Direction & Electric Field Dependence of Dielectric Constant

"Dielectric Constant" is often used to represent the relative permittivity. It is defined as

\begin{align} \epsilon_{r}=\frac{\epsilon_{s}}{\epsilon_{0}} \end{align}

It can be related with the electric susceptibility, $\chi_{e}$, $\epsilon_{r}=1+\chi_{e}$. Then, we could know that the dielectric constant is related with how easily a dipole polarizes in response to an electric field and how big is the dipole moment. While, both of the dipole moment and the electric field have directions, which must results in isotropic and anisotropic dielectric constant. The isotropic dielectric constant is difficult to see because the real measurement measures the sum of the dielectric constant of a collection of dipole moments, and the dipole moments are randomly oriented. Here, two specific methods are illustrated to prepare the anisotropic dielectric properties.

Other than the direction dependence, the dielectric constant also depends on the electric field that applied, especially for the ferroelectric materials. With the increasing of the electric field above the coercive field, more and more dipoles are poled, and less and less dipoles are left which can be poled, so the dielectric constant decreases with the electric field. Finally, the whole ferroelectric material saturate, and the dielectric constant goes to minimum.

1. Anisotropic Dielectric Constant

1.1 Orientation Dependence of Ferroelectric Ba1-xSrxTiO3 Thin Films

Even though the anisotropic dielectric properties is simple, it is difficult to make an anisotropic sample and measure the anisotropic properties macroscopically. The first sample is made in a positive logic thinking, where you grow an epitaxial direction.

Recently, ferroelectric thin films have received much attention for application in tunable microwave devices such as the phase shifter, delay line, resonator, tunable filter, and varator. (Ba1-xSrx)TiO3 (BST) thin films especially have been extensively investigated as promising due to their high dielectric constant tunability and low dielectric loss.

In order to study the orientation effect on the electrical properties of BST films, orientation engineered BST films, whose surface normal orientations are (001), (011), and (111), have been epitaxially grown on MgO (001), (011), and (111) single crystals, respectively, by pulsed laser deposition (PLD).[1] The PLD conditions are critically chosen, so that the epitaxial films are ensured without secondary orientations. For example, Epitaxial BST film growth on a (001) MgO substrate was achieved with in a
broad range of deposition conditions: heater temperature higher than 650 °C and oxygen pressure between 50 to 500 mTorr. At 825 °C, epitaxial (011) and (111) BST films with 200 mTorr of oxygen pressure were grown successfully without other intensities.

The XRD patterns of (001), (0110, and (1110 oriented BST films on (001), (011), and (111) MgO substrates, respectively, are shown in Figure 1.1. Secondary orientations are not seen in any of the three XRD patterns, which suggests that each BST film is oriented along one direction and has a single phase.


Figure 1.1 X-ray 2_Theta Diffraction Patterns of (001), (011), and (111) oriented BST films deposited on (001), (011), and (111) MgO Substrates.

Figure 1.2 shows the measured capacitances as a function of the frequency with dc bias from 0 to 40 V for IDT devices
based on (001), (011), and (111) oriented BST films deposited on (001), (011), and (111) MgO substrates, respectively. It can be seen that the dielectric constant with the surface orientation of (011) is higher than that of (001) and (111).


Figure 1.2 Measured capacitances as a function of the frequency with dc bias from 0 to 40 V for IDT devices

based on (001), (011), and (111) oriented BST films.

The dielectric quality factors (Q) of the IDT device based on (001), (011), and (111) oriented BST films as a function of dc bias from 0 to 40 V are shown in Fig. 1.3. The calculated dielectric Q values of the IDT capacitors based on (001), (011), and (111) oriented BST films at 9 GHz with no dc bias were about 12, 14, and 21, respectively. This indicates that dielectric loss in the ferroelectric BST film has been reduced without losing dielectric tunability by engineering the orientation of the BST films.


Figure 1.3 Dielectric Q for IDT devices based on (001), (011), and (111) oriented BST films.

1.2 Anisotropic Dielectric Constants Through Porosity in Titania

The second sample is made in a negative logic thinking, where the properties are deduced by porosity, and the pores are oriented. This is achieved by a technique called Macrofabrication by extrusion (MFCX).

Microfabrication by coextrusion (MFCX) involves multiple extrusions to achieve the desired dimensional reduction.[2] It is a
powerful method for controlling pore morphology. A large-scale unit cell can be reduced in size by MFCX, to create structured porosity. MFCX can create very-finescale structures and voids useful for decreasing the dielectric constant of a material. When combined with the thermoplastic green machining and assembly technique, three-dimensional (3-D) dielectric texturization from a single material can be fabricated.


