History
The phenomenon of ferroelectricity was discovered in 1921 by J. Valasek[1] who was investigating the dielectric properties of Rochelle salt (NaKC4H4O6.4H2O). Barium titanate (BaTiO3) was discovered to be ferroelectric in 1944 by A von Hippel[2,3] and is perhaps the most commonly though of material when one thinks of ferroelectricity. While there are some 250+ materials that exhibit ferroelectric properties, some of the more common/significant materials include:
- Lead titanate, PbTiO3
- Lead zirconate titanate (PZT)
- Lead lanthanum zirconate titanate (PLZT)
Figure.1. The perovskite structure ABO3, shown here for PbTiO3 which has a cubic structure in the paraelectric phase and tetragonal structure in the ferroelectric phase.
Crystal symmetry and physical properties
Twenty of the 32 crystal classes are so-called piezoelectric, and crystals belonging to one of these classes (point groups) display piezoelectricity. All 21 piezoelectric classes lack a center of symmetry (Note that Cubic 432 has higher symmetry). Any material develops a dielectric polarization when an electric field is applied, but a substance which has such a natural charge separation even in the absence of a field is called a polar material. Whether or not a material is polar is determined solely by its crystal structure. Only 10 of the 32 point groups are polar. All polar crystals are pyroelectric, so the 10 polar crystal classes are sometimes referred to as the pyroelectric classes[III].
Figure.2. The relationship between the piezoelectric, pyroelectric and ferroelectric materials
Below the Curie temperature, there is a dipole moment created in the structure. Therefore, a spontaneous polarization is found. One of the characteristics of ferroelectric materials is the initially rapid rise of their spontaneous polarization as their temperature drops below the critical value (Curie temperature), Tc. In crystalline ferroelectrics, this initial rapid rise in spontaneous polarization is due to the elementary dipoles in the material interacting with each other, producing an internal field which lines up the dipoles, eventually yielding saturation of the spontaneous polarization.
Figure.3. The spontaneous polarization vs. temperature plot of the PbTiO3
Ferroelectricity
We have often commented that ionic susceptibility is not sensitive to variations in temperature. Although this is true for most substance, there is a class of materials which exhibits a marked departure from this rule: the ferroelectric materials. In these substances, the static dielectric dielectric constant changes with temperature according to the relation:
(1)where B and C are constants independent of temperature. This relation is known as the Curie-Weiss law, and the parameters C and TC are called the Curie constant and Curie temperature, respectively.
A phase transition occurs at the temperature TC. Above the transition temperature, the substance is in the paraelectric phase, in which the elementary dipoles of the various unit cells in the crystal are oriented randomly. And below the transition temperature, the substance is in the ferroelectric phase [4].
Figure.4. (P-E) hysteresis loops of paraelectric phase (left) and ferroelectric phase
Ferroelectricity is a spontaneous electric polarization of a material that can be reversed by the application of an external electric field. Ferroelectricity is the phenomenon, by virtue of which some materials exhibit spontaneous electric polarization, even in the absence of any externally applied field. Ferroelectric crystals possess regions with uniform polarization called ferroelectric domains. Within a domain, all the electric dipoles are aligned in the same direction. There may be many domains in a crystal separated by interfaces called domain walls. A ferroelectric single crystal, when grown, has multiple ferroelectric domains. A single domain can be obtained by domain wall motion made possible by the application of an appropriate electric field. A very strong field could lead to the reversal of the polarization in the domain, known as domain switching. The polarization reversal can be observed by measuring the ferroelectric hysteresis as shown in Figure.5.
Figure.5. The ferroelectric (P-E) hysteresis loop. Circles with arrows represent the polarization state of the material at the indicated fields. The actual loop is measured on a (111)-oriented 1.3 μm thick sol-gel Pb(Zr0.45Ti0.55)O3 film.
