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Ferroelectrics investigation

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Ferroelectrics investigation

1. Introduction

A dielectric is understood as a material where the electric field induces an electric momentum.

Let’s consider a vacuum capacitor made of two planar metallic electrodes biased with a DC voltage, U. The capacitance C0 is given by:

U

C0 = Q0 , (1)

where Q0 denotes the electric charge gathered on the capacitor electrodes.

The electric capacitance is related to the geometric dimensions of the planar capacitor in the following approximate way:

d

C0 = ε0S (2) - the electrode area,

- the distance between the electrodes,

- dielectric permittivity of vacuum 0 = 8.85418781710-12 F/m).

The electric field intensity inside the capacitor is given by:

d

E =U (3)

When the capacitor is filled with a dielectric, the capacity increases up to:

d C S

Cr 00εr , (4) where εr is the relative dielectric permittivity.

Part of the charge brought to the capacitor is compensated by charge gathered in the electrodes (Fig.1). This charge is called the bound charge. The surface free charge density remains unchanged. That is why when the dielectric is inserted into the biased capacitor the electric field intensity does not change.

The electric induction is defined for linear dielectrics as follows:

E

D=ε , (5)

where ε =ε0εr is called the dielectric permittivity of a medium. The electric field intensity inside the capacitor is connected to the surface free charge density. Whereas the electric induction is equal to the total surface charge density.

The electric induction inside the dielectric can be expressed as:

P D

D = 0 + , (6)

where P denotes the medium polarization, and D00E . The medium polarization (P) is equal to the surface bound charge density.

The polarization is a linear function of the electric field density for most of dielectrics in moderately weak electric fields:

E

P0χ (7)

where χ denotes the medium electric susceptibility.

Using the notation mentioned above, Eqn. (6) may be rewritten in a form:

+ +

+

+

+

+

- -

-

- -

- - -

- +

+

+

Elektrody Dielektryk Dielectric

Electrodes

Figure 1 Thecapacitor

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2 E

E E

P0χ =ε00χ . (8) It can come of Eqn. (7) that:

ε = 1+χ and

E P

0

1 ε

ε = + . (9)

There are a few mechanisms of a dielectric polarization.

1. Electron polarization

The electric field deforms electronic shells of atoms so that the gravity centers of the nucleus electric charge and those of the electronic shells do no coincide, forming electric dipoles. The electric dipole diagram is shown in Fig. 2.

Dipole moment

=gl

µ (10)

The electric polarization is the dipole moment of a unit volume

=

i i

P V1 µ (11)

2. Atomic (ionic) polarization

In ionic materials there is a shift of atoms (or ions) due to the electric field that causes the dipole moment creation.

After the electric field is switched off the atomic and ionic polarization decays very fast and this is why this polarization is called elastic.

3. Dipole polarization.

Some dielectrics e.g. water, are built out of particles showing the dipole moments. Thermal motion causes that the dipole orientation is random, and in total the polarization is zero. An electric bias causes that the dipoles in a dielectric are ordered, and the polarization related to the process is called the orientation one.

4. Polarization connected to the pace charge.

Dielectrics can exhibit a non-uniform distribution of the space charge due to the production process or an intentional process after the production. The electric field due to the space charge cause the dielectric polarization called the space-charge polarization. This polarization is used in electrets.

The material electric polarization is a sum of all above.

2. Dielectric in the alternating electric field

Let’s consider a dielectric placed in an electric field, oscillating according to:

( ) t E e

i t

E =

0 ω (12)

where E0 denotes the electric field amplitude, - - circular frequency

(

ω=2πf

)

−1

=

i , t – time

+ _

L

q q

Figure 2 An electric dipole

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3 The induction can be delayed with respect to the electric field in the real dielectrics. The delay reason is a finite time needed for the dipole orientation change. The induction is changed according to the formula:

(ω ϕ)

= D e

i t

D

0 (13)

where D0 denotes the amplitude of the electric induction, –is the phase shift between the induction and the field.

Based on the dielectric permittivity definition (Eqn.(4)) we can rewrite

( )

ϕ ϕ

i 0 0 t

i 0

t i

0 e

E D e

E e D E

D

=

=

= (14)

The Euler’s formula says:

ϕ ϕ

ϕ cos isin

ei = −

therefore

"

i ' E sin

iD E cos

D

0 0 0

0 − = −

= ϕ ϕ (15)

where:

ϕ E cos ' D

0

= 0 and sinϕ

E

" D

0

= 0 (16)

Quantities ’ and ” denote the real and imaginary part of the dielectric permittivity.

