 Phenomenology of ferroelectrics

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 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.

2.3 The copolymer (VDF-TrFE)

Before the
copolymer was in use the polymer from VDF was studied as a ferroelectric
polymer. The polymer crystallized as a semi-crystalline plastic with a
crystallinity up to 50%. The crystal parts which are the parts which can be
switched are chains having an axis along the carbon chain call chain axis and
in the crystal parts the chain axes are parallel to each other. Furthermore,
the on each carbon atom added substituent can rotate between certain stable positions
around the chain axis. The states where the chain is stable are called
conformations and the material crystallizes in the same conformation called a
phase 15. It should be noted that the material can crystallize in different
conformation at the same time. PVDF has 4 distinct phases called ?, ?, ? and ? phase.
The ?, ? and ? phase are
ferroelectric and of these phases the ? phase is the phase with the highest remanent
polarization because the substituent of hydrogen on one carbon atom and
fluorine on the next carbon atom in the chain build up dipoles with the same direction
over the whole chain. This leads to that only the full chain of PVDF can rotate
if an electric field is applied, which is causing the ability to switch the
dipoles. PVDF doesn’t crystallize in a ferroelectric phase from the melt under
normal conditions. Through different treatments of the material the phases can
be changed e.g. the ? phase can
be reached through stretching the material. The supplementation of the VDF
monomers with TrFE monomers now forces the crystal parts of the material to
crystallize in a conformation similar to the ? phase
from the melt and the copolymer is ferroelectric below the Curie point. As a
copolymer the two monomers are mixable in any ratio and crystallized as a
semi-crystalline plastic with crystallinity up to 90%. The copolymer
polymerized in carbon chains with random distribution of the monomers. The two
important factors for the value of the remnant polarization for the copolymer
are the crystallinity and the VDF content 15.

 

The best
ratio for the VDF content is between 50% and 80% 15 and the crystalline can
be increased through annealing the material at a temperature above the Curie
point and lower the melting point 15. Another for this work important
ferroelectric properties are the dielectric nonlinearities ? which will be
modeled after 13 as complex because of the conductivity of the amorphous
parts and nonlinearity at the crystal parts. The amorphous parts are chains in
between the parallel chains, but having no order 13. This leads to that the
dipoles of the chain are distributed in a way that the force caused by the
dipoles cancel each other out. In addition, the amorphous part makes no
contribution to the remnant polarization still the amorphous parts are needed
in small amounts. Furthermore, the samples must be thin films with a thickness
not stronger 1?m. The
reason is simply the needed electric fields are too high to work with bulk
material.

2.4 Theory of Polymer-Based Ferroelectric Multilayer Systems

The growth of
superlattices is a promising approach to create artificial materials with
unique properties. Like described by Koehler 6 this is done by the
combination of two or more materials with diffeerent electronic properties and
lattice parameters in an alternating layer system while each layer has a
thickness of several nanometers.

 As stated by Lichtensteiger et al. 7 it has
been focused on perovskites during the past years due to their well known
ferroelectric properties. Material combinations that has been researched on are
for example BaTiO3/SrTiO3, KNbO3/KTaO3, PbTiO3/SrTiO3, PbTiO3/PbZrO3 and a so
called tricolor superlattice of SrTiO3/BaTiO3/CaTiO3. The layers of the
materials are grown by epitaxial deposition techniques such as pulsed laser
deposition, sputtering or oxide molecular beam epitaxy.

 The created unique properties are varying dramatically
between the different material combinations. In the case of BaCuO2/SrCuO2
superconductivity can be seen although none of the two materials is
superconducting 8. The same can be observed in SrZrO3/SrTiO3 superlattices
with respect to ferroelectricity 9. The above mentioned tricolor structure of
SrTiO3/BaTiO3/CaTiO3 breaks the inversion symmetry which allows
ferroelectricity 10. By varying the composition of the tricolor superlattice
the ferroelectric properties are tunable. Superlattices consisting of more than
one ferroelectric, for example PbTiO3/BaTiO3, are considered to be useful to
study polarisation switching processes and domain dynamics 11, especially
when the polarisation of the used ferroelectrics exhibits similar magnitudes,
while the polarisation switching behaves differently. LaAlO3/SrTiO3
superlattices possess charge discontinuities at the LaO/TiO2. Theoretical Model
of Electrostatic Coupling and Interface Intermixing interface, which form a
conduction channel between the single layers consisting of nonferroelectric
insulators. Combining a ferroelectric insulator and another insulator resulting
in a similar conduction channel could lead to the tunability of ferroelectric
coupling between superlattice layers 12.

It becomes obvious
that there are multiple ways to combine a wide range of materials in
superlattices with a variety of different effects that can be studied.
Moreover, they have an immense potential of being used for innovative
applications. In the next passages the focus lies on superlattices where
ferroelectric and paraelectric materials are combined since these multilayer
systems seem to be realizable most easily with the well studied polymer PVDF
and its copolymer P(VDF-TrFE).

As a matter of fact,
the behaviour of materials in superlattices differs from that in bulk material.
The neighboring layers in influence each other in an electrical and a
mechanical way. Additionally, intermixed layers can form at the interfaces
between the layers. The formation of these intermixed layers can hardly be
controlled and their properties are different from the adjacent layers, which
may affect the behavior of superlattices.