Polarization mechanism in ferroelectric materials:
Sources of polarizability[IV]
Let us examine more closely the physical process which gives rise to polarizability. Basically, polarizability is a consequence of the fact that the molecules, which are the building blocks of all substances, are composed of both positive charges (nuclei) and negative charges (electrons). When a field acts on a molecule, the positive charges are displayed in a direction opposite to that of the field. The effect is therefore to pull the opposite charges apart, i.e., to polarize the molecule [5].
There are different types of polarization processes, depending on the structure of the molecules which constitute the solid (for details please see the Yuan's webpage [VI]). Basically, there are three types polarization mechanisms contribute to the total polarizability. In general, therefore, we may write for the total polarizability:
(2)which is the sum of the various contributions; $\alpha_{e}$, $\alpha_{i}$, and $\alpha_{d}$ are the electronic, ionic, and dipolar polarizabilities, respectively. The electronic contribution is present in any type of substances, but the presence of the other two terms depends on the material under consideration. Thus the term $\alpha_{i}$ is present in ionic covalent crystals such as Si and Ge, which are nonionic and nondipolar, the polarizability is entirely electronic in nature.
The relative magnitudes of the various contributions in (eq.2) are such that in nondipolar, ionic substances the electronic part is often of the same order as the ionic. In dipolar substances, however, the greatest contribution comes from the dipolar part.
The various polarizabilities may be segregated from each other because each contribution has its own characteristic features which distinguish it from the others. Dipolar polarizability, for instance, exhibits strong dependence on temperature, while the other two contributions are essentially temperature independent [5].
Figure.6. Total polarizability $\alpha$ versus frequency ω for a dipolar substance.
Landau theory of ferroelectric phase transformations [II,V]
Based on Ginzburg–Landau theory, the free energy of a ferroelectric material, in the absence of an electric field and applied stress may be written as a Taylor expansion in terms of the order parameter, P. If a sixth order expansion is used (i.e. 8th order and higher terms truncated), the free energy is given by:
$\Delta G=\frac{1}{2}\alpha_{0}(T-T_{0})(P_{x}^{2}+P_{y}^{2}+P_{z}^{2})+\frac{1}{4}\alpha_{11}(P_{x}^{4}+P_{y}^{4}+P_{z}^{4})+\frac{1}{2}\alpha_{12}(P_{x}^{2}P_{y}^{2}+P_{y}^{2}P_{z}^{2}+P_{z}^{2}P_{x}^{2})+\frac{1}{6}\alpha_{111}(P_{x}^{6}+P_{y}^{6}+P_{z}^{6})+\frac{1}{2}\alpha_{112}(P_{x}^{4}(P_{y}^{2}+P_{z}^{2})+P_{y}^{4}(P_{x}^{2}+P_{z}^{2})+P_{z}^{4}(P_{x}^{2}+P_{y}^{2}))+\frac{1}{2}\alpha_{123}P_{x}^{2}P_{y}^{2}P_{z}^{2}$
where Px, Py, and Pz are the components of the polarization vector in the x, y, and z directions respectively, and the coefficients, and the coefficients, $\alpha_{i}$, $\alpha_{ij}$, $\alpha_{ijk}$ must be consistent with the crystal symmetry. To investigate domain formation and other phenomena in ferroelectrics, these equations are often used in the context of a phase field model. Typically, this involves adding a gradient term, an electrostatic term and an elastic term to the free energy. The equations are then discretized onto a grid using the finite difference method and solved subject to the constraints of Gauss's law and Linear elasticity [II].
In all known ferroelectrics, $\alpha_{0}$ > 0 and $\alpha_{111}$ > 0. These coefficients may be obtained experimentally or from ab-initio simulations. For ferroelectrics with a first order phase transition, $\alpha_{11}$ < 0 and $\alpha_{11}$ > 0 for a second order phase transition.
Figure.7. The relationship between crystal structure and polarization.