The dielectric permittivity can thus be described as an imaginary number. In practice the tangent of the loss angle (defined below) is used to describe dielectric properties

ϕ ϕ ϕ tg cos sin '

"

tg = = = (17)

3. Dielectric permittivity measurement

Schering bridge is most frequently used for the dielectric permittivity measurements (Fig 3).

The bridge is fed with the AC. A variable capacitor C2 is placed in one of the bridge arms and parallel to it there is an adjustable resistor R2. The real capacitor (Cx*) is substituted by an ideal capacitor (Cx) and a parallel resistor (Rx). The Rx is responsible for the current flowing through the real capacitor.

R

x

C

x

R

4

R

3

R

2

Z

4

Z

3

Z

2

C

2

I

2

I

1

Z

1

C*

X

Figure 3 Schering bridge diagram and the equivalent circuit of the real capacitor, Cx

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4 Admittance of the capacitor in question is:

x x

x x

1

1 i C

R 1 C i

1 1 R

1 Z

Y = 1 = + = + (18)

Similarly

2 2

2

2 i C

Z 1 Z

Y = 1 = + (19)

The bridge undergoes the condition of equilibrium when:

4 2 2 1

3 2 1 1

Z I Z I

Z I Z I

=

= (20)

where Z denotes the impedance of the respective arm of the bridge.

Dividing the equations by sides we can obtain:

4 3 2 1

Z Z

ZZ = (21)

thus

2 3 4

1 Z

1 Z Z Z

1 = ⋅ (22)

Substituting Eqn.(22) into (18) and (19) we can obtain:

+

=

+ 2

2 3 x 4 x

C R i

1 R C R R i

1 (23)

Eqn. (23) is fulfilled when the respective parts: real and imaginary are equal to each other, so:

2 3 4

x R

1 R R R

1 = ⋅ and 2

3

x 4 C

R

C = R (24)

Based on the complex form of the dielectric permittivity the capacitor impedance can be written as:

( )

0

0 i 'i "C 1 C

i Z 1

= −

= (25)

or admittance

0 0 "C C

' Z i

1 = + (26)

Therefore

0

x = 'C and 0

x

C

"

R

1 = ⋅ (27)

From Eqn.(27)

0 2 3

4 0

x

0 2 3 4 0 x

C R R

R C

R

" 1

C C R R C ' C

=

=

=

=

(28)

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5 Dividing by sides of Eqns. (28) and based on Eqn. (27) we can obtain:

2 2 0

2 3 4

0 2 3

4

C R

1 C

C R R

C R R

R ' tg

" =

=

= (29)

We can easily calculate C0, and thus ’, when we know geometric dimensions of the capacitor:

0 x

C

'=C (30)

Most of automatic bridges make it possible to read Cx and tg . The measurements should consider of the wiring capacitance, Cd. This capacitance should be subtracted from the measured one.

4.Basic terms related to ferroelectrics

Some crystals possess certain polarization even under absence of the electric field. The polarization is called the spontaneous polarization and labeled as Ps.

It is found that the spontaneous polarization can only occur in crystals with polar axes of symmetry. The crystals are called pyroelectrics. A temperature dependence of the polarization is linear for pyroelectrics.

dT

dPs = (T) (31)

Factor is called the pyroelectric coefficient.

When the polarization direction can be changed by an external electric field the crystal is called a ferroelectric (analogous to a ferromagnetic).

A temperature of the polar symmetry axis decay (so the spontaneous polarization decay) is called the phase transition temperature.

This is a transition from the ferroelectric phase to the paraelectric one. The phase transition is a process of the crystal structure change related to its symmetry reconstruction. There also are phase transitions between phases with polar symmetry axes which does not cause a decay of the spontaneous polarization. Both phases keep the spontaneous polarization but its direction or its value may change. There are some other phase transitions, but we shall restrict ourselves to transitions from the paraelectric (non-polar) phase to the ferroelectric one.

5. Ferroelectric phase transitions – the thermodynamic description

The crystal state can be described with the Gibbs free energy which is defined as:

ST X Y U− K K

= (32)

where: U –the crystal internal energy,

YK –the electric field intensity or the mechanical strain,

XK –the electric induction, the polarization or the deformation, S –the entropy,

T –the absolute temperature.