Figure 1.4 Microfabrication by coextrusion (MFCX)

Here, two stocks of materials need to be prepared for the microcellular structure. The first stock consists of titania and carbon thermoplastic, which are compounded in an acrylate binder using a high shear mixer (Brabender) to form the outer shell of a feed rod. The second stock is created by compounding carbon with titania at 15% volume ratio to form the partially sacrificial core of the feed rod. The shells are then warm bonded around the cores to create large-scale master feed rods with a diameter of 15.75 mm. These master feed rods (a white shell around a black core) are extruded through a reduction die to achieve 3.25-mm-diameter rods. They are then arrayed together in a microcellular structure and undergo a final coextrusion. The 15.75-mm rods are reduced to less than 1 mm in diameter. This fabrication process flow is illustrated in Figure 1.4.[3]

In order to fabricate anisotropic materials, long uniform anisotropic coextrusion fibers are used. The uniformity of the orientation of the filbers are confirmed by microcomputed tomography (Micro CT), in Figure 1.5.


Figure 1.5 Micro CT and SEM images of the anisotropic material. (a) (left) 3-D Micro CT view of the cross sections (right) 2-D views of the cross sections. (b) SEM close-up of the x-y 2-D view in (a). (c) Further close-up of (b).

Two microcellular-structure-based anisotropic titania samples are tested. The first sample’s dimensions are 10.03 mm, 10.03 mm and 3.64 mm. The coextruded fibers are aligned along one of the 10.03 mm lateral sides. The second sample’s dimensions are 8.20 mm, 7.60 mm and 3.85 mm. The coextruded fibers are aligned along the 8.20 mm lateral side.The effective dielectric constant along the fiber is extracted to be 88.66 and the one perpendicular to the fiber is extracted to be 11.8. This indicates that dielectric constant has been changed by the coextrusion with fibers.

2. Nonlinear Dielectric Constant in Ferroelectric Polymers

2.1 Ferroelectric Polymer: Poly(vinylidene Fluoride) and Its Copolymers

Among many ferroelectric polymers, the family of PVDF and its copolymers exhibits the strongest ferroelectric activity, and thus has attracted substantial research interest.

The ferroelectricity in PVDF comes from its highly polar bond between carbon and fluorine.[4] When PVDF organized into ordered crystalline structures, the dipole moment (perpendicular to chains) can be either maintained or cancelled in the crystal, depending on the chain conformation and unit cell structure, as shown in Figure 2.1. Five crystalline phases for PVDF have been well-studied; (I), (II or IIa), (IIp), (IIIp), and (IIIa).[4]~[10] The unit cell of α-crystal has two chains in the tg+tg- configuration. The dipole moment of one chain is antiparallel to the next chain, and the dipole moment of an α-crystal is neutralized (Figure 2.1 a). If one of the two chains in the unit cell rotates by 180° around its chain axes, the two dipole moments pointing in the same direction result in a polar δ crystalline structure (Figure 2.1 b). The most polar structure is the β crystalline form (Figure 2.1 c), where the chain has nearly all trans conformation and all the dipoles pointing to the b-axis direction.

Survival of either polar or nonpolar features in macroscopic specimens requires specified crystallization and processing conditions, as summarized in Figure 2.1.[4], [5] The formation of nonpolar α crystalline phase is kinetically favorable so that normal crystallization from the melt or solution-casting from non-polar solvent results in the β phase. However, β crystalline phase is thermodynamically more stable. Therefore, mechanical stretching can transform the α phase into the β phase. Additionally, the α phase can be transformed to the polar δ and then the β phase by electric poling.[7], [8] The γ phase can be directly acquired by solution-casting from polar solvents. Crystallization under high pressure facilitates the growth of β-PVDF, because it can be packed more densely in a smaller unit cell.[11] The transition of PVDF from the polar β phase into its nonpolar α phase is close to or even higher than the melting point.


Figure 2.1 Phase transitions among polymorphs of PVDF.

2.2 Measurements of Nonlinear Dielectric Constant

For a dielectric containing free-rotating dipoles, the electric displacement tends to saturate at high electric fields according to the Langevin functions, $L(x)=coth(x)-1/x$, as shown in Figure 2.2.[12] This results in a slow down polarization.


Figure 2.2 Langevin function (red line), compared with $tanh(x/3)$ (blue line).

On the other hand, the dipoles in ferroelectric materials align cooperatively in the direction of an applies electric field, resulting in an enhanced polarization. These two factors play together and give the ferroelectric polymers a complicate nonlinear dielectric behavior. Measurement and understanding the nonlinear dielectric behavior is important to the understanding of the physical origin underlying the polymer ferroelectricity, and is helpful for the improvement of the application of the ferroelectric polymers as the capacitor material.