The spontaneous polarization, Ps of a ferroelectric for a cubic to tetragonal phase transition may be obtained by considering the 1D expression of the free energy which is:
(3)This free energy has the shape of a double well potential with two free energy minima at $P=\pm P_{s}$, where Ps is the spontaneous polarization. At these two minima, the derivative of the free energy is zero, i.e.:
(4)Since Px = 0 corresponds to a free energy maxima in the ferroelectric phase, the spontaneous polarization, Ps, is obtained from the solution of the equation:
(6)which is:
(7)and elimination of solutions yielding a negative square root (for either the first or second order phase transitions) gives:
(8)If $\alpha_{111}$ = 0, using the same approach as above, the spontaneous polarization may be obtained as:
(9)Applications for Ferroelectric Materials
The biggest use of ferroelectric ceramics have been in the areas such as dielectric ceramics for capacitor applications, ferroelectric thin films for non volatile memories, piezoelectric materials for medical ultrasound imaging and actuators, and electro-optic materials for data storage and displays [VII].
*Capacitors
*Non-volatile memory
*Piezoelectrics for ultrasound imaging and actuators
*Electro-optic materials for data storage applications
*Thermistors
*Switches known as transchargers or transpolarizers
*Oscillators and filters
*Light deflectors, modulators and displays
Reference
1. J. Valasek (1920). "Piezoelectric and allied phenomena in Rochelle salt". Physical Review
15: 537. and J. Valasek (1921). Physical Review 17: 475
2. A. von Hippel and coworkers, NDRC Reports 14-300 (1944), 14-540 (1945); A. von Hippel,
R. G. Breckenridge, F. G. Chesley, and L. Tisza, J. Ind. Eng. Chem. 38, 1097 (1946).
3. B. Matthias * and A. von Hippel Phys. Rev. 73, 1378 - 1384 (1948)
4. M. Ali Omar. "Elementary Solid State Physics"p.408. (ISBN 0- 201-60733-6)
5. M. Ali Omar. "Elementary Solid State Physics"p.381. (ISBN 0- 201-60733-6)
Reference websites
I. http://www.azom.com/details.asp?ArticleID=3593
II. http://en.wikipedia.org/wiki/Ferroelectric
III. http://en.wikipedia.org/wiki/Crystal_lattice
IV. http://mysite.du.edu/~jcalvert/phys/polariza.htm
V. http://en.wikipedia.org/wiki/Ginzburg%E2%80%93Landau_theory
VI. http://electrons.wikidot.com/polarization-of-ceramic-materials
VII. http://4engr.com/dictionary/catalog/1322/index.html
Good work BB.
Just curious to know, is it because of polarization in the materials we see ferroelectric behavior or because the material is ferroelectric it gets polarized.
Thank you Dharma, I think for the major difference between a ferroelectric material and a non-ferroelectric material (such as paraelectric or anti-ferroelectric materials) is that ferroelectric material has spontaneous polarization behavior. Please see fig.5. And for your question, as far as I know that every dieletric material can be "polarized" (with a external force, heat, electric field…), but not all of them has the spontaneous polarization.
If you still have further quetion about this, please let me know.
How one measure the ferroelectricity, I mean instrument used to measure it.
It depends on which property you are interesting in. If you are looking for a PE hysteresis curve, most people construct a simple circuit to measure hysteresis using a function generator and an oscilloscope. This approach was first used by Sawyer and Tower in 1929 to measure one of the earliest known ferroelectric materials, Rochelle Salt. Like the figure.
If you are interesing in the electrical properties, there are several different equipments to measure that. For example, a lots of researchers measure their ferroelectrics electrical properties by LCR meter. This is also the one I used to measure my sample's electrical properties (inductance, capacitance, resistance). The one I used is LCR meter 4284A
I am doing both test with polymer and polymer/nanocomposite materials in EIRC, locating at the ground floor of IMS. If you are interested, I am very glad to share my experience.
Hi,
I am wondering if you are using any reference capacitor like in the figure when you are characterizing the ferroelectric capacitor? From the capacitance information how do you extract the polarization (P)? And last question is what do you do with inductance of a ferroelectric capacitor, does it tell anything?
thanks.