Let us assume that the mechanical strain is constant.

The free energy is thus a function of the temperature and the electric field intensity.

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6

( )

T,E =UEPST

= (33)

The energies of the both phases are equal to each other at the phase transition temperature

P

F = (34)

where: F - the free energy of the ferroelectric phase,

P- the free energy of the paraelectric phase.

A phase transition is of the n-th order when the (n-1)-order partial derivatives of the free energy are equal each other, and the n-th order partial derivatives are not.

The first-order transition

T T

E E

P F

P P F

F

≠ ∂

≠ ∂

= ∂ (35)

but

E =−P

∂ Φ

∂ and S

T =−

∂ (36)

There is a jump in polarization and entropy of the crystal at the phase transition temperature.

The second-order transition

T T

E E

P F

P P F

F

= ∂

= ∂

= ∂ (37)

and

2 P 2 2

F 2 2

P 2 2

F 2

E E

T

T ∂

≠ ∂

≠ ∂

but

E P E

T C T S

T 2

p 2 2

2 =−

−∂

∂ =

− ∂

∂ =

−∂

∂ =

∂ (38)

where Cp – the specific heat.

The ferroelectric phase polarization and entropy are equal to those of the paraelectric phase (a continuous change). The specific heat Cp and the electric susceptibility change in a jump- like style.

Let us consider a transition from the ferroelectric phase to the paraelectric phase in more detail. Additionally let us assume that the spontaneous polarization occurs in one direction only. Devonshire, based on the basic theory of phase transitions derived by Landau, suggested to develop the free energy into a power series with respect to the polarization. Since the crystal energy does not change when the spontaneous polarization direction changes, even powers of Ps should be considered only

EP ...

6CP BP 1 4 AP 1 2

1 2 4 6

0 + + + + −

= (39)

Component EP stands for the external electric field interaction energy,

0 - the free energy of the paraelectric phase, A, B, C are the development coefficients.

Coefficient A is a linear temperature function

(

T Tc

)

A= − (40)

where Tc is the Curie-Weiss temperature.

(7)

7 The relation comes from the experiment (Curie-Weiss law). Coefficients B and C weakly depend on temperature, so we omit these relations.

In the case of the second-order phase transitions we can omit P6 and higher expressions in Eqn.(29). But it cannot be, however, omitted in the case of the first-order phase transitions.

Our considerations will be restricted to the second-order phase transitions only because:

1) triglycine sulphate (TGS) will be a subject of the investigation (the crystal shows a second-order phase transition),

2) the considerations are more educational.

The crystal stable state corresponds to a minimum of the free energy, therefore E

BP AP P 0

3− +

=

∂ =

∂ (41)

From Eqn.(41)

BP3

AP

E= + (42)

Figure 4 Ferroelectric hysteresis loop

A relation between the polarization and the electric field intensity is shown in a figure 4 according to Eqn.(42). Ps denotes the spontaneous polarization (the polarization at E = 0), and Ec is the coercion field, i.e. an electric field intensity required for the polarization orientation change. The curve shown in the figure 4 is called a hysteresis loop.

A ferroelectric crystal consists of domains, i.e. regions with various polarization directions.

The crystal partitioning into domains is caused by a high depolarization energy, the energy related to a uniform surface charge distribution. A rise of the intermediate layer energy, called as domain walls, counteracts this partitioning process (by analogy to a surface tension in liquids).

An equilibrium is established and this state corresponds to the energy minimum. At equilibrium the macroscopic polarization equals zero. A change in the macroscopic polarization consists in an increase of a domain volume with one polarization direction at expense of domains with an opposite orientation. In the extreme case the entire crystal constitutes a single domain (a uniform polarization).

Let us come back to the thermodynamic considerations.

When we assume E = 0 in Eqn. (42), then we get:

0 BP

AP+ 3 = (43)

Hence:

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8 B

P A or

0

P= =± − (44)

The case with P = 0 corresponds to the paraelectric phase, whereas P ≠ 0 to the ferroelectric one. Substituting Eqn.(40) into Eqn.(44) we get:

( )

B T

P=± − T− c (45)

In order that Eqn.(45) has a physical meaning, the following conditions must be fulfilled:

α > 0, B > 0 (T-TC < 0 for the ferroelectric phase). A temperature dependence derived from Eqn.(45) is shown in the figure 5.