Here, two methods are introduced to measure the nonlinear dielectric constant for the ferroelectric polymers. The results of poly(vinylidene fluoride) (PVDF) are given as a sample.[13]

Figure 2.3 gives the schematic diagram of the experiment set up for the nonlinear dielectric constant measurement. Basically, one or two sinusoidal waves are generated and amplified, then they are applied to the films under test. The charge generated on the sample is amplified (D(t)) and input into the data acquisition unit together with the applied wave signal (E(t)).


Figure 2.3 Schematic diagram of the experiment set up for the nonlinear dielectric constant measurement.

The electric displacement D(t) for a nonlinear dielectric is expanded in powers of the electric field E(t),

\begin{align} D=P_{0}+\varepsilon_{1}E+\varepsilon_{2}E^2+\varepsilon_{3}E^3+...... \end{align}
\begin{align} E(t)=E_{0}\cos\omega t \end{align}
\begin{align} D(t)=D_{0}+D_{1}\cos\omega t+D_{2}\cos2\omega t+D_{3}\cos3\omega t+...... \end{align}

Put equation (3) into equation (2),

\begin{align} D_{0}=P_{0}+\frac{1}{2}\varepsilon_{2}E_{0}^2+\frac{3}{8}\varepsilon_{4}E_{0}^4+...... \end{align}
\begin{align} D_{1}=\varepsilon_{1}E_{0}+\frac{3}{4}\varepsilon_{3}E_{0}^3+\frac{10}{16}\varepsilon_{5}E_{0}^5+...... \end{align}
\begin{align} D_{2}=\frac{1}{2}\varepsilon_{2}E_{0}^2+\frac{1}{2}\varepsilon_{4}E_{0}^4+\frac{15}{32}\varepsilon_{6}E_{0}^6+...... \end{align}
\begin{align} D_{3}=\frac{1}{4}\varepsilon_{3}E_{0}^3+\frac{5}{16}\varepsilon_{5}E_{0}^5+\frac{21}{64}\varepsilon_{7}E_{0}^7+...... \end{align}

The input electric field applied to the sample E(t), and the output electric displacement D(t) are shown in the up two part of Figure 2.4. Then D(t) is decomposed to D1 D2 and D3.


Figure 2.4 Plots of sampled data, E(t) and D(t), and decomposed components, D1 D2 and D3,

as a function of time for a single frequency measurement.

The decomposed D1 D2 and D3 as the function of E0 for the unpoled and poled PVDF films are shown in Figure 2.5 and Figure 2.6, respectively.


Figure 2.5 Dependence of fundamental and higher harmonic electric displacements D1, D2, and D3

on the amplitude of applied electric field of E0 for unpoled PVDF.


Figure 2.6 Dependence of fundamental and higher harmonic electric displacements D1, D2, and D3

on the amplitude of applied electric field of E0 for poled PVDF.

The nearly linear relationship between D1 and E0 in Figure 2.5 and Figure 2.6 indicates that the first term is dominant in Equation (6). The initial slope yields the linear permittivity \varepsilon1 for unpoled and poled PVDF, as listed in Table 2.1. The larger \varepsilon1 in poled PVDF may be an indication of increased fluctuations of crystalline dipoles.


Figure 2.7 Plots of D2 vs E02 and D3 vs E03 for poled and unpoled PVDF respectively.

Figure 2.7 shows plots of D2 and D3 against E02 and E03, respectively. The linear relationship again indicates that the first term is dominant in Equation (7) Equation (8). From the initial slope, \varepsilon2 and \varepsilon3 are obtained and listed in Table 2.1.

ε1 (F/m) ε2 (F/V) ε3 (Fm/V2)
Unpoled $1.10 \times 10^{-10}$ ~0 $3.47 \times 10^{-27}$
Poled $1.22 \times 10^{-10}$ $-6.20 \times 10^{-20}$ $7.60 \times 10^{-27}$

Table 2.1 Linear and nonlinear dielectric constant of unpoled and poled PVDF films.

It is found that if PVDF is positively poled, \varepsilon2 is negative and and if negatively poled, \varepsilon2 is positive. The positive \varepsilon3 implies that the molecular dipoles of PVDF tend to align cooperatively even at relatively low fields and predicts ferroelectric hysteresis behavior occurring at much higher fields.

An alternative experiment for nonlinear dielectricity was made by using a double wave consisting of a high-frequency low-field superimposed on a low-frequency high field as shown in Figure 2.8. As a result, nonlinear dielectricity in terms of the dependence of the linear dielectric constant on the high bias field is obtained.