Nice work BB. I was just wandering, what defines the "thickness" of the hysteresis loop (= the electric field that is necessary to apply in order to have zero polarization)?
Thanks Maria, I think I still don't quite understand about the describe of "the electric field that is necessary to apply in order to have zero polarization". Do you mean about the phenomena of how much external electric field to reach maximum polarization. Or maybe it just only I have not hear this yet, but I will keep lookng for the answer for your question. And post on the webpage when I found some anwser for this.
Actually, I was wandering about what defines how big the electric field will be for zero polarization (it corresponds to point F and its symmetric point in fig.5). Is it the material's structure or it depends on the electronic properties of the atoms of the lattice? For example, how does the field corresponding to zero polarizability change from PbTiO3 to BTiO3?
In the case of dipolar interaction, where does the greater contribution come from?
There seems to some inconsistency between the text and fig.6.
Good point, Maria. I think this figure is not so accurate for every situation. The magnitudes of different polarizations depend on frequency, temperature and material itself. Like I mentioned in the paragraph, if you measure the polarization of a dipolar substance at low frequency (w<109), then the dipolar polarization should be the dominant polarization provider in the material. In other word, if you measure it at high frequency, then the electronic should be the dominant one.
And I will try to find another better plot of the polarization V.S. frequency. Thanks.
hey Maria, sorry I can not find the more proper figure on other websites, so I draw the figure (follow the fig. 8.10 in the text book)
I hope it looks a little bit better.
Hi, Ching-Chang,
Good job! I am doing some ferroelectric properties study with polymers for my own research and I do find a lot of useful informations from the ferroelectric ceramic world. And there are some concepts that I am always confusing. They are paraelectricity and/or anti-ferroelectricity. Do you know how to differentiate them? Thank you!
hello Fangxiao,
Paraelectricity is the ability of many materials (specifically ceramic crystals) to become polarized under an applied electric field. Unlike Ferroelectricity; this can happen even if there is no permanent electric dipole that exists in the material, and removal of the fields results in the polarization in the material returning to zero. And Antiferroelectricity is a physical property of certain materials. It is closely related to ferroelectricity; In an antiferroelectric, unlike a ferroelectric, the total, macroscopic spontaneous polarization is zero, since the adjacent dipoles cancel each other out.
In another explaination, Paraelectricity occurs in crystal phases in which electric dipoles are unaligned (i.e. unordered domains that are electrically charged) and thus have the potential to align in an external electric field and strengthen it. In comparison to the ferroelectric phase, the domains are unordered and the internal field is weak. And for anti-ferroelectric materials, An antiferroelectric material consists of an ordered (crystalline) array of electric dipoles (from the ions and electrons in the material), but with adjacent dipoles oriented in opposite (antiparallel) directions (the dipoles of each orientation form interpenetrating sublattices, loosely analogous to a checkerboard pattern). This can be contrasted with a ferroelectric, in which the dipoles all point in the same direction.
By following figure, I hope it can provide you more useful information between their relationship.
Hi, Ching-Chang,
Thank you very much! This do help me a lot!
Hi Ching-Chang Chung,
I am wondering if the hysteresis loops show deviations as a ferroelectric capacitor is electrically stressed continuously (endurance) and a very long time after they are stressed (retention). Do you thing this properties are enough to use them as non-volatile memories which requires more than 106 cycles and more than 10 years of retention.
Thanks.
Hi Gokhan,
I think Two of the major reliability concerns hampering the use of ferroelectric thin films in non-volatile memory devices are fatigue and imprint. Fatigue consists in the loss of remnant polarization with cumulative switching of the capacitor. If the remnant polarization falls below the detection threshold of the device, the memory cell becomes unreadable.
And the hysteresis loop will show deviations when it under stressed for a long time, The fatigue in ferroelectric films is mainly due to the pinning of domain walls by space charge near the boundaries of electrodes and structural defects such as microcracking and porosity. So, if you wanna have a good non-volatile memories you do have to consider the fatigue effect on the ferroelectric thin film.