Figure 5 Temperature dependence of the spontaneous polarization for ferroelectrics with second-order phase transition.

Let us calculate a derivative from Eqn.(42)

3BP2

1 A P

E = = +

∂ (46)

P = 0 for the paraelectric phase, so:

(

T Tc

)

1 =A= − (47)

>> 1 for ferroelectrics, so we can assume Eqn. (8) and:

≈ (48)

hence:

(

T Tc

)

1= − (49)

This is the Curie-Weiss law. Eqn.(49) makes it possible to determine a temperature relation of coefficient A. The Curie-Weiss law is also written in a form:

Tc

T '

= − (50)

where: ’ – the Curie constant,

Tc – the Curie-Weiss temperature.

In the ferroelectric phase:

(9)

9

(

T Tc

)

2 B 2A

3B A

1 =A+ − =− =− − (51)

The Curie constant can be calculated from a temperature relation of 1/ε in the paraelectric phase. And knowing α we can determine coefficient B from a temperature dependence of Ps2.

Figure 6 Temperature dependence: of the relative dielectric permittivity for ferroelectrics with the second-order phase transitions (a), and of the reciprocal dielectric permittivity (b).

Relations of ε and 1/ε obtained from the considerations conducted above are shown in the figure 6. The Curie temperature TC can be determined from the temperature dependence of 1/ε .

6. Sawyer-Tower polarization measurement

There are a few known methods of the spontaneous polarization measurements in ferroelectrics. A method elaborated by Sawyer and Tower in 1930 is one of the most frequently used. The Sawyer-Tower circuit is shown in the figure, where Cx* denotes the measured sample, and Cn – the reference capacitor.

The current intensity in the circuit is as follows:

dt SdP

I= (52)

where S denotes the electrode surface area of the sample.

Charge gathered in capacitor Cn

=

=

= 1

P

0 t

0 t

0

dP S dtdt S dP Idt

Q (53)

Voltage on capacitor Cn

=

= P

n 0 n

n dP

C S C

U Q (54)

Polarization P can be calculated when we know the electrode surface area S, the capacity Cn, and when we measure the voltage Un. For simplicity the capacitor Cn is chosen so that the condition of Cn >> Cx is satisfied, and then Un << Ux. Voltage on capacitor Cx is practically equal to the feed voltage. We use an oscilloscope to measure voltages U and Un (see Fig. 7).

The oscilloscope does not overload the measurement circuit as voltmeters could do. A vertical

(10)

10 spot deflection is proportional to the voltage on capacitor Cn, thus to the polarization. Whereas a horizontal spot deflection is proportional to the sample bias, therefore to the electric field intensity E = Ux/d

The hysteresis loop as a relation of the polarization upon the electric field intensity is displayed on the oscilloscope screen. Often the hysteresis saturation voltage is too high for the oscilloscope circuitry, then a voltage divider Dn is employed to reduce the voltage. In order to compensate the phase shift due to the ferroelectric losses a shunt resistor is connected to capacitor Cn. The measurement diagram is shown in the figure 7.

Figure 7 Diagram of Sawyer - Tower Circuit

The spontaneous polarization and the coercion field can be determined from the hysteresis loop dimensions (see figure 4).

The spontaneous polarization can be calculated from the formula:

S C

Ps =Uy n (55)

where Uy is the vertical spot deflection voltage for E = 0, S denotes the electrode surface area of the sample, Cn is the reference capacitor capacity.

The coercion field:

kd

Ec = Ux (56)

where k is the voltage divider factor, Ux denotes the voltage of the shift between the hysteresis and the OX axis, and d is the sample thichness.

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11

7. Measurement task

1. Determination of the sample capacity and the sample tg with respect to temperature.

2. Determination of the oscilloscope voltage sensitivity along the X-direction.

3. Determination of the temperature relation of the spontaneous polarization by means of the Sawyer-Tower circuit.

8. Results in detail

1. Calculate and plot relations of ε and 1/ε versus temperature.

2. Read the Curie temperature and calculate the Curie-Weiss constant (from the paraelectric phase!) from 1/ε plot.

3. Plot a temperature relation of tg .

4. Plot a temperature dependence of the spontaneous polarization Ps. 5. Plot a temperature dependence of Ps2.

6. Determine coefficient B from Ps2(T) relation.

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