Figure 2.8 Plots of sampled data, E(t) and D(t), and decomposed components, D0 and Dn

as a function of time for a double frequency measurement.

Figure 2.9 shows the D-E and \varepsilon/\varepsilon0-E hysteresis loops for PVDF. It seems as though the gradient of the D-E hysteresis loop predicts the shape of the \varepsilon/\varepsilon0 - E loop. However this is not the case quantitatively, because the gradient is much greater than the value of \varepsilon/\varepsilon0. The D contains contributions from both ferroelectric and dielectric polarizations. The contribution from the latter was calculated by integrating \varepsilon(E) dE and the results were shown by the dashed line in Figure 2.9. The remaining contribution from the ferroelectric polarization is shown by the solid line in the same figure. The decrease in D due to the decrease in E from its maximum to zero is much diminished but
there remains a 15% decrease in D which is associated with the switchback of the ferroelectric polarization.


Figure 2.9 Comparison of D-E and \varepsilon/\varepsilon0-E hysteresis loops for PVDF.

The dashed line indicates contributions from \varepsilon/\varepsilon0 to the D-E loop

and the solid line corresponds to purely irreversible hysteresis associated with ferroelectric polarization reversal.

In summation of Section 2.2, single frequency and double frequency measurements are carried out for the ferroelectric polymer, PVDF. \varepsilon1, \varepsilon2 and \varepsilon3 are obtained respectively. It is proposed that \varepsilon1 represents the amplitude of the dipole fluctuation, \varepsilon2 represents the polarity, and \varepsilon3 represents the cooperative interaction among the dipoles.

3. References

1. Moon, S.E.; Kim, E.K.; Kwak, M.H.; Ryu, H.C.; Kim, Y.T.; Kang, K.Y.; Lee, S.J.; Kim, W.J. "Orientation Dependent Microwave Dielectric Properties of Ferroelectric Ba1-xSrxTiO3 Thin Films", Appl. Phys. Lett. 2003, 83, 2166-2168.

2. Crumm, A.T.; Halloran, J.W. "Fabrication of Microconfigured Multicomponent Ceramics", J. Amer. Ceram. Soc. 1998, 81, 1053-1057.

3. Gong, X.; She, W.H.; Hoppenjans, E.E.; Wing, Z.N.; Geyer, R.G.; Halloran, J.W.; Chappell, W.J. "Tailored and Anisotropic Dielectric Constants Through Porosity in Ceramic Components", IEEE Trans. Microw. Theo. Tech. 2005, 53, 3638-3647.

4. Lovinger, A. J. "Ferroelectric Polymers", Science 1983, 220, 1115-1121.

5. Tashiro, K. "Crystal Structure and Phase Transition of PVDF and Related Copolymers", In Ferroelectric polymers, chemistry, physics, and applications, 1 ed.; Nalwa, H. S., Ed. Dekker: New York, 1995; 63-182.

6. Lovinger, A. J. "Annealing of Poly(Vinylidene Fluoride) and Formation of a Fifth Phase", Macromolecules 1982, 15, 40-44.

7. Dradi, E.; Casiraghi, G.; Sartori, G.; Casnati, G. "Foramtion of a New Crystal Form (αp) of Poly(Vinylidene Fluoride) Under Electric Field", Macromolecules 1978, 11, 1297-1298.

8. Davis, G. T.; McKinney, J. E.; Broadhurst, M. G.; Roth, S. C. "Electric-Field-Induced Phase Changes in Poly(Vinylidene Fluoride)", J. Appl. Phys. 1978, 49, 4998-5002.

9. Su, H. B.; Strachan, A.; Goddard, W. A. "Density Functional Theory and Molecular Dynamics Studies of the Energetics and kinetics of Electroactive Polymers: PVDF and P(VDF-TrFE)", Phys. Rev. B 2004, 70, 064101.

10. Lovinger, A. J. "Unit Cell of the γ Phase of Poly(Vinylidene Fluoride)", Macromolecules 1981, 14, 322-325.

11. Lovinger, A. J. "Radiation Effects on the Structure and Properties of Poly(Vinylidene Fluoride) and Its Ferroelectric Copolymers", In Radiation effects on polymers, Clough, R. L.; Shalaby, W., Eds. American Chemical Society: Washington, DC, 1991; 84-100.

12. Brillouin_Langevin_functions:

13. Furukawa, T.; Nakajima, K.; Koizumi, T.; Date, M. "Measurements of Nonlinear Dielectric in Ferroelectric Polymers", Japa. J. Appl. Phys. 1987, 26, 1039-1045